Prove or refute a Page transition in the purification complexity of evaporating random circuits

Problem statement

Does the radiation of an efficiently generated evaporating quantum circuit acquire a parametrically larger purification complexity near the Page point? Settle this question for the explicit benchmark below, without assuming a holographic complexity formula. The motivation is arXiv:2607.21734, especially Sections 4.3 and 5.2. Its bounds allow a Page-time increase of radiation purification complexity but do not supply a post-Page lower bound. A microscopic result would test whether the proposed island-volume contribution reflects the cost of preparing the radiation state. The benchmark here makes additional choices that the paper leaves unspecified; it is a precise version of its shrinking-interior idea, not a claim that the paper already defines this ensemble. For each integer $n\geq3$, start with $n$ qubits in $|0\rangle^{\otimes n}$, all in an active register $B$, and an empty radiation register $R$. At an emission round with $m\geq2$ active qubits, apply $$g_m(n)=\left\lceil100m(\log n)^2\right\rceil$$ successive gates. Each gate acts on a uniformly chosen unordered pair of active qubits and is independently Haar distributed in $U(4)$; pair choices and gates are independent across rounds. Then select one active qubit uniformly and move it to the end of $R$. Moving the qubit is a change of register membership, not a measurement or an additional unitary. No later evolution gate acts on any emitted qubit. When $m=1$, emit the remaining qubit without applying a gate, so $g_1(n)=0$. All logarithms are natural. The coefficient and schedule are fixed parts of this benchmark. After $k$ emissions, write the pure joint state as $|\psi_k\rangle_{R_kB_k}$ and define $$\rho_R(k)=\operatorname{Tr}_{B_k}|\psi_k\rangle\langle\psi_k|,\qquad G_n(k)=\sum_{m=n-k+1}^{n}g_m(n).$$ Here $|R_k|=k$, $|B_k|=n-k$, and $G_n(k)$ is the actual accumulated gate count, not a proven minimum. In particular, $G_n(k)=\Theta(n^2(\log n)^2)$ when $k/n$ is bounded below by a positive constant. Use the fixed error $\varepsilon=10^{-2}$ and trace distance $D(\rho,\sigma)=\frac12\|\rho-\sigma\|_1$. Define $C_P^\varepsilon(\rho_R(k))$ as the minimum number of arbitrary one- or two-qubit unitary gates in any circuit $U$ satisfying $$D\!\left(\operatorname{Tr}_{A} U|0\rangle\langle0|^{\otimes(k+a)}U^\dagger,\rho_R(k)\right)\leq\varepsilon.$$ The minimum is over every finite ancilla count $a\geq0$ and every such circuit on $R_kA$, with unrestricted pair connectivity. Every gate counts as one operation; all ancillas start in $|0\rangle$. No measurements, postselection, initially mixed ancillas, or free correlated randomness are allowed. The final ancilla trace is free. These restrictions define the preparation task only. A candidate preparation circuit may act on all its output and ancilla qubits; it need not follow the evaporation history. Thus the actual history gives the deterministic upper bound $C_P^\varepsilon(\rho_R(k))\leq G_n(k)$. The target is a typical Page transition at the accumulated-circuit-cost scale. Determine whether there are constants $c,C>0$, a finite constant $p\geq0$, an explicitly bounded integer window $w(n)=o(n)$, and an explicit failure bound $q(n)\to0$, such that, with probability at least $1-q(n)$ over a whole evaporation trajectory, both of the following hold simultaneously: $$C_P^\varepsilon(\rho_R(k))\leq Cn \quad\text{for every }0\leq k\leq\lfloor n/2\rfloor-w(n),$$ $$\frac{cG_n(k)}{(\log n)^p}\leq C_P^\varepsilon(\rho_R(k))\leq G_n(k) \quad\text{for every }\lceil n/2\rceil+w(n)\leq k\leq n.$$ All constants must be independent of $n$, $k$, and the circuit draw. Establish or refute these statements using the preparation minimum itself. The lower bound remains superlinear in $n$ for every fixed finite $p$. This is a sharpened hypothesis for this benchmark. A slower but still superlinear rise could refute the accumulated-cost claim while leaving a weaker Page-like transition possible. Also determine whether the same ensemble has the leading entropy Page law $$\sup_{0\leq k\leq n} \left|S(\rho_R(k))-\min(k,n-k)\log2\right|=o(n)$$ with high probability. This identifies whether the proposed complexity transition is centered at the entropy Page point in this benchmark. The problem admits a negative resolution. A rigorous typical failure of the proposed scaling or transition location, together with an alternative asymptotic law or incompatible bound, would identify a limitation of the circuit/island analogy. A specially chosen easy circuit of vanishing probability is not such a resolution. A result here would address a microscopic assumption behind the holographic picture; it would not by itself establish or refute an exact gravity/circuit duality.

