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All sourcesNASA NTRSDOE OSTI
DOE OSTI2023

Variational quantum simulation of the critical Ising model with symmetry averaging

Here we investigate the use of deep multiscale entanglement renormalization ansatz (DMERA) circuits as a variational ansatz. We use the exactly solvable one-dimensional critical transverse-field Ising model as a test bed. Numerically exact simulation of the quantum circuit ansatz can in this case be carried out to hundreds of qubits by exploiting efficient classical algorithms for simulating matchgate circuits. We find that, for this system, the DMERA strongly outperforms a standard quantum approximate optimization algorithm (QAOA)–style ansatz, and that a major source of systematic error in correlation functions approximated using the DMERA is the breaking of the translational and Kramers-Wannier symmetries of the transverse-field Ising model. We are able to reduce this error by up to four orders of magnitude by symmetry averaging, without incurring additional cost in qubits or circuit depth. Here, we propose that this technique for mitigating systematic error could be applied to noisy intermediate-scale quantum (NISQ) simulations of physical systems with other symmetries.

1-dimensional spin chains↗
DOE OSTI2024

Quantum-classical tradeoffs and multi-controlled quantum gate decompositions in variational algorithms

The computational capabilities of near-term quantum computers are limited by the noisy execution of gate operations and a limited number of physical qubits. Hybrid variational algorithms are well-suited to near-term quantum devices because they allow for a wide range of tradeoffs between the amount of quantum and classical resources used to solve a problem. This paper investigates tradeoffs available at both the algorithmic and hardware levels by studying a specific case – applying the Quantum Approximate Optimization Algorithm (QAOA) to instances of the Maximum Independent Set (MIS) problem. We consider three variants of the QAOA which offer different tradeoffs at the algorithmic level in terms of their required number of classical parameters, quantum gates, and iterations of classical optimization needed. Since MIS is a constrained combinatorial optimization problem, the QAOA must respect the problem constraints. This can be accomplished by using many multi-controlled gate operations which must be decomposed into gates executable by the target hardware. We study the tradeoffs available at this hardware level, combining the gate fidelities and decomposition efficiencies of different native gate sets into a single metric called the gate decomposition cost .

Tomesh, Teague↗
DOE OSTI2023

Exploiting Kronecker structure in exponential integrators: Fast approximation of the action of φ $-$functions of matrices via quadrature

Here, in this article, we propose an algorithm for approximating the action of φ $-$ functions of matrices against vectors, which is a key operation in exponential time integrators. In particular, we consider matrices with Kronecker sum structure, which arise from problems admitting a tensor product representation. The method is based on quadrature approximations of the integral form of the φ $-$ functions combined with a scaling and modified squaring method. Owing to the Kronecker sum representation, only actions of 1D matrix exponentials are needed at each quadrature node and assembly of the full matrix can be avoided. Additionally, we derive a priori bounds for the quadrature error, which show that, as expected by classical theory, the rate of convergence of our method is supergeometric. Guided by our analysis, we construct a fast and robust method for estimating the optimal scaling factor and number of quadrature nodes that minimizes the total cost for a prescribed error tolerance. We investigate the performance of our algorithm by solving several linear and semilinear time-dependent problems in 2D and 3D. The results show that our method is accurate and orders of magnitude faster than the current state-of-the-art.

97 MATHEMATICS AND COMPUTING↗
DOE OSTI2021

Empirical performance bounds for quantum approximate optimization

The quantum approximate optimization algorithm (QAOA) has been put forth as a method for near-term quantum computers to solve optimization problems. However, assessments of QAOA performance have mostly focused on small structured problem instances while performance on more general instances is less clear. Here, we numerically simulate QAOA pure state dynamics for every instance of MaxCut on non-isomorphic unweighted graphs with nine or fewer vertices with depth parameters p ≤ 3. We find the approximation ratios and optimized circuit parameters concentrate across graphs of a given size and empirically show increases in concentration as graph size increases. The parameter concentration leads to two median-angle heuristics that overcome difficulties in QAOA parameter optimization and obtain mean approximation ratios within 3% and 0.2% of the optimal. We also analyze the probability to measure an optimal solution and find increasing variations between graphs as depth increases, in stark contrast to the approximation ratios which concentrate as depth increases. Furthermore, the resulting benchmark data set gives empirical bounds for on-going experimental realizations and lays groundwork for theoretical extensions to greater problem sizes and depths where QAOA may prove important for practically relevant problems.

