Lyapunov controlinspired strategies for quantum combinatorial optimization
Alicia B. Magann, Kenneth M. Rudinger, Matthew D. Grace, Mohan Sarovar
The prospect of using quantum computers to solve combinatorial optimization problems via the quantum approximate optimization algorithm (QAOA) has attracted considerable interest in recent years. However, a key limitation associated with QAOA is the need to classically optimize over a set of quantum circuit parameters. This classical optimization can have significant associated costs and challenges. Here, we provide an expanded description of Lyapunov controlinspired strategies for quantum optimization, as first presented in arXiv:2103.08619, that do not require any classical optimization effort. Instead, these strategies utilize feedback from qubit measurements to assign values to the quantum circuit parameters in a deterministic manner, such that the combinatorial optimization problem solution improves monotonically with the quantum circuit depth. Numerical analyses are presented that investigate the utility of these strategies towards MaxCut on weighted and unweighted 3regular graphs, both in ideal implementations and also in the presence of measurement noise. We also discuss how how these strategies may be used to seed QAOA optimizations in order to improve performance for nearterm applications, and explore connections to quantum annealing.
arXiv:2108.05945

Unifying and benchmarking stateoftheart quantum error mitigation techniques
Daniel Bultrini, Max Hunter Gordon, Piotr Czarnik, Andrew Arrasmith, Patrick J. Coles, Lukasz Cincio
Error mitigation is an essential component of achieving practical quantum advantage in the near term, and a number of different approaches have been proposed. In this work, we recognize that many stateoftheart error mitigation methods share a common feature: they are datadriven, employing classical data obtained from runs of different quantum circuits. For example, Zeronoise extrapolation (ZNE) uses variable noise data and Clifforddata regression (CDR) uses data from nearClifford circuits. We show that Virtual Distillation (VD) can be viewed in a similar manner by considering classical data produced from different numbers of state preparations. Observing this fact allows us to unify these three methods under a general datadriven error mitigation framework that we call UNIfied Technique for Error mitigation with Data (UNITED). In certain situations, we find that our UNITED method can outperform the individual methods (i.e., the whole is better than the individual parts). Specifically, we employ a realistic noise model obtained from a trapped ion quantum computer to benchmark UNITED, as well as stateoftheart methods, for problems with various numbers of qubits, circuit depths and total numbers of shots. We find that different techniques are optimal for different shot budgets. Namely, ZNE is the best performer for small shot budgets (10^{5}), while Cliffordbased approaches are optimal for larger shot budgets (10^{6}−10^{8}), and for our largest considered shot budget (10^{10}), UNITED gives the most accurate correction. Hence, our work represents a benchmarking of current error mitigation methods, and provides a guide for the regimes when certain methods are most useful.
arXiv:2107.13470

How to perform the coherent measurement of a curved phase space by continuous isotropic measurement. I. Spin and the Krausoperator geometry of SL(2,C)
Christopher S. Jackson, Carlton M. Caves
Recently it was reported that the spincoherent state (SCS) positiveoperatorvalued measure (POVM) can be performed for any spin system by continuous isotropic measurement of the three total spin components [E. Shojaee, C. S. Jackson, C. A. Riofrío, A. Kalev, and I. H. Deutsch, Phys. Rev. Lett. 121, 130404 (2018)]. The outcome probability distribution of the SCS POVM for an input quantum state is the generalized Qfunction, which is defined on the 2sphere phase space of SCSs. This article develops the theoretical details of the continuous isotropic measurement and places it within the general context of applying curvedphasespace correspondences to quantum systems, indicating their experimental utility by explaining how to analyze this measurement's performance. The analysis is in terms of the Kraus operators that develop over the course of a continuous isotropic measurement. The Kraus operators represent elements of the Lie group SL(2,C), a complex version of the usual unitary operators that represent elements of SU(2). Consequently, the associated POVM elements represent points in the 3hyperboloid SU(2)∖SL(2,C). Three equivalent stochastic techniques, path integral, diffusion (FokkerPlanck) equation, and stochastic differential equations, are applied to show that the POVM quickly limits to the SCS POVM. We apply two basic mathematical tools to the Kraus operators, the MaurerCartan form, modified for stochastic applications, and the Cartan decomposition associated with the symmetric pair SU(2)⊂SL(2,C). Informed by these tools, the three stochastic techniques are applied directly to the Kraus operators in a representation independent, and thus geometric, way (independent of any spectral information about the spin components).
arXiv:2107.12396

