Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Multi-qubit controlled gate with optimal T-count

Soichiro Yamazaki, Seiseki Akibue

Abstract

Controlled gates are key components in various quantum algorithms. Improving on the prior work of Gosset et al., we show that, for an allowed error $\varepsilon$, $3\log_2(1/\varepsilon) + o(\log(1/\varepsilon))$ $T$ gates are sufficient to approximate most multi-qubit controlled SU(2)s. We also show that this T-count matches the lower bound when the use of an almost controlled gate is prohibited. As an application, general controlled gate synthesis and efficient SU(4) gate synthesis are given.

Multi-qubit controlled gate with optimal T-count

Abstract

Controlled gates are key components in various quantum algorithms. Improving on the prior work of Gosset et al., we show that, for an allowed error , gates are sufficient to approximate most multi-qubit controlled SU(2)s. We also show that this T-count matches the lower bound when the use of an almost controlled gate is prohibited. As an application, general controlled gate synthesis and efficient SU(4) gate synthesis are given.
Paper Structure (15 sections, 10 theorems, 84 equations)

This paper contains 15 sections, 10 theorems, 84 equations.

Table of Contents

  1. Introduction
  2. Applications
  3. Discussion
  4. Preliminaries
  5. Basic notations
  6. Asymptotic notation
  7. Clifford+T gate synthesis
  8. $n$-qubit controlled gate synthesis
  9. Circuit construction
  10. T-count analysis
  11. Ancilla-free controlled gate synthesis
  12. Applications
  13. Even-number T-count SU(2) synthesis
  14. Correspondence between SU(4) and SO(6)
  15. Diamond distance between target SU(4) and its approximation

Key Result

Theorem 1

For $n\geq1$, $n$-qubit controlled gates whose diagonal blocks are composed of SU(2) can be implemented up to error $\varepsilon$ by a Clifford+T circuit using T gates and ancillae. Furthermore, when $n$ is fixed, no Clifford+T circuit with a diagonal block form can use asymptotically fewer T gates.

Theorems & Definitions (20)

  • Theorem 1: $\mathrm{SU}(2)^{\oplus 2^n}$ controlled gate synthesis with optimal T-count
  • Theorem 2: $\mathrm{SU}(2)\oplus\mathrm{SU(2)}$ ancilla-free controlled gate synthesis
  • Theorem 3: Controlled gate synthesis
  • Theorem 4: $\mathrm{SU}(2)^{\oplus 2^n}$ ancilla-free controlled gate synthesis
  • Theorem 5: SU(4) gate synthesis
  • Definition 6: Diamond distance
  • Lemma 7: Boolean oracle
  • proof : Proof of Theorem \ref{['thm:su2cnt']}.
  • Lemma 8: Bound on T-count of SU(2) Syntheses
  • proof : Proof of Lemma \ref{['lmm:bound']}.
  • ...and 10 more