Encoding One Logical Qubit Into Six Physical Qubits

TL;DR

This paper presents two methods for encoding a single logical qubit into six physical qubits, each capable of correcting any single-qubit error, with one method utilizing entanglement assistance.

Contribution

The paper introduces a degenerate six-qubit quantum error-correcting code and a six-qubit CSS entanglement-assisted code, both correcting single-qubit errors and minimal in size.

Findings

The degenerate code saturates the subsystem Singleton bound.

The entanglement-assisted code is equivalent to the Steane code.

Six qubits are the minimum for a CSS code correcting single errors.

Abstract

We discuss two methods to encode one qubit into six physical qubits. Each of our two examples corrects an arbitrary single-qubit error. Our first example is a degenerate six-qubit quantum error-correcting code. We explicitly provide the stabilizer generators, encoding circuit, codewords, logical Pauli operators, and logical CNOT operator for this code. We also show how to convert this code into a non-trivial subsystem code that saturates the subsystem Singleton bound. We then prove that a six-qubit code without entanglement assistance cannot simultaneously possess a Calderbank-Shor-Steane (CSS) stabilizer and correct an arbitrary single-qubit error. A corollary of this result is that the Steane seven-qubit code is the smallest single-error correcting CSS code. Our second example is the construction of a non-degenerate six-qubit CSS entanglement-assisted code. This code uses one bit of entanglement (an ebit) shared between the sender and the receiver and corrects an arbitrary single-qubit error. The code we obtain is globally equivalent to the Steane seven-qubit code and thus corrects an arbitrary error on the receiver's half of the ebit as well. We prove that this code is the smallest code with a CSS structure that uses only one ebit and corrects an arbitrary single-qubit error on the sender's side. We discuss the advantages and disadvantages for each of the two codes.