One of the founding results of lattice based cryptography is a quantum reduction from the Short Integer Solution (SIS) problem to the Learning with Errors (LWE) problem introduced by Regev. It has recently been pointed out by Chen, Liu and Zhandry [Chen et al., 2022] that this reduction can be made more powerful by replacing the LWE problem with a quantum equivalent, where the errors are given in quantum superposition. In parallel, Regev’s reduction has recently been adapted in the context of code-based cryptography by Debris, Remaud and Tillich [Debris-Alazard et al., 2023], who showed a reduction between the Short Codeword Problem and the Decoding Problem (the DRT reduction). This motivates the study of the Quantum Decoding Problem (QDP), which is the Decoding Problem but with errors in quantum superposition and see how it behaves in the DRT reduction.

The purpose of this paper is to introduce and to lay a firm foundation for QDP. We first show QDP is likely to be easier than classical decoding, by proving that it can be solved in quantum polynomial time in a large regime of noise whereas no non-exponential quantum algorithm is known for the classical decoding problem. Then, we show that QDP can even be solved (albeit not necessarily efficiently) beyond the information theoretic Shannon limit for classical decoding. We give precisely the largest noise level where we can solve QDP giving in a sense the information theoretic limit for this new problem. Finally, we study how QDP can be used in the DRT reduction. First, we show that our algorithms can be properly used in the DRT reduction showing that our quantum algorithms for QDP beyond Shannon capacity can be used to find minimal weight codewords in a random code. On the negative side, we show that the DRT reduction cannot be, in all generality, a reduction between finding small codewords and QDP by exhibiting quantum algorithms for QDP where this reduction entirely fails. Our proof techniques include the use of specific quantum measurements, such as q-ary unambiguous state discrimination and pretty good measurements as well as strong concentration bounds on weight distribution of random shifted dual codes, which we relate using quantum Fourier analysis.