79 ASTRONOMY AND ASTROPHYSICS↗
DOE OSTI2022

Constrained quantum optimization for extractive summarization on a trapped-ion quantum computer

Abstract Realizing the potential of near-term quantum computers to solve industry-relevant constrained-optimization problems is a promising path to quantum advantage. In this work, we consider the extractive summarization constrained-optimization problem and demonstrate the largest-to-date execution of a quantum optimization algorithm that natively preserves constraints on quantum hardware. We report results with the Quantum Alternating Operator Ansatz algorithm with a Hamming-weight-preserving XY mixer (XY-QAOA) on trapped-ion quantum computer. We successfully execute XY-QAOA circuits that restrict the quantum evolution to the in-constraint subspace, using up to 20 qubits and a two-qubit gate depth of up to 159. We demonstrate the necessity of directly encoding the constraints into the quantum circuit by showing the trade-off between the in-constraint probability and the quality of the solution that is implicit if unconstrained quantum optimization methods are used. We show that this trade-off makes choosing good parameters difficult in general. We compare XY-QAOA to the Layer Variational Quantum Eigensolver algorithm, which has a highly expressive constant-depth circuit, and the Quantum Approximate Optimization Algorithm. We discuss the respective trade-offs of the algorithms and implications for their execution on near-term quantum hardware.

97 MATHEMATICS AND COMPUTING↗
DOE OSTI2022

Numerical gate synthesis for quantum heuristics on bosonic quantum processors

There is a recent surge of interest and insights regarding the interplay of quantum optimal control and variational quantum algorithms. We study the framework in the context of qudits which are, for instance, definable as controllable electromagnetic modes of a superconducting cavity system coupled to a transmon. By employing recent quantum optimal control approaches described in (Petersson and Garcia, 2021), we showcase control of single-qudit operations up to eight states, and two-qutrit operations, mapped respectively onto a single mode and two modes of the resonator. We discuss the results of numerical pulse engineering on the closed system for parametrized gates useful to implement Quantum Approximate Optimization Algorithm (QAOA) for qudits. The results show that high fidelity ( > 0.99) is achievable with sufficient computational effort for most cases under study, and extensions to multiple modes and open, noisy systems are possible. The tailored pulses can be stored and used as calibrated primitives for a future compiler in circuit quantum electrodynamics (cQED) systems.

71 CLASSICAL AND QUANTUM MECHANICS, GENERAL PHYSIC↗
DOE OSTI2021

Globally Optimizing QAOA Circuit Depth for Constrained Optimization Problems

We develop a global variable substitution method that reduces n-variable monomials in combinatorial optimization problems to equivalent instances with monomials in fewer variables. We apply this technique to 3-SAT and analyze the optimal quantum unitary circuit depth needed to solve the reduced problem using the quantum approximate optimization algorithm. For benchmark 3-SAT problems, we find that the upper bound of the unitary circuit depth is smaller when the problem is formulated as a product and uses the substitution method to decompose gates than when the problem is written in the linear formulation, which requires no decomposition.

