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arXiv:2104.15001

Hot-Start Optimization for Variational Quantum Eigensolver

Belozerova Polina, Shangareev Arthur, Zotov Yuriy, Yung Manhong, Dingshun Lv

Published Apr 30, 2021• 2 citations
arXivHTMLPDF

Abstract

The Variational Quantum Eigensolver (VQE) is one the most perspective algorithms for simulation of quantum many body physics that have recently attached a lot of attention and believed would be practical for implementation on the near term quantum devices. However, its feasibility and accuracy critically depend on the ansatz structure, which can be defined in different ways and appropriate choosing the structure presents a bottleneck of the protocol. In our work we present the optimization approach allowing shallow circuit solution. The major achievement of our Hot-Start method is the requiring gates number restriction by several times. In result it increases the fidelity of the final results what is important for the current noise quantum devices implementation. We suggest Hot-Start optimization to be a good new methods beginning for the modern quantum computational devices functionality development.

DOI
10.48550/arxiv.2104.15001

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The Variational Quantum Eigensolver (VQE) is one the most perspective algorithms for simulation of quantum many body physics that have recently attached a lot of attention and believed would be practical for implementation on the near term quantum devices. However, its feasibility and accuracy critically depend on the ansatz structure, which can be defined in different ways and appropriate choosing the structure presents a bottleneck of the protocol. In our work we present the optimization approach allowing shallow circuit solution. The major achievement of our Hot-Start method is the requiring gates number restriction by several times. In result it increases the fidelity of the final results what is important for the current noise quantum devices implementation. We suggest Hot-Start optimization to be a good new methods beginning for the modern quantum computational devices functionality development.
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.
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.
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.
The variational quantum eigensolver (VQE) is one of the most promising quantum algorithms for the near-term noisy intermediate-scale quantum (NISQ) devices. The VQE typically involves finding the minimum energy of a quantum Hamiltonian through classical optimization of a parametrized quantum ansatz. One of the bottlenecks in VQEs is the number of quantum circuits to be measured. In this work, we propose a physically intuitive truncation technique that starts the optimization procedure with a truncated Hamiltonian and then gradually transitions to the optimization for the original Hamiltonian via an operator classification method. This strategy allows us to reduce the required number of evaluations for the expectation value of Hamiltonian on a quantum computer. The reduction in required quantum resources for our strategy is substantial and likely scales with the system size. With numerical simulations, we demonstrate our method for various molecular systems.
The Variational Quantum Eigensolver (VQE) has emerged as a transformative approach in quantum computing, providing a practical means to tackle complex quantum systems. This chapter delves into the essential components and strategies for enhancing VQE performance, focusing on key areas such as hybrid optimization techniques, quantum circuit design, and measurement efficiency. By integrating classical and quantum resources, hybrid optimization methods significantly improve convergence rates while reducing the computational burden on quantum devices. The chapter explores the design of quantum circuits, emphasizing the importance of selecting appropriate variational ansätze to achieve optimal results. Challenges related to Hamiltonian simulation and measurement are critically examined, alongside innovative solutions to mitigate noise and enhance measurement fidelity. Through these comprehensive insights, this chapter aims to illuminate the potential of VQE in advancing quantum machine learning applications across diverse scientific domains.
Abstract The variational quantum eigensolver (VQE) is a method that uses a hybrid quantum-classical computational approach to find eigenvalues of a Hamiltonian. VQE has been proposed as an alternative to fully quantum algorithms such as quantum phase estimation (QPE) because fully quantum algorithms require quantum hardware that will not be accessible in the near future. VQE has been successfully applied to solve the electronic Schrödinger equation for a variety of small molecules. However, the scalability of this method is limited by two factors: the complexity of the quantum circuits and the complexity of the classical optimization problem. Both of these factors are affected by the choice of the variational ansatz used to represent the trial wave function. Hence, the construction of an efficient ansatz is an active area of research. Put another way, modern quantum computers are not capable of executing deep quantum circuits produced by using currently available ansatzes for problems that map onto more than several qubits. In this review, we present recent developments in the field of designing efficient ansatzes that fall into two categories—chemistry–inspired and hardware–efficient—that produce quantum circuits that are easier to run on modern hardware. We discuss the shortfalls of ansatzes originally formulated for VQE simulations, how they are addressed in more sophisticated methods, and the potential ways for further improvements.