Encoding one logical qubit into six physical qubits

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 controlled NOT operator for this code. We also show how to convert this code into a nontrivial 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 nondegenerate 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.

Figure 1

Encoding circuit for the first six-qubit code. The $H$ gate is a Hadamard gate. For example, we apply a Hadamard on qubit four followed by a CNOT with qubit four as the control qubit and qubit six as the target qubit.

Figure 2

Logical CNOT for the six-qubit quantum code. The first six qubits represent a logical source qubit and the last six represent a logical target qubit. For example we begin the circuit by applying a controlled-$Z$ gate from source qubit two to target qubit seven.

Figure 3

Encoding circuit for the [[6,1,3;1]] code. The “$H$” gate is a Hadamard gate.