Asymptotic bond-dimension scaling and reliability of internal error checks

Determine whether the bond dimension required by the Sweeping RTM algorithm for fixed probability-error threshold ε_p, denoted χ(T, ε_p, N), grows exponentially, subexponentially, or polynomially with circuit time, and establish whether the algorithm’s internal convergence checks suffice to guarantee the stated error.

Background

The paper presents numerical and heuristic evidence that, for one-dimensional chaotic brick-wall circuits and fixed relative precision, the bond dimension required by the Sweeping RTM algorithm grows subexponentially over the accessible time window. However, the observed finite-time behavior does not establish the asymptotic complexity of the method. The authors explicitly identify the classification of the growth as exponential, subexponential, or polynomial as unresolved.

The algorithm uses consistency checks based on variations across spatial cuts, sweeps, and bond dimensions. Earlier in the paper, the authors state that these checks do not provide rigorous error bounds and do not prove that passing them guarantees the prescribed relative amplitude error or that the sweep optimization always converges. The open problem therefore also concerns the rigorous validity of the proposed error-certification procedure.

References

Several questions remain open. The present evidence is numerical and heuristic, therefore rigorous bounds need to be found. In particular, it remains to be fully established whether the growth of the bond dimension at a given thresold $\epsilon_p$, i.e. $\chi(T,\epsilon_p,N)$, is exponential, subexponential, or purely polynomial and if the internal checks for the error provided here are enough.

— Strong Simulation of 1D Quantum Circuits via Reduced Transition Matrices  (2610.02082 - Grassi et al., 1 Oct 2026) in Outlook

How the bond dimension needed to pass scales with $n$ is not settled by sizes up to ten.

— Classical Verification of a Remote Quantum Processor via Degenerate Concordant Computations: An Application to Randomness Generation  (2610.08185 - Das et al., 6 Oct 2026) in Section 7, paragraph “Instances and measurements”