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Transversal gates for probabilistic implementation of multi-qubit Pauli rotations

Nobuyuki Yoshioka, Alireza Seif, Andrew Cross, Ali Javadi-Abhari

TL;DR

This work develops a general theory of weak transversal gates that implement logical Pauli rotations probabilistically via local physical unitaries, syndrome measurements, and recovery, enabling input-state–independent operations inside a code block. It proves a constructive path for CSS codes to realize multi-qubit Pauli rotations and extends discussions to non-CSS codes through numerical evidence, culminating in a partially fault-tolerant Clifford+$\phi$ architecture that performs in-place Pauli rotations through a repeat-until-success strategy. Phenomenological simulations show that a $0.003$-radian rotation can achieve a logical error of $9.5\times 10^{-5}$ on a surface code of distance $d=7$ at a physical error rate of $10^{-4}$, without magic-state overhead. Resource estimation for a $N=108$ qubit Trotter-like circuit demonstrates tens-to-hundredsfold runtime improvements over Clifford+T due to natural rotation parallelism, in both surface and gross codes, signaling a practical paradigm for scalable logical operations beyond conventional transversal gates.

Abstract

We introduce a general framework for weak transversal gates -- probabilistic implementation of logical unitaries realized by local physical unitaries -- and propose a novel partially fault-tolerant quantum computing architecture that surpasses the standard Clifford+T architecture on workloads with million-scale Clifford+T gate counts. First, we prove the existence of weak transversal gates on the class of Calderbank-Shor-Steane codes, covering high-rate qLDPC and topological codes such as surface code or color codes, and present an efficient algorithm to determine the physical multi-qubit Pauli rotations required for the desired logical rotation. Second, we propose a partially fault-tolerant Clifford+$φ$ architecture that performs in-place Pauli rotations via a repeat-until-success strategy; phenomenological simulations indicate that a rotation of 0.003 attains logical error of $9.5\times10^{-5}$ on a surface code with $d=7$ at physical error rate of $10^{-4}$, while avoiding the spacetime overheads of magic state factories, small angle synthesis, and routing. Finally, we perform resource estimation on surface and gross codes for a Trotter-like circuit with $N=108$ logical qubits to show that the Clifford+$φ$ architecture outperforms the conventional Clifford+T approach by a factor of tens to a hundred in runtime due to natural rotation-gate parallelism. This work open a novel paradigm for realizing logical operations beyond the constraints of conventional design.

Transversal gates for probabilistic implementation of multi-qubit Pauli rotations

TL;DR

This work develops a general theory of weak transversal gates that implement logical Pauli rotations probabilistically via local physical unitaries, syndrome measurements, and recovery, enabling input-state–independent operations inside a code block. It proves a constructive path for CSS codes to realize multi-qubit Pauli rotations and extends discussions to non-CSS codes through numerical evidence, culminating in a partially fault-tolerant Clifford+ architecture that performs in-place Pauli rotations through a repeat-until-success strategy. Phenomenological simulations show that a -radian rotation can achieve a logical error of on a surface code of distance at a physical error rate of , without magic-state overhead. Resource estimation for a qubit Trotter-like circuit demonstrates tens-to-hundredsfold runtime improvements over Clifford+T due to natural rotation parallelism, in both surface and gross codes, signaling a practical paradigm for scalable logical operations beyond conventional transversal gates.

Abstract

We introduce a general framework for weak transversal gates -- probabilistic implementation of logical unitaries realized by local physical unitaries -- and propose a novel partially fault-tolerant quantum computing architecture that surpasses the standard Clifford+T architecture on workloads with million-scale Clifford+T gate counts. First, we prove the existence of weak transversal gates on the class of Calderbank-Shor-Steane codes, covering high-rate qLDPC and topological codes such as surface code or color codes, and present an efficient algorithm to determine the physical multi-qubit Pauli rotations required for the desired logical rotation. Second, we propose a partially fault-tolerant Clifford+ architecture that performs in-place Pauli rotations via a repeat-until-success strategy; phenomenological simulations indicate that a rotation of 0.003 attains logical error of on a surface code with at physical error rate of , while avoiding the spacetime overheads of magic state factories, small angle synthesis, and routing. Finally, we perform resource estimation on surface and gross codes for a Trotter-like circuit with logical qubits to show that the Clifford+ architecture outperforms the conventional Clifford+T approach by a factor of tens to a hundred in runtime due to natural rotation-gate parallelism. This work open a novel paradigm for realizing logical operations beyond the constraints of conventional design.

Paper Structure

This paper contains 25 sections, 19 theorems, 88 equations, 11 figures, 4 tables, 2 algorithms.

