An ensemble of quantum states can be described by a Hermitian, positive semidefinite and unit trace matrix called density matrix. Thus, the study of methods for optimizing a certain function (energy, entropy) over the set of density matrices has a direct application to important problems in quantum information and computation. We propose a projected gradient method for solving such problems. By exploiting the geometry of the feasible set, which is the intersection of the cone of Hermitian positive semidefinite matrices with the hyperplane defined by the unit trace constraint, we describe an efficient procedure to compute the projection onto this set using the Frobenius norm. Some important applications, such as quantum state tomography, are described and numerical experiments illustrate the effectiveness of the method when compared to previous methods based on fixed-point iterations or semidefinite programming.
An ensemble of quantum states can be described by a Hermitian, positive semidefinite and unit trace matrix called density matrix. Thus, the study of methods for optimizing a certain function (energy, entropy) over the set of density matrices has a direct application to important problems in quantum information and computation. We propose a projected gradient method for solving such problems. By exploiting the geometry of the feasible set, which is the intersection of the cone of Hermitian positive semidefinite matrices with the hyperplane defined by the unit trace constraint, we describe an efficient procedure to compute the projection onto this set using the Frobenius norm. Some important applications, such as quantum state tomography, are described and numerical experiments illustrate the effectiveness of the method when compared to previous methods based on fixed-point iterations or semidefinite programming.
The density matrix is a positive semidefinite operator of trace 1 characterizing the state of a quantum system. We consider the inverse problem to reconstruct such density matrices from indirect measurements, also known as quantum state tomography. To solve such inverse problems in high or infinite dimensional settings, we study variational regularization using the quantum relative entropy as penalty functional. Quantum relative entropy is an analog of the well-known maximum entropy functional with compositions of functions replaced by the spectral functional calculus. The main aim of this paper is to establish the regularizing property of this scheme. As a crucial intermediate step, we establish lower semi-compactness of the penalty functional with respect to the weak-∗-topology. Moreover, we compute the subgradient, proximal operator, and conjugate functional of the quantum relative entropy on finite dimensional spaces. This enables us to apply iterative algorithms from convex optimization to solve the regularized problems numerically. To show the validity and practical value of our results, we apply our theory to the examples of Photon-Induced Near-field Electron Microscopy (PINEM) and to optical homodyne tomography.
We consider the problem of constructing quantum operations or channels, if they exist, that transform a given set of quantum states {ρ1,…,ρk} to another such set {ρ^1,…,ρ^k}. In other words, we must find a {\em completely positive linear map}, if it exists, that maps a given set of density matrices to another given set of density matrices. This problem, in turn, is an instance of a positive semi-definite feasibility problem, but with highly structured constraints. The nature of the constraints makes projection based algorithms very appealing when the number of variables is huge and standard interior point-methods for semi-definite programming are not applicable. We provide emperical evidence to this effect. We moreover present heuristics for finding both high rank and low rank solutions. Our experiments are based on the \emph{method of alternating projections} and the \emph{Douglas-Rachford} reflection method.
Abstract Quantum relative entropy optimization refers to a class of convex problems in which a linear functional is minimized over an affine section of the epigraph of the quantum relative entropy function. Recently, the self-concordance of a natural barrier function was proved for this set, and various implementations of interior-point methods have been made available to solve this class of optimization problems. In this paper, we show how common structures arising from applications in quantum information theory can be exploited to improve the efficiency of solving quantum relative entropy optimization problems using interior-point methods. First, we show that the natural barrier function for the epigraph of the quantum relative entropy composed with positive linear operators is self-concordant, even when these linear operators map to singular matrices. Compared to modeling problems using the full quantum relative entropy cone, this allows us to remove redundant log-determinant expressions from the barrier function and reduce the overall barrier parameter. Second, we show how certain slices of the quantum relative entropy cone exhibit useful properties which should be exploited whenever possible to perform certain key steps of interior-point methods more efficiently. We demonstrate how these methods can be applied to applications in quantum information theory, including quantifying quantum key rates, quantum rate-distortion functions, quantum channel capacities, and the ground state energy of Hamiltonians. Our numerical results show that these techniques improve computation times by up to several orders of magnitude, and allow previously intractable problems to be solved.
