Classical algorithm uses quantum chaos to estimate scrambling in one-dimensional circuits

A classical computer may not need to follow every detail of a chaotic quantum system to calculate useful measures of its behavior. In an arXiv preprint, Yunchao Liu, Yihui Quek, Shreya Vardhan and Tianci Zhou present an algorithm for estimating two such measures in individual one-dimensional random quantum circuits.

Their results address a proposed route to quantum advantage: calculating observables that a quantum processor can measure but conventional classical methods struggle to reproduce. The scope matters, however. The study concerns particular observables in typical one-dimensional circuits, not arbitrary quantum computations, and it does not settle whether recent two-dimensional experiments demonstrate quantum advantage.

Measuring how quantum influence spreads

The quantities are out-of-time-ordered correlators, or OTOCs. They combine local operators with forward and backward time evolution to probe scrambling: how an initially local operator spreads across a quantum system. The researchers study the standard four-point OTOC and a more detailed eight-point version.

Averaging the four-point quantity over random circuits is already straightforward. Estimating its value for a particular circuit is a different task. Individual circuits fluctuate around the average, and those variations contain information that an average alone cannot capture.

The study targets the region near the advancing front of operator spreading, where circuit-to-circuit fluctuations are substantial. This is also the regime the paper identifies as accessible to sample-efficient quantum experiments. Simply predicting a nearly constant value outside that region would not meet the same challenge.

A simpler description, plus short corrections

Rather than representing the full quantum state, the algorithm rewrites each OTOC using multiple mathematical copies of the circuit. In that representation, important contributions can be organized around different pairings between forward- and backward-evolving copies.

Boundaries between regions with different pairings behave like domain walls. For the four-point calculation near the chain’s edge, the simplest relevant configurations contain at most one wall. For the eight-point calculation, the chosen configurations contain up to three elementary walls.

The method, called a membrane cluster expansion, adds corrections describing excursions outside these simple configurations. The researchers call these excursions “bubbles.” Their central physical argument is that chaotic dynamics cause fluctuating phases to cancel, strongly suppressing bubbles that persist for a long time.

The algorithm therefore retains bubbles only up to a chosen duration. It computes the necessary short-interval amplitudes using tensor-network contractions, then combines them through dynamic programming, storing and reusing intermediate results instead of enumerating every possible history.

In one dimension, the number of retained simple configurations grows only polynomially with circuit depth. Under bubble-suppression and an additional convexity-type assumption, the authors prove an error bound for the bulk transition amplitudes. They argue that this supports polynomial-time OTOC estimation at inverse-polynomial accuracy in the target regime. This is a conditional efficiency argument, supported by theory and numerics—not an unconditional guarantee for every circuit.

What the benchmarks show

The main numerical tests used 400 circuit samples at depth 40 for the four-point OTOC and 200 samples at depth 24 for the eight-point OTOC. Estimates were compared with values obtained through exact tensor-network contraction.

For the four-point quantity, increasing the retained bubble duration produced an approximately exponential decrease in mean absolute error and increase in signal-to-noise ratio. Tests with the initial operator near the edge and deep inside the chain showed comparable accuracy.

The eight-point calculation was more demanding. Its signal-to-noise ratio reached about six, indicating that the approximation tracked circuit-specific fluctuations. But its raw estimates were generally biased upward: restricting the simple configurations also omitted contributions to the ensemble average.

The authors separately calibrated those estimates by shifting their sample mean to an ensemble mean calculated with a matrix-product-state method. For this calibrated data, mean absolute errors fell roughly four- to fivefold as the bulk cutoff increased from one to six layers. Those calibrated errors measure accuracy around the mean; they should not be confused with the uncorrected method’s total error.

Practical costs remained substantial. For all observable locations of one circuit instance at the benchmark sizes, the reported implementation took approximately three minutes for the edge four-point calculation, one hour for the interior four-point calculation, and nine hours for the edge eight-point calculation, using a single NVIDIA H200 GPU.

The two-dimensional question remains open

The paper reports higher signal-to-noise ratios than cited Google two-dimensional experiments at comparable or greater depths. That is not a like-for-like demonstration of classical superiority: the circuit geometries differ.

In higher dimensions, the retained configurations become much harder to sum. Extending the algorithm efficiently remains unresolved, as does its applicability to dynamics with conservation laws. The preprint’s central contribution is narrower: evidence that chaos can simplify selected classical calculations even while the underlying quantum dynamics remain complex.