Entanglement-assisted quantum convolutional coding

We show how to protect a stream of quantum information from decoherence induced by a noisy quantum communication channel. We exploit preshared entanglement and a convolutional coding structure to develop a theory of entanglement-assisted quantum convolutional coding. Our construction produces a Calderbank-Shor-Steane (CSS) entanglement-assisted quantum convolutional code from two arbitrary classical binary convolutional codes. The rate and error-correcting properties of the classical convolutional codes directly determine the corresponding properties of the resulting entanglement-assisted quantum convolutional code. We explain how to encode our CSS entanglement-assisted quantum convolutional codes starting from a stream of information qubits, ancilla qubits, and shared entangled bits.

Figure 1

The operation of a general stabilizer code. Thin lines denote quantum information and thick lines denote classical information. Slanted bars denote multiple qubits. A sender employs a unitary encoding operation $U$ to encode a set of information qubits in the state $|\psi \rangle$ with the help of ancilla qubits each in the state $|0\rangle$. The sender transmits the encoded qubits over the noisy quantum communication channel. The receiver performs quantum measurements to diagnose which error has occurred. He finally performs a recovery operation $R$ to reverse the error from the channel.

Figure 2

The operation of a general entanglement-assisted stabilizer code. The sender encodes a set of information qubits with the help of ancilla qubits and her half of a set of shared ebits. She sends her encoded qubits over a noisy quantum communication channel. The entanglement-assisted communication paradigm assumes that the receiver’s half of the shared ebits remain noiseless throughout this process. The receiver combines the noisy qubits with his half of the shared ebits. He performs measurements on all of the qubits to diagnose which error occurs and reverses the effect of this error by performing a recovery operation.

Figure 3

An entanglement-assisted quantum convolutional code operates on a stream of qubits partitioned into a countable number of frames. The sender encodes the frames of information qubits, ancilla qubits, and half of shared ebits with a repeated, overlapping encoding circuit $U$. The noisy channel affects the sender’s encoded qubits but does not affect the receiver’s half of the shared ebits. The receiver performs overlapping measurements on both the encoded qubits and his half of the shared ebits. These measurements produce an error syndrome which the receiver can process to determine the most likely error. The receiver reverses the errors on the noisy qubits from the sender. The final decoding circuit operates on all qubits in a frame and recovers the original stream of information qubits.

Figure 4

The finite-depth encoding circuit for the entanglement-assisted quantum convolutional code in Example 1. The above operations in reverse order give a valid decoding circuit.

Figure 5

An example of an infinite-depth operation. A sequence of C-NOT gates acts on the third qubit of every frame. This infinite-depth operation effectively multiplies the third column of the “X” side of the binary polynomial matrix by the rational polynomial $1/(1+D)$ and multiplies the third column of the “Z” side of the binary polynomial matrix by $1+D^{-1}$.

Figure 6

Another example of an infinite-depth operation. An infinite-depth operation acts on qubit $i$ in every frame. This particular infinite-depth operation multiplies column $i$ on the “X” side of the binary polynomial matrix by $1/(1+D+D^3)$ and multiplies column $i$ on the “Z” side of the binary polynomial matrix by $1+D^{-1}+D^{-3}$.

Figure 7

The encoding and decoding circuits for the entanglement-assisted quantum convolutional code in Example 2. The third series of gates in the above encoding circuit is an infinite-depth operation. The other operations in the encoding circuit are finite-depth operations. The decoding circuit has finite-depth operations only.