Real-Valued Somewhat-Pseudorandom Unitaries
arxiv(2024)
摘要
We explore a very simple distribution of unitaries: random (binary) phase –
Hadamard – random (binary) phase – random computational-basis permutation. We
show that this distribution is statistically indistinguishable from random Haar
unitaries for any polynomial set of orthogonal input states (in any basis) with
polynomial multiplicity. This shows that even though real-valued unitaries
cannot be completely pseudorandom (Haug, Bharti, Koh, arXiv:2306.11677), we can
still obtain some pseudorandom properties without giving up on the simplicity
of a real-valued unitary.
Our analysis shows that an even simpler construction: applying a random
(binary) phase followed by a random computational-basis permutation, would
suffice, assuming that the input is orthogonal and flat (that is, has
high min-entropy when measured in the computational basis).
Using quantum-secure one-way functions (which imply quantum-secure
pseudorandom functions and permutations), we obtain an efficient cryptographic
instantiation of the above.
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