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arXiv:2110.02864

Digital quantum simulation of strong correlation effects with iterative quantum phase estimation over the variational quantum eigensolver algorithm: $\mathrm{H_4}$ on a circle as a case study

Dipanjali Halder, V. S. Prasannaa, Valay Agarawal, Rahul Maitra

Published Oct 6, 2021• 1 citation
arXivHTMLPDF

Abstract

The iterative quantum phase estimation algorithm, applied to calculating the ground state energies of quantum chemical systems, is theoretically appealing in its wide scope of being able to handle both weakly and strongly correlated regimes. However, the goodness of the initial state that is sent as an input to the algorithm could strongly decide the quality of the results obtained. In this work, we generate the initial state by using the classical-quantum hybrid variational quantum eigensolver algorithm with unitary coupled cluster ansatz. We apply the procedure to obtain the ground state energies of the H4 molecule on a circle, as the system exhibits an interplay of dynamic as well as static correlation effects at different geometries. Furthermore, we argue on the importance of static correlation in construction of the reference determinant, and propose a minimally parametrized unitary coupled cluster ansatz, which drastically reduces number of variational parameters while incorporating the static correlation effects in the wavefunction. We demonstrate that a carefully and appropriately prepared initial state can greatly reduce the effects of noise due to sampling in the estimation of the desired eigenphase.

DOI
10.48550/arxiv.2110.02864

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Digital quantum simulation of strong correlation effects with iterative quantum phase estimation over the variational quantum eigensolver algorithm: H4\mathrm{H_4}H4​ on a circle as a case study
The iterative quantum phase estimation algorithm, applied to calculating the ground state energies of quantum chemical systems, is theoretically appealing in its wide scope of being able to handle both weakly and strongly correlated regimes. However, the goodness of the initial state that is sent as an input to the algorithm could strongly decide the quality of the results obtained. In this work, we generate the initial state by using the classical-quantum hybrid variational quantum eigensolver algorithm with unitary coupled cluster ansatz. We apply the procedure to obtain the ground state energies of the H4 molecule on a circle, as the system exhibits an interplay of dynamic as well as static correlation effects at different geometries. Furthermore, we argue on the importance of static correlation in construction of the reference determinant, and propose a minimally parametrized unitary coupled cluster ansatz, which drastically reduces number of variational parameters while incorporating the static correlation effects in the wavefunction. We demonstrate that a carefully and appropriately prepared initial state can greatly reduce the effects of noise due to sampling in the estimation of the desired eigenphase.
Abstract The iterative quantum phase estimation algorithm (IQPE) is theoretically appealing in its wide scope of being able to handle electronic correlation. However, the quality of the initial input state strongly enhances the probability of landing on the desired eigenstate. In this work, we systematically study two different parametrization schemes of the unitary coupled cluster (UCC) ansätz in the variational quantum eigensolver (VQE) framework toward the reference state preparation for IQPE. The efficacy of the UCC variants toward an appropriate state preparation is studied with prototypical H4 molecule on a circle. While the conventional UCC ansätz can lead to high success probability across various degrees of electronic complexity, a resource efficient minimally parametrized UCC ansätz consisting of active space excitations is shown to incorporate the essential static correlation in the reference state description. We demonstrate that such a carefully prepared initial state can significantly reduce the effects of noise due to sampling in the estimation of the desired eigenphase.
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.
Variational algorithms for strongly correlated chemical and materials systems are one of the most promising applications of near-term quantum computers. We present an extension to the variational quantum eigensolver that approximates the ground state of a system by solving a generalized eigenvalue problem in a subspace spanned by a collection of parametrized quantum states. This allows for the systematic improvement of a logical wavefunction ansatz without a significant increase in circuit complexity. To minimize the circuit complexity of this approach, we propose a strategy for efficiently measuring the Hamiltonian and overlap matrix elements between states parametrized by circuits that commute with the total particle number operator. This strategy doubles the size of the state preparation circuits but not their depth, while adding a small number of additional two-qubit gates relative to standard variational quantum eigensolver. We also propose a classical Monte Carlo scheme to estimate the uncertainty in the ground state energy caused by a finite number of measurements of the matrix elements. We explain how this Monte Carlo procedure can be extended to adaptively schedule the required measurements, reducing the number of circuit executions necessary for a given accuracy. We apply these ideas to two model strongly correlated systems, a square configuration of H4 and the π-system of hexatriene (C6H8).
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.
Quantum simulation of chemical systems is one of the most promising near-term applications of quantum computers. The variational quantum eigensolver, a leading algorithm for molecular simulations on quantum hardware, has a serious limitation in that it typically relies on a pre-selected wavefunction ansatz that results in approximate wavefunctions and energies. Here we present an arbitrarily accurate variational algorithm that, instead of fixing an ansatz upfront, grows it systematically one operator at a time in a way dictated by the molecule being simulated. This generates an ansatz with a small number of parameters, leading to shallow-depth circuits. We present numerical simulations, including for a prototypical strongly correlated molecule, which show that our algorithm performs much better than a unitary coupled cluster approach, in terms of both circuit depth and chemical accuracy. Our results highlight the potential of our adaptive algorithm for exact simulations with present-day and near-term quantum hardware.
We present a quantum information-inspired ansatz for the variational quantum eigensolver and demonstrate its efficacy in calculating ground-state energies of atomic systems. Instead of adopting a heuristic approach, we start with an approximate multiqubit target state and utilize two quantum information-theoretic quantities, i.e., von Neumann entropy and quantum mutual information, to construct our ansatz. The quantum information encoded in the target state helps us design unique blocks and identify qubit pairs that exhibit maximal quantum correlations within the multi-qubit system, thereby enabling us to deterministically place two-qubit entanglers in the suitably constructed parametrized quantum circuit. We find that our approach offers the advantage of reduced circuit depth compared to the physics-inspired unitary coupled-cluster (UCC) ansatz, while yielding accurate results. To test the performance of our ansatz, we apply it to compute ground-state energies of atomic systems in the relativistic regime. Our results show that for up to 12-qubits (12-spinorbitals) noiseless calculations, our ansatz yields energies within chemical accuracy ( ∼ 1.6 mhartree) relative to the complete active space configuration interaction values, while utilizing only a few blocks that contain in the best case 98 % fewer two-qubit gates when compared with the UCC ansatz in the singles and doubles approximation.