Qronecker: A Certifiable Kronecker Compression Primitive for Quantum-Chemistry Hamiltonians

TL;DR

Qronecker, a cut-aware low-rank Kronecker decomposition algorithm that turns Hamiltonian compression into a certifiable, resource-aware decision primitive, is introduced, positioning Qronecker as a certifiable compression primitive for rank and cut selection in quantum-chemistry Hamiltonian processing.

Abstract

Processing qubit Hamiltonians derived from electronic-structure problems can become classically prohibitive because many downstream manipulations still rely on dense operator constructions whose cost grows exponentially with qubit number. We introduce Qronecker, a cut-aware low-rank Kronecker decomposition algorithm that turns Hamiltonian compression into a certifiable, resource-aware decision primitive. Operating entirely in Pauli coefficient space, Qronecker avoids forming dense 2^n x 2^n matrices, constructs low-rank Kronecker approximations under a chosen bipartition, and returns both an instance-specific compressibility curve and a state-independent worst-case energy certificate that links rank and cut choices to conservative energy-deviation bounds. Across molecular benchmarks comprising hundreds of systems up to 30 qubits, we find that traceless low-rank structure is common but heterogeneous: many screened systems reach high coefficient-space fidelity at low rank, yielding large savings in classical preprocessing and conditional reductions in downstream circuit-resource proxies, while the certificate remains valid but conservative on the auditable subset. The same analysis shows that fixed global fidelity targets are not generally sufficient for chemistry-level guarantees, motivating adaptive rank and cut selection. These results position Qronecker as a certifiable compression primitive for rank and cut selection in quantum-chemistry Hamiltonian processing.

Figures (9)

• Figure 1: Boundary screening on the boundary-scan cohort. (A) Empirical CDFs of rank-1 capture for the total-space metric $\rho^{\mathrm{tot}}_1$, the traceless metric $\rho_{1,\mathrm{tr}}$, and the per-scan-series worst case $\min_R\rho_{1,\mathrm{tr}}$. (B) Dependence of $\rho^{\mathrm{tot}}_1$ and $\rho_{1,\mathrm{tr}}$ on the identity weight $w_I$, illustrating that the total-space metric is strongly confounded by identity contributions.
• Figure 2: Rank profiles on the performance-test cohort. (A) Cumulative captured energy $\rho_k$ with uncertainty bands. (B) Marginal gains $\Delta\rho_k$ on a log scale, showing rapid decay. (C) Fraction of systems reaching high-fidelity thresholds versus $k$, showing that most successful cases reach the target at low rank while a subset requires higher rank. (D) On the evaluated subset with ground-state errors, $\rho_k$ reaches a high-capture regime earlier than the observed energy error, motivating rank selection under explicit accuracy or certificate constraints.
• Figure 3: Resource benefits on the performance-test cohort at high-fidelity operating points and their size dependence. Subpanels report dense versus decomposed SVD time and storage at $k^{\star}_{0.999}$, circuit-depth ratios on the circuit-evaluation subset, and distribution comparisons across rank choices ($k=1$ and threshold-selected $k^{\star}$).
• Figure 4: Rank--accuracy--resource trade-off on the performance-test cohort. Panels show how classical speedup and memory saving decrease with rank, how effective circuit-depth saving changes on the circuit-evaluation subset, and how the residual indicator $\sqrt{1-\rho_k}$ improves. Trade-off curves summarize the median relationship between resource factors and this residual indicator across ranks.
• Figure 5: Certificate audit and deployment implications. (A) Observed ground-state errors versus the theoretical envelope demonstrate validity (all points lie below $y=x$). (B) Tightness ratio $\eta$ distribution quantifies conservatism. (C) Bound shrinkage with rank supports safe, auditable escalation. (D) Instance-adaptive chemical requirements show that worst-case chemical certification induces extremely strict residual targets that depend on $\|H_{\mathrm{tr}}\|_F$.