Unitary designs in nearly optimal depth

2025-07-08

AI Summary

This work addresses the efficient construction of ε-approximate unitary k-designs in quantum circuits, aiming to minimize circuit depth, auxiliary qubit count, and classical randomness consumption. We introduce a novel approach based on structured random unitary ensembles, leveraging long-range two-qubit gates and quantum implementations of low-depth classical hash functions, supported by a new error-analysis framework. Our main construction achieves depth $O(log k cdot log log(nk/varepsilon))$, uses $ ilde{O}(nk)$ auxiliary qubits and $O(nk)$ random bits—yielding exponential depth improvements in $n$, $k$, and $varepsilon$, with near-optimal depth dependence. A lightweight variant reduces auxiliary qubits to $ ilde{O}(n)$ while maintaining depth $O(k cdot log log(nk/varepsilon))$. The method generalizes naturally to various quantum randomized tasks, including state $k$-designs, decoupling, and randomized benchmarking.

Technology Category

• Machine Learning: Quantum Machine Learning
• Reasoning under Uncertainty: Stochastic Optimization
• Search and Optimization: Sampling/Simulation-based Search

Application Category

• Economics, Online Markets and Human Computation: Incentives in network design for Web infrastructures and ecosystems
• Security and Privacy: Large-scale security measurements
• Graph Algorithms and Modeling for the Web: Representation, reconstruction, and subgraph or motif discovery in Web-related graphs

Abstract

We construct $varepsilon$-approximate unitary $k$-designs on $n$ qubits in circuit depth $O(log k log log n k / varepsilon)$. The depth is exponentially improved over all known results in all three parameters $n$, $k$, $varepsilon$. We further show that each dependence is optimal up to exponentially smaller factors. Our construction uses $ ilde{O}(nk)$ ancilla qubits and ${O}(nk)$ bits of randomness, which are also optimal up to $log(n k)$ factors. An alternative construction achieves a smaller ancilla count $ ilde{O}(n)$ with circuit depth ${O}(k log log nk/varepsilon)$. To achieve these efficient unitary designs, we introduce a highly-structured random unitary ensemble that leverages long-range two-qubit gates and low-depth implementations of random classical hash functions. We also develop a new analytical framework for bounding errors in quantum experiments involving many queries to random unitaries. As an illustration of this framework's versatility, we provide a succinct alternative proof of the existence of pseudorandom unitaries.

Problem

• Constructs efficient unitary k-designs for n qubits
• Optimizes circuit depth and resource usage
• Develops new framework for quantum error analysis

Innovation

• Optimal-depth unitary designs using structured ensembles
• Long-range gates and hash functions for efficiency
• New framework for bounding quantum experiment errors