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All sourcesNASA NTRSDOE OSTI
DOE OSTI2022

Quantum Local Search with the Quantum Alternating Operator Ansatz

We present a new hybrid, local search algorithm for quantum approximate optimization of constrained combinatorial optimization problems. We focus on the Maximum Independent Set problem and demonstrate the ability of quantum local search to solve large problem instances on quantum devices with few qubits. This hybrid algorithm iteratively finds independent sets over carefully constructed neighborhoods and combines these solutions to obtain a global solution. We study the performance of this algorithm on 3-regular, Community, and Erdős-Rényi graphs with up to 100 nodes.

Tomesh, Teague↗
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 OSTI2019

O(N) Quantum Molecular Dynamics (QMD) Algorithms, Code Requirements, Release Notes, and Library Deployment

The Exascale Computing Project (ECP) Co-design Center for Particle Applications (CoPA) is addressing the challenges for particle applications to run on upcoming exascale computing architectures. This includes short- and long-range molecular dynamics (MD), particle-in-cell (PIC), and quantum molecular dynamics (QMD). This document addresses $\mathcal{O}(N)$ QMD. The goal of this CoPA team is to develop $\mathcal{O}(N)$ linear scaling methods for QMD optimized for new and next generation architectures. This will provide the capability to simulate larger systems and run for longer timeframes than is possible today. Focus is initially on the density matrix construction. This requires efficient matrix formats and linear algebra operations for distributed architectures. A significant reduction in the wallclock time for an MD timestep is necessary to achieve practical large scale QMD. Equally challenging is to develop scalable distributed implementations. This requires the exploration of efficient data formats (dense, sparse, etc.), data decomposition and distribution across compute nodes using block- and graph-based methods. Traditional programming models such as MPI/OpenMP, hybrid MPI/OpenMP/Accelerator, as well as asynchronous task-based models will be explored to achieve maximum parallelism. Parallelism at the level of a matrix operation, solver, and/or application needs to be explored to gain the most benefit on new and emerging architectures. This report serves as a “living document” of the additions and enhancements throughout the ECP CoPA project for $\mathcal{O}(N)$ QMD. The following sections address the applications we are working with, the quantum chemistry libraries (BML and PROGRESS), and our co-design exploration through the use of a proxy application.

36 MATERIALS SCIENCE↗
DOE OSTI2023

Convex Optimization for Nonequilibrium Steady States on a Hybrid Quantum Processor

Finding the transient and steady state properties of open quantum systems is a central problem in various fields of quantum technologies. Here, in this work, we present a quantum-assisted algorithm to determine the steady states of open system dynamics. By reformulating the problem of finding the fixed point of Lindblad dynamics as a feasibility semidefinite program, we bypass several well-known issues with variational quantum approaches to solving for steady states. We demonstrate that our hybrid approach allows us to estimate the steady states of higher dimensional open quantum systems and discuss how our method can find multiple steady states for systems with symmetries.

97 MATHEMATICS AND COMPUTING↗
DOE OSTI2019

A Hybrid Approach for Solving Optimization Problems on Small Quantum Computers

Solving larger-sized problems is an important area of research in quantum computing. Designing hybrid quantum-classical algorithms is a promising approach to solving this. We discuss decomposition-based hybrid approaches for solving optimization problems and demonstrate them for applications related to community detection.

97 MATHEMATICS AND COMPUTING↗
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 OSTI2019

Locality and Digital Quantum Simulation of Power-Law Interactions

The propagation of information in nonrelativistic quantum systems obeys a speed limit known as a Lieb-Robinson bound. We derive a new Lieb-Robinson bound for systems with interactions that decay with distance r as a power law, 1 / r α . The bound implies an effective light cone tighter than all previous bounds. Our approach is based on a technique for approximating the time evolution of a system, which was first introduced as part of a quantum simulation algorithm by Haah et al ., FOCS’18. To bound the error of the approximation, we use a known Lieb-Robinson bound that is weaker than the bound we establish. This result brings the analysis full circle, suggesting a deep connection between Lieb-Robinson bounds and digital quantum simulation. In addition to the new Lieb-Robinson bound, our analysis also gives an error bound for the Haah et al. quantum simulation algorithm when used to simulate power-law decaying interactions. In particular, we show that the gate count of the algorithm scales with the system size better than existing algorithms when α > 3 D (where D is the number of dimensions).

