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arXiv:1904.11188 · IEEE International Symposium on Information Theory (ISIT)

A Blahut-Arimoto Type Algorithm for Computing Classical-Quantum Channel Capacity

Haobo Li, Ning Cai

Published Jul 1, 2019• 20 citations
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Abstract

Based on Arimoto's work in 1978 [1], we propose an iterative algorithm for computing the capacity of a discrete memoryless classical-quantum channel with a finite input alphabet and a finite dimensional output, which we call the Blahut-Arimoto algorithm for classical-quantum channel, and an input cost constraint is considered. We show that to reach ε accuracy, the iteration complexity of the algorithm is up bounded by log n log ε ε where n is the size of the input alphabet. In particular, when the output state {ρx}x∈Xis linearly independent in complex matrix space, the algorithm has a geometric convergence. We also show that the algorithm reaches an ε accurate solution with a complexity of O(m3log n log ε/ε), and O(m3log ε log(1-δ)D(ε/p*||pN(0))) in the special case, where m is the output dimension and D(p*||pN(0)) is the relative entropy of two distributions and δ is a positive number.

DOI
10.1109/isit.2019.8849608

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Based on Arimoto's work in 1978 [1], we propose an iterative algorithm for computing the capacity of a discrete memoryless classical-quantum channel with a finite input alphabet and a finite dimensional output, which we call the Blahut-Arimoto algorithm for classical-quantum channel, and an input cost constraint is considered. We show that to reach ε accuracy, the iteration complexity of the algorithm is up bounded by log n log ε ε where n is the size of the input alphabet. In particular, when the output state {ρx}x∈Xis linearly independent in complex matrix space, the algorithm has a geometric convergence. We also show that the algorithm reaches an ε accurate solution with a complexity of O(m3log n log ε/ε), and O(m3log ε log(1-δ)D(ε/p*||pN(0))) in the special case, where m is the output dimension and D(p*||pN(0)) is the relative entropy of two distributions and δ is a positive number.
Based on Arimoto’s work in 1972, we propose an iterative algorithm for computing the capacity of a discrete memoryless classical-quantum channel with a finite input alphabet and a finite dimensional output, which we call the Blahut–Arimoto algorithm for classical-quantum channel, and an input cost constraint is considered. We show that, to reach ε accuracy, the iteration complexity of the algorithm is upper bounded by log n log ε ε where n is the size of the input alphabet. In particular, when the output state { ρ x } x ∈ X is linearly independent in complex matrix space, the algorithm has a geometric convergence. We also show that the algorithm reaches an ε accurate solution with a complexity of O ( m 3 log n log ε ε ) , and O ( m 3 log ε log ( 1 − δ ) ε D ( p * | | p N 0 ) ) in the special case, where m is the output dimension, D ( p * | | p N 0 ) is the relative entropy of two distributions, and δ is a positive number. Numerical experiments were performed and an approximate solution for the binary two-dimensional case was analysed.
In our previous work, we presented a Blahut-Arimoto type algorithm for computing the discrete memoryless (DM) classical-quantum channel capacity. And the speed of convergence is analyzed. In this paper, we present numerical experiment to show that the algorithm converged much faster than the theoretical prediction. Also we present an explicit approximate solution for the binary inputs, two dimensional outputs classical-quantum channel, which has an error within 3×10−43\times10^{-4}3×10−4
We propose an iterative method for approximating the capacity of classical-quantum channels with a discrete input alphabet and a finite-dimensional output under additional constraints on the input distribution. Based on duality of convex programming, we derive explicit upper and lower bounds for the capacity. To provide an additive ε-close estimate to the capacity, the presented algorithm requires O((N ν M)M3log(N)1/2ε-1) steps, where N denotes the input alphabet size and M denotes the output dimension. We then generalize the method to the task of approximating the capacity of classical-quantum channels with a bounded continuous input alphabet and a finite-dimensional output. This, using the idea of a universal encoder, allows us to approximate the Holevo capacity for channels with a finite-dimensional quantum mechanical input and output. In particular, we show that the problem of approximating the Holevo capacity can be reduced to a multi-dimensional integration problem. For certain families of quantum channels, we prove that the complexity to derive an additive ε-close solution to the Holevo capacity is subexponential or even polynomial in the problem size. We provide several examples to illustrate the performance of the approximation scheme in practice.
We study the computation of the ααα-Rényi capacity of a classical-quantum (c-q) channel for α∈(0,1)α\in(0,1)α∈(0,1). We propose an exponentiated-gradient (mirror descent) iteration that generalizes the Blahut-Arimoto algorithm. Our analysis establishes relative smoothness with respect to the entropy geometry, guaranteeing a global sublinear convergence of the objective values. Furthermore, under a natural tangent-space nondegeneracy condition (and a mild spectral lower bound in one regime), we prove local linear (geometric) convergence in Kullback-Leibler divergence on a truncated probability simplex, with an explicit contraction factor once the local curvature constants are bounded.
The capacity of noisy quantum channels characterizes the highest rate at which information can be reliably transmitted and it is therefore of practical as well as fundamental importance. Capacities of classical channels are computed using alternating optimization schemes, called Blahut-Arimoto algorithms. In this work, we generalize classical Blahut-Arimoto algorithms to the quantum setting. In particular, we give efficient iterative schemes to compute the capacity of channels with classical input and quantum output, the quantum capacity of less noisy channels, the thermodynamic capacity of quantum channels, as well as the entanglement-assisted capacity of quantum channels. We give rigorousa priorianda posterioribounds on the estimation error by employing quantum entropy inequalities and demonstrate fast convergence of our algorithms in numerical experiments.
The generalized quantum Arimoto--Blahut (QAB) algorithm is a powerful derivative-free iterative method in quantum information theory. A key obstacle to its broader use is that existing convergence guarantees typically rely on analytical conditions that are either overly restrictive or difficult to verify for concrete problems. We address this issue by introducing an a posteriori certification viewpoint: instead of requiring fully a priori verifiable assumptions, we provide convergence and error guarantees that can be validated directly from the iterates produced by the algorithm. Specifically, we prove a generalized global convergence theorem showing that, under convexity and a substantially weaker numerically verifiable condition, the QAB iteration converges to the global minimizer. This theorem yields a practical certification procedure: by checking explicit inequalities along the computed trajectory, one can certify global optimality and bound the suboptimality of the obtained value. As an application, we develop a certified iterative scheme for computing the quantum relative entropy of channels, a fundamental measure of distinguishability in quantum dynamics. This quantity is notoriously challenging to evaluate numerically: gradient-based methods are impeded by the complexity of matrix functions such as square roots and logarithms, while recent semidefinite programming approaches can become computationally and memory intensive at high precision. Our method avoids these bottlenecks by combining the QAB iteration with a posteriori certification, yielding an efficient and scalable algorithm. Numerical experiments demonstrate rapid convergence and improved scalability and adaptivity compared with SDP-based approaches.