Utility-Constrained Pauli–Weyl Randomization for Robust Quantum α-z Rényi Privacy

Nguyen Hoang Minh, Trung Hoa Dinh, Le Cong Trinh

Published Jul 3, 2026• 0 citations

Abstract

Abstract Random-unitary noise is an implementable way to hide quantum information, since it can be realized by applying unitary gates sampled from a classical random key. However, the unconstrained minimization of pairwise quantum $\alpha$-$z$ R\'enyi divergences over all mixed-unitary channels is degenerate: the completely depolarizing channel is a Weyl average and maps every input state to $I_d/d$. This paper develops a non-erasing privacy--utility framework based on Pauli--Weyl randomization. For a finite protected ensemble $\mathcal{E}=\{\rho_1,\ldots,\rho_m\}$, a compact admissible set $K$ of Weyl probability distributions, and a fidelity-type utility threshold $\eta$, we study the robust radius $$R^\eta_{\alpha,z}(\mathcal{E};K) = \inf_{p\in K:\,\mathcal{U}_{\mathcal{E}}(\Phi_p)\ge\eta} \max_{i\ne j} D_{\alpha,z}\bigl(\Phi_p(\rho_i)\Vert\Phi_p(\rho_j)\bigr). $$ We prove well-posedness, a zero-radius characterization, and active-pair optimality conditions. The main exact result is a Pauli reduction theorem for equal-purity binary qubit ensembles: in adapted Bloch coordinates, optimization over all Pauli channels reduces to a two-parameter contraction problem separating the sensitive direction from the common direction. This yields an explicit two-eigenvalue formula for the reduced $\alpha$-$z$ divergence and a closed-form optimal mechanism on the fidelity line $\alpha=z=1/2$ for arbitrary pure binary qubit states. A complementary commuting result solves antipodal binary pairs throughout the data-processing region. For diagonal qudit ensembles, Weyl randomization reduces to cyclic convolution; the utility has a Fourier autocorrelation representation, and the minimax problem admits convex global formulations for $0<\alpha<1$ and $\alpha=2$. Finally, the radius is shown to be stable under post-processing and measurement, to imply binary discrimination bounds, and to support hardware-aware calibration certificates. MSC Classification: Primary 81P45 , 94A17; Secondary 81P50 , 47N50 , 49K35 , 90C25 ,90C30

DOI: 10.21203/rs.3.rs-10148794/v1

Related papers

Barycentric and pairwise Rényi quantum leakage with application to privacy-utility trade-off

Farhad Farokhi

Proceedings of the Royal Society A Mathematical Physical and Engineering SciencesDec 1, 20241 citation

Maximal Classicalization of Finite-Group Quantum Reference-Frame Noise

Maxim V. Churilov

quant-pharXiv:2607.12968Jul 15, 20260 citations

Barycentric and Pairwise Renyi Quantum Leakage

Farhad Farokhi

arXiv:2402.06156Feb 9, 20241 citation

Improved Quantum Algorithms for Rényi Entropies Estimation

Zongxing Xiong, Chengshen Gao, Lvzhou Li

quant-pharXiv:2610.19936Oct 15, 20260 citations

When Is History Truncation Safe for Quantum-Code Optimization? Projective-Unitary Path Collapse and Distribution-Robust Optimizer Reversal

Byoungwoo Lee

Jul 30, 20260 citations

Efficient information recovery from Pauli noise via classical shadow

Yifei Chen, Yu, Zhan, Chenghong Zhu +1 more

arXiv:2305.04148May 6, 20231 citation

Utility-Constrained Pauli–Weyl Randomization for Robust Quantum α-z Rényi Privacy

Nguyen Hoang Minh, Trung Hoa Dinh, Le Cong Trinh

Published Jul 3, 2026• 0 citations

Publisher

Abstract

\alpha

z

R'enyi divergences over all mixed-unitary channels is degenerate: the completely depolarizing channel is a Weyl average and maps every input state to

I_d/d

. This paper develops a non-erasing privacy--utility framework based on Pauli--Weyl randomization. For a finite protected ensemble

\mathcal{E}=\{\rho_1,\ldots,\rho_m\}

, a compact admissible set

K

of Weyl probability distributions, and a fidelity-type utility threshold

\eta

, we study the robust radius

R^\eta_{\alpha,z}(\mathcal{E};K) = \inf_{p\in K:\,\mathcal{U}_{\mathcal{E}}(\Phi_p)\ge\eta} \max_{i\ne j} D_{\alpha,z}\bigl(\Phi_p(\rho_i)\Vert\Phi_p(\rho_j)\bigr).

We prove well-posedness, a zero-radius characterization, and active-pair optimality conditions. The main exact result is a Pauli reduction theorem for equal-purity binary qubit ensembles: in adapted Bloch coordinates, optimization over all Pauli channels reduces to a two-parameter contraction problem separating the sensitive direction from the common direction. This yields an explicit two-eigenvalue formula for the reduced

\alpha

z

divergence and a closed-form optimal mechanism on the fidelity line

\alpha=z=1/2

for arbitrary pure binary qubit states. A complementary commuting result solves antipodal binary pairs throughout the data-processing region. For diagonal qudit ensembles, Weyl randomization reduces to cyclic convolution; the utility has a Fourier autocorrelation representation, and the minimax problem admits convex global formulations for

0<\alpha<1

and

\alpha=2

. Finally, the radius is shown to be stable under post-processing and measurement, to imply binary discrimination bounds, and to support hardware-aware calibration certificates. MSC Classification: Primary 81P45 , 94A17; Secondary 81P50 , 47N50 , 49K35 , 90C25 ,90C30