71 CLASSICAL AND QUANTUM MECHANICS, GENERAL PHYSIC↗
DOE OSTI2024

Determining the N -Representability of a Reduced Density Matrix via Unitary Evolution and Stochastic Sampling

The N-representability problem consists in determining whether, for a given p-body matrix, there exists at least one N-body density matrix from which the p-body matrix can be obtained by contraction, that is, if the given matrix is a p-body reduced density matrix (p-RDM). The knowledge of all necessary and sufficient conditions for a p-body matrix to be N-representable allows the constrained minimization of a many-body Hamiltonian expectation value with respect to the p-body density matrix and, thus, the determination of its exact ground state. However, the number of constraints that complete the N-representability conditions grows exponentially with system size, and hence, the procedure quickly becomes intractable for practical applications. This work introduces a hybrid quantum-stochastic algorithm to effectively replace the N-representability conditions. The algorithm consists of applying to an initial N-body density matrix a sequence of unitary evolution operators constructed from a stochastic process that successively approaches the reduced state of the density matrix on a p-body subsystem, represented by a p-RDM, to a target p-body matrix, potentially a p-RDM. The generators of the evolution operators follow the well-known adaptive derivative-assembled pseudo-Trotter method (ADAPT), while the stochastic component is implemented by using a simulated annealing process. The resulting algorithm is independent of any underlying Hamiltonian, and it can be used to decide whether a given p-body matrix is N-representable, establishing a criterion to determine its quality and correcting it. We apply the proposed hybrid ADAPT algorithm to alleged reduced density matrices from a quantum chemistry electronic Hamiltonian, from the reduced Bardeen–Cooper–Schrieffer model with constant pairing, and from the Heisenberg XXZ spin model. In all cases, the proposed method behaves as expected for 1-RDMs and 2-RDMs, evolving the initial matrices toward different targets.

37 INORGANIC, ORGANIC, PHYSICAL, AND ANALYTICAL CH↗
DOE OSTI2011

Adaptive schemes for incomplete quantum process tomography

We propose an iterative algorithm for incomplete quantum process tomography with the help of quantum state estimation. The algorithm, which is based on the combined principles of maximum likelihood and maximum entropy, yields a unique estimator for an unknown quantum process when one has less than a complete set of linearly independent measurement data to specify the quantum process uniquely. We apply this iterative algorithm adaptively in various situations and so optimize the amount of resources required to estimate a quantum process with incomplete data.

71 CLASSICAL AND QUANTUM MECHANICS, GENERAL PHYSIC↗
DOE OSTI2023

The learnability of Pauli noise

Recently, several quantum benchmarking algorithms have been developed to characterize noisy quantum gates on today’s quantum devices. A fundamental issue in benchmarking is that not everything about quantum noise is learnable due to the existence of gauge freedom, leaving open the question what information is learnable and what is not, which is unclear even for a single CNOT gate. Here we give a precise characterization of the learnability of Pauli noise channels attached to Clifford gates using graph theoretical tools. Our results reveal the optimality of cycle benchmarking in the sense that it can extract all learnable information about Pauli noise. We experimentally demonstrate noise characterization of IBM’s CNOT gate up to 2 unlearnable degrees of freedom, for which we obtain bounds using physical constraints. In addition, we show that an attempt to extract unlearnable information by ignoring state preparation noise yields unphysical estimates, which is used to lower bound the state preparation noise.

97 MATHEMATICS AND COMPUTING↗
DOE OSTI2019

Polynomial time algorithms for estimating spectra of adiabatic Hamiltonians

Much research regarding quantum adiabatic optimization has focused on stoquastic Hamiltonians with Hamming-symmetric potentials, such as the well-studied “spike” example. Due to the large amount of symmetry in these potentials such problems are readily open to analysis both analytically and computationally. However, more realistic potentials do not have such a high degree of symmetry and may have many local minima. Here we present a somewhat more realistic class of problems consisting of many individually Hamming-symmetric potential wells. For two or three such wells we demonstrate that such a problem can be solved exactly in time polynomial in the number of qubits and wells. For greater than three wells, we present a tight-binding approach with which to efficiently analyze the performance of such Hamiltonians in an adiabatic computation. Here, we provide several basic examples designed to highlight the usefulness of this toy model and to give insight into using the tight-binding approach to examining it, including (1) an adiabatic unstructured search with a transverse field driver and a prior guess to the marked item and (2) a scheme for adiabatically simulating the ground states of small collections of strongly interacting spins, with an explicit demonstration for an Ising-model Hamiltonian.

