# Ryan Babbush

Ryan leads the Quantum Algorithm & Applications Team at Google. The mandate of this team is to develop new and more efficient quantum algorithms, research potential applications of quantum computing, build and open source tools for accelerating quantum algorithms research, and to execute experiments on existing quantum devices. Ryan's individual contributor research focuses on developing algorithms for fault-tolerant quantum computers, especially with applications in chemistry, physics, and machine learning.

Authored Publications

Google Publications

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Dynamics of magnetization at infinite temperature in a Heisenberg spin chain

Trond Andersen

Rhine Samajdar

Andre Petukhov

Jesse Hoke

Dmitry Abanin

ILYA Drozdov

Xiao Mi

Alexis Morvan

Charles Neill

Rajeev Acharya

Richard Ross Allen

Kyle Anderson

Markus Ansmann

Frank Arute

Kunal Arya

Juan Atalaya

Gina Bortoli

Alexandre Bourassa

Leon Brill

Michael Broughton

Bob Buckley

Tim Burger

Nicholas Bushnell

Juan Campero

Hung-Shen Chang

Jimmy Chen

Benjamin Chiaro

Desmond Chik

Josh Cogan

Roberto Collins

Paul Conner

William Courtney

Alex Crook

Ben Curtin

Agustin Di Paolo

Andrew Dunsworth

Clint Earle

Lara Faoro

Edward Farhi

Reza Fatemi

Vinicius Ferreira

Ebrahim Forati

Austin Fowler

Brooks Foxen

Gonzalo Garcia

Élie Genois

William Giang

Dar Gilboa

Raja Gosula

Alejo Grajales Dau

Steve Habegger

Michael Hamilton

Monica Hansen

Sean Harrington

Paula Heu

Gordon Hill

Trent Huang

Ashley Huff

Bill Huggins

Sergei Isakov

Justin Iveland

Zhang Jiang

Cody Jones

Pavol Juhas

Mostafa Khezri

Marika Kieferova

Alexei Kitaev

Andrey Klots

Alexander Korotkov

Fedor Kostritsa

John Mark Kreikebaum

Dave Landhuis

Pavel Laptev

Kim Ming Lau

Lily Laws

Joonho Lee

Kenny Lee

Yuri Lensky

Alexander Lill

Wayne Liu

Salvatore Mandra

Orion Martin

Steven Martin

Seneca Meeks

Amanda Mieszala

Shirin Montazeri

Ramis Movassagh

Wojtek Mruczkiewicz

Ani Nersisyan

Michael Newman

JiunHow Ng

Murray Ich Nguyen

Tom O'Brien

Seun Omonije

Alex Opremcak

Rebecca Potter

Leonid Pryadko

David Rhodes

Charles Rocque

Negar Saei

Kannan Sankaragomathi

Henry Schurkus

Christopher Schuster

Mike Shearn

Aaron Shorter

Noah Shutty

Vladimir Shvarts

Vlad Sivak

Jindra Skruzny

Clarke Smith

Rolando Somma

George Sterling

Doug Strain

Marco Szalay

Doug Thor

Alfredo Torres

Guifre Vidal

Benjamin Villalonga

Cheng Xing

Jamie Yao

Ping Yeh

Juhwan Yoo

Grayson Young

Yaxing Zhang

Ningfeng Zhu

Jeremy Hilton

Anthony Megrant

Yu Chen

Vadim Smelyanskiy

Vedika Khemani

Sarang Gopalakrishnan

Tomaž Prosen

Science, vol. 384 (2024), pp. 48-53

Preview abstract
Understanding universal aspects of quantum dynamics is an unresolved problem in statistical mechanics. In particular, the spin dynamics of the one-dimensional Heisenberg model were conjectured as to belong to the Kardar-Parisi-Zhang (KPZ) universality class based on the scaling of the infinite-temperature spin-spin correlation function. In a chain of 46 superconducting qubits, we studied the probability distribution of the magnetization transferred across the chain’s center, P(M). The first two moments of P(M) show superdiffusive behavior, a hallmark of KPZ universality. However, the third and fourth moments ruled out the KPZ conjecture and allow for evaluating other theories. Our results highlight the importance of studying higher moments in determining dynamic universality classes and provide insights into universal behavior in quantum systems.
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Drug Design on Quantum Computers

Raffaele Santagati

Alán Aspuru-Guzik

Matthias Degroote

Leticia Gonzalez

Elica Kyoseva

Nikolaj Moll

Markus Oppel

Robert Parrish

Michael Streif

Christofer Tautermann

Horst Weiss

Nathan Wiebe

Clemens Utschig-Utschig

Nature Physics (2024)

Preview abstract
The promised industrial applications of quantum computers often rest on their anticipated ability to perform accurate, efficient quantum chemical calculations. Computational drug discovery relies on accurate predictions of how candidate drugs interact with their targets in a cellular environment involving several thousands of atoms at finite temperatures. Although quantum computers are still far from being used as daily tools in the pharmaceutical industry, here we explore the challenges and opportunities of applying quantum computers to drug design. We discuss where these could transform industrial research and identify the substantial further developments needed to reach this goal.
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Stable quantum-correlated many-body states through engineered dissipation

Xiao Mi

Alexios Michailidis

Sara Shabani

Jerome Lloyd

Rajeev Acharya

Igor Aleiner

Trond Andersen

Markus Ansmann

Frank Arute

Kunal Arya

Juan Atalaya

Gina Bortoli

Alexandre Bourassa

Leon Brill

Michael Broughton

Bob Buckley

Tim Burger

Nicholas Bushnell

Jimmy Chen

Benjamin Chiaro

Desmond Chik

Charina Chou

Josh Cogan

Roberto Collins

Paul Conner

William Courtney

Alex Crook

Ben Curtin

Alejo Grajales Dau

Dripto Debroy

Agustin Di Paolo

ILYA Drozdov

Andrew Dunsworth

Lara Faoro

Edward Farhi

Reza Fatemi

Vinicius Ferreira

Ebrahim Forati

Austin Fowler

Brooks Foxen

Élie Genois

William Giang

Dar Gilboa

Raja Gosula

Steve Habegger

Michael Hamilton

Monica Hansen

Sean Harrington

Paula Heu

Trent Huang

Ashley Huff

Bill Huggins

Sergei Isakov

Justin Iveland

Zhang Jiang

Cody Jones

Pavol Juhas

Kostyantyn Kechedzhi

Mostafa Khezri

Marika Kieferova

Alexei Kitaev

Andrey Klots

Alexander Korotkov

Fedor Kostritsa

John Mark Kreikebaum

Dave Landhuis

Pavel Laptev

Kim Ming Lau

Lily Laws

Joonho Lee

Kenny Lee

Yuri Lensky

Alexander Lill

Wayne Liu

Orion Martin

Amanda Mieszala

Shirin Montazeri

Alexis Morvan

Ramis Movassagh

Wojtek Mruczkiewicz

Charles Neill

Ani Nersisyan

Michael Newman

JiunHow Ng

Murray Ich Nguyen

Tom O'Brien

Alex Opremcak

Andre Petukhov

Rebecca Potter

Leonid Pryadko

Charles Rocque

Negar Saei

Kannan Sankaragomathi

Henry Schurkus

Christopher Schuster

Mike Shearn

Aaron Shorter

Noah Shutty

Vladimir Shvarts

Jindra Skruzny

Clarke Smith

Rolando Somma

George Sterling

Doug Strain

Marco Szalay

Alfredo Torres

Guifre Vidal

Benjamin Villalonga

Cheng Xing

Jamie Yao

Ping Yeh

Juhwan Yoo

Grayson Young

Yaxing Zhang

Ningfeng Zhu

Jeremy Hilton

Anthony Megrant

Yu Chen

Vadim Smelyanskiy

Dmitry Abanin

Science, vol. 383 (2024), pp. 1332-1337

Preview abstract
Engineered dissipative reservoirs have the potential to steer many-body quantum systems toward correlated steady states useful for quantum simulation of high-temperature superconductivity or quantum magnetism. Using up to 49 superconducting qubits, we prepared low-energy states of the transverse-field Ising model through coupling to dissipative auxiliary qubits. In one dimension, we observed long-range quantum correlations and a ground-state fidelity of 0.86 for 18 qubits at the critical point. In two dimensions, we found mutual information that extends beyond nearest neighbors. Lastly, by coupling the system to auxiliaries emulating reservoirs with different chemical potentials, we explored transport in the quantum Heisenberg model. Our results establish engineered dissipation as a scalable alternative to unitary evolution for preparing entangled many-body states on noisy quantum processors.
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Analyzing Prospects for Quantum Advantage in Topological Data Analysis

Dominic W. Berry

Yuan Su

Casper Gyurik

Robbie King

Joao Basso

Abhishek Rajput

Nathan Wiebe

Vedran Djunko

PRX Quantum, vol. 5 (2024), pp. 010319

Preview abstract
Lloyd et al. were first to demonstrate the promise of quantum algorithms for computing Betti numbers in persistent homology (a way of characterizing topological features of data sets). Here, we propose, analyze, and optimize an improved quantum algorithm for topological data analysis (TDA) with reduced scaling, including a method for preparing Dicke states based on inequality testing, a more efficient amplitude estimation algorithm using Kaiser windows, and an optimal implementation of eigenvalue projectors based on Chebyshev polynomials. We compile our approach to a fault-tolerant gate set and estimate constant factors in the Toffoli complexity. Our analysis reveals that super-quadratic quantum speedups are only possible for this problem when targeting a multiplicative error approximation and the Betti number grows asymptotically. Further, we propose a dequantization of the quantum TDA algorithm that shows that having exponentially large dimension and Betti number are necessary, but insufficient conditions, for super-polynomial advantage. We then introduce and analyze specific problem examples for which super-polynomial advantages may be achieved, and argue that quantum circuits with tens of billions of Toffoli gates can solve some seemingly classically intractable instances.
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Evaluating the Evidence for Exponential Quantum Advantage in Ground-State Quantum Chemistry

Seunghoon Lee

Joonho Lee

Huanchen Zhai

Yu Tong

Alexander Dalzell

Ashutosh Kumar

Phillip Helms

Johnnie Gray

Zhi-Hao Cui

Michael Kastoryano

John Preskill

David Reichman

Earl Campbell

Edward Valeev

Lin Lin

Garnet Chan

Nature Communications, vol. 14 (2023)

Preview abstract
Due to intense interest in the potential applications of quantum computing, it is critical to understand the basis for potential exponential quantum advantage in quantum chemistry. Here we gather the evidence for this case in the most common task in quantum chemistry, namely, ground-state energy estimation, for generic chemical problems where heuristic quantum state preparation might be assumed to be efficient. The availability of exponential quantum advantage then centers on whether features of the physical problem that enable efficient heuristic quantum state preparation also enable efficient solution by classical heuristics. Through numerical studies of quantum state preparation and empirical complexity analysis (including the error scaling) of classical heuristics, in both ab initio and model Hamiltonian settings, we conclude that evidence for such an exponential advantage across chemical space has yet to be found. While quantum computers may still prove useful for ground-state quantum chemistry through polynomial speedups, it may be prudent to assume exponential speedups are not generically available for this problem.
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Quantum computation of stopping power for inertial fusion target design

Dominic Berry

Alina Kononov

Alec White

Joonho Lee

Andrew Baczewski

arXiv preprint (2023)

Preview abstract
Stopping power is the rate at which a material absorbs the kinetic energy of a charged particle passing through it - one of many properties needed over a wide range of thermodynamic conditions in modeling inertial fusion implosions. First-principles stopping calculations are classically challenging because they involve the dynamics of large electronic systems far from equilibrium, with accuracies that are particularly difficult to constrain and assess in the warm-dense conditions preceding ignition. Here, we describe a protocol for using a fault-tolerant quantum computer to calculate stopping power from a first-quantized representation of the electrons and projectile. Our approach builds upon the electronic structure block encodings of Su et al. [PRX Quantum 2, 040332 2021], adapting and optimizing those algorithms to estimate observables of interest from the non-Born-Oppenheimer dynamics of multiple particle species at finite temperature. We also work out the constant factors associated with a novel implementation of a high order Trotter approach to simulating a grid representation of these systems. Ultimately, we report logical qubit requirements and leading-order Toffoli costs for computing the stopping power of various projectile/target combinations relevant to interpreting and designing inertial fusion experiments. We estimate that scientifically interesting and classically intractable stopping power calculations can be quantum simulated with
roughly the same number of logical qubits and about one hundred times more Toffoli gates than is required for state-of-the-art quantum simulations of industrially relevant molecules such as FeMoCo or P450.
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Measurement-induced entanglement and teleportation on a noisy quantum processor

