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arxiv: 2604.15427 · v1 · submitted 2026-04-16 · 🪐 quant-ph

Tensor Networks with Belief Propagation Cannot Feasibly Simulate Google's Quantum Echoes Experiment

Pablo Bermejo , Benjamin Villalonga , Brayden Ware , Guifre Vidal , Aaron Szasz This is my paper

Pith reviewed 2026-05-10 10:57 UTC · model grok-4.3

classification 🪐 quant-ph
keywords tensor networksbelief propagationout-of-time-order correlatorsquantum simulationentanglementclassical simulationrandom circuitsquantum advantage
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The pith

Tensor networks with belief propagation cannot simulate the OTOC circuits from Google's quantum echoes experiment.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper establishes that the random circuits used to measure out-of-time-order correlators produce states with entanglement that grows too rapidly for tensor network representations to compress efficiently. Belief propagation on the dense 2D connectivity of the processor adds a second barrier that prevents accurate classical reproduction of the measured values. If this holds, classical methods of this type cannot match the speed or precision achieved by the quantum processor. The result strengthens the case that the experiment demonstrates a regime where quantum hardware outperforms classical simulation for this class of dynamical quantities.

Core claim

The OTOC circuits generate enough entanglement that the quantum states are largely incompressible by tensor network methods, and the dense connectivity of the processor makes belief propagation ineffective, so TNBP cannot feasibly simulate the full experiment to the accuracy reached by the quantum device.

What carries the argument

Entanglement growth in the OTOC random circuits, which overwhelms the compression power of tensor network states when belief propagation is applied on a lattice with dense connections.

If this is right

  • Other tensor network methods that evolve states in the Schrödinger picture will encounter the same incompressibility barrier for these circuits.
  • Classical simulation of the echoes experiment via TNBP is ruled out at the reported scale and accuracy.
  • The quantum processor's measured speedup over the tested classical methods extends to this additional simulation approach.
  • Similar highly entangled dynamics in random circuits will remain hard for TNBP-style methods.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • The incompressibility argument may apply to other measurement protocols that rely on out-of-time-order correlators in chaotic quantum systems.
  • Verification of such experiments may increasingly require either quantum hardware or entirely different classical algorithms that do not rely on state compression.
  • If entanglement continues to grow without compressible patterns, it could bound the reach of tensor-network techniques for many classes of quantum dynamics experiments.

Load-bearing premise

Entanglement scaling and numerical behavior observed on smaller or simplified circuits continue without new compressible structure appearing at the full size and connectivity of the Google experiment.

What would settle it

A TNBP calculation that reproduces the full experiment's OTOC values to the same accuracy as the quantum processor within feasible classical resources would disprove the claim.

Figures

Figures reproduced from arXiv: 2604.15427 by Aaron Szasz, Benjamin Villalonga, Brayden Ware, Guifre Vidal, Pablo Bermejo.

Figure 1
Figure 1. Figure 1: FIG. 1 [PITH_FULL_IMAGE:figures/full_fig_p003_1.png] view at source ↗
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Figure 23
Figure 23. Figure 23: , the remaining operators C have a tensor net￾work width that scales as the square root of the circuit depth. Specifically, the time-direction width of the irre￾ducible core, ℓ (P ) (shown in [PITH_FULL_IMAGE:figures/full_fig_p033_23.png] view at source ↗
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read the original abstract

In the recent quantum echoes experiment, Google Quantum AI showed that out-of-time-order correlators (OTOCs) for random-circuit time evolution can be measured using a quantum processor more than 10,000x faster than they can be computed to similar accuracy via classical computation. This claim was substantiated by comparison with a variety of state-of-the-art classical simulation methods. One classical simulation method that was not explicitly tested was tensor networks with belief propagation (TNBP). TNBP should be poorly suited to simulating Google's echoes experiment: the states involved are highly entangled, a challenge for tensor network states; and the Willow chip has dense 2D connectivity, a challenge for belief propagation. Here we confirm, via a combination of theoretical scaling arguments and explicit numerical simulation, the intuition that TNBP is unable to simulate the quantum echoes experiment. We show that the OTOC circuits generate enough entanglement that they are largely incompressible, implying that other approaches in which OTOCs are computed by evolving a tensor network state in the Schr\"odinger picture will also fail. Our results further reinforce the claim that the quantum echoes experiment cannot be reproduced by classical computation.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

