# Craig Michael Gidney

Software engineer on the quantum computing team.

### Research Areas

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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

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

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

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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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

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

Cody Jones

Pavol Juhas

Kostyantyn Kechedzhi

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

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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Suppressing quantum errors by scaling a surface code logical qubit

Anthony Megrant

Cody Jones

Jeremy Hilton

Jimmy Chen

Juan Atalaya

Kenny Lee

Michael Newman

Vadim Smelyanskiy

Yu Chen

Nature (2023)

Preview abstract
Practical quantum computing will require error rates that are well below what is achievable with
physical qubits. Quantum error correction [1, 2] offers a path to algorithmically-relevant error rates
by encoding logical qubits within many physical qubits, where increasing the number of physical
qubits enhances protection against physical errors. However, introducing more qubits also increases
the number of error sources, so the density of errors must be sufficiently low in order for logical
performance to improve with increasing code size. Here, we report the measurement of logical qubit
performance scaling across multiple code sizes, and demonstrate that our system of superconducting
qubits has sufficient performance to overcome the additional errors from increasing qubit number.
We find our distance-5 surface code logical qubit modestly outperforms an ensemble of distance-3
logical qubits on average, both in terms of logical error probability over 25 cycles and logical error
per cycle (2.914%±0.016% compared to 3.028%±0.023%). To investigate damaging, low-probability
error sources, we run a distance-25 repetition code and observe a 1.7 × 10−6 logical error per round
floor set by a single high-energy event (1.6 × 10−7 when excluding this event). We are able to
accurately model our experiment, and from this model we can extract error budgets that highlight
the biggest challenges for future systems. These results mark the first experimental demonstration
where quantum error correction begins to improve performance with increasing qubit number, and
illuminate the path to reaching the logical error rates required for computation.
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Direct Measurement of Nonlocal Interactions in the Many-Body Localized Phase

Amit Vainsencher

Andrew Dunsworth

Anthony Megrant

Ben Chiaro

Brooks Foxen

Charles Neill

Dave Landhuis

Fedor Kostritsa

Frank Carlton Arute

Jimmy Chen

John Martinis

Josh Mutus

Kostyantyn Kechedzhi

Kunal Arya

Rami Barends

Roberto Collins

Trent Huang

Vadim Smelyanskiy

Yu Chen

Physical Review Research, vol. 4 (2022), pp. 013148

Preview abstract
The interplay of interactions and strong disorder can lead to an exotic quantum many-body localized (MBL) phase of matter. Beyond the absence of transport, the MBL phase has distinctive signatures, such as slow dephasing and logarithmic entanglement growth; they commonly result in slow and subtle modifications of the dynamics, rendering their measurement challenging. Here, we experimentally characterize these properties of the MBL phase in a system of coupled superconducting qubits. By implementing phase sensitive techniques, we map out the structure of local integrals of motion in the MBL phase. Tomographic reconstruction of single and two-qubit density matrices allows us to determine the spatial and temporal entanglement growth between the localized sites. In addition, we study the preservation of entanglement in the MBL phase. The interferometric protocols implemented here detect affirmative quantum correlations and exclude artifacts due to the imperfect isolation of the system. By measuring elusive MBL quantities, our work highlights the advantages of phase sensitive measurements in studying novel phases of matter.
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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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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

Benjamin Chiaro

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

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

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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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

Bálint Pató

Benjamin Chiaro

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

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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Removing leakage-induced correlated errors in superconducting quantum error correction