Jesse Hoke

Matteo Ippoliti

Dmitry Abanin

Rajeev Acharya

Trond Andersen

Markus Ansmann

Frank Arute

Kunal Arya

Juan Atalaya

Gina Bortoli

Alexandre Bourassa

Leon Brill

Michael Broughton

Bob Buckley

Tim Burger

Nicholas Bushnell

Jimmy Chen

Benjamin Chiaro

Desmond Chik

Josh Cogan

Roberto Collins

Paul Conner

William Courtney

Alex Crook

Ben Curtin

Alejo Grajales Dau

Agustin Di Paolo

ILYA Drozdov

Andrew Dunsworth

Daniel Eppens

Edward Farhi

Reza Fatemi

Vinicius Ferreira

Ebrahim Forati

Austin Fowler

Brooks Foxen

William Giang

Dar Gilboa

Raja Gosula

Steve Habegger

Michael Hamilton

Monica Hansen

Paula Heu

Trent Huang

Ashley Huff

Bill Huggins

Sergei Isakov

Justin Iveland

Zhang Jiang

Cody Jones

Pavol Juhas

Kostyantyn Kechedzhi

Mostafa Khezri

Marika Kieferova

Alexei Kitaev

Andrey Klots

Alexander Korotkov

Fedor Kostritsa

John Mark Kreikebaum

Dave Landhuis

Pavel Laptev

Kim Ming Lau

Lily Laws

Joonho Lee

Kenny Lee

Yuri Lensky

Alexander Lill

Wayne Liu

Orion Martin

Amanda Mieszala

Shirin Montazeri

Alexis Morvan

Ramis Movassagh

Wojtek Mruczkiewicz

Charles Neill

Ani Nersisyan

Michael Newman

JiunHow Ng

Murray Ich Nguyen

Tom O'Brien

Seun Omonije

Alex Opremcak

Andre Petukhov

Rebecca Potter

Leonid Pryadko

Charles Rocque

Negar Saei

Kannan Sankaragomathi

Henry Schurkus

Christopher Schuster

Mike Shearn

Aaron Shorter

Noah Shutty

Vladimir Shvarts

Jindra Skruzny

Clarke Smith

Rolando Somma

George Sterling

Doug Strain

Marco Szalay

Alfredo Torres

Guifre Vidal

Benjamin Villalonga

Cheng Xing

Jamie Yao

Ping Yeh

Juhwan Yoo

Grayson Young

Yaxing Zhang

Ningfeng Zhu

Jeremy Hilton

Anthony Megrant

Yu Chen

Vadim Smelyanskiy

Xiao Mi

Vedika Khemani

Nature, vol. 622 (2023), 481–486

Preview abstract
Measurement has a special role in quantum theory: by collapsing the wavefunction, it can enable phenomena such as teleportation and thereby alter the ‘arrow of time’ that constrains unitary evolution. When integrated in many-body dynamics, measurements can lead to emergent patterns of quantum information in space–time that go beyond the established paradigms for characterizing phases, either in or out of equilibrium. For present-day noisy intermediate-scale quantum (NISQ) processors, the experimental realization of such physics can be problematic because of hardware limitations and the stochastic nature of quantum measurement. Here we address these experimental challenges and study measurement-induced quantum information phases on up to 70 superconducting qubits. By leveraging the interchangeability of space and time, we use a duality mapping to avoid mid-circuit measurement and access different manifestations of the underlying phases, from entanglement scaling to measurement-induced teleportation. We obtain finite-sized signatures of a phase transition with a decoding protocol that correlates the experimental measurement with classical simulation data. The phases display remarkably different sensitivity to noise, and we use this disparity to turn an inherent hardware limitation into a useful diagnostic. Our work demonstrates an approach to realizing measurement-induced physics at scales that are at the limits of current NISQ processors.
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Fault-Tolerant Quantum Simulation of Materials Using Bloch Orbitals

Dominic Berry

Alec White

Eugene DePrince III

Sabrina Sicolo

Michael Kuehn

Michael Kaicher

Joonho Lee

PRX Quantum, vol. 4 (2023), pp. 040303

Preview abstract
The simulation of chemistry is among the most promising applications of quantum computing. However, most prior work exploring algorithms for block encoding, time evolving, and sampling in the eigenbasis of electronic structure Hamiltonians has either focused on modeling finite-sized systems, or has required a large number of plane-wave basis functions. In this work, we extend methods for quantum simulation with Bloch orbitals constructed from symmetry-adapted atom-centered orbitals so that one can model periodic ab initio Hamiltonians using only a modest number of basis functions. We focus on adapting existing algorithms based on combining qubitization with tensor factorizations of the Coulomb operator. Significant modifications of those algorithms are required to obtain an asymptotic speedup leveraging translational (or, more broadly, Abelian) symmetries. We implement block encodings using known tensor factorizations and a new Bloch orbital form of tensor hypercontraction. Finally, we estimate the resources required to deploy our algorithms to classically challenging model materials relevant to the chemistry of lithium nickel oxide battery cathodes within the surface code. We find that even with these improvements, the quantum runtime of these algorithms is on the order of thousands of days and further algorithmic improvements are required to make realistic quantum simulation of materials practical.
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Quantum Error Mitigation

Zhenyu Cai

Simon Benjamin

Suguru Endo

William J. Huggins

Ying Li

Thomas E O'Brien

Reviews of Modern Physics, vol. 95 (2023), pp. 045005

Preview abstract
For quantum computers to successfully solve real-world problems, it is necessary to tackle the challenge of noise: the errors that occur in elementary physical components due to unwanted or imperfect interactions. The theory of quantum fault tolerance can provide an answer in the long term, but in the coming era of noisy intermediate-scale quantum machines one must seek to mitigate errors rather than completely eliminate them. This review surveys the diverse methods that have been proposed for quantum error mitigation, assesses their in-principle efficacy, and describes the hardware demonstrations achieved to date. Commonalities and limitations among the methods are identified, while mention is made of how mitigation methods can be chosen according to the primary type of noise present, including algorithmic errors. Open problems in the field are identified, and the prospects for realizing mitigation-based devices that can deliver a quantum advantage with an impact on science and business are discussed.
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Purification-Based Quantum Error Mitigation of Pair-Correlated Electron Simulations

Thomas E O'Brien

Gian-Luca R. Anselmetti

Fotios Gkritsis

Vincent Elfving

Stefano Polla

William J. Huggins

Oumarou Oumarou

Kostyantyn Kechedzhi

Dmitry Abanin

Rajeev Acharya

Igor Aleiner

Richard Ross Allen

Trond Ikdahl Andersen

Kyle Anderson

Markus Ansmann

Frank Carlton Arute

Kunal Arya

Juan Atalaya

Michael Blythe Broughton

Bob Benjamin Buckley

Alexandre Bourassa

Leon Brill

Tim Burger

Nicholas Bushnell

Jimmy Chen

Yu Chen

Benjamin Chiaro

Desmond Chun Fung Chik

Josh Godfrey Cogan

Roberto Collins

Paul Conner

William Courtney

Alex Crook

Ben Curtin

Ilya Drozdov

Andrew Dunsworth

Daniel Eppens

Lara Faoro

Edward Farhi

Reza Fatemi

Ebrahim Forati

Austin Fowler

Brooks Riley Foxen

William Giang

Dar Gilboa

Alejandro Grajales Dau

Steve Habegger

Michael C. Hamilton

Sean Harrington

Jeremy Patterson Hilton

Trent Huang

Ashley Anne Huff

Sergei Isakov

Justin Thomas Iveland

Cody Jones

Pavol Juhas

Mostafa Khezri

Marika Kieferova

Andrey Klots

Alexander Korotkov

Fedor Kostritsa

John Mark Kreikebaum

Dave Landhuis

Pavel Laptev

Kim Ming Lau

Lily MeeKit Laws

Joonho Lee

Kenny Lee

Alexander T. Lill

Wayne Liu

Orion Martin

Trevor Johnathan Mccourt

Anthony Megrant

Xiao Mi

Masoud Mohseni

Shirin Montazeri

Alexis Morvan

Ramis Movassagh

Wojtek Mruczkiewicz

Charles Neill

Ani Nersisyan

Michael Newman

Jiun How Ng

Murray Nguyen

Alex Opremcak

Andre Gregory Petukhov

Rebecca Potter

Kannan Aryaperumal Sankaragomathi

Christopher Schuster

Mike Shearn

Aaron Shorter

Vladimir Shvarts

Jindra Skruzny

Vadim Smelyanskiy

Clarke Smith

Rolando Diego Somma

Doug Strain

Marco Szalay

Alfredo Torres

Guifre Vidal

Benjamin Villalonga

Jamie Yao

Ping Yeh

Juhwan Yoo

Grayson Robert Young

Yaxing Zhang

Ningfeng Zhu

Christian Gogolin

Nature Physics (2023)

Preview abstract
An important measure of the development of quantum computing platforms has been the simulation of increasingly complex physical systems. Prior to fault-tolerant quantum computing, robust error mitigation strategies are necessary to continue this growth. Here, we study physical simulation within the seniority-zero electron pairing subspace, which affords both a computational stepping stone to a fully correlated model, and an opportunity to validate recently introduced ``purification-based'' error-mitigation strategies. We compare the performance of error mitigation based on doubling quantum resources in time (echo verification) or in space (virtual distillation), on up to 20 qubits of a superconducting qubit quantum processor. We observe a reduction of error by one to two orders of magnitude below less sophisticated techniques (e.g. post-selection); the gain from error mitigation is seen to increase with the system size. Employing these error mitigation strategies enables the implementation of the largest variational algorithm for a correlated chemistry system to-date. Extrapolating performance from these results allows us to estimate minimum requirements for a beyond-classical simulation of electronic structure. We find that, despite the impressive gains from purification-based error mitigation, significant hardware improvements will be required for classically intractable variational chemistry simulations.
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Quantum Simulation of Realistic Materials in First Quantization Using Non-local Pseudopotentials

Dominic Berry

Ahmed Elnabawy

Gabriele Ahlers

Albert Eugene DePrince III

Joonho Lee

Christian Gogolin

arXiv:2312.07654 (2023)

Preview abstract
This paper improves and demonstrates the usefulness of the first quantized plane-wave algorithms for the quantum simulation of electronic structure, developed by Babbush et al. and Su et al. We describe the first quantum algorithm for first quantized simulation that accurately includes pseudopotentials. We focus on the Goedecker-Tetter-Hutter (GTH) pseudopotential, which is among the most accurate and widely used norm-conserving pseudopotentials enabling the removal of core electrons from the simulation. The resultant screened nuclear potential regularizes cusps in the electronic wavefunction so that orders of magnitude fewer plane waves are required for a chemically accurate basis. Despite the complicated form of the GTH pseudopotential, we are able to block encode the associated operator without significantly increasing the overall cost of quantum simulation. This is surprising since simulating the nuclear potential is much simpler without pseudopotentials, yet is still the bottleneck. We also generalize prior methods to enable the simulation of materials with non-cubic unit cells, which requires nontrivial modifications. Finally, we combine these techniques to estimate the block-encoding costs for commercially relevant instances of heterogeneous catalysis (e.g. carbon monoxide adsorption on transition metals) and compare to the quantum resources needed to simulate materials in second quantization. We conclude that for computational cells with many particles, first quantization often requires meaningfully less spacetime volume.
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Exponential Quantum Speedup in Simulating Coupled Classical Oscillators