Behavior of Analog Quantum Algorithms
Lucas T. Brady, Lucas Kocia, Przemyslaw Bienias, Aniruddha Bapat, Yaroslav Kharkov, Alexey V. Gorshkov
Analog quantum algorithms are formulated in terms of Hamiltonians rather than unitary gates and include quantum adiabatic computing, quantum annealing, and the quantum approximate optimization algorithm (QAOA). These algorithms are promising candidates for nearterm quantum applications, but they often require fine tuning via the annealing schedule or variational parameters. In this work, we explore connections between these analog algorithms, as well as limits in which they become approximations of the optimal procedure. Notably, we explore how the optimal procedure approaches a smooth adiabatic procedure but with a superposed oscillatory pattern that can be explained in terms of the interactions between the ground state and first excited state that effect the coherent error cancellation of diabatic transitions. Furthermore, we provide numeric and analytic evidence that QAOA emulates this optimal procedure with the length of each QAOA layer equal to the period of the oscillatory pattern. Additionally, the ratios of the QAOA bangs are determined by the smooth, nonoscillatory part of the optimal procedure. We provide arguments for these phenomena in terms of the product formula expansion of the optimal procedure. With these arguments, we conclude that different analog algorithms can emulate the optimal protocol under different limits and approximations. Finally, we present a new algorithm for better approximating the optimal protocol using the analytic and numeric insights from the rest of the paper. In practice, numerically, we find that this algorithm outperforms standard QAOA and naive quantum annealing procedures.
arXiv:2107.01218

Feedbackbased quantum optimization
Alicia B. Magann, Kenneth M. Rudinger, Matthew D. Grace, Mohan Sarovar
Quantum computers are expected to offer advantages over classical computers for combinatorial optimization. Here, we introduce a feedbackbased strategy for quantum optimization, where the results of qubit measurements are used to constructively assign values to quantum circuit parameters. We show that this procedure results in an estimate of the combinatorial optimization problem solution that improves monotonically with the depth of the quantum circuit. Importantly, the measurementbased feedback enables approximate solutions to the combinatorial optimization problem without the need for any classical optimization effort, as would be required for the quantum approximate optimization algorithm (QAOA). Numerical analyses are presented that investigate the utility of this feedbackbased protocol for the graphpartitioning problem MaxCut.
arXiv:2103.08619

Electronic Structure in a Fixed Basis is QMAcomplete
Bryan O’Gorman, Sandy Irani, James Whitfield, Bill Fefferman
Finding the ground state energy of electrons subject to an external electric field is a fundamental problem in computational chemistry. We prove that this electronicstructure problem, when restricted to a fixed singleparticle basis and fixed number of electrons, is QMAcomplete. Schuch and Verstraete have shown hardness for the electronicstructure problem with an additional sitespecific external magnetic field, but without the restriction to a fixed basis. In their reduction, a local Hamiltonian on qubits is encoded in the sitespecific magnetic field. In our reduction, the local Hamiltonian is encoded in the choice of spatial orbitals used to discretize the electronicstructure Hamiltonian. As a step in their proof, Schuch and Verstraete show a reduction from the antiferromagnetic Heisenberg Hamiltonian to the FermiHubbard Hamiltonian. We combine this reduction with the fact that the antiferromagnetic Heisenberg Hamiltonian is QMAhard to observe that the FermiHubbard Hamiltonian on generic graphs is QMAhard, even when all the hopping coefficients have the same sign. We then reduce from FermiHubbard by showing that an instance of FermiHubbard can be closely approximated by an instance of the ElectronicStructure Hamiltonian in a fixed basis. Finally, we show that estimating the energy of the lowestenergy Slaterdeterminant state (i.e., the HartreeFock state) is NPcomplete for the ElectronicStructure Hamiltonian in a fixed basis.
arXiv:2103.08215

