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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 &#x03B5; accuracy, the iteration complexity of the algorithm is up bounded by log n log &#x03B5; &#x03B5; where n is the size of the input alphabet. In particular, when the output state {&#x03C1;x}x&#x2208;X is linearly independent in complex matrix space, the algorithm has a geometric convergence. We also show that the algorithm reaches an &#x03B5; accurate solution with a complexity of O(m3 log n log &#x03B5;/&#x03B5;), and O(m3 log &#x03B5; log(1-&#x03B4;) D(&#x03B5;/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 &#x03B4; is a positive number.

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