Measurement-free code-switching for low overhead quantum computation using permutation invariant codes

TL;DR

This work addresses universal fault-tolerant quantum computation without measurements by introducing a measurement-free code-switching protocol that alternates between a stabiliser code (transversal Clifford gates) and permutation-invariant (PI) codes (transversal non-Clifford gates). The approach relies on geometric phase gates mediated by a bosonic mode to implement cross-code entangling operations and prepares PI ancilla states via optimized linear GPG sequences. Key contributions include (i) a concrete code-switching circuit between stabiliser and PI codes, (ii) a rich family of PI codes with tunable distance enabling transversal $Z( heta)$ gates and even the binary icosahedral group structure, and (iii) quantitative analysis of gate counts, fault-tolerance properties, and performance showing substantial overhead reductions compared to measurement-based or triorthogonal-code schemes. The results indicate a viable path to more resource-efficient universal quantum computation in near-term architectures employing collective, permutation-invariant encodings and cavity-mediated interactions, with extensions to super-golden gate approximations and fault-tolerant protocols discussed for future work.

Abstract

Transversal gates on quantum error correction codes have been a promising approach for fault-tolerant quantum computing, but are limited by the Eastin-Knill no-go theorem. Existing solutions like gate teleportation and magic state distillation are resource-intensive. We present a measurement-free code-switching protocol for universal quantum computation, switching between a stabiliser code for transversal Cliffords and a permutation-invariant (PI) code for transversal non-Cliffords that are logical $Z$ rotations for any rational multiple of $π$. The novel non-Clifford gates enabled by this code-switching protocol provide for a lower gate count implementation of a universal gate set relative to the Clifford$+T$ gate set. To achieve this, we present a protocol for performing controlled-NOTs between the codes using near-term quantum control operations that employ a catalytic bosonic mode. We also present a new class of PI codes with tunable code distance, supporting transversal non-Clifford gates, and demonstrate their reduced gate count overhead relative to a comparable stabilizer code to stabilizer code switching scheme.

Figures (2)

• Figure 1: (a) Two registers of qubits are coupled to a common cavity mode with strength $g$. The cavity mode decays at a rate $\kappa$, while the spins decay from their excited states at a rate $\gamma$. The cavity acts as a mediator of interactions between the qubits, allowing the two registers to influence one another through their coupling to the same cavity mode. (b) Illustration of a closed trajectory in phase space of a coherent state $\alpha(t)$ of the cavity mode associated to an eigenspace of the spin operator $\hat{w}_{\Gamma}$ for a Linear GPG. After the gate, the mode and the spins are disentangled and the phase accumulated on the eigenspace depends on the area of the trajectory.
• Figure 2: Process infidelity $1-F_{\rm pro}(\mathcal{E}_{\overline{H}},\overline{\mathcal{H}})$ for implementing the logical Hadamard gate (orange, triangles) on the PI-11 code, and state infidelity $1-\langle+_{\rm pi11}|\rho|+_{\rm pi11}\rangle$ for preparing the logical $|+_{\rm pi11}\rangle$ state (blue,circles), using l-GPGs as a function of cooperativity $C$. In both cases $P=10$ gate sequences are used for the state preparation steps.

Theorems & Definitions (2)

• Lemma 1
• proof