DOI: 10.21203/rs.3.rs-10148794/v1

Related papers

Barycentric and pairwise Rényi quantum leakage with application to privacy-utility trade-off

Farhad Farokhi

Barycentric and pairwise quantum Rényi leakages are proposed as two measures of information leakage for privacy and security analysis in quantum computing and communication systems. These quantities both require minimal assumptions on the eavesdropper, i.e. they do not make any assumptions on the eavesdropper’s attack strategy or the statistical prior on the secret or private classical data encoded in the quantum system. They also satisfy important properties of positivity, independence, post-processing inequality and unitary invariance. The barycentric quantum Rényi leakage can be computed by solving a semi-definite program; the pairwise quantum Rényi leakage possesses an explicit formula. The barycentric and pairwise quantum Rényi leakages form upper bounds on the maximal quantum leakage, the sandwiched quantum α -mutual information, the accessible information and the Holevo’s information. Furthermore, differentially private quantum channels are shown to bound these measures of information leakage. Global and local depolarizing channels, that are common models of noise in quantum computing and communication, restrict private or secure information leakage. Finally, a privacy-utility trade-off formula in quantum machine learning using variational circuits is developed. The privacy guarantees can only be strengthened, i.e. information leakage can only be reduced, if the performance degradation grows larger and vice versa .Barycentric and pairwise quantum Rényi leakages are proposed as two measures of information leakage for privacy and security analysis in quantum computing and communication systems. These quantities both require minimal assumptions on the eavesdropper, i.e. they do not make any assumptions on the eavesdropper’s attack strategy or the statistical prior on the secret or private classical data encoded in the quantum system. They also satisfy important properties of positivity, independence, post-processing inequality and unitary invariance. The barycentric quantum Rényi leakage can be computed by solving a semi-definite program; the pairwise quantum Rényi leakage possesses an explicit formula. The barycentric and pairwise quantum Rényi leakages form upper bounds on the maximal quantum leakage, the sandwiched quantum α -mutual information, the accessible information and the Holevo’s information. Furthermore, differentially private quantum channels are shown to bound these measures of information leakage. Global and local depolarizing channels, that are common models of noise in quantum computing and communication, restrict private or secure information leakage. Finally, a privacy-utility trade-off formula in quantum machine learning using variational circuits is developed. The privacy guarantees can only be strengthened, i.e. information leakage can only be reduced, if the performance degradation grows larger and vice versa .

Proceedings of the Royal Society A Mathematical Physical and Engineering SciencesDec 1, 20241 citation

Maximal Classicalization of Finite-Group Quantum Reference-Frame Noise

Maxim V. Churilov

A finite quantum reference token with group-valued misalignment induces a random-unitary channel, but optimal degradation is generally an optimization over all quantum post-processings. For a unitary representation U of a finite group G, we prove that the following conditions are equivalent: U contains every irreducible type; one ancilla-assisted input has an orthonormal G-orbit; signed group measures embed isometrically into channels in diamond norm; and, for every pair of noise laws p,q, $\inf_{Λ\in\mathrm{CPTP}} \frac{1}{2}\|Φ_q^U-Λ\circΦ_p^U\|_\diamond =\min_{r\in\mathcal{P}(G)}\frac{1}{2}\|q-r*p\|_1$ . Thus representation completeness is the exact carrier condition for universal reduction of quantum post-processing to classical convolution. We determine the minimum ancilla dimensions for an orthogonal orbit and for an invariant calibration seed. For an incomplete carrier, with visible Plancherel dimension S(U), we derive the exact conditional-expectation distance $\frac{1}{2}\|\operatorname{id}-Φ_u^U\|_\diamond=1-1/S(U)$ and an explicit quantum--classical deficiency gap. For irreducible carriers the deficiency is obtained in closed form; the faithful two-dimensional representation of $S_3$ yields an exact ten-percent reduction relative to classical convolution. We also characterize law identifiability through the conjugation representation, provide finite linear programs and decision witnesses, and establish both a finite-dimensional obstruction and stable visible-band reconstruction for infinite compact groups. Deterministic ancillary code reproduces the finite-group examples and numerical regression checks.A finite quantum reference token with group-valued misalignment induces a random-unitary channel, but optimal degradation is generally an optimization over all quantum post-processings. For a unitary representation U of a finite group G, we prove that the following conditions are equivalent: U contains every irreducible type; one ancilla-assisted input has an orthonormal G-orbit; signed group measures embed isometrically into channels in diamond norm; and, for every pair of noise laws p,q, $\inf_{Λ\in\mathrm{CPTP}} \frac{1}{2}\|Φ_q^U-Λ\circΦ_p^U\|_\diamond =\min_{r\in\mathcal{P}(G)}\frac{1}{2}\|q-r*p\|_1$. Thus representation completeness is the exact carrier condition for universal reduction of quantum post-processing to classical convolution. We determine the minimum ancilla dimensions for an orthogonal orbit and for an invariant calibration seed. For an incomplete carrier, with visible Plancherel dimension S(U), we derive the exact conditional-expectation distance $\frac{1}{2}\|\operatorname{id}-Φ_u^U\|_\diamond=1-1/S(U)$ and an explicit quantum--classical deficiency gap. For irreducible carriers the deficiency is obtained in closed form; the faithful two-dimensional representation of $S_3$ yields an exact ten-percent reduction relative to classical convolution. We also characterize law identifiability through the conjugation representation, provide finite linear programs and decision witnesses, and establish both a finite-dimensional obstruction and stable visible-band reconstruction for infinite compact groups. Deterministic ancillary code reproduces the finite-group examples and numerical regression checks.

