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

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 (3)

• 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 \ref{['alg:FD-thermal-alg']}. The state $|\psi_{T_{1}}\rangle\!\langle\psi_{T_{1}}|$ of the control qumode is defined in \ref{['eq:control-state-alg-FD']}, 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 $0$ if $p\geq0$ and $1$ otherwise.
• Figure 2: Quantum circuit for simulating the evolution in \ref{['eq:simulated-evol-DME']} for the case $n=3$, as detailed in Algorithm \ref{['alg:DME-alg']}.
• Figure 3: Quantum circuit for estimating the Hessian matrix element $\left(i,j\right)$, as detailed in Algorithm \ref{['alg:hessian-est-alg']}. “ Had” stands for the qubit Hadamard gate. The classical values $k_{1}$, $k_{2}$, and $t$ are sampled from the probability distributions $\left|\alpha_{i,k_{1}}\right|/\left\Vert \alpha_{i}\right\Vert _{1}$, $\left|\alpha_{j,k_{2}}\right|/\left\Vert \alpha_{j}\right\Vert _{1}$, and $\gamma(t)$ in \ref{['eq:high-peak-tent-prob-dens']}, 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 \ref{['alg:hessian-est-alg']} for more details.

Theorems & Definitions (47)

• Definition 1
• Proposition 3
• proof
• Definition 4
• Proposition 5
• proof
• Remark 6
• Definition 7: Fermi--Dirac thermal measurement
• Proposition 8
• proof