Quantum Markov Chain Monte Carlo with Digital Dissipative Dynamics on Quantum Computers
Mekena Metcalf, Emma Stone, Katherine Klymko, Alexander F. Kemper, Mohan Sarovar, Wibe A. de Jong
Modeling the dynamics of a quantum system connected to the environment is critical for advancing our understanding of complex quantum processes, as most quantum processes in nature are affected by an environment. Modeling a macroscopic environment on a quantum simulator may be achieved by coupling independent ancilla qubits that facilitate energy exchange in an appropriate manner with the system and mimic an environment. This approach requires a large, and possibly exponential number of ancillary degrees of freedom which is impractical. In contrast, we develop a digital quantum algorithm that simulates interaction with an environment using a small number of ancilla qubits. By combining periodic modulation of the ancilla energies, or spectral combing, with periodic reset operations, we are able to mimic interaction with a large environment and generate thermal states of interacting manybody systems. We evaluate the algorithm by simulating preparation of thermal states of the transverse Ising model. Our algorithm can also be viewed as a quantum Markov chain Monte Carlo (QMCMC) process that allows sampling of the Gibbs distribution of a multivariate model. To demonstrate this we evaluate the accuracy of sampling Gibbs distributions of simple probabilistic graphical models using the algorithm.
arXiv:2103.03207

Qubitefficient exponential suppression of errors
Piotr Czarnik, Andrew Arrasmith, Lukasz Cincio, Patrick J. Coles
Achieving a practical advantage with nearterm quantum computers hinges on having effective methods to suppress errors. Recent breakthroughs have introduced methods capable of exponentially suppressing errors by preparing multiple noisy copies of a state and virtually distilling a more purified version. Here we present an alternative method, the ResourceEfficient Quantum Error Suppression Technique (REQUEST), that adapts this breakthrough to much fewer qubits by making use of active qubit resets, a feature now available on commercial platforms. Our approach exploits a space/time tradeoff to achieve a similar error reduction using only 2N+1 qubits as opposed to MN+1 qubits, for M copies of an N qubit state. Additionally, we propose a method based on nearClifford circuits to find the optimal number of these copies in the presence of realistic noise, which limits this error suppression. We perform a numerical comparison between the original method and our qubitefficient version with a realistic trappedion noise model. We find that REQUEST can reproduce the exponential suppression of errors of the virtual distillation approach for simple noise models while outperforming virtual distillation when fewer than 3N+1 qubits are available.
arXiv:2102.06056

Improved Strong Simulation of Universal Quantum Circuits
Lucas Kocia
We find a scaling reduction in the stabilizer rank of the twelvequbit tensored T gate magic state. This lowers its asymptotic bound to 2^{∼0.463t} for multiPauli measurements on t magic states, improving over the best previously found bound of 2^{∼0.468t}. We numerically demonstrate this reduction. This constructively produces the most efficient strong simulation algorithm of the Clifford+T gateset to relative or multiplicative error. We then examine the cost of Pauli measurement in terms of its Gauss sum rank, which is a slight generalization of the stabilizer rank and is a lower bound on its asymptotic scaling. We demonstrate that this lower bound appears to be tight at low tcounts, which suggests that the stabilizer rank found at the twelvequbit state can be lowered further to 2^{∼0.449t} and we prove and numerically show that this is the case for singlePauli measurements. Our construction directly shows how the reduction at 12 qubits is iteratively based on the reduction obtained at 6, 3, 2, and 1 qubits. This explains why novel reductions are found at tensor factors for these number of qubit primitives, an explanation lacking previously in the literature. Furthermore, in the process we observe an interesting relationship between the T gate magic state's stabilizer rank and decompositions that are Cliffordisomorphic to a computational subbasis tensored with singlequbit states that produce minimal unique stabilizer state inner products  the same relationship that allowed for finding minimal numbers of unique Gauss sums in the odddimensional qudit Wigner formulation of Pauli measurements.
arXiv:2012.11739

