A few observations: we don't yet know how to perform long-term qbit storage, so all multi-day storage at present is classic bits.
If the proposal is to hash all of the possible lattice points using irreversable computation on Earth in a reasonable amount of time, for a cryptographically useful lattice, the waste heat would literally boil the oceans. That's limitation doesn't hold for quantum computers, but results need to be converted into classic bits for storage. Presumably rainbow-table-like operations could reduce the storage space and waste heat necessary in storing these bits. Maybe quantum rainbow tables are a fruitful area of research.
If we're not pre-hashing all possible lattice points, then either we has the public value and we have some heuristic that allows us to hash only some small subset of lattice points for this particular value (... but then why use a hash? ... just compare points in your heuristic sequence) or we have some inverse hash operation where we hash the public value, then explore the "un-hash space" of lattice points corresponding to the LSH for the point in question.
This last general line of inquiry (trying to find search-space reducing heuristics) is probably where most of the research effort is going, though perhaps not specifically using LSH. In order to be useful for the lattice problem, the distance metrics for the input and output spaces of the LSH need to have specific relations to the lattice distance metric.
So... yea... maybe it helps to re-cast the problem in terms of LSH, but not in any obvious way. Also, any search-space reducing heuristic could probably be re-interpreted as a LSH if you squint at it hard enough.
If the proposal is to hash all of the possible lattice points using irreversable computation on Earth in a reasonable amount of time, for a cryptographically useful lattice, the waste heat would literally boil the oceans. That's limitation doesn't hold for quantum computers, but results need to be converted into classic bits for storage. Presumably rainbow-table-like operations could reduce the storage space and waste heat necessary in storing these bits. Maybe quantum rainbow tables are a fruitful area of research.
If we're not pre-hashing all possible lattice points, then either we has the public value and we have some heuristic that allows us to hash only some small subset of lattice points for this particular value (... but then why use a hash? ... just compare points in your heuristic sequence) or we have some inverse hash operation where we hash the public value, then explore the "un-hash space" of lattice points corresponding to the LSH for the point in question.
This last general line of inquiry (trying to find search-space reducing heuristics) is probably where most of the research effort is going, though perhaps not specifically using LSH. In order to be useful for the lattice problem, the distance metrics for the input and output spaces of the LSH need to have specific relations to the lattice distance metric.
So... yea... maybe it helps to re-cast the problem in terms of LSH, but not in any obvious way. Also, any search-space reducing heuristic could probably be re-interpreted as a LSH if you squint at it hard enough.