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Mark M. Wilde

Publications and source records attributed to Mark M. Wilde.

At least 37 records · Page 2Linked to original sources
arXiv2026

Fundamentals of quantum Boltzmann machine learning with visible and hidden units

One of the primary applications of classical Boltzmann machines is generative modeling, wherein the goal is to tune the parameters of a model distribution so that it closely approximates a target distribution. Training relies on estimating the gradient of the relative entropy between the target and model distributions, a task that is well understood when the classical Boltzmann machine has both visible and hidden units. For some years now, it has been an obstacle to generalize this finding to quantum state learning with quantum Boltzmann machines that have both visible and hidden units. In this paper, I derive an analytical expression for the gradient of the quantum relative entropy between a target quantum state and the reduced state of the visible units of a quantum Boltzmann machine. Crucially, this expression is amenable to estimation on a quantum computer, as it involves modular-flow-generated unitary rotations reminiscent of those appearing in my prior work on rotated Petz recovery maps. This leads to a quantum algorithm for gradient estimation in this setting. I then specialize the setting to quantum visible units and classical hidden units, and vice versa; I also provide analytical expressions for the gradients, along with quantum algorithms for estimating them. Finally, I replace the quantum relative entropy objective function with the Petz-Tsallis relative entropy; here I develop an analytical expression for the gradient and sketch a quantum algorithm for estimating it, as an application of an independent derivation of a formula for the derivative of the matrix power function, which also involves modular-flow-generated unitary rotations. Ultimately, this paper demarcates progress in training quantum Boltzmann machines with visible and hidden units for generative modeling and quantum state learning.

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arXiv2026

Unifying quantum measurement constructions via a relative-entropy minimum change principle

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.

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arXiv2026

Accelerated optimization of measured relative entropies

The measured relative entropy and measured Rényi relative entropy are quantifiers of the distinguishability of two quantum states $ρ$ and $σ$. They are defined as the maximum classical relative entropy or Rényi relative entropy realizable by performing a measurement on $ρ$ and $σ$, and they have interpretations in terms of asymptotic quantum hypothesis testing. Crucially, they can be rewritten in terms of variational formulas involving the optimization of a concave or convex objective function over the set of positive definite operators. In this paper, we establish foundational properties of these objective functions by analyzing their matrix gradients and Hessian superoperators; namely, we prove that these objective functions are $β$-smooth and $γ$-strongly convex / concave, where $β$ and $γ$ depend on the max-relative entropies of $ρ$ and $σ$. A practical consequence of these properties is that we can conduct Nesterov accelerated projected gradient descent / ascent, a well known classical optimization technique, to calculate the measured relative entropy and measured Rényi relative entropy to arbitrary precision. These algorithms are generally more memory efficient than our previous algorithms based on semi-definite optimization [Huang and Wilde, arXiv:2406.19060], and for well conditioned states $ρ$ and $σ$, these algorithms are notably faster.

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arXiv2026

Information geometry of bosonic Gaussian thermal states

Bosonic Gaussian thermal states form a fundamental class of states in quantum information science. This paper explores the information geometry of these states, focusing on characterizing the distance between two nearby states and the geometry induced by a parameterization in terms of their mean vectors and Hamiltonian matrices. In particular, for the family of bosonic Gaussian thermal states, we derive expressions for their Fisher-Bures, Kubo-Mori, and $α$-$z$ information matrices with respect to their mean vectors and Hamiltonian matrices. An important application of our formulas consists of fundamental limits on how well one can estimate these parameters. We additionally establish formulas for the derivatives and the symmetric logarithmic derivatives of bosonic Gaussian thermal states. The former could have applications in gradient descent algorithms for quantum machine learning when using bosonic Gaussian thermal states as an ansatz, and the latter in formulating optimal strategies for single parameter estimation of bosonic Gaussian thermal states. Finally, the expressions for the aforementioned information matrices could have additional applications in natural gradient descent algorithms when using bosonic Gaussian thermal states as an ansatz.

