Sample-Based Quantum Diagonalization

• Sample-Based Quantum Diagonalization is a hybrid method that uses a quantum circuit to sample key determinants and classical diagonalization to approximate the ground-state energy of many-body systems.
• The approach leverages parameterized circuits (like UCJ) and post-processing algorithms to manage exhaustive measurement costs and optimize convergence on noisy intermediate-scale quantum hardware.
• Empirical studies show that SQD scales steeply (up to L^7) and that careful basis engineering, circuit design, and error mitigation are critical for accurate and efficient performance.

Sample-Based Quantum Diagonalization (SQD) is a hybrid quantum–classical algorithm that leverages quantum processors as sampling engines to construct a selected configuration subspace, followed by classical diagonalization of the many-body Hamiltonian within this reduced subspace. SQD has emerged as a prominent approach for quantum chemistry and strongly correlated systems on noisy intermediate-scale quantum (NISQ) and early fault-tolerant hardware. Unlike expectation-value-based methods, SQD delegates the intensive problems of circuit optimization and measurement cost to classical post-processing, relying on hardware to efficiently identify highly weighted electronic configurations. However, the convergence and scalability of SQD depend critically on the interplay between circuit design, sampling complexity, orbital basis engineering, hardware connectivity, and noise characteristics (Wray et al., 4 Dec 2025). This article provides a detailed, methodology-focused review of SQD, its variants, and empirical convergence properties, with primary reference to the study of variable-length cuprate chains.

1. Algorithmic Structure and Mathematical Principles

SQD aims to approximate the ground-state energy and wavefunction of a molecular or lattice Hamiltonian $H$ by sampling a subset of Slater determinants (configurations) using a parameterized quantum circuit, and then diagonalizing $H$ in the span of these determinants (selected configuration interaction, or selected-CI). NISQ-compatible circuits such as the unitary cluster Jastrow (UCJ) enable sampling of determinants corresponding to states with high excitation order relative to a chosen reference (e.g., Hartree–Fock).

The SQD workflow is:

• Prepare a parameterized quantum circuit $\ket{\psi(\theta)}$ (e.g. UCJ or LUCJ, expansion order $r$), targeting $\ket{\psi(\theta)} \approx \ket{\psi_0}$.
• Execute the circuit for $N_{\text{shots}}$ measurements in the computational basis, obtaining a multiset of bitstrings, each representing a Slater determinant.
• Extract the unique determinants and construct a basis $\{\ket{i_1},...,\ket{i_M}\}$.
• Classically compute all matrix elements $\langle i_p | H | i_q \rangle$ and diagonalize the resulting $M\times M$ matrix (e.g., Davidson algorithm).
• Optionally update circuit parameters (analogous to VQE), or increase $N_{\text{shots}}$, until convergence is reached.

The variational subspace energy is

$$E_{\mathrm{CI}} = \sum_{p,q=1}^M c_p^* c_q \langle i_p | H | i_q \rangle,$$

where $\{c_p\}$ are the eigenvector components extracted from the CI diagonalization (Wray et al., 4 Dec 2025).

2. Ansatz Architecture and Series Expansion Order

The UCJ ansatz is central to SQD’s performance for correlated electron systems, especially when high-excitation configurations are important. The UCJ operator is constructed via a factorized exponential of two-body operators: $$U_{\mathrm{UCJ}}(\theta) = \prod_{k=1}^R \exp(i \theta_k J_k) \approx \prod_{k=1}^R \left(1 + i \theta_k J_k + \frac{(i\theta_k J_k)^2}{2!} + \cdots + \frac{(i\theta_k J_k)^r}{r!}\right),$$ where each $J_k$ is a two-body operator diagonal in a (possibly non-canonical) single-particle basis. The parameter $r$ controls the order of the expansion; increasing $r$ systematically reduces the ansatz error. Empirically, raising $r$ from 1 to 5 can reduce the required shot count for a given accuracy by up to two orders of magnitude (Wray et al., 4 Dec 2025).

Error from truncating the order scales approximately as $$\|\exp(-i\theta J)-U_r\| \sim \mathcal{O}(\|\theta J\|^{r+1}/(r+1)!)$$. The trade-off is that circuit depth grows linearly with $r$.

3. Hardware Connectivity: All-to-All vs. Limited Circuits

• UCJ (All-to-All Connectivity): All controlled-phase (CP) gates are retained, allowing all two-body interactions and rapid convergence, at the expense of increased two-qubit gate count ($N_{\rm op} \propto L^{2.2}$, where $L$ is the number of fragments or orbitals).
• LUCJ (Limited Connectivity): CP gates are pruned to enforce device connectivity (e.g., honeycomb for IBM hardware), usually dropping weaker interaction terms (~20–25%), which leads to a further slowdown in shot convergence.

