Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Efficient unitary designs with a system-size independent number of non-Clifford gates

Jonas Haferkamp, Felipe Montealegre-Mora, Markus Heinrich, Jens Eisert, David Gross, Ingo Roth

TL;DR

The paper addresses the difficulty of generating random unitaries by showing that a polynomial-depth Clifford circuit supplemented with a system-size-independent number of non-Clifford gates can realize an additive ε-approximate unitary t-design, with the non-Clifford budget scaling as $O(t^{4}\log^{2} t\log(1/\varepsilon))$. The approach combines a refined Schur-Weyl duality analysis of the Clifford commutant with restricted spectral-gap bounds to quantify the impact of non-Clifford gates, and it analyzes two circuit models: K-interleaved Clifford circuits and local Clifford circuits. The main contributions include explicit depth and gate-count bounds for achieving approximate t-designs, relative ε-approximate designs, and convergence results for local designs, all with resource costs that do not scale with system size. These results have practical implications for efficient, scalable randomized benchmarking, decoupling, and quantum information tasks in fault-tolerant settings, while also shedding light on the structure of Clifford-only versus universal gate sets for unitary designs.

Abstract

Many quantum information protocols require the implementation of random unitaries. Because it takes exponential resources to produce Haar-random unitaries drawn from the full $n$-qubit group, one often resorts to $t$-designs. Unitary $t$-designs mimic the Haar-measure up to $t$-th moments. It is known that Clifford operations can implement at most $3$-designs. In this work, we quantify the non-Clifford resources required to break this barrier. We find that it suffices to inject $O(t^{4}\log^{2}(t)\log(1/\varepsilon))$ many non-Clifford gates into a polynomial-depth random Clifford circuit to obtain an $\varepsilon$-approximate $t$-design. Strikingly, the number of non-Clifford gates required is independent of the system size -- asymptotically, the density of non-Clifford gates is allowed to tend to zero. We also derive novel bounds on the convergence time of random Clifford circuits to the $t$-th moment of the uniform distribution on the Clifford group. Our proofs exploit a recently developed variant of Schur-Weyl duality for the Clifford group, as well as bounds on restricted spectral gaps of averaging operators.

Efficient unitary designs with a system-size independent number of non-Clifford gates

TL;DR

The paper addresses the difficulty of generating random unitaries by showing that a polynomial-depth Clifford circuit supplemented with a system-size-independent number of non-Clifford gates can realize an additive ε-approximate unitary t-design, with the non-Clifford budget scaling as . The approach combines a refined Schur-Weyl duality analysis of the Clifford commutant with restricted spectral-gap bounds to quantify the impact of non-Clifford gates, and it analyzes two circuit models: K-interleaved Clifford circuits and local Clifford circuits. The main contributions include explicit depth and gate-count bounds for achieving approximate t-designs, relative ε-approximate designs, and convergence results for local designs, all with resource costs that do not scale with system size. These results have practical implications for efficient, scalable randomized benchmarking, decoupling, and quantum information tasks in fault-tolerant settings, while also shedding light on the structure of Clifford-only versus universal gate sets for unitary designs.

Abstract

Many quantum information protocols require the implementation of random unitaries. Because it takes exponential resources to produce Haar-random unitaries drawn from the full -qubit group, one often resorts to -designs. Unitary -designs mimic the Haar-measure up to -th moments. It is known that Clifford operations can implement at most -designs. In this work, we quantify the non-Clifford resources required to break this barrier. We find that it suffices to inject many non-Clifford gates into a polynomial-depth random Clifford circuit to obtain an -approximate -design. Strikingly, the number of non-Clifford gates required is independent of the system size -- asymptotically, the density of non-Clifford gates is allowed to tend to zero. We also derive novel bounds on the convergence time of random Clifford circuits to the -th moment of the uniform distribution on the Clifford group. Our proofs exploit a recently developed variant of Schur-Weyl duality for the Clifford group, as well as bounds on restricted spectral gaps of averaging operators.

Paper Structure

This paper contains 25 sections, 35 theorems, 148 equations, 1 figure.

Table of Contents

  1. Results
  2. Approximate $t$-designs with few non-Clifford gates
  3. Local random Clifford circuits for Clifford and unitary designs
  4. Technical preliminaries
  5. Operators and superoperators
  6. Commutant of the diagonal representation of the Clifford group
  7. Approximate unitary $t$-designs
  8. Convergence to higher moments of the Clifford group
  9. Singling out the Clifford group
  10. Finite case.
  11. Infinite case.
  12. Proofs
  13. Proof of overlap lemmas
  14. Proof of Lemmas for Theorem \ref{['thm:Tcountdesign']}
  15. Proof of Haar symmetrization Lemma \ref{['lemma:haarsymmetrization']}
  16. ...and 10 more sections

Key Result

Theorem 1

Let $K \in U(2)$ be a non-Clifford unitary. There are constants $C_1(K), C_2(K)$ such that for any $k\geq C_1(K)\log^2(t)(t^{4}+t\log(1/\varepsilon))$, a $K$-interleaved Clifford circuit with depth $k$ acting on $n$ qubits is an additive $\varepsilon$-approximate $t$-design for all $n\geq C_2(K)t^2$

Figures (1)

  • Figure 1: $K$-interleaved Clifford circuits: We consider a model where random Clifford operations are alternated with a non-Clifford gate $K$ or its inverse $K^\dagger$.

Theorems & Definitions (65)

  • Definition 1: Approximate unitary design
  • Definition 2: $K$-interleaved Clifford circuits
  • Theorem 1: Unitary designs with few non-Clifford gates
  • Definition 3: Relative $\varepsilon$-approximate $t$-design
  • Corollary 1: $K$-interleaved Clifford circuits as relative $\varepsilon$-approximate $t$-designs
  • Proposition 1: Haar-interleaved Clifford circuits as additive $\varepsilon$-approximate $t$-designs
  • Definition 4: Local random Clifford circuit
  • Definition 5: Relative approximate Clifford $t$-designs
  • Theorem 2: Local random Clifford designs
  • Corollary 2: Local random unitary design
  • ...and 55 more