Verifying quantum circuit equivalences using ProveIt, an interactive theorem proving assistant
Joaquín E. Madrid Larrañaga, Wayne M. Witzel
It is challenging to ensure that a quantum program is a faithful implementation of a quantum algorithm after it is tailored and optimized for specific quantum hardware. While quantum computation literature often focuses on the general problem of proving that two arbitrary quantum programs are equivalent, which has exponential complexity with respect to the number of qubits, we consider the simpler, practical problem of verifying each step in a series of small transformations to certify that a final program is equivalent to the original one. We use quantum circuits as an intuitive visual representation of quantum programs and extend ProveIt (a Pythonbased, Sandiadeveloped interactive theorem prover) with quantum circuit proof capabilities. This tool could be interfaced with an automated quantum compiler/optimizer or be used interactively by a quantum information expert for exploratory purposes. Using a small number of basic facts about circuit equivalences, ProveIt will be able to solve the verification problem for most practical purposes with linear time complexity.
Computer Science Research Institute Summer Proceedings 2020

ProveIt: A Proof Assistant for Organizing and Verifying General Mathematical Knowledge
Wayne M. Witzel, Warren D. Craft, Robert D. Carr, Joaquín E. Madrid Larrañaga
We introduce ProveIt, a Pythonbased generalpurpose interactive theoremproving assistant designed with the goal of making formal theorem proving as easy and natural as informal theorem proving (with moderate training). ProveIt uses a highlyflexible Jupyter notebookbased user interface that documents interactions and proof steps using LaTeX. We review ProveIt's highly expressive representation of expressions, judgments, theorems, and proofs; demonstrate the system by constructing a traditional proofbycontradiction that √2 ∉ ℚ; and discuss how the system avoids inconsistencies such as Russell's and Curry's paradoxes. Extensive documentation is provided in the appendices about core elements of the system. Current development and future work includes promising applications to quantum circuit manipulation and quantum algorithm verification.
arXiv:2012.10987

Consistency testing for robust phase estimation
Antonio E. Russo, William M. Kirby, Kenneth M. Rudinger, Andrew D. Baczewski, Shelby Kimmel
We present an extension to the robust phase estimation protocol, which can identify incorrect results that would otherwise lie outside the expected statistical range. Robust phase estimation is increasingly a method of choice for applications such as estimating the effective process parameters of noisy hardware, but its robustness is dependent on the noise satisfying certain threshold assumptions. We provide consistency checks that can indicate when those thresholds have been violated, which can be difficult or impossible to test directly. We test these consistency checks for several common noise models, and identify two possible checks with high accuracy in locating the point in a robust phase estimation run at which further estimates should not be trusted. One of these checks may be chosen based on resource availability, or they can be used together in order to provide additional verification.
arXiv:2011.13442
Phys. Rev. A 103, 042609 (2021)

Optimizing parametrized quantum circuits via noiseinduced breaking of symmetries
Enrico Fontana, M. Cerezo, Andrew Arrasmith, Ivan Rungger, Patrick J. Coles
Very little is known about the cost landscape for parametrized Quantum Circuits (PQCs). Nevertheless, PQCs are employed in Quantum Neural Networks and Variational Quantum Algorithms, which may allow for nearterm quantum advantage. Such applications require good optimizers to train PQCs. Recent works have focused on quantumaware optimizers specifically tailored for PQCs. However, ignorance of the cost landscape could hinder progress towards such optimizers. In this work, we analytically prove two results for PQCs: (1) We find an exponentially large symmetry in PQCs, yielding an exponentially large degeneracy of the minima in the cost landscape. (2) We show that noise (specifically nonunital noise) can break these symmetries and lift the degeneracy of minima, making many of them local minima instead of global minima. Based on these results, we introduce an optimization method called Symmetrybased Minima Hopping (SYMH), which exploits the underlying symmetries in PQCs to hop between local minima in the cost landscape. The versatility of SYMH allows it to be combined with local optimizers (e.g., gradient descent) with minimal overhead. Our numerical simulations show that SYMH improves the overall optimizer performance.
arXiv:2011.08763

