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arXiv:2008.06637

A Quantum-Inspired Algorithm for the Factorized Form of Unitary Coupled Cluster Theory

Jia Chen, Hai‐Ping Cheng, James K. Freericks

Published Aug 15, 2020• 0 citations
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Abstract

The factorized form of unitary coupled cluster theory (UCC) is a promising wave-function ansatz for the variational quantum eigensolver algorithm. Here, we present a quantum inspired algorithm for UCC based on an exact operator identity for the individual UCC factors. We implement this algorithm for calculations of the H$_{10}$ linear chain and the H$_2$O molecule with single and double $ζ$ basis sets to provide insights about UCC as a wave-function ansatz. We find that as an electronic structure method, UCC could potentially be valuable for strongly correlated systems, for which conventional coupled cluster theory has difficulties. On quantum computers, the factorized form of UCC can also serve as an initial state preparation method for the quantum phase-estimation algorithm, since it yields higher overlap with the ground state than many other common variational ansatzes. This classical algorithm for UCC also enables widespread testing of the technique, which will be useful for benchmarking quantum computations.

DOI
10.48550/arxiv.2008.06637

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The factorized form of unitary coupled cluster theory (UCC) is a promising wave-function ansatz for the variational quantum eigensolver algorithm. Here, we present a quantum inspired algorithm for UCC based on an exact operator identity for the individual UCC factors. We implement this algorithm for calculations of the H10_{10}10​ linear chain and the H2_22​O molecule with single and double ζζζ basis sets to provide insights about UCC as a wave-function ansatz. We find that as an electronic structure method, UCC could potentially be valuable for strongly correlated systems, for which conventional coupled cluster theory has difficulties. On quantum computers, the factorized form of UCC can also serve as an initial state preparation method for the quantum phase-estimation algorithm, since it yields higher overlap with the ground state than many other common variational ansatzes. This classical algorithm for UCC also enables widespread testing of the technique, which will be useful for benchmarking quantum computations.
The factorized form of unitary coupled cluster theory (UCC) is a promising wave-function ansatz for the variational quantum eigensolver algorithm. Here, we present a quantum-inspired classical algorithm for UCC based on an exact operator identity for the individual UCC factors. We implement this algorithm for calculations of the H 10 linear chain and the H 2 O molecule with single and double ζ basis sets to provide insights into UCC as a wave-function ansatz. We find that for weakly correlated molecules, the factorized form of the UCC provides similar accuracy to conventional coupled cluster theory (CC); for strongly correlated molecules, where CC often breaks down, UCC significantly outperforms the configuration interaction (CI) ansatz. As a result, the factorized form of the UCC is an accurate, efficient, and reliable electronic structure method in both the weakly and strongly correlated regions. This classical algorithm now allows robust benchmarking of anticipated results from quantum computers and application of coupled-cluster techniques to more strongly correlated molecules.
The unitary coupled cluster (UCC) approximation is one of the more promising wave function ansätzes for electronic structure calculations on quantum computers via the variational quantum eigensolver algorithm. However, for large systems with many orbitals, the required number of UCC factors still leads to very deep quantum circuits, which can be challenging to implement. Based on the observation that most UCC amplitudes are small for both weakly correlated and strongly correlated molecules, we devise an algorithm that employs a Taylor expansion in the small amplitudes, trading off circuit depth for extra measurements. Strong correlations can be taken into account by performing the expansion about a small set of UCC factors, which are treated exactly. Near equilibrium, the Taylor series expansion often works well without the need to include any exact factors; as the molecule is stretched and correlations increase, we find only a small number of factors need to be treated exactly.
