Here we list publications reporting on project output and also software packages that are associated to OVER-QC.

**Publications**

### Simultaneous Measurements of Noncommuting Observables. Positive Transformations and Instrumental Lie Groups

Christopher S. Jackson, Carlton M. Caves

We formulate a general program for analyzing continuous, differential weak, simultaneous measurements of noncommuting observables, which focuses on describing the measuring instrument autonomously, without states. The Kraus operators of such measuring processes are time-ordered products of fundamental differential positive transformations, which generate nonunitary transformation groups that we call instrumental Lie groups. The temporal evolution of the instrument is equivalent to the diffusion of a Kraus-operator distribution function defined relative to the invariant measure of the instrumental Lie group. This way of considering instrument evolution we call the Instrument Manifold Program. We relate the Instrument Manifold Program to state-based stochastic master equations. We then explain how the Instrument Manifold Program can be used to describe instrument evolution in terms of a universal cover, the universal instrumental Lie group, which is independent of Hilbert space. The universal instrument is generically infinite dimensional, in which situation the instrument’s evolution is chaotic. Special simultaneous measurements have a finite-dimensional universal instrument, in which case the instrument is considered to be principal and can be analyzed within the universal instrumental Lie group. Principal instruments belong at the foundation of quantum mechanics. We consider the three most fundamental examples: measurement of a single observable, of position and momentum, and of the three components of angular momentum. These measurements limit to strong simultaneous measurements. For a single observable, this gives the standard decay of coherence between inequivalent irreps; for the latter two, it gives a collapse within each irrep onto the canonical or spherical phase space, locating phase space at the boundary of these instrumental Lie groups.

### Simultaneous Momentum and Position Measurement and the Instrumental Weyl-Heisenberg Group

Christopher S. Jackson, Carlton M. Caves

The canonical commutation relation, [Q,P]=iℏ, stands at the foundation of quantum theory and the original Hilbert space. The interpretation of P & Q as observables has always relied on the analogies that exist between the unitary transformations of Hilbert space and the canonical (a.k.a. contact) transformations of classical phase space. Now that the theory of quantum measurement is essentially complete (this took a while), it is possible to revisit the canonical commutation relation in a way that sets the foundation of quantum theory not on unitary transformations, but on positive transformations. This paper shows how the concept of simultaneous measurement leads to a fundamental differential geometric problem whose solution shows us the following: The simultaneous P & Q measurement (SPQM) defines a universal measuring instrument, which takes the shape of a 7-dimensional manifold, a universal covering group we call the Instrumental Weyl-Heisenberg Group, IWH. The group IWH connects the identity to classical phase space in unexpected ways that are significant enough that the positive-operator-valued measure (POVM) offers a complete alternative to energy quantization. Five of the dimensions define processes that can be easily recognized and understood. The other two dimensions, the normalization and phase in the center of IWH, are less familiar. The normalization, in particular, requires special handling in order to describe and understand the SPQM instrument.

### Basis set generation and optimization in the NISQ era with Quiqbox.jl

Weishi Wang, James D. Whitfield

In the noisy intermediate-scale quantum era, ab initio computation of the electronic structure problem has become one of the major benchmarks for identifying the boundary between classical and quantum computational power. The single-particle basis set plays a key role in the electronic structure methods implemented on both classical and quantum devices. To investigate the consequences of the single-particle basis set, we propose a framework for more customizable basis set generation and basis set optimization. This framework allows configurations of composite Gaussian-type basis functions to go beyond typical Gaussian-type basis set frameworks such as the atomic orbitals and floating basis sets. Such basis set generations set the stage for more flexible variational optimization of basis set parameters. To realize this framework, we have developed an open-source electronic structure package named “Quiqbox” in the Julia programming language. Both the Hartree–Fock procedure and Gaussian-based electronic integral computations are implemented in this package. We compare Quiqbox with the basis set optimization package DiffiQult and find faster convergence of the basis set optimization with lower run time. We also demonstrate the additional customizability Quiqbox brings for more systematic basis set research with an example of constructing and optimizing delocalized orbitals.

