Recursive Encoding and Decoding of Noiseless Subsystem and Decoherence Free Subspace

TL;DR

This paper presents quantum circuits for collective error correction codes using recursive relations among noiseless subsystems and decoherence free subspaces, enabling efficient encoding of multiple logical qubits with minimal gates.

Contribution

It introduces simple recursive quantum circuits for implementing noiseless subsystems and decoherence free subspaces for small numbers of qubits, optimizing the encoding process.

Findings

Single logical qubit encoded with n=3 and n=4 qubits

Two logical qubits encoded with n=5 qubits

Gate count increases linearly with number of logical qubits

Abstract

When the environmental disturbace to a quantum system has a wavelength much larger than the system size, all qubits localized within a small area are under action of the same error operators. Noiseless subsystem and decoherence free subspace are known to correct such collective errors. We construct simple quantum circuits, which implement these collective error correction codes, for a small number $n$ of physical qubits. A single logical qubit is encoded with $n=3$ and $n=4$, while two logical qubits are encoded with $n=5$. The recursive relations among the subspaces employed in noiseless subsystem and decoherence free subspace play essential r\^oles in our implementation. The recursive relations also show that the number of gates required to encode $m$ logical qubits increases linearly in $m$.