Resource-efficient Quantum Algorithms for Selected Hamiltonian Subspace Diagonalization

TL;DR

This paper introduces resource-efficient quantum algorithms for Hamiltonian subspace diagonalization, including a novel QSCI in the CIM framework, with improved qubit scaling and error mitigation, tested on molecules with promising results.

Contribution

The paper presents the first QSCI algorithm in the CIM framework, a new error mitigation technique, and an augmented QSCI method to match classical HCI performance.

Findings

Achieved similar accuracy to SQD with fewer quantum resources.

CIM-QSCI has optimal qubit scaling of $ ceil \log_2 (N_{CSF}) ceil$.

Augmented QSCI (QSHCI) performs comparably to classical HCI.

Abstract

Quantum algorithms for selecting a subspace of Hamiltonians to diagonalize have emerged as a promising alternative to variational algorithms in the NISQ era. So far, such algorithms, which include the quantum selected configuration interaction (QSCI) and sample-based quantum diagonalization (SQD), have been formulated in second-quantization in Fock space, which leads to inefficient usage of qubit resources. We introduce the first QSCI algorithm developed in the CI-matrix (CIM) framework, which is known to have optimal qubit scaling of exactly $\lceil \log_2 (N_{CSF}) \rceil$ where $N$ is the size of the CIM. In addition, we introduce a novel single-bit flip error mitigation which comes at the overhead of a single qubit and we combine this with a stochastic approximate Trotterization evolution adapted from qDRIFT. Simulating benchmark N$_2$ and naphthalene molecules on quantum hardware, our results achieved similar accuracy as SQD methods but with significantly less quantum resources. However, our CIM-QSCI algorithm and SQD methods could not match the performance of classical heat-bath CI (HCI) for the same task. Hence, we introduce an augmented version of QSCI called quantum selected heat-bath CI (QSHCI). This variant replaces classical heat-bath sampling with quantum sampling from QSCI to achieve performance comparable to HCI. We note that a current drawback of our approach is the preprocessing cost of $\mathcal{O}(N^2\log N)$ for constructing the CIM and performing the Pauli decomposition. This can be further improved by considering efficient CIM access models for the stochastic Trotter evolution.