Linear-time classical approximate optimization of cubic-lattice classical spin glasses: upper bounds on optimality gaps of quantum speedups

TL;DR

The paper addresses quantum speedups for approximate optimization of cubic-lattice Ising spin glasses by defining TT$\varepsilon$ (time-to-ε) with $\varepsilon=(E-E_{gs})/|E_{gs}|$ and proposing a linear-time subsystem-optimization meta-heuristic that yields an upper bound $\varepsilon_{lin}$ on the optimality gap where quantum speedups might arise. It implements a tensor-network-based subsystem optimizer that contracts only local fragments to compute exact marginals, enabling linear-time scaling in the number of spins. In the cubic-lattice tile-planting $F_6$ class, the method indicates $\varepsilon_{lin}$ around $3\%$ asymptotically (with finite-size data near $7.5\%$ at $L\approx30$), and shows that simulated annealing and parallel tempering with isoenergetic cluster moves exhibit superlinear scaling for $\varepsilon>\varepsilon_{lin}$, thereby constraining the plausible quantum-speedup region to $0\leq\varepsilon\leq\varepsilon_{lin}$. The work provides a practical, scalable bound on the quantum-speedup search space and motivates fixed-scaling classical heuristics and tensor-network-based subsystem optimizers for spin-glass problems, with implications for hardware acceleration strategies.

Abstract

Demonstrating quantum speedup for approximate optimization of classical spin glasses is of current interest. Such a demonstration must be done with respect to the best-known scaling of classical heuristics at a given optimality gap of a given problem. For cubic-lattice classical Ising spin glasses, recent theoretical and experimental developments open the possibility of showing quantum speedup for approximate optimization with quantum annealing. It is therefore desirable to understand the optimality-gap range over which such a speedup should be searched for. Here we show that on cubic-lattice tile-planting models, classical meta-heuristics that are linear-time by construction can reach optimality gaps at which simulated annealing and parallel tempering exhibit super-linear scaling. This implies that the optimality gaps achieved by linear-time classical meta-heuristics can serve as useful upper bounds for the optimality-gap range over which quantum speedups in approximate optimization should be searched for. We also explain how classical heuristics with fixed scaling that is beyond-cubic can provide upper bounds to optimality-gap ranges for beyond-quadratic quantum speedups in approximate optimization. These results encourage the development of classical heuristics with fixed scaling that achieve optimality gaps as small as possible.

Figures (8)

• Figure 1: (color online). Hypothetical example of polynomial scaling exponent ($\alpha$) vs. optimality gap for quantum and classical approximate optimization algorithms. As illustrated, in some cases $\alpha$ of a quantum algorithm compared to a classical algorithm can be inferior at smaller optimality gaps ($\varepsilon$) but superior at larger optimality gaps. This was found to be the case in the experiments in Ref. bauza2025scaling, for example. Larger values of $\varepsilon$ can therefore be relevant in the search for quantum speedup, and ruling them out with a fixed-scaling heuristic is therefore useful.
• Figure 2: (color online). (a) A large disc (blue) with two legs represents a two-index tensor of Boltzmann weights. A small disc (black) with $n$ legs represents an $n$-index kronecker delta function. The index dimensions are equal to the number of possible single-spin configurations (in this case two, corresponding to Ising spins). (b) Tensor network representation of the partition function for a square-lattice classical Ising model. The delta functions are located at the sites of the spin lattice; they can alternatively be treated as hyperindices. The joining of legs from different tensors represents a contraction of the tensors along a common (hyper) index. The contraction of the entire network yields the partition function.
• Figure 3: (color online). Local energy minimization bootstrapping procedure. (a) Adding an open leg to a single black disc turns it into a kronecker delta function such that the contraction of the entire network yields a vector that is the (unnormalized) unconditional marginal for the corresponding spin. In the present algorithm, the approximation is made to compute the marginals by contracting only a local fragment, defined by the tensors within the fuzzy (green) rounded-square boundary (tensors with legs that cross the boundary are not included in the fragment). The spin is decimated by choosing its most probable configuration according to this approximate marginal. (b) The decimation is graphically denoted by legs with dashed lines and an up or down arrow (denoting $+1$ or $-1$). The decimation is internally accomplished by selecting the appropriate value of the corresponding index. (c) Adding an open leg to a different black disc after decimating previous spins yields the (unnormalized) marginal for the corresponding different spin. The marginal for this spin is conditional upon the configuration of the previously decimated spins that lie within the fragment. If $\beta$ is sufficiently large, decimating spins in this manner results in an approximate local energy minimization. (d) Performing sequential decimations by overlapping the fragment with at least some of the previously decimated spins yields a bootstrapping of approximate local energy minimizations over the whole lattice. Using multiple fragments simultaneously (not shown) yields parallelization.
• Figure 4: (color online). Cubic-lattice tile-planting model (linear size $L=30$), classes gallus_26 (blue, discs) and gallus_46 (green, triangles), linear-time meta-heuristic ($l=5$ and $\beta=2$): optimality gap vs. $p_6$. Twenty instances at each $p_6$. While Ref. hamze2018near confirms monotonically increasing MCMC hardness (for exact optimization) with increasing $p_6$ between $p_6=0.8$ and $p_6=1$ (red shaded region), the error of our linear-time heuristic monotonically decreases.
• Figure 5: (color online). Cubic-lattice tile-planting model, $F_6$ class (linear size $L$), linear-time meta-heuristic: optimality gap vs. linear system size ($L$) at $\beta=2$. Data computed with cubic fragments of linear size $l=5$ spins. Twenty instances at each $L$. The fit assumes that the finite-size contribution to $\varepsilon_{\textrm{lin}}$ is due to the boundaries.