Resource-efficient Quantum Algorithms for Selected Hamiltonian Subspace Diagonalization

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.

Figures (5)

• Figure 1: Overview of the QSCI algorithms based on the CIM framework for diagonalising a large general $N \times N$ matrix. For molecular chemistry problems, this matrix corresponds to the full CI-matrix.
• Figure 2: Simulated potential energy curve of N$_2$ molecule (upper panel) and absolute error of the methods (lower panel) relative to exact diagonalization for different bond lengths. The results of our CIM-QSCI are shown as brown, green, and purple solid lines with circles, corresponding to 40%, 60%, and 80% of the subspace Hamiltonian, respectively. These results are compared to CCSD, LUCJ-SQD with 2 LUCJ repetitions (on emulator), and exact diagonalization computed using a Lanczos method. All error curves (bottom panel) are plotted against chemical accuracy (0.0016 Hartree) shown as a blue dotted line.
• Figure 3: Absolute error of simulated N$_2$ ground state energy relative to exact diagonalization (upper panel) and resulting sizes of subspace Hamiltonian (lower panel) for different bond lengths. The results of our CIM-QSCI using a subspace Hamiltonian of 80% are shown as a solid purple line with circles; QSHCI with a variance factor of 1.0 is shown as the red solid line with circles; and QSHCI with a variance factor of 100 is shown as the red dashed line with circles. These results are compared with HCI using tolerances of $5\times10^{-5}$ and $5\times10^{-3}$ (Subfig. \ref{['fig:fig1-CIM_QSHCI_N2_Results']}) and $8\times10^{-4}$ (Subfig. \ref{['fig:fig2-CIM_QSHCI_N2_Results']}). All error curves (top panel) are plotted against the chemical accuracy (0.0016 Hartree), indicated by a blue dotted line.
• Figure 4: Absolute error of simulated ground state energy of Naphthalene computed using several methods. The CIM-QSCI: red, orange, and brown circles with solid lines, CIM-QSHCI: red and brown circles with dot-dashed lines, using LUCJ-SQD with 2 LUCJ repetitions: cyan, using LUCJ-SQD with 1 LUCJ repetition: blue triangles, using SqDRIFT: green triangles. The $n_a$ and $r$ indicate the number of Hamiltonian terms and repetitions used in the evolution. The superscript $\dagger$ indicates data from piccinelli_quantum_2025. The X-axis shows the size of the subspace Hamiltonian as a percentage of the full Hamiltonian size. These quantum algorithms are compared with the classical CCSD and HCI methods. Chemical accuracy (0.0016 Hartree) is shown as the blue dotted line.
• Figure 5: Error of N$_2$ ground state energy for different bond lengths using the (10,10) active space. CIM-QSCI results are shown in green using 60 % of the subspace Hamiltonian. The degree of the approximate-transpiliation was varied as: 1.00; solid line with circles, 0.75; dashed line with crosses, 0.50; dotted line with triangles. Chemical accuracy (0.0016 Hartree) is shown as a blue dotted line.