71 CLASSICAL AND QUANTUM MECHANICS, GENERAL PHYSIC↗
DOE OSTI2023

Inventory of Public Key Cryptography in US Electric Vehicle Charging

Electric vehicles (EVs) and charging infrastructure are networked systems, which employ high-level communications in support of charging and grid service decisions. Public key cryptography (PKC) underlies much of the security and privacy protections of the information exchange. We are entering a new epoch where quantum computing threats must be seriously considered. A sufficiently large quantum computer, so named Cryptographically Relevant Quantum Computer (QRQC), will be able to perform the mathematical operations to efficiently attack the underpinnings of traditional PKC, thus jeopardizing the digital foundations for trust, communications security, and data security. Estimates suggest a QRQC can break public key encryption and digital signatures in the manner of tens to hundreds of hours, compared to traditional computing that would demand more than 10 18 years in a brute force-style attack. A consensus belief of quantum theorists, quantum experimenters, and cryptographers suggest that the quantum threat will be likely realized in the next twenty years. To address the threat, post-quantum cryptography, which is cryptosystems that are designed to be secure against both traditional and quantum computing threats, must be adopted. Migration from traditional PKC to quantum-resilient cryptography is a global undertaking and likely represents the largest transition in computing history. The nascent state of EV public key infrastructure, combined with limited adoption of the vehicle secure charging features, presents an opportunity to establish a preference for quantum-resistant cryptography as a step on the migration path. Delays will stunt the efforts as rapidly accelerating EVs sales and huge infrastructure investments will create large growing bases of long-lived vehicles and infrastructure. Migration preparations can commence while NIST continues the process to standardize post-quantum cryptography (PQC), which are quantum-resilient algorithms designed to be secure against traditional and quantum computing threats. The first step in preparing EV charging is to identify the presence of traditional public key cryptography algorithms and applications. With this objective in mind, this report is intended to advise the vehicle manufacturers, charging station manufacturers, charging station operators, charge network providers and other EV charging stakeholders with information on traditional PKC application and the potential risks when PKC becomes insecure. This report, the first in a series of reports discussing the topics existing at the confluence of post-quantum cryptography adoption and EV charging, identifies traditional public key applications employed and identifies potential consequences of leaving EV charging infrastructure vulnerable to quantum computing. The focus remains squarely on the of EV charging and infrastructure with respect to PKC and is believed by the authors to complement the NIST SP 1800-38 Migration to Post-Quantum Cryptography. While the report is centered on infrastructure, there are implications to vehicles.

33 ADVANCED PROPULSION SYSTEMS↗
DOE OSTI2025

Quantum approximate multi-objective optimization

The goal of multi-objective optimization is to understand optimal trade-offs between competing objective functions by finding the Pareto front, that is, the set of all Pareto-optimal solutions, where no objective can be improved without degrading another one. Multi-objective optimization can be challenging classically, even if the corresponding single-objective optimization problems are efficiently solvable. Thus, multi-objective optimization represents a compelling problem class to analyze with quantum computers. Here we use a low-depth quantum approximate optimization algorithm to approximate the optimal Pareto front of certain multi-objective weighted maximum-cut problems. We demonstrate its performance on an IBM Quantum computer, as well as with matrix product state numerical simulation, and show its potential to outperform classical approaches.

71 CLASSICAL AND QUANTUM MECHANICS, GENERAL PHYSIC↗
DOE OSTI2022

Numerical simulations of noisy quantum circuits for computational chemistry

The opportunities afforded by near-term quantum computers to calculate the ground-state properties of small molecules depend on the structure of the computational ansatz as well as the errors induced by device noise. Here we investigate the behavior of these noisy quantum circuits using numerical simulations to estimate the accuracy and fidelity of the prepared quantum states relative to the ground truth obtained by conventional means. We implement several different types of ansatz circuits derived from unitary coupled cluster theory for the purposes of estimating the ground-state energy of sodium hydride using the variational quantum eigensolver algorithm. We show how relative error in the energy and the fidelity scale with the levels of gate-based noise, the internuclear configuration, the ansatz circuit depth, and the parameter optimization methods.