Dominic Berry

Rolando Somma

Nathan Wiebe

Physical Review X, vol. 13 (2023), pp. 041041

Preview abstract
We present a quantum algorithm for simulating the classical dynamics of 2^n coupled oscillators (e.g., 2^n masses coupled by springs). Our approach leverages a mapping between the Schrodinger equation and Newton's equations for harmonic potentials such that the amplitudes of the evolved quantum state encode the momenta and displacements of the classical oscillators. When individual masses and spring constants can be efficiently queried, and when the initial state can be efficiently prepared, the complexity of our quantum algorithm is polynomial in n, almost linear in the evolution time, and sublinear in the sparsity. As an example application, we apply our quantum algorithm to efficiently estimate the kinetic energy of an oscillator at any time, for a specification of the problem that we prove is \BQP-complete. Thus, our approach solves a potentially practical application with an exponential speedup over classical computers. Finally, we show that under similar conditions our approach can efficiently simulate more general classical harmonic systems with 2^n modes.
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Quantum Simulation of Exact Electron Dynamics can be more Efficient than Classical Mean-Field Methods

William J. Huggins

Dominic W. Berry

Shu Fay Ung

Andrew Zhao

David Reichman

Andrew Baczewski

Joonho Lee

Nature Communications, vol. 14 (2023), pp. 4058

Preview abstract
Quantum algorithms for simulating electronic ground states are slower than popular classical mean-field algorithms such as Hartree-Fock and density functional theory, but offer higher accuracy. Accordingly, quantum computers have been predominantly regarded as competitors to only the most accurate and costly classical methods for treating electron correlation. However, here we tighten bounds showing that certain first quantized quantum algorithms enable exact time evolution of electronic systems with exponentially less space and polynomially fewer operations in basis set size than conventional real-time time-dependent Hartree-Fock and density functional theory. Although the need to sample observables in the quantum algorithm reduces the speedup, we show that one can estimate all elements of the k-particle reduced density matrix with a number of samples scaling only polylogarithmically in basis set size. We also introduce a more efficient quantum algorithm for first quantized mean-field state preparation that is likely cheaper than the cost of time evolution. We conclude that quantum speedup is most pronounced for finite temperature simulations and suggest several practically important electron dynamics problems with potential quantum advantage.
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Nearly Optimal Quantum Algorithm for Estimating Multiple Expectation Values

William J. Huggins

Kianna Wan

Thomas E O'Brien

Nathan Wiebe

Physical Review Letters, vol. 129 (2022), pp. 240501

Preview abstract
Many quantum algorithms involve the evaluation of expectation values. Optimal strategies for estimating a single expectation value are known, requiring a number of iterations that scales with the target error $\epsilon$ as $\mathcal{O}(\epsilon^{-1})$. In this paper we address the task of estimating the expectation values of \(M\) different observables, each to within an error \(\epsilon\), with the same \(\epsilon^{-1}\) dependence. We describe an approach that leverages Gily\'{e}n \emph{et al.}'s~quantum gradient estimation algorithm to achieve $\mathcal{O}\sqrt{M}\epsilon^{-1})$ scaling up to logarithmic factors, regardless of the commutation properties of the $M$ observables.
We prove that this scaling is optimal in the worst case, even when the operators are mutually commuting. We highlight the flexibility of our approach by presenting several generalizations, including a strategy for accelerating the estimation of a collection of dynamic correlation functions.
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Reliably Assessing the Electronic Structure of Cytochrome P450 on Today’s Classical Computers and Tomorrow’s Quantum Computers

Joshua Goings

Alec White

Joonho Lee

Christofer Tautermann

Matthias Degroote

Toru Shiozaki

PNAS, vol. 119 (2022)

Preview abstract
An accurate assessment of how quantum computers can be used for chemical simulation, especially their potential computational advantages, provides important context on how to deploy these future devices. To perform this assessment reliably, quantum resource estimates must be coupled with classical computations attempting to answer relevant chemical questions and to define the classical algorithms simulation frontier. Herein, we explore the quantum computation and classical computation resources required to assess the electronic structure of cytochrome P450 enzymes (CYPs) and thus define a classical–quantum advantage boundary. This is accomplished by analyzing the convergence of density matrix renormalization group plus n-electron valence state perturbation theory (DMRG+NEVPT2) and coupled-cluster singles doubles with noniterative triples [CCSD(T)] calculations for spin gaps in models of the CYP catalytic cycle that indicate multireference character. The quantum resources required to perform phase estimation using qubitized quantum walks are calculated for the same systems. Compilation into the surface code provides runtime estimates to compare directly to DMRG runtimes and to evaluate potential quantum advantage. Both classical and quantum resource estimates suggest that simulation of CYP models at scales large enough to balance dynamic and multiconfigurational electron correlation has the potential to be a quantum advantage problem and emphasizes the important interplay between classical computations and quantum algorithms development for chemical simulation.
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Efficient Quantum Computation of Molecular Forces and Other Energy Gradients

Thomas E O'Brien

Michael Streif

Raffaele Santagati

Yuan Su

William J. Huggins

Joshua Goings

Nikolaj Moll

Elica Kyoseva

Matthias Degroote

Christofer Tautermann

Joonho Lee

Dominic Berry

Nathan Wiebe

Physical Review Research, vol. 4 (2022), pp. 043210

Preview abstract
While most work on the quantum simulation of chemistry has focused on computing energy surfaces, a similarly important application requiring subtly different algorithms is the computation of energy derivatives. Almost all molecular properties can be expressed an energy derivative, including molecular forces, which are essential for applications such as molecular dynamics simulations. Here, we introduce new quantum algorithms for computing molecular energy derivatives with significantly lower complexity than prior methods. Under cost models appropriate for noisy-intermediate scale quantum devices we demonstrate how low rank factorizations and other tomography schemes can be optimized for energy derivative calculations. We perform numerics revealing that our techniques reduce the number of circuit repetitions required by many orders of magnitude for even modest systems. In the context of fault-tolerant algorithms, we develop new methods of estimating energy derivatives with Heisenberg limited scaling incorporating state-of-the-art techniques for block encoding fermionic operators. Our results suggest that the calculation of forces on a single nuclei may be of similar cost to estimating energies of chemical systems, but that further developments are needed for quantum computers to meaningfully assist with molecular dynamics simulations.
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Noise-resilient Majorana Edge Modes on a Chain of Superconducting Qubits

Alejandro Grajales Dau

Alex Crook

Alex Opremcak

Alexa Rubinov

Alexander Korotkov

Alexandre Bourassa

Alexei Kitaev

Alexis Morvan

Andre Gregory Petukhov

Andrew Dunsworth

Andrey Klots

Anthony Megrant

Ashley Anne Huff

Austin Fowler

Benjamin Chiaro

Benjamin Villalonga

Bernardo Meurer Costa

Bob Benjamin Buckley

Brooks Foxen

Charles Neill

Christopher Schuster

Cody Jones

Daniel Eppens

Dar Gilboa

Dave Landhuis

Dmitry Abanin

Doug Strain

Ebrahim Forati

Edward Farhi

Emily Mount

Fedor Kostritsa

Frank Carlton Arute

Guifre Vidal

Igor Aleiner

Jamie Yao

Jeremy Patterson Hilton

Joao Basso

John Mark Kreikebaum

Joonho Lee

Juan Atalaya

Juhwan Yoo

Justin Thomas Iveland

Kannan Aryaperumal Sankaragomathi

Kenny Lee

Kim Ming Lau

Kostyantyn Kechedzhi

Kunal Arya

Lara Faoro

Leon Brill

Marco Szalay

Masoud Mohseni

Michael Blythe Broughton

Michael Newman

Michel Henri Devoret

Mike Shearn

Nicholas Bushnell

Orion Martin

Paul Conner

Pavel Laptev

Ping Yeh

Rajeev Acharya

Rebecca Potter

Reza Fatemi

Roberto Collins

Sergei Isakov

Shirin Montazeri

Steve Habegger

Thomas E O'Brien

Trent Huang

Trond Ikdahl Andersen

Vadim Smelyanskiy

Vladimir Shvarts

Wayne Liu

William Courtney

William Giang

William J. Huggins

Wojtek Mruczkiewicz

Xiao Mi

Yaxing Zhang

Yu Chen

Yuan Su

Zhang Jiang

Zijun Chen

Science (2022) (to appear)

Preview abstract
Inherent symmetry of a quantum system may protect its otherwise fragile states. Leveraging such protection requires testing its robustness against uncontrolled environmental interactions. Using 47 superconducting qubits, we implement the kicked Ising model which exhibits Majorana edge modes (MEMs) protected by a $\mathbb{Z}_2$-symmetry. Remarkably, we find that any multi-qubit Pauli operator overlapping with the MEMs exhibits a uniform decay rate comparable to single-qubit relaxation rates, irrespective of its size or composition. This finding allows us to accurately reconstruct the exponentially localized spatial profiles of the MEMs. Spectroscopic measurements further indicate exponentially suppressed hybridization between the MEMs over larger system sizes, which manifests as a strong resilience against low-frequency noise. Our work elucidates the noise sensitivity of symmetry-protected edge modes in a solid-state environment.
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Simulating Challenging Correlated Molecules and Materials on the Sycamore Quantum Processor

Ruslan Tazhigulov

Shi-Ning Sun

Reza Haghshenas

Huanchen Zhai

Adrian Tan

Austin Minnich

Garnet Kin-Lic Chan

PRX Quantum, vol. 3 (2022), pp. 040318

Preview abstract
Simulating complex molecules and materials is an anticipated application of quantum devices. With strong quantum advantage demonstrated in artificial tasks, we examine how such advantage translates into modeling physical problems, and in particular, strongly correlated electronic structure. We simulate
static and dynamical electronic structure on a superconducting quantum processor derived from Google’s Sycamore architecture for two representative correlated electron problems: the nitrogenase iron-sulfur molecular clusters, and α-ruthenium trichloride, a proximate spin-liquid material. To do so, we simplify the electronic structure into low-energy spin models that fit on the device. With extensive error mitigation and assistance from classically simulated data, we achieve quantitatively meaningful results deploying about 1/5 of the gate resources used in artificial quantum advantage experiments
on a similar architecture. This increases to over 1/2 of the gate resources when choosing a model that suits the hardware. Our work serves to convert artificial measures of quantum advantage into a physically relevant setting.
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Quantum Computation of Molecular Structure using Data from Challenging-to-Classically-Simulate Nuclear Magnetic Resonance Experiments

Thomas E O'Brien

Yuan Su

David Fushman

Vadim Smelyanskiy

PRX Quantum, vol. 3 (2022)

Preview abstract
We propose a quantum algorithm for inferring the molecular nuclear spin Hamiltonian from time-resolved measurements of spin-spin correlators, which can be obtained via nuclear magnetic resonance (NMR). We focus on learning the anisotropic dipolar term of the Hamiltonian, which generates dynamics that are challenging to classically simulate in some contexts. We demonstrate the ability to directly estimate the Jacobian and Hessian of the corresponding learning problem on a quantum computer, allowing us to learn the Hamiltonian parameters. We develop algorithms for performing this computation on both noisy near-term and future fault-tolerant quantum computers. We argue that the former is promising as an early beyond-classical quantum application since it only requires evolution of a local spin Hamiltonian. We investigate the example of a protein (ubiquitin) confined on a membrane as a benchmark of our method. We isolate small spin clusters, demonstrate the convergence of our learning algorithm on one such example, and then investigate the learnability of these clusters as we cross the ergodic to nonergodic phase transition by suppressing the dipolar interaction. We see a clear correspondence between a drop in the multifractal dimension measured across many-body eigenstates of these clusters, and a transition in the structure of the Hessian of the learning cost function (from degenerate to learnable). Our hope is that such quantum computations might enable the interpretation and development of new NMR techniques for analyzing molecular structure.
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Optimal Scaling Quantum Linear Systems Solver via Discrete Adiabatic Theorem

