We analyze a generalization of Huffman coding to the quantum case. In particular, we notice various difficulties in using instantaneous codes for quantum communication. Nevertheless, for the storage of quantum information, we have succeeded in constructing a Huffman coding inspired quantum scheme. The number of computational steps in the encoding and decoding processes of N quantum signals can be made to be of polylogarithmic depth by a massively parallel implementation of a quantum gate array. This is to be compared with the O(N/sup 3/) computational steps required in the sequential implementation by Cleve and DiVincenzo (see Phys. Rev., vol.A54, p.2636, 1996) of the well-known quantum noiseless block-coding scheme of Schumacher. We also show that O(N/sup 2/(log N)/sup a/) sequential computational steps are needed for the communication of quantum information using another Huffman coding inspired scheme where the sender must disentangle her encoding device before the receiver can perform any measurements on his signals.
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