What are the transversal gates of the [[5,1,3]] code?

[quantumadmin]

The [[5,1,3]] code is a specific type of quantum error-correcting code used in quantum error correction. It's a stabilizer code that encodes one logical qubit using five physical qubits, providing protection against certain types of errors. In the context of the [[5,1,3]] code, a transversal gate is a logical gate that acts on each of the physical qubits individually, without directly interacting with the other qubits.

For the [[5,1,3]] code, the logical Pauli operators X_L and Z_L are defined as follows:

X_L = X_1 ⊗ X_2 ⊗ X_3 ⊗ X_4 ⊗ X_5
Z_L = Z_1 ⊗ Z_2 ⊗ Z_3 ⊗ Z_4 ⊗ Z_5

Where X_i and Z_i are the Pauli X and Z operators on the ith physical qubit.

Transversal gates are gates that act on each physical qubit with a Pauli gate (X, Y, or Z) individually. In the case of the [[5,1,3]] code, the transversal gates correspond to logical Pauli operators acting individually on each of the five physical qubits. Specifically:

The logical X gate (X_L) corresponds to applying a transversal X gate (Pauli X gate) to each of the five physical qubits.
The logical Z gate (Z_L) corresponds to applying a transversal Z gate (Pauli Z gate) to each of the five physical qubits.
Transversal gates have the desirable property that they commute with the stabilizer generators of the code, which helps to preserve the code's error-correcting properties. This makes them important for fault-tolerant quantum error correction.

It's important to note that while transversal gates are advantageous in terms of error correction, not all quantum error-correcting codes have transversal gates for all logical gates. The presence of transversal gates depends on the specific code being used.