### The photodetector, the heterodyne instrument, and the principle of instrument autonomy

Christopher S. Jackson

Measuring instruments, especially ones that observe continually over time, have a reality to them that is independent of the states that stimulate their senses. This is the Principle of Instrument Autonomy. Although the mathematical concept of an instrument implicitly embodies this principle, the conventional analysis of continual observation has become overwhelmingly focused on state evolution rather than on descriptions of instruments themselves. Because of this, it can be hard to appreciate that an instrument that observes for a finite amount of time has an evolution of its own, a stochastic evolution that precedes the Born rule and Schrödinger equation of the measured system. In this article, the two most established of the continually observing instruments, the Srinivas-Davies photodetector and the Goetsch-Graham-Wiseman heterodyne detector, are reviewed with an emphasis on the autonomous instrument evolution they define, made explicit by application of the recently introduced Kraus-operator distribution function. It is then pointed out how the heterodyne instrument evolution is a complete alternative to the original idea of energy quantization, where the usual ideas of *temperature* and *energy* of a *stat*e are replaced by the *time* and *positivity* of the *instrument*.

### Self-Healing of Trotter Error in Digital Adiabatic State Preparation

Lucas K. Kovalsky, Fernando A. Calderon-Vargas, Matthew D. Grace, Alicia B. Magann, James B. Larsen, Andrew D. Baczewski, Mohan Sarovar

Adiabatic time evolution can be used to prepare a complicated quantum many-body state from one that is easier to synthesize and Trotterization can be used to implement such an evolution digitally. The complex interplay between non-adiabaticity and digitization influences the infidelity of this process. We prove that the first-order Trotterization of a complete adiabatic evolution has a cumulative infidelity that scales as O(T^{-2} (dt)^{2}) instead of O(T(dt)) expected from general Trotter error bounds, where dt is the time step and T is the total time. This result explains why, despite increasing T, infidelities for fixed-dt digitized evolutions still decrease for a wide variety of Hamiltonians. It also establishes a correspondence between the Quantum Approximate Optimization Algorithm (QAOA) and digitized quantum annealing.

arXiv:2209.06242 | Phys. Rev. Lett. 131, 060602 (2023)

### Machine-learning Kohn-Sham potential from dynamics in time-dependent Kohn-Sham systems

Jun Yang, James D Whitfield

A machine learning method to develop the energy functional and the Kohn-Sham potential of a time-dependent Kohn-Sham(TDKS) system is proposed. The method is based on the dynamics of the Kohn-Sham system, so no data of the exact Kohn-Sham potential is required for training the model. We demonstrate the results of our method with a 1d harmonic oscillator example and a 1d two-electron example. We show that the machine-learned Kohn-Sham potential matches the exact Kohn-Sham potential in the absence of memory effect. In the case where there is a memory effect, our method can still capture the dynamics of the Kohn-Sham system. The machine learning method developed in this article provides an insight into making a better approximation to the energy functional and the Kohn-Sham potential in the TDKS system.

### The battle of clean and dirty qubits in the era of partial error correction

Daniel Bultrini, Samson Wang, Piotr Czarnik, Max Hunter Gordon, M. Cerezo, Patrick J. Coles, Lukasz Cincio

When error correction becomes possible it will be necessary to dedicate a large number of physical qubits to each logical qubit. Error correction allows for deeper circuits to be run, but each additional physical qubit can potentially contribute an exponential increase in computational space, so there is a trade-off between using qubits for error correction or using them as noisy qubits. In this work we look at the effects of using noisy qubits in conjunction with noiseless qubits (an idealized model for error-corrected qubits), which we call the “clean and dirty” setup. We employ analytical models and numerical simulations to characterize this setup. Numerically we show the appearance of Noise-Induced Barren Plateaus (NIBPs), i.e., an exponential concentration of observables caused by noise, in an Ising model Hamiltonian variational ansatz circuit. We observe this even if only a single qubit is noisy and given a deep enough circuit, suggesting that NIBPs cannot be fully overcome simply by error-correcting a subset of the qubits. On the positive side, we find that for every noiseless qubit in the circuit, there is an exponential suppression in concentration of gradient observables, showing the benefit of partial error correction. Finally, our analytical models corroborate these findings by showing that observables concentrate with a scaling in the exponent related to the ratio of dirty-to-total qubits.

