can be applied in [*parallel*]{}, the encoding and decoding of quantum Huffman coding can be done efficiently. While a sequential implementation of quantum source [*block*]{} coding [@schumacher; @jozsa; @fuchs] for $N$ signals requires $O (N^3)$ computational steps [@cleve], a parallel implementation of quantum Huffman coding has only $O ( ({\log N})^a)$ depth for some positive integer $a$.

We will now describe our coding scheme for the storage of quantum signals. As before, we consider a quantum source emitting a sequence of independent identically distributed quantum signals with a density matrix for each signal shown in Eq. (1) where $p_i$’s are the eigenvalues. Considering Huffman coding for a classical source with probabilities $p_i$’s allows one to construct a one-to-one correspondence between Huffman codewords $h_i $ and the eigenstates $|
\phi_i \rangle $. For parallel implementation, we find it useful to represent $| \phi_i \rangle$ by two pieces,[^6] the first being the Huffman codeword, padded by the appropriate number of zeroes to make it into constant length,[^7] $|
0 \cdots 0 h_i\rangle$, the second being the length of the Huffman codeword, $| l_i \rangle$, where $l_i = {\rm length} ( h_i) $. We also pad zeroes to the second piece so that 