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arXiv2026

Canonical quantization of neurons

Canonical quantization provides a systematic procedure for constructing quantum models from classical Hamiltonians. Here, we apply this principle to a fundamental computational primitive of machine learning: the neuron. Specifically, by viewing a neuron as a composition of an energy function and an activation function, we quantize this model by replacing the energy function with a quantum Hamiltonian and applying the activation function to it through matrix functional calculus. This results in an activation observable that can be measured on an input quantum state. We investigate the use of these quantized neurons for function approximation, where the objective is to learn an unknown observable from labeled quantum data. For this purpose, we develop hybrid quantum-classical algorithms for training and evaluation, including procedures for measuring the activation observable and estimating gradients of the squared loss error. Our algorithms for gradient estimation rely on basic primitives like classical random sampling, the Hadamard test, and Hamiltonian simulation, and those for measuring an activation observable rely on quantum algorithms known as the power of one qumode and Schroedingerization. Numerical experiments demonstrate that our quantized neurons exhibit enhanced expressive capabilities relative to corresponding classical neurons on representative learning tasks. Our work establishes canonical quantization as a principled framework for constructing quantum machine learning primitives and provides a foundation for developing neural architectures tailored to quantum data.

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arXiv2026

Quantum Spectral Anomaly Detection

A core task in quantum anomaly detection is to compute an anomaly score that quantifies how strongly a test quantum state deviates from a given quantum dataset assumed to be normal. Classically, principal component analysis (PCA) for centered data computes the anomaly score by evaluating the test sample relative to the subspace spanned by the selected leading eigenvectors. However, for quantum data that lack a standard centering, explicitly recovering principal eigenvectors, constructing full Gram matrices, or loading quantum-random-access-memory-style data can be more costly than estimating the anomaly score itself. To avoid these costs, we propose Quantum Spectral Anomaly Detection (QSPADE), which computes PCA-like anomaly scores directly from the spectrum of the average state of the normal dataset. By replacing hard PCA rank selection with a smooth, temperature-controlled spectral threshold, QSPADE makes near-threshold spectral components contribute partially to the anomaly score. This makes the score vary continuously rather than jump when a borderline component is included or excluded, and makes it less sensitive to noise or arbitrary hard cutoffs near the threshold. In the zero-temperature limit, QSPADE recovers the hard-projector PCA score. The proposed measurement-based quantum detector can be calibrated with a sample complexity independent of the data dimension. Numerical simulations show that QSPADE behaves like kernel-PCA on encoded classical data and detects changes across a transverse-field Ising transition without predefined order parameters. Consequently, QSPADE gives an efficient framework for both quantum-kernel anomaly detection on encoded classical data and the monitoring of quantum-native systems where diagnostic observables are unknown.

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arXiv2026

Retrocausal capacity of a quantum channel: Communicating through noisy closed timelike curves

We study the capacity of a quantum channel for retrocausal communication, where messages are transmitted backward in time, from a sender in the future to a receiver in the past, through a noisy postselected closed timelike curve mathematically represented by the channel. We completely characterize the one-shot retrocausal quantum and classical capacities, and we show that the corresponding asymptotic capacities are equal to the average and sum, respectively, of the channel's max-information and its regularized Doeblin information. This endows these information measures with a novel operational interpretation. Furthermore, our characterization can be generalized beyond quantum channels to all completely positive maps. This imposes information-theoretic limits on transmitting messages via postselected-teleportation-like mechanisms with arbitrary initial- and final-state boundary conditions, including those considered in various black-hole final-state models.

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arXiv2026

Improved sample complexity bound for sample-based Lindbladian simulation

We establish improved sample-complexity bounds for sample-based Lindbladian simulation based on the Wave Matrix Lindbladization (WML) algorithm. For a jump operator $L$ with dimension $d$, we derive an explicit non-asymptotic sample complexity bound $n_d^*(t,\varepsilon) \le \left( \frac{2d+3}{8} \right) \|L\|_\infty^2 \left( \frac{t^2}{\varepsilon} \right)$, holding for simulation time $t$ and error $\varepsilon$. This refines the dimension dependence of the best previously known bound, $O(d^2 t^2/\varepsilon)$, from [Go et al., Quantum Sci. Tech. 10, 045058 (2025)]. Remarkably, we show that this dimensional overhead can be entirely avoided when $\| L\|_\infty^2 = O(1/d)$, a condition satisfied with high probability for random Lindblad operators, yielding a typical-case sample complexity of $O(t^2/\varepsilon)$. On the other hand, in the worst case, we show that WML necessarily requires $Ω(dt^2/\varepsilon)$ samples by constructing an explicit example with a rank-one Lindblad operator. Our results reveal a sharp dichotomy between typical and adversarial sample complexities in Lindbladian simulation, thereby strengthening the theoretical foundations of sample-based quantum algorithms.