### Dynamical simulation via quantum machine learning with provable generalization

Joe Gibbs, Zoë Holmes, Matthias C. Caro, Nicholas Ezzell, Hsin-Yuan Huang, Lukasz Cincio, Andrew T. Sornborger, Patrick J. Coles

Much attention has been paid to dynamical simulation and quantum machine learning (QML) independently as applications for quantum advantage, while the possibility of using QML to enhance dynamical simulations has not been thoroughly investigated. Here we develop a framework for using QML methods to simulate quantum dynamics on near-term quantum hardware. We use generalization bounds, which bound the error a machine learning model makes on unseen data, to rigorously analyze the training data requirements of an algorithm within this framework. This provides a guarantee that our algorithm is resource-efficient, both in terms of qubit and data requirements. Our numerics exhibit efficient scaling with problem size, and we simulate 20 times longer than Trotterization on IBMQ-Bogota.

### Establishing trust in quantum computations

Timothy Proctor, Stefan Seritan, Erik Nielsen, Kenneth Rudinger, Kevin Young, Robin Blume-Kohout, Mohan Sarovar

Real-world quantum computers have grown sufficiently complex that they can no longer be simulated by classical supercomputers, but their computational power remains limited by errors. These errors corrupt the results of quantum algorithms, and it is no longer always feasible to use classical simulations to directly check the correctness of quantum computations. Without practical methods for quantifying the accuracy with which a quantum algorithm has been executed, it is difficult to establish trust in the results of a quantum computation. Here we solve this problem, by introducing a simple and efficient technique for measuring the fidelity with which an as-built quantum computer can execute an algorithm. Our technique converts the algorithm’s quantum circuits into a set of closely related circuits whose success rates can be efficiently measured. It enables measuring the fidelity of quantum algorithm executions both in the near-term, with algorithms run on hundreds or thousands of physical qubits, and into the future, with algorithms run on logical qubits protected by quantum error correction.

### Improving the efficiency of learning-based error mitigation

Piotr Czarnik, Michael McKerns, Andrew T. Sornborger, Lukasz Cincio

Error mitigation will play an important role in practical applications of near-term noisy quantum computers. Current error mitigation methods typically concentrate on correction quality at the expense of frugality (as measured by the number of additional calls to quantum hardware). To fill the need for highly accurate, yet inexpensive techniques, we introduce an error mitigation scheme that builds on Clifford data regression (CDR). The scheme improves the frugality by carefully choosing the training data and exploiting the symmetries of the problem. We test our approach by correcting long range correlators of the ground state of XY Hamiltonian on IBM Toronto quantum computer. We find that our method is an order of magnitude cheaper while maintaining the same accuracy as the original CDR approach. The efficiency gain enables us to obtain a factor of 10 improvement on the unmitigated results with the total budget as small as 2⋅10^{5} shots.

### Surrogate-based optimization for variational quantum algorithms

Ryan Shaffer, Lucas Kocia, Mohan Sarovar

Variational quantum algorithms are a class of techniques intended to be used on near-term quantum computers. The goal of these algorithms is to perform large quantum computations by breaking the problem down into a large number of shallow quantum circuits, complemented by classical optimization and feedback between each circuit execution. One path for improving the performance of these algorithms is to enhance the classical optimization technique. Given the relative ease and abundance of classical computing resources, there is ample opportunity to do so. In this work, we introduce the idea of learning surrogate models for variational circuits using few experimental measurements, and then performing parameter optimization using these models as opposed to the original data. We demonstrate this idea using a surrogate model based on kernel approximations, through which we reconstruct local patches of variational cost functions using batches of noisy quantum circuit results. Through application to the quantum approximate optimization algorithm and preparation of ground states for molecules, we demonstrate the superiority of surrogate-based optimization over commonly-used optimization techniques for variational algorithms.

### Covariance matrix preparation for quantum principal component analysis

Max Hunter Gordon, M. Cerezo, Lukasz Cincio, Patrick J. Coles

Principal component analysis (PCA) is a dimensionality reduction method in data analysis that involves diagonalizing the covariance matrix of the dataset. Recently, quantum algorithms have been formulated for PCA based on diagonalizing a density matrix. These algorithms assume that the covariance matrix can be encoded in a density matrix, but a concrete protocol for this encoding has been lacking. Our work aims to address this gap. Assuming amplitude encoding of the data, with the data given by the ensemble {pi,|ψi⟩}, then one can easily prepare the ensemble average density matrix ρ=∑ipi|ψi⟩⟨ψi|. We first show that ρ is precisely the covariance matrix whenever the dataset is centered. For quantum datasets, we exploit global phase symmetry to argue that there always exists a centered dataset consistent with ρ, and hence ρ can always be interpreted as a covariance matrix. This provides a simple means for preparing the covariance matrix for arbitrary quantum datasets or centered classical datasets. For uncentered classical datasets, our method is so-called “PCA without centering”, which we interpret as PCA on a symmetrized dataset. We argue that this closely corresponds to standard PCA, and we derive equations and inequalities that bound the deviation of the spectrum obtained with our method from that of standard PCA. We numerically illustrate our method for the MNIST handwritten digit dataset. We also argue that PCA on quantum datasets is natural and meaningful, and we numerically implement our method for molecular ground-state datasets.

