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Exact and Approximate Unitary 2-Designs: Constructions and Applications

Christoph Dankert, Richard Cleve, Joseph Emerson, Etera Livine

TL;DR

This work addresses the inefficiency of Haar-random unitaries by constructing exact and approximate unitary 2-designs for multi-qubit systems. It provides an exact construction via the Clifford group with circuit size O(n^2) and an effective approximate design with in-place circuits of size O(n log 1/ε) and depth O(log n log 1/ε). These designs enable scalable fidelity estimation of quantum channels using only in-place circuits, eliminating the need for ancillas. The results advance practical deployment of unitary designs in quantum information tasks and open pathways for extending to higher t-designs.

Abstract

We develop the concept of a unitary t-design as a means of expressing operationally useful subsets of the stochastic properties of the uniform (Haar) measure on the unitary group U(2^n) on n qubits. In particular, sets of unitaries forming 2-designs have wide applicability to quantum information protocols. We devise an O(n)-size in-place circuit construction for an approximate unitary 2-design. We then show that this can be used to construct an efficient protocol for experimentally characterizing the fidelity of a quantum process on n qubits with quantum circuits of size O(n) without requiring any ancilla qubits, thereby improving upon previous approaches.

Exact and Approximate Unitary 2-Designs: Constructions and Applications

TL;DR

This work addresses the inefficiency of Haar-random unitaries by constructing exact and approximate unitary 2-designs for multi-qubit systems. It provides an exact construction via the Clifford group with circuit size O(n^2) and an effective approximate design with in-place circuits of size O(n log 1/ε) and depth O(log n log 1/ε). These designs enable scalable fidelity estimation of quantum channels using only in-place circuits, eliminating the need for ancillas. The results advance practical deployment of unitary designs in quantum information tasks and open pathways for extending to higher t-designs.

Abstract

We develop the concept of a unitary t-design as a means of expressing operationally useful subsets of the stochastic properties of the uniform (Haar) measure on the unitary group U(2^n) on n qubits. In particular, sets of unitaries forming 2-designs have wide applicability to quantum information protocols. We devise an O(n)-size in-place circuit construction for an approximate unitary 2-design. We then show that this can be used to construct an efficient protocol for experimentally characterizing the fidelity of a quantum process on n qubits with quantum circuits of size O(n) without requiring any ancilla qubits, thereby improving upon previous approaches.

Paper Structure

This paper contains 5 sections, 3 theorems, 20 equations, 3 figures.

Table of Contents

  1. Introduction
  2. Definitions and summary of results
  3. Exact construction
  4. Approximate construction
  5. Application to fidelity estimation

Key Result

Theorem 1

The uniform distribution over the Clifford group on $n$ qubits is a unitary $2$-design with $D=2^n$.

Figures (3)

  • Figure 1: The uniformization procedure used in the approximate construction.
  • Figure 2: Conjugating the first qubit by a random XOR is conjugation by a randomly generated circuit of the above form, where a numerical label associated with a gate indicates that it occurs with probability $3/4$ (with probability $1/4$ there is no gate). That is, for each $k \in \{2,\dots,n\}$, with independent probability $3/4$, there is a CNOT gate with the first qubit as target and qubit $k$ as control.
  • Figure 3: Part (A) shows a circuit consisting of $n=7$ CNOT gates with common target; part (B) shows as equivalent circuit (based on a binary tree of additions modulo 2) whose depth is bounded by $2\log n$ and size is bounded $2n$.

Theorems & Definitions (3)

  • Theorem 1
  • Theorem 2
  • Theorem 3