3-SAT↗
DOE OSTI2024

Multistart algorithm for identifying all optima of nonconvex stochastic functions

Here, we propose a multistart algorithm to identify all local minima of a constrained, nonconvex stochastic optimization problem. The algorithm uniformly samples points in the domain and then starts a local stochastic optimization run from any point that is the "probabilistically best" point in its neighborhood. Under certain conditions, our algorithm is shown to asymptotically identify all local optima with high probability; this holds even though our algorithm is shown to almost surely start only finitely many local stochastic optimization runs. We demonstrate the performance of an implementation of our algorithm on nonconvex stochastic optimization problems, including identifying optimal variational parameters for the quantum approximate optimization algorithm.

97 MATHEMATICS AND COMPUTING↗
DOE OSTI2024

Grover-QAOA for 3-SAT: quadratic speedup, fair-sampling, and parameter clustering

Abstract The SAT problem is a prototypical NP-complete problem of fundamental importance in computational complexity theory with many applications in science and engineering; as such, it has long served as an essential benchmark for classical and quantum algorithms. This study shows numerical evidence for a quadratic speedup of the Grover Quantum Approximate Optimization Algorithm (G-QAOA) over random sampling for finding all solutions to 3-SAT (All-SAT) and Max-SAT problems. G-QAOA is less resource-intensive and more adaptable for these problems than Grover’s algorithm, and it surpasses conventional QAOA in its ability to sample all solutions. We show these benefits by classical simulations of many-round G-QAOA on thousands of random 3-SAT instances. We also observe G-QAOA advantages on the IonQ Aria quantum computer for small instances, finding that current hardware suffices to determine and sample all solutions. Interestingly, a single-angle-pair constraint that uses the same pair of angles at each G-QAOA round greatly reduces the classical computational overhead of optimizing the G-QAOA angles while preserving its quadratic speedup. We also find parameter clustering of the angles. The single-angle-pair protocol and parameter clustering significantly reduce obstacles to classical optimization of the G-QAOA angles.

Zhang, Zewen (ORCID:000000032258613X)↗
DOE OSTI2020

Quantum approximate optimization of the long-range Ising model with a trapped-ion quantum simulator

Quantum computers and simulators may offer significant advantages over their classical counterparts, providing insights into quantum many-body systems and possibly improving performance for solving exponentially hard problems, such as optimization and satisfiability. Here, we report the implementation of a low-depth Quantum Approximate Optimization Algorithm (QAOA) using an analog quantum simulator. We estimate the ground-state energy of the Transverse Field Ising Model with long-range interactions with tunable range, and we optimize the corresponding combinatorial classical problem by sampling the QAOA output with high-fidelity, single-shot, individual qubit measurements. We execute the algorithm with both an exhaustive search and closed-loop optimization of the variational parameters, approximating the ground-state energy with up to 40 trapped-ion qubits. We benchmark the experiment with bootstrapping heuristic methods scaling polynomially with the system size. We observe, in agreement with numerics, that the QAOA performance does not degrade significantly as we scale up the system size and that the runtime is approximately independent from the number of qubits. We finally give a comprehensive analysis of the errors occurring in our system, a crucial step in the path forward toward the application of the QAOA to more general problem instances.

71 CLASSICAL AND QUANTUM MECHANICS, GENERAL PHYSIC↗
DOE OSTI2024

Quantum Complexity of the Kronecker Coefficients

Whether or not the Kronecker coefficients of the symmetric group count some set of combinatorial objects is a longstanding open question. In this work we show that a given Kronecker coefficient is proportional to the rank of a projector that can be measured efficiently using a quantum computer. In other words a Kronecker coefficient counts the dimension of the vector space spanned by the accepting witnesses of a QMA verifier, where QMA is the quantum analogue of NP . This implies that approximating the Kronecker coefficients to within a given relative error is not harder than a certain natural class of problems that captures the complexity of estimating thermal properties of quantum many-body systems. A second consequence is that deciding positivity of Kronecker coefficients is contained in QMA , complementing a recent NP -hardness result of Ikenmeyer, Mulmuley, and Walter. We obtain similar results for the related problem of approximating row sums of the character table of the symmetric group. Finally, we discuss an efficient quantum algorithm that approximates normalized Kronecker coefficients to inverse-polynomial additive error. Published by the American Physical Society 2024