arXiv:2204.03495 | PRX Quantum 3, 030334 (2022)

### Quantum computing 2022

James D. Whitfield, Jun Yan, Weishi Wang, Joshuah T. Heath, Brent Harrison

Quantum technology is full of figurative and literal noise obscuring its promise. In this overview, we will attempt to provide a sober assessment of the promise of quantum technology with a focus on computing. We provide a tour of quantum computing and quantum technology that is aimed to be comprehensible to scientists and engineers without becoming a popular account. The goal is not a comprehensive review nor a superficial introduction but rather to serve as a useful map to navigate the hype, the scientific literature, and upcoming press releases about quantum technology and quantum computing. We have aimed to cite the most recent topical reviews, key results, and guide the reader away from fallacies and towards active discussions in the current quantum computing literature. The goal of this article was to be pedantic and introductory without compromising on the science.

### Quantum simulation of weak-field light-matter interactions

Steve M. Young, Mohan Sarovar

Simulation of the interaction of light with matter, including at the few-photon level, is important for understanding the optical and optoelectronic properties of materials, and for modeling next-generation non-linear spectroscopies that use entangled light. At the few-photon level the quantum properties of the electromagnetic field must be accounted for with a quantized treatment of the field, and then such simulations quickly become intractable, especially if the matter subsystem must be modeled with a large number of degrees of freedom, as can be required to accurately capture many-body effects and quantum noise sources. Motivated by this we develop a quantum simulation framework for simulating such light-matter interactions on platforms with controllable bosonic degrees of freedom, such as vibrational modes in the trapped ion platform. The key innovation in our work is a scheme for simulating interactions with a continuum field using only a few discrete bosonic modes, which is enabled by a Green’s function (response function) formalism. We develop the simulation approach, sketch how the simulation can be performed using trapped ions, and then illustrate the method with numerical examples. Our work expands the reach of quantum simulation to important light-matter interaction models and illustrates the advantages of extracting dynamical quantities such as response functions from quantum simulations.

arXiv:2112.07177 | Phys. Rev. Research 5, 013027 (2023)

### Can Error Mitigation Improve Trainability of Noisy Variational Quantum Algorithms?

Samson Wang, Piotr Czarnik, Andrew Arrasmith, M. Cerezo, Lukasz Cincio, Patrick J. Coles

Variational Quantum Algorithms (VQAs) are widely viewed as the best hope for near-term quantum advantage. However, recent studies have shown that noise can severely limit the trainability of VQAs, e.g., by exponentially flattening the cost landscape and suppressing the magnitudes of cost gradients. Error Mitigation (EM) shows promise in reducing the impact of noise on near-term devices. Thus, it is natural to ask whether EM can improve the trainability of VQAs. In this work, we first show that, for a broad class of EM strategies, exponential cost concentration cannot be resolved without committing exponential resources elsewhere. This class of strategies includes as special cases Zero Noise Extrapolation, Virtual Distillation, Probabilistic Error Cancellation, and Clifford Data Regression. Second, we perform analytical and numerical analysis of these EM protocols, and we find that some of them (e.g., Virtual Distillation) can make it harder to resolve cost function values compared to running no EM at all. As a positive result, we do find numerical evidence that Clifford Data Regression (CDR) can aid the training process in certain settings where cost concentration is not too severe. Our results show that care should be taken in applying EM protocols as they can either worsen or not improve trainability. On the other hand, our positive results for CDR highlight the possibility of engineering error mitigation methods to improve trainability.

### Lyapunov control-inspired 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 control-inspired 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 3-regular 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 near-term applications, and explore connections to quantum annealing.

arXiv:2108.05945 | Phys. Rev. A 106, 062414 (2022)

### Unifying and benchmarking state-of-the-art 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 state-of-the-art error mitigation methods share a common feature: they are data-driven, employing classical data obtained from runs of different quantum circuits. For example, Zero-noise extrapolation (ZNE) uses variable noise data and Clifford-data regression (CDR) uses data from near-Clifford 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 data-driven 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 state-of-the-art 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 Clifford-based 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.