quant-pharXiv:2607.12968Jul 15, 20260 citations

Barycentric and Pairwise Renyi Quantum Leakage

Farhad Farokhi

Barycentric and pairwise quantum Renyi leakages are proposed as two measures of information leakage for privacy and security analysis in quantum computing and communication systems. These quantities both require minimal assumptions on the eavesdropper, i.e., they do not make any assumptions on the eavesdropper's attack strategy or the statistical prior on the secret or private classical data encoded in the quantum system. They also satisfy important properties of positivity, independence, post-processing inequality, and unitary invariance. The barycentric quantum Renyi leakage can be computed by solving a semi-definite program and the pairwise quantum Renyi leakage possesses an explicit formula. The barycentric and pairwise quantum Renyi leakages form upper bounds on the maximal quantum leakage, the sandwiched quantum $α$ -mutual information, the accessible information, and the Holevo's information. Furthermore, differentially-private quantum channels are shown to bound these measures of information leakage. Global and local depolarizing channels, that are common models of noise in quantum computing and communication, restrict private or secure information leakage. Finally, a privacy-utility trade-off formula in quantum machine learning using variational circuits is developed. The privacy guarantees can only be strengthened, i.e., information leakage can only be reduced, if the performance degradation grows larger and vice versa.Barycentric and pairwise quantum Renyi leakages are proposed as two measures of information leakage for privacy and security analysis in quantum computing and communication systems. These quantities both require minimal assumptions on the eavesdropper, i.e., they do not make any assumptions on the eavesdropper's attack strategy or the statistical prior on the secret or private classical data encoded in the quantum system. They also satisfy important properties of positivity, independence, post-processing inequality, and unitary invariance. The barycentric quantum Renyi leakage can be computed by solving a semi-definite program and the pairwise quantum Renyi leakage possesses an explicit formula. The barycentric and pairwise quantum Renyi leakages form upper bounds on the maximal quantum leakage, the sandwiched quantum $α$-mutual information, the accessible information, and the Holevo's information. Furthermore, differentially-private quantum channels are shown to bound these measures of information leakage. Global and local depolarizing channels, that are common models of noise in quantum computing and communication, restrict private or secure information leakage. Finally, a privacy-utility trade-off formula in quantum machine learning using variational circuits is developed. The privacy guarantees can only be strengthened, i.e., information leakage can only be reduced, if the performance degradation grows larger and vice versa.

arXiv:2402.06156Feb 9, 20241 citation

Improved Quantum Algorithms for Rényi Entropies Estimation

Zongxing Xiong, Chengshen Gao, Lvzhou Li

Entropy estimation of discrete probability distributions is a fundamental problem in statistical inference and quantum property estimation. In this paper, we study the problem of estimating Rényi entropy $H_α(\mathbf{p})$ for any distribution $\mathbf{p} = (p_i)_{i=1}^n$ given its purified quantum query-access. For non-integer $α> 1$ , our algorithm estimates $H_α(\mathbf{p})$ to within additive error $ε$ using only $\widetilde{\mathcal{O}}\big(n^{\max\{1/2,1-1/α\}}/ε\big)$ queries, which improves the best known upper bound for dependence on both $n$ and $1/ε$ . For $0<α<1$ , our algorithm has query complexity $\widetilde{\mathcal{O}}\big(n^{1/(2α)}/ε^{\max\{1,1/(2α)\}}\big)$ , which improves the dependence on $1/ε$ of the current best upper bound without incresing its $n$ dependence. Our approach adapts the multi-level estimation framework to relative approximation of the power sums and leverage binary search to remove the requirement of prior knowledge of them. We also strengthen the precision-dependent lower bound to $Ω\big(1/ε^{\max\{1,1/(2α)\}}\big)$ for $0<α<1$ , which identifies the optimal precision dependence of our algorithm.Entropy estimation of discrete probability distributions is a fundamental problem in statistical inference and quantum property estimation. In this paper, we study the problem of estimating Rényi entropy $H_α(\mathbf{p})$ for any distribution $\mathbf{p} = (p_i)_{i=1}^n$ given its purified quantum query-access. For non-integer $α> 1$, our algorithm estimates $H_α(\mathbf{p})$ to within additive error $ε$ using only $\widetilde{\mathcal{O}}\big(n^{\max\{1/2,1-1/α\}}/ε\big)$ queries, which improves the best known upper bound for dependence on both $n$ and $1/ε$. For $0<α<1$, our algorithm has query complexity $\widetilde{\mathcal{O}}\big(n^{1/(2α)}/ε^{\max\{1,1/(2α)\}}\big)$, which improves the dependence on $1/ε$ of the current best upper bound without incresing its $n$ dependence. Our approach adapts the multi-level estimation framework to relative approximation of the power sums and leverage binary search to remove the requirement of prior knowledge of them. We also strengthen the precision-dependent lower bound to $Ω\big(1/ε^{\max\{1,1/(2α)\}}\big)$ for $0<α<1$, which identifies the optimal precision dependence of our algorithm.

quant-pharXiv:2610.19936Oct 15, 20260 citations

When Is History Truncation Safe for Quantum-Code Optimization? Projective-Unitary Path Collapse and Distribution-Robust Optimizer Reversal