Unified approach to datadriven quantum error mitigation
Angus Lowe, Max Hunter Gordon, Piotr Czarnik, Andrew Arrasmith, Patrick J. Coles, Lukasz Cincio
Achieving nearterm quantum advantage will require effective methods for mitigating hardware noise. Datadriven approaches to error mitigation are promising, with popular examples including zeronoise extrapolation (ZNE) and Clifford data regression (CDR). Here we propose a novel, scalable error mitigation method that conceptually unifies ZNE and CDR. Our approach, called variablenoise Clifford data regression (vnCDR), significantly outperforms these individual methods in numerical benchmarks. vnCDR generates training data first via nearClifford circuits (which are classically simulable) and second by varying the noise levels in these circuits. We employ a noise model obtained from IBM's Ourense quantum computer to benchmark our method. For the problem of estimating the energy of an 8qubit Ising model system, vnCDR improves the absolute energy error by a factor of 33 over the unmitigated results and by factors 20 and 1.8 over ZNE and CDR, respectively. For the problem of correcting observables from random quantum circuits with 64 qubits, vnCDR improves the error by factors of 2.7 and 1.5 over ZNE and CDR, respectively.
arXiv:2011.01157
Phys. Rev. Research, 3, 033098 (2021)

Custom fermionic codes for quantum simulation
Riley W. Chien, James D. Whitfield
Simulating a fermionic system on a quantum computer requires encoding the anticommuting fermionic variables into the operators acting on the qubit Hilbert space. The most familiar of which, the JordanWigner transformation, encodes fermionic operators into nonlocal qubit operators. As nonlocal operators lead to a slower quantum simulation, recent works have proposed ways of encoding fermionic systems locally. In this work, we show that locality may in fact be too strict of a condition and the size of operators can be reduced by encoding the system quasilocally. We give examples relevant to lattice models of condensed matter and systems relevant to quantum gravity such as SYK models. Further, we provide a general construction for designing codes to suit the problem and resources at hand and show how one particular class of quasilocal encodings can be thought of as arising from truncating the state preparation circuit of a local encoding. We end with a discussion of designing codes in the presence of device connectivity constraints.
arXiv:2009.11860

From pulses to circuits and back again: A quantum optimal control perspective on variational quantum algorithms
Alicia B. Magann, Christian Arenz, Matthew D. Grace, TakSan Ho, Robert L. Kosut, Jarrod R. McClean, Herschel A. Rabitz, Mohan Sarovar
The last decade has witnessed remarkable progress in the development of quantum technologies. Although faulttolerant devices likely remain years away, the noisy intermediatescale quantum devices of today may be leveraged for other purposes. Leading candidates are variational quantum algorithms (VQAs), which have been developed for applications including chemistry, optimization, and machine learning, but whose implementations on quantum devices have yet to demonstrate improvements over classical capabilities. In this Perspective, we propose a variety of ways that progress toward this potential crossover point could be informed by quantum optimal control theory. To set the stage, we identify VQAs and quantum optimal control as formulations of variational optimization at the circuit level and pulse level, respectively, where these represent just two levels in a broader hierarchy of abstractions that we consider. In this unified picture, we suggest several ways that the different levels of abstraction may be connected, in order to facilitate the application of quantum optimal control theory to VQA challenges associated with ansatz selection, optimization landscapes, noise, and robustness. A major theme throughout is the need for sufficient control resources in VQA implementations; we discuss different ways this need can manifest, outline a variety of open questions, and conclude with a look to the future.
arXiv:2009.06702
PRX Quantum (Perspective), 2, 010101 (2020)

Quantum state verification in the quantum linear systems problem
Rolando D. Somma, Yigit Subasi
We analyze the complexity of quantum state verification in the context of solving systems of linear equations of the form Ax =b . We show that any quantum operation that verifies whether a given quantum state is within a constant distance from the solution of the quantum linear systems problem requires q=Ω(κ) uses of a unitary that prepares a quantum state ∣ b⟩, proportional to b , and its inverse in the worst case. Here, κ is the condition number of the matrix A. For typical instances, we show that q=Ω(√κ) with high probability. These lower bounds are almost achieved if quantum state verification is performed using known quantum algorithms for the quantum linear systems problem. We also analyze the number of copies of ∣ b⟩ required by verification procedures of the prepare and measure type. In this case, the lower bounds are quadratically worse, being Ω(κ^{2}) in the worst case and Ω(κ) in typical instances with high probability. We discuss the implications of our results to known variational and related approaches to this problem, where state preparation, gate, and measurement errors will need to decrease rapidly with κ for worstcase and typical instances if error correction is not used, and present some open problems.
arXiv:2007.15698
PRX Quantum, 2, 010315 (2021)