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arXiv2026

Quantum principal component analysis without eigenvector recovery

Principal component analysis (PCA) is traditionally implemented through a covariance or kernel matrix, leading-eigenvector extraction, and hard rank-$k$ projection. These steps can be computationally costly in high-dimensional and quantum-data settings, sensitive to small eigengaps, and unnecessary when downstream tasks only require principal-subspace scores. Such score-based objectives are important in applications such as anomaly detection, spectral-energy profiling, and other postselection tasks. To address these needs, we introduce a measurement-based soft PCA framework replacing the hard top-$k$ projector with an entropy-regularized Fermi--Dirac filter. This filter is the unique optimizer of an entropy-regularized variational formulation of PCA and converges to the classical PCA projector in the zero-temperature limit. This filter has a direct interpretation as a quantum measurement, which naturally suggests a quantum approach. For centered covariance operators represented by quantum feature states, a single fixed circuit, together with threshold calibration, accesses all optimal filters for different rank budgets or retained-variance levels without rank-dependent circuit updates or eigenvector recovery. For new inputs, the same calibrated quantum circuit yields soft principal subspace scores, spectral energy profiles, and postselected filtered states. The required centering of both training and test data is performed coherently inside the quantum protocol, which is particularly important for quantum data where no classical feature vectors or centered Gram matrix are directly available. By reframing PCA as a calibrated measurement task, this framework bypasses the need for iterative eigenvector extraction and achieves a dimension-independent sample complexity $O(η^{-2})$ for normalized fractional-rank or retained variance scoring at additive accuracy $η$.

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arXiv2026

Bose-Einstein thermal operators for semidefinite optimization

We establish that semidefinite programs (SDPs) over the unbounded positive semidefinite cone are mathematically equivalent to thermodynamic systems of independent bosonic modes: the eigenvalues of the optimization variable play the role of expected occupation numbers, the linear objective plays the role of total expected energy, and the linear equality constraints play the role of conserved non-commuting charges. Building on this perspective, we recast general SDPs as bosonic free-energy minimization problems at strictly positive temperature, regularized by the Bose-Einstein entropy; the original SDP is recovered in the zero-temperature limit. The optimal primal variable takes the form of a Bose-Einstein thermal operator parametrized by the dual variables. We prove an approximation-error bound that depends on the ground-space degeneracy and the spectral gap of the dual slack operator, improving on the linear-in-dimension worst-case duality gap of interior-point methods. We also introduce the Bose-Einstein quantum relative entropy as a Bregman divergence on the unbounded positive semidefinite cone, generated by the negative Bose-Einstein entropy. We propose it as a natural divergence for unnormalized positive operators, for which the standard Umegaki relative entropy can become negative, and we show that it satisfies a restricted monotonicity property under affine maps modeling bosonic Gaussian channels. Finally, we develop hybrid quantum-classical algorithms for the regularized SDP using only Hamiltonian simulation, Hadamard tests, and classical sampling, and bound their runtime in closed form. Unlike existing quantum SDP solvers, whose runtimes scale polynomially with an a priori upper bound on the primal trace, our framework operates directly on the unbounded cone, replacing this bound with a dependence on the spectral structure of the dual slack operator.

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arXiv2026

Fermi-Dirac machines as quantizations of neurons

Fermi-Dirac machines were proposed recently as an approach to solving semidefinite optimization problems on quantum computers. Here, we reinterpret them as canonical quantizations of classical neurons. By viewing a classical neuron as an activation function applied to a parameterized classical Hamiltonian, we quantize this model by replacing classical variables with operators whose eigenvalues encode their possible values. This follows the standard approach to canonical quantization in quantum mechanics. Crucially, when the Hamiltonian consists of commuting operators, our construction reduces exactly to a classical neuron. More generally, our approach yields an activation observable, defined as an activation function applied to a parameterized quantum Hamiltonian. The output of this quantized neuron is a random variable with expectation value equal to that of the activation observable with respect to an input state. We develop efficient hybrid quantum-classical algorithms for evaluating outputs and gradients of our quantized neurons, enabling evaluation and training. These algorithms rely on basic primitives that include random sampling, Hamiltonian simulation, and the Hadamard test. We also quantize a whole host of other activation functions, including the smooth rectified linear unit (ReLU), sigmoid linear unit, Gaussian-smoothed ReLU, and Gaussian error linear unit (GeLU), which are known to be useful for deep learning applications. Numerical experiments indicate that neurons based on quantum Hamiltonians can learn functions that classical neurons cannot. We further define a computational decision problem based on Fermi-Dirac neurons and prove that it is BQP-complete, providing complexity-theoretic evidence against efficient classical simulation. Finally, we generalize our approach to continuous quantum variables and sketch two different ways of composing these neurons into networks.