Byoungwoo Lee

## Overview This paper studies when a reduction of a finite quantum error-history model preserves the local direction of quantum-code optimization, and when it reverses that direction. For finite coactive spectral branches, the paper first proves a second-order local truncation-safety certificate. A strict optimizer of a truncated model persists when the full-versus-truncated effective-jet discrepancy is smaller than the coercivity margin of the truncated objective. Opposite leading jets provide a certificate of local optimizer reversal. The paper then identifies an exact boundary between genuine temporal structure and removable path redundancy. For a finite projective unitary error group, any open-loop path operator has the form $W_\gamma=e^{i\theta_\gamma}U_{c(\gamma)},$ where $c(\gamma)$ is the final group product. Consequently, an arbitrary temporally correlated path law is isometrically equivalent, for complete complementary leakage and for every code, to the single-shot distribution of the final projective product: $\mathcal L_{\mathrm{path}}(V)=\mathcal L_{\pi}(V).$ For Pauli trajectories, this remains true even when ordered errors do not commute, because different orderings can differ only by projective phases after contraction to the final channel. Thus open-loop final-channel leakage cannot identify temporal organization beyond the final projective Pauli-product distribution. Intermediate measurements, controls, syndrome extraction, or adaptive recovery are required for genuinely intervention-resolved multi-time code geometry. ## Distribution-robust optimizer reversal For an arbitrary three-location bit-flip trajectory law, let $\pi$ denote the induced final-parity distribution on $\mathbb F_2^3$ , let $c=(1,1,1)$ , and define $\Lambda_\pi:=\sum_{t\in\mathbb F_2^3}\sqrt{\pi_t\pi_{t\oplus c}}.$ For the three-qubit repetition code $E=\operatorname{span}\{|000\rangle,|111\rangle\},$ the complete complementary leakage is exactly $\mathcal L_\pi(E)=\Lambda_\pi.$ For every fixed full-support final-parity law, the intrinsic Fourier-syndrome graph coordinate $w$ satisfies $\mathcal L_\pi(V(w))=\Lambda_\pi-2\Lambda_\pi|w|^2+o(|w|^2).$ Hence the repetition code is a strict local maximum throughout this two-real-dimensional graph chart for every full-support final-parity distribution, independently of temporal correlations, stationarity, roundwise independence, identical error rates, or cyclic spatial symmetry. By contrast, a conditioned zero-and-one-fault truncation with positive coarse single-location weights has $\mathcal L_{\leq 1}(E)=0$ and strictly positive leakage at every nonzero sufficiently nearby graph code. The same reference code is therefore a strict local minimum for the truncated model and a strict local maximum for the complete process. The central conclusion is: > Exact aggregation of paths that implement the same projective physical operator preserves the entire code-optimization landscape, whereas deletion of higher-fault history fibers can reverse the local optimizer geometry. ## Scope The results concern random-unitary, open-loop finite-history models and complete complementary-channel leakage. The optimizer reversal is proved in a specified Fourier-syndrome graph chart around the three-qubit repetition code. The paper does not claim: - a full-Grassmannian classification for the three-qubit model;- a decoder or recovery-circuit construction;- a logical-error threshold;- a circuit-level hardware noise model;- collapse of an intervention-resolved process tensor;- equivalence in the presence of intermediate syndrome measurements, controls, or adaptive recovery. These intervention-resolved settings form the natural next boundary of the theory. ## Relation to the preceding research program This paper is a concluding bridge theorem paper connected to a six-paper exact finite-history QEC and spectral-minimax research line. ### 1. Compression-geometric foundation **Sharp Complementarity, Complete Attainable Region, and Metric Rigidity of Rank-Two Quantum Codes under Collective and Differential Dephasing** Version v0.5r2b DOI: 10.5281/zenodo.21523950 This paper determined the complete attainable compression-defect region for rank-two codes under collective and differential two-qubit dephasing, classified the equality geometry, and established sharp metric rigidity. ### 2. Balanced finite-history minimax **Environment-Record Leakage for Finite Quantum Error Histories: Balanced Global Minimax, Complete Minimizer Geometry, and Distributional Stability** Version v0.5r2 DOI: 10.5281/zenodo.21560397 This paper introduced the complete environment-record leakage objective for finite error histories, solved the balanced two-step minimax problem globally over $\operatorname{Gr}(2,4)$ , classified all balanced minimizers, and established distributional stability. ### 3. Bias-induced geometric selection **Bias-Induced Non-Isoclinic Optimality in Finite Quantum Error Histories: Exact Separation, Thresholdless Bifurcation, and Near-Balanced All-Code Minimax** Version v0.3r2 DOI: 10.5281/zenodo.21598820 This paper proved exact non-isoclinic separation, an open non-isoclinic parameter region, thresholdless loss of isoclinic optimality under arbitrarily small bias, and the leading full-Grassmannian near-balanced minimax selector. ### 4. Fourth-order overlap selection **Flat-Interior Persistence and Singular Overlap Selection in Near-Balanced Quantum Error Histories: Fourth-Order Transition Curves and Full-Grassmannian Minimax Closure** Version v0.1r3 DOI: 10.5281/zenodo.21622029 This sequel resolved the higher-order overlap degeneracy, established flat-interior persistence, derived singular boundary-layer transition curves, and closed the near-balanced full-Grassmannian minimax expansion through fourth order. ### 5. Abstract reflected spectral-jet framework **Reflected Spectral-Jet Realization for Degenerate Minimax Problems: Hard-Normal Selection, Flat Tangential Persistence, and Coactive Boundary Transitions** Version v0.2r1 DOI: 10.5281/zenodo.21636022 This paper extracted a reusable abstract framework for hard-normal isolation, soft tangential persistence, reflected active branches, and coactive boundary transitions in degenerate spectral minimax problems. ### 6. Finite-orbit generalization and first QEC truncation reversal **Beyond Reflected Pairs: Finite-Orbit Spectral Jets, Quadratic Isolation, and Multi-Branch Coactivity in Degenerate Minimax Problems** Version v0.4r1 DOI: 10.5281/zenodo.21644364 This paper generalized the reflected-pair theory to finite transitive orbits, developed multi-branch coactivity and representation-theoretic jet obstructions, and exhibited an exact three-location QEC realization in which restoring omitted multi-fault histories reverses the local optimizer geometry. ### 7. Present bridge theorem The present paper determines which open-loop temporal path information is exactly removable and which history reductions remain unsafe. It extends the earlier single-process reversal from independent or cyclic models to arbitrary correlated projective-unitary paths and arbitrary full-support final-parity laws. The resulting research progression is $\text{compression geometry}\longrightarrow\text{balanced global minimax}\longrightarrow\text{bias-induced selection}\longrightarrow\text{fourth-order closure}\longrightarrow\text{reflected spectral jets}\longrightarrow\text{finite-orbit coactivity}\longrightarrow\text{projective path collapse and distribution-robust reversal}.$ ## Reproducibility package The accompanying bundle contains: - the manuscript PDF;- the LaTeX source;- an exact symbolic verification script;- a machine-readable verification report;- a README describing the proof-artifact contract;- a SHA-256 manifest;- version and package-audit records. The verifier performs 72 exact symbolic checks, including projective Pauli phase identities, phase-covariant path compression, arbitrary-parity polar factors, translation orthogonality, Fourier-syndrome graph compressions, and coefficient bookkeeping for the distribution-robust local expansion. The verifier deliberately rejects optimized Python mode. PASS decisions do not rely on floating-point tolerances. Compactness, maximizing-direction localization, trace-norm differentiability, and uniform asymptotic assembly remain analytic parts of the manuscript proof rather than delegated computational claims.