A unitary coupled cluster (UCC) form for the wave function in the variational quantum eigensolver has been suggested as a systematic way to go beyond the mean-field approximation and include electron correlation in solving quantum chemistry problems on a quantum computer. Although being exact in the limit of including all possible coupled cluster excitations, practically, the accuracy of this approach depends on the number and type of terms are included in the wave function parametrization. Another difficulty of UCC is a growth of the number of simultaneously entangled qubits even at the fixed Fermionic excitation rank. Not all quantum computing architectures can cope with this growth. To address both problems, we introduce a qubit coupled cluster (QCC) method that starts directly in the qubit space and uses energy response estimates for ranking the importance of individual entanglers for the variational energy minimization. Also, we provide an exact factorization of a unitary rotation of more than two qubits to a product of two-qubit unitary rotations. Thus, the QCC method with the factorization technique can be limited to only two-qubit entanglement gates and allows for very efficient use of quantum resources in terms of the number of coupled cluster operators. The method performance is illustrated by calculating ground-state potential energy curves of H 2 and LiH molecules with chemical accuracy, ≤1 kcal/mol, and a symmetric water dissociation curve.
The Coupled Cluster (CC) method is used to compute the electronic correlation energy in atoms and molecules and often leads to highly accurate results. However, due to its single-reference nature, standard CC in its projected form fails to describe quantum states characterized by strong electronic correlations and multi-reference projective methods become necessary. On the other hand, quantum algorithms for the solution of many-electron problems have also emerged recently. The quantum unitary variant of CC (UCC) with singles and doubles (q-UCCSD) is a popular wavefunction Ansatz for the variational quantum eigensolver algorithm. The variational nature of this approach can lead to significant advantages compared to its classical equivalent in the projected form, in particular, for the description of strong electronic correlation. However, due to the large number of gate operations required in q-UCCSD, approximations need to be introduced in order to make this approach implementable in a state-of-the-art quantum computer. In this work, we evaluate several variants of the standard q-UCCSD Ansatz in which only a subset of excitations is included. In particular, we investigate the singlet and pair q-UCCD approaches combined with orbital optimization. We show that these approaches can capture the dissociation/distortion profiles of challenging systems, such as H4, H2O, and N2 molecules, as well as the one-dimensional periodic Fermi–Hubbard chain. These results promote the future use of q-UCC methods for the solution of challenging electronic structure problems in quantum chemistry.
The variational quantum eigensolver is one of the most promising algorithms for near-term quantum computers. It has the potential to solve quantum chemistry problems involving strongly correlated electrons with relatively low-depth circuits, which are otherwise difficult to solve on classical computers. The variational eigenstate is constructed from a number of factorized unitary coupled-cluster terms applied onto an initial (single-reference) state. Current algorithms for applying one of these operators to a quantum state require a number of operations that scale exponentially with the rank of the operator. We exploit a hidden SU(2) symmetry to allow us to employ the linear combination of unitaries approach, Our Prepare subroutine uses n+2 ancilla qubits for a rank-n operator. Our Select(U^) scheme uses O(n)Cnot gates. This results in a full algorithm that scales like the cube of the rank of the operator n3, a significant reduction in complexity for rank five or higher operators. This approach, when combined with other algorithms for lower-rank operators (when compared to the standard implementation), will make the factorized form of the unitary coupled-cluster approach much more efficient to implement on all types of quantum computers.
Quantum chemistry calculations of large, strongly correlated systems are typically limited by the computation cost that scales exponentially with the size of the system. Quantum algorithms, designed specifically for quantum computers, can alleviate this, but the resources required are still too large for today's quantum devices. Here, we present a quantum algorithm that combines a localization of multireference wave functions of chemical systems with quantum phase estimation (QPE) and variational unitary coupled cluster singles and doubles (UCCSD) to compute their ground-state energy. Our algorithm, termed "local active space unitary coupled cluster" (LAS-UCC), scales linearly with the system size for certain geometries, providing a polynomial reduction in the total number of gates compared with QPE, while providing accuracy above that of the variational quantum eigensolver using the UCCSD ansatz and also above that of the classical local active space self-consistent field. The accuracy of LAS-UCC is demonstrated by dissociating (H2)2 into two H2 molecules and by breaking the two double bonds in trans-butadiene, and resource estimates are provided for linear chains of up to 20 H2 molecules.