Fault-tolerant multi-qubit gates in Parity Codes

TL;DR

The paper addresses fault-tolerant realization of multi-qubit gates in parity codes by exploiting parity qubits to enable long-range, locally implemented logical operations. It introduces a universal fault-tolerant gate set including transversal CNOTs and logical $R_{ZZ}(\pi/2)$ and $R_{ZZ}(\pi/4)$ via magic-state teleportation, and a protocol for arbitrary $\bar{R}_{ZZ}(\alpha)$ rotations using a protected parity-qubit copy. It also defines parity-controlled-NOT gates as parity-based sequences of CNOTs and analyzes the fault-tolerance requirements for such operations across different encodings. These methods enable highly parallelizable, lattice-surgery-free computation that can be integrated with surface codes or bias-preserving encodings, offering practical pathways to scalable quantum computation for applications in Hamiltonian simulation and optimization.

Abstract

We present a set of efficiently implementable logical multi-qubit gates in concatenated quantum error correction codes using parity qubits. In particular, we show how fault-tolerant high-weight rotation gates of arbitrary angle can be implemented on single physical qubits of a classical stabilizer code, or on localized regions of full quantum error correction codes. Similarly, we show how transversal CNOT gates can implement logical parity-controlled-NOT operations between arbitrarily many logical qubits. Both operation types can be implemented and in many cases parallelized without the use of lattice surgery or the need for complicated routing operations.

Figures (3)

• Figure 1: Equivalence of stabilizer condition and relation between physical $Z$ operators. If two physical qubits have already been assigned a logical operator (indicated by their label), in this case $Z_{\{1\}}=\bar{Z}_1$ and $Z_{\{2\}}=\bar{Z}_2$, then the third qubit must have a mapping to their product and we label it accordingly, $Z_{\{1,2\}} =Z_{\{1\}}Z_{\{2\}}= \bar{Z}_1 \bar{Z}_2$.
• Figure 2: Schematic implementation of a fault-tolerant multi-qubit rotation on the example of $\bar{R}_{Z_1 Z_3}(\alpha)$ in the LHZ layout. First, a second instance of parity qubit $13$ is added to the code (blue square). This "copy" should be protected on its own (e.g. by encoding it within another classical code, or by increasing the Z-distance of the existing underlying encoding) and not rely on the remaining parity code anymore. Multiple rounds of stabilizer measurements might be needed to ensure the protection. Once the parity qubit copy is sufficiently protected, the stabilizer connecting it to the rest of the parity code can be excluded from further syndrome extraction rounds and a decomposition of the rotation $R_Z (\alpha)$ be applied to the copy (fault-tolerantly regarding it's own encoding). After the decomposition is complete, the connecting stabilizer to the parity code can be activated again or the copy be removed directly.
• Figure 3: Example of physical-to-logical operator mapping for the encoding circuit of a three-body $Z$ stabilizer. After encoding, a physical $Z$ on the added qubit has the effect of a $Z$-product on the two logical qubits (labelled 1 and 2). The two original physical qubits retain a direct mapping to the corresponding logical qubits.