Physics↗
DOE OSTI2021

Revisiting the ODE Method for Recursive Algorithms: Fast Convergence Using Quasi Stochastic Approximation

Several decades ago, Profs. Sean Meyn and Lei Guo were postdoctoral fellows at ANU, where they shared interest in recursive algorithms. It seems fitting to celebrate Lei Guo's 60th birthday with a review of the ODE Method and its recent evolution. The method has been regarded as a technique for algorithm analysis. It is argued that this viewpoint is backwards: The original stochastic approximation method was surely motivated by an ODE, and tools for analysis came much later (based on establishing robustness of Euler approximations). The paper presents a brief survey of recent research in machine learning that shows the power of algorithm design in continuous time, following by careful approximation to obtain a practical recursive algorithm. While these methods are usually presented in a stochastic setting, this is not a prerequisite. In fact, recent theory shows that rates of convergence can be dramatically accelerated by applying techniques inspired by quasi Monte-Carlo. Subject to conditions, the optimal rate of convergence can be obtained by applying the averaging technique of Polyak and Ruppert. The conditions are not universal, but theory suggests alternatives to achieve acceleration. The theory is illustrated with applications to gradient-free optimization, and policy gradient algorithms for reinforcement learning.

learning and adaptive systems in artificial intell↗
DOE OSTI2023

Quantum Gauge Networks: A New Kind of Tensor Network

Although tensor networks are powerful tools for simulating low-dimensional quantum physics, tensor network algorithms are very computationally costly in higher spatial dimensions. We introduce quantum gauge networks: a different kind of tensor network ansatz for which the computation cost of simulations does not explicitly increase for larger spatial dimensions. We take inspiration from the gauge picture of quantum dynamics, which consists of a local wavefunction for each patch of space, with neighboring patches related by unitary connections. A quantum gauge network (QGN) has a similar structure, except the Hilbert space dimensions of the local wavefunctions and connections are truncated. We describe how a QGN can be obtained from a generic wavefunction or matrix product state (MPS). All 2k-point correlation functions of any wavefunction for M many operators can be encoded exactly by a QGN with bond dimension O(M k ). In comparison, for just k = 1, an exponentially larger bond dimension of 2 M/6 is generically required for an MPS of qubits. We provide a simple QGN algorithm for approximate simulations of quantum dynamics in any spatial dimension. The approximate dynamics can achieve exact energy conservation for time-independent Hamiltonians, and spatial symmetries can also be maintained exactly. We benchmark the algorithm by simulating the quantum quench of fermionic Hamiltonians in up to three spatial dimensions.

71 CLASSICAL AND QUANTUM MECHANICS, GENERAL PHYSIC↗
DOE OSTI2021

Optimizing FPGA-based Accelerator Design for Large-Scale Molecular Similarity Search (Special Session Paper)

Molecular similarity search has been widely used in drug discovery to rapidly identify structurally similar compounds from large molecular databases. With the increasing size of chemical libraries, there is growing interest in the efficient ac- celeration of large-scale similarity search. Existing works mainly focus on CPU and GPU to accelerate the computation of Tatimoto coefficient in measuring the pairwise similarity between different molecular fingerprints. In this paper, we propose and optimize an FPGA-based accelerator design on exhaustive and approximate search algorithms. On exhaustive search using BitBound & fold- ing, we analyze the similarity cutoff and folding level relationship with search speedup and accuracy, and propose a scalable on- the-fly query engine on FPGAs to reduce the resource utilization and pipeline interval. We achieve a 450 million compounds-per- second processing throughput for a single query engine. On approximate search using hierarchical navigable small world (HNSW), a popular algorithm with high recall and query speed, we propose an FPGA-based graph traversal engine to utilize high throughput register array based priority queue and fine- grained distance calculation engine to increase the processing capability. Experimental results show that the proposed FPGA- based HNSW implementation achieves a 35× speedup than existing works on CPU. To the best of our knowledge, our FPGA- based implementation is the first attempt to accelerate molecular similarity search on FPGA and has the highest performance among existing approaches.

