Quantum-shift-register circuits

A quantum-shift-register circuit acts on a set of input qubits and memory qubits, outputs a set of output qubits and updated memory qubits, and feeds the memory back into the device for the next cycle (similar to the operation of a classical shift register). Such a device finds application as an encoding and decoding circuit for a particular type of quantum error-correcting code called a quantum convolutional code. Building on the Ollivier-Tillich and Grassl-Rötteler encoding algorithms for quantum convolutional codes, I present a method to determine a quantum-shift-register encoding circuit for a quantum convolutional code. I also determine a formula for the amount of memory that a Calderbank-Shor-Steane (CSS) quantum convolutional code requires. I then detail primitive quantum-shift-register circuits that realize all of the finite- and infinite-depth transformations in the shift-invariant Clifford group (the class of transformations important for encoding and decoding quantum convolutional codes). The memory formula for a CSS quantum convolutional code then immediately leads to a formula for the memory required by a CSS entanglement-assisted quantum convolutional code.

Figure 1

The above figure depicts a simple CNOT transformation conditional on the bit $f_0$. The circuit does not apply the gate if $f_0=0$ and applies it if $f_0=1$.

Figure 2

A quantum-shift-register device that incorporates one frame of memory qubits.

Figure 3

The circuit in the above figure combines the circuits in Figs. 1, 2.

Figure 4

The circuit in the above figure implements a two-delay CNOT transformation.

Figure 5

The above circuit connects the outputs of the circuit in Fig. 3 to the inputs of the circuit in Fig. 4.

Figure 6

The above circuit reduces the amount of memory required to implement the transformation in Eq. (5).

Figure 7

The circuit to the left implements the set of transformations outlined in Eqs. (8, 9).

Figure 8

The figure to the left depicts a simplification of the circuit in Fig. 7.

Figure 9

The above figure depicts a simplified version of the circuit in Fig. 8 where we have removed the second unnecessary quantum-shift-register circuit. The above circuit uses less memory than the one in Fig. 8, while still effecting the same transformation.

Figure 10

The circuit in the above figure is a quantum-shift-register encoding circuit for the CSS quantum convolutional code in Eq. (7).

Figure 11

A circuit that implements a simple delay operation on the first qubit in each frame.

Figure 12

The circuit in the above figure implements the transformation in Eq. (10).

Figure 13

The circuit in the above figure implements the transformation in Eq. (11). Comparing the circuit in the above figure to the one in Fig. 12 reveals that it merely flips the direction of the CNOT gates.

Figure 14

The circuit in the above figure implements the transformation in Eq. (13).

Figure 15

The circuit in the above figure implements the transformation in Eq. (15).

Figure 16

The above circuit encodes the Forney-Grassl-Guha code from Ref. [23].

Figure 17

The quantum-shift-register circuit in the above figure implements the transformation in Eq. (17).

Figure 18

The quantum-shift-register circuit in the above figure implements the transformation in Eq. (20).