A Finite-Window Recovery Hierarchy for Local Quantum Memory

Abstract: When quantum information initially stored in a local qubit disappears, it need not be lost: it may have moved into nearby degrees of freedom or become inaccessible to shallow local control. We introduce finite-window recoverability as an operational channel benchmark that separates these possibilities. It compares optimal recovery from the target site, recovery by a bounded-depth decoder on a finite window, and the unrestricted optimum for that window. Its operational component, local variational recovery, uses local state preparation, window-local control, and target-qubit Pauli readout to certify recoverable memory beyond the target and quantify how much of the same-window advantage is accessible to shallow control. In a disordered kicked-Ising Floquet chain, a depth-6 decoder on a five-site window realizes $Q<sup>{\mathrm{opt}}_0&lt;Q<sup>{\mathrm{shallow}}_2&lt;Q<sup>{\mathrm{opt}}_2$ across the crossover regime, with positive certified gain for most disorder realizations and substantial shallow-accessibility fractions. The signal differs from target-site persistence and reconstructed coherent-information increments. Positive radius-2 gain also persists when the task is embedded in longer open chains using an independent tensor-network backend. Guided by this hierarchy, we test a carrier-deletion task in which the original target register is reset after the dynamics. A depth-8 decoder repairs the input from a radius-3 surrounding halo with held-out median $F_{\mathrm{avg}}=0.758$, above the single-qubit classical benchmark $2/3$, and outperforms optimal one-, two-, and three-site halo-subwindow counterfactuals. These results establish finite-window recovery as a local-control benchmark for off-site quantum memory, diagnosing both where local quantum information remains and whether bounded-depth control can refocus it.

• The paper introduces a locality-constrained recovery hierarchy that compares on-site recovery, shallow-circuit recovery from a finite window, and the unrestricted window optimum.
• In a disordered kicked-Ising chain, depth-6 local variational recovery produces statistically positive certified gains across the disorder scan, while shallow accessibility rises to about 0.94.
• A carrier-deletion test achieves median channel fidelity of 0.758, exceeding the classical 2/3 benchmark and outperforming all tested one-, two-, and three-site halo subwindows.

Overview

The paper introduces finite-window recoverability as an operational benchmark for local quantum memory in interacting many-body systems (2608.12803). The central question is: when quantum information initially stored on a local qubit ceases to be recoverable from that qubit, has it been destroyed, or does it remain recoverable from nearby degrees of freedom — and if so, can bounded-depth local control actually retrieve it? The authors formalize this as a locality-constrained channel-recovery task and propose local variational recovery (LVR), a shallow-circuit protocol requiring only local state preparation, window-local control, and target-qubit Pauli readout. The framework is validated on a disordered kicked-Ising Floquet chain, both with exact diagonalization at $N=12$ and with an independent MPS/TEBD backend for $N=24$–$40$, culminating in a carrier-deletion repair task that converts the diagnostic into an explicit local repair stress test.

Operational framework

The access model is deliberately narrow. A target qubit is prepared in one of six cardinal states $$\mathcal{S}_6 = \{|0\rangle, |1\rangle, |\pm\rangle, |\pm i\rangle\}$$ against a fixed Néel product-state background; the system evolves under a many-body unitary $U$; and a layered nearest-neighbor circuit $V_\theta$ of depth $L$ acts only on the radius-$r$ window $\Lambda_r(i)$. Three channel-level quantities are compared:

• $Q^{\mathrm{opt}}_0$: the optimal CPTP recovery score from the target-site output alone;
• $N=24$0: what the depth-$N=24$1 decoder actually recovers from the window;
• $N=24$2: the unrestricted same-window optimum, computed via Choi-matrix SDP.

Two derived quantities carry the certification logic. The certified gain $N=24$3 is an existence witness: if positive, recoverable information lies outside the target site but inside the window. This follows because if $N=24$4 were a CPTP post-processing of $N=24$5, no decoder could beat $N=24$6; a stability version shows that any gain exceeding $N=24$7 rules out even approximate post-processing explanations within diamond-norm distance $N=24$8. The shallow-accessibility fraction $N=24$9 quantifies how much of the same-window advantage is reachable by bounded-depth control. The six-state probe set forms a qubit state 2-design, so the estimator equals the Haar-averaged channel fidelity $40$0, placing the score on standard channel-fidelity footing; the classical measure-and-prepare bound of $40$1 provides an absolute reference for the reset-erasure task.

The authors are explicit that "local quantum information" here means recoverability under the specified probe ensemble, background, and access model — not a background-independent property of the dynamics.

Recovery hierarchy in the kicked-Ising chain

The benchmark model is a periodically driven disordered kicked-Ising chain with couplings $40$2, $40$3, target site $40$4, radius $40$5, decoder depth $40$6, and 100 disorder realizations per point on a 12-point grid $40$7. Across the crossover regime the paper establishes the hierarchy