Pedro C. S. Costa

Dong An

Yuval Sanders

Yuan Su

Dominic W. Berry

PRX Quantum, vol. 3 (2022), pp. 040303

Preview abstract
Recently, several approaches to solving linear systems on a quantum computer have been formulated in terms of the quantum adiabatic theorem for a continuously varying Hamiltonian. Such approaches enabled near-linear scaling in the condition number $\kappa$ of the linear system, without requiring a complicated variable-time amplitude amplification procedure. However, the most efficient of those procedures is still asymptotically sub-optimal by a factor of $\log(\kappa)$. Here, we prove a rigorous form of the adiabatic theorem that bounds the error in terms of the spectral gap for intrinsically discrete time evolutions. We use this discrete adiabatic theorem to develop a quantum algorithm for solving linear systems that is asymptotically optimal, in the sense that the complexity is strictly linear in $\kappa$, matching a known lower bound on the complexity. Our $\mathcal{O}(\kappa\log(1/\epsilon))$ complexity is also optimal in terms of the combined scaling in $\kappa$ and the precision $\epsilon$. Compared to existing suboptimal methods, our algorithm is simpler and easier to implement. Moreover, we determine the constant factors in the algorithm, which would be suitable for determining the complexity in terms of gate counts for specific applications.
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Realizing topologically ordered states on a quantum processor

Y.-J. Liu

A. Smith

C. Knapp

M. Newman

N. C. Jones

Z. Chen

X. Mi

A. Dunsworth

I. Aleiner

F. Arute

K. Arya

J. Atalaya

R. Barends

J. Basso

M. Broughton

B. B. Buckley

N. Bushnell

B. Chiaro

R. Collins

W. Courtney

A. R Derk

D. Eppens

L. Faoro

E. Farhi

B. Foxen

A. Greene

S. D. Harrington

J. Hilton

T. Huang

W. J. Huggins

S. V. Isakov

Z. Jiang

K. Kechedzhi

A. N. Korotkov

F. Kostritsa

D. Landhuis

P. Laptev

O. Martin

M. Mohseni

S. Montazeri

W. Mruczkiewicz

J. Mutus

C. Neill

T. E. O'Brien

A. Opremcak

B. Pato

A. Petukhov

V. Shvarts

D. Strain

M. Szalay

B. Villalonga

Z. Yao

P. Yeh

J. Yoo

A. Megrant

Y. Chen

V. Smelyanskiy

A. Kitaev

M. Knap

F. Pollmann

Science, vol. 374 (2021), pp. 1237-1241

Preview abstract
The discovery of topological order has revolutionized the understanding of quantum matter in modern physics and provided the theoretical foundation for many quantum error correcting codes. Realizing topologically ordered states has proven to be extremely challenging in both condensed matter and synthetic quantum systems. Here, we prepare the ground state of the emblematic toric code Hamiltonian using an efficient quantum circuit on a superconducting quantum processor. We measure a topological entanglement entropy of Stopo ≈ −0.95 × ln 2 and simulate anyon interferometry to extract the braiding statistics of the emergent excitations. Furthermore, we investigate key aspects of the surface code, including logical state injection and the decay of the non-local order parameter. Our results illustrate the topological nature of these states and demonstrate their potential for implementing the surface code.
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Fault-Tolerant Quantum Simulations of Chemistry in First Quantization

Yuan Su

Dominic W. Berry

Nathan Wiebe

PRX Quantum, vol. 2 (2021), pp. 040332

Preview abstract
Quantum simulations of chemistry in first quantization offer important advantages over approaches in second quantization including faster convergence to the continuum limit and the opportunity for practical simulations outside the Born-Oppenheimer approximation. However, as all prior work on quantum simulation in first quantization has been limited to asymptotic analysis, it has been impossible to compare the resources required for these approaches to those for more commonly studied algorithms in second quantization. Here, we analyze and optimize the resources required to implement two first quantized quantum algorithms for chemistry from Babbush et al that realize block encodings for the qubitization and interaction picture frameworks of Low et al. The two algorithms we study enable simulation with gate complexities O(η^{8/3} N^{1/3} t+η^{4/3} N^{2/3} t) and O(η^{8/3} N^{1/3} t) where η is the number of electrons, N is the number of plane wave basis functions, and t is the duration of time-evolution (t is inverse to target precision when the goal is to estimate energies). In addition to providing the first explicit circuits and constant factors for any first quantized simulation and introducing improvements which reduce circuit complexity by about a thousandfold over naive implementations for modest sized systems, we also describe new algorithms that asymptotically achieve the same scaling in a real space representation. We assess the resources required to simulate various molecules and materials and conclude that the qubitized algorithm will often be more practical than the interaction picture algorithm. We demonstrate that our qubitized algorithm often requires much less surface code spacetime volume for simulating millions of plane waves than the best second quantized algorithms require for simulating hundreds of Gaussian orbitals.
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Focus Beyond Quadratic Speedups for Error-Corrected Quantum Advantage

Michael Newman

PRX Quantum, vol. 2 (2021), pp. 010103

Preview abstract
In this perspective we discuss conditions under which it would be possible for a modest fault-tolerant quantum computer to realize a runtime advantage by executing a quantum algorithm with only a small polynomial speedup over the best classical alternative. The challenge is that the computation must finish within a reasonable amount of time while being difficult enough that the small quantum scaling advantage would compensate for the large constant factor overheads associated with error correction. We compute several examples of such runtimes using state-of-the-art surface code constructions under a variety of assumptions. We conclude that quadratic speedups will not enable quantum advantage on early generations of such fault-tolerant devices unless there is a significant improvement in how we realize quantum error correction. While this conclusion persists even if we were to increase the rate of logical gates in the surface code by more than an order of magnitude, we also repeat this analysis for speedups by other polynomial degrees and find that quartic speedups look significantly more practical.
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Unbiasing Fermionic Quantum Monte Carlo with a Quantum Computer

William J. Huggins

Bryan O'Gorman

David Reichman

Joonho Lee

Nature (2021)

Preview abstract
Interacting many-electron problems pose some of the greatest computational challenges in science, with essential applications across many fields. The solutions to these problems will offer accurate predictions of chemical reactivity and kinetics, and other properties of quantum systems. Fermionic quantum Monte Carlo (QMC) methods which use a statistical sampling of the ground state, are among the most powerful approaches to these problems. Controlling the fermionic sign problem with constraints ensures the efficiency of QMC at the expense of potentially significant biases owing to the limited flexibility of classical computation. Here we propose an approach that combines constrained QMC with quantum computation to reduce such biases. We implement our scheme experimentally using up to 16 qubits to unbias constrained QMC calculations performed on chemical systems with as many as 120 orbitals. These experiments represent the largest chemistry simulations performed with the help of quantum computers, while achieving accuracy that is competitive with state-of-the-art classical methods without burdensome error mitigation. Compared with the popular variational quantum eigensolver our hybrid quantum-classical computational model offers an alternative path towards achieving a practical quantum advantage for the electronic structure problem without demanding exceedingly accurate preparation and measurement of the ground-state wavefunction.
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Efficient and Noise Resilient Measurements for Quantum Chemistry on Near-Term Quantum Computers

William Huggins

Zhang Jiang

Nathan Wiebe

K. Birgitta Whaley

Nature Quantum Information, vol. 7 (2021)

Preview abstract
Variational algorithms are a promising paradigm for utilizing near-term quantum devices for modeling electronic states of molecular systems. However, previous bounds on the measurement time required have suggested that the application of these techniques to larger molecules might be infeasible. We present a measurement strategy based on a low-rank factorization of the two-electron integral tensor. Our approach provides a cubic reduction in term groupings over prior state-of-the-art and enables measurement times three orders of magnitude smaller than those suggested by commonly referenced bounds for the largest systems we consider. Although our technique requires execution of a linear-depth circuit prior to measurement, this is compensated for by eliminating challenges associated with sampling nonlocal Jordan–Wigner transformed operators in the presence of measurement error, while enabling a powerful form of error mitigation based on efficient postselection. We numerically characterize these benefits with noisy quantum circuit simulations for ground-state energies of strongly correlated electronic systems.
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Even More Efficient Quantum Computations of Chemistry through Tensor Hyper-Contraction

Joonho Lee

Dominic W. Berry

William J. Huggins

Nathan Wiebe

PRX Quantum, vol. 2 (2021), pp. 030305

Preview abstract
We describe quantum circuits with only $\widetilde{\cal O}(N)$ Toffoli complexity that block encode the spectra of quantum chemistry Hamiltonians in a basis of $N$ arbitrary (e.g., molecular) orbitals. With ${\cal O}(\lambda / \epsilon)$ repetitions of these circuits one can use phase estimation to sample in the molecular eigenbasis, where $\lambda$ is the 1-norm of Hamiltonian coefficients and $\epsilon$ is the target precision. This is the lowest complexity that has been shown for quantum computations of chemistry within an arbitrary basis. Furthermore, up to logarithmic factors, this matches the scaling of the most efficient prior block encodings that can only work with orthogonal basis functions diagonalizing the Coloumb operator (e.g., the plane wave dual basis). Our key insight is to factorize the Hamiltonian using a method known as tensor hypercontraction (THC) and then to transform the Coulomb operator into an isospectral diagonal form with a non-orthogonal basis defined by the THC factors. We then use qubitization to simulate the non-orthogonal THC Hamiltonian, in a fashion that avoids most complications of the non-orthogonal basis. We also reanalyze and reduce the cost of several of the best prior algorithms for these simulations in order to facilitate a clear comparison to the present work. In addition to having lower asymptotic scaling spacetime volume, compilation of our algorithm for challenging finite-sized molecules such as FeMoCo reveals that our method requires the least fault-tolerant resources of any known approach.
By laying out and optimizing the surface code resources required of our approach we show that FeMoCo can be simulated using about four million physical qubits and under four days of runtime, assuming $1 \, \mu {\rm s}$ cycle times and physical gate error rates no worse than $0.1\%$.
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Low-Depth Mechanisms for Quantum Optimization

Masoud Mohseni

Zhang Jiang

Vadim Smelyanskiy

PRX Quantum, vol. 3 (2021), pp. 030312

Preview abstract
One of the major application areas of interest for both near-term and fault-tolerant quantum computers is the optimization of classical objective functions. In this work, we develop intuitive constructions for a large class of these algorithms based on connections to simple dynamics of quantum systems, quantum walks, and classical continuous relaxations. We focus on developing a language and tools connected with kinetic energy on a graph for understanding the physical mechanisms of success and failure to guide algorithmic improvement. This physical language, in combination with uniqueness results related to unitarity, allow us to identify some potential pitfalls from kinetic energy fundamentally opposing the goal of optimization. This is connected to effects from wavefunction confinement, phase randomization, and shadow defects lurking in the objective far away from the ideal solution. As an example, we explore the surprising deficiency of many quantum methods in solving uncoupled spin problems and how this is both predictive of performance on some more complex systems while immediately suggesting simple resolutions. Further examination of canonical problems like the Hamming ramp or bush of implications show that entanglement can be strictly detrimental to performance results from the underlying mechanism of solution in approaches like QAOA. Kinetic energy and graph Laplacian perspectives provide new insights to common initialization and optimal solutions in QAOA as well as new methods for more effective layerwise training. Connections to classical methods of continuous extensions, homotopy methods, and iterated rounding suggest new directions for research in quantum optimization. Throughout, we unveil many pitfalls and mechanisms in quantum optimization using a physical perspective, which aim to spur the development of novel quantum optimization algorithms and refinements.
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Quantum advantage in learning from experiments