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arXiv2026

Quantum Doeblin Coefficients: Interpretations and Applications

In classical information theory, the Doeblin coefficient of a classical channel provides an efficiently computable upper bound on the total-variation contraction coefficient of the channel, leading to what is known as a strong data-processing inequality. Here, we investigate quantum Doeblin coefficients as a generalization of the classical concept. In particular, we define various new quantum Doeblin coefficients, one of which has several desirable properties, including concatenation and multiplicativity, in addition to being efficiently computable. We also develop various interpretations of two of the quantum Doeblin coefficients, including representations as minimal singlet fractions, exclusion values, reverse max-mutual and oveloH informations, reverse robustnesses, and hypothesis testing reverse mutual and oveloH informations. Our interpretations of quantum Doeblin coefficients as either entanglement-assisted or unassisted exclusion values are particularly appealing, indicating that they are proportional to the best possible error probabilities one could achieve in state-exclusion tasks by making use of the channel. We also outline various applications of quantum Doeblin coefficients, ranging from limitations on quantum machine learning algorithms that use parameterized quantum circuits (noise-induced barren plateaus), on error mitigation protocols, on the sample complexity of noisy quantum hypothesis testing, and on mixing, distinguishability, and decoupling times of time-varying channels. All of these applications make use of the fact that quantum Doeblin coefficients appear in upper bounds on various trace-distance contraction coefficients of a channel. Furthermore, in all of these applications, our analysis using Doeblin coefficients provides improvements of various kinds over contributions from prior literature, both in terms of generality and being efficiently computable.

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arXiv2026

Central Limit Theorem for Bosonic Quantum Channels

In this paper, we develop an extension of the Central Limit Theorem (CLT) to the setting of bosonic quantum channels. This extension provides a deeper understanding of Gaussian bosonic channels as extremal objects. Using our CLT for bosonic quantum channels, we recover both the classical CLT and the CLT for bosonic quantum states, thereby offering a unified perspective that connects classical probability theory with continuous-variable quantum systems. Moreover, using our result, we can provide necessary uncertainty relations that every physical (possibly non-Gaussian) bosonic quantum channel must satisfy. As another application of our limit theorems, we derive tight lower bounds on the energy-constrained quantum capacity of linear bosonic channels by relating it to the capacity of their associated Gaussian bosonic channels, further reinforcing the role of Gaussian channels as extremal.

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arXiv2026

Performance Guarantees for Quantum Neural Estimation of Entropies

Estimating quantum entropies and divergences is an important problem in quantum physics, information theory, and machine learning. Quantum neural estimators (QNEs), which utilize a hybrid classical-quantum architecture, have recently emerged as an appealing computational framework for estimating these measures. Such estimators combine classical neural networks with parametrized quantum circuits, and their deployment typically entails tedious tuning of hyperparameters controlling the sample size, network architecture, and circuit topology. This work initiates the study of formal guarantees for QNEs of measured (Rényi) relative entropies in the form of non-asymptotic error risk bounds. We further establish exponential tail bounds showing that the error is sub-Gaussian and thus sharply concentrates about the ground truth value. For an appropriate sub-class of density operator pairs on a space of dimension $d$ with bounded Thompson metric, our theory establishes a copy complexity of $O(|Θ(\mathcal{U})|d/ε^2)$ for QNE with a quantum circuit parameter set $Θ(\mathcal{U})$, which has minimax optimal dependence on the accuracy $ε$. Additionally, if the density operator pairs are permutation invariant, we improve the dimension dependence above to $O(|Θ(\mathcal{U})|\mathrm{polylog}(d)/ε^2)$. Our theory aims to facilitate principled implementation of QNEs for measured relative entropies and guide hyperparameter tuning in practice.

