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  • Open Access

Designs from Magic-Augmented Clifford Circuits

Yuzhen Zhang1,*, Sagar Vijay1,†, Yingfei Gu2,‡, and Yimu Bao3,§

  • *Contact author: yuzhenzhang@ucsb.edu
  • †Contact author: sagarvijay@ucsb.edu
  • ‡Contact author: guyingfei@tsinghua.edu.cn
  • §Contact author: baoyimu@gmail.com
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PRX Quantum 7, 010344 – Published 4 March, 2026

DOI: https://doi.org/10.1103/myrb-nyhf

Abstract

We introduce magic-augmented Clifford circuits—architectures in which Clifford circuits are preceded and/or followed by constant-depth circuits of non-Clifford (“magic”) gates—as a resource-efficient way to realize approximate k-designs, with reduced circuit depth and usage of magic. We prove that shallow-Clifford circuits, when augmented with constant-depth circuits of magic gates, can generate approximate unitary and state k-designs with ε relative error. The total circuit depth for these constructions on N qubits is O(log(N/ε))+2O(klogk) in one dimension and O(loglog(N/ε))+2O(klogk) in all-to-all circuits using ancillas, which improves upon previous results for small k≥4. Furthermore, our construction of relative-error state k-designs only involves states with strictly local magic. The required number of magic gates is parametrically reduced when considering k-designs with bounded additive error. As an example, we show that shallow-Clifford circuits followed by O(k2) single-qubit magic gates, independent of system size, can generate an additive-error state k-design. We develop a classical statistical mechanics description of our random circuit architectures, which provides a quantitative understanding of the required depth and number of magic gates for additive-error state k-designs. We also prove no-go theorems for various architectures to generate designs with bounded relative error.

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Popular Summary

Random quantum circuits that reproduce the statistical properties of global random unitaries have broad applications in quantum information science and high-energy physics. Yet, existing architectures capable of generating such highly random operations either involve an unlimited use of generic local gates or require circuit depths that scale linearly with system size—both of which are challenging for current quantum platforms.

In this work, we introduce magic-augmented Clifford circuits—architectures consisting of Clifford circuits preceded and followed by low-depth magic gates—to generate approximate k-designs, a standard notion of random unitary in quantum information science. Our architecture is motivated by realistic quantum hardware operating over error-correcting units (based on stabilizer codes), where Clifford gates are easy to implement, and the main resources to optimize are circuit depth and the number of magic gates. Our architecture is proven to excel on both fronts.

We prove that our construction can generate approximate k-designs with a lower circuit depth than any existing constructions for k≥4, including those that allow unlimited use of magic gates. We establish results for k-designs with various approximation measures, including the most stringent notion, called bounded relative error. At the same time, the magic cost remains modest. For example, we show that a state k-design with bounded additive error can be realized in a logarithmic-depth circuit with a system-size independent number of magic gates.

Our results demonstrate that approximate designs can arise from surprisingly shallow circuits with limited use of magic gates, offering a resource-efficient path toward generating random unitary on realistic quantum devices.

Article Text

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PRX Quantum 7, 010344– Published 4 March, 2026
  • Received 15 August 2025
  • Revised 24 November 2025
  • Accepted 20 January 2026
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