Hsin-Yuan (Robert) Huang

Michael Blythe Broughton

Jordan Cotler

Sitan Chen

Jerry Li

Masoud Mohseni

Richard Kueng

John Preskill

Science, vol. 376 (2021), pp. 1182 - 1186

Preview abstract
Quantum technology has the potential to revolutionize how we acquire and process experimental data to learn about the physical world. An experimental setup that transduces data from a physical system to a stable quantum memory, and processes that data using a quantum computer, could have significant advantages over conventional experiments in which the physical system is measured and the outcomes are processed using a classical computer. We prove that, in various tasks, quantum machines can learn from exponentially fewer experiments than those required in conventional experiments. The exponential advantage holds in predicting properties of physical systems, performing quantum principal component analysis on noisy states, and learning approximate models of physical dynamics. In some tasks, the quantum processing needed to achieve the exponential advantage can be modest; for example, one can simultaneously learn about many noncommuting observables by processing only two copies of the system. Conducting experiments with up to 40 superconducting qubits and 1300 quantum gates, we demonstrate that a substantial quantum advantage can be realized using today's relatively noisy quantum processors. Our results highlight how quantum technology can enable powerful new strategies to learn about nature.
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What the foundations of quantum computer science teach us about chemistry

Joonho Lee

Thomas E O'Brien

William J. Huggins

Hsin-Yuan Huang

Journal of Chemical Physics, vol. 155 (2021), pp. 150901

Preview abstract
With the rapid development of quantum technology, one of the leading applications that has been identified is the simulation of chemistry. Interestingly, even before full scale quantum computers are available, quantum computer science has exhibited a remarkable string of results that directly impact what is possible in chemical simulation, even with a quantum computer. Some of these results even impact our understanding of chemistry in the real world. In this perspective, we take the position that direct chemical simulation is best understood as a digital experiment. While on one hand this clarifies the power of quantum computers to extend our reach, it also shows us the limitations of taking such an approach too directly. Leveraging results that quantum computers cannot outpace the physical world, we build to the controversial stance that some chemical problems are best viewed as problems for which no algorithm can deliver their solution in general, known in computer science as undecidable problems. This has implications for the predictive power of thermodynamic models and topics like the ergodic hypothesis. However, we argue that this perspective is not defeatist, but rather helps shed light on the success of existing chemical models like transition state theory, molecular orbital theory, and thermodynamics as models that benefit from data. We contextualize recent results showing that data-augmented models are more powerful rote simulation. These results help us appreciate the success of traditional chemical theory and anticipate new models learned from experimental data. Not only can quantum computers provide data for such models, but they can extend the class and power of models that utilize data in fundamental ways. These discussions culminate in speculation on new ways for quantum computing and chemistry to interact and our perspective on the eventual roles of quantum computers in the future of chemistry.
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Power of data in quantum machine learning

Hsin-Yuan (Robert) Huang

Michael Blythe Broughton

Masoud Mohseni

Nature Communications, vol. 12 (2021), pp. 2631

Preview abstract
The use of quantum computing for machine learning is among the most exciting prospective applications of quantum technologies. However, machine learning tasks where data is provided can be considerably different than commonly studied computational tasks. In this work, we show that some problems that are classically hard to compute can be easily predicted by classical machines learning from data. Using rigorous prediction error bounds as a foundation, we develop a methodology for assessing potential quantum advantage in learning tasks. The bounds are tight asymptotically and empirically predictive for a wide range of learning models. These constructions explain numerical results showing that with the help of data, classical machine learning models can be competitive with quantum models even if they are tailored to quantum problems. We then propose a projected quantum model that provides a simple and rigorous quantum speed-up for a learning problem in the fault-tolerant regime. For near-term implementations, we demonstrate a significant prediction advantage over some classical models on engineered data sets designed to demonstrate a maximal quantum advantage in one of the largest numerical tests for gate-based quantum machine learning to date, up to 30 qubits.
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Low Rank Representations for Quantum Simulations of Electronic Structure

Mario Motta

Erika Ye

Zhendong Li

Austin Minnich

Garnet Kin-Lic Chan

NPJ Quantum Information, vol. 7 (2021)

Preview abstract
The quantum simulation of quantum chemistry is a promising application of quantum computers. However, for N molecular orbitals, the O(N^4) gate complexity of performing Hamiltonian and unitary Coupled Cluster Trotter steps makes simulation based on such primitives challenging. We substantially reduce the gate complexity of such primitives through a two-step low-rank factorization of the Hamiltonian and cluster operator, accompanied by truncation of small terms. Using truncations that incur errors below chemical accuracy, we are able to perform Trotter steps of the arbitrary basis electronic structure Hamiltonian with O(N^3) gate complexity in small simulations, which reduces to O(N^2 log N) gate complexity in the asymptotic regime, while our unitary Coupled Cluster Trotter step has O(N^3) gate complexity as a function of increasing basis size for a given molecule. In the case of the Hamiltonian Trotter step, these circuits have O(N^2) depth on a linearly connected array, an improvement over the O(N^3) scaling assuming no truncation. As a practical example, we show that a chemically accurate Hamiltonian Trotter step for a 50 qubit molecular simulation can be carried out in the molecular orbital basis with as few as 4,000 layers of parallel nearest-neighbor two-qubit gates, consisting of fewer than 100,000 non-Clifford rotations. We also apply our algorithm to iron-sulfur clusters relevant for elucidating the mode of action of metalloenzymes.
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The Fermionic Quantum Emulator

Klaas Gunst

Alec White

Leon Freitag

Kyle Throssell

Garnet Kin-Lic Chan

Toru Shiozaki

Quantum, vol. 5 (2021), pp. 568

Preview abstract
The fermionic quantum emulator (FQE) is a collection of protocols for emulating quantum dynamics of fermions efficiently taking advantage of common symmetries present in chemical, materials, and condensed-matter systems. The library is fully integrated with the OpenFermion software package and serves as the simulation backend. The FQE reduces memory footprint by exploiting number and spin symmetry along with custom evolution routines for sparse and dense Hamiltonians, allowing us to study significantly larger quantum circuits at modest computational cost when compared against qubit state vector simulators. This release paper outlines the technical details of the simulation methods and key advantages.
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Error Mitigation via Verified Phase Estimation

Thomas E O'Brien

Stefano Polla

Bill Huggins

Sam Connor McArdle

PRX Quantum, vol. 2 (2021)

Preview abstract
We present a novel error mitigation technique for quantum phase estimation. By post-selecting the system register to be in the starting state, we convert all single errors prior to final measurement to a time-dependent decay (up to on average exponentially small corrections), which may be accurately corrected for at the cost of additional measurement. This error migitation can be built into phase estimation techniques that do not require control qubits. By separating the observable of interest into a linear combination of fast-forwardable Hamiltonians and measuring those components individually, we can convert this decay into a constant offset. Using this technique, we demonstrate the estimation of expectation values on numerical simulations of moderately-sized quantum circuits with multiple orders of magnitude improvement over unmitigated estimation at near-term error rates. We further combine verified phase estimation with the optimization step in a variational algorithm to provide additional mitigation of control error. In many cases, our results demonstrate a clear signature that the verification technique can mitigate against any single error occurring in an instance of a quantum computation: the error $\epsilon$ in the expectation value estimation scales with $p^2$, where $p$ is the probability of an error occurring at any point in the circuit. Further numerics indicate that our scheme remains robust in the presence of sampling noise, though different classical post-processing methods may lead to up to an order of magnitude error increase in the final energy estimates.
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Virtual Distillation for Quantum Error Mitigation

William J. Huggins

Sam Connor McArdle

Thomas E O'Brien

Joonho Lee

Birgitta Whaley

Physical Review X, vol. 11 (2021), pp. 041036

Preview abstract
Contemporary quantum computers have relatively high levels of noise, making it difficult to use them to perform useful calculations, even with a large number of qubits. Quantum error correction is expected to eventually enable fault-tolerant quantum computation at large scales, but until then it will be necessary to use alternative strategies to mitigate the impact of errors. We propose a near-term friendly strategy to mitigate errors by entangling and measuring \(M\) copies of a noisy state \(\rho\). This enables us to estimate expectation values with respect to a state with dramatically reduced error, \(\rho^M/ \tr(\rho^M)\), without explicitly preparing it, hence the name ``virtual distillation''. As \(M\) increases, this state approaches the closest pure state to \(\rho\), exponentially quickly. We analyze the effectiveness of virtual distillation and find that it is governed in many regimes by the behaviour of this pure state (corresponding to the dominant eigenvector of \(\rho\)). We numerically demonstrate that virtual distillation is capable of suppressing errors by multiple orders of magnitude and explain how this effect is enhanced as the system size grows. Finally, we show that this technique can improve the convergence of randomized quantum algorithms, even in the absence of device noise.
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Variational Quantum Algorithms

Marco Cerezo

Andrew Arrasmith

Simon Benjamin

Suguro Endo

Keisuke Fujii

Kosuke Mitarai

Xiao Yuan

Lukasz Cincio

Patrick Coles

Nature Reviews Physics (2021)

Preview abstract
Applications such as simulating large quantum systems or solving large-scale linear algebra problems are immensely challenging for classical computers due their extremely high computational cost. Quantum computers promise to unlock these applications, although fault-tolerant quantum computers will likely not be available for several years. Currently available quantum devices have serious constraints, including limited qubit numbers and noise processes that limit circuit depth. Variational Quantum Algorithms (VQAs), which employ a classical optimizer to train a parametrized quantum circuit, have emerged as a leading strategy to address these constraints. VQAs have now been proposed for essentially all applications that researchers have envisioned for quantum computers, and they appear to the best hope for obtaining quantum advantage. Nevertheless, challenges remain including the trainability, accuracy, and efficiency of VQAs. In this review article we present an overview of the field of VQAs. Furthermore, we discuss strategies to overcome their challenges as well as the exciting prospects for using them as a means to obtain quantum advantage.
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Quantum Approximate Optimization of Non-Planar Graph Problems on a Planar Superconducting Processor

Kevin Jeffery Sung

Frank Carlton Arute

Kunal Arya

Juan Atalaya

Rami Barends

Michael Blythe Broughton

Bob Benjamin Buckley

Nicholas Bushnell

Jimmy Chen

Yu Chen

Ben Chiaro

Roberto Collins

William Courtney

Andrew Dunsworth

Austin Fowler

Brooks Riley Foxen

Rob Graff

Steve Habegger

Sergei Isakov

Zhang Jiang

Cody Jones

Kostyantyn Kechedzhi

Alexander Korotkov

Fedor Kostritsa

Dave Landhuis

Pavel Laptev

Martin Leib

Mike Lindmark

Orion Martin

John Martinis

Anthony Megrant

Xiao Mi

Masoud Mohseni

Wojtek Mruczkiewicz

Josh Mutus

Charles Neill

Florian Neukart

Thomas E O'Brien

Bryan O'Gorman

A.G. Petukhov

Harry Putterman

Andrea Skolik

Vadim Smelyanskiy

Doug Strain

Michael Streif

Marco Szalay

Amit Vainsencher

Jamie Yao

Leo Zhou

Edward Farhi

Nature Physics (2021)

Preview abstract
Faster algorithms for combinatorial optimization could prove transformative for diverse areas such as logistics, finance and machine learning. Accordingly, the possibility of quantum enhanced optimization has driven much interest in quantum technologies. Here we demonstrate the application of the Google Sycamore superconducting qubit quantum processor to combinatorial optimization problems with the quantum approximate optimization algorithm (QAOA). Like past QAOA experiments, we study performance for problems defined on the planar connectivity graph native to our hardware; however, we also apply the QAOA to the Sherrington–Kirkpatrick model and MaxCut, non-native problems that require extensive compilation to implement. For hardware-native problems, which are classically efficient to solve on average, we obtain an approximation ratio that is independent of problem size and observe that performance increases with circuit depth. For problems requiring compilation, performance decreases with problem size. Circuits involving several thousand gates still present an advantage over random guessing but not over some efficient classical algorithms. Our results suggest that it will be challenging to scale near-term implementations of the QAOA for problems on non-native graphs. As these graphs are closer to real-world instances, we suggest more emphasis should be placed on such problems when using the QAOA to benchmark quantum processors.
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Exponential suppression of bit or phase flip errors with repetitive quantum error correction