### How to perform the coherent measurement of a curved phase space by continuous isotropic measurement. I. Spin and the Kraus-operator geometry of SL(2,C)

Christopher S. Jackson, Carlton M. Caves

Recently it was reported that the spin-coherent state (SCS) positive-operator-valued 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 Q-function, which is defined on the 2-sphere phase space of SCSs. This article develops the theoretical details of the continuous isotropic measurement and places it within the general context of applying curved-phase-space 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 3-hyperboloid SU(2)∖SL(2,C). Three equivalent stochastic techniques, path integral, diffusion (Fokker-Planck) 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 Maurer-Cartan 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).

### 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 near-term 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, non-oscillatory 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.

### Feedback-based 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 feedback-based 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 measurement-based 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 feedback-based protocol for the graph-partitioning problem MaxCut.

arXiv:2103.08619 | Phys. Rev. Lett. 129, 250502 (2022)

### Intractability of Electronic Structure in a Fixed Basis

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. While the theory of QMA-completeness has been instrumental in understanding the complexity of finding ground states in many-body quantum systems, prior to this work it has been unknown whether or not the special form of the Hamiltonian for the electronic structure of molecules can be exploited to find ground states efficiently or whether the problem remains hard for this special case. We prove that the electronic-structure problem, when restricted to a fixed single-particle basis and a fixed number of electrons, is QMA-complete. In our proof, the local Hamiltonian is encoded in the choice of spatial orbitals used to discretize the electronic-structure Hamiltonian. In contrast, Schuch and Verstraete have proved hardness for the electronic-structure problem with an additional site-specific external *magnetic* field, but without the restriction to a fixed basis, by encoding a local Hamiltonian on qubits in the site-specific magnetic field. We also show that estimation of the energy of the lowest-energy Slater-determinant state (i.e., the Hartree-Fock state) is nondeterministic polynomial time (NP)-complete for the electronic-structure Hamiltonian in a fixed basis.

arXiv:2103.08215 | PRX Quantum 3, 020322 (2022)

### 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 many-body 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 | Quantum Sci. Technol. 7, 025017 (2022)

### Qubit-efficient exponential suppression of errors

Piotr Czarnik, Andrew Arrasmith, Lukasz Cincio, Patrick J. Coles

Achieving a practical advantage with near-term 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 Resource-Efficient 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 trade-off 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 near-Clifford 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 qubit-efficient version with a realistic trapped-ion noise model. We find that REQUEST can reproduce the exponential suppression of errors of the virtual distillation approach for simple noise models while out-performing virtual distillation when fewer than 3N+1 qubits are available.

### Variational Quantum Algorithms

M. Cerezo, A. Arrasmith, R. Babbush, S. C. Benjamin, S. Endo, K. Fujii, J. R. McClean, K. Mitarai, X. Yuan, L. Cincio, P. J. Coles

Applications such as simulating complicated quantum systems or solving large-scale linear algebra problems are very challenging for classical computers due to the extremely high computational cost. Quantum computers promise a solution, although fault-tolerant quantum computers will likely not be available in the near future. Current quantum devices have serious constraints, including limited numbers of qubits and noise processes that limit circuit depth. Variational Quantum Algorithms (VQAs), which use a classical optimizer to train a parametrized quantum circuit, have emerged as a leading strategy to address these constraints. VQAs have now been proposed for essentially all applications that researchers have envisioned for quantum computers, and they appear to the best hope for obtaining quantum advantage. Nevertheless, challenges remain including the trainability, accuracy, and efficiency of VQAs. Here we overview the field of VQAs, discuss strategies to overcome their challenges, and highlight the exciting prospects for using them to obtain quantum advantage.

arXiv:2012.09265 | Nature Reviews Physics **3**, 625-644 (2021)

### Improved Strong Simulation of Universal Quantum Circuits

Lucas Kocia

We find a scaling reduction in the stabilizer rank of the twelve-qubit tensored T gate magic state. This lowers its asymptotic bound to 2^{∼0.463t} for multi-Pauli 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 t-counts, which suggests that the stabilizer rank found at the twelve-qubit state can be lowered further to 2^{∼0.449t} and we prove and numerically show that this is the case for single-Pauli 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 Clifford-isomorphic to a computational sub-basis tensored with single-qubit states that produce minimal unique stabilizer state inner products — the same relationship that allowed for finding minimal numbers of unique Gauss sums in the odd-dimensional qudit Wigner formulation of Pauli measurements.

