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arXiv:2610.29035

Optimization versus fair sampling in quantum annealing: An exactly characterized frustrated Ising benchmark

Masaru Hitomi, Masayuki Ohzeki

quant-phSubmitted Oct 21, 2026• Announced Oct 22, 2026• 0 citations
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Abstract

Reaching a degenerate ground-state manifold and sampling it uniformly are distinct properties of a quantum annealer. We quantify this distinction using an Ising ring with additional couplings between maximally separated sites. For antiferromagnetic benchmark couplings and $N \geq 5$, we derive the exact ground-state energy, degeneracy, and defect structure for each congruence class of $N$ modulo four. This classification separates ground-state hit probability from conditional sampling bias. We embed instances with up to 162 logical spins on a D-Wave Advantage processor using length-two chains and compare its output with simulated annealing on the embedded and logical problems. For a frustrated $N=83$ instance, the processor visits 164 of the 166 exact ground states, yet the conditional distributions over states and defect sectors remain nonuniform. For $N=160$, the processor has a larger ground-state hit probability than logical simulated annealing (0.869 versus 0.519), but also a larger total-variation distance from uniformity (0.317 versus 0.136). A pause during the anneal raises the ground-state hit probability from 0.42 to 0.86 without restoring uniform sampling. These results identify a relaxation window and demonstrate that optimization success and sampling quality need not improve together, providing an exactly characterized benchmark for open-system quantum annealing.