## Overview This paper studies when a reduction of a finite quantum error-history model preserves the local direction of quantum-code optimization, and when it reverses that direction. For finite coactive spectral branches, the paper first proves a second-order local truncation-safety certificate. A strict optimizer of a truncated model persists when the full-versus-truncated effective-jet discrepancy is smaller than the coercivity margin of the truncated objective. Opposite leading jets provide a certificate of local optimizer reversal. The paper then identifies an exact boundary between genuine temporal structure and removable path redundancy. For a finite projective unitary error group, any open-loop path operator has the form $$W_\gamma=e^{i\theta_\gamma}U_{c(\gamma)},$$ where $c(\gamma)$ is the final group product. Consequently, an arbitrary temporally correlated path law is isometrically equivalent, for complete complementary leakage and for every code, to the single-shot distribution of the final projective product: $$\mathcal L_{\mathrm{path}}(V)=\mathcal L_{\pi}(V).$$ For Pauli trajectories, this remains true even when ordered errors do not commute, because different orderings can differ only by projective phases after contraction to the final channel. Thus open-loop final-channel leakage cannot identify temporal organization beyond the final projective Pauli-product distribution. Intermediate measurements, controls, syndrome extraction, or adaptive recovery are required for genuinely intervention-resolved multi-time code geometry. ## Distribution-robust optimizer reversal For an arbitrary three-location bit-flip trajectory law, let $\pi$ denote the induced final-parity distribution on $\mathbb F_2^3$, let $c=(1,1,1)$, and define $$\Lambda_\pi:=\sum_{t\in\mathbb F_2^3}\sqrt{\pi_t\pi_{t\oplus c}}.$$ For the three-qubit repetition code $$E=\operatorname{span}\{|000\rangle,|111\rangle\},$$ the complete complementary leakage is exactly $$\mathcal L_\pi(E)=\Lambda_\pi.$$ For every fixed full-support final-parity law, the intrinsic Fourier-syndrome graph coordinate $w$ satisfies $$\mathcal L_\pi(V(w))=\Lambda_\pi-2\Lambda_\pi|w|^2+o(|w|^2).$$ Hence the repetition code is a strict local maximum throughout this two-real-dimensional graph chart for every full-support final-parity distribution, independently of temporal correlations, stationarity, roundwise independence, identical error rates, or cyclic spatial symmetry. By contrast, a conditioned zero-and-one-fault truncation with positive coarse single-location weights has $$\mathcal L_{\leq 1}(E)=0$$ and strictly positive leakage at every nonzero sufficiently nearby graph code. The same reference code is therefore a strict local minimum for the truncated model and a strict local maximum for the complete process. The central conclusion is: > Exact aggregation of paths that implement the same projective physical operator preserves the entire code-optimization landscape, whereas deletion of higher-fault history fibers can reverse the local optimizer geometry. ## Scope The results concern random-unitary, open-loop finite-history models and complete complementary-channel leakage. The optimizer reversal is proved in a specified Fourier-syndrome graph chart around the three-qubit repetition code. The paper does not claim: - a full-Grassmannian classification for the three-qubit model;- a decoder or recovery-circuit construction;- a logical-error threshold;- a circuit-level hardware noise model;- collapse of an intervention-resolved process tensor;- equivalence in the presence of intermediate syndrome measurements, controls, or adaptive recovery. These intervention-resolved settings form the natural next boundary of the theory. ## Relation to the preceding research program This paper is a concluding bridge theorem paper connected to a six-paper exact finite-history QEC and spectral-minimax research line. ### 1. Compression-geometric foundation **Sharp Complementarity, Complete Attainable Region, and Metric Rigidity of Rank-Two Quantum Codes under Collective and Differential Dephasing** Version v0.5r2b DOI: 10.5281/zenodo.21523950 This paper determined the complete attainable compression-defect region for rank-two codes under collective and differential two-qubit dephasing, classified the equality geometry, and established sharp metric rigidity. ### 2. Balanced finite-history minimax **Environment-Record Leakage for Finite Quantum Error Histories: Balanced Global Minimax, Complete Minimizer Geometry, and Distributional Stability** Version v0.5r2 DOI: 10.5281/zenodo.21560397 This paper introduced the complete environment-record leakage objective for finite error histories, solved the balanced two-step minimax problem globally over $\operatorname{Gr}(2,4)$, classified all balanced minimizers, and established distributional stability. ### 3. Bias-induced geometric selection **Bias-Induced Non-Isoclinic Optimality in Finite Quantum Error Histories: Exact Separation, Thresholdless Bifurcation, and Near-Balanced All-Code Minimax** Version v0.3r2 DOI: 10.5281/zenodo.21598820 This paper proved exact non-isoclinic separation, an open non-isoclinic parameter region, thresholdless loss of isoclinic optimality under arbitrarily small bias, and the leading full-Grassmannian near-balanced minimax selector. ### 4. Fourth-order overlap selection **Flat-Interior Persistence and Singular Overlap Selection in Near-Balanced Quantum Error Histories: Fourth-Order Transition Curves and Full-Grassmannian Minimax Closure** Version v0.1r3 DOI: 10.5281/zenodo.21622029 This sequel resolved the higher-order overlap degeneracy, established flat-interior persistence, derived singular boundary-layer transition curves, and closed the near-balanced full-Grassmannian minimax expansion through fourth order. ### 5. Abstract reflected spectral-jet framework **Reflected Spectral-Jet Realization for Degenerate Minimax Problems: Hard-Normal Selection, Flat Tangential Persistence, and Coactive Boundary Transitions** Version v0.2r1 DOI: 10.5281/zenodo.21636022 This paper extracted a reusable abstract framework for hard-normal isolation, soft tangential persistence, reflected active branches, and coactive boundary transitions in degenerate spectral minimax problems. ### 6. Finite-orbit generalization and first QEC truncation reversal **Beyond Reflected Pairs: Finite-Orbit Spectral Jets, Quadratic Isolation, and Multi-Branch Coactivity in Degenerate Minimax Problems** Version v0.4r1 DOI: 10.5281/zenodo.21644364 This paper generalized the reflected-pair theory to finite transitive orbits, developed multi-branch coactivity and representation-theoretic jet obstructions, and exhibited an exact three-location QEC realization in which restoring omitted multi-fault histories reverses the local optimizer geometry. ### 7. Present bridge theorem The present paper determines which open-loop temporal path information is exactly removable and which history reductions remain unsafe. It extends the earlier single-process reversal from independent or cyclic models to arbitrary correlated projective-unitary paths and arbitrary full-support final-parity laws. The resulting research progression is $$\text{compression geometry}\longrightarrow\text{balanced global minimax}\longrightarrow\text{bias-induced selection}\longrightarrow\text{fourth-order closure}\longrightarrow\text{reflected spectral jets}\longrightarrow\text{finite-orbit coactivity}\longrightarrow\text{projective path collapse and distribution-robust reversal}.$$ ## Reproducibility package The accompanying bundle contains: - the manuscript PDF;- the LaTeX source;- an exact symbolic verification script;- a machine-readable verification report;- a README describing the proof-artifact contract;- a SHA-256 manifest;- version and package-audit records. The verifier performs 72 exact symbolic checks, including projective Pauli phase identities, phase-covariant path compression, arbitrary-parity polar factors, translation orthogonality, Fourier-syndrome graph compressions, and coefficient bookkeeping for the distribution-robust local expansion. The verifier deliberately rejects optimized Python mode. PASS decisions do not rely on floating-point tolerances. Compactness, maximizing-direction localization, trace-norm differentiability, and uniform asymptotic assembly remain analytic parts of the manuscript proof rather than delegated computational claims.