Alan Derk

Alan Ho

Alex Opremcak

Alexander Korotkov

Alexandre Bourassa

Andre Gregory Petukhov

Andrew Dunsworth

Anthony Megrant

Austin Fowler

Bálint Pató

Benjamin Chiaro

Benjamin Villalonga

Brooks Riley Foxen

Charles Neill

Cody Jones

Daniel Eppens

Dave Landhuis

Doug Strain

Edward Farhi

Eric Ostby

Fedor Kostritsa

Frank Carlton Arute

Igor Aleiner

Jamie Yao

Jeremy Patterson Hilton

Jimmy Chen

Josh Mutus

Juan Atalaya

Kostyantyn Kechedzhi

Kunal Arya

Marco Szalay

Masoud Mohseni

Matt Trevithick

Michael Broughton

Michael Newman

Nicholas Bushnell

Nicholas Redd

Orion Martin

Pavel Laptev

Ping Yeh

Rami Barends

Roberto Collins

Sean Harrington

Sergei Isakov

Thomas E O'Brien

Trent Huang

Trevor Mccourt

Vadim Smelyanskiy

Vladimir Shvarts

William Courtney

Wojtek Mruczkiewicz

Xiao Mi

Yu Chen

Zhang Jiang

Nature (2021)

Preview abstract
Realizing the potential of quantum computing will require achieving sufficiently low logical error rates. Many applications call for error rates below 10^-15, but state-of-the-art quantum platforms typically have physical error rates near 10^-3. Quantum error correction (QEC) promises to bridge this divide by distributing quantum logical information across many physical qubits so that errors can be corrected. Logical errors are then exponentially suppressed as the number of physical qubits grows, provided that the physical error rates are below a certain threshold. QEC also requires that the errors are local, and that performance is maintained over many rounds of error correction, a major outstanding experimental challenge. Here, we implement 1D repetition codes embedded in a 2D grid of superconducting qubits which demonstrate exponential suppression of bit or phase-flip errors, reducing logical error per round by more than 100x when increasing the number of qubits from 5 to 21. Crucially, this error suppression is stable over 50 rounds of error correction. We also introduce a method for analyzing error correlations with high precision, and characterize the locality of errors in a device performing QEC for the first time. Finally, we perform error detection using a small 2D surface code logical qubit on the same device, and show that the results from both 1D and 2D codes agree with numerical simulations using a simple depolarizing error model. These findings demonstrate that superconducting qubits are on a viable path towards fault tolerant quantum computing.
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Tuning Quantum Information Scrambling on a 53-Qubit Processor

Alan Derk

Alan Ho

Alex Opremcak

Alexander Korotkov

Alexandre Bourassa

Andre Gregory Petukhov

Andrew Dunsworth

Anthony Megrant

Austin Fowler

Bálint Pató

Benjamin Chiaro

Benjamin Villalonga

Brooks Riley Foxen

Charles Neill

Cody Jones

Daniel Eppens

Dave Landhuis

Doug Strain

Edward Farhi

Eric Ostby

Fedor Kostritsa

Frank Carlton Arute

Igor Aleiner

Jamie Yao

Jeffrey Marshall

Jeremy Patterson Hilton

Jimmy Chen

Josh Mutus

Juan Atalaya

Kostyantyn Kechedzhi

Kunal Arya

Marco Szalay

Masoud Mohseni

Matt Trevithick

Michael Blythe Broughton

Michael Newman

Nicholas Bushnell

Nicholas Redd

Orion Martin

Pavel Laptev

Ping Yeh

Rami Barends

Roberto Collins

Salvatore Mandra

Sean Harrington

Sergei Isakov

Thomas E O'Brien

Trent Huang

Trevor Mccourt

Vadim Smelyanskiy

Vladimir Shvarts

William Courtney

Wojtek Mruczkiewicz

Xiao Mi

Yu Chen

Zhang Jiang

arXiv (2021)

Preview abstract
As entanglement in a quantum system grows, initially localized quantum information is spread into the exponentially many degrees of freedom of the entire system. This process, known as quantum scrambling, is computationally intensive to study classically and lies at the heart of several modern physics conundrums. Here, we characterize scrambling of different quantum circuits on a 53-qubit programmable quantum processor by measuring their out-of-time-order correlators (OTOCs). We observe that the spatiotemporal spread of OTOCs, as well as their circuit-to-circuit fluctuation, unravel in detail the time-scale and extent of quantum scrambling. Comparison with numerical results indicates a high OTOC measurement accuracy despite the large size of the quantum system. Our work establishes OTOC as an experimental tool to diagnose quantum scrambling at the threshold of being classically inaccessible.
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Compilation of Fault-Tolerant Quantum Heuristics for Combinatorial Optimization

Yuval Sanders

Dominic W. Berry

Pedro C. S. Costa

Louis W. Tessler

Nathan Wiebe

PRX Quantum, vol. 1 (2020), pp. 020312

Preview abstract
Here we explore which heuristic quantum algorithms for combinatorial optimization might be most practical to try out on a small fault-tolerant quantum computer. We compile circuits for several variants of quantum-accelerated simulated annealing including those using qubitization or Szegedy walks to quantize classical Markov chains and those simulating spectral-gap-amplified Hamiltonians encoding a Gibbs state. We also optimize fault-tolerant realizations of the adiabatic algorithm, quantum-enhanced population transfer, the quantum approximate optimization algorithm, and other approaches. Many of these methods are bottlenecked by calls to the same subroutines; thus, optimized circuits for those primitives should be of interest regardless of which heuristic is most effective in practice. We compile these bottlenecks for several families of optimization problems and report for how long and for what size systems one can perform these heuristics in the surface code given a range of resource budgets. Our results discourage the notion that any quantum optimization heuristic realizing only a quadratic speedup achieves an advantage over classical algorithms on modest superconducting qubit surface code processors without significant improvements in the implementation of the surface code. For instance, under quantum-favorable assumptions (e.g., that the quantum algorithm requires exactly quadratically fewer steps), our analysis suggests that quantum-accelerated simulated annealing requires roughly a day and a million physical qubits to optimize spin glasses that could be solved by classical simulated annealing in about 4 CPU-minutes.
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Improved Fault-Tolerant Quantum Simulation of Condensed-Phase Correlated Electrons via Trotterization

Ian Kivlichan

Dominic Berry

Wei Sun

Zhang Jiang

Austin Fowler

Alán Aspuru-Guzik

Quantum, vol. 4 (2020), pp. 296

Preview abstract
Recent work has deployed linear combinations of unitaries techniques to significantly reduce the cost of performing fault-tolerant quantum simulations of correlated electron models. Here, we show that one can sometimes improve over those results with optimized implementations of Trotter-Suzuki-based product formulas. We show that low-order Trotter methods perform surprisingly well when used with phase estimation to compute relative precision quantities (e.g. energy per unit cell), as is often the goal for condensed-phase (e.g. solid-state) systems. In this context, simulations of the Hubbard model and plane wave electronic structure models with $N < 10^5$ fermionic modes can be performed with roughly O(1) and O(N^2) T complexities. We also perform numerics that reveal tradeoffs between the error of a Trotter step and Trotter step gate complexity across various implementations; e.g., we show that split-operator techniques have less Trotter error than popular alternatives. By compiling to surface code fault-tolerant gates using lattice surgery and assuming error rates of one part in a thousand, we show that one can error-correct quantum simulations of interesting, classically intractable instances with only a few hundred thousand physical qubits.
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Decoding Quantum Errors Using Subspace Expansions

Zhang Jiang

Nature Communications, vol. 11 (2020), pp. 636

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With the rapid developments in quantum hardware comes a push towards the first practical applications on these devices. While fully fault-tolerant quantum computers may still be years away, one may ask if there exist intermediate forms of error correction or mitigation that might enable practical applications before then. In this work, we consider the idea of post-processing error decoders using existing quantum codes, which are capable of mitigating errors on encoded logical qubits using classical post-processing with no complicated syndrome measurements or additional qubits beyond those used for the logical qubits. This greatly simplifies the experimental exploration of quantum codes on near-term devices, removing the need for locality of syndromes or fast feed-forward, allowing one to study performance aspects of codes on real devices. We provide a general construction equipped with a simple stochastic sampling scheme that does not depend explicitly on a number of terms that we extend to approximate projectors within a subspace. This theory then allows one to generalize to the correction of some logical errors in the code space, correction of some physical unencoded Hamiltonians without engineered symmetries, and corrections derived from approximate symmetries. In this work, we develop the theory of the method and demonstrate it on a simple example with the perfect [[5,1,3]] code, which exhibits a pseudo-threshold of p≈0.50 under a single qubit depolarizing channel applied to all qubits. We also provide a demonstration under the application of a logical operation and performance on an unencoded hydrogen molecule, which exhibits a significant improvement over the entire range of possible errors incurred under a depolarizing channel.
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Increasing the Representation Accuracy of Quantum Simulations of Chemistry without Extra Quantum Resources

Tyler Takeshita

Zhang Jiang

Eunseok Lee

Physical Review X, vol. 10 (2020), pp. 011004

Preview abstract
Proposals for near-term experiments in quantum chemistry on quantum computers leverage the ability to target a subset of degrees of freedom containing the essential quantum behavior, sometimes called the active space. This approximation allows one to treat more difficult problems using fewer qubits and lower gate depths than would otherwise be possible. However, while this approximation captures many important qualitative features, it may leave the results wanting in terms of absolute accuracy (basis error) of the representation. In traditional approaches, increasing this accuracy requires increasing the number of qubits and an appropriate increase in circuit depth as well. Here we introduce a technique requiring no additional qubits or circuit depth that is able to remove much of this approximation in favor of additional measurements. The technique is constructed and analyzed theoretically, and some numerical proof of concept calculations are shown. As an example, we show how to achieve the accuracy of a 20 qubit representation using only 4 qubits and a modest number of additional measurements for a simple hydrogen molecule. We close with an outlook on the impact this technique may have on both near-term and fault-tolerant quantum simulations.
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TensorFlow Quantum: A Software Framework for Quantum Machine Learning

Michael Broughton

Guillaume Verdon

Trevor McCourt

Antonio J. Martinez

Jae Hyeon Yoo

Sergei V. Isakov

Philip Massey

Ramin Halavati

Alexander Zlokapa

Evan Peters

Owen Lockwood

Andrea Skolik

Sofiene Jerbi

Vedran Djunko

Martin Leib

Michael Streif

David Von Dollen

Hongxiang Chen

Chuxiang Cao

Roeland Wiersema

Hsin-Yuan Huang

Alan K. Ho

Masoud Mohseni

(2020)

Preview abstract
We introduce TensorFlow Quantum (TFQ), an open source library for the rapid prototyping of hybrid quantum-classical models for classical or quantum data. This framework offers high-level abstractions for the design and training of both discriminative and generative quantum models under TensorFlow and supports high-performance quantum circuit simulators. We provide an overview of the software architecture and building blocks through several examples and review the theory of hybrid quantum-classical neural networks. We illustrate TFQ functionalities via several basic applications including supervised learning for quantum classification, quantum control, simulating noisy quantum circuits, and quantum approximate optimization. Moreover, we demonstrate how one can apply TFQ to tackle advanced quantum learning tasks including meta-learning, layerwise learning, Hamiltonian learning, sampling thermal states, variational quantum eigensolvers, classification of quantum phase transitions, generative adversarial networks, and reinforcement learning. We hope this framework provides the necessary tools for the quantum computing and machine learning research communities to explore models of both natural and artificial quantum systems, and ultimately discover new quantum algorithms which could potentially yield a quantum advantage.
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Demonstrating a Continuous Set of Two-qubit Gates for Near-term Quantum Algorithms

Brooks Riley Foxen

Charles Neill

Andrew Dunsworth

Ben Chiaro

Anthony Megrant

Jimmy Chen

Rami Barends

Frank Carlton Arute

Kunal Arya

Yu Chen

Roberto Collins

Edward Farhi

Austin Fowler

Rob Graff

Trent Huang

Sergei Isakov

Zhang Jiang

Kostyantyn Kechedzhi

Alexander Korotkov

Fedor Kostritsa

Dave Landhuis

Xiao Mi

Masoud Mohseni

Josh Mutus

Vadim Smelyanskiy

Amit Vainsencher

Jamie Yao

John Martinis

arXiv:2001.08343 (2020)

