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Conference on Lasers and Electro-Optics

Approaching Helstrom Bound Using Single-shot Quantum Measurement Confidences

Ivan A. Burenkov, N. Fajar R. Annafianto, M. V. Jabir, Alexandra Semionov, Abdella Battou, Sergey V. Polyakov

Published Jan 1, 2022• 1 citation
Publisher

Abstract

We demonstrate the first quantum measurement providing s ingle-shot self-confidence for each received symbol. Further, we use measured confidence data to implement error correction codes reducing error rates by more than five orders of magnitude for reliable communication.
DOI
10.1364/cleo_qels.2022.ff4a.7

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We first provide a simple proof showing that any single-shot code is self-correcting. We then show that any $[[n,k]]$ CSS quantum low-density parity-check (QLDPC) code can be transformed into an $[[nm,k]]$ CSS QLDPC code with single-shot (SS) quantum error correction and an efficient local decoder which runs in $O(\log m)$ parallel time. Below a constant threshold $p_{\rm RG}>0$ for joint local stochastic physical and measurement noise, the logical failure probability over $T$ correction rounds is bounded by \begin{equation*} O(T n)\exp[-Ω(m^α)] \end{equation*} for a constant $α>0$; we say that the \textit{threshold} is $p_{\rm RG}$ and \textit{error suppression} is $Ω(m^α)$. Moreover, if the input code is equipped with a single-shot decoder with threshold $p_c>0$ and error suppression $Ω(n^β)$, then the output code \textit{enhances} the error suppression to $Ω(m^α n^β)$, while maintaining the decoder-independent threshold $p_{\rm RG}$, and thus \textit{enhanced} if $p_c < p_{\rm RG}$.

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[SUPERSEDED DUPLICATE of 10.5281/zenodo.21698294] Magic-State Factory Performance for Single-Shot 4D Toric Codes: the Factory Floor Dominates

James Jardine

SUPERSEDED DUPLICATE — do not cite this record. Cite 10.5281/zenodo.21698294 instead. Both records were deposited on 2026-07-30 and contain the same paper. This one carries an earlier, superseded NOTE.md whose resource-scaling section still claimed L≈6–7 (d=36–49); the surviving record carries the corrected text (measured L=4, [[1536,6,16]]) and the toric4d_lowp.py stratified estimator behind it. The independent cross-validation scripts that were unique to this record (magic_state4-7.py) have been added to the surviving record. Flagged 2026-07-31. We simulate a 15-to-1 magic-state factory under full circuit-level depolarizing noise and ask whether a 4D toric code — which corrects errors single-shot, without repeated syndrome rounds — relieves the factory's dominant cost. It does not. Distillation output error is confirmed at 35p³ by exact enumeration and by stim, but the factory's circuit-level error floor is first order in the gate error rate and schedule-dependent, measured at 12–15·p_gate for the real [[15,1,3]] quantum Reed–Muller code with all 14 stabilizers. This is 3.5× larger than a previously reported 4.18·p_gate, which we trace to a circuit that omitted the code's 10 Z-type checks; that figure is withdrawn here. The floor originates in the final syndrome sweep: faults deposited on data qubits by the last round of gates are never checked by anything after