Unconditional Pseudorandomness against Shallow Quantum Circuits

2025-07-24

AI Summary

This work studies unconditionally secure constructions of quantum pseudorandomness, designed to withstand attacks by shallow quantum circuits—including QNC⁰ and AC⁰∘QNC⁰—without relying on any computational complexity assumptions. Methodologically, it establishes, for the first time, the indistinguishability (against auxiliary-input shallow quantum circuits) between quantum state 2-designs and random phase subspace states, under no assumptions. Key technical ingredients include: construction of phases via 4-wise independent functions, explicit unitary 2-designs, and fine-grained complexity analysis tailored to shallow circuits. The core contribution is the first unconditional demonstration of quantum pseudorandomness and pseudoentanglement: ensembles of 2-design states are computationally indistinguishable from truly random states to any constant-depth quantum circuit. This result opens a new pathway toward achieving quantum pseudorandomness without computational hardness assumptions.

Technology Category

• Machine Learning: Quantum Machine Learning
• Reasoning under Uncertainty: Other Foundations of Reasoning under Uncertainty
• Knowledge Representation and Reasoning: Qualitative Reasoning

Application Category

• Security and Privacy: Large-scale security measurements
• Economics, Online Markets and Human Computation: Fairness, privacy, and diversity in economic environments
• Responsible Web: Measurement, analysis, and circumvention of Web censorship

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: $ullet$ Any quantum state 2-design yields unconditional pseudorandomness against both $mathsf{QNC}^0$ circuits with arbitrarily many ancillae and $mathsf{AC}^0circmathsf{QNC}^0$ circuits with nearly linear ancillae. $ullet$ Random phased subspace states, where the phases are picked using a 4-wise independent function, are unconditionally pseudoentangled against the above circuit classes. $ullet$ 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.

Problem

• Unconditional pseudorandomness for shallow quantum circuits
• Pseudorandom constructions without complexity assumptions
• Security against restricted quantum adversaries

Innovation

• Quantum state 2-designs for pseudorandomness against QNC^0 circuits
• Random phased subspace states with 4-wise independence
• Unitary 2-designs secure against geometrically local QNC^0 adversaries