Comments
12 pages, 4 figures

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Reaching a degenerate ground-state manifold and sampling it uniformly are distinct properties of a quantum annealer. We quantify this distinction using an Ising ring with additional couplings between maximally separated sites. For antiferromagnetic benchmark couplings and N≥5N \geq 5N≥5, we derive the exact ground-state energy, degeneracy, and defect structure for each congruence class of NNN modulo four. This classification separates ground-state hit probability from conditional sampling bias. We embed instances with up to 162 logical spins on a D-Wave Advantage processor using length-two chains and compare its output with simulated annealing on the embedded and logical problems. For a frustrated N=83N=83N=83 instance, the processor visits 164 of the 166 exact ground states, yet the conditional distributions over states and defect sectors remain nonuniform. For N=160N=160N=160, the processor has a larger ground-state hit probability than logical simulated annealing (0.869 versus 0.519), but also a larger total-variation distance from uniformity (0.317 versus 0.136). A pause during the anneal raises the ground-state hit probability from 0.42 to 0.86 without restoring uniform sampling. These results identify a relaxation window and demonstrate that optimization success and sampling quality need not improve together, providing an exactly characterized benchmark for open-system quantum annealing.
Quantum annealing generally samples degenerate ground states with nonuniform probabilities. This unfair sampling depends on how the driver Hamiltonian couples these states near the end of annealing. For combinatorial optimization problems with a cardinality constraint, the penalty method and constrained quantum annealing (CQA) differ in these couplings. The transverse-field driver of the penalty method does not directly couple feasible ground states, whereas the fully connected XY driver of CQA couples pairs of them related by a single spin exchange. We investigate how this difference affects ground-state sampling, using closed-system simulations of random graph bipartitioning instances together with degenerate perturbation theory. For the instances studied, the conditional ground-state distribution of penalty-based quantum annealing approaches that of CQA at sufficiently large penalties and long annealing times. This convergence can be explained by the effective Hamiltonian of penalty-based quantum annealing, in which virtual single-spin flips to infeasible states generate a second-order interaction proportional to the CQA driver projected onto the feasible ground-state manifold, which is the leading term of the effective Hamiltonian of CQA. In the numerical simulations, the Jensen--Shannon divergence between the conditional ground-state distributions of penalty-based quantum annealing and CQA decreases approximately as the inverse square of the penalty-to-objective energy ratio, consistent with the perturbative analysis. This correspondence between the effective Hamiltonians extends to equal-weight fixed-magnetization constraints under the same perturbative conditions, providing a common mechanism for sampling bias in the two implementations.
We study the performance of the D-Wave 2X quantum annealing machine on systems with well-controlled ground-state degeneracy. While obtaining the ground state of a spin-glass benchmark instance represents a difficult task, the gold standard for any optimization algorithm or machine is to sample all solutions that minimize the Hamiltonian with more or less equal probability. Our results show that while naive transverse-field quantum annealing on the D-Wave 2X device can find the ground-state energy of the problems, it is not well suited in identifying all degenerate ground-state configurations associated with a particular instance. Even worse, some states are exponentially suppressed, in agreement with previous studies on toy model problems [New J. Phys. 11, 073021 (2009)NJOPFM1367-263010.1088/1367-2630/11/7/073021]. These results suggest that more complex driving Hamiltonians are needed in future quantum annealing machines to ensure a fair sampling of the ground-state manifold.
Quantum annealing (QA) is a promising approach for solving combinatorial optimization problems; however, it is known to exhibit unfair sampling, in which degenerate ground states are not sampled with equal probability even for sufficiently long annealing times. Fair sampling is important in applications such as solution diversity assessment and combinatorial counting, yet the mechanism of unfair sampling remains poorly understood, particularly in constrained combinatorial optimization problems. In this work, we investigate unfair sampling of QA in weighted graph bipartitioning problems (GBP), a representative constrained optimization problem. We study how the penalty coefficient in the penalty method affects sampling fairness. Through numerical simulations and experiments on the D-Wave Advantage2 system, we show that increasing the penalty coefficient reduces unfair sampling in a representative single instance, and that this qualitative behavior is also observed on actual hardware. A scaling analysis over randomly generated instances with up to 12 spins reveals that, while this trend does not hold universally, more than 70% of instances exhibit monotonically increasing sampling fairness as the penalty coefficient increases, even at the largest system size studied. These results show that increasing the penalty coefficient improves sampling fairness, though at the cost of ground-state probability under practical annealing conditions, and call for a deeper theoretical understanding of unfair sampling in constrained optimization problems.
In applications where multiple optimal solutions are needed, transverse-field quantum annealing (QA) is known to sample degenerate ground states in a strongly biased manner. Despite extensive empirical observations, it remains unclear which features of degenerate ground states are preferentially sampled and why by QA. Here we analyze the final states using degenerate perturbation theory to characterize the preference among them. In this analysis, the adjacency matrix of the graph composed by the ground states naturally emerges, and we can predict the eigenvector centralities (one of the node centralities) are related to the probabilities of these states. We verify this prediction on toy models where degeneracy is lifted at first and second order, and we show that second-order weights encode local barrier information, relating sampling fairness to the flatness of the local energy landscape. Finally, this perspective suggests two practical routes toward fair sampling -- promoting connectivity of the graph and reducing heterogeneity of centralities -- and we illustrate consistency with higher-order drivers and minor-embedding transformations.
Recent advances in quantum technologies have intensified the search for algo rithms that can use quantum resources effectively. Quantum annealing and hybrid quantum–classical variational algorithms are two prominent approaches in this set ting. Conventional quantum annealing follows a prescribed Hamiltonian interpo lation and may therefore be limited by small spectral gaps along that path. This Thesis investigates how algorithmic performance changes when three elements are varied: the annealing schedule (Chapter 2), the independent parameters of a digi tized evolution (Chapter 3), and the encoding of the problem (Chapter 4). For weighted MaxCut instances with anomalously small annealing gaps, opti mized smooth schedules outperform linear digitized annealing at equal circuit depth. Rather than restoring adiabaticity or enlarging the gap, the dynamics deliberately leaves the instantaneous ground state and later returns the population to the target state. Smoothness allows these digital solutions to retain their quality as continuous time controls and when transferred between related instances. The schedule restriction is then removed for a frustrated Ising ring whose anneal ing gap closes exponentially with system size. Optimizing all circuit angles inde pendently yields numerically exact ground-state preparation at the critical depth Pcr 1 =(N2−1)/4. This quadratic threshold is explained by the dimension and sym metries of the reachable free-fermion manifold, demonstrating that controllability, rather than the gap of a particular interpolation, determines the required number of digital controls. Finally, the same ring family is reformulated through the parity mapping. For the original frustrated ring, Parity-QAOA reaches the exact ground state using only single-qubit rotations, without applying the constraint unitary. The number of layers equals the number of distinct coupling values and does not grow with system size. To clarify the role of constraints, an equal-magnitude signed ring is also studied, for which the constraint unitary is essential for preparing the target ground state. Together, these results identify the shape, controllability, and encoding of a quan tum protocol as resources for overcoming limitations imposed by more restricted routes to ground-state preparation.
Sampling all ground states of a Hamiltonian with equal probability is a desired feature of a sampling algorithm, but recent studies indicate that common variants of transverse field quantum annealing sample the ground state subspace unfairly. In this note, we present perturbation theory arguments suggesting that this deficiency can be corrected by employing reverse annealing-inspired paths. We confirm that this conclusion holds in simulations of previously studied small instances with degeneracy, as well as larger instances on quantum annealing hardware.