It is proved that the accuracy of any quantum-computing inference procedure is upper bounded by the maximal quantum leakage from the classical data through its quantum encoding, establishing leakage as a universal, task-agnostic quality measure for encoders.
Abstract
Optimal encoding of classical data for quantum-assisted statistical inference is investigated from an information-theoretic perspective. We prove that the accuracy of any quantum-computing inference procedure is upper bounded by the maximal quantum leakage from the classical data through its quantum encoding, establishing leakage as a universal, task-agnostic quality measure for encoders. This demonstrates that the maximal quantum leakage is a universal measure of the quality of the encoding strategy for statistical inference as it only depends on the quantum encoding of the data and not the inference task itself. The optimal universal encoding strategy, i.e., an encoding strategy that maximizes the maximal quantum leakage, is proved to be attained by pure states. When there are enough qubits, basis encoding is proved to be universally optimal. However, when the dimension of the system is small, phase encoding is optimal. For the latter, any tight frame, any ensemble whose average state is the maximally mixed state, is in fact optimal. Within tight frames, equiangular tight frames (ETFs) are distinguished as the uniquely symmetric optimal encodings, i.e., they saturate the Welch lower bound on pairwise overlaps and possess a self-referential optimal measurement. Prominent special cases are the qubit trine, the regular simplex, and symmetric informationally complete positive operator-valued measures (SIC-POVMs), for which the ETF structure and explicit codeword constructions are provided. Numerical examples are presented to validate the theoretical predictions.
The minimum change principle provides an information-theoretic characterization of the Bayes reversal channel in classical probability theory and has recently been proposed as a framework for extending Bayes'rule to quantum information theory. Using quantum relative entropy, we investigate a minimum change principle for the setting of quantum statistical inference. Specifically, we consider a forward process based on a classical-to-quantum preparation channel and a reverse process based on a quantum-to-classical measurement channel. We establish a closed-form characterization of measurements that are optimal for this principle, and this optimal measurement can be found via a dual formulation involving a single unconstrained Hermitian variable. This perspective allows us to recover some notable measurements within the same framework, including pretty good measurements and Fermi-Dirac thermal measurements, and we use it to discover a novel family that we call softmin thermal measurements. We further show that softmin thermal measurements arise as optimal solutions to entropy-regularized semidefinite optimization problems, demonstrating that they play a role for measurements analogous to that of thermal states in statistical mechanics. Finally, we prove an additivity property for the relative-entropy minimum change principle and investigate the performance of Fermi-Dirac thermal measurements for quantum hypothesis testing.
Quantum algorithms are conventionally presented with their input state supplied for free. When the input is classical data, this convention conceals a cost that is frequently larger than the algorithm it precedes. We review what the three standard encodings, such as basis encoding, amplitude encoding, and Grover--Rudolph distribution loading, actually cost once transpiled to a hardware gate set, and argue that the resulting $\Theta(N)$ bound is a counting theorem rather than an engineering limitation that improved hardware will remove. Measured gate counts for a representative loading task are reported: an optimal library implementation requires $247$ CNOT gates at $n=8$ qubits and doubles with each additional qubit, while the classical preprocessing that produces the rotation angles requires reading the entire input vector. We show how this cost eliminates the quadratic advantage of quantum amplitude estimation for Monte Carlo integration, and argue that the same accounting constrains quantum machine learning more broadly: the strong input models that make quantum algorithms fast on classical data also enable classical dequantization, and quantum kernel methods carry a $\Theta(M^2)$ state-preparation cost for the Gram matrix that does not amortize. We explain that the efficiently preparable states, device-generated distributions, variationally learned loading, and amortized preparation are required to get advantage from quantum machine learning and close with a checklist for evaluating input-dependent advantage claims. Executable notebooks reproducing every construction and measurement discussed here are available.
Quantum state ensembles are important in quantum information processing. For example, quantum $t$-designs model highly entangled states in complex systems, while projected ensembles appear in generative quantum machine learning and studies of thermalization. With their sample state accompanied by a classical label, these ensembles contain operational information beyond their average density operators. Yet an ensemble differs from a classical-quantum state because it is invariant under permutations of labels. We formulate binary hypothesis testing between finite quantum ensembles and derive fundamental limits on error probability. Given an observed label pattern, we show that the joint sampled state can be described by power-weighted ensemble moments. This yields the Bayes-optimal measurement and exact finite-sample error, revealing that discrimination is governed by the full moment hierarchy up to the number of samples. In the many-sample limit, we derive Chernoff bounds and obtain exact error exponents for finite uniform pure-state ensembles. We apply these results to optical communication and $t$-designs. For finite uniform pure-state $t$-designs with large $t$, the maximal discrimination exponent scales sharply as $\sim t^{-2}$, while equal-prior fixed-error testing requires $\sim t^2$ samples.
Maximal quantum leakage (MQL) is a worst-case information leakage measure that quantifies an adversary's inference advantage gained from accessing quantum encoding of classical data with arbitrary measurements. While MQL admits an exact characterization for a given ensemble of quantum states, its robustness to implementation imperfections has not been systematically studied. In this paper, we analyze the sensitivity of maximal quantum leakage under perturbations of the quantum encoding. We establish a continuity bound in terms of the trace distance between ideal and perturbed quantum states, and show, via an example, that this bound is attainable. We further derive fidelity-based and relative-entropy-based sufficient conditions for bounding the variation of maximal quantum leakage, and illustrate numerically that these conditions can be loose.
This work presents a Lean 4 library for quantum information, designed as a reusable formal infrastructure for theoretical analysis, and formalizes the DPI for the sandwiched R\'enyi relative entropy for positive semidefinite operators on finite-dimensional quantum systems.
Kazumi Kasaura, Kei Tsukamoto, Kento Mori et al.· 2 citations
Quantum information theory (QIT) characterizes the capabilities and fundamental limits of quantum information processing, underpinning quantum communication, computation, and error correction. Formalizing its coding theorems requires connecting finite-block protocols, analytic inequalities, and asymptotic limits within a unified machine-checked framework. Existing developments, however, lack a reusable operational layer that defines codes, error criteria, achievable rates, and capacities independently of their information-theoretic characterizations. In this work, we present LeanQIT, a Lean 4 library for finite-dimensional QIT. It provides composable, kernel-checked interfaces for quantum states and channels, source and channel codes, finite-block performance criteria, hypothesis testing, one-shot quantities, and asymptotic rate constructions. Using this infrastructure, we formalize Schumacher's quantum source-coding theorem, the Holevo--Schumacher--Westmoreland classical-capacity theorem, and the entanglement-assisted classical-capacity theorem together with its strong converse. By separating operational definitions from analytic characterizations and exposing reusable achievability, converse, and asymptotic components, Lean-QIT provides a machine-readable foundation for formal QIT and a compositional knowledge substrate for emerging AI-assisted formalization, automated proof search, and agentic reasoning in quantum information and computation.
Chengkai Zhu, Ziao Tang, Guocheng Zhen et al.· 2 citations