Efficiently constructing a quantum uniform superposition over bit strings near a binary linear code

We demonstrate that a high fidelity approximation to $| \Psi_b \rangle$, the quantum superposition over all bit strings within Hamming distance $b$ of the codewords of a dimension-$k$ linear code over $\mathbb{Z}_2^n$, can be efficiently constructed by a quantum circuit for large values of $n$, $b$ and $k$ which we characterize. We do numerical experiments at $n=1000$ which back up our claims. The achievable radius $b$ is much larger than the distance out to which known classical algorithms can efficiently find the nearest codeword. Hence, these states cannot be prepared by quantum constuctions that require uncomputing to find the codeword nearest a string. Unlike the analogous states for lattices in $\mathbb{R}^n$, $|\Psi_b \rangle$ is not a useful resource for bounded distance decoding because the relevant overlap falls off too quickly with distance and known classical algorithms do better. Furthermore the overlap calculation can be dequantized. Perhaps these states could be used to solve other code problems. The technique used to construct these states is of interest and hopefully will have applications beyond codes.