### Verifying quantum circuit equivalences using Prove-It, 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 Prove-It (a Python-based, Sandia-developed 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, Prove-It will be able to solve the verification problem for most practical purposes with linear time complexity.

Computer Science Research Institute Summer Proceedings 2020

### Prove-It: 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 Prove-It, a Python-based general-purpose interactive theorem-proving assistant designed with the goal of making formal theorem proving as easy and natural as informal theorem proving (with moderate training). Prove-It uses a highly-flexible Jupyter notebook-based user interface that documents interactions and proof steps using LaTeX. We review Prove-It’s highly expressive representation of expressions, judgments, theorems, and proofs; demonstrate the system by constructing a traditional proof-by-contradiction 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.

### 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 noise-induced 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 near-term quantum advantage. Such applications require good optimizers to train PQCs. Recent works have focused on quantum-aware 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 non-unital 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 Symmetry-based 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.

### Unified approach to data-driven quantum error mitigation

Angus Lowe, Max Hunter Gordon, Piotr Czarnik, Andrew Arrasmith, Patrick J. Coles, Lukasz Cincio

Achieving near-term quantum advantage will require effective methods for mitigating hardware noise. Data-driven approaches to error mitigation are promising, with popular examples including zero-noise 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 variable-noise Clifford data regression (vnCDR), significantly outperforms these individual methods in numerical benchmarks. vnCDR generates training data first via near-Clifford 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 8-qubit 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 anti-commuting fermionic variables into the operators acting on the qubit Hilbert space. The most familiar of which, the Jordan-Wigner transformation, encodes fermionic operators into non-local qubit operators. As non-local 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 quasi-locally. 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 quasi-local 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.

### From pulses to circuits and back again: A quantum optimal control perspective on variational quantum algorithms

Alicia B. Magann, Christian Arenz, Matthew D. Grace, Tak-San 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 fault-tolerant devices likely remain years away, the noisy intermediate-scale 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 worst-case and typical instances if error correction is not used, and present some open problems.

arXiv:2007.15698 | PRX Quantum, 2, 010315 (2021)

### Noise-Induced 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 Intermediate-Scale 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 noise-induced barren plateaus (NIBPs) are conceptually different from noise-free 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 | Nature Communications 12, 6961 (2021)

### Quantum proportional-integral (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 proportional-integral 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 two-qubit 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 Hartree-Fock with quantum resources

Sahil Gulania, James Daniel Whitfield

The Hartree-Fock 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 Hartree-Fock problem is a natural target. While quantum computers and quantum simulation offer many prospects for the future of modern chemistry, the Hartree-Fock 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 low-lying 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 noise-resilient quantum circuits

Lukasz Cincio, Kenneth Rudinger, Mohan Sarovar, Patrick J. Coles

Noise mitigation and reduction will be crucial for obtaining useful answers from near-term 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 noise-aware circuit learning (NACL), applies to circuits designed to compute a unitary transformation, prepare a set of quantum states, or estimate an observable of a many-qubit 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 task-specific cost function over circuit depths and circuit structures. To demonstrate NACL, we construct circuits resilient to a fine-grained noise model derived from gate set tomography on a superconducting-circuit quantum device, for applications including quantum state overlap, quantum Fourier transform, and W-state preparation.

arXiv:2007.01210 | PRX Quantum, 2, 010324 (2021)

### Quantum simulation of the qubit-regularized O(3)-sigma model

Alexander J. Buser, Tanmoy Bhattacharya, Lukasz Cincio, Rajan Gupta

Recently, Singh and Chandrasekharan showed that fixed points of the classical non-linear 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 weak-coupling and quantum-critical regimes and use shadow tomography to measure the dynamics of local observables. We then present and analyze a quantum algorithm based on non-unitary randomized simulation methods that may yield an approach suitable for intermediate-term noisy quantum devices.

arXiv:2006.15746 | Phys. Rev. D, 102, 114514 (2020)

