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Low-overhead Magic State Circuits with Transversal CNOTs

Nicholas Fazio, Mark Webster, Zhenyu Cai

TL;DR

The paper develops a phase-rotation parallelisation framework tailored to architectures with transversal CNOTs to dramatically reduce the overhead of fault-tolerant magic-state circuits. By expressing magic-state protocols as products of multi-qubit phase rotations and conjugating them with carefully synthesized CNOT blocks, the authors achieve low CNOT depth, small qubit counts, and maximal $T$-parallelisation for CCZ, CS, and $T$-state circuits. The method introduces a CNOT-synthesis greedy heuristic and partitions rotations into independent sets via parity matrices, enabling scalable optimization without auxiliary qubits. Compared to lattice-surgery approaches, the resulting circuits offer substantial spacetime savings (factor of 5–7.5 in reported regimes) and sharpen practicality for near-term fault-tolerant quantum computing, especially on hardware with improved connectivity or quasi-3D layouts.

Abstract

With the successful demonstration of transversal CNOTs in many recent experiments, it is the right moment to examine its implications on one of the most critical parts of fault-tolerant computation -- magic state preparation. Using an algorithm that can recompile and simplify a circuit of consecutive multi-qubit phase rotations, we manage to construct fault-tolerant circuits for CCZ, CS and T states with much lower CNOT depths and qubit counts than before and minimal T-depth for the given workspace. These circuits can play crucial roles in fault-tolerant computation with transversal CNOTs, and we hope that the algorithms and methods developed in this paper can be used to further simplify other protocols in similar contexts.

Low-overhead Magic State Circuits with Transversal CNOTs

TL;DR

The paper develops a phase-rotation parallelisation framework tailored to architectures with transversal CNOTs to dramatically reduce the overhead of fault-tolerant magic-state circuits. By expressing magic-state protocols as products of multi-qubit phase rotations and conjugating them with carefully synthesized CNOT blocks, the authors achieve low CNOT depth, small qubit counts, and maximal -parallelisation for CCZ, CS, and -state circuits. The method introduces a CNOT-synthesis greedy heuristic and partitions rotations into independent sets via parity matrices, enabling scalable optimization without auxiliary qubits. Compared to lattice-surgery approaches, the resulting circuits offer substantial spacetime savings (factor of 5–7.5 in reported regimes) and sharpen practicality for near-term fault-tolerant quantum computing, especially on hardware with improved connectivity or quasi-3D layouts.

Abstract

With the successful demonstration of transversal CNOTs in many recent experiments, it is the right moment to examine its implications on one of the most critical parts of fault-tolerant computation -- magic state preparation. Using an algorithm that can recompile and simplify a circuit of consecutive multi-qubit phase rotations, we manage to construct fault-tolerant circuits for CCZ, CS and T states with much lower CNOT depths and qubit counts than before and minimal T-depth for the given workspace. These circuits can play crucial roles in fault-tolerant computation with transversal CNOTs, and we hope that the algorithms and methods developed in this paper can be used to further simplify other protocols in similar contexts.
Paper Structure (24 sections, 1 theorem, 8 equations, 15 figures, 1 algorithm)

This paper contains 24 sections, 1 theorem, 8 equations, 15 figures, 1 algorithm.

Table of Contents

  1. Introduction
  2. Motivating example: fault-tolerant CCZ
  3. Algorithms for Phase Rotation Parallelisation
  4. Notation for Phase Rotation and CNOT Operators
  5. Parallelisation of Phase Rotation Operators
  6. Partition of Phase Rotations into Independent Sets
  7. Simplification of CNOT Circuit Blocks
  8. Fault-tolerant CCZ Example
  9. More magic state circuits
  10. Fault-tolerant CS gate
  11. T gate distillation circuit
  12. Resource Analysis
  13. Discussion
  14. Circuit Compilation Tricks
  15. Off-loading $Z$ Rotations
  16. ...and 9 more sections

Key Result

Lemma 1

Let $\prod_{0 \le i < n}Z^{\theta_i}_{\mathbf{u}_i}$ be a series of independent phase rotation operators on $n$ qubits and let $U$ be the invertible binary matrix with columns $\mathbf{u}_i$. The product of the phase rotation operators can be written as:

Figures (15)

  • Figure 1: Circuit for implementing CC($i$Z).
  • Figure 2: Circuit for fault-tolerant CCZ. The CCZs here can be implemented using \ref{['fig:CCiZfromT']} and its conjugate, thus there are two additional auxiliary qubits used that are not shown here.
  • Figure 3: Circuit for distilling CCZ states from $T$/$T^\dagger$ gates. The red and blue dashed boxes denote the $T$ gates that are used for implementing a particular CC($i$Z) gate.
  • Figure 4: Circuit for distilling CCZ states using multi-qubit $\pi/8$ phase rotations. The index over each phase rotation denotes which $T$ gate qubit it corresponds to in \ref{['fig:JonesCirc']}.
  • Figure 5: Circuit for fault-tolerant CCZ. Note that the first layer of CNOTs only has depth $3$. The CNOT gates in the circuit can be implemented using 2D nearest-neighbour connectivity.
  • ...and 10 more figures

Theorems & Definitions (1)

  • Lemma 1: parallelisation of independent phase rotation operators