# Zhang Jiang

Zhang is part of Google's quantum AI team. He worked at NASA Ames research center before joining Google. His main interests include quantum control, quantum simulation, and quantum optimization.

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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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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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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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Efficient and Noise Resilient Measurements for Quantum Chemistry on Near-Term Quantum Computers

William Huggins

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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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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Low-Depth Mechanisms for Quantum Optimization

Masoud Mohseni

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 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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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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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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Creating and manipulating a Laughlin-type ν=1/3 fractional quantum Hall state on a quantum computer with linear depth circuits

Armin Rahmani

Kevin J. Sung

Harald Putterman

Pouyan Ghaemi

PRX Quantum, vol. 1 (2020), pp. 020309

Preview abstract
Here we present an efficient quantum algorithm to generate an equivalent many-body state to Laughlin’s ν= 1/3 fractional quantum Hall state on a digitized quantum computer. Our algorithm only uses quantum gates acting on neighboring qubits in a quasi one-dimensional setting, and its circuit depth is linear in the number of qubits, i.e., the number of Landau levels in the second quantized picture. We identify correlation functions that serve as signatures of the Laughlin state and discuss how to obtain them on a quantum computer. We also discuss a generalization of the algorithm for creating quasiparticles in the Laughlin state. This paves the way for several important studies, including quantum simulation of non-equilibrium dynamics and braiding of quasiparticles in quantum Hall states.
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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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Using Models to Improve Optimizers for Variational Quantum Algorithms

Kevin Jeffery Sung

Jiahao Yao

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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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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Decoding Quantum Errors Using Subspace Expansions

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

Preview abstract
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

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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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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Learning to learn with quantum neural networks via classical neural networks

Guillaume Verdon

Michael Broughton

Kevin Jeffery Sung

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

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

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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

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

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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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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

Martin Head-Gordon

K. Birgitta Whaley

arXiv:1809.05523 (2018)

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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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Characterizing Quantum Supremacy in Near-Term Devices

Sergei Isakov

Vadim Smelyanskiy

Michael J. Bremner

John Martinis

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

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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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Understanding Quantum Tunneling through Quantum Monte Carlo Simulations

Sergei Isakov

Guglielmo Mazzola

Vadim Smelyanskiy

Matthias Troyer

PRL (2016)

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The tunneling between the two ground states of an Ising ferromagnet is a typical example of many-body tunneling processes between two local minima, as they occur during quantum annealing. Performing quantum Monte Carlo (QMC) simulations we find that the QMC tunneling rate displays the same scaling with system size, as the rate of incoherent tunneling. The scaling in both cases is O(Δ2), where Δ is the tunneling splitting. An important consequence is that QMC simulations can be used to predict the performance of a quantum annealer for tunneling through a barrier. Furthermore, by using open instead of periodic boundary conditions in imaginary time, equivalent to a projector QMC algorithm, we obtain a quadratic speedup for QMC, and achieve linear scaling in Δ. We provide a physical understanding of these results and their range of applicability based on an instanton picture.
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