Unconditional Pseudorandomness against Shallow Quantum Circuits

Abstract

Quantum computational pseudorandomness has emerged as a fundamental notion that spans connections to complexity theory, cryptography and fundamental physics. However, all known constructions of efficient quantum-secure pseudorandom objects rely on complexity theoretic assumptions. In this work, we establish the first unconditionally secure efficient pseudorandom constructions against shallow-depth quantum circuit classes. We prove that: $\bullet$ Any quantum state 2-design yields unconditional pseudorandomness against both $\mathsf{QNC}^0$ circuits with arbitrarily many ancillae and $\mathsf{AC}^0\circ\mathsf{QNC}^0$ circuits with nearly linear ancillae. $\bullet$ Random phased subspace states, where the phases are picked using a 4-wise independent function, are unconditionally pseudoentangled against the above circuit classes. $\bullet$ Any unitary 2-design yields unconditionally secure parallel-query pseudorandom unitaries against geometrically local $\mathsf{QNC}^0$ adversaries, even with limited $\mathsf{AC}^0$ postprocessing. Our indistinguishability results for 2-designs stand in stark contrast to the standard setting of quantum pseudorandomness against $\mathsf{BQP}$ circuits, wherein they can be distinguishable from Haar random ensembles using more than two copies or queries. Our work demonstrates that quantum computational pseudorandomness can be achieved unconditionally for natural classes of restricted adversaries, opening new directions in quantum complexity theory.