Real-Valued Somewhat-Pseudorandom Unitaries

TL;DR

This paper demonstrates that a simple distribution of real-valued unitaries, constructed from random phases and permutations, exhibits pseudorandom properties similar to Haar unitaries for polynomially many orthogonal states, with cryptographic applications.

Contribution

It introduces a simple, real-valued unitary construction that achieves pseudorandomness similar to Haar unitaries, expanding the understanding of pseudorandom quantum operations.

Findings

Distribution is statistically indistinguishable from Haar unitaries for polynomial input sets.

A simpler construction with phases and permutations suffices under certain input conditions.

Provides a cryptographic instantiation using quantum-secure one-way functions.

Abstract

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 \emph{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.