Numerically, for a $L=6$ cuprate chain system, convergence to 90% ground-state coverage requires:

• Ideal SQD: $N_{\text{shots}} \sim 10^4$
• UCJ ($r=1$): $N_{\text{shots}} \sim 10^7$
• LUCJ ($r=1$): $N_{\text{shots}}$ increases by 1–2 orders of magnitude beyond UCJ (Wray et al., 4 Dec 2025).

Shot complexity for ideal SQD empirically scales as $L^5$, and factoring in gate count leads to an overall scaling of $L^7$.

4. Molecular Orbital Basis Engineering and Sampling Bottlenecks

Choice of orbital basis is crucial for balancing sampling efficiency against the classical cost of diagonalization:

• HF (Hartree–Fock) basis: Reference single-particle eigenstates.
• Kin basis: Eigenstates of the kinetic energy operator only.
• HF+: An “over-corrected” mean-field basis obtained by diagonalizing $2M-\mathrm{diag}(M)$, interpolating between HF and Kin in terms of compactness and convergence.

Benchmarks demonstrate that the Kin basis achieves fastest shot convergence for ideal SQD, but produces the largest determinant subspace, inflating classical diagonalization cost. The HF+ basis provides a sweet spot, offering reduced shot requirements relative to HF, yet with only moderate increases in the number of determinants.

Empirical scaling for number of unique determinants (for ideal SQD) is between $L^{4.5}$ and $L^{7}$; for HF+ and UCJ, the basis size and shot number both remain manageable up to moderately large $L$ (Wray et al., 4 Dec 2025).

5. Numerical Convergence, Bottlenecks, and Scaling Analysis

Shot Convergence and Excitation Structure

• For small chains ($L=2$), all protocols achieve rapid, monotonic convergence; for larger systems ($L=4,6$), UCJ/LUCJ exhibit plateau behavior, with UCJ ($r=1$) stalling at a 20% missing weight for $10^6$ shots.
• High-excitation-number sectors ($n_{\text{ex}}>2$) are progressively harder to sample with low-order ansätze, contributing significant weight ($\gtrsim10\%$) in moderate chains.

Scaling Summary

Protocol	Determinant Scaling	Shot Scaling	Notes
Ideal SQD	$L^{4.5}$–$L^7$	$\propto L^5$	No device encoding
UCJ/LUCJ ($r=1$)	Slightly higher	$1$–$2$ orders slower	Limited connectivity slows further

No strict low-order polynomial exists for either basis size or shot complexity, reflecting inherent many-body complexity. Scaling is ultimately bottlenecked by the requirement to sample all significant determinants.

6. Device Experiments and Noise-Assisted Sampling

Experiments on the Quantinuum H2 trapped-ion device (up to 48 qubits) use UCJ with $r=1,2$ and implement error-mitigation by flipping bits to correct spin-sector populations. Data indicate:

• $\sim35\%$ of raw bitstrings are correct for electron count; error mitigation increases the effective set of unique error-free determinants by nearly $8\times$.
• Mitigated SQD outperforms noise-free emulator for equivalent shot count and expansion order, attributing to beneficial sampling-noise “diversity.”

Noise-assisted sampling thus sometimes offers faster convergence than idealized, noiseless circuits, as device imperfections can actually enhance determinant diversity within the subspace (Wray et al., 4 Dec 2025).

7. Practical Guidelines and Outlook for NISQ SQD

Several guidelines are distilled from comprehensive numerical and hardware analysis:

• Use moderate UCJ expansion order ($r=4–6$) on all-to-all devices for systems with $L\le8$, combining with the HF+ basis to minimize shot count and maintain modest determinant subspace sizes.
• On limited-connectivity hardware, employ LUCJ ($r\ge3$) and aggressive error mitigation to compensate for connectivity bottlenecks.
• Monitor excitation-number convergence to ensure high-order correlation effects are sampled.
• Aggressive post-selection and bit-flip correction are recommended error-mitigation strategies.

SQD achieves competitive $L^7$ scaling for moderately correlated chains, with dominant costs shifting from quantum hardware (shots, circuit depth) to classical post-processing (diagonalization of large subspaces). For even larger systems (more than a handful of correlated fragments), exponential growth in determinant support will ultimately bound scalability, motivating research into interleaved sampling and on-the-fly compression (Wray et al., 4 Dec 2025).

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References:

• "Convergence of sample-based quantum diagonalization on a variable-length cuprate chain" (Wray et al., 4 Dec 2025)