Structured near-optimal channel-adapted quantum error correction

We present a class of numerical algorithms which adapt a quantum error correction scheme to a channel model. Given an encoding and a channel model, it was previously shown that the quantum operation that maximizes the average entanglement fidelity may be calculated by a semidefinite program (SDP), which is a convex optimization. While optimal, this recovery operation is computationally difficult for long codes. Furthermore, the optimal recovery operation has no structure beyond the completely positive trace-preserving constraint. We derive methods to generate structured channel-adapted error recovery operations. Specifically, each recovery operation begins with a projective error syndrome measurement. The algorithms to compute the structured recovery operations are more scalable than the SDP and yield recovery operations with an intuitive physical form. Using Lagrange duality, we derive performance bounds to certify near-optimality.

Figure 1

Quantum error correction block diagram. For channel-adapted recovery, the encoding isometry $U_C$ and the channel ${\mathcal{E}}^{\prime }$ are considered as a fixed operation $\mathcal{E}$ and the recovery $\mathcal{R}$ is chosen according to the design criteria.

Figure 2

(Color online) Fidelity contribution of EIGQER recovery operators for the amplitude-damping channel $\left(\gamma =.09\right)$ and the Steane code. Notice that the QEC performance is equaled with only eight operator elements, and the relative benefit of additional operators goes nearly to zero after 30.

Figure 3

(Color) EIGQER and optimal QER for the amplitude-damping channel and the five-qubit stabilizer code. EIGQER nearly duplicates the optimal channel-adapted performance, especially for lower-noise channels (small $\gamma$).

Figure 4

(Color) EIGQER and optimal QER for the pure state rotation channel with $\theta =5\pi /12$ and the five-qubit stabilizer code. EIGQER nearly duplicates the optimal channel-adapted performance, especially for lower-noise channels (small $\varphi$).

Figure 5

(Color) EIGQER and standard QEC recovery performance for the five-, seven-, and nine-qubit codes and the amplitude-damping channel. Note that generic QEC performance decreases for longer codes, as multiple-qubit errors become more likely. While the EIGQER performance for the nine-qubit Shor code is excellent, the seven-qubit Steane code shows only modest improvement, with performance similar to the generic five-qubit QEC recovery.

Figure 6

(Color) EIGQER and standard QEC recovery performance for the five-, seven-, and nine-qubit codes and the pure state rotation channel with $\theta =5\pi /12$. Despite the least redundancy, the five-qubit code has the best channel-adapted performance. The Steane code appears particularly poor for this channel: both the generic QEC and the adapted recovery have lower fidelity than the other codes.

Figure 7

(Color) BLOCKEIGQER performance for the five-qubit code and the pure state rotation channel with $\theta =5\pi /12$. BLOCKEIGQER is computed with fixed block lengths of 2, 4, and 8. In (a) we compare the entanglement fidelity to the EIGQER recovery, standard QEC recovery, and single-qubit baseline. The different block lengths have nearly indistinguishable performance from EIGQER. In (b), we compute the fidelity relative to the EIGQER recovery and show that the fidelity improves by less than 4% for the displayed region. We can note, however, that longer block lengths tend to better performance.

Figure 8

(Color) BLOCKEIGQER for the amplitude-damping channel and a random [6,2] code. We compare the BLOCKEIGQER algorithm for block sizes of 2, 4, and 8 with the EIGQER algorithm. We see significant performance improvement for larger block sizes, at the cost of computational and recovery complexity. Baseline in this case is the entanglement fidelity for two qubits input to the channel without error correction.

Figure 9

(Color) ORDERQER recovery for the seven-qubit Steane code and the amplitude-damping channel. We compare the recovery fidelity of the first-order error to the standard QEC performance. The performance of the first- and second-order recoveries together are comparable to the EIGQER recovery, especially as $\gamma$ approaches 0.

Figure 10

(Color) Geršgorin and SVD dual bound for the amplitude-damping channel and the five-qubit stabilizer code. The Geršogrin bound is clearly not very useful as in some cases it is greater than 1. The SVD dual bound clearly tracks the optimal performance, although the departure from optimal of the bound exceeds the EIGQER recovery.

Figure 11

(Color) Dual bound comparison for the amplitude-damping channel and the five-qubit code. The iterated dual initialized with the BLOCKEIGQER algorithm with $M=2$ is essentially indistinguishable from the optimal recovery performance, thus producing a very tight bound. Included for comparison are the EIGQER performance, the SVD dual bound, and both a channel-adapted recovery and associated bound derived by Barnum and Knill in [10].

Figure 12

(Color) Dual bound comparison for the amplitude-damping channel and the nine-qubit Steane code. The iterated dual bound initialized with the BLOCKEIGQER recovery with $M=2$ produces a bound that is tight to the EIGQER recovery operation. This demonstrates that the EIGQER recovery operation is essentially optimal in this case. Notice that the iterated bound initialized with the EIGQER recovery operation does not generate a tight bound.

Figure 13

(Color) Dual bound comparison for the pure state rotation channel with $\theta =5\pi /12$ and the seven-qubit Steane code. Note that the iterated bounds are generally, though not universally, better than the SVD dual bound. We also see that the shorter block lengths for the BLOCKEIGQER algorithm generally produce a tighter bound, despite slightly poorer recovery performance.