Dataset for the paper "Quantum Error Correction Protects Quantum Search Algorithms Against Decoherence", Panagiotis Botsinis, Zunaira Babar, Dimitrios Alanis, Hung Nguyen, Daryus Chandra, Soon Xin Ng and Lajos Hanzo, Scientific Reports (submitted). Results may reproduced using the Graphics Layout Engine (GLE) and MATLAB.
Abstract: When quantum computing becomes a wide-spread commercial reality, Quantum Search Algorithms (QSA) and especially Grover's QSA will inevitably be one of their main applications, constituting their cornerstone. Most of the literature assumes that the quantum circuits are free from decoherence. Practically, decoherence will remain unavoidable as is the Gaussian noise of classic circuits imposed by the Brownian motion of electrons, hence it may have to be mitigated. In this contribution, we investigate the effect of quantum noise on the performance of QSAs, in terms of their success probability as a function of the database size to be searched, when decoherence is modelled by depolarizing channels' deleterious effects imposed on the quantum gates. Moreover, we employ quantum error correction codes for limiting the effects of quantum noise and for correcting quantum flips. More specifically, we demonstrate that, when we search for a single solution in a database having 4096 entries using Grover's QSA at an aggressive depolarizing probability of 10^{-3}, the success probability of the search is 0.22 when no quantum coding is used, which is improved to 0.96 when Steane's quantum error correction code is employed.
Acknowledgements: The financial support of the European Research Council under the Advanced Fellow Grant, that of the Royal Society’s Wolfson Research Merit Award and that of the Engineering and Physical Sciences Research Council under Grant EP/L018659/1 is gratefully acknowledged. The use of the IRIDIS High Performance Computing Facility at the University of Southampton is also acknowledged.
* Figure 3: SN1N-channel1000.gle, SN1N-channel0100.gle, SN1N-channel1100.gle and SN1N-channel1111.gle plot the success probability of Grover's QSA, when depolarizing channels occur in various locations on Grover's operator's circuit, shown in Fig. 1, with respect to the depolarizing probability of the channels, when the architecture of Fig. 2 was used. Randomly generated databases with different sizes N were used, while a single solution S=1 was present in a random position in the database in each search problem.
* Figure 4: Success-1e-4.fig, Success-1e-3.fig, Success-5e-3.fig and Success-1e-2.fig plot the success probability of Grover's QSA, when a depolarizing channel appears only right before the Oracle in Grover's operator's circuit, shown in Fig. 1, with respect to the depolarizing probability of the channel p_1, the size of the database N and the number of Grover iterations L. The circuit architecture of Fig. 2 was employed. Randomly generated databases with different sizes N were used, while a single solution S=1 was present in a random position in the database in each search problem. The white curve marked by the diamonds indicates the optimal number of Grover iterations L_{opt} that would be required in the ideal Grover's QSA, associated with p_1=0, while the cyan curve marked by the circles represents the actual optimal number of Grover iterations for the specific value of depolarizing probability p_1.
* Figure 6: MSN1N-channel1000-Steane.gle, SN1N-channel0100-Steane.gle, SN1N-channel1100-Steane.gle and SN1N-channel1111-Steane-gray.gle plot the success probability of Grover's QSA, when quantum noise occurs in various locations on Grover's operator's circuit, shown in Fig. 1, with respect to the depolarizing probability of the channels. Steane code with rate R=1/7 has been employed for improving the performance of Grover's QSA. Randomly generated databases with different sizes N were used, while a single solution S=1 was present in a random position in the database in each search problem.
* Figure 7: S1N1024-channel1100-p-Steane.gle plots the success probability of Grover's QSA in a database with N=1024 entries and S=1 solution, with respect to the number of applications L of Grover's operator \mathcal{G}, when depolarizing channels appear before and after the Oracle operator of Fig. 1, associated with $p_3=p_4=0$. The architecture of Fig. 2 is used. Two different depolarizing probabilities are assumed, namely p_1=p_2=0.001 and p_1=p_2=0.003, while the performance when Steane code is employed for cancelling the channels' effects is compared to that of an uncoded system. The performance of the ideal Grover's QSA is also attached as a reference.
* Figure 8: S1N128-channel1100-p-Steane-QBCH.gle plots the success probability of Grover's QSA in a database with N=128 entries and S=1 solution, with respect to the number of applications L of Grover's operator \mathcal{G}, when depolarizing channels exist before and after the Oracle operator of Fig. 1, associated with p_3=p_4=0, when the architecture of Fig. 2 is used. Two different depolarizing probabilities are assumed, namely p_1=p_2=0.003 and p_1=p_2=0.005, while the performance when QBCH[15,7] code is employed for cancelling the channels' effects is compared to that of the Steane code, as well as that of an uncoded system. The performance of the ideal Grover's QSA is also attached as a reference.