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arXiv2026

Onset of superactivation of quantum capacity

Superactivation of quantum capacity is the phenomenon whereby two quantum channels, each with zero quantum capacity, can exhibit a strictly positive capacity when used in tandem. In this work, we explore superactivation in the previously unexplored non-asymptotic regime of finitely many channel uses. We give a definition of finite-blocklength superactivation and propose numerical methods that can certify it. Then, focusing on the 50% erasure and positive-partial-transpose channels considered in the original work on superactivation, we show that as few as 17 uses of the joint channel already enable qubit transmission with a fidelity unattainable by any number of uses of either channel alone, demonstrating a strong finite-blocklength form of superactivation and opening the door to experimental demonstration.

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arXiv2026

Cost of quantum secret key

In this paper, we develop the resource theory of quantum secret key. Operating under the assumption that entangled states with zero distillable key do not exist, we define the key cost of a quantum state, and device. We study its properties through the lens of a quantity that we call the key of formation. The main result of our paper is that the regularized key of formation is an upper bound on the key cost of a quantum state. The core protocol underlying this result is privacy dilution, which converts states containing ideal privacy into ones with diluted privacy. Next, we show that the key cost is bounded from below by the regularized relative entropy of entanglement, which implies the irreversibility of the privacy creation-distillation process for a specific class of states. We further focus on mixed-state analogues of pure quantum states in the domain of privacy, and we prove that a number of entanglement measures are equal to each other for these states, similar to the case of pure entangled states. The privacy cost and distillable key in the single-shot regime exhibit a yield-cost relation, and basic consequences for quantum devices are also provided. Importantly, our results presented here will remain valid even if entangled states with zero distillable key were shown to exist.

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arXiv2026

Constrained free energy minimization for the design of thermal states and stabilizer thermodynamic systems

A quantum thermodynamic system is described by a Hamiltonian and a list of conserved, non-commuting charges, and a fundamental goal is to determine the minimum energy of the system subject to constraints on the charges. Recently, [Liu et al., arXiv:2505.04514] proposed first- and second-order classical and hybrid quantum-classical algorithms for solving a dual chemical potential maximization problem, and they proved that these algorithms converge to global optima by means of gradient-ascent approaches. In this paper, we benchmark these algorithms on several problems of interest in thermodynamics, including one- and two-dimensional quantum Heisenberg models with nearest- and next-nearest neighbor interactions and with the charges set to the total x, y, and z magnetizations. We also offer an alternative compelling interpretation of these algorithms as methods for designing ground and thermal states of controllable Hamiltonians, with potential applications in molecular and material design. Furthermore, we introduce stabilizer thermodynamic systems as thermodynamic systems based on stabilizer codes, with the Hamiltonian constructed from a given code's stabilizer operators and the charges constructed from the code's logical operators. We benchmark the aforementioned algorithms on several examples of stabilizer thermodynamic systems, including those constructed from the one-to-three-qubit repetition code, the perfect one-to-five-qubit code, and the two-to-four-qubit error-detecting code. Finally, we observe that the aforementioned hybrid quantum-classical algorithms, when applied to stabilizer thermodynamic systems, can serve as alternative methods for encoding quantum information into stabilizer codes at a fixed temperature, and we provide an effective method for warm-starting these encoding algorithms whenever a single qubit is encoded into multiple physical qubits.

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arXiv2026

Dual-VQE: A quantum algorithm to lower bound the ground-state energy

The variational quantum eigensolver (VQE) is a hybrid quantum-classical variational algorithm that produces an upper-bound estimate of the ground-state energy of a Hamiltonian. As quantum computers become more powerful and go beyond the reach of classical brute-force simulation, it is important to assess the quality of solutions produced by them. Here we propose a dual variational quantum eigensolver (dual-VQE) that produces a lower-bound estimate of the ground-state energy. As such, VQE and dual-VQE can serve as quality checks on their solutions; in the ideal case, the VQE upper bound and the dual-VQE lower bound form an interval containing the true optimal value of the ground-state energy. The idea behind dual-VQE is to employ semidefinite programming duality to rewrite the ground-state optimization problem as a constrained maximization problem, which itself can be bounded from below by an unconstrained optimization problem to be solved by a variational quantum algorithm. When using a convex combination ansatz in conjunction with a classical generative model, the quantum computational resources needed to evaluate the objective function of dual-VQE are no greater than those needed for that of VQE. We also show that the problem is well suited for classical pretraining using matrix product states and these methods help warm-start the optimization. We simulated the performance of dual-VQE on the transverse-field Ising model with and without pretraining and found that, for the example considered, while dual-VQE training is slower and noisier than VQE, it approaches the true value with an error of order $10^{-2}$.

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