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arXiv:2009.11001

Quantum Assisted Eigensolver

Kishor Bharti

Published Sep 23, 2020• 19 citations
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Abstract

We propose a hybrid quantum-classical algorithm for approximating the ground state and ground state energy of a Hamiltonian. Once the Ansatz has been decided, the quantum part of the algorithm involves the calculation of two overlap matrices. The output from the quantum part of the algorithm is utilized as input for the classical computer. The classical part of the algorithm is a quadratically constrained quadratic program with a single quadratic equality constraint. Unlike the variational quantum eigensolver algorithm, our algorithm does not have any classical-quantum feedback loop. Using convex relaxation techniques, we provide an efficiently computable lower bound to the classical optimization program. Furthermore, using results from Bar-On et al. (Journal of Optimization Theory and Applications, 82(2):379--386, 1994), we provide a sufficient condition for a local minimum to be a global minimum. A solver can use such a condition as a stopping criterion.

DOI
10.48550/arxiv.2009.11001

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We propose a hybrid quantum-classical algorithm for approximating the ground state and ground state energy of a Hamiltonian. Once the Ansatz has been decided, the quantum part of the algorithm involves the calculation of two overlap matrices. The output from the quantum part of the algorithm is utilized as input for the classical computer. The classical part of the algorithm is a quadratically constrained quadratic program with a single quadratic equality constraint. Unlike the variational quantum eigensolver algorithm, our algorithm does not have any classical-quantum feedback loop. Using convex relaxation techniques, we provide an efficiently computable lower bound to the classical optimization program. Furthermore, using results from Bar-On et al. (Journal of Optimization Theory and Applications, 82(2):379--386, 1994), we provide a sufficient condition for a local minimum to be a global minimum. A solver can use such a condition as a stopping criterion.
The task of estimating the ground state of Hamiltonians is an important problem in physics with numerous applications ranging from solid-state physics to combinatorial optimization. We provide a hybrid quantum-classical algorithm for approximating the ground state of a Hamiltonian that builds on the powerful Krylov subspace method in a way that is suitable for current quantum computers. Our algorithm systematically constructs the ansatz by using any given choice of the initial state and the unitaries describing the Hamiltonian. The only task of the quantum computer is to measure overlaps and no feedback loops are required. The measurements can be performed efficiently on current quantum hardware without requiring any complicated measurements such as the Hadamard test. Finally, a classical computer solves a well-characterized quadratically constrained optimization program. Our algorithm can reuse previous measurements to calculate the ground state of a wide range of Hamiltonians without requiring additional quantum resources. Further, we demonstrate our algorithm for solving a class of problems with thousands of qubits. The algorithm works for almost every random choice of the initial state and circumvents the barren plateau problem.
The variational quantum eigensolver (VQE) is a hybrid quantum-classical variational algorithm that produces an upper-bound estimate of the ground-state energy of a Hamiltonian. As quantum computers become more powerful and go beyond the reach of classical brute-force simulation, it is important to assess the quality of solutions produced by them. Here we propose a dual variational quantum eigensolver (dual-VQE) that produces a lower-bound estimate of the ground-state energy. As such, VQE and dual-VQE can serve as quality checks on their solutions; in the ideal case, the VQE upper bound and the dual-VQE lower bound form an interval containing the true optimal value of the ground-state energy. The idea behind dual-VQE is to employ semidefinite programming duality to rewrite the ground-state optimization problem as a constrained maximization problem, which itself can be bounded from below by an unconstrained optimization problem to be solved by a variational quantum algorithm. When using a convex combination ansatz in conjunction with a classical generative model, the quantum computational resources needed to evaluate the objective function of dual-VQE are no greater than those needed for that of VQE. We also show that the problem is well suited for classical pretraining using matrix product states and these methods help warm-start the optimization. We simulated the performance of dual-VQE on the transverse-field Ising model with and without pretraining and found that, for the example considered, while dual-VQE training is slower and noisier than VQE, it approaches the true value with an error of order 10−210^{-2}10−2.
