Avoiding symmetry roadblocks and minimizing the measurement overhead of adaptive variational quantum eigensolvers

Abstract

Quantum simulation of strongly correlated systems is potentially the most feasible useful application of near-term quantum computers. Minimizing quantum computational resources is crucial to achieving this goal. A promising class of algorithms for this purpose consists of variational quantum eigensolvers (VQEs). Among these, problem-tailored versions such as ADAPT-VQE that build variational ansätze step by step from a predefined operator pool perform particularly well in terms of circuit depths and variational parameter counts. However, this improved performance comes at the expense of an additional measurement overhead compared to standard VQEs. Here, we show that this overhead can be reduced to an amount that grows only linearly with the number $n$ of qubits, instead of quartically as in the original ADAPT-VQE. We do this by proving that operator pools of size $2n-2$ can represent any state in Hilbert space if chosen appropriately. We prove that this is the minimal size of such "complete" pools, discuss their algebraic properties, and present necessary and sufficient conditions for their completeness that allow us to find such pools efficiently. We further show that, if the simulated problem possesses symmetries, then complete pools can fail to yield convergent results, unless the pool is chosen to obey certain symmetry rules. We demonstrate the performance of such symmetry-adapted complete pools by using them in classical simulations of ADAPT-VQE for several strongly correlated molecules. Our findings are relevant for any VQE that uses an ansatz based on Pauli strings.

15+10 pages, 7 figures

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• Towards Efficient Quantum Computing for Quantum Chemistry: Reducing Circuit Complexity with Transcorrelated and Adaptive Ansatz Techniques
• Shortcut to Chemically Accurate Quantum Computing via Density-based Basis-set Correction
• Non-Iterative Disentangled Unitary Coupled-Cluster based on Lie-algebraic structure
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• Mitigating the measurement overhead of ADAPT-VQE with optimised informationally complete generalised measurements
• Shot-Efficient ADAPT-VQE via Reused Pauli Measurements and Variance-Based Shot Allocation
• Hamiltonian-reconstruction distance as a success metric for the Variational Quantum Eigensolver