Fermi-Dirac thermal measurements: A framework for quantum hypothesis testing and semidefinite optimization

Nana Liu; Mark M. Wilde

quant-phMar 4, 2026

Fermi-Dirac thermal measurements: A framework for quantum hypothesis testing and semidefinite optimization

Authors: Nana Liu, Mark M. Wilde

Organizations: Institute of Natural Sciences, School of Mathematical Sciences, Ministry of Education Key Laboratory in Scientific and Engineering Computing, and Global College, Shanghai Jiao Tong University, Shanghai 200240, China · School of Electrical and Computer Engineering, Cornell University, Ithaca, New York 14850, USA

Abstract

Quantum measurements are the means by which we recover messages encoded into quantum states. They are at the forefront of quantum hypothesis testing, wherein the goal is to perform an optimal measurement for arriving at a correct conclusion. Mathematically, a measurement operator is Hermitian with eigenvalues in [0,1]. By noticing that this constraint on each eigenvalue is the same as that imposed on fermions by the Pauli exclusion principle, we interpret every eigenmode of a measurement operator as an independent effective fermionic mode. Under this perspective, various objective functions in quantum hypothesis testing can be viewed as the total expected energy associated with these fermionic occupation numbers. By instead fixing a temperature and minimizing the total expected fermionic free energy, we find that optimal measurements for these modified objective functions are Fermi-Dirac thermal measurements, wherein their eigenvalues are specified by Fermi-Dirac distributions. In the low-temperature limit, their performance closely approximates that of optimal measurements for quantum hypothesis testing, and we show that their parameters can be learned by classical or hybrid quantum-classical optimization algorithms. This leads to a new quantum machine-learning model, termed Fermi-Dirac machines, consisting of parameterized Fermi-Dirac thermal measurements-an alternative to quantum Boltzmann machines based on thermal states. Beyond hypothesis testing, we show how general semidefinite optimization problems can be solved using this approach, leading to a novel paradigm for semidefinite optimization on quantum computers, in which the goal is to implement thermal measurements rather than prepare thermal states. Finally, we propose quantum algorithms for implementing Fermi-Dirac thermal measurements, and we also propose second-order hybrid quantum-classical optimization algorithms.

Figures & tables

Figure 1: Quantum circuit for realizing a Fermi–Dirac thermal measurement

\left(M_{T}(A),I-M_{T}(A)\right)

of temperature

T=T_{1}T_{2}

, where

T_{1},T_{2}>0

, as detailed in Algorithm 19 . The state

|\psi_{T_{1}}\rangle\!\langle\psi_{T_{1}}|

of the control qumode is defined in ( 111 ), and

\rho

is the input state on which we would like to perform the desired Fermi–Dirac thermal measurement. The measurement of the control qumode at the end is a momentum-quadrature measurement giving an outcome

p\in\mathbb{R}

. The algorithm finally outputs

0

p\geq 0

and

1

otherwise.

Figure 2: Quantum circuit for simulating the evolution in ( 116 ) for the case

n=3

, as detailed in Algorithm 21 .

Figure 3: Quantum circuit for estimating the Hessian matrix element

\left(i,j\right)

, as detailed in Algorithm 22 . “Had” stands for the qubit Hadamard gate. The classical values

k_{1}

k_{2}

, and

t

are sampled from the probability distributions

\left|\alpha_{i,k_{1}}\right|/\left\|\alpha_{i}\right\|_{1}

\left|\alpha_{j,k_{2}}\right|/\left\|\alpha_{j}\right\|_{1}

, and

\gamma(t)

in ( 79 ), respectively. The measurements at the end consist of a computational basis measurement on the control qubit and the Fermi–Dirac thermal measurement

\left(M_{T}(\mu),I-M_{T}(\mu)\right)

on both data registers. See Algorithm 22 for more details.

Quantum Machine Learning

Quantum machine learning uses quantum circuits and quantum data to build learning algorithms and study their computational and statistical limits.

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May 23, 2026J/K move · Enter open · S save

Fermi-Dirac machines as quantizations of neurons

Alexander He, Nana Liu, Mark M. Wilde

Department of Physics, Cornell University, Ithaca, New York 14850, USA · Institute of Natural Sciences, Shanghai Jiao Tong University, Shanghai 200240, China · School of Mathematical Sciences, Shanghai Jiao Tong University, Shanghai 200240, China+3

Hybrid Quantum-Classical ML Quantum Neural Networks

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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.

Abstract

Quantum Machine Learning

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