Optimized quantum error-correction codes for experiments

We identify gauge freedoms in quantum error correction (QEC) codes and introduce strategies for optimal control algorithms to find the gauges which allow the easiest experimental realization. Hereby, the optimal gauge depends on the underlying physical system and the available means to manipulate it. The final goal is to obtain optimal decompositions of QEC codes into elementary operations which can be realized with high experimental fidelities. In the first part of this paper, this subject is studied in a general fashion, while in the second part, a system of trapped ions is treated as a concrete example. A detailed optimization algorithm is explained and various decompositions are presented for the three qubit code, the five qubit code, and the seven qubit Steane code.

Figure 1

Example circuit for the three qubit quantum error correction code without measurement. The top three horizontal lines represent the physical qubits used to encode the logical qubit, while the two horizontal lines at the bottom represent two auxiliary qubits. The dotted vertical line was included to separate the two functional parts of the code: The part to the left of the dotted line maps the information of a possible bit-flip error onto the auxiliary qubits. The part on the right uses this information to correct the error. After the error is corrected, the state of the auxiliary qubits is of no further interest and might be reset (wavy line) for the next run. Therefore, an arbitrary unitary operation $U$ on the auxiliary system can be included.

Figure 2

Example circuit for a three qubit quantum error correction code based on measurement. As in Fig. 1, the top three lines represent the physical qubits used to encode the logical qubit, while the bottom line represents an auxiliary qubit. After the first measurement, the auxiliary qubit is reset to reuse it for the second measurement. The measurement results $\left(00\right)$, $\left(01\right)$, $\left(10\right)$, and $\left(11\right)$ correspond to the four cases: no error, bit-flip on the first qubit, bit-flip on the second qubit, and bit-flip on the third qubit.

Figure 3

The symbols used for the graphical representation of $X\left(\Theta \right),Y\left(\Theta \right),z_j\left(\Theta \right),X^2\left(\Theta \right)$, and $Y^2\left(\Theta \right)$ (31) in a quantum circuit. The width of the symbols is proportional to the absolute value of $\Theta$, where the width belonging to $\Theta =\pi$ equals the vertical distance of the qubits in the quantum circuit. The symbols for the entangling Mølmer-Sørensen gates $X^2\left(\Theta \right)$ and $Y^2\left(\Theta \right)$ are extended over all qubits, while the symbols for the nonentangling collective operations $X\left(\Theta \right)$ and $Y\left(\Theta \right)$ are (vertically) repeated such that all qubits are covered. The symbol for the local $z_j\left(\Theta \right)$ is placed on the $j\mathrm{th}$ qubit only. Currently, $z_j\left(\Theta \right)$ with negative values for $\Theta$ are experimentally realized as $z_j(-|\Theta \left|\right)=z_j(2\pi -|\Theta \left|\right)$. Therefore, the symbols for negative $z_j$ are replaced by their longer positive counterparts, as well.

Figure 4

Measurement of the error syndrome for the three qubit bit-flip error correction code (33). The first three qubits encode the logical qubit. After the first measurement, the auxiliary qubit is reset (wavy line).

Figure 5

Measurement of the error syndrome for the three qubit phase-flip error correction code (34). The first three qubits encode the logical qubit. After the first measurement, the auxiliary qubit is reset (wavy line).

Figure 6

Coherent three qubit bit-flip error correction code (35), which corrects a possible error on the logical qubit (encoded in the first three physical qubits) without the need of a measurement. To remove the entropy, the auxiliary qubit has to be reset (wavy line). Because of its length, the sequence is split into two parts.

Figure 7

Sequence to generate the logical five qubit codeword $|0_L\rangle$ (37).

Figure 8

Sequence to generate the logical superposition state $sin\left(\alpha \right)|0_L\rangle +cos\left(\alpha \right)|1_L\rangle$ for the five qubit code (38). The length $\Theta$ of the second last $z_1\left(\Theta \right)$ operation depends on the angle $\alpha$, i.e., $z_1(\Theta =2\alpha )$.

Figure 9

Sequence to generate the logical superposition state $cos\left(\alpha \right)|0_L\rangle +sin\left(\alpha \right)|1_L\rangle$ for the Steane code (40). The length $\Theta$ of the second $z_7\left(\Theta \right)$ operation depends on the angle $\alpha$, i.e., $z_7(\Theta =2\alpha -\frac{\pi }{2})$.

Figure 10

Sequence to generate the logical Steane codeword $|0_L\rangle$ (41).

Figure 11

Sequence to generate the logical Steane codeword $|0_L\rangle$ (42). The third Mølmer-Sørensen gate only acts on the qubits number 1, 3, 5, and 7.

Figure 12

Measurement of the stabilizer $\left(XZZXI\right)$ for the five qubit code (43).

Figure 13

Measurement of all four stabilizers for the five qubit code (44). Because of its length, the sequence is split into two parts. After the first three measurements, the auxiliary qubit is reset.

Figure 14

Measurement of the first stabilizer $\left(IIIXXXX\right)$ for the Steane code (45).

Figure 15

Measurement of the fourth stabilizer $\left(IIIZZZZ\right)$ for the Steane code (47).

Figure 16

Logical Hadamard gate for the five qubit code (48).

Figure 17

Logical $\frac{\pi }{8}$ gate for the Steane code (49).