them. Whichever check type is measured last, that sector's faults escape, so rescheduling shifts the floor between sectors and interleaving shaves roughly 20%, but cannot remove it. Reordering, redundant checks and flag qubits leave it first order; decoding weight-1 errors instead of discarding them makes it 5× worse. We construct the 4D toric code on an L⁴ torus from its cubical chain complex and verify [[486,6,9]] at L=3 (k=6 exactly, explicit weight-9 logicals in both sectors, d=4 exact at L=2 by exhaustive search), and measure a code-capacity threshold between 5% and 7% under BP+OSD. Single-shot repair using the 80 independent metachecks recovers near-perfect-syndrome performance, but only while syndrome noise stays roughly 10–30× below the data error rate: it improves logical error by 195× at q=10⁻³ and by only 4.5× at q=10⁻². Crucially, metachecks constrain the syndrome, not the data, and cannot see faults deposited directly on data qubits by the final sweep — which is precisely where the factory floor originates. Single-shot therefore buys latency, not factory fidelity. Because the factory's Cliffords act on encoded blocks, p_gate is the logical error rate of the hosting code. The correction moves the surface-code requirement from d=15 to d=17 (578 physical qubits per patch); on the 4D toric side, the smallest sufficient block is L=4, [[1536,6,16]], with [[486,6,9]] missing the fidelity demand by four to five orders of magnitude. We situate the result against Xu et al. (arXiv:2407.18490), whose parallel distillation across qLDPC patches reduces space–time cost but leaves the per-factory floor untouched. A corrections log records eight claims withdrawn or corrected during this work, including our own earlier 4.18·p_gate figure and a d=15 requirement that rested on a single error event. Section 6 states what is not established: these are code-capacity numbers for the 4D code, the distance requirement is quoted at p=3–5% rather than 10⁻³, d=L² is verified exactly only at L=2, the Λ extrapolations are long, and no end-to-end injection simulation was run. Correction, 2026-07-31. The files deposited with this record on 2026-07-30 predate a low-p re-measurement of the resource-scaling section. The originally deposited text stated that hosting the factory requires L≈6–7 (d=36–49, n≈7800–14400); that figure came from fitting a single suppression factor Λ to near-threshold points at p=3–5%. The single-Λ ansatz does not hold for this code: the measured suppression per two units of distance runs from 10.9 (d=4→9 at p=0.02) to 793 (d=9→16 at p=0.01). Re-measuring at the factory's actual operating regime with a weight-stratified estimator gives d ≥ 14.5 at p=0.02 and d ≥ 9.7 at p=0.01, so the smallest sufficient 4D block is L=4, [[1536,6,16]], not L=6–7. The L≈6–7 figure is withdrawn. These are point estimates whose upper confidence bands remain wide at the smallest p, and should be read as order-of-magnitude. The corrected text and the estimator script (toric4d_lowp.py) are not in this record's file set; replacing files requires a new version, which will carry the corrected note. Nothing in Sections 1–4 — the 35p³ distillation result, the 12–15·p_gate factory floor, the last-sweep mechanism, or the d=17 surface-code requirement — is affected.

