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IEEE International Conference on Manipulation, Manufacturing and Measurement on the Nanoscale (3M-NANO)

Combinatorial optimization with variational approaches on noisy quantum devices

Tsukasa Miki, Ryo Okita, Moe Shimada, Jun‐ichi Shirakashi

Published Aug 2, 2021• 1 citation
Publisher

Abstract

Variational quantum eigensolver (VQE), a kind of quantum-classical hybrid method, has latent powers to leverage noisy quantum devices by effectively managing a limited number of qubits with finite coherent times. We implemented VQE with real quantum processors through IBM Quantum Experience and applied it to combinatorial optimizations. Notably, a short-depth quantum circuit consisting of a lower number of quantum gate than that in the quantum approximate optimization algorithm can search the ground states of diagonal Hamiltonians encoding the problems. Since the simplest case of the quantum circuit for VQE requires only single-qubit gates, it can be realized using current quantum computers.

DOI
10.1109/3m-nano49087.2021.9599740

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Variational quantum eigensolver (VQE), a kind of quantum-classical hybrid method, has latent powers to leverage noisy quantum devices by effectively managing a limited number of qubits with finite coherent times. We implemented VQE with real quantum processors through IBM Quantum Experience and applied it to combinatorial optimizations. Notably, a short-depth quantum circuit consisting of a lower number of quantum gate than that in the quantum approximate optimization algorithm can search the ground states of diagonal Hamiltonians encoding the problems. Since the simplest case of the quantum circuit for VQE requires only single-qubit gates, it can be realized using current quantum computers.
The variational quantum eigensolver (VQE), which is a quantum–classical hybrid approach, has latent powers to leverage near-term quantum devices by effectively managing a limited number of qubits with finite coherent lifetimes. While it is generally argued that the quantum approximate optimization algorithm (QAOA), which is a special case of VQE with a variational Ansatz based on the adiabatic theorem, may enable practical applications of noisy quantum devices for classical combinatorial optimizations, the strategy to improve the performance of this algorithm by increasing the circuit depth conflicts with the limited coherence time of near-term quantum devices. Here, we introduce strategies involving the VQE to reduce the circuit resources required for solving combinatorial optimizations. Our concept of a parameterized quantum circuit allows the Ansatz preparation to be achieved by only single-qubit operation. We find that the variational Ansatz without controlled X-gates leads to quick convergence in a classical subroutine used to determine the variational parameters. In addition, the variational Ansatz with optimized parameters maintains performance over the problem sizes both on the numerical simulation and IBM 27-qubit processor “ibm_kawasaki.” Therefore, the variational Ansatz introduced in this study has several advantages considering the total calculation time and performance scaling over the problem sizes. We also show that the variational Ansatz consisting of a lower number of gate operations than that of QAOA can approximate the eigenstates of diagonal Hamiltonians with high accuracy. We illustrate our ideas with a maximum-cut problem and show that near-term quantum applications may be feasible using short-depth circuits.
Combinatorial optimization on near-term quantum devices is a promising path to demonstrating quantum advantage. However, the capabilities of these devices are constrained by high noise or error rates. In this article, inspired by the variational quantum eigensolver (VQE), we propose an iterative layer VQE (L-VQE) approach. We present a large-scale numerical study, simulating circuits with up to 40 qubits and 352 parameters, that demonstrates the potential of the proposed approach. We evaluate quantum optimization heuristics on the problem of detecting multiple communities in networks, for which we introduce a novel qubit-frugal formulation. We numerically compare L-VQE with the quantum approximate optimization algorithm (QAOA) and demonstrate that QAOA achieves lower approximation ratios while requiring significantly deeper circuits. We show that L-VQE is more robust to finite sampling errors and has a higher chance of finding the solution as compared with standard VQE approaches. Our simulation results show that L-VQE performs well under realistic hardware noise.
Currently developed quantum devices have the potential to show a computational advantage if quantum applications exploit the devices efficiently. Variational quantum algorithms (VQAs) combine a quantum processor with a classical computer to minimize the expectation value of a problem Hamiltonian. An important element of VQAs is the state-preparation circuit, referred to as variational quantum circuit. To improve the ability for solving combinatorial optimizations with VQAs, we show that the variational quantum circuit with single-qubit gates only can find the ground state more quickly. We demonstrate our VQA by using real quantum processors through IBM Quantum. As a result, the variational circuit with single-qubit gates takes four times smaller numbers of function evaluations than that with entangling gates, such as controlled-X gates.
Variational quantum algorithms have promising applications in noisy intermediate-scale quantum (NISQ) devices. These algorithms rely on a classical optimization outer loop that minimizes a parameterized quantum circuit function. The optimization in variational quantum eigensolver (VQE) is NP-hard, meaning that finding the optimal solution is infeasible in the worst-case scenario. One way to address this challenge is through parallel optimization of parameters using multiple-parameterized quantum circuits. However, this approach is unsuitable for cloud-based quantum processing unit utilization due to the increased number of quantum circuit executions. Although NISQ devices have limitations in terms of gate depth, their size has been growing in recent years. Therefore, implementing multiple-parameterized quantum circuits in NISQ devices can suppress the increase in the number of executions. In this study, we propose a parallel-VQE, which leverages the parallel execution of parameterized quantum circuits to perform parallel parameter optimization in VQE, achieving convergence to solutions closer to the ground state. We validate the effectiveness of parallel-VQE in solving the random weighted max-cut problem using numerical simulations and a real quantum device. We present the results of running up to six circuits in parallel (120 qubits) and demonstrate the advantages of using multiple units to improve computational accuracy. This study provides a potential method for solving eigenvalue problems and combinatorial optimization problems for future quantum devices.
Abstract Current gate-based quantum computers have the potential to provide a computational advantage if algorithms use quantum hardware efficiently. To make combinatorial optimization more efficient, we introduce the filtering variational quantum eigensolver which utilizes filtering operators to achieve faster and more reliable convergence to the optimal solution. Additionally we explore the use of causal cones to reduce the number of qubits required on a quantum computer. Using random weighted MaxCut problems, we numerically analyze our methods and show that they perform better than the original VQE algorithm and the quantum approximate optimization algorithm. We also demonstrate the experimental feasibility of our algorithms on a Quantinuum trapped-ion quantum processor powered by Honeywell.
Recent practical approaches for the use of current generation noisy quantum devices in the simulation of quantum many-body problems have been dominated by the use of a variational quantum eigensolver (VQE). These coupled quantum-classical algorithms leverage the ability to perform many repeated measurements to avoid the currently prohibitive gate depths often required for exact quantum algorithms, with the restriction of a parametrized circuit to describe the states of interest. In this work, we show how the calculation of zero-temperature dynamic correlation functions defining the linear response characteristics of quantum systems can also be recast into a modified VQE algorithm, which can be incorporated into the current variational quantum infrastructure. This allows for these important physical expectation values describing the dynamics of the system to be directly converged on the frequency axis, and they approach exactness over all frequencies as the flexibility of the parametrization increases. The frequency resolution hence does not explicitly scale with gate depth, which is approximately twice as deep as a ground-state VQE. We apply the method to compute the single-particle Green's function of ab initio dihydrogen and lithium hydride molecules, and demonstrate the use of a practical active space embedding approach to extend to larger systems. While currently limited by the fidelity of two-qubit gates, whose number is increased compared to the ground-state algorithm on current devices, we believe the approach shows potential for the extraction of frequency dynamics of correlated systems on near-term quantum processors.