Jul 30, 20260 citations

Efficient information recovery from Pauli noise via classical shadow

Yifei Chen, Yu, Zhan, Chenghong Zhu +1 more

The rapid advancement of quantum computing has led to an extensive demand for effective techniques to extract classical information from quantum systems, particularly in fields like quantum machine learning and quantum chemistry. However, quantum systems are inherently susceptible to noises, which adversely corrupt the information encoded in quantum systems. In this work, we introduce an efficient algorithm that can recover information from quantum states under Pauli noise. The core idea is to learn the necessary information of the unknown Pauli channel by post-processing the classical shadows of the channel. For a local and bounded-degree observable, only partial knowledge of the channel is required rather than its complete classical description to recover the ideal information, resulting in a polynomial-time algorithm. This contrasts with conventional methods such as probabilistic error cancellation, which requires the full information of the channel and exhibits exponential scaling with the number of qubits. We also prove that this scalable method is optimal on the sample complexity and generalise the algorithm to the weight contracting channel. Furthermore, we demonstrate the validity of the algorithm on the 1D anisotropic Heisenberg-type model via numerical simulations. As a notable application, our method can be severed as a sample-efficient error mitigation scheme for Clifford circuits.The rapid advancement of quantum computing has led to an extensive demand for effective techniques to extract classical information from quantum systems, particularly in fields like quantum machine learning and quantum chemistry. However, quantum systems are inherently susceptible to noises, which adversely corrupt the information encoded in quantum systems. In this work, we introduce an efficient algorithm that can recover information from quantum states under Pauli noise. The core idea is to learn the necessary information of the unknown Pauli channel by post-processing the classical shadows of the channel. For a local and bounded-degree observable, only partial knowledge of the channel is required rather than its complete classical description to recover the ideal information, resulting in a polynomial-time algorithm. This contrasts with conventional methods such as probabilistic error cancellation, which requires the full information of the channel and exhibits exponential scaling with the number of qubits. We also prove that this scalable method is optimal on the sample complexity and generalise the algorithm to the weight contracting channel. Furthermore, we demonstrate the validity of the algorithm on the 1D anisotropic Heisenberg-type model via numerical simulations. As a notable application, our method can be severed as a sample-efficient error mitigation scheme for Clifford circuits.