Mirror descent is a powerful framework to solve nonlinear optimization problems by taking advantage of the geometry of the space in which the problem lives [155]. Mirror descent relies on the same concepts as projected (sub)gradient descent, with the significant variation that the descent takes place in a “mirror” space, which is the vector space dual to the original space. When dealing with optimization problems over the cone of positive semidefinite matrices, it is possible to apply mirror descent using the quantum relative entropy as the mirror map, resulting in an iterative optimization scheme for semidefinite programming that simultaneously operates in two spaces: the space of primal iterates, which are positive semidefinite matrices expressed as matrix exponentials, and the space of (vector) dual iterates (i.e., the mirror space), which are the matrix logarithms of the primal iterates. This choice of the mirror map is natural in the context of quantum algorithms that want to take advantage of the ability of quantum computers to efficiently compute Gibbs states, i.e., matrix exponentials of a certain kind; see Sect. 7.2.4 and in particular Prop. 7.38. Since the mirror descent framework has proven very fruitful for the development of quantum optimization algorithms, in this chapter we discuss its main components, with an emphasis on those that (almost) directly lead to quantum algorithms with an expectation of quantum advantage of some type.
Let Sm be the set of all m×m density matrices (Hermitian positively semi-definite matrices of unit trace). Consider a problem of estimation of an unknown density matrix ρ∈Sm based on outcomes of n measurements of observables X1,…,Xn∈Hm (Hm being the space of m×m Hermitian matrices) for a quantum system identically prepared n times in state ρ. Outcomes Y1,…,Yn of such measurements could be described by a trace regression model in which Eρ(Yj∣Xj)=tr(ρXj),j=1,…,n. The design variables X1,…,Xn are often sampled at random from the uniform distribution in an orthonormal basis {E1,…,Em2} of Hm (such as Pauli basis). The goal is to estimate the unknown density matrix ρ based on the data (X1,Y1),…,(Xn,Yn). Let Z^:=nm2∑j=1nYjXj and let ρˇ be the projection of Z^ onto the convex set Sm of density matrices. It is shown that for estimator ρˇ the minimax lower bounds in classes of low rank density matrices (established earlier) are attained up logarithmic factors for all Schatten p-norm distances, p∈[1,∞] and for Bures version of quantum Hellinger distance. Moreover, for a slightly modified version of estimator ρˇ the same property holds also for quantum relative entropy (Kullback-Leibler) distance between density matrices.
The cone of positive-semidefinite (PSD) matrices is fundamental in convex optimization, and we extend this notion to tensors, defining PSD tensors, which correspond to separable quantum states. We study the convex optimization problem over the PSD tensor cone. While this convex cone admits a smooth reparameterization through tensor factorizations (analogous to the matrix case), it is not self-dual. Moreover, there are currently no efficient algorithms for projecting onto or testing membership in this cone, and the semidefinite tensor optimization problem, although convex, is NP-hard. To address these challenges, we develop methods for computing lower and upper bounds on the optimal value of the problem. We propose a general-purpose iterative refinement algorithm that combines a lifted alternating direction method of multipliers with a cutting-plane approach. This algorithm exploits PSD tensor factorizations to produce heuristic solutions and refine the solutions using cutting planes. Since the method requires a linear minimization oracle over PSD tensors, we design a spatial branch-and-bound algorithm based on convex relaxations and valid inequalities. Our framework allows us to study the white-noise mixing threshold, which characterizes the entanglement properties of quantum states. Numerical experiments on benchmark instances demonstrate the effectiveness of the proposed methods.