Soundness of the explicit hidden-parity construction

Prove that, for circuits generated by the specified three-qubit construction with linear depth, block factors W and W CZ, affine permutations, Haar-random local frames, public measurement basis M unequal to U_L, uniformly random inputs, and secret weights, no polynomial-time classical algorithm given the public data can pass both verification tests at the prescribed round counts except with negligible probability.

Background

The protocol's ideal-model soundness theorem assumes that the server's view is independent of the hidden data. In the explicit construction, however, the compiled gates are generated using the secret-dependent degeneracy structure, so the public circuit may potentially leak information.

The unresolved issue is whether the concrete circuit remains as hard to spoof as the ideal model against polynomial-time classical servers. The paper provides simulations and diagnostics against several known strategies but does not prove this computational claim.

References

For circuits generated by the recipe of Appendix~\ref{app:construction}, with $\ell=3$, $L=\Theta(n)$, block factors $W$ and $W\mathrm{CZ}$, affine permutations, Haar frames, $M\neq U_L$, uniform inputs and secret weights, no classical algorithm running in time polynomial in $n$, given the public data, passes both tests at the round counts of Eq.~eq:rounds except with probability $\varepsilon(n)$ negligible in $n$.

— 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 Conjecture 2, Section 7, “Conjectures”

For instances of the construction, no classical algorithm running within time $T_{\max}$, given the public data and an input $x$, outputs a list of $J\le2\eta$ labels with $\frac1J\sum_iD\,p(y_i|x)\ge1+\delta$, except with negligible probability.

— 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 Conjecture 3, Section 8, “Randomness”

At the sizes studied here it is reproduced by truncated simulations of large bond dimension, which is why the collision test is used; whether it remains sound on its own against efficient simulation as $n$ grows, so that the collision test could be dropped, is the most direct question.

— 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 10, Discussion

Independently, shallower circuits raise the fidelity and so lower the number of rounds; the depth used here was chosen conservatively, and the smallest depth that keeps the properties of Sec.~\ref{sec:soundness} is an open question about the construction.

— 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 10, Discussion

And the effect of hardware noise, in particular coherent errors in the local frames, on both statistics has not been analyzed; testing the instances on hardware is the next step.

— 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 10, Discussion