NoiseInduced Barren Plateaus in Variational Quantum Algorithms
Samson Wang, Enrico Fontana, M. Cerezo, Kunal Sharma, Akira Sone, Lukasz Cincio, Patrick J. Coles
Variational Quantum Algorithms (VQAs) may be a path to quantum advantage on Noisy IntermediateScale Quantum (NISQ) computers. A natural question is whether the noise on NISQ devices places any fundamental limitations on the performance of VQAs. In this work, we rigorously prove a serious limitation for noisy VQAs, in that the noise causes the training landscape to have a barren plateau (i.e., vanishing gradient). Specifically, for the local Pauli noise considered, we prove that the gradient vanishes exponentially in the number of layers L. This implies exponential decay in the number of qubits n when L scales as poly(n), for sufficiently large coefficients in the polynomial. These noiseinduced barren plateaus (NIBPs) are conceptually different from noisefree barren plateaus, which are linked to random parameter initialization. Our result is formulated for an abstract ansatz that includes as special cases the Quantum Alternating Operator Ansatz (QAOA) and the Unitary Coupled Cluster Ansatz, among others. In the case of the QAOA, we implement numerical heuristics that confirm the NIBP phenomenon for a realistic hardware noise model.
arXiv:2007.14384

Quantum proportionalintegral (PI) control
Hui Chen, Hanhan Li, Felix Motzoi, Leigh S. Martin, K. Birgitta Whaley, Mohan Sarovar
Feedback control is an essential component of many modern technologies and provides a key capability for emergent quantum technologies. We extend existing approaches of direct feedback control in which the controller applies a function directly proportional to the output signal (P feedback), to strategies in which feedback determined by an integrated output signal (I feedback), and to strategies in which feedback consists of a combination of P and I terms. The latter quantum PI feedback constitutes the analog of the widely used proportionalintegral feedback of classical control. All of these strategies are experimentally feasible and require no complex state estimation. We apply the resulting formalism to two canonical quantum feedback control problems, namely, stabilization of a harmonic oscillator under thermal noise, and generation of an entangled state of two remote qubits under conditions of arbitrary measurement efficiency. These two problems allow analysis of the relative benefits of P, I, and PI feedback control. We find that P feedback shows the best performance for harmonic state stabilization when actuation of both position and momentum feedback is possible, while when only actuation of position is available, I feedback consistently shows the best performance, although feedback delay is shown to improve the performance of a P strategy here. In contrast, for the twoqubit remote entanglement generation we find that the best strategy can be a combined PI strategy, when measurement efficiency is not one.
arXiv:2007.13853
New J. Phys. 22, 113014 (2020)

Limitations of HartreeFock with quantum resources
Sahil Gulania, James Daniel Whitfield
The HartreeFock problem provides the conceptual and mathematical underpinning of a large portion of quantum chemistry. As efforts in quantum technology aim to enhance computational chemistry algorithms, the fundamental HartreeFock problem is a natural target. While quantum computers and quantum simulation offer many prospects for the future of modern chemistry, the HartreeFock problem is not a likely candidate. We highlight this fact from a number of perspectives including computational complexity, practical examples, and the full characterization of the energy landscapes for simple systems.
arXiv:2007.09806
J. Chem. Phys., 154, 044112 (2021)

Evaluating energy differences on a quantum computer with robust phase estimation
A.E. Russo, K.M. Rudinger, B.C.A. Morrison, A.D. Baczewski
We adapt the robust phase estimation algorithm to the evaluation of energy differences between two eigenstates using a quantum computer. This approach does not require controlled unitaries between auxiliary and system registers or even a single auxiliary qubit. As a proof of concept, we calculate the energies of the ground state and lowlying electronic excitations of a hydrogen molecule in a minimal basis on a cloud quantum computer. The denominative robustness of our approach is then quantified in terms of a high tolerance to coherent errors in the state preparation and measurement. Conceptually, we note that all quantum phase estimation algorithms ultimately evaluate eigenvalue differences.
arXiv:2007.08697
Phys. Rev. Lett. 126, 210501 (2021)

