Tailored Codes for Small Quantum Memories

We demonstrate that small quantum memories, realized via quantum error correction in multiqubit devices, can benefit substantially by choosing a quantum code that is tailored to the relevant error model of the system. For a biased noise model, with independent bit and phase flips occurring at different rates, we show that a single code greatly outperforms the well-studied Steane code across the full range of parameters of the noise model, including for unbiased noise. In fact, this tailored code performs almost optimally when compared with 10 000 randomly selected stabilizer codes of comparable experimental complexity. Tailored codes can even outperform the Steane code with realistic experimental noise, and without any increase in the experimental complexity, as we demonstrate by comparison in the observed error model in a recent seven-qubit trapped ion experiment.

Figure 1

(a) The probability of a logical error for different seven-qubit stabilizer codes under a biased Pauli error model as the bias is varied for two choices of physical error rate, $p=0.01$ and 0.001. The blue line and green line denote the performance of the Steane code and the cyclic code of Eq. (2), respectively. The red shows the performance of the best identified random code for that error rate. (b) The probability of a logical error for different seven-qubit stabilizer codes under a biased Pauli error model as a function of $p$ for two values of the bias.

Figure 2

(a) The superoperator representation, in the Pauli basis, of the channel estimated from process tomography data collected from the Innsbruck experiment. The tomography experiment consists of preparing and then immediately measuring a state (that is, a wait time of $\tau =0$), as described in the Supplemental Material [27]. This Hinton diagram depicts each positive (negative) matrix element by a white (black) square, where the size of each square indicates the relative magnitude of the respective matrix elements. The presence of non-negligible off-diagonal elements, such as $(XY,XX)$, indicates that the estimated channel cannot be entirely explained as a convex combination of the action of Pauli operators. The first row and column indicate that the estimated channel is trace preserving, but not unital. (b) The probability of a logical error following the application of a recovery operation for the Innsbruck tomography data compared to the time over which the state is left to decohere. These data include a high rate of correlated two-qubit $Z$ errors. This figure demonstrates that the use of a tailored stabilizer code with an optimal decoding that corrects for two qubit $Z$ errors greatly exceeds the correction ability of an optimally decoded Steane code.