Preview abstract
Quantum algorithms offer a dramatic speedup for computational problems in machine learning, material science, and chemistry. However, any near-term realizations of these algorithms will need to be heavily optimized to fit within the finite resources offered by existing noisy quantum hardware. Here, taking advantage of the strong adjustable coupling of gmon qubits, we demonstrate a continuous two qubit gate set that can provide a 5x reduction in circuit depth. We implement two gate families: an iSWAP-like gate to attain an arbitrary swap angle, $\theta$, and a CPHASE gate that generates an arbitrary conditional phase, $\phi$. Using one of each of these gates, we can perform an arbitrary two qubit gate within the excitation-preserving subspace allowing for a complete implementation of the so-called Fermionic Simulation, or fSim, gate set. We benchmark the fidelity of the iSWAP-like and CPHASE gate families as well as 525 other fSim gates spread evenly across the entire fSim($\theta$, $\phi$) parameter space achieving purity-limited average two qubit Pauli error of $3.8 \times 10^{-3}$ per fSim gate.
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Accurately computing electronic properties of materials using eigenenergies

Alan Derk

Alan Ho

Alex Opremcak

Alexander Korotkov

Andre Gregory Petukhov

Andrew Dunsworth

Anthony Megrant

Austin Fowler

Bálint Pató

Benjamin Chiaro

Benjamin Villalonga

Bob Benjamin Buckley

Brooks Riley Foxen

Charles Neill

Cody Jones

Daniel Eppens

Dave Landhuis

Doug Strain

Edward Farhi

Eric Ostby

Fedor Kostritsa

Frank Carlton Arute

Igor Aleiner

Jamie Yao

Jeremy Patterson Hilton

Jimmy Chen

Josh Mutus

Juan Atalaya

Juan Campero

Kostyantyn Kechedzhi

Kunal Arya

Marco Szalay

Masoud Mohseni

Matt Jacob-Mitos

Matt Trevithick

Michael Blythe Broughton

Michael Newman

Nicholas Bushnell

Nicholas Redd

Orion Martin

Pavel Laptev

Ping Yeh

Rami Barends

Roberto Collins

Sean Harrington

Sergei Isakov

Thomas E O'Brien

Trent Huang

Trevor Mccourt

Vadim Smelyanskiy

Vladimir Shvarts

William Courtney

William J. Huggins

Wojtek Mruczkiewicz

Xiao Mi

Yu Chen

Zhang Jiang

arXiv preprint arXiv:2012.00921 (2020)

Preview abstract
A promising approach to study quantum materials is to simulate them on an engineered quantum platform. However, achieving the accuracy needed to outperform classical methods has been an outstanding challenge. Here, using superconducting qubits, we provide an experimental blueprint for a programmable and accurate quantum matter simulator and demonstrate how to probe fundamental electronic properties. We illustrate the underlying method by reconstructing the single-particle band-structure of a one-dimensional wire. We demonstrate nearly complete mitigation of decoherence and readout errors and arrive at an accuracy in measuring energy eigenvalues of this wire with an error of ~0.01 radians, whereas typical energy scales are of order 1 radian. Insight into this unprecedented algorithm fidelity is gained by highlighting robust properties of a Fourier transform, including the ability to resolve eigenenergies with a statistical uncertainty of 1e-4 radians. Furthermore, we synthesize magnetic flux and disordered local potentials, two key tenets of a condensed-matter system. When sweeping the magnetic flux, we observe avoided level crossings in the spectrum, a detailed fingerprint of the spatial distribution of local disorder. Combining these methods, we reconstruct electronic properties of the eigenstates where we observe persistent currents and a strong suppression of conductance with added disorder. Our work describes an accurate method for quantum simulation and paves the way to study novel quantum materials with superconducting qubits.
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Discontinuous Galerkin Discretization for Quantum Simulation of Chemistry

Fabian M. Faulstich

Qinyi Zhu

Bryan O'Gorman

Yiheng Qiu

Steven White

Lin Lin

New Journal of Physics, vol. 22 (2020)

Preview abstract
Methods for electronic structure based on Gaussian and molecular orbital discretizations offer a well established, compact representation that forms much of the foundation of correlated quantum chemistry calculations on both classical and quantum computers. Despite their ability to describe essential physics with relatively few basis functions, these representations can suffer from a quartic growth of the number of integrals. Recent results have shown that, for some quantum and classical algorithms, moving to representations with diagonal two-body operators can result in dramatically lower asymptotic costs, even if the number of functions required increases significantly. We introduce a way to interpolate between the two regimes in a systematic and controllable manner, such that the number of functions is minimized while maintaining a block diagonal structure of the two-body operator and desirable properties of an original, primitive basis. Techniques are analyzed for leveraging the structure of this new representation on quantum computers. Empirical results for hydrogen chains suggest a scaling improvement from O(N^4.5) in molecular orbital representations to O(N^2.6) in our representation for quantum evolution in a fault-tolerant setting, and exhibit a constant factor crossover at 15 to 20 atoms. Moreover, we test these methods using modern density matrix renormalization group methods classically, and achieve excellent accuracy with respect to the complete basis set limit with a speedup of 1-2 orders of magnitude with respect to using the primitive or Gaussian basis sets alone. These results suggest our representation provides significant cost reductions while maintaining accuracy relative to molecular orbital or strictly diagonal approaches for modest-sized systems in both classical and quantum computation for correlated systems.
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Using Models to Improve Optimizers for Variational Quantum Algorithms

Kevin Jeffery Sung

Jiahao Yao

Zhang Jiang

Lin Lin

Quantum Science and Technology, vol. 5 (2020), pp. 044008

Preview abstract
Variational quantum algorithms are a leading candidate for early applications on noisy intermediate-scale quantum computers. These algorithms depend on a classical optimization outer-loop that minimizes some function of a parameterized quantum circuit. In practice, finite sampling error and gate errors make this a stochastic optimization with unique challenges that must be addressed at the level of the optimizer. The sharp trade-off between precision and sampling time in conjunction with experimental constraints necessitates the development of new optimization strategies to minimize overall wall clock time in this setting. We introduce an optimization method and numerically compare its performance with common methods in use today. The method is a simple surrogate model-based algorithm designed to improve reuse of collected data. It does so by estimating the gradient using a least-squares quadratic fit of sampled function values within a moving trusted region. To make fair comparisons between optimization methods, we develop experimentally relevant cost models designed to balance efficiency in testing and accuracy with respect to cloud quantum computing systems. The results here underscore the need to both use relevant cost models and optimize hyperparameters of existing optimization methods for competitive performance. We compare tuned methods using cost models presented by superconducting devices accessed through cloud computing platforms. The method introduced here has several practical advantages in realistic experimental settings, and has been used successfully in a separately published experiment on Google's Sycamore device.
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Hartree-Fock on a Superconducting Qubit Quantum Computer

Frank Carlton Arute

Kunal Arya

Rami Barends

Michael Blythe Broughton

Bob Benjamin Buckley

Nicholas Bushnell

Yu Chen

Jimmy Chen

Benjamin Chiaro

Roberto Collins

William Courtney

Andrew Dunsworth

Edward Farhi

Austin Fowler

Brooks Riley Foxen

Rob Graff

Steve Habegger

Alan Ho

Trent Huang

William J. Huggins

Sergei Isakov

Zhang Jiang

Cody Jones

Kostyantyn Kechedzhi

Alexander Korotkov

Fedor Kostritsa

Dave Landhuis

Pavel Laptev

Mike Lindmark

Orion Martin

John Martinis

Anthony Megrant

Xiao Mi

Masoud Mohseni

Wojtek Mruczkiewicz

Josh Mutus

Charles Neill

Thomas E O'Brien

Eric Ostby

Andre Gregory Petukhov

Harry Putterman

Vadim Smelyanskiy

Doug Strain

Kevin Jeffery Sung

Marco Szalay

Tyler Y. Takeshita

Amit Vainsencher

Nathan Wiebe

Jamie Yao

Ping Yeh

Science, vol. 369 (2020), pp. 6507

Preview abstract
As the search continues for useful applications of noisy intermediate scale quantum devices, variational simulations of fermionic systems remain one of the most promising directions. Here, we perform a series of quantum simulations of chemistry which involve twice the number of qubits and more than ten times the number of gates as the largest prior experiments. We model the binding energy of ${\rm H}_6$, ${\rm H}_8$, ${\rm H}_{10}$ and ${\rm H}_{12}$ chains as well as the isomerization of diazene. We also demonstrate error-mitigation strategies based on $N$-representability which dramatically improve the effective fidelity of our experiments. Our parameterized ansatz circuits realize the Givens rotation approach to free fermion evolution, which we variationally optimize to prepare the Hartree-Fock wavefunction. This ubiquitous algorithmic primitive corresponds to a rotation of the orbital basis and is required by many proposals for correlated simulations of molecules and Hubbard models. Because free fermion evolutions are classically tractable to simulate, yet still generate highly entangled states over the computational basis, we use these experiments to benchmark the performance of our hardware while establishing a foundation for scaling up more complex correlated quantum simulations of chemistry.
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Learning to learn with quantum neural networks via classical neural networks

Guillaume Verdon

Michael Broughton

Kevin Jeffery Sung

Zhang Jiang

Masoud Mohseni

arXiv:1907.05415 (2019)

Preview abstract
Quantum Neural Networks (QNNs) are a promising variational learning paradigm with applications to near-term quantum processors, however they still face some significant challenges. One such challenge is finding good parameter initialization heuristics that ensure rapid and consistent convergence to local minima of the parameterized quantum circuit landscape. In this work, we train classical neural networks to assist in the quantum learning process, also know as meta-learning, to rapidly find approximate optima in the parameter landscape for several classes of quantum variational algorithms. Specifically, we train classical recurrent neural networks to find approximately optimal parameters within a small number of queries of the cost function for the Quantum Approximate Optimization Algorithm (QAOA) for MaxCut, QAOA for Sherrington-Kirkpatrick Ising model, and for a Variational Quantum Eigensolver for the Hubbard model. By initializing other optimizers at parameter values suggested by the classical neural network, we demonstrate a significant improvement in the total number of optimization iterations required to reach a given accuracy. We further demonstrate that the optimization strategies learned by the neural network generalize well across a range of problem instance sizes. This opens up the possibility of training on small, classically simulatable problem instances, in order to initialize larger, classically intractably simulatable problem instances on quantum devices, thereby significantly reducing the number of required quantum-classical optimization iterations.
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Majorana Loop Stabilizer Codes for Error Mitigation in Fermionic Quantum Simulations

Zhang Jiang

Physical Review Applied, vol. 12 (2019), pp. 064041

Preview abstract
Fermion-to-qubit mappings that preserve geometric locality are especially useful for simulating lattice fermion models (e.g., the Hubbard model) on a quantum computer. They avoid the overhead associated with geometric nonlocal parity terms in mappings such as the Jordan-Wigner transformation and the Bravyi-Kitaev transformation. As a result, they often provide quantum circuits with lower depth and gate complexity. In such encodings, fermionic states are encoded in the common +1 eigenspace of a set of stabilizers, akin to stabilizer quantum error-correcting codes. Here, we discuss several known geometric locality-preserving mappings and their abilities to correct and detect single-qubit errors. We introduce a geometric locality-preserving map, whose stabilizers correspond to products of Majorana operators on closed paths of the fermionic hopping graph. We show that our code, which we refer to as the Majorana loop stabilizer code (MLSC) can correct all single-qubit errors on a two-dimensional square lattice, while previous geometric locality-preserving codes can only detect single-qubit errors on the same lattice. Compared to existing codes, the MLSC maps the relevant fermionic operators to lower-weight qubit operators despite having higher code distance. Going beyond lattice models, we demonstrate that the MLSC is compatible with state-of-the-art algorithms for simulating quantum chemistry, and can offer those simulations the same error-correction properties without additional asymptotic overhead. These properties make the MLSC a promising candidate for error-mitigated quantum simulations of fermions on near-term devices
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Quantum Supremacy using a Programmable Superconducting Processor