Machine learning of noiseresilient quantum circuits
Lukasz Cincio, Kenneth Rudinger, Mohan Sarovar, Patrick J. Coles
Noise mitigation and reduction will be crucial for obtaining useful answers from nearterm quantum computers. In this work, we present a general framework based on machine learning for reducing the impact of quantum hardware noise on quantum circuits. Our method, called noiseaware circuit learning (NACL), applies to circuits designed to compute a unitary transformation, prepare a set of quantum states, or estimate an observable of a manyqubit state. Given a task and a device model that captures information about the noise and connectivity of qubits in a device, NACL outputs an optimized circuit to accomplish this task in the presence of noise. It does so by minimizing a taskspecific cost function over circuit depths and circuit structures. To demonstrate NACL, we construct circuits resilient to a finegrained noise model derived from gate set tomography on a superconductingcircuit quantum device, for applications including quantum state overlap, quantum Fourier transform, and Wstate preparation.
arXiv:2007.01210
PRX Quantum, 2, 010324 (2021)

Quantum simulation of the qubitregularized O(3)sigma model
Alexander J. Buser, Tanmoy Bhattacharya, Lukasz Cincio, Rajan Gupta
Recently, Singh and Chandrasekharan showed that fixed points of the classical nonlinear O(3)sigma model can be reproduced near a quantum phase transition of a spin model with just two qubits per lattice site. In this paper, we demonstrate how to prepare the ground state of this model and measure a dynamical quantity of interest, the O(3) Noether charge, on a quantum computer. In particular, we apply Trotter methods to obtain results for the complexity of adiabatic ground state preparation in both the weakcoupling and quantumcritical regimes and use shadow tomography to measure the dynamics of local observables. We then present and analyze a quantum algorithm based on nonunitary randomized simulation methods that may yield an approach suitable for intermediateterm noisy quantum devices.
arXiv:2006.15746
Phys. Rev. D, 102, 114514 (2020)

Error mitigation with Clifford quantumcircuit data
Piotr Czarnik, Andrew Arrasmith, Patrick J. Coles, Lukasz Cincio
Achieving nearterm quantum advantage will require accurate estimation of quantum observables despite significant hardware noise. For this purpose, we propose a novel, scalable errormitigation method that applies to gatebased quantum computers. The method generates training data {X^{noisy}_{i},X^{exact}_{i}} via quantum circuits composed largely of Clifford gates, which can be efficiently simulated classically, where X^{noisy}_{i} and X^{exact}_{i} are noisy and noiseless observables respectively. Fitting a linear ansatz to this data then allows for the prediction of noisefree observables for arbitrary circuits. We analyze the performance of our method versus the number of qubits, circuit depth, and number of nonClifford gates. We obtain an orderofmagnitude error reduction for a groundstate energy problem on a 16qubit IBMQ quantum computer and on a 64qubit noisy simulator.
arXiv:2005.10189

Variational Quantum State Eigensolver
M. Cerezo, Kunal Sharma, Andrew Arrasmith, Patrick J. Coles
Extracting eigenvalues and eigenvectors of exponentially large matrices will be an important application of nearterm quantum computers. The Variational Quantum Eigensolver (VQE) treats the case when the matrix is a Hamiltonian. Here, we address the case when the matrix is a density matrix ρ. We introduce the Variational Quantum State Eigensolver (VQSE), which is analogous to VQE in that it variationally learns the largest eigenvalues of ρ as well as a gate sequence V that prepares the corresponding eigenvectors. VQSE exploits the connection between diagonalization and majorization to define a cost function C=Tr(ρ̃H) where H is a nondegenerate Hamiltonian. Due to Schurconcavity, C is minimized when ρ̃=VρV† is diagonal in the eigenbasis of H. VQSE only requires a single copy of ρ (only n qubits), making it amenable for nearterm implementation. We demonstrate two applications of VQSE: (1) Principal component analysis, and (2) Error mitigation.
arXiv:2004.01372