Jimmy Chen

Juan Atalaya

Frank Carlton Arute

Kunal Arya

Bob Benjamin Buckley

Nicholas Bushnell

Benjamin Chiaro

Roberto Collins

Andrew Dunsworth

Brooks Riley Foxen

Trent Huang

Kostyantyn Kechedzhi

Fedor Kostritsa

Pavel Laptev

Anthony Megrant

Xiao Mi

Josh Mutus

Charles Neill

Alexandru Paler

Nick Redd

Jamie Yao

Ping Yeh

Yu Chen

Vadim Smelyanskiy

John Martinis

Alexander Korotkov

Andre Gregory Petukhov

Rami Barends

Nature Communications, vol. 12 (2021), pp. 1761

Preview abstract
Quantum computing becomes scalable through error correction, but logical error rates only decrease with system size when physical errors are sufficiently uncorrelated. During computation, the unused high energy states of the qubits can become excited. In weakly nonlinear qubits, such as the superconducting transmon, these leakage states are long-lived and mobile, opening a path to errors that are correlated in space and time. The effects of leakage and its mitigation during quantum error correction remain an open question. Here, we report a reset protocol that returns a qubit to the ground state from all relevant higher level states. It requires no additional hardware and combines speed, fidelity, and resilience to noise. We test its performance with the bit-flip stabilizer code, a simplified version of the surface code scheme for quantum error correction. We investigate the accumulation and dynamics of leakage during the stabilizer codes. Using this protocol, we find lower rates of logical errors, and an improved scaling and stability of error suppression with qubits. This demonstration provides a key step on the path towards scalable quantum computing.
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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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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

Brooks Riley Foxen

Rob Graff

Steve Habegger

Sergei Isakov

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

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

Alan Derk

Alan Ho

Alex Opremcak

Alexander Korotkov

Alexandre Bourassa

Andre Gregory Petukhov

Andrew Dunsworth

Anthony Megrant

Bálint Pató

Benjamin Chiaro

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

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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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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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

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

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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Accurately computing electronic properties of materials using eigenenergies

Alan Derk

Alan Ho

Alex Opremcak

Alexander Korotkov

Andre Gregory Petukhov

Andrew Dunsworth

Anthony Megrant

Bálint Pató

Benjamin Chiaro

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

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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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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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

Rob Graff

Trent Huang

Sergei Isakov

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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Improved Fault-Tolerant Quantum Simulation of Condensed-Phase Correlated Electrons via Trotterization

Ian Kivlichan

Dominic Berry

Wei Sun

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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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

Brooks Riley Foxen

Rob Graff

Steve Habegger

Alan Ho

Trent Huang

William J. Huggins

Sergei Isakov

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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A 28nm Bulk-CMOS 4-to-8GHz <2mW Cryogenic Pulse Modulator for Scalable Quantum Computing

Trent Huang

Rami Barends

Kunal Arya

Ben Chiaro

Jimmy Chen

Yu Chen

Andrew Dunsworth

Brooks Foxen

Rob Graff

Josh Mutus

Anthony Megrant

Charles Neill

Amit Vainsencher

John Martinis

Proceedings of the 2019 International Solid State Circuits Conference, IEEE, pp. 456-458

Preview abstract
Future quantum computing systems will require cryogenic integrated circuits to control and measure millions of qubits. In this paper, we report design and measurement of a prototype cryogenic CMOS integrated circuit that has been optimized for the control of transmon qubits. The circuit has been integrated into a quantum measurement setup and its performance has been validated through multiple quantum control experiments.
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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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Design and Characterization of a 28-nm Bulk-CMOS Cryogenic Quantum Controller Dissipating Less than 2 mW at 3 K

