Digital-analog variational neural-simulation hybrid eigensolver with classical shadow

Abstract Variational quantum eigensolvers (VQEs) are garnering significant attention for their ability to estimate the ground-state energy of target Hamiltonians. In particular, digital-analog quantum computation (DAQC) stands out as a crucial platform for VQEs in the noisy intermediate-scale quantum era, leveraging single-qubit operations and natural time evolution governed by parameterized Hamiltonians. On the other hand, the variational quantum-neural hybrid eigensolver is also a vital technique that enhances the accuracy of estimating expectation values, utilizing parameterized quantum circuits combined with neural networks. This hybrid approach typically requires the implementation of several controlled gates before measurements and post-processing. However, implementing controlled gates within the DAQC framework poses challenges. To address this, we propose a hybrid architecture that integrates DAQC with neural networks, substituting controlled gates with the classical shadow method. Our approach maintains a polynomial measurement cost as long as the number of qubits with Pauli X or Y operators in each term of the target Hamiltonian is constant or <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" overflow="scroll"> <mml:mi>O</mml:mi> <mml:mo stretchy="false">(</mml:mo> <mml:mi>log</mml:mi> <mml:msub> <mml:mrow> <mml:mi>N</mml:mi> </mml:mrow> <mml:mrow> <mml:mi mathvariant="normal">q</mml:mi> </mml:mrow> </mml:msub> <mml:mo stretchy="false">)</mml:mo> </mml:math> , where N q denotes the total number of qubits. We demonstrate the efficacy of our method through numerical simulations, applying it to estimate the ground-state energy of the H 2 molecule.