36 MATERIALS SCIENCE↗
DOE OSTI2017

Delayed Slater determinant update algorithms for high efficiency quantum Monte Carlo

Within ab initio Quantum Monte Carlo simulations, the leading numerical cost for large systems is the computation of the values of the Slater determinants in the trial wavefunction. Each Monte Carlo step requires finding the determinant of a dense matrix. This is most commonly iteratively evaluated using a rank-1 Sherman-Morrison updating scheme to avoid repeated explicit calculation of the inverse. The overall computational cost is therefore formally cubic in the number of electrons or matrix size. To improve the numerical efficiency of this procedure, we propose a novel multiple rank delayed update scheme. This strategy enables probability evaluation with application of accepted moves to the matrices delayed until after a predetermined number of moves, K . The accepted events are then applied to the matrices en bloc with enhanced arithmetic intensity and computational efficiency via matrix-matrix operations instead of matrix-vector operations. Here this procedure does not change the underlying Monte Carlo sampling or its statistical efficiency. For calculations on large systems and algorithms such as diffusion Monte Carlo where the acceptance ratio is high, order of magnitude improvements in the update time can be obtained on both multi- core CPUs and GPUs.

97 MATHEMATICS AND COMPUTING↗
DOE OSTI2024

Solving the Hele–Shaw flow using the Harrow–Hassidim–Lloyd algorithm on superconducting devices: A study of efficiency and challenges

The development of quantum processors for practical fluid flow problems is a promising yet distant goal. Recent advances in quantum linear solvers have highlighted their potential for classical fluid dynamics. In this study, we evaluate the Harrow–Hassidim–Lloyd (HHL) quantum linear systems algorithm (QLSA) for solving the idealized Hele–Shaw flow. Our focus is on the accuracy and computational cost of the HHL solver, which we find to be sensitive to the condition number, scaling exponentially with problem size. This emphasizes the need for preconditioning to enhance the practical use of QLSAs in fluid flow applications. Moreover, we perform shots-based simulations on quantum simulators and test the HHL solver on superconducting quantum devices, where noise, large circuit depths, and gate errors limit performance. Error suppression and mitigation techniques improve accuracy, suggesting that such fluid flow problems can benchmark noise mitigation efforts. Finally, our findings provide a foundation for future, more complex application of QLSAs in fluid flow simulations.

75 CONDENSED MATTER PHYSICS, SUPERCONDUCTIVITY AND↗
DOE OSTI2017

Quantum Circuit Designs of Integer Division Optimizing T-Count and T-Depth

Quantum circuits for basic mathematical functions such as division are required to implement scientific computing algorithms on quantum computers. In this work, we propose two designs for quantum integer division. The designs are based on quantum Clifford+T gates and are optimized for T-count and T-depth. Quantum circuits that are based on Clifford+T gates can be made fault tolerant in nature but the T gate is very costly to implement. As a result, reducing T-count and T-depth have become important optimization goals. Existing quantum hardware is limited in terms of number of available qubits. Thus, ancillary qubits are a circuit overhead that needs to be kept to a minimum. We propose two quantum integer division circuits. The first quantum integer division circuit is based on the non-restoring division algorithm. The proposed non-restoring division circuit is optimized for total quantum hardware (T-count and T-depth) cost but requires 2* n + 1 ancillary qubits. We also propose a quantum integer division circuit based on the restoring division algorithm. The proposed restoring division circuit is optimized for total qubits. The design requires only n ancillary qubits but will need more quantum hardware than the non-restoring division circuit. Both proposed quantum circuits are based on (i) a new quantum conditional addition circuit, (ii) a new quantum adder-subtractor and (iii) a new quantum subtraction circuit. Further, both designs are compared and shown to be superior to existing work in terms of T-count and T-depth. The proposed quantum non-restoring integer division circuit has a 96% improvement in terms of T-count and a 93% improvement in terms of T-depth compared to existing work. The proposed quantum restoring integer division circuit has a 91% improvement in terms of T-count and a 86% improvement in terms of T-count compared to the existing work.

Thapliyal, Himanshu↗