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arXiv:2401.06984 · Electronic Structure

Perturbative variational quantum algorithms for material simulations

Jie Liu, Zhenyu Li, Jinlong Yang

Published Jan 25, 2024• 7 citations
arXivHTMLPDF

Abstract

Abstract Reducing circuit depth is essential for implementing quantum simulations of electronic structure on near-term quantum devices. In this work, we propose a variational quantum eigensolver (VQE) based perturbation theory (PT) algorithm to accurately simulate electron correlation of periodic materials with shallow ansatz circuits, which are generated from adaptive derivative-assembled pseudo-trotter or qubit-excitation-based VQE calculations using a loose convergence criteria. Here, the major part of the electron correlation is described using the VQE ansatz circuit and the remaining correlation energy is described by either multireference or similarity transformation-based PT. Numerical results demonstrate that the new algorithms are able to accurately describe electron correlation of the LiH crystal with only one circuit parameter, in contrast with ~30 parameters required in the adaptive VQE to achieve the same accuracy. Meanwhile, for fixed-depth ansätze, e.g. unitary coupled cluster, we demonstrate that the VQE-based PT provides an appealing scheme to improve their accuracy.

DOI
10.1088/2516-1075/ad2277

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Abstract Reducing circuit depth is essential for implementing quantum simulations of electronic structure on near-term quantum devices. In this work, we propose a variational quantum eigensolver (VQE) based perturbation theory (PT) algorithm to accurately simulate electron correlation of periodic materials with shallow ansatz circuits, which are generated from adaptive derivative-assembled pseudo-trotter or qubit-excitation-based VQE calculations using a loose convergence criteria. Here, the major part of the electron correlation is described using the VQE ansatz circuit and the remaining correlation energy is described by either multireference or similarity transformation-based PT. Numerical results demonstrate that the new algorithms are able to accurately describe electron correlation of the LiH crystal with only one circuit parameter, in contrast with ~30 parameters required in the adaptive VQE to achieve the same accuracy. Meanwhile, for fixed-depth ansätze, e.g. unitary coupled cluster, we demonstrate that the VQE-based PT provides an appealing scheme to improve their accuracy.
Materials design stands to be one of the most promising applications of quantum computing. However, the presence of noise in near-term quantum devices restricts quantum simulations of materials to shallow circuits. In this work, we present circuit-efficient variational quantum eigensolver (VQE) simulations of periodic materials using qubit-encoded wave functions based on Adaptive Derivative-Assembled Pseudo-Trotter (ADAPT) VQE. To iteratively construct accurate wave functions for periodic systems, we introduce operator pools comprising a complete set of anti-Hermitian one- and two-body qubit excitation/flipping operators. Numerical results demonstrate that these qubit-encoded algorithms can accurately predict the ground-state energy of periodic systems while significantly reducing circuit depth compared to Fermion-encoded algorithms. Additionally, we integrate the variance extrapolation technique with ADAPT-VQE algorithms to enhance the accuracy of ground-state energy estimations. This strategy further reduces the required circuit depth, enabling scalable and precise simulations of periodic systems.
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.
The electronic structure is an anticipated application for quantum computers. However, quantum circuits required to represent the highly entangled electronic wave functions within the variational quantum eigensolver (VQE) framework are far beyond the capacity of current quantum devices. To adapt the VQE algorithms to near-term quantum hardware, it has been suggested to incorporate a part of the electronic correlation into an effective Hamiltonian, leaving the wave function in a less entangled form. We propose a new scheme to construct the effective Hamiltonian with the transformation in the form of a product of linear combinations of excitation operators. This new scheme promises a quadratic multiplicative growth of the effective Hamiltonian. We integrate this effective Hamiltonian method into the adaptive VQE algorithms to maintain constant-size quantum circuits. The new computational scheme is assessed by performing numerical simulations for small molecules. A milli-Hartree accuracy in a minimal basis is achieved with a much shallower circuit depth.
The variational quantum eigensolver (VQE) is a method of choice to solve the electronic structure problem for molecules on near-term gate-based quantum computers.However, the circuit depth is expected to grow significantly with the problem size.Increased depth can both degrade the accuracy of the results and reduce trainability.In this work, we propose an approach to reduce ansatz circuit depth.Our approach, called "PermVQE," adds an additional optimization loop to the VQE that permutes qubits in order to solve for the qubit Hamiltonian that maximally localizes correlations in the ground state.The choice of permutations is based on mutual information, which is a measure of interaction between electrons and/or holes in spin-orbitals.Encoding strongly entangled spin-orbitals into proximal qubits on a quantum chip naturally reduces the circuit depth needed to prepare the ground state.For representative molecular systems, LiH, H 2 , (H 2 ) 2 , H = 4 , H + 3 , and N 2 , we demonstrate that placing entangled qubits in close proximity leads to shallower depth circuits required to reach a given eigenvalue-eigenvector accuracy.The approach is designed for hardware-efficient ansatz of any qubit connectivity, and examples are demonstrated for linear and twodimensional grid architectures.The main ideas can also be applied to simulate molecules with other ansatz as well as variational quantum algorithms beyond the VQE.In particular, we demonstrate the beneficial effect of qubit permutations to build fermionic-adaptive derivative assembled pseudo-Trotter ansatz on a linear qubit connectivity architecture with nearly a twofold reduction of the number of controlled NOT gates.
We demonstrate the use of the Variational Quantum Eigensolver (VQE) to simulate solid state crystalline materials. We adapt the Unitary Coupled Cluster ansatz to periodic boundary conditions in real space and momentum space representations and directly map complex cluster operators to a quantum circuit ansatz to take advantage of the reduced number of excitation operators and Hamiltonian terms due to momentum conservation. To further reduce required quantum resources, such as the number of UCCSD amplitudes, circuit depth, required number of qubits and number of measurement circuits, we investigate a translational Quantum Subspace Expansion method (TransQSE) for the localized representation of the periodic Hamiltonian. Additionally, we also demonstrate an extension of the point group symmetry based qubit tapering method to periodic systems. We compare accuracy and computational costs for a range of geometries for 1D chains of dimerized hydrogen, helium and lithium hydride with increasing number of momentum space grid points and also demonstrate VQE calculations for 2D and 3D hydrogen and helium lattices. Our presented strategies enable the use of near-term quantum hardware to perform solid state simulation with variational quantum algorithms.
The use of the variational quantum eigensolver (VQE) for quantum chemistry is one of the most promising applications for noisy intermediate-scale quantum (NISQ) devices. A major limitation is represented by the need to build compact and shallow circuit ansatzes having the variational flexibility to catch the complexity of the electronic structure problem. To alleviate this drawback, we introduce a modified VQE scheme in which the form of the molecular Hamiltonian is adapted to the circuit ansatz through an optimization procedure. Exploiting the invariance of the Hamiltonian by molecular orbital rotations, we can optimize it using gradients that can be calculated without significant computational overload. The proposed method, named Wavefunction Adapted Hamiltonian Through Orbital Rotation (WAHTOR), has been applied to small molecules in numerical state vector simulations. The results demonstrate that, at variance with standard VQE, the method is less dependent on circuit topology and less prone to be trapped into high-energy local minima. It is able to recover a significant amount of electron correlation even with only empirical ansatzes with shallow circuit depth. Noisy calculations demonstrate the robustness and feasibility of the proposed methodology and indicate the hardware requirements to effectively apply the procedure using forthcoming NISQ devices.