Sample-based Quantum Diagonalization (SQD)

• SQD is a hybrid quantum–classical technique that samples computational basis states and constructs a subspace for efficient Hamiltonian diagonalization.
• It leverages symmetry correction and Pauli grouping to reduce measurement overhead, achieving sub-milliHartree accuracy in accessing ground and excited states.
• The method scales to tens of qubits and large active spaces, making it competitive with classical configuration interaction approaches.

Sample-based Quantum Diagonalization (SQD) is a hybrid quantum–classical paradigm that systematically combines quantum sampling of computational-basis configurations with classical subspace diagonalization. Unlike conventional quantum algorithms that directly optimize or measure energy expectation values (e.g., VQE), SQD leverages the ability of near-term quantum devices to efficiently draw samples from highly correlated wavefunctions in large active spaces and transfers the burden of Hamiltonian diagonalization to classical post-processing. The method provides direct access to ground and low-lying excited states, achieves accuracy competitive with classical configuration interaction (CI) approaches, and is inherently well-suited to the architectural limitations of noisy intermediate-scale quantum (NISQ) devices (Hossain et al., 17 Sep 2025).

1. Mathematical Framework and Algorithmic Structure

At its core, SQD targets an $n$-qubit molecular Hamiltonian written in Pauli operator decomposition,

$$\hat H = \sum_i c_i P_i,\quad P_i\in\{I,X,Y,Z\}^{\otimes n}.$$

The protocol proceeds as follows:

• Quantum Sample Generation: A classical reference or ansatz wavefunction $|\psi\rangle$ (e.g., LUCJ or UCCSD) is prepared and measured in the computational ($Z$) basis. $M$ raw bitstrings $\{x\}$ are obtained, approximately distributed as $p(x) = |\langle x|\psi\rangle|^2$.
• Symmetry Correction: Each measured $x$ is checked for quantum number validity (particle number, $S_z$, point-group irreps). Violations are corrected (“tapered”) by bit flips or excluded, yielding a filtered pool $\{\tilde x\}$.
• Subspace Construction: The $N_s$ most probable, distinct bitstrings $\{\ket{x_i}\}$ are retained to span the working subspace $S = \mathrm{span}(\ket{x_i})$.
• Hamiltonian Matrix Elements: For each $(i,j)$, matrix elements are computed as

$$(H_S)_{ij} = \langle x_i|\hat H|x_j\rangle = \sum_k c_k\,\langle x_i|P_k|x_j\rangle,$$

with nonzero contributions only when $|x_j\rangle = P_k|x_i\rangle$.

• Generalized Eigenproblem: The subspace Hamiltonian is classically diagonalized:

$H_S \mathbf{c} = E S \mathbf{c},$

where $$S_{ij} = \langle x_i|x_j\rangle \approx \delta_{ij}$$ due to orthogonality.

• Self-consistency and Iteration: Optionally, the ansatz or the subspace itself is updated based on the leading eigenvector and the process is iterated until convergence in energy $E$.
• Excited States: Higher eigenroots provide immediate access to excited-state energies and wavefunctions within the constructed subspace.

This algorithmic cycle is defined such that $N_s \ll 2^n$, enabling tractable diagonalization for active spaces previously unreachable by exact or classical methods (Hossain et al., 17 Sep 2025).

2. Measurement Reduction, Symmetry Tapering, and Sampling Strategy

To minimize both quantum circuit depth and measurement overhead:

• Symmetry Tapering: Known $\mathbb{Z}_2$ symmetries (e.g., total parity, spin-parity) are used to project out redundant degrees of freedom, decreasing the effective number of qubits and reducing the number of required Pauli terms in $\hat H$.
• Pauli Grouping: Commuting Pauli terms are aggregated into groups $G \ll N_p$ (“Pauli grouping”), allowing simultaneous measurement in a single basis and drastically decreasing the number of measurement circuits.
• Sampling Requirements and Scaling: For $n$ qubits and subspace size $N_s$, total shots $M \sim \sum_{i,j=1}^{N_s} M_{ij}$ are necessary to estimate all relevant matrix elements. For target precision $\epsilon$, $M_{ij}\sim O(1/\epsilon^2)$, leading to $M_\text{tot}\sim O(N_s^2/\epsilon^2)$. The dominant classical cost is matrix diagonalization, $O(N_s^3)$, but for electronic structure scenarios, sparsity of the ground state in the computational basis means $N_s\sim \mathrm{poly}(n)$ suffices for chemical accuracy (Hossain et al., 17 Sep 2025).

This architecture naturally mitigates measurement noise by batch-wise redundancy and exploits classical error suppression at the matrix-construction and overlap-renormalization stages.

3. Implementation Workflow and Pseudocode

A NISQ-oriented implementation of the SQD protocol involves the following concrete sequence:

1. Select ansatz $U(\theta)$ and initialize parameters $\theta$.
2. Prepare $|\psi\rangle = U(\theta)|0\rangle^{\otimes n}$.
3. Measure in the computational basis, collecting $M$ raw bitstrings.
4. Apply symmetry correction, yielding valid $\{\tilde x\}$.
5. Select $N_s$ highest-probability bitstrings to define the subspace.
6. For each $(i,j)$, group Pauli strings connecting $\ket{x_i}$ and $\ket{x_j}$; measure appropriate circuits to estimate $(H_S)_{ij}$.
7. Form $H_S$ and overlap $S$; solve the generalized eigenproblem.
8. If convergence in $E$ or subspace norm is not reached, update the ansatz or add further configurations, returning to step 2.
9. Output the full set of eigenvalues $\{E_p\}$ and eigenvectors $\{\mathbf{c}^{(p)}\}$.

