Convergence of sample-based quantum diagonalization on a variable-length cuprate chain

TL;DR

The paper investigates convergence bottlenecks in sample-based quantum diagonalization (SQD) applied to a variable-length cuprate spin chain modeled with a minimal two-band basis. By comparing UCJ and locally truncated LUCJ circuits, and by varying the operator expansion order $r$ and molecular orbital bases, the authors demonstrate strategies that overcome sampling plateaus and accelerate convergence toward chemical accuracy, including all-to-all connectivity and higher $r$. Hardware experiments on a Quantinuum H2-2 device reveal that error mitigation can further improve energy estimates beyond noise-free simulations, underscoring a beneficial interaction between hardware noise and quantum sampling. Collectively, the work suggests scalable SQD pathways for strongly entangled, highly correlated systems and provides actionable tradeoffs between connectivity, fidelity, and classical post-processing for near-term quantum devices.

Abstract

Sample-based quantum diagonalization (SQD) is an algorithm for hybrid quantum-classical molecular simulation that has been of broad interest for application with noisy intermediate scale quantum (NISQ) devices. However, SQD does not always converge on a practical timescale. Here, we explore scaling of the algorithm for a variable-length molecule made up of 2 to 6 copper oxide plaquettes with a minimal molecular orbital basis. The results demonstrate that enabling all-to-all connectivity, instituting a higher expansion order for the SQD algorithm, and adopting a non-Hartree-Fock molecular orbital basis can all play significant roles in overcoming sampling bottlenecks, though with tradeoffs that need to be weighed against the capabilities of quantum and classical hardware. Additionally, we find that noise on a real quantum computer, the Quantinuum H2 trapped ion device, can improve energy convergence beyond expectations based on noise-free statevector simulations.

Figures (7)

• Figure 1: Cuprate chain model system: (a) Diagram of the charge-neutral chain molecule used for Hatree-Fock calculations. (b) The simplified and extended cuprate chain with L=6 copper oxide plaquettes. (c) Charge density distribution in Cu $\text{3d}_{s^2-y^2}$ orbitals as a function of chain length.
• Figure 2: SQD versus system size: (a-c) Fraction of the ground state projecting onto SQD determinants as a function of shot number for (a) L=2, (b) L=4, and (c) L=6 plaquette length chains. (d-f) Blue bars show the full ground state decomposed by excitation number on the Hartree-Fock basis. The fraction accounted for within the three quantum-selected bases is shown in red, for shot numbers indicated with large dots in panels (a-c). All plots show expectation values rather than single-experiment outcomes, as described in Section III of the SM.
• Figure 3: UCJ operator expansion order: The performance of SQD with different orders of double-factorized operator expansion is shown for an L=6 site chain. The (a) number of determinants and (b) fraction of the ground state lost when projecting to these determinants are plotted against shot number. (c) A histogram of cp gate amplitudes, which represent Coulomb interaction terms. (d) Ground state energy predictions from SQD. Data from a 7000 shot run of the r=1 UCJ cirquit on Quantinuum H2 hardware are included for (higher energy) unmodified and (lower energy) error mitigated versions of the output. Emulated circuit results in panels (a,b,d) represent mult-run averages and expectation values rather than single-experiment outcomes, as described in Section III of the SM.
• Figure 4: Computational resources for chemical accuracy: (a, left) Energy error versus shot number for emulated UCJ on the L=4 chain with (light orange) kinetic orbitals, and (dark orange) HF+ orbitals is compared with (blue) ideal SQD with HF orbitals. (a, right) Energy error as a function of the number of unique Slater determinants. (b, top) Number of unique determinants in each orbital basis at chemical accuracy for length L=2, 4, and 6 chains with r=5 operator expansions. Ideal SQD is plotted with a larger point size for visibility. Black brackets indicate the vertical spacings expected for $L^{4.5}$ and $L^7$ growth rates, which roughly match gaps between the ideal SQD data points. (b, bottom) Shots required to reach chemical accuracy. Black brackets indicate vertical spacings expected for $L^5$ complexity, and are aligned with HF+ basis data points for ideal SQD. (c) The number of two qubit operations (XX+YY and CP gates) involved in each calculation.
• Figure S1: Quantum circuit for LUCJ. The quantum circuit for LUCJ on the L=2 chain with 8 spin-orbitals. A detailed tutorial for simulating UCJ and LUCJ circuits in qiskit can be found in Ref. UCJ_tutorial.