Trent Huang

Sayan Das

Anthony Megrant

Rami Barends

Kunal Arya

Ben Chiaro

Zijun Chen

Yu Chen

Andrew Dunsworth

Brooks Foxen

Rob Graff

Josh Mutus

Charles Neill

Amit Vainsencher

John Martinis

IEEE Journal of Solid State Circuits, vol. 54(11) (2019), pp. 3043 - 3060

Preview abstract
Implementation of an error corrected quantum computer is believed to require a quantum processor with on the order of a million or more physical qubits and, in order to run such a processor, a quantum control system of similar scale will be required. Such a controller will need to be integrated within the cryogenic system and in close proximity with the quantum processor in order to make such a system practical. Here, we present a prototype cryogenic CMOS quantum controller designed in a 28-nm bulk CMOS process and optimized to implement a 4-bit XY gate instruction set for transmon qubits. After introducing the transmon qubit, including a discussion of how it is controlled, design considerations are discussed, with an emphasis on error rates and scalability. The circuit design is then discussed. Cryogenic performance of the underlying technology is presented and the results of several quantum control experiments carried out using the integrated controller are described. The paper ends with a comparison to the state of the art. It has been shown that the quantum control IC achieves comparable
performance with a conventional rack mount control system while dissipating less than 2mW of total AC and DC power and requiring a digital data stream of less than 500 Mb/s.
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Diabatic gates for frequency-tunable superconducting qubits

Rami Barends

A.G. Petukhov

Yu Chen

Kostyantyn Kechedzhi

Roberto Collins

Frank Carlton Arute

Kunal Arya

Jimmy Chen

Ben Chiaro

Andrew Dunsworth

Brooks Foxen

Rob Graff

Trent Huang

Fedor Kostritsa

Dave Landhuis

Anthony Megrant

Xiao Mi

Josh Mutus

Charles Neill

Eric Ostby

Amit Vainsencher

Jamie Yao

Ping Yeh

Vadim Smelyanskiy

John Martinis

Physical Review Letters, vol. 123 (2019), pp. 210501

Preview abstract
We demonstrate diabatic two-qubit gates with Pauli error rates down to 4.3(2)*10^{-3} in as fast as 18 ns using frequency-tunable superconducting qubits. This is achieved by synchronizing the entangling parameters with minima in the leakage channel. The synchronization shows a landscape in gate parameter space that agrees with model predictions and facilitates robust tune-up. We test both iSWAP-like and CPHASE gates with cross-entropy benchmarking. The presented approach can be extended to multibody operations as well.
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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

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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Fluctuations of Energy-Relaxation Times in Superconducting Qubits

Jimmy Chen

Anthony Megrant

Rami Barends

Kunal Arya

Ben Chiaro

Yu Chen

Andrew Dunsworth

Brooks Foxen

Rob Graff

Trent Huang

Josh Mutus

Charles Neill

Amit Vainsencher

Jim Wenner

Vadim Smelyanskiy

John Martinis

Physical Review Letters, vol. 121 (2018), pp. 090502

Preview abstract
Superconducting qubits are an attractive platform for quantum computing since they have demonstrated high-fidelity quantum gates and extensibility to modest system sizes. Nonetheless, an outstanding challenge is stabilizing their energy-relaxation times, which can fluctuate unpredictably in frequency and time. Here, we use qubits as spectral and temporal probes of individual two-level-system defects to provide direct evidence that they are responsible for the largest fluctuations. This research lays the foundation for stabilizing qubit performance through calibration, design and fabrication.
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Encoding Electronic Spectra in Quantum Circuits with Linear T Complexity

Dominic W. Berry

Nathan Wiebe

Alexandru Paler

Physical Review X, vol. 8 (2018), pp. 041015

Preview abstract
We construct quantum circuits which exactly encode the spectra of correlated electron models up to errors from rotation synthesis. By invoking these circuits as oracles within the recently introduced "qubitization" framework, one can use quantum phase estimation to sample states in the Hamiltonian eigenbasis with optimal query complexity O(lambda / epsilon) where lambda is an absolute sum of Hamiltonian coefficients and epsilon is target precision. For both the Hubbard model and electronic structure Hamiltonian in a second quantized basis diagonalizing the Coulomb operator, our circuits have T gate complexity O(N + \log (1/epsilon)) where N is number of orbitals in the basis. Compared to prior approaches, our algorithms are asymptotically more efficient in gate complexity and require fewer T gates near the classically intractable regime. Compiling to surface code fault-tolerant gates and assuming per gate error rates of one part in a thousand reveals that one can error-correct phase estimation on interesting instances of these problems beyond the current capabilities of classical methods using only a few times more qubits than would be required for magic state distillation.
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Quantum Simulation of Electronic Structure with Linear Depth and Connectivity

