A tensor network scheme developed at the Flatiron Institute in New York City has outperformed D-Wave's Advantage2 quantum annealer on simulations of Ising spin glass dynamics, delivering lower errors in two-point correlators on cylindrical and diamond lattices and errors of the same order on cubic lattices. The work, published in Science by Joseph Tindall and colleagues, directly challenges a recent claim by quantum computing researchers that classical machines could not match the annealer's results on this problem.

The contest matters for R&D managers tracking quantum computing investments. Many-particle quantum systems are a standard computational benchmark because their complexity grows exponentially with particle count. Quantum hardware encodes spin states naturally in superposed qubits, while classical machines must represent the same physics in bits. Whether that theoretical advantage translates into measured results on problems of practical size is exactly the question the Flatiron team interrogated.

The Ising spin glass model describes spins on a lattice pointing in random directions, with disordered nearest-neighbor interactions that differ for each pair. Difficulty scales with system size, which has made the model a recurring benchmark in the classical-versus-quantum comparison.

How the method works. Tindall's group built a tensor network representation of both the Ising spin glass Hamiltonian and its wave function — the first specifying spin interactions, the second describing the spin state. Tensors act like interlocking building blocks: each tensor hosts a site with its physical attributes, and "legs" (bond indices) join through contraction to form a network encoding particle interactions and entanglement. Widening the bond dimension captures correlations more accurately, at higher computational cost.

That cost is the catch. As a tensor network wave function evolves forward in time, entanglement builds up between parts of the system, forcing bond indices to widen and making ground-state computation at each time step harder — sometimes impossible. Conventional tensor network schemes stall before reaching the timescales relevant to quantum annealing.

Belief propagation. To push past that barrier, the team paired tensor networks with a belief propagation (BP) message-passing approach for the contraction. Instead of accounting for every contribution exactly, each tensor receives a compact summary of its effective environment — what the rest of the network "looks like" from its position. Tindall describes it as a mean-field approximation on each tensor's environment: information arriving from neighboring tensors is assumed to be independent. That assumption buys tractability at the price of exactness, a trade-off managers should keep in mind when weighing such results against exact methods.

With the TN-BP approach in place, the classical implementation evolved the system to much longer times than conventional tensor network schemes — far enough to reach the regime in which the quantum annealer operates.

Measured results. The researchers computed two-point correlators — how the spin at one site relates to spins elsewhere — at various times and system sizes, on two- and three-dimensional lattices with cylindrical, diamond and cubic geometries. On the cylindrical lattice, once the bond dimension was large enough, the classical error came in markedly below the annealer's across different annealing schedules. On the diamond lattice, the classical error again fell under the annealer's. On the cubic lattice, the errors were of the same order at a fixed annealing time.

These are measured comparisons against the annealer's output, not projections. The caveat is the method's own approximation: belief propagation assumes independence of environmental messages, and the accuracy depends on bond dimension. The study did not report error bars or sample sizes in the summary available, and readers should weigh the results accordingly.

What comes next. Tindall and colleagues plan to apply TN-BP to interacting electronic systems such as the Hubbard model and to extend the scheme to finite-temperature problems. The group also maintains an open-source tensor network quantum simulator library on GitHub, which lowers the barrier for labs wanting to benchmark the method against their own workloads.

The result shows how quickly classical simulation of quantum physics is advancing, and how the competition between classical and quantum computation is itself driving that progress. For now, claims of quantum advantage on spin glass dynamics will need to clear a bar that classical tensor networks keep raising.