Variational quantum approaches have shown great promise in finding near-optimal solutions to computationally challenging tasks, including solving optimization problems. Nonetheless, optimization problems with constraints may not have been handled in a disciplined fashion thus far. To address this gap, this work proposes a hybrid quantum-classical algorithmic paradigm termed the variational quantum eigensolver with constraints (VQEC) that extends the celebrated VQE to handle optimization with constraints. As with the standard VQE, the vector of optimization variables is captured by the state of a variational quantum circuit (VQC). To deal with constraints, VQEC optimizes a Lagrangian function classically over both the VQC parameters as well as the dual variables associated with constraints. To comply with the quantum setup, variables are updated via a perturbed primal-dual method leveraging the parameter shift rule. Among a wide gamut of potential applications, we showcase how VQEC can approximately solve quadratically constrained binary optimization problems, find stochastic binary policies satisfying quadratic constraints on the average and in probability, and solve large-scale linear programs over the probability simplex. Under an assumption on the error for the VQC to approximate an arbitrary probability mass function, we provide bounds on the optimality gap attained by a VQC. Numerical tests on a quantum simulator investigate the effect of various parameters and corroborate that VQEC can generate high-quality solutions.
Variational quantum approaches have shown great promise in finding near-optimal solutions to computationally challenging tasks. Nonetheless, enforcing constraints in a disciplined fashion has been largely unexplored. To address this gap, this work proposes a hybrid quantum-classical algorithmic paradigm termed VQEC that extends the celebrated VQE to handle optimization with constraints. As with the standard VQE, the vector of optimization variables is captured by the state of a variational quantum circuit (VQC). To deal with constraints, VQEC optimizes a Lagrangian function classically over both the VQC parameters as well as the dual variables associated with constraints. To comply with the quantum setup, variables are updated via a perturbed primal-dual method leveraging the parameter shift rule. Among a wide gamut of potential applications, we showcase how VQEC can approximately solve quadratically-constrained binary optimization (QCBO) problems, find stochastic binary policies satisfying quadratic constraints on the average and in probability, and solve large-scale linear programs (LP) over the probability simplex. Under an assumption on the error for the VQC to approximate an arbitrary probability mass function (PMF), we provide bounds on the optimality gap attained by a VQC. Numerical tests on a quantum simulator investigate the effect of various parameters and corroborate that VQEC can generate high-quality solutions.
In this study, we study the projective quantum eigensolver (PQE) approach, a hybrid quantum-classical algorithm that, by optimizing a unitary coupled cluster wave function, aims at computing the ground state of many-body systems. Instead of trying to minimize the energy of the system like the variational quantum eigensolver (VQE), PQE uses projections of the Schrödinger equation to efficiently bring the trial state closer to an eigenstate of the Hamiltonian. In this study, we provide a mathematical study of the algorithm. We derive a bound relating off-diagonal coefficients (residues) of the Hamiltonian to the energy error of the algorithm and the overlap achieved by the obtained wavefunction. These bounds not only give formal guarantees to PQE, but they also allow us to formulate a well-informed convergence criterion for residue-based optimizers. We then study the classical optimization itself and derive convergence guarantees under certain conditions. We propose a new residue-based optimizer, with numerical evidence of the superiority of this new approach for H4, H6, BeH2, and LiH dissociation curves over both the optimization introduced in Stair and Evangelista [PRX Quantum 2, 030301 (2021)] and VQE optimized using the Broyden-Fletcher-Goldfarb-Shanno method.
Simulating the dynamics of many-body quantum systems is believed to be one of the first fields that quantum computers can show a quantum advantage over classical computers. Noisy intermediate-scale quantum (NISQ) algorithms aim at effectively using the currently available quantum hardware. For quantum simulation, various types of NISQ algorithms have been proposed with individual advantages as well as challenges. In this work, we propose a new algorithm, truncated Taylor quantum simulator (TQS), that shares the advantages of existing algorithms and alleviates some of the shortcomings. Our algorithm does not have any classical-quantum feedback loop and bypasses the barren plateau problem by construction. The classical part in our hybrid quantum-classical algorithm corresponds to a quadratically constrained quadratic program (QCQP) with a single quadratic equality constraint, which admits a semidefinite relaxation. The QCQP based classical optimization was recently introduced as the classical step in quantum assisted eigensolver (QAE), a NISQ algorithm for the Hamiltonian ground state problem. Thus, our work provides a conceptual unification between the NISQ algorithms for the Hamiltonian ground state problem and the Hamiltonian simulation. We recover differential equation-based NISQ algorithms for Hamiltonian simulation such as quantum assisted simulator (QAS) and variational quantum simulator (VQS) as particular cases of our algorithm. We test our algorithm on some toy examples on current cloud quantum computers. We also provide a systematic approach to improve the accuracy of our algorithm.