Ian D. Kivlichan

Nathan Wiebe

Alán Aspuru-Guzik

Garnet Kin-Lic Chan

Physical Review Letters, vol. 120 (2018), pp. 110501

Preview abstract
As physical implementations of quantum architectures emerge, it is increasingly important to consider the cost of algorithms for practical connectivities between qubits. We show that by using an arrangement of gates that we term the fermionic swap network, we can simulate a Trotter step of the electronic structure Hamiltonian in exactly N depth and with N^2/2 two-qubit entangling gates, and prepare arbitrary Slater determinants in at most N/2 depth, all assuming only a minimal, linearly connected architecture. We conjecture that no explicit Trotter step of the electronic structure Hamiltonian is possible with fewer entangling gates, even with arbitrary connectivities. These results represent significant practical improvements on the cost of all current proposed algorithms for both variational and phase estimation based simulation of quantum chemistry.
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Improved Techniques for Preparing Eigenstates of Fermionic Hamiltonians

Dominic W. Berry

Mária Kieferová

Artur Scherer

Yuval Sanders

Guang Hao Low

Nathan Wiebe

npj Quantum Information, vol. 4 (2017), pp. 22

Preview abstract
Modeling low energy eigenstates of fermionic systems can provide insight into chemical reactions and material properties and is one of the most anticipated applications of quantum computing. We present three techniques for reducing the cost of preparing fermionic Hamiltonian eigenstates using phase estimation. First, we report a polylogarithmic-depth quantum algorithm for antisymmetrizing the initial states required for simulation of fermions in first quantization. This is an exponential improvement over the previous state-of-the-art. Next, we show how to reduce the overhead due to repeated state preparation in phase estimation when the goal is to prepare the ground state to high precision and one has knowledge of an upper bound on the ground state energy that is less than the excited state energy (often the case in quantum chemistry). Finally, we explain how one can perform the time evolution necessary for the phase estimation based preparation of Hamiltonian eigenstates with exactly zero error by using the recently introduced qubitization procedure.
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OpenFermon: The Electronic Structure Package for Quantum Computers

Ian D. Kivlichan

Kevin Sung

Damian Steiger

Yudong Cao

Chengyu Dai

E. Schuyler Fried

Brendan Gimby

Thomas Häner

Tarini Hardikar

Vojtĕch Havlíček

Cupjin Huang

Zhang Jiang

Thomas O'Brien

Isil Ozfidan

Jhonathan Romero

Nicolas Sawaya

Kanav Setia

Sukin Sim

Mark Steudtner

Wei Sun

Fang Zhang

Quantum Science and Technology, vol. 5 (2017), pp. 034014

Preview abstract
Quantum simulation of chemistry and materials is predicted to be a key application for both near-term and fault-tolerant quantum devices. However, at present, developing and studying algorithms for these problems can be difficult due to the prohibitive amount of domain knowledge required in both the area of chemistry and quantum algorithms. To help bridge this gap and open the field to more researchers, we have developed the OpenFermion software package (www.openfermion.org). OpenFermion is an open-source software library written largely in Python under an Apache 2.0 license, aimed at enabling the simulation of fermionic models and quantum chemistry problems on quantum hardware. Beginning with an interface to common electronic structure packages, it simplifies the translation between a molecular specification and a quantum circuit for solving or studying the electronic structure problem on a quantum computer, minimizing the amount of domain expertise required to enter the field. Moreover, the package is designed to be extensible and robust, maintaining high software standards in documentation and testing. This release paper outlines the key motivations for design choices in OpenFermion and discusses some of the basic OpenFermion functionality available for the initial release of the package, which we believe will aid the community in the development of better quantum algorithms and tools for this exciting area.
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