arXiv:2305.04148May 6, 20231 citation

\alpha

z

R'enyi divergences over all mixed-unitary channels is degenerate: the completely depolarizing channel is a Weyl average and maps every input state to

I_d/d

. This paper develops a non-erasing privacy--utility framework based on Pauli--Weyl randomization. For a finite protected ensemble

\mathcal{E}=\{\rho_1,\ldots,\rho_m\}

, a compact admissible set

K

of Weyl probability distributions, and a fidelity-type utility threshold

\eta

, we study the robust radius

R^\eta_{\alpha,z}(\mathcal{E};K) = \inf_{p\in K:\,\mathcal{U}_{\mathcal{E}}(\Phi_p)\ge\eta} \max_{i\ne j} D_{\alpha,z}\bigl(\Phi_p(\rho_i)\Vert\Phi_p(\rho_j)\bigr).

\alpha

z

divergence and a closed-form optimal mechanism on the fidelity line

\alpha=z=1/2

0<\alpha<1

and

\alpha=2

\inf_{Λ\in\mathrm{CPTP}} \frac{1}{2}\|Φ_q^U-Λ\circΦ_p^U\|_\diamond =\min_{r\in\mathcal{P}(G)}\frac{1}{2}\|q-r*p\|_1

. Thus representation completeness is the exact carrier condition for universal reduction of quantum post-processing to classical convolution. We determine the minimum ancilla dimensions for an orthogonal orbit and for an invariant calibration seed. For an incomplete carrier, with visible Plancherel dimension S(U), we derive the exact conditional-expectation distance

\frac{1}{2}\|\operatorname{id}-Φ_u^U\|_\diamond=1-1/S(U)

and an explicit quantum--classical deficiency gap. For irreducible carriers the deficiency is obtained in closed form; the faithful two-dimensional representation of

S_3

yields an exact ten-percent reduction relative to classical convolution. We also characterize law identifiability through the conjugation representation, provide finite linear programs and decision witnesses, and establish both a finite-dimensional obstruction and stable visible-band reconstruction for infinite compact groups. Deterministic ancillary code reproduces the finite-group examples and numerical regression checks.

α

-mutual information, the accessible information, and the Holevo's information. Furthermore, differentially-private quantum channels are shown to bound these measures of information leakage. Global and local depolarizing channels, that are common models of noise in quantum computing and communication, restrict private or secure information leakage. Finally, a privacy-utility trade-off formula in quantum machine learning using variational circuits is developed. The privacy guarantees can only be strengthened, i.e., information leakage can only be reduced, if the performance degradation grows larger and vice versa.

H_α(\mathbf{p})

for any distribution

\mathbf{p} = (p_i)_{i=1}^n

given its purified quantum query-access. For non-integer

α> 1

, our algorithm estimates

H_α(\mathbf{p})

to within additive error

ε

using only

\widetilde{\mathcal{O}}\big(n^{\max\{1/2,1-1/α\}}/ε\big)

queries, which improves the best known upper bound for dependence on both

n

and

1/ε

. For

0<α<1

, our algorithm has query complexity

\widetilde{\mathcal{O}}\big(n^{1/(2α)}/ε^{\max\{1,1/(2α)\}}\big)

, which improves the dependence on

1/ε

of the current best upper bound without incresing its

n

dependence. Our approach adapts the multi-level estimation framework to relative approximation of the power sums and leverage binary search to remove the requirement of prior knowledge of them. We also strengthen the precision-dependent lower bound to

Ω\big(1/ε^{\max\{1,1/(2α)\}}\big)

for

0<α<1

, which identifies the optimal precision dependence of our algorithm.

W_\gamma=e^{i\theta_\gamma}U_{c(\gamma)},

where

c(\gamma)

is the final group product. Consequently, an arbitrary temporally correlated path law is isometrically equivalent, for complete complementary leakage and for every code, to the single-shot distribution of the final projective product:

\mathcal L_{\mathrm{path}}(V)=\mathcal L_{\pi}(V).

For Pauli trajectories, this remains true even when ordered errors do not commute, because different orderings can differ only by projective phases after contraction to the final channel. Thus open-loop final-channel leakage cannot identify temporal organization beyond the final projective Pauli-product distribution. Intermediate measurements, controls, syndrome extraction, or adaptive recovery are required for genuinely intervention-resolved multi-time code geometry. ## Distribution-robust optimizer reversal For an arbitrary three-location bit-flip trajectory law, let

\pi

denote the induced final-parity distribution on

\mathbb F_2^3

, let

c=(1,1,1)

, and define

\Lambda_\pi:=\sum_{t\in\mathbb F_2^3}\sqrt{\pi_t\pi_{t\oplus c}}.

For the three-qubit repetition code

E=\operatorname{span}\{|000\rangle,|111\rangle\},

the complete complementary leakage is exactly

\mathcal L_\pi(E)=\Lambda_\pi.

For every fixed full-support final-parity law, the intrinsic Fourier-syndrome graph coordinate

w

satisfies

\mathcal L_\pi(V(w))=\Lambda_\pi-2\Lambda_\pi|w|^2+o(|w|^2).

