bicycle_cliffords

Following Cross et al. arXiv:2407.18393, Section 4.3, the 12 logical qubits of a gross or two-gross qLDPC module are divided into one pivot qubit and 11 data qubits. Here we implement a scheme that allows:

• implementation of arbitrary 11-qubit Clifford gates on the data qubits,
• and, implementation of multi-module Pauli rotations $\exp(i \phi (P_1 \otimes P_2 \otimes ...))$ or multi-module Pauli measurements $(P_1 \otimes P_2 \otimes ...)$, for any 11-qubit $P_i$ on each module.

This is achieved by synthesizing Clifford Pauli rotations $\exp(i \phi Q)$ for 11-qubit Pauli matrices $Q$. These are synthesized as follows:

• The LPU can perform some family of Pauli measurements $\mathcal{P}_\text{LPU}$, and some family of Clifford unitaries can be implemented by automorphisms $\mathcal{C}_\text{aut}$. The native measurements are defined as the set ${ CPC^\dagger \text{ for } P \in \mathcal{P}_\text{LPU} \text{, for } C \in \mathcal{C}_\text{aut} }$ of 12-qubit Pauli matrices.
• The sequence of measurements $(X \otimes I_{11})$, $(Y \otimes Q), (Z \otimes I_{11})$, for an 11-qubit Pauli $Q$ synthesizes $\exp(i \frac{\pi}{4} (I \otimes Q))$ up to Pauli corrections. Here the measurements on the first qubit were $X,Y,Z$, but any ordering of these works. Furthermore, $(X \otimes I_{11}),(Y \otimes I_{11}),(Z \otimes I_{11})$ are all native from the construction of the LPU. Hence we can implement $\exp(i \frac{\pi}{4} (I \otimes Q))$ whenever $I \otimes Q$ is native. We call such 11-qubit $Q$ native rotations.
• Suppose $P$ and $Q$ are native rotations, and $PQ = -QP$. Then $\exp(i \frac{\pi}{4}Q)\exp(i \frac{\pi}{4}P)\exp(-i \frac{\pi}{4}Q) = \exp(i \frac{\pi}{4}PQ)$. Hence we can construct more Pauli rotations by multiplying the Pauli matrices of existing anticommuting ones together.

This module implements a search algorithm that computes efficient implementations of all $4^{11}$ Pauli matrices in terms of native rotations, see decomposition.rs. Which Pauli matrices are native depends entirely on the automorphism actions, since the LPU design is mostly the same in Cross et al. arXiv:2407.18393 and Tour de Gross arXiv:2506.03094. The automorphism actions are tabulated in native_measurement.rs, and can be computed using notebooks/gross_code_automorphisms.ipynb.