Generalization of the output of variational quantum eigensolver by parameter interpolation with low-depth ansatz

Abstract

The variational quantum eigensolver (VQE) is an attracting possible application of near-term quantum computers. Originally, the aim of the VQE is to find a ground state for a given specific Hamiltonian. It is achieved by minimizing the expectation value of the Hamiltonian with respect to an ansatz state by tuning parameters \(\bmθ\) on a quantum circuit which constructs the ansatz. Here we consider an extended problem of the VQE, namely, our objective in this work is to "generalize" the optimized output of the VQE just like machine learning. We aim to find ground states for a given set of Hamiltonians \(\{H(\bm{x})\}\), where \(\bm{x}\) is a parameter which specifies the quantum system under consideration, such as geometries of atoms of a molecule. Our approach is to train the circuit on the small number of \(\bm{x}\)'s. Specifically, we employ the interpolation of the optimal circuit parameter determined at different \(\bm{x}\)'s, assuming that the circuit parameter \(\bmθ\) has simple dependency on a hidden parameter \(\bm{x}\) as \(\bmθ(\bm{x})\). We show by numerical simulations that, using an ansatz which we call the Hamiltonian-alternating ansatz, the optimal circuit parameters can be interpolated to give near-optimal ground states in between the trained \(\bm{x}\)'s. The proposed method can greatly reduce, on a rough estimation by a few orders of magnitude, the time required to obtain ground states for different Hamiltonians by the VQE. Once generalized, the ansatz circuit can predict the ground state without optimizing the circuit parameter \(\bmθ\) in a certain range of \(\bm{x}\).

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• Expanding the horizon of automated metamaterials discovery via quantum annealing
• Near-Term Quantum Computing Techniques: Variational Quantum Algorithms, Error Mitigation, Circuit Compilation, Benchmarking and Classical Simulation
• Constructing a virtual two-qubit gate by sampling single-qubit operations
• The Meta-Variational Quantum Eigensolver (Meta-VQE): Learning energy profiles of parameterized Hamiltonians for quantum simulation
• NISQ Computers: A Path to Quantum Supremacy
• Theory of analytical energy derivatives for the variational quantum eigensolver
• Reduced Density Matrix Sampling: Self-consistent Embedding and Multiscale Electronic Structure on Current Generation Quantum Computers
• Adaptive pruning-based optimization of parameterized quantum circuits
• Computation of molecular excited states on IBM quantum computers using a discriminative variational quantum eigensolver
• Optimizing quantum circuits with Riemannian gradient flow
• Preserving Symmetries for Variational Quantum Eigensolvers in the Presence of Noise
• Improving the variational quantum eigensolver using variational adiabatic quantum computing
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• Exponential Qubit Reduction in Optimization for Financial Transaction Settlement
• Optimized numerical gradient and Hessian estimation for variational quantum algorithms
• Hot-Start Optimization for Variational Quantum Eigensolver
• Mitigating the measurement overhead of ADAPT-VQE with optimised informationally complete generalised measurements
• Evaluating Ground State Energies of Chemical Systems with Low-Depth Quantum Circuits and High Accuracy