Concluding Remarks
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We have successfully constructed a Huffman-coding inspired scheme for the storage of quantum information. Our scheme is highly efficient. The encoding and decoding processes of $N$ quantum signals can be done [*in parallel*]{} with depth polynomial in $\log N$. (If parallel machines are unavailable, as shown in subsection \[ss:sequent\] our encoding scheme will still take only $O \left(N (\log N)^a \right)$ computational steps for a sequential implementation. In contrast, a naive implementation of Schumacher’s scheme will require $O(N^3)$ computational steps.) This massive parallelism is possible because we explicitly use another tape to store the length information of the individual signals. The storage space needed is asymptotically $N ( \bar{L} + \delta + \lceil \log l_{\rm max} \rceil )$ where $\bar{L}$ is the average length of the corresponding classical Huffman coding problem for the density matrix in the diagonal form, $\delta $ is an arbitrary small positive number and $ l_{\rm max}$ is the length of the longest Huffman codeword.

We also considered the problem of using quantum Huffman coding for communication in which case Alice encodes the signals one-by-one. $N
( \bar{L} + \delta +\lceil \log l_{\rm max} \rceil ) + O(\log N)$ qubits are needed. With 