1 major / 0 minor

Summary. The manuscript claims that tensor networks with belief propagation (TNBP) cannot feasibly simulate Google's quantum echoes experiment on out-of-time-order correlators (OTOCs). It combines theoretical scaling arguments based on entanglement growth with explicit numerical simulations on smaller circuits to argue that the OTOC circuits produce volume-law entanglement making them incompressible, and the dense connectivity challenges BP. This implies failure for other Schrödinger-picture TN methods and reinforces the quantum advantage claim.

Significance. If the result holds, it is significant as it rules out TNBP as a classical simulator for this experiment, a method potentially suited to entangled systems but hindered here by entanglement volume and connectivity. This provides additional support for the infeasibility of classical computation of the OTOCs to the accuracy achieved experimentally.

major comments (1)
  1. The simulations are performed on smaller or simplified circuits. The extrapolation to the full-size Google experiment with its specific random-circuit + OTOC protocol and dense 2D connectivity assumes that no additional compressible structure emerges at scale, but this is not demonstrated or tested, which is load-bearing for the infeasibility conclusion.

Simulated Author's Rebuttal

1 responses · 0 unresolved

We thank the referee for their careful reading and constructive feedback on our manuscript. We address the major comment below.

read point-by-point responses

  1. Referee: The simulations are performed on smaller or simplified circuits. The extrapolation to the full-size Google experiment with its specific random-circuit + OTOC protocol and dense 2D connectivity assumes that no additional compressible structure emerges at scale, but this is not demonstrated or tested, which is load-bearing for the infeasibility conclusion.

    Authors: We agree that our numerical simulations are necessarily restricted to smaller circuit sizes, as exact classical simulation of the full Google experiment is computationally prohibitive. Our primary argument, however, rests on theoretical scaling analysis of entanglement growth under the random-circuit + OTOC protocol, which establishes volume-law entanglement that increases with system size and precludes compressible structure. The dense 2D connectivity of the Willow chip is a fixed architectural property independent of circuit depth or width, and belief propagation is known to fail on such graphs due to short loops. We have revised the manuscript to include an expanded discussion (new paragraph in Section 4 and strengthened language in the conclusion) explaining why the random, scrambling nature of the circuits makes additional compressible features at scale unlikely, with supporting citations to entanglement scaling results in random quantum circuits. revision: partial

Circularity Check

0 steps flagged

No circularity: scaling arguments and small-circuit simulations are independent of the target claim

full rationale

The paper's central argument relies on standard entanglement scaling (linear growth implying exponential bond-dimension cost) plus explicit TNBP simulations on smaller or simplified OTOC circuits. These inputs are drawn from general tensor-network theory and direct computation rather than from fitting parameters to the full Google experiment or from self-citations that define the result. No equation reduces the infeasibility conclusion to a quantity defined by the target data, and the extrapolation step is presented as an assumption rather than a derived identity. The derivation chain therefore remains self-contained against external benchmarks.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

The central claim rests on standard quantum information assumptions about entanglement growth in random circuits and the limitations of belief propagation on dense graphs; no new entities or fitted parameters are introduced in the abstract.

axioms (2)
  • domain assumption OTOC circuits on the Willow chip generate entanglement that exceeds the compressibility threshold of tensor networks.
    Invoked to conclude incompressibility from scaling arguments.
  • domain assumption Belief propagation cannot efficiently handle the dense 2D connectivity of the chip for these states.
    Used to explain why TNBP is poorly suited.

pith-pipeline@v0.9.0 · 5510 in / 1287 out tokens · 37495 ms · 2026-05-10T10:57:39.816156+00:00 · methodology

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    The physi- cal lightcone is closer to circular. As shown in Fig. 20(b), the physical lightcone velocity varies by only about 10% from one direction to another. The result is that the physical lightcone spreads nearly as fast as the geometric lightcone along the di- agonal directions (vd/cd ≈0.95), and more slowly relative to the geometric lightcone alongˆ...

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