Frank Arute

Kunal Arya

Rami Barends

Rupak Biswas

Fernando Brandao

David Buell

Yu Chen

Jimmy Chen

Ben Chiaro

Roberto Collins

William Courtney

Andrew Dunsworth

Edward Farhi

Brooks Foxen

Austin Fowler

Rob Graff

Keith Guerin

Steve Habegger

Michael Hartmann

Alan Ho

Trent Huang

Travis Humble

Sergei Isakov

Zhang Jiang

Kostyantyn Kechedzhi

Sergey Knysh

Alexander Korotkov

Fedor Kostritsa

Dave Landhuis

Mike Lindmark

Dmitry Lyakh

Salvatore Mandrà

Anthony Megrant

Xiao Mi

Kristel Michielsen

Masoud Mohseni

Josh Mutus

Charles Neill

Eric Ostby

Andre Petukhov

Eleanor G. Rieffel

Vadim Smelyanskiy

Kevin Jeffery Sung

Matt Trevithick

Amit Vainsencher

Benjamin Villalonga

Z. Jamie Yao

Ping Yeh

John Martinis

Nature, vol. 574 (2019), 505–510

Preview abstract
The promise of quantum computers is that certain computational tasks might be executed exponentially faster on a quantum processor than on a classical processor. A fundamental challenge is to build a high-fidelity processor capable of running quantum algorithms in an exponentially large computational space. Here we report the use of a processor with programmable superconducting qubits to create quantum states on 53 qubits, corresponding to a computational state-space of dimension 2^53 (about 10^16). Measurements from repeated experiments sample the resulting probability distribution, which we verify using classical simulations. Our Sycamore processor takes about 200 seconds to sample one instance of a quantum circuit a million times-our benchmarks currently indicate that the equivalent task for a state-of-the-art classical supercomputer would take approximately 10,000 years. This dramatic increase in speed compared to all known classical algorithms is an experimental realization of quantum supremacy for this specific computational task, heralding a much-anticipated computing paradigm.
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Quantum Simulation of the Sachdev-Ye-Kitaev Model by Asymmetric Qubitization

Dominic W. Berry

Physical Review A Rapid Communication, vol. 99 (2019), 040301(R)

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We show that one can quantum simulate the dynamics of a Sachdev-Ye-Kitaev model with $N$ Majorana modes for time $t$ to precision $\epsilon$ with gate complexity ${\cal O}(N^{7/2} t + N^{5/2} \log(1 / \epsilon) / \log\log(1/\epsilon))$. In addition to scaling sublinearly in the number of Hamiltonian terms, this gate complexity represents an exponential improvement in $1/\epsilon$ and large polynomial improvement in $N$ and $t$ over prior state-of-the-art algorithms which scale as ${\cal O}(N^{10} t^2 / \epsilon)$. Our approach involves a variant of the qubitization technique in which we encode the Hamiltonian $H$ as an asymmetric projection of a signal oracle $U$ onto two different signal states prepared by distinct state oracles, $A\ket{0} \mapsto \ket{A}$ and $B\ket{0} \mapsto \ket{B}$, such that $H = \bra{B} U \ket{A}$. Our strategy for applying this method to the Sachdev-Ye-Kitaev model involves realizing $B$ using only Hadamard gates and realizing $A$ as a random quantum circuit.
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Quantum Simulation of Chemistry with Sublinear Scaling in Basis Size

Dominic W. Berry

NPJ Quantum Information, vol. 5 (2019)

Preview abstract
We present a quantum algorithm for simulating quantum chemistry with gate complexity Õ(η^{8/3}N^{1/3}),where η is the number of electrons and N is the number of plane wave orbitals. In comparison, the most efficient prior algorithms for simulating electronic structure using plane waves (which are at least as efficient as algorithms using any other basis) have complexity Õ(η^{2/3}N^{8/3}). We achieve our scaling in first quantization by performing simulation in the rotating frame of the kinetic operator using interaction picture techniques. Our algorithm is far more efficient than all prior approaches when N ≫ η, as is needed to suppress discretization error when representing molecules in the plane wave basis, or when simulating without the Born-Oppenheimer approximation.
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Qubitization of Arbitrary Basis Quantum Chemistry Leveraging Sparsity and Low Rank Factorization

Dominic W. Berry

Mario Motta

Quantum, vol. 3 (2019), pp. 208

Preview abstract
Recent work has dramatically reduced the gate complexity required to quantum simulate chemistry by using linear combinations of unitaries based methods to exploit structure in the plane wave
basis Coulomb operator. Here, we show that one can achieve similar scaling even for arbitrary basis
sets (which can be hundreds of times more compact than plane waves) by using qubitized quantum
walks in a fashion that takes advantage of structure in the Coulomb operator, either by directly
exploiting sparseness, or via a low rank tensor factorization. We provide circuits for several variants
of our algorithm (which all improve over the scaling of prior methods) including one with O(N^{3/2} λ)
T complexity, where N is number of orbitals and λ is the 1-norm of the chemistry Hamiltonian. We
deploy our algorithms to simulate the FeMoco molecule (relevant to Nitrogen fixation) and obtain
circuits requiring almost one thousand times less surface code spacetime volume than prior quantum
algorithms for this system, despite us using a larger and more accurate active space.
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Barren Plateaus in Quantum Neural Network Training Landscapes

Vadim Smelyanskiy

Nature Communications, vol. 9 (2018), pp. 4812

Preview abstract
Many experimental proposals for noisy intermediate scale quantum devices involve training a parameterized quantum circuit with a classical optimization loop. Such hybrid quantum-classical algorithms are popular for applications in quantum simulation, optimization, and machine learning. Due to its simplicity and hardware efficiency, random circuits are often proposed as initial guesses for exploring the space of quantum states. We show that the exponential dimension of Hilbert space and the gradient estimation complexity make this choice unsuitable for hybrid quantum-classical algorithms run on more than a few qubits. Specifically, we show that for a wide class of reasonable parameterized quantum circuits, the probability that the gradient along any reasonable direction is non-zero to some fixed precision is exponentially small as a function of the number of qubits. We argue that this is related to the 2-design characteristic of random circuits, and that solutions to this problem must be studied.
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Strategies for Quantum Computing Molecular Energies Using the Unitary Coupled Cluster Ansatz

Jhonathan Romero Fontalvo

Cornelius Hempel

Peter J. Love

Alán Aspuru-Guzik

Quantum Science and Technology, vol. 4 (2018), pp. 14008

Preview abstract
The variational quantum eigensolver (VQE) algorithm combines the ability of quantum computers to efficiently compute expectations values with a classical optimization routine in order to approximate ground state energies of quantum systems. In this paper, we study the application of VQE to the simulation of molecular energies using the unitary coupled cluster (UCC) ansatz. We introduce new strategies to reduce the circuit depth for the implementation of UCC and improve the optimization of the wavefunction based on efficient classical approximations of the cluster amplitudes. Additionally, we propose a method to compute the energy gradient within the VQE approach. We illustrate our methodology with numerical simulations for a system of four hydrogen atoms that exhibit strong correlation and show that the cost of the circuit depth and execution time of VQE using a UCC ansatz can be reduced without introducing significant loss of accuracy in the final wavefunctions and energies.
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Quantum Chemistry Calculations on a Trapped-Ion Quantum Simulator

Cornelius Hempel

Christine Maier

Jhonathan Romero

Thomas Monz

Heng Shen

Petar Jurcevic

Ben Lanyon

Peter J. Love

Alán Aspuru-Guzik

Rainer Blatt

Christian Roos

Physical Review X, vol. 8 (2018), pp. 031022

Preview abstract
Quantum-classical hybrid algorithms are emerging as promising candidates for near-term practical applications of quantum information processors in a wide variety of fields ranging from chemistry to physics and materials science. We report on the experimental implementation of such an algorithm to solve a quantum chemistry problem, using a digital quantum simulator based on trapped ions. Specifically, we implement the variational quantum eigensolver algorithm to calculate the molecular ground-state energies of two simple molecules and experimentally demonstrate and compare different encoding methods using up to four qubits. Furthermore, we discuss the impact of measurement noise as well as mitigation strategies and indicate the potential for adaptive implementations focused on reaching chemical accuracy, which may serve as a cross-platform benchmark for multiqubit quantum simulators.
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Characterizing Quantum Supremacy in Near-Term Devices

Sergei Isakov

Vadim Smelyanskiy

Zhang Jiang

Michael J. Bremner

John Martinis

Nature Physics, vol. 14 (2018), 595–600

Preview abstract
A critical question for quantum computing in the near future is whether quantum devices without error correction can perform a well-defined computational task beyond the capabilities of supercomputers. Such a demonstration of what is referred to as quantum supremacy requires a reliable evaluation of the resources required to solve tasks with classical approaches. Here, we propose the task of sampling from the output distribution of random quantum circuits as a demonstration of quantum supremacy. We extend previous results in computational complexity to argue that this sampling task must take exponential time in a classical computer. We introduce cross-entropy benchmarking to obtain the experimental fidelity of complex multiqubit dynamics. This can be estimated and extrapolated to give a success metric for a quantum supremacy demonstration. We study the computational cost of relevant classical algorithms and conclude that quantum supremacy can be achieved with circuits in a two-dimensional lattice of 7 × 7 qubits and around 40 clock cycles. This requires an error rate of around 0.5% for two-qubit gates (0.05% for one-qubit gates), and it would demonstrate the basic building blocks for a fault-tolerant quantum computer
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Postponing the Orthogonality Catastrophe: Efficient State Preparation for Electronic Structure Simulations on Quantum Devices

Norman Tubman

Carlos Mejuto Zaera

Jeffrey Epstein

Diptarka Hait

Daniel Levine

William Huggins

Zhang Jiang

Martin Head-Gordon

K. Birgitta Whaley

arXiv:1809.05523 (2018)

Preview abstract
Despite significant work on resource estimation for quantum simulation of electronic systems, the challenge of preparing states with sufficient ground state support has so far been largely neglected. In this work we investigate this issue in several systems of interest, including organic molecules, transition metal complexes, the uniform electron gas, Hubbard models, and quantum impurity models arising from embedding formalisms such as dynamical mean-field theory. Our approach uses a state-of-the-art classical technique for high-fidelity ground state approximation. We find that easy-to-prepare single Slater determinants such as the Hartree-Fock state often have surprisingly robust support on the ground state for many applications of interest. For the most difficult systems, single-determinant reference states may be insufficient, but low-complexity reference states may suffice. For this we introduce a method for preparation of multi-determinant states on quantum computers.
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Low-Depth Quantum Simulation of Materials

Nathan Wiebe

James McClain

Garnet Chan

Physical Review X, vol. 8 (2018), pp. 011044

Preview abstract
Quantum simulation of the electronic structure problem is one of the most researched applications of quantum computing. The majority of quantum algorithms for this problem encode the wavefunction using $N$ molecular orbitals, leading to Hamiltonians with ${\cal O}(N^4)$ second-quantized terms. To avoid this overhead, we introduce basis functions which diagonalize the periodized Coulomb operator, providing Hamiltonians for condensed phase systems with $N^2$ second-quantized terms. Using this representation we can implement single Trotter steps of the Hamiltonians with gate depth of ${\cal O}(N)$ on a planar lattice of qubits -- a quartic improvement over prior methods. Special properties of our basis allow us to apply Trotter based simulations with planar circuit depth in $\widetilde{\cal O}(N^{7/2} / \epsilon^{1/2})$ and Taylor series methods with circuit size $\widetilde{\cal O}(N^{11/3})$, where $\epsilon$ is target precision. Variational algorithms also require significantly fewer measurements to find the mean energy using our representation, ameliorating a primary challenge of that approach. We conclude with a proposal to simulate the uniform electron gas (jellium) using a linear depth variational ansatz realizable on near-term quantum devices with planar connectivity. From these results we identify simulation of low-density jellium as an ideal first target for demonstrating quantum supremacy in electronic structure.
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