### Error mitigation with Clifford quantum-circuit data

Piotr Czarnik, Andrew Arrasmith, Patrick J. Coles, Lukasz Cincio

Achieving near-term quantum advantage will require accurate estimation of quantum observables despite significant hardware noise. For this purpose, we propose a novel, scalable error-mitigation method that applies to gate-based 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 noise-free observables for arbitrary circuits. We analyze the performance of our method versus the number of qubits, circuit depth, and number of non-Clifford gates. We obtain an order-of-magnitude error reduction for a ground-state energy problem on a 16-qubit IBMQ quantum computer and on a 64-qubit noisy simulator.

arXiv:2005.10189 | Quantum 5, 592 (2021)

### 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 near-term 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 non-degenerate Hamiltonian. Due to Schur-concavity, 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 near-term implementation. We demonstrate two applications of VQSE: (1) Principal component analysis, and (2) Error mitigation.

### 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 non-trivial reductions on its upper bound for t=1, t=2, t=3, and t=6 with odd-prime-qudits. 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 non-trivial reductions by >10^{10^4} and extend the bounds to t=14 for qutrits. This is accomplished by using the Wigner-Weyl-Moyal formalism to algebraically find bounds instead of relying on numerics. We find a new reduction in the upper bound from the 12-qutrit magic state of 3^{∼0.469t,} which improves on the bound obtained from the 6-qutrit 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

Optimally-shaped 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 quantum-computing 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 optimally-shaped fields to control molecular systems. We introduce a hybrid algorithm that utilizes a quantum computer for simulating the field-induced 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 optimally-shaped fields could be used to elucidate the mechanisms of energy transfer in light-harvesting complexes; resource estimates are provided for the latter task.

arXiv:2002.12497 | Phys. Rev. Research, 3, 023165 (2021)

### Cost-Function-Dependent 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 large-scale 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 early-stage applications for quantum computers. Current state-of-the-art 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)

### An Adaptive Optimizer for Measurement-Frugal Variational Algorithms

Jonas M. Kübler, Andrew Arrasmith, Lukasz Cincio, Patrick J. Coles

Variational hybrid quantum-classical algorithms (VHQCAs) have the potential to be useful in the era of near-term quantum computing. However, recently there has been concern regarding the number of measurements needed for convergence of VHQCAs. Here, we address this concern by investigating the classical optimizer in VHQCAs. We introduce a novel optimizer called individual Coupled Adaptive Number of Shots (iCANS). This adaptive optimizer frugally selects the number of measurements (i.e., number of shots) both for a given iteration and for a given partial derivative in a stochastic gradient descent. We numerically simulate the performance of iCANS for the variational quantum eigensolver and for variational quantum compiling, with and without noise. In all cases, and especially in the noisy case, iCANS tends to out-perform state-of-the-art optimizers for VHQCAs. We therefore believe this adaptive optimizer will be useful for realistic VHQCA implementations, where the number of measurements is limited.

arXiv:1909.09083 | Quantum **4** 263 (2020)

### Variational Quantum Linear Solver

Carlos Bravo-Prieto, Ryan LaRose, M. Cerezo, Yigit Subasi, Lukasz Cincio, Patrick J. Coles

Solving linear systems of equations is central to many engineering and scientific fields. Several quantum algorithms have been proposed for linear systems, where the goal is to prepare |x⟩ such that A|x⟩∝|b⟩. While these algorithms are promising, the time horizon for their implementation is long due to the required quantum circuit depth. In this work, we propose a variational hybrid quantum-classical algorithm for solving linear systems, with the aim of reducing the circuit depth and doing much of the computation classically. We propose a cost function based on the overlap between |b⟩ and A|x⟩, and we derive an operational meaning for this cost in terms of the solution precision ϵ. We also introduce a quantum circuit to estimate this cost, while showing that this cost cannot be efficiently estimated classically. Using Rigetti’s quantum computer, we successfully implement our algorithm up to a problem size of 32×32. Furthermore, we numerically find that the complexity of our algorithm scales efficiently in both 1/ϵ and κ, with κ the condition number of A. Our algorithm provides a heuristic for quantum linear systems that could make this application more near term.

