Conditional-correlation decay in amplitude-damped random circuits

Problem

Do computational-basis outputs of one-dimensional Haar-random circuits with fixed amplitude damping have conditional mutual information that decays exponentially with separation, uniformly in circuit depth?

Consider an \(n\)-qubit nearest-neighbor Haar-random brickwork circuit on a line, starting from \(|0^n\rangle\). After each layer, apply amplitude damping of fixed strength \(\gamma\in(0,1)\) independently to every qubit, with

\begin{equation} K_0=|0\rangle\langle0|+\sqrt{1-\gamma}|1\rangle\langle1|, \qquad K_1=\sqrt\gamma|0\rangle\langle1|. \tag{1} \end{equation}

Let \(P_U\) be the computational-basis output distribution. For disjoint \(A,B,C\), define

\begin{equation} I_{P_U}(A:C\mid B)=H_{P_U}(AB)+H_{P_U}(BC)-H_{P_U}(B)-H_{P_U}(ABC), \tag{2} \end{equation}

and let \(r(A,C)\) be the chain distance between nonempty \(A\) and \(C\). For every fixed \(\gamma\), do constants \(a,b,c>0\) exist, independent of system size, depth, and the subsets, such that

\begin{equation} \mathbb E_U I_{P_U}(A:C\mid B)\leq an^b e^{-c r(A,C)}? \tag{3} \end{equation}

Equation (3) concerns the distribution produced using Eq. (1) and the classical conditional mutual information in Eq. (2).

Source

This precise formulation is editor wording based on the unresolved direction and limitations documented in the cited primary literature [Lee25][Mele26][Shravan26]; it is not presented as a verbatim conjecture of those authors.

Progress

Lee and coauthors identify this average approximate-Markov condition as sufficient for classical sampling. In one dimension, their Theorem 3 gives runtime \(\operatorname{poly}(n,1/\varepsilon,1/\delta)\) and output error \(\|P_U-Q_U\|_1\leq\varepsilon\), except on a fraction \(\delta\) of circuits, provided the condition holds uniformly over depth. Their amplitude-damping evidence is numerical, not a general proof. [Lee25]
Mele and coauthors prove effective-depth bounds for expectation values under nonunital noise. For a bounded observable \(O\), the effect of discarding all but the last \(m\) noisy layers is bounded on average by

\begin{equation} O\!\left(\|O\|_\infty e^{-\alpha_\gamma m}\right), \qquad \alpha_\gamma>0. \tag{4} \end{equation}

Their 2026 publication explicitly distinguishes these results from the still-open general sampling problem. [Mele26]

The displayed definitions, constraints, and target bounds are recorded in Eqs. (4).
The April 2026 IQP simulation paper explicitly describes the nonunital conditional-mutual-information evidence as numerical. Its polynomial-time result for amplitude-damped IQP circuits at \(d=\Omega(\log n)\) uses a different, diagonal-gate structure and does not prove the Haar-circuit inequality above. No later proof of this inequality was located. [Shravan26]

Comment

This would give a rigorous route from dissipative loss of conditional correlations to efficient full-distribution sampling at arbitrary depth. Efficient local-observable estimation, decay of ordinary two-point correlations, and a quantum-state conditional-mutual-information bound are not interchangeable with the classical inequality asked here.

References

[Lee25]: S.-u. Lee, S. Ghosh, C. Oh, K. Noh, B. Fefferman, and L. Jiang, "Classical simulation of noisy random circuits from exponential decay of correlation," arXiv preprint (2025), version 1, 7 October 2025.arXiv
[Mele26]: A. A. Mele, A. Angrisani, S. Ghosh, S. Khatri, J. Eisert, D. Stilck França, and Y. Quek, "Noise-induced shallow circuits and the absence of barren plateaus," Nature Physics 22, 751–756 (2026).DOI arXiv
[Shravan26]: S. Shravan, M. Raza, and A. Shlosberg, "Efficient simulation of noisy IQP circuits with amplitude-damping noise," arXiv preprint (2026), version 2, 22 April 2026.arXiv

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