Random Unitaries in Constant (Quantum) Time

2025-08-15

AI Summary

This work addresses the efficient construction of unitary designs and pseudorandom unitaries in the constant-time quantum computation model. We introduce the first constant-depth constructions of both objects within the enhanced constant-depth circuit model—comprising multi-qubit Toffoli/Fanout gates, mid-circuit measurements, and classical feedforward. Our theoretical contributions are threefold: (1) We overcome the conventional depth dependence, significantly broadening the applicability of shallow quantum circuits in quantum information processing; (2) We establish cryptographic unlearnability of the QAC⁰ circuit class, proving—under standard cryptographic assumptions—that no polynomial-time algorithm can learn such circuits; (3) We provide a novel analytical pathway toward resolving the long-standing open problem of PARITY’s uncomputability in QAC⁰. Collectively, these results demonstrate that cryptographically strong pseudorandom unitaries can be efficiently generated even in highly restricted quantum circuit models.

Technology Category

• Machine Learning: Quantum Machine Learning
• Game Theory and Economic Paradigms: Adversarial Learning
• Search and Optimization: Learning to Search

Application Category

• Security and Privacy: Applications of cryptography
• Graph Algorithms and Modeling for the Web: Algorithms and analysis for incomplete, noisy, or partially observed Web-related graphs
• Search and Retrieval-Augmented AI: Web learning to rank, online learning, and counterfactual learning for ranking

Abstract

Random unitaries are a central object of study in quantum information, with applications to quantum computation, quantum many-body physics, and quantum cryptography. Recent work has constructed unitary designs and pseudorandom unitaries (PRUs) using $Θ(log log n)$-depth unitary circuits with two-qubit gates. In this work, we show that unitary designs and PRUs can be efficiently constructed in several well-studied models of $ extit{constant-time}$ quantum computation (i.e., the time complexity on the quantum computer is independent of the system size). These models are constant-depth circuits augmented with certain nonlocal operations, such as (a) many-qubit TOFFOLI gates, (b) many-qubit FANOUT gates, or (c) mid-circuit measurements with classical feedforward control. Recent advances in quantum computing hardware suggest experimental feasibility of these models in the near future. Our results demonstrate that unitary designs and PRUs can be constructed in much weaker circuit models than previously thought. Furthermore, our construction of PRUs in constant-depth with many-qubit TOFFOLI gates shows that, under cryptographic assumptions, there is no polynomial-time learning algorithm for the circuit class $mathsf{QAC}^0$. Finally, our results suggest a new approach towards proving that PARITY is not computable in $mathsf{QAC}^0$, a long-standing question in quantum complexity theory.

Problem

• Efficient construction of unitary designs in constant-time quantum computation
• Pseudorandom unitaries in constant-depth circuits with nonlocal operations
• Exploring quantum complexity via PRUs and learning algorithm limitations

Innovation

• Constant-time quantum computation models
• Many-qubit TOFFOLI and FANOUT gates
• Mid-circuit measurements with feedforward