### A comparison of three ways to measure time-dependent densities with quantum simulators

Jun Yang, James Brown, James Daniel Whitfield

Quantum algorithms are touted as a way around some classically intractable problems such as the simulation of quantum mechanics. At the end of all quantum algorithms is a quantum measurement whereby classical data is extracted and utilized. In fact, many of the modern hybrid-classical approaches are essentially quantum measurements of states with short quantum circuit descriptions. Here, we compare and examine three methods of extracting the time-dependent one-particle probability density from a quantum simulation: direct Z-measurement, Bayesian phase estimation and harmonic inversion. We have tested these methods in the context of the potential inversion problem of time-dependent density functional theory. Our test results suggest that direct measurement is the preferable method. We also highlight areas where the other two methods may be useful and report on tests using Rigetti’s quantum virtual device. This study provides a starting point for imminent applications of quantum computing.

arXiv:1909.03078 | Frontiers in Physics, 9, 118 (2021)

### Engineered thermalization of quantum many-body systems

Mekena Metcalf, Jonathan E. Moussa, Wibe A. de Jong, Mohan Sarovar

We develop a scheme for engineering genuine thermal states in analog quantum simulation platforms by coupling local degrees of freedom to driven, dissipative ancilla pseudospins. We demonstrate the scheme in a many-body quantum spin lattice simulation setting. A Born-Markov master equation describing the dynamics of the many-body system is developed, and we show that if the ancilla energies are periodically modulated, with a carefully chosen hierarchy of timescales, one can effectively thermalize the many-body system. Through analysis of the time-dependent dynamical generator, we determine the conditions under which the true thermal state is an approximate dynamical fixed point for general system Hamiltonians. Finally, we evaluate the thermalization protocol through numerical simulation and discuss prospects for implementation on current quantum simulation hardware.

arXiv:1909.02023 | Physical Review Research, **2** 023214 (2020)

### Noise Resilience of Variational Quantum Compiling

Kunal Sharma, Sumeet Khatri, M. Cerezo, Patrick J. Coles

Variational hybrid quantum-classical algorithms (VHQCAs) are near-term algorithms that leverage classical optimization to minimize a cost function, which is efficiently evaluated on a quantum computer. Recently VHQCAs have been proposed for quantum compiling, where a target unitary U is compiled into a short-depth gate sequence V. In this work, we report on a surprising form of noise resilience for these algorithms. Namely, we find one often learns the correct gate sequence V (i.e., the correct variational parameters) despite various sources of incoherent noise acting during the cost-evaluation circuit. Our main results are rigorous theorems stating that the optimal variational parameters are unaffected by a broad class of noise models, such as measurement noise, gate noise, and Pauli channel noise. Furthermore, our numerical implementations on IBM’s noisy simulator demonstrate resilience when compiling the quantum Fourier transform, Toffoli gate, and W-state preparation. Hence, variational quantum compiling, due to its robustness, could be practically useful for noisy intermediate-scale quantum devices. Finally, we speculate that this noise resilience may be a general phenomenon that applies to other VHQCAs such as the variational quantum eigensolver.

arXiv:1908.04416 | New Journal of Physics **22** 043006 (2020)

### Young frames for quantum chemistry

Sahil Gulania, James Daniel Whitfield

Quantum chemistry often considers atoms and molecules with non-zero spin. In such cases, the need for proper spin functions results in the theory of configuration state functions. Here, we consider the construction of such wavefunctions using the symmetric group and more specifically Young projectors. We discuss the formalism and detail an example to illustrate the theory. Additionally, we consider the pros and cons of specific implementations of spin symmetry in quantum simulation.

### Solver for the electronic V-representation problem of time-dependent density functional theory

James Brown, Jun Yang, James D Whitfield

One route to numerically propagating quantum systems is time-dependent density functional theory (TDDFT). The application of TDDFT to a particular system’s time evolution is predicated on V-representability which we have analyzed in a previous publication. Here we describe a newly developed solver for the scalar time-dependent Kohn-Sham potential. We present and interpret the force-balance equation central to our numerical method, describe details of its implementation, and present illustrative numerical results for one- and two-electron systems. A new characterization of V-representability for one-electron systems is also included along with possible improvements and future directions.

arXiv:1904.10958 | J. Chem. Theory Comput. 16, 6014 (2020)

## Software

ProveIt: A tool for proving and organizing general theorems using Python.

pyQAOA: Python code for optimized simulation of quantum approximate optimization algorithm circuits.

pyRPE: Python implementation of the robust phase estimation algorithm, including consistency checks developed in arXiv:2011.13442.

StrongSim: C code for fast strong simulation of quantum circuits using Gauss sums, as developed in arXiv:2012.11739.

sbovqaopt: Python package providing a surrogate-based optimizer for variational quantum algorithms as introduced in arXiv:2204.05451.