Hence the repetition code is a strict local maximum throughout this two-real-dimensional graph chart for every full-support final-parity distribution, independently of temporal correlations, stationarity, roundwise independence, identical error rates, or cyclic spatial symmetry. By contrast, a conditioned zero-and-one-fault truncation with positive coarse single-location weights has

\mathcal L_{\leq 1}(E)=0

and strictly positive leakage at every nonzero sufficiently nearby graph code. The same reference code is therefore a strict local minimum for the truncated model and a strict local maximum for the complete process. The central conclusion is: > Exact aggregation of paths that implement the same projective physical operator preserves the entire code-optimization landscape, whereas deletion of higher-fault history fibers can reverse the local optimizer geometry. ## Scope The results concern random-unitary, open-loop finite-history models and complete complementary-channel leakage. The optimizer reversal is proved in a specified Fourier-syndrome graph chart around the three-qubit repetition code. The paper does not claim: - a full-Grassmannian classification for the three-qubit model;- a decoder or recovery-circuit construction;- a logical-error threshold;- a circuit-level hardware noise model;- collapse of an intervention-resolved process tensor;- equivalence in the presence of intermediate syndrome measurements, controls, or adaptive recovery. These intervention-resolved settings form the natural next boundary of the theory. ## Relation to the preceding research program This paper is a concluding bridge theorem paper connected to a six-paper exact finite-history QEC and spectral-minimax research line. ### 1. Compression-geometric foundation **Sharp Complementarity, Complete Attainable Region, and Metric Rigidity of Rank-Two Quantum Codes under Collective and Differential Dephasing** Version v0.5r2b DOI: 10.5281/zenodo.21523950 This paper determined the complete attainable compression-defect region for rank-two codes under collective and differential two-qubit dephasing, classified the equality geometry, and established sharp metric rigidity. ### 2. Balanced finite-history minimax **Environment-Record Leakage for Finite Quantum Error Histories: Balanced Global Minimax, Complete Minimizer Geometry, and Distributional Stability** Version v0.5r2 DOI: 10.5281/zenodo.21560397 This paper introduced the complete environment-record leakage objective for finite error histories, solved the balanced two-step minimax problem globally over

\operatorname{Gr}(2,4)

, classified all balanced minimizers, and established distributional stability. ### 3. Bias-induced geometric selection **Bias-Induced Non-Isoclinic Optimality in Finite Quantum Error Histories: Exact Separation, Thresholdless Bifurcation, and Near-Balanced All-Code Minimax** Version v0.3r2 DOI: 10.5281/zenodo.21598820 This paper proved exact non-isoclinic separation, an open non-isoclinic parameter region, thresholdless loss of isoclinic optimality under arbitrarily small bias, and the leading full-Grassmannian near-balanced minimax selector. ### 4. Fourth-order overlap selection **Flat-Interior Persistence and Singular Overlap Selection in Near-Balanced Quantum Error Histories: Fourth-Order Transition Curves and Full-Grassmannian Minimax Closure** Version v0.1r3 DOI: 10.5281/zenodo.21622029 This sequel resolved the higher-order overlap degeneracy, established flat-interior persistence, derived singular boundary-layer transition curves, and closed the near-balanced full-Grassmannian minimax expansion through fourth order. ### 5. Abstract reflected spectral-jet framework **Reflected Spectral-Jet Realization for Degenerate Minimax Problems: Hard-Normal Selection, Flat Tangential Persistence, and Coactive Boundary Transitions** Version v0.2r1 DOI: 10.5281/zenodo.21636022 This paper extracted a reusable abstract framework for hard-normal isolation, soft tangential persistence, reflected active branches, and coactive boundary transitions in degenerate spectral minimax problems. ### 6. Finite-orbit generalization and first QEC truncation reversal **Beyond Reflected Pairs: Finite-Orbit Spectral Jets, Quadratic Isolation, and Multi-Branch Coactivity in Degenerate Minimax Problems** Version v0.4r1 DOI: 10.5281/zenodo.21644364 This paper generalized the reflected-pair theory to finite transitive orbits, developed multi-branch coactivity and representation-theoretic jet obstructions, and exhibited an exact three-location QEC realization in which restoring omitted multi-fault histories reverses the local optimizer geometry. ### 7. Present bridge theorem The present paper determines which open-loop temporal path information is exactly removable and which history reductions remain unsafe. It extends the earlier single-process reversal from independent or cyclic models to arbitrary correlated projective-unitary paths and arbitrary full-support final-parity laws. The resulting research progression is

\text{compression geometry}\longrightarrow\text{balanced global minimax}\longrightarrow\text{bias-induced selection}\longrightarrow\text{fourth-order closure}\longrightarrow\text{reflected spectral jets}\longrightarrow\text{finite-orbit coactivity}\longrightarrow\text{projective path collapse and distribution-robust reversal}.

## Reproducibility package The accompanying bundle contains: - the manuscript PDF;- the LaTeX source;- an exact symbolic verification script;- a machine-readable verification report;- a README describing the proof-artifact contract;- a SHA-256 manifest;- version and package-audit records. The verifier performs 72 exact symbolic checks, including projective Pauli phase identities, phase-covariant path compression, arbitrary-parity polar factors, translation orthogonality, Fourier-syndrome graph compressions, and coefficient bookkeeping for the distribution-robust local expansion. The verifier deliberately rejects optimized Python mode. PASS decisions do not rely on floating-point tolerances. Compactness, maximizing-direction localization, trace-norm differentiability, and uniform asymptotic assembly remain analytic parts of the manuscript proof rather than delegated computational claims.