Peng, Hongwu↗
DOE OSTI2024

A Novel Noise-Aware Classical Optimizer for Variational Quantum Algorithms

A key component of variational quantum algorithms (VQAs) is the choice of classical optimizer employed to update the parameterization of an ansatz. It is well recognized that quantum algorithms will, for the foreseeable future, necessarily be run on noisy devices with limited fidelities. Thus, the evaluation of an objective function (e.g., the guiding function in the quantum approximate optimization algorithm (QAOA) or the expectation of the electronic Hamiltonian in variational quantum eigensolver (VQE)) required by a classical optimizer is subject not only to stochastic error from estimating an expected value but also to error resulting from intermittent hardware noise. Model-based derivative-free optimization methods have emerged as popular choices of a classical optimizer in the noisy VQA setting, based on empirical studies. However, these optimization methods were not explicitly designed with the consideration of noise. In this work we adapt recent developments from the “noise-aware numerical optimization” literature to these commonly used derivative-free model-based methods. We introduce the key defining characteristics of these novel noise-aware derivative-free model-based methods that separate them from standard model-based methods. In conclusion, we study an implementation of such noise-aware derivative-free model-based methods and compare its performance on demonstrative VQA simulations to classical solvers packaged in scikit-quant.

classical optimizers↗
DOE OSTI2021

Adaptive variational quantum eigensolvers for highly excited states

Highly excited states of quantum many-body systems are central objects in the study of quantum dynamics and thermalization that challenge classical computational methods due to their volume-law entanglement content. In this work, we explore the potential of variational quantum algorithms to approximate such states. We propose an adaptive variational quantum eigensolver (VQE) for excited states (X) that self-generates a variational ansatz for arbitrary eigenstates of a many-body Hamiltonian H by attempting to minimize the energy variance with respect to H. We benchmark the method by applying it to an Ising spin chain with integrable and nonintegrable regimes, where we calculate various quantities of interest, including the total energy, magnetization density, and entanglement entropy. We also compare the performance of adaptive VQE-X to an adaptive variant of the folded-spectrum method. For both methods, we find a strong dependence of the algorithm's performance on the choice of operator pool used for the adaptive construction of the ansatz. In particular, an operator pool including long-range two-body gates accelerates the convergence of both algorithms in the nonintegrable regime. Here, we also study the scaling of the number of variational parameters with system size, finding that an exponentially large number of parameters may be necessary to approximate individual highly excited states. Nevertheless, we argue that these methods lay a foundation for the use of quantum algorithms to study finite-energy-density properties of many-body systems.

75 CONDENSED MATTER PHYSICS, SUPERCONDUCTIVITY AND↗
DOE OSTI2020

On the Solution of ℓ 0 -Constrained Sparse Inverse Covariance Estimation Problems

The sparse inverse covariance matrix is used to model conditional dependencies between variables in a graphical model to fit a multivariate Gaussian distribution. Estimating the matrix from data are well known to be computationally expensive for large-scale problems. Sparsity is employed to handle noise in the data and to promote interpretability of a learning model. Although the use of a convex ℓ 1 regularizer to encourage sparsity is common practice, the combinatorial ℓ 0 penalty often has more favorable statistical properties. In this paper, we directly constrain sparsity by specifying a maximally allowable number of nonzeros, in other words, by imposing an ℓ 0 constraint. Here, we introduce an efficient approximate Newton algorithm using warm starts for solving the nonconvex ℓ 0 -constrained inverse covariance learning problem. Numerical experiments on standard data sets show that the performance of the proposed algorithm is competitive with state-of-the-art methods.

$\ell_0$-Constrained↗