Table of Contents

  1. Introduction
  2. Theory of weak transversal operation
  3. Preliminaries and setup
  4. Weak transversal Z rotation for odd distance
  5. Single-qubit Y rotation
  6. General Pauli rotation
  7. General Pauli rotation in non-CSS codes
  8. In-place Logical Pauli rotation using weak transversal gates
  9. Overview
  10. Distribution of angles
  11. Repeat-until-success strategy
  12. Resource estimation
  13. Surface code
  14. Clifford+T compilation
  15. Clifford+$\phi$ compilation
  16. ...and 10 more sections

Key Result

Theorem 1

(Weak transversal Z rotation gate on odd-distance code) Let a logical QECC patch be an $[[n, k, d_x, d_z]]$ CSS code. Assume that following conditions are satisfied: Then, $U \coloneqq \prod_{j \in {\rm supp}(\bar{Z}_{\rm target})} e^{i \theta Z_j}$ is a weak transversal Z rotation gate.

Figures (11)

  • Figure 1: (a) Concept of a weak transversal gate. If a set of given local unitaries followed by syndrome measurement with result $\bm s$ and correction operation $C_{\bm s}$ results in a logical unitary operation $\bar{U}_{\bm s}$ that is independent of the input state, then we call such an operation weakly transversal. (b) Comparison with existing works that study weak transversal gates bravyi2018correctingsuzuki2017efficientcheng2024emergentPhysRevLett.131.060603behrends2024statistical. Our main contribution is to prove that Pauli rotation gates in various classes of CSS codes, including topological codes such as surface codes and also qLDPC codes such as the gross code bravyi2024highthreshold, allow weak transversal implementation of multi-body Pauli rotation gates (Sec. \ref{['sec:theory']}). Furthermore, we propose a partially fault-tolerant quantum computing architecture that employs weak transversal gate to implement ancilla-free in-place Pauli rotations via a repeat-until-success strategy (Sec. \ref{['sec:rus']}), which results in an order-of-magnitude reduction of runtime in a Trotter-like circuit of $N=108$ logical qubits (Sec. \ref{['sec:resource-est']}).
  • Figure 2: Numerical verification of weak transversal gates in non-CSS codes. The stabilizers are all taken from Ref. codetable. The color of the cells (and the inset integers) indicate the code distance. For all the non-CSS codes shown in this figure, weak transversal implementation of arbitrary Pauli rotation $e^{i \theta \bar{P}}~(\bar{P} \in \{\bar{I}, \bar{X}, \bar{Y}, \bar{Z}\}^{\otimes k})$ is admitted. The gray line shows the boundary of $k=d$, corresponding to whether there is any $Q$-body Pauli operator with $Q > d$. The missing cells are either CSS codes or invalid code parameters without logical operator.
  • Figure 3: Schematic description of repeat-until-success strategy for in-place Pauli rotation gate.
  • Figure 4: Output distribution of logical angles $\bar{\theta}$. Physical angles are drawn from (a) homogeneous distribution $\theta \sim [-\pi/2, \pi/2],$ (b) binomial distribution $\theta \sim \mathcal{N}(-\pi/4, 0.04) + \mathcal{N}(\pi/4, 0.04)$, (c) sharp-peaked binomial distribution $\theta \sim \mathcal{N}(-\pi/4, 4\cdot10^{-6}) + \mathcal{N}(\pi/4, 4\cdot10^{-6})$. Here, $\mathcal{N}(\mu, \sigma^2)$ is the normal distribution with mean value of $\mu$ and standard deviation of $\sigma$.
  • Figure 5: Numerical evaluation of repeat-until-success strategy using $M=3, 5$ rotation gates under phenomenological noise. The physical error rate is assumed to be $p_{\rm phys}=10^{-4}$. (a) Diamond distance between the target rotation gate of $\bar{\theta}=10^{-2}$ and the average output from repeat-until-success strategy using $M=3$ rotation gates. (b) Diamond distance for various target angles using $M=3$ physical rotations. The blue triangle is the one with repeat-until-success, and yellow stars correspond to diluted version. For comparison, we also show the accumulation of logical error in rotated surface code of distance 5 and 7 after 30 code cycles. Panels (c) and (d) are similar plots for $M=5$ physical rotations.
  • ...and 6 more figures

Theorems & Definitions (33)

  • Definition 1
  • Definition 2
  • Definition 3
  • Theorem 1
  • Lemma 1
  • Lemma 2
  • proof
  • Lemma 3
  • Lemma 4
  • proof
  • ...and 23 more