Hamiltonian operator approximation for energy measurement and ground state preparation

Abstract

The Hamiltonian operator plays a central role in quantum theory being a generator of unitary quantum dynamics. Its expectation value describes the energy of a quantum system. Typically being a non-unitary operator, the action of the Hamiltonian is either encoded using complex ancilla-based circuits, or implemented effectively as a sum of Pauli string terms. Here, we show how to approximate the Hamiltonian operator as a sum of propagators using a differential representation. The proposed approach, named Hamiltonian operator approximation (HOA), is designed to benefit analog quantum simulators, where one has direct access to simulation of quantum dynamics, but measuring separate circuits is not possible. We describe how to use this strategy in the hybrid quantum-classical workflow for performing energy measurements. Benchmarking the measurement scheme, we discuss the relevance of the discretization step size, stencil order, number of shots, and noise. We also use HOA to prepare ground states of complex material science models with direct iteration and quantum filter diagonalization, finding the lowest energy for the 12-qubit Hamiltonian of hydrogen chain H${}_6$ with $10^{-5}$ Hartree precision using $11$ time-evolved reference states. The approach is compared to the variational quantum eigensolver, proving HOA beneficial for systems at increasing size corresponding to noisy large scale quantum devices. We find that for Heisenberg model with twelve or more spins our approach may outperform variational methods, both in terms of the gate depth and the total number of measurements.

v3: VQE and QPEA comparison extended; close to the published version

References in corpus (18)

• Supplementary information for "Quantum supremacy using a programmable superconducting processor"
• Simulated Quantum Computation of Molecular Energies
• Predicting Many Properties of a Quantum System from Very Few Measurements
• Cost Function Dependent Barren Plateaus in Shallow Parametrized Quantum Circuits
• The Kernel Polynomial Method
• Simulating Hamiltonian dynamics with a truncated Taylor series
• Power of data in quantum machine learning
• Hartree-Fock on a superconducting qubit quantum computer
• Hybrid quantum-classical algorithms and quantum error mitigation
• A Theory of Trotter Error
• Quantum Approximate Optimization of Non-Planar Graph Problems on a Planar Superconducting Processor
• Layerwise learning for quantum neural networks
• Exploring entanglement and optimization within the Hamiltonian Variational Ansatz
• Adding control to arbitrary unknown quantum operations
• Higher Order Derivatives of Quantum Neural Networks with Barren Plateaus
• Yao.jl: Extensible, Efficient Framework for Quantum Algorithm Design
• Quantum Power Method by a Superposition of Time-Evolved States
• Destructive Error Interference in Product-Formula Lattice Simulation

Cited by in corpus (22)

• Quantum Krylov subspace algorithms for ground and excited state energy estimation
• Iterative Quantum Assisted Eigensolver
• Generalized quantum circuit differentiation rules
• Accessing ground state and excited states energies in many-body system after symmetry restoration using quantum computers
• Quantum Kernel Methods for Solving Differential Equations
• Randomizing multi-product formulas for Hamiltonian simulation
• Noisy intermediate-scale quantum algorithm for semidefinite programming
• Quantum simulations employing connected moments expansions
• Qubit-Efficient Randomized Quantum Algorithms for Linear Algebra
• Spatial, spin, and charge symmetry projections for a Fermi-Hubbard model on a quantum computer
• Assessment of the variational quantum eigensolver: application to the Heisenberg model
• Quantum Gaussian filter for exploring ground-state properties
• What can we learn from quantum convolutional neural networks?
• Analysis of quantum Krylov algorithms with errors
• Measurement-efficient quantum Krylov subspace diagonalisation
• Improved iterative quantum algorithm for ground-state preparation
• Assisted quantum simulation of open quantum systems
• Quantum topological data analysis via the estimation of the density of states
• Solving lattice gauge theories using the quantum Krylov algorithm and qubitization
• Quantum subspace expansion approach for simulating dynamical response functions of Kitaev spin liquids
• NISQ algorithm for the matrix elements of a generic observable
• Co-Designing Spectral Transformation Oracles with Hybrid Oscillator-Qubit Quantum Processors: From Algorithms to Compilation