Computing ClassicalQuantum Channel Capacity Using Blahutxe2x80x93Arimoto Type Algorithm: A Theoretical and Numerical Analysis Based on Arimotoxe2x80x99s work in 1972, we propose an iterative algorithm for computing the capacity of a discrete memoryless classicalquantum channel with a finite input alphabet and a finite dimensional output, which we call the Blahutxe2x80x93Arimoto algorithm for classicalquantum channel, and an input cost constraint is considered. We show that, to reach    e    accuracy, the iteration complexity of the algorithm is upper bounded by      log n log e  e     where n is the size of the input alphabet. In particular, when the output state        xcfx81 x     x xe2x88x88 X      is linearly independent in complex matrix space, the algorithm has a geometric convergence. We also show that the algorithm reaches an    e    accurate solution with a complexity of     O (    m 3  log n log e  e  )    , and     O (  m 3  log e  log  ( 1  xcexb4 )    e  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    xcexb4    is a positive number. Numerical experiments were performed and an approximate solution for the binary twodimensional case was analysed.

Span programs and quantum space complexity While quantum computers hold the promise of significant computational speedups, the limited size of early quantum machines motivates the study of spacebounded quantum computation. We relate the quantum space complexity of computing a function f with onesided error to the logarithm of its span program size, a classical quantity that is wellstudied in attempts to prove formula size lower  the :59,more natural bounded error model, we show that the amount of space needed for a unitary quantum algorithm to compute f with bounded (twosided) error is lower bounded by the logarithm of its approximate span program size. Approximate span programs were introduced in the field of quantum algorithms but not studied classically. However, the approximate span program size of a function is a natural generalization of its span program  :130,no nontrivial lower bound is known on the span program size (or approximate span program size) of any concrete function, a number of lower bounds are known on the monotone span program size. We show that the approximate monotone span program size of f is a lower bound on the space needed by quantum algorithms of a particular form, called monotone phase estimation algorithms, to compute f. We then give the first nontrivial lower bound on the approximate span program size of an explicit function.