All mitigation steps (“symmetry verification,” “overlap renormalization” via the Gram matrix $S$, bootstrap error estimation) are performed in post-processing and do not demand additional quantum resources or ancilla qubits (Hossain et al., 17 Sep 2025).

4. Accuracy, Scaling, and Benchmark Results

SQD achieves chemical accuracy on a broad class of realistic active-space Hamiltonians where standard VQE or FCI approaches are infeasible due to exponential scaling:

• In simulations of LiPF$_6$, NaPF$_6$, LiFSI, and NaFSI salts with $n$ up to 32 qubits, subspaces with a few hundred determinants yielded sub-milliHartree agreement with exact CASCI (subspace-FCI) references. For example,

| Method | LiPF$_6$ | NaPF$_6$ | LiFSI | NaFSI | |-----------------|---------------|---------------|---------------|---------------| | Exact (CASCI) | –945.081427 | –(1099.48501)5 | –(1355.14883)4 | –(1509.55578)0 | | SQD (32 qubits) | –945.081406<br>(0.021 mHa) | –(1099.48500)4<br>(0.011 mHa) | –(1355.14760)7<br>(1.227 mHa) | –(1509.55575)6<br>(0.024 mHa) |

By contrast, VQE–UCCSD on reduced 10-qubit active spaces gave errors up to 2.4 mHa (LiFSI) (Hossain et al., 17 Sep 2025).

• Multiple excited states are extracted in a single diagonalization, in contrast to VQE–qEOM which requires additional variational layers for each root.

Such results confirm that SQD can be scaled to tens of qubits and active spaces with million-plus determinants using only shallow circuits, moderate sampling, and classical post-processing (Hossain et al., 17 Sep 2025).

5. Error Mitigation and Symmetry Restoration

Error mitigation in SQD leverages post-measurement techniques:

• Symmetry Verification and Correction: Bitstrings violating particle number or spin projections are rejected or minimally corrected (“tapered” or bit-flipped) to enforce physical subspace constraints.
• Overlap Renormalization: The measured Gram (overlap) matrix $S$ is used to orthonormalize the sampled subspace prior to diagonalization, correcting for any inadvertent redundancy or near-linearity among sampled determinants.
• Bootstrap Resampling: Statistical error estimates on $E$ are obtained by resampling measurement batches, providing reliable error bars without additional quantum effort.

No explicit tomography or ancillary-qubit protocols are required for effective error suppression and recovery; all corrections are “zero-cost” in the sense of not increasing circuit depth or quantum shot count (Hossain et al., 17 Sep 2025).

6. Distinction from VQE and Other Quantum Algorithms

SQD fundamentally differs from the Variational Quantum Eigensolver and other quantum-classical algorithms:

• Parallelization: Rather than optimizing a single variational energy expectation value, SQD samples and diagonalizes an entire subspace, extracting multiple eigenstates without variational updates for each.
• Measurement Overhead: Standard VQE requires measuring hundreds to thousands of non-commuting Pauli terms per ansatz parameter set; SQD uses symmetry-informed grouping and subspace projection, which significantly lowers this cost.
• Ground and Excited States: SQD’s classical diagonalization step directly yields low-lying excited states, in contrast to the sequential or iterative approaches required by VQE–qEOM.

The method is thus systematic, transparent, and ideally fits the hybrid capacity of NISQ-era hardware, remaining within feasible depths and readout windows (Hossain et al., 17 Sep 2025).

7. Limitations, Generalizations, and Future Prospects

While demonstrating high accuracy and resource efficiency, several points are noted:

• Subspace Construction: For systems with extremely delocalized or strongly correlated ground states, the number of sampled determinants needed to achieve completeness may grow rapidly, potentially stressing classical resources.
• Classical Bottleneck: Although $N_s \ll 2^n$, classical diagonalization becomes the limiting factor as subspace size increases, especially for excited-state spectroscopy or larger materials (Hossain et al., 17 Sep 2025).
• Extensions: The methodology can incorporate advances such as adaptive sampling, more sophisticated ansätze, or embedded fragmentation (e.g., DMET), further enhancing efficiency. Extensions to lattice models, excited-state algorithms, and quantum-centric embedding frameworks are natural avenues.
• Experimental Feasibility: All mitigation and correction steps are device-agnostic, and conjugate measurement circuits can be designed for different hardware topologies, further promoting practical scalability (Hossain et al., 17 Sep 2025).

Ongoing developments include incorporating more aggressive error mitigation, self-adaptive subspace selection, and embedding SQD as a quantum subsolver within multi-scale quantum–classical workflows.

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References

• Quantum Simulations of Battery Electrolytes with VQE-qEOM and SQD: Active-Space Design, Dissociation, and Excited States of LiPF$_6$, NaPF$_6$, and FSI Salts (Hossain et al., 17 Sep 2025)