Filter-assisted quantum subspace diagonalization via wavefunction sparsity engineering

Abstract: Subspace diagonalization techniques based on quantum sampling, such as quantum selected configuration interaction (QSCI) and sample-based quantum diagonalization (SQD), have recently emerged as promising quantum-centric approaches for approximating ground-state energies of many-body systems. However, their performance is fundamentally limited by an intrinsic trade-off between sampling efficiency and the sparsity of the ground-state wavefunction, which becomes particularly severe in strongly correlated systems. Here, we introduce a filter-assisted SQD protocol that engineers wavefunction sparsity via a quantum filter, i.e., a unitary transformation of the Hamiltonian designed to concentrate the ground-state weight onto a small number of computational basis states. Using the Gini coefficient as a robust sparsity measure, we establish a quantitative relationship between wavefunction sparsity and the resource requirements of SQD, providing theoretical bounds on the required subspace dimension and sampling cost. To realize the quantum filter, we employ a tensor-network-based circuit-encoding algorithm that maps target states to quantum circuits with controllable fidelity. We benchmark our approach on the quantum Ising model with transverse and longitudinal fields using both numerical simulations and quantum hardware experiments. Our results demonstrate that, compared with standard SQD, the proposed protocol significantly enhances wavefunction sparsity, reduces ground-state energy estimation errors by orders of magnitude, and substantially lowers sampling overhead. These findings establish filter-assisted subspace diagonalization as a powerful and scalable framework for quantum many-body calculations in the strongly correlated regime.

• The paper introduces a filter-assisted SQD protocol that uses quantum filtering to engineer ground-state sparsity and lower computational resources.
• It employs MPS-based circuit encoding to optimize local two-qubit unitaries, achieving near-maximum sparsity and polynomial resource scaling.
• Experimental benchmarks on quantum hardware and the quantum Ising model validate enhanced sampling efficiency and sub-microhartree energy accuracy.

Filter-Assisted Quantum Subspace Diagonalization via Wavefunction Sparsity Engineering

Introduction and Motivation

Subspace diagonalization methods enable quantum-classical hybrid algorithms to approximate ground-state energies in many-body physics, leveraging quantum state sampling to construct effective truncated Hilbert spaces. Conventional sample-based approaches such as SQD (sample-based quantum diagonalization) and QSCI (Quantum Selected Configuration Interaction) are fundamentally limited by the trade-off between wavefunction sparsity and sampling efficiency, especially in strongly correlated regimes. This work introduces a filter-assisted SQD (FSQD) protocol, which engineers ground-state sparsity via quantum filters, transforming the Hamiltonian such that its ground state becomes strongly concentrated in the computational basis, thus reducing resource requirements.

Figure 1: Schematic workflow comparison: standard SQD versus filter-assisted SQD, showing the role of a quantum filter circuit to enhance sparsity.

Quantitative Wavefunction Sparsity and Resource Scaling

Wavefunction sparsity is formally quantified by the Lorenz curve $\mathcal{L}(x)$ and the Gini coefficient $G$, capturing statistical concentration in measurement distributions across computational basis states. The authors derive explicit bounds relating the required sampled subspace dimension $N_R$ and measurement shots $N_S$ to $\mathcal{L}(x)$ and $G$. Specifically, in standard SQD, exponential or power-law scaling in $(1-G)N$ leads to intractable resource overhead as system size $n$ increases. However, by applying a quantum filter, FSQD reshapes the sampling distribution, ideally producing a system-size independent $G$ and thus polynomial scaling in resources.

Figure 2: Lorenz curve visualization demonstrating the graphical interpretation of wavefunction sparsity and Gini coefficient.

Figure 3: System-size scaling of $1-G$, with filter circuits approaching theoretical maximum sparsity as $G$0 increases.

MPS-Based Circuit Encoding for Quantum Filtering

The quantum filter is realized via classical tensor-network (specifically MPS) circuit encoding: an approximate ground state is computed classically and mapped to a shallow quantum circuit comprised of local two-qubit unitaries. The encoding algorithm alternates forward and backward sweeps to optimize each local unitary, maximizing overlap with the target MPS.

Figure 4: Diagram of the circuit-encoding bidirectional optimization sweeps over two-qubit local unitaries.

Figure 5: Optimization trajectories of infidelity per site for MPS-based circuit encoding, varying layers and system sizes.

Numerical Benchmarks: Quantum Ising Model

Benchmarking is performed on the quantum Ising chain with transverse and longitudinal fields. Classical simulations verify that, after filtering, the ground state is highly concentrated on the $G$1 computational basis state—enabling reduced sampled-subspace dimension and shot count at fixed target accuracy. The filtering effect is shown to be layer-dependent: more circuit layers achieve higher sparsity, pushing $G$2 to near-unity and lowering the scaling exponent $G$3 in $G$4.

Figure 6: Hamiltonian matrices before and after filtering, showing dominance of the $G$5 sector.

Figure 7: Energy estimation error $G$6 versus number of shots $G$7, contrasting standard SQD and FSQD.

FSQD yields estimation errors several orders of magnitude smaller than SQD at comparable $G$8, with decay exponents $G$9 (in $N_R$0) remaining large and nearly independent of $N_R$1. Energy-variance extrapolation further improves accuracy for filtered samplers, attaining sub-microhartree error for $N_R$2.

Figure 8: Energy-estimation error versus energy variance, supporting variance extrapolation for improved estimation.

Experimental Demonstration on Quantum Hardware

FSQD and SQD are implemented on IBM Quantum's Heron R2 processor. Experimental samplers include both directly filtered, sequentially projected, and directly circuit-encoded projected states. The projected sampler constructions consistently outperform standard SQD, though hardware noise and imperfect sampler preparation degrade the filter advantage at large $N_R$3. Bitstring recovery rates and the fraction of target bitstrings indicate that hardware noise broadens the sampling distribution, but does not fundamentally alter FSQD's scaling advantage.

Figure 9: Hardware experimental results for 20, 50, 100 qubits comparing four sampler constructions, showing sampling efficiency and error scaling.

Implications and Future Directions

FSQD fundamentally alters the scaling bottleneck in sample-based quantum subspace diagonalization by transforming the sampling distribution, rather than relying on aggressive truncation strategies. Practically, this enables scalable quantum many-body calculations (e.g., spin systems, quantum chemistry) in regimes where conventional SQD is exponentially costly. Theoretically, the protocol connects quantum state engineering (via tensor-network encoding and filtering) directly to algorithmic resource requirements, suggesting new hybrid architectures for state preparation and error mitigation.

Future research directions include improved filter circuit ans\"atze, symmetrization and adaptive filtering for quantum chemistry, integration with classical post-processing (e.g., configuration recovery), and hardware-specific optimizations to combat noise-induced broadening.

Conclusion

Filter-assisted quantum subspace diagonalization via wavefunction sparsity engineering presents a scalable framework for quantum many-body ground-state energy estimation, validated by both classical and quantum hardware benchmarks. By tailoring the sampling distribution through classical-quantum hybrid tensor-network encoding, FSQD achieves superior sampling efficiency, weak system-size dependence, and robust accuracy in strongly correlated regimes, with clear implications for future quantum hardware and algorithm design.