$40$8

with positive paired-disorder certified gain throughout the scan. The statistical support is strong: one-sided paired Wilcoxon $40$9-values range from $$\mathcal{S}_6 = \{|0\rangle, |1\rangle, |\pm\rangle, |\pm i\rangle\}$$0 to $$\mathcal{S}_6 = \{|0\rangle, |1\rangle, |\pm\rangle, |\pm i\rangle\}$$1, bootstrap intervals for the median gain remain strictly positive at every disorder point, and the positive fraction reaches $$\mathcal{S}_6 = \{|0\rangle, |1\rangle, |\pm\rangle, |\pm i\rangle\}$$2 near its strongest points. The median shallow-accessibility fraction rises from roughly $$\mathcal{S}_6 = \{|0\rangle, |1\rangle, |\pm\rangle, |\pm i\rangle\}$$3 at $$\mathcal{S}_6 = \{|0\rangle, |1\rangle, |\pm\rangle, |\pm i\rangle\}$$4 to about $$\mathcal{S}_6 = \{|0\rangle, |1\rangle, |\pm\rangle, |\pm i\rangle\}$$5 by $$\mathcal{S}_6 = \{|0\rangle, |1\rangle, |\pm\rangle, |\pm i\rangle\}$$6, reaching about $$\mathcal{S}_6 = \{|0\rangle, |1\rangle, |\pm\rangle, |\pm i\rangle\}$$7 near the certified-gain peak at $$\mathcal{S}_6 = \{|0\rangle, |1\rangle, |\pm\rangle, |\pm i\rangle\}$$8. The implication is twofold: loss of on-site recoverability does not imply loss from the nearby window, and a depth-6 decoder already accesses a substantial portion of the locally available memory.

Robustness controls address the obvious confounds. Changing the product-state background shifts absolute losses but preserves the regime ordering; multi-start optimization shows the signal is not an artifact of poor minima; layer scans show the crossover structure stabilizes by $$\mathcal{S}_6 = \{|0\rangle, |1\rangle, |\pm\rangle, |\pm i\rangle\}$$9; and finite-shot ($U$0) and gate-miscalibration models indicate the protocol tolerates realistic imperfections. A free-fermion Floquet reference shows that off-site recoverable information is not unique to interacting dynamics — but the interaction changes the disorder profile, producing a pronounced intermediate-regime maximum and higher accessibility fractions (about $U$1 versus $U$2 near $U$3).

Carrier-deletion repair

Guided by the hierarchy, the authors impose a stricter test: after evolution, the target register is reset via $U$4, so all input-dependent information must reside in the surrounding halo $U$5. At a frozen held-out working point ($U$6, $U$7, $U$8, $U$9), selected on disjoint realizations, a depth-8 decoder achieves median $V_\theta$0, i.e., $V_\theta$1, above the single-qubit classical benchmark $V_\theta$2, with 95% of held-out realizations exceeding it. Critically, the recovery exceeds optimal CPTP decoders restricted to every one-, two-, and three-site halo subwindow, with paired gains $V_\theta$3, $V_\theta$4, $V_\theta$5, all with strictly positive bootstrap intervals. The three-site separation is modest but statistically clean, ruling out explanations based on any small displaced subregion. This demonstrates that the halo retains a distributed channel remnant writable back into the reset target under locality and depth constraints — though the authors caution this remains a finite-size, single-qubit, fixed-background result and not a scalable error-correcting code.

Distinction from existing diagnostics

The certified gain is shown to be operationally distinct from two familiar quantities. Target-site magnetization retention grows nearly monotonically with disorder, tracking persistence rather than the intermediate-regime peak of $V_\theta$6. The tomographic coherent-information increment $V_\theta$7 counts channel content anywhere in the window without asking whether bounded decoding can refocus it. An operator shell-weight analysis makes the distinction microscopic: the Heisenberg-evolved weight $V_\theta$8 continues rising to strong disorder while $V_\theta$9 peaks near the crossover and falls to about half its peak by $L$0. Local operator support inside the window is therefore necessary but not sufficient for shallow recovery; decodability under constrained control is the missing ingredient.

Longer-chain embedding

To exclude an artifact of the $L$1 exact-diagonalization setting, the identical radius-2 task is embedded in open-boundary chains of $L$2 using an MPS/TEBD backend ($L$3, cutoff $L$4). Positive certified gain persists, peaking in the crossover-to-strong-disorder window at each size, and subset-level accessibility fractions of roughly $L$5–$L$6 ($L$7) and $L$8–$L$9 ($r$0) confirm substantial shallow access. The claim is deliberately restricted: the same-window optimum is evaluated only on a subset of $r$1 points, so the larger-system statement covers persistence of positive gain plus subset-level accessibility, not full-hierarchy scaling.

Limitations and open questions

The paper concedes several boundaries plainly. The study uses a single input qubit, a fixed product-state background, and a specific CZ-plus-Euler ansatz; the recovery radius is part of the task definition rather than an optimized resource. The carrier-deletion result is a frozen-working-point benchmark, not evidence for asymptotic decoder-complexity behavior, and the resource map over $r$2 explicitly disclaims any system-size scaling claim. In the tensor-network deployment, $r$3 is not evaluated on the full grid, leaving the complete hierarchy unverified at larger sizes. Open questions include how the hierarchy extends to multi-qubit input blocks — sketched blockwise in the discussion but not computed — and how the tradeoff among recovery radius, decoder depth, and evolution time behaves as a joint resource map beyond the fixed $r$4 illustration.

Conclusion

The paper establishes finite-window recoverability as a locality-constrained channel-recovery benchmark that separates three operationally distinct questions: what survives on site, what exists in a finite neighborhood, and what bounded-depth control can refocus onto the target. The exact kicked-Ising results — a strict three-level hierarchy with strongly significant paired gains, a held-out carrier-deletion repair above the classical benchmark that defeats all tested subwindow counterfactuals, and persistence of the witness in longer tensor-network chains — collectively demonstrate that the benchmark diagnoses not merely where local quantum information resides, but whether it is actively recoverable under specified locality, depth, and time resources.