Improved Simulation of Quantum Circuits by Fewer Gaussian Eliminations
Lucas Kocia, Mohan Sarovar
We show that the cost of strong simulation of quantum circuits using t T gate magic states exhibits nontrivial reductions on its upper bound for t=1, t=2, t=3, and t=6 with oddprimequdits. This agrees with previous numerical bounds found for qubits. We define simulation cost by the number of terms that require Gaussian elimination of a t×t matrix and so capture the cost of simulation methods that proceed by computing stabilizer inner products or evaluating quadratic Gauss sums. Prior numerical searchs for qubits were unable to converge beyond t=7. We effectively increase the space searched for these nontrivial reductions by >10^{10^4} and extend the bounds to t=14 for qutrits. This is accomplished by using the WignerWeylMoyal formalism to algebraically find bounds instead of relying on numerics. We find a new reduction in the upper bound from the 12qutrit magic state of 3^{∼0.469t,} which improves on the bound obtained from the 6qutrit magic state of 3^{∼0.482t}.
arXiv:2003.01130
Phys. Rev. A, 103, 022603 (2021)

Digital quantum simulation of molecular dynamics and control
Alicia B. Magann, Matthew D. Grace, Herschel A Rabitz, Mohan Sarovar
Optimallyshaped electromagnetic fields have the capacity to coherently control the dynamics of quantum systems, and thus offer a promising means for controlling molecular transformations relevant to chemical, biological, and materials applications. Currently, advances in this area are hindered by the prohibitive cost of the quantum dynamics simulations needed to explore the principles and possibilities of molecular control. However, the emergence of nascent quantumcomputing devices suggests that efficient simulations of quantum dynamics may be on the horizon. In this article, we study how quantum computers could be leveraged to design optimallyshaped fields to control molecular systems. We introduce a hybrid algorithm that utilizes a quantum computer for simulating the fieldinduced quantum dynamics of a molecular system in polynomial time, in combination with a classical optimization approach for updating the field. Qubit encoding methods relevant for molecular control problems are described, and procedures for simulating the quantum dynamics and obtaining the simulation results are discussed. Numerical illustrations are then presented that explicitly treat paradigmatic vibrational and rotational control problems, and also consider how optimallyshaped fields could be used to elucidate the mechanisms of energy transfer in lightharvesting complexes; resource estimates are provided for the latter task.
arXiv:2002.12497
Phys. Rev. Research, 3, 023165 (2021)

CostFunctionDependent Barren Plateaus in Shallow Quantum Neural Networks
M. Cerezo, Akira Sone, Tyler Volkoff, Lukasz Cincio, Patrick J. Coles
Variational quantum algorithms (VQAs) optimize the parameters θ of a quantum neural network V(θ) to minimize a cost function C. While VQAs may enable practical applications of noisy quantum computers, they are nevertheless heuristic methods with unproven scaling. Here, we rigorously prove two results. Our first result states that defining C in terms of global observables leads to an exponentially vanishing gradient (i.e., a barren plateau) even when V (θ) is shallow. This implies that several VQAs in the literature must revise their proposed cost functions. On the other hand, our second result states that defining C with local observables leads to a polynomially vanishing gradient, so long as the depth of V(θ) is O(log n). Taken together, our results establish a precise connection between locality and trainability. Finally, we illustrate these ideas with largescale simulations, up to 100 qubits, of a particular VQA known as quantum autoencoders.
arXiv:2001.00550
Nature Communications 12, 1791 (2021)

Reducing qubit requirements for quantum simulation using molecular point group symmetries
Kanav Setia, Richard Chen, Julia E. Rice, Antonio Mezzacapo, Marco Pistoia, James Whitfield
Simulating molecules is believed to be one of the earlystage applications for quantum computers. Current stateoftheart quantum computers are limited in size and coherence, therefore optimizing resources to execute quantum algorithms is crucial. In this work, we develop a formalism to reduce the number of qubits required for simulating molecules using spatial symmetries, by finding qubit representations of irreducible symmetry sectors. We present our results for various molecules and elucidate a formal connection of this work with a previous technique that analyzed generic Z_{2} Pauli symmetries.
arXiv:1910.14644
J. Chem. Theory Comput. 16, 6091 (2020)
