Hybrid Quantum-Classical Eigensolver Without Variation or Parametric Gates

Abstract

The use of near-term quantum devices that lack quantum error correction, for addressing quantum chemistry and physics problems, requires hybrid quantum-classical algorithms and techniques. Here we present a process for obtaining the eigenspectrum of electronic quantum systems. This is achieved by projecting the Hamiltonian of a quantum system onto a limited effective Hilbert space specified by a set of computational basis. From this projection an effective Hamiltonian is obtained. Furthermore, a process for preparing short depth quantum circuits to measure the corresponding diagonal and off-diagonal terms of the effective Hamiltonian is given, whereby quantum entanglement and ancilla qubits are used. The effective Hamiltonian is then diagonalized on a classical computer using numerical algorithms in order to obtain the eigenvalues. The use case of this approach is demonstrated for ground-sate and excited states of BeH2 and LiH molecules, and the density of states, which agrees well with exact solutions.

Noisy intermediate-scale quantum algorithms

“…Proposed implementations Variational quantum eigensolver (VQE) and related solvers VQE (McClean et al, 2016;Peruzzo et al, 2014;Wecker et al, 2015) Adaptive VQE (Grimsley et al, 2019b;Ryabinkin et al, 2018b;Sim et al, 2020) IQAE (Bharti, 2020;) Krylov approaches (Huggins et al, 2020;Jouzdani and Bringuier, 2020;Stair et al, 2020) Imaginary time evolution McArdle et al, 2019a;Motta et al, 2020;Sun et al, 2020c) VQE for excited states Folded spectrum (Peruzzo et al, 2014;Ryabinkin et al, 2018a) Orthogonally constrained VQE (Higgott et al, 2019;Kottmann et al, 2020b; Quantum kernel methods (Havlíček et al, 2019;Kusumoto et al, 2019;Schuld et al, 2020b; Variational quantum classifiers (VQC) (Farhi and Neven, 2018;Lloyd et al, 2020;Mitarai et al, 2018;Pérez-Salinas et al, 2020a;Schuld et al, 2020a,c;Vidal and Theis, 2019) Encoding strategies in VQA (Cervera-Lierta et al, 2020; Quantum reservoir computing (Chien and Whitfield, 2020;Fujii and Nakajima, 2017;Ghosh et al, 2019;Mitarai et al, 2018;Nakajima et al, 2019;Negoro et al, 2018 Quantum metrology (Beckey et al, 2020;Kaubruegger et al, 2019;Koczor et al, 2020;Ma et al, 2020; Fidelity estimation Fidelity estimation Quantum error correction (QEC) Quantum variational error corrector (QVECTOR)…”