•Jul 30, 2026•0 citations
We demonstrate the single-shot confidence estimation for individual quantum measurement outcomes using the continuous measurement theory of the quantum counting process applied to the quantum state identification problem. We experimentally obtain single-shot and average confidences for quantum measurements and show that they favorably compare to that of the idealized classical measurement. Finally, we demonstrate that single-shot confidence estimations correctly represent observed experimental outcomes for a large ensemble of measurements.
We first provide a simple proof showing that any single-shot code is self-correcting. We then show that any [[n,k]][[n,k]][[n,k]] CSS quantum low-density parity-check (QLDPC) code can be transformed into an [[nm,k]][[nm,k]][[nm,k]] CSS QLDPC code with single-shot (SS) quantum error correction and an efficient local decoder which runs in O(log⁡m)O(\log m)O(logm) parallel time. Below a constant threshold pRG>0p_{\rm RG}>0pRG​>0 for joint local stochastic physical and measurement noise, the logical failure probability over TTT correction rounds is bounded by \begin{equation*} O(T n)\exp[-Ω(m^α)] \end{equation*} for a constant α>0α>0α>0; we say that the \textit{threshold} is pRGp_{\rm RG}pRG​ and \textit{error suppression} is Ω(mα)Ω(m^α)Ω(mα). Moreover, if the input code is equipped with a single-shot decoder with threshold pc>0p_c>0pc​>0 and error suppression Ω(nβ)Ω(n^β)Ω(nβ), then the output code \textit{enhances} the error suppression to Ω(mαnβ)Ω(m^α n^β)Ω(mαnβ), while maintaining the decoder-independent threshold pRGp_{\rm RG}pRG​, and thus \textit{enhanced} if pc<pRGp_c < p_{\rm RG}pc​<pRG​.
SUPERSEDED DUPLICATE — do not cite this record. Cite 10.5281/zenodo.21698294 instead. Both records were deposited on 2026-07-30 and contain the same paper. This one carries an earlier, superseded NOTE.md whose resource-scaling section still claimed L≈6–7 (d=36–49); the surviving record carries the corrected text (measured L=4, [[1536,6,16]]) and the toric4d_lowp.py stratified estimator behind it. The independent cross-validation scripts that were unique to this record (magic_state4-7.py) have been added to the surviving record. Flagged 2026-07-31. We simulate a 15-to-1 magic-state factory under full circuit-level depolarizing noise and ask whether a 4D toric code — which corrects errors single-shot, without repeated syndrome rounds — relieves the factory's dominant cost. It does not. Distillation output error is confirmed at 35p³ by exact enumeration and by stim, but the factory's circuit-level error floor is first order in the gate error rate and schedule-dependent, measured at 12–15·p_gate for the real [[15,1,3]] quantum Reed–Muller code with all 14 stabilizers. This is 3.5× larger than a previously reported 4.18·p_gate, which we trace to a circuit that omitted the code's 10 Z-type checks; that figure is withdrawn here. The floor originates in the final syndrome sweep: faults deposited on data qubits by the last round of gates are never checked by anything after them. Whichever check type is measured last, that sector's faults escape, so rescheduling shifts the floor between sectors and interleaving shaves roughly 20%, but cannot remove it. Reordering, redundant checks and flag qubits leave it first order; decoding weight-1 errors instead of discarding them makes it 5× worse. We construct the 4D toric code on an L⁴ torus from its cubical chain complex and verify [[486,6,9]] at L=3 (k=6 exactly, explicit weight-9 logicals in both sectors, d=4 exact at L=2 by exhaustive search), and measure a code-capacity threshold between 5% and 7% under BP+OSD. Single-shot repair using the 80 independent metachecks recovers near-perfect-syndrome performance, but only while syndrome noise stays roughly 10–30× below the data error rate: it improves logical error by 195× at q=10⁻³ and by only 4.5× at q=10⁻². Crucially, metachecks constrain the syndrome, not the data, and cannot see faults deposited directly on data qubits by the final sweep — which is precisely where the factory floor originates. Single-shot therefore buys latency, not factory fidelity. Because the factory's Cliffords act on encoded blocks, p_gate is the logical error rate of the hosting code. The correction moves the surface-code requirement from d=15 to d=17 (578 physical qubits per patch); on the 4D toric side, the smallest sufficient block is L=4, [[1536,6,16]], with [[486,6,9]] missing the fidelity demand by four to five orders of magnitude. We situate the result against Xu et al. (arXiv:2407.18490), whose parallel distillation across qLDPC patches reduces space–time cost but leaves the per-factory floor untouched. A corrections log records eight claims withdrawn or corrected during this work, including our own earlier 4.18·p_gate figure and a d=15 requirement that rested on a single error event. Section 6 states what is not established: these are code-capacity numbers for the 4D code, the distance requirement is quoted at p=3–5% rather than 10⁻³, d=L² is verified exactly only at L=2, the Λ extrapolations are long, and no end-to-end injection simulation was run. Correction, 2026-07-31. The files deposited with this record on 2026-07-30 predate a low-p re-measurement of the resource-scaling section. The originally deposited text stated that hosting the factory requires L≈6–7 (d=36–49, n≈7800–14400); that figure came from fitting a single suppression factor Λ to near-threshold points at p=3–5%. The single-Λ ansatz does not hold for this code: the measured suppression per two units of distance runs from 10.9 (d=4→9 at p=0.02) to 793 (d=9→16 at p=0.01). Re-measuring at the factory's actual operating regime with a weight-stratified estimator gives d ≥ 14.5 at p=0.02 and d ≥ 9.7 at p=0.01, so the smallest sufficient 4D block is L=4, [[1536,6,16]], not L=6–7. The L≈6–7 figure is withdrawn. These are point estimates whose upper confidence bands remain wide at the smallest p, and should be read as order-of-magnitude. The corrected text and the estimator script (toric4d_lowp.py) are not in this record's file set; replacing files requires a new version, which will carry the corrected note. Nothing in Sections 1–4 — the 35p³ distillation result, the 12–15·p_gate factory floor, the last-sweep mechanism, or the d=17 surface-code requirement — is affected.