Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Catalytic $z$-rotations in constant $T$-depth

Isaac H. Kim

TL;DR

The paper addresses the time overhead of fault-tolerant quantum circuits by aiming to minimize the $T$-depth of single-qubit $z$-rotations. It introduces the Primitive Polynomial Method, leveraging a primitive polynomial over $\\mathbb{F}_2$ to define a Clifford unitary $U_f$ whose phase-kickback on a carefully prepared catalyst state yields rotations with constant $T$-depth, specifically $T$-depth $3$. For an $\\epsilon$-approximation, a catalyst state of size poly$(\\log(1/\\epsilon))$ suffices, and the catalyst can be prepared in time poly$(\\log(1/\\epsilon))$, enabling a finite universal gate set Clifford+$T$ for $\\mathsf{QNC}^0_f/\\mathsf{qpoly}$. This approach implies that important subroutines such as multi-qubit Toffoli gates, adders, and the quantum Fourier transform can be implemented at constant $T$-depth given enough catalyst resources. The paper also provides two polynomial-time methods to prepare the catalyst states, via discrete logarithm and Frobenius endomorphism, establishing a concrete foundation for catalytic constant-$T$-depth rotations with broad implications for fault-tolerant quantum computation.

Abstract

We show that the $T$-depth of any single-qubit $z$-rotation can be reduced to $3$ if a certain catalyst state is available. To achieve an $ε$-approximation, it suffices to have a catalyst state of size polynomial in $\log(1/ε)$. This implies that $\mathsf{QNC}^0_f/\mathsf{qpoly}$ admits a finite universal gate set consisting of Clifford+$T$. In particular, there are catalytic constant $T$-depth circuits that approximate multi-qubit Toffoli, adder, and quantum Fourier transform arbitrarily well. We also show that the catalyst state can be prepared in time polynomial in $\log (1/ε)$.

Catalytic $z$-rotations in constant $T$-depth

TL;DR

The paper addresses the time overhead of fault-tolerant quantum circuits by aiming to minimize the -depth of single-qubit -rotations. It introduces the Primitive Polynomial Method, leveraging a primitive polynomial over to define a Clifford unitary whose phase-kickback on a carefully prepared catalyst state yields rotations with constant -depth, specifically -depth . For an -approximation, a catalyst state of size poly suffices, and the catalyst can be prepared in time poly, enabling a finite universal gate set Clifford+ for . This approach implies that important subroutines such as multi-qubit Toffoli gates, adders, and the quantum Fourier transform can be implemented at constant -depth given enough catalyst resources. The paper also provides two polynomial-time methods to prepare the catalyst states, via discrete logarithm and Frobenius endomorphism, establishing a concrete foundation for catalytic constant--depth rotations with broad implications for fault-tolerant quantum computation.

Abstract

We show that the -depth of any single-qubit -rotation can be reduced to if a certain catalyst state is available. To achieve an -approximation, it suffices to have a catalyst state of size polynomial in . This implies that admits a finite universal gate set consisting of Clifford+. In particular, there are catalytic constant -depth circuits that approximate multi-qubit Toffoli, adder, and quantum Fourier transform arbitrarily well. We also show that the catalyst state can be prepared in time polynomial in .

Paper Structure

This paper contains 9 sections, 17 equations, 4 figures.

Table of Contents

  1. Introduction
  2. Primitive Polynomial Method
  3. Clifford Interpretation
  4. Phase Kickback
  5. Resource Estimate
  6. Efficient Preparation of the Catalyst State
  7. Algorithm Based on Discrete Logarithm
  8. Algorithm Based on Frobenius Endomorphism
  9. Discussion

Figures (4)

  • Figure 1: A depth-$2$ circuit consisting of $\mathsf{SWAP}$ gates for implementing $\mathsf{S}$. The red dots are ancillary qubits. In the right figure, a periodic boundary condition in the $x$-direction is used.
  • Figure 2: Fanout gate can be realized by a sequence of $\mathsf{CX}$ gates.
  • Figure 3: Top: Depth-$2$ implementation of controlled $\mathsf{S}$. Bottom: Relation between controlled-$\mathsf{SWAP}$ and Toffoli.
  • Figure 4: Using unbounded fanout, controlled $\mathsf{CX}_{n-1 \to Q_f}$ can be implemented by a circuit with a Toffoli depth of $1$.