Quantum Nested Ranking and Selection under a Fixed Budget

Mingjie HU, Jian-Qiang Hu, Tianxiang Wang

Hongyu Wu

Ordinal optimization selects designs that are good enough rather than optimal when evaluation is expensive and the budget is small. The alignment probability measures the chance that the selected set contains at least a prescribed number of truly good designs. We formulate quantum ordinal optimization as a class of problems that maximize this probability under a fixed oracle budget. We study exact, coherent pairwise comparisons with one selected design as a first instance, with the budget counted in comparison queries. For every classical comparison budget smaller than the number of designs, we prove that a knockout among uniformly sampled designs attains the optimum. The sample is drawn without replacement and contains one more design than the number of comparisons. For a uniformly random order, the selected design's rank has a polynomial probability law. At the same comparison budget, the maximum quantum degree is at most twice the classical maximum, a factor attained whenever there are more than twice as many designs as comparisons, so a quantum algorithm can realize rank laws that no classical algorithm with the same budget can. We determine the quantum optimum with one comparison for every number of designs and every good-enough set. For selecting the best of n designs, the optimal success probability with one quantum comparison is approximately 3.30/n for large n, compared with 2/n for one classical comparison. When the best design is sought, one quantum comparison gives a higher alignment probability than two classical comparisons whenever there are more than nine designs.

Francesco Vicari

Digital dynamical decoupling (DD) of an n-qubit register assigns each qubit a Walsh switching function. Static pairwise ZZ crosstalk is removed to first order by any set of distinct indices (orthogonal arrays). When Ising couplings fluctuate, the products of pair switching functions must also be filtered. Nested sequences achieve this but scale exponentially. We treat the index selection under a fixed slot budget (~4n) as a code-design problem. The infidelity is additive: a spectral cost per single-qubit function plus one per pair product. This cost ranks fidelities with Spearman ρ = 0.97–0.98. With static couplings, the maximal-sequency assignment reaches the computed optimum. With fluctuating couplings, the maximal-sequency sequence collapses (F = 0.302 vs 0.700 at n = 10, 64 slots). The optimum moves to a fast subspace closed under XOR. At fixed total time the code outperforms nested Uhrig decoupling (n = 5: F = 0.929 with 221 pulses versus 0.576 with 1023). In a 6-qubit density-matrix check with T1 and finite pulses it is the best design in all four conditions tested, with paired differences over the natural Hadamard order all above zero. Scripts and data: doi:10.5281/zenodo.23661132.

Yuan-Chieh Chen

In molecular variational quantum eigensolver (VQE) calculations for a fixed electron-number sector, the projected-sector energy alone does not certify that a prepared state is physically valid. We show that a state can have zero projected-sector error while almost all of its probability mass lies outside the target particle-number sector. We therefore evaluate ansatz families using three quantities: projected-sector error, leakage outside the physical sector and parameter count. Our noiseless experiments show that particle-preserving UCC-style families eliminate leakage induced by ansatze, but do not guarantee ground-state accuracy. In the parameter-budget comparison experiments, the one-shot ranking rules select identical generator sequences and yield identical errors. The adaptive hybrid procedure achieves lower mean projected-sector errors at selected budgets. Using established Kraus invariant-subspace criteria, we also express channel-induced leakage through a positive operator and discuss global and local depolarizing noise analytically. This gives a practical diagnostic framework for comparing ansatze beyond energy alone.

Qingle Wang, Yuancheng Li, Chaohang Yu +2 more

Vojtěch Novák, Ivan Zelinka, Swagatam Das +1 more

We benchmark classical optimization of a fixed low-depth Quantum Approximate Optimization Algorithm (QAOA) ansatz across four cost-Hamiltonian families at $N=12$, $p=3$, and $D=6$. Ten optimizers are compared over 25 independent runs under common ceilings of 10\,000 and 30\,000 function evaluations (FEs), first with exact statevector objectives and then with two additive observation-noise levels. Exact objectives favor multistart BFGS and multistart CMA-ES. Under noisy feedback, adaptive population methods become more competitive, but the ranking depends on whether performance is measured by the best exact point visited or by the point selected from noisy observations. A targeted extension over all three pre-screened instances per family confirms this regime change while showing that named adaptive-DE winners are instance dependent: jSO-lite leads low-noise oracle search, iL-SHADE high-noise oracle search, and L-SRTDE high-noise selected solutions in the equal-instance summaries. Bootstrap analysis quantifies a non-negligible high-noise search--selection gap, and a retrospective fixed-budget verification proxy shows that reserving a small measurement budget for final re-evaluation improves selected quality across all ten methods at 30\,000 FEs. A supplementary structure-aware study further shows that QAOA cross-depth restriction and continuous basin refinement are more useful in these conditions than standalone Monte Carlo tree-search selection. Overall, optimizer choice depends jointly on landscape structure, observation noise, and final-point identification.