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arXiv:2609.36541

Exponential Advantage of Quantum over Classical References in Leakage Detection

Zheng An

quant-phSubmitted Sep 29, 2026• Announced Sep 30, 2026• 0 citations
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Abstract

Leakage from a known encoding subspace can be detected by projection. However, the corresponding projector is unavailable when the encoding subspace is not classically specified. Here we show how independent quantum references prepared by the same encoder enable leakage detection without a classical description of the encoding. A coherent measurement on the references and a single message leaves every state within the two-dimensional encoding subspace, including its entanglement with a remote system, exactly unchanged. We derive the exact detection law for $M$ ideal references and prove optimality among tests with zero false alarm for every encoding. For orthogonal leakage, the miss probability is asymptotic to $4/M$, independent of the ambient dimension $d$. Measuring all references first, even collectively, gives zero detection at every finite budget under the same zero-false-alarm requirement. For a fixed detection target between zero and one, the optimal measurement-first cost is $Θ(d/ε)$ at sufficiently small tolerance $ε$ on normal false alarm and conditional disturbance; a dimension-independent coherent budget suffices. For one logical qubit encoded in three physical qubits, seven coherent references achieve at least $50\%$ orthogonal-leakage detection, whereas any measurement-first receiver needs at least $588$ under the same $1\%$ normal tolerances. Quantum references thus support leakage checks without classical reconstruction, with a sample advantage exponential in the number of physical qubits.

Comments
25 pages, 4 figures; Supplemental Material included

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Leakage from a known encoding subspace can be detected by projection. However, the corresponding projector is unavailable when the encoding subspace is not classically specified. Here we show how independent quantum references prepared by the same encoder enable leakage detection without a classical description of the encoding. A coherent measurement on the references and a single message leaves every state within the two-dimensional encoding subspace, including its entanglement with a remote system, exactly unchanged. We derive the exact detection law for MMM ideal references and prove optimality among tests with zero false alarm for every encoding. For orthogonal leakage, the miss probability is asymptotic to 4/M4/M4/M, independent of the ambient dimension ddd. Measuring all references first, even collectively, gives zero detection at every finite budget under the same zero-false-alarm requirement. For a fixed detection target between zero and one, the optimal measurement-first cost is Θ(d/ε)Θ(d/ε)Θ(d/ε) at sufficiently small tolerance εεε on normal false alarm and conditional disturbance; a dimension-independent coherent budget suffices. For one logical qubit encoded in three physical qubits, seven coherent references achieve at least 50%50\%50% orthogonal-leakage detection, whereas any measurement-first receiver needs at least 588588588 under the same 1%1\%1% normal tolerances. Quantum references thus support leakage checks without classical reconstruction, with a sample advantage exponential in the number of physical qubits.
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Standard quantum inference converts quantum data into classical outputs. We study an alternative inference setting in which the desired output is quantum. Such settings include quantum purity amplification (QPA), random purification and approximate cloning, and density matrix exponentiation. We show that such protocols can achieve exponentially lower sample complexity than incoherent, measurement-mediated protocols, establishing, to the best of our knowledge, the first nontrivial separations between dimension-independent coherent and dimension-dependent incoherent sample complexity. For QPA with principal-eigenstate targets and ddd-dimensional inputs, at fixed output number and spectral gap, coherent processing achieves error ε\varepsilonε using O(1/ε)O(1/\varepsilon)O(1/ε) copies, whereas any incoherent protocol requires Ω(d/ε)Ω(d/\varepsilon)Ω(d/ε) copies. Together, these separations motivate a unified picture of coherent quantum inference, in which an entanglement-breaking limit identifies the optimal incoherent counterpart.