A polynomial-time perfect sampler for the Q-Ising with local fields (アルゴリズムと計算機科学の数理的基盤とその応用--RIMS研究集会報告集)

Victor Chulaevsky, Yuri Suhov

Soo-Jong Rey

I study the mixing time of quantum Gibbs samplers (Davies / Chen– Kastoryano–Gilyén generators) at low temperature. I prove a locality lower bound: any local, KMS–detailed-balanced sampler whose target has an order- parameter bottleneck has an Arrhenius-small spectral gap, by a quantum Cheeger argument, and hence mixing time eΩ(β∆F). The bound is a quantum Cheeger argument and holds for every KMS–detailed-balanced sampler. The coherence cannot lift an order-parameter bottleneck, thereby the obstruction survives quantization. It is not, however, intrinsic to the target. I give a 3-local, detailed-balanced Lindbladian on an enlarged spin+bond register whose spin marginal realizes the Fortuin–Kasteleyn cluster update and which, for commuting ferromagnets, has an inverse-polynomial gap at every temperature and prepares σ in poly(n)log(1/ε) gates. This gives an unconditional separation from the local-reversible class in which locality (not coherence) is the operative resource. As a conjecture, I isolate the genuinely quantum question about cluster updates for noncommuting targets, where no nonnegative bond representation exists and the cluster must be grown coherently.

Chinonso Onah, Kristel Michielsen

When does a noisy quantum sampler yield an end-to-end polynomial-time optimization algorithm with performance guarantees? Building on finite-depth and finite-shot guarantees for Constraint-Enhanced QAOA, we show that inverse-polynomial ideal probability on the optimal set, together with independent sampling, polynomial-time feasibility repair, and scoring, produces an exact-hit fully polynomial randomized approximation scheme, which we call an FPRASq. This guarantee survives device noise within an instance-dependent window. For effective circuit depth linear in the product of layer count and problem size, preserving an inverse-depth fraction of the ideal optimal mass increases the required shot complexity by one power of the problem size. Beyond this window, deterministic repair guarantees feasibility and provides an instance-dependent approximation guarantee whenever the induced objective inflation is controlled. The resulting NP-HQ algorithm fits the Chen-Cotler-Huang-Li oracle model. On any NP-hard kernel-admissible promise family, reproducing its inverse-polynomial optimal overlap with a polynomial-time classical sampler would imply that NP is contained in BPP, even with identical repair and perfect access to the constraint structure. Thus, the separation lies in generating the sampling distribution. We further introduce Heavy-Hitter QAOA, which preserves these conditional guarantees while reducing the retained candidate set and classical post-processing cost by one power of the problem size. Hardware experiments on IBM Eagle r3 processors cover instances with up to one hundred logical variables and match or improve every tested QOptlib reference tour.

Chih-Chieh Chen, Kodai Shiba, Masaru Sogabe +2 more

Jon Nelson, Joel Rajakumar, Chao Yin +2 more

Developing classical simulation algorithms for noisy quantum circuits is essential to delineating the limits of quantum advantage. Existing classical sampling approaches for general circuits require circuit depths to grow logarithmically with system size, so that noise drives the global output state close to a trivial state. Here we show that local accumulation of noise at a depth independent of system size is already sufficient. Specifically, we prove that any geometrically local circuit composed of unital operations and interspersed with single-qubit depolarizing noise of strength $p$ after $Θ(p^{-1} \log p^{-1})$ depth can be approximately sampled from a polynomial-time classical computer. This generalizes existing results that impose anticoncentration assumptions or non-universal gate sets in order to obtain a classical sampler at such shallow depths. Our proof maps the output state of the circuit to a polymer model in statistical mechanics, and combines a convergent cluster expansion with hypercontractivity of the depolarizing channel.

Jaron Kent-Dobias, James P. Sethna