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Designs from magic-augmented Clifford circuits

Yuzhen Zhang, Sagar Vijay, Yingfei Gu, Yimu Bao

TL;DR

The paper shows that shallow Clifford circuits augmented with fixed-depth magic gates can realize approximate $k$-designs with reduced depth and limited magic usage. It develops both relative- and additive-error frameworks for state and unitary designs, providing explicit depth bounds in 1D and all-to-all layouts, and introduces a unifying statistical-mechanics picture to explain when and why these designs emerge. No-go theorems delineate fundamental limitations for certain architectures to achieve bounded relative error, while constructive results demonstrate feasible additive-error designs with a system-size-independent magic budget. The work thereby advances resource-efficient routes to pseudo-randomness in quantum circuits, with implications for benchmarking, quantum chaos studies, and fault-tolerant architectures, and links circuit design to a tractable physical interpretation via a stat-mech mapping.

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 $O(\log (N/ε)) +2^{O(k\log k)}$ in one dimension and $O(\log\log(N/ε))+2^{O(k\log k)}$ in all-to-all circuits using ancillas, which improves upon previous results for small $k \geq 4$. 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 $O(k^2)$ 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.

Designs from magic-augmented Clifford circuits

TL;DR

The paper shows that shallow Clifford circuits augmented with fixed-depth magic gates can realize approximate -designs with reduced depth and limited magic usage. It develops both relative- and additive-error frameworks for state and unitary designs, providing explicit depth bounds in 1D and all-to-all layouts, and introduces a unifying statistical-mechanics picture to explain when and why these designs emerge. No-go theorems delineate fundamental limitations for certain architectures to achieve bounded relative error, while constructive results demonstrate feasible additive-error designs with a system-size-independent magic budget. The work thereby advances resource-efficient routes to pseudo-randomness in quantum circuits, with implications for benchmarking, quantum chaos studies, and fault-tolerant architectures, and links circuit design to a tractable physical interpretation via a stat-mech mapping.

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 -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 -designs with relative error. The total circuit depth for these constructions on qubits is in one dimension and in all-to-all circuits using ancillas, which improves upon previous results for small . Furthermore, our construction of relative-error state -designs only involves states with strictly local magic. The required number of magic gates is parametrically reduced when considering -designs with bounded additive error. As an example, we show that shallow Clifford circuits followed by single-qubit magic gates, independent of system size, can generate an additive-error state -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 -designs. We also prove no-go theorems for various architectures to generate designs with bounded relative error.

Paper Structure

This paper contains 33 sections, 30 theorems, 155 equations, 2 tables.

Table of Contents

  1. Introduction
  2. Main results
  3. State designs with bounded relative error
  4. Unitary designs with bounded relative error
  5. State designs with bounded additive error
  6. Unitary designs with bounded additive error
  7. Statistical mechanics picture
  8. Reader's guide
  9. Preliminaries
  10. Approximate unitary designs
  11. Approximate state designs
  12. Commutant of Clifford tensor powers
  13. Uniformity condition
  14. Designs with bounded relative error
  15. State designs with bounded relative error
  16. ...and 18 more sections

Key Result

Theorem 1

Consider a state ensemble $\mathcal{E} = \{UV\ket{0}^{\otimes N}\}$ generated by the circuit architecture below Here, $U$ is a product of random unitaries drawn from exact unitary $k$-designs acting on disjoint qubit clusters of size $\ell$. The two-layer Clifford unitary $V$ consists of Clifford ga

Theorems & Definitions (58)

  • Theorem : Relative-error state designs from low-depth Clifford unitaries followed by constant depth magic gates
  • Theorem
  • Theorem : Relative-error unitary designs from low-depth Clifford unitaries sandwiched by constant depth magic gates
  • Corollary
  • Theorem
  • Theorem : Additive-error state designs from low-depth Clifford unitaries followed by constant magic
  • Theorem : Additive-error stabilizer state designs from low-depth Clifford circuits
  • Corollary
  • Theorem : Additive-error unitary designs from circuits with magic gates over constant-number of qubits
  • Corollary
  • ...and 48 more