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arXiv:2406.07364

Chebyshev Approximated Variational Coupled Cluster for Quantum Computing

Luca Erhart, Yuichiro Yoshida, Viktor E. Khinevich, Wataru Mizukami

Published Jun 11, 2024• 0 citations
arXivHTMLPDF

Abstract

We propose an approach to approximately implement the variational coupled cluster (VCC) theory on quantum computers, which struggles with exponential scaling of computational costs on classical computers. To this end, we employ expanding the exponential cluster operator using Chebyshev polynomials and introduce two methods: the Chebyshev approximated VCC (C$^d$-VCC) and the Hermitian-part Chebyshev approximated VCC (HC$^d$-VCC), where $d$ indicates the maximum degree of the Chebyshev polynomials. The latter method decomposes the cluster operator into anti-Hermitian and Hermitian parts, with the anti-Hermitian part represented by the disentangled unitary coupled cluster ansatz and the Hermitian part approximated using Chebyshev expansion. We illustrate the implementation of the HC$^d$-VCC in a quantum circuit using the quantum singular value transformation technique. Numerical simulations show that the C$^d$-VCC rapidly converges to the exact VCC with increasing truncation degree $d$, and the HC$^d$-VCC effectively reduces the Chebyshev expansion error compared to the C$^d$-VCC. The HC$^d$-VCC method for realizing non-unitary coupled cluster wave functions on quantum computers is expected to be useful for initial state preparation on quantum computers and efficient tomography of the quantum state for post-processing on classical computers after quantum computations.

DOI
10.48550/arxiv.2406.07364

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Abstract The variational quantum eigensolver (VQE) algorithm combines the ability of quantum computers to efficiently compute expectation values with a classical optimization routine in order to approximate ground state energies of quantum systems. In this paper, we study the application of VQE to the simulation of molecular energies using the unitary coupled cluster (UCC) ansatz. We introduce new strategies to reduce the circuit depth for the implementation of UCC and improve the optimization of the wavefunction based on efficient classical approximations of the cluster amplitudes. Additionally, we propose an analytical method to compute the energy gradient that reduces the sampling cost for gradient estimation by several orders of magnitude compared to numerical gradients. We illustrate our methodology with numerical simulations for a system of four hydrogen atoms that exhibit strong correlation and show that the circuit depth of VQE using a UCC ansatz can be reduced without introducing significant loss of accuracy in the final wavefunctions and energies.
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Unitary coupled cluster (UCC), originally developed as a variational alternative to the popular traditional coupled cluster method, has seen a resurgence as a functional form for use on quantum computers. However, the number of excitors present in the Ansatz often presents a barrier to implementation on quantum computers. Given the natural sparsity of wavefunctions obtained from quantum Monte Carlo methods, we consider here a stochastic solution to the UCC problem. Using the coupled cluster Monte Carlo framework, we develop cluster selection schemes that capture the structure of the UCC wavefunction, as well as its Trotterized approximation, and use these to solve the corresponding projected equations. Due to the fast convergence of the equations with order in the cluster expansion, this approach scales polynomially with the size of the system. Unlike traditional UCC implementations, our approach naturally produces a non-variational estimator for the energy in the form of the projected energy. For unitary coupled cluster singles and doubles (UCCSD) in small systems, we find that this agrees well with the expectation value of the energy and, in the case of two electrons, with full configuration interaction results. For the larger N2 system, the two estimators diverge, with the projected energy approaching the coupled cluster result, while the expectation value is close to results from traditional UCCSD.
Many quantum algorithms rely on a quality initial state for optimal performance. Preparing an initial state for specific applications can considerably reduce the cost of probabilistic algorithms such as the well studied quantum phase estimation (QPE). Fortunately, in the application space of quantum chemistry, generating approximate wave functions for molecular systems is well studied, and quantum computing algorithms stand to benefit from importing these classical methods directly into a quantum circuit. In this work, we propose a state preparation method based on coupled cluster (CC) theory, which is a pillar of quantum chemistry on classical computers, by incorporating midcircuit measurements into the circuit construction. Currently, the most well studied state preparation method for quantum chemistry on quantum computers is the variational quantum eigensolver (VQE) with a unitary-CC with single- and double-electron excitation terms (UCCSD) ansatz whose operations are limited to unitary gates. We verify the accuracy of our state preparation protocol using midcircuit measurements by performing energy evaluation and state overlap computation for a set of small chemical systems. We further demonstrate that our approach leads to a reduction of the classical computation overhead, and the number of CNOT and T gates by 28 and 57% on average when compared against the standard VQE-UCCSD protocol.