Current state

Concepcion, Nomura, Ritchie and Weiss (arXiv:2607.21734v1) propose generalized subregion CV for an evaporating AdS black hole and compare its island-induced volume transition with a shrinking random circuit. Section 5.2 explicitly acknowledges that the change between $C_P\leq C_S+C_B$ and $C_P\leq C_{\rm full}$ permits a jump but does not prove it. The present challenge fixes the missing microscopic ensemble and demands a lower bound after emission has made the radiation the larger subsystem. Fan, Hunter-Jones, Karch and Mittal (arXiv:2510.18832v2, Section 4 and Appendix C) prove subsystem-complexity bounds in random circuits with fixed partitions, including growth above half-system size and equilibration below it. Haah and Stanford (arXiv:2510.18805v4, Sections 2-4) establish related results in one-dimensional random circuits using a local-channel complexity definition. Their architectures, evolving registers, and complexity conventions must be compared explicitly with the benchmark here. Neither a fixed-partition theorem nor Haar typicality alone supplies the stated lower bound for a growing register that no longer evolves after emission. Agon, Headrick and Swingle (arXiv:1804.01561) distinguish purification, spectrum, and basis complexity. This distinction matters because a hard eigenbasis or a hard decoding task need not imply the required preparation lower bound when ancillary purifications are optimized over. Island-related volume transitions and generalized complexity already have important antecedents: Bhattacharya et al. (arXiv:2010.04134, Section 2.3.2) calculate radiation-volume jumps in multiboundary wormhole models, and Hernandez, Myers and Ruan (arXiv:2010.16398, Section 4) discuss island-volume and quantum-field contributions in double holography. The new target is the microscopic typical-complexity statement above, rather than the existence of a geometric island-volume jump. Zhao (arXiv:1912.00909v3, Section 3) already gives a circuit interpretation of island formation when computations occur only on the remaining black-hole register. Its argument assigns parts of the circuit to subsystems using undoability and scrambling. This is relevant prior motivation, but it does not establish the fixed-error preparation lower bound over all purifications required here. The original paper and the cited primary results were checked, and targeted primary-literature searches were performed on 2026-09-07. No resolution of this explicit benchmark was identified. This is a bounded literature assessment, not a claim that every related result has been excluded.

Resolution criteria

A public research paper or technical report resolves this challenge by giving a complete, checkable argument for either the positive or the negative route below. The ensemble, gate counting, ancilla convention, and fixed trace-distance error in the problem statement must be preserved. 1. Positive route: prove the two complexity inequalities simultaneously along a typical trajectory, with explicit constants or valid bounds on them, a specified $w(n)=o(n)$, and a quantified $q(n)\to0$. Prove the leading entropy Page law for the same ensemble with an explicit sublinear error bound and a probability tending to one. The post-Page bound must apply to the minimum over all permitted purifications and establish the accumulated-gate-count scale up to a fixed power of $\log n$. 2. Negative route: rigorously refute the asserted typical transition or the accompanying entropy law for this same benchmark. Supply a quantitative alternative asymptotic scaling, transition location, or bound that is incompatible with the positive route and holds with a stated non-vanishing probability as $n\to\infty$; prove the relevant probability statement. Refuting the transition claim must address its existential constants and sublinear window, rather than one arbitrarily chosen candidate window or coefficient. Distinguish failure of the entropy model from failure of the complexity scaling. For either route, identify which claim in the circuit interpretation of arXiv:2607.21734 is supported or requires qualification. Clearly state all probabilistic, asymptotic, and approximation assumptions. Any computation used as part of the evidence must include reproducible code, parameters, and error control. An entropy curve alone, replacement of the circuit by a globally Haar-random state, a larger preparation upper bound, decoding hardness without a valid reduction, an atypical easy circuit, or a finite-size numerical fit alone does not meet these criteria. No new holographic calculation or exact CV duality is required to resolve this microscopic challenge.

Citations

1. V. Concepcion, Y. Nomura, K. Ritchie, and S. Weiss, Page transition for the complexity of an evaporating black hole, arXiv:2607.21734 (2026), Sections 4.3 and 5.2.
2. Y. Fan, N. Hunter-Jones, A. Karch, and S. Mittal, Sharp Transitions for Subsystem Complexity, arXiv:2510.18832 (2025), Section 4 and Appendix C; version 2 (2026).
3. J. Haah and D. Stanford, Growth and collapse of subsystem complexity under random unitary circuits, arXiv:2510.18805 (2025), Sections 2-4; version 4 (2026).
4. C. A. Agon, M. Headrick, and B. Swingle, Subsystem Complexity and Holography, JHEP 02 (2019) 145, arXiv:1804.01561.
5. A. Bhattacharya, A. Chanda, S. Maulik, C. Northe, and S. Roy, Topological shadows and complexity of islands in multiboundary wormholes, JHEP 02 (2021) 152, arXiv:2010.04134, Section 2.3.2.
6. J. Hernandez, R. C. Myers, and S.-M. Ruan, Quantum Extremal Islands Made Easy, Part III: Complexity on the Brane, JHEP 02 (2021) 173, arXiv:2010.16398, Section 4.
7. Y. Zhao, A quantum circuit interpretation of evaporating black hole geometry, JHEP 07 (2020) 139, arXiv:1912.00909, Section 3.

Linked papers

• Page transition for the complexity of an evaporating black holerelated7 Sept 2026
• Purification complexity in evaporating random circuits: a Page-transition benchmarkdeclared at submission7 Sept 2026