Suppressing systematic control errors to high orders

P. Bažant, H. Frydrych, G. Alber, and I. Jex

Phys. Rev. A 92, 022325 – Published 10 August 2015

Abstract

Dynamical decoupling is a powerful method for protecting quantum information against unwanted interactions with the help of open-loop control pulses. Realistic control pulses are not ideal and may introduce additional systematic errors. We introduce a class of self-stabilizing pulse sequences capable of suppressing such systematic control errors efficiently in qubit systems. Embedding already known decoupling sequences into these self-stabilizing sequences offers powerful means to achieve robustness against unwanted external perturbations and systematic control errors. As these self-stabilizing sequences are based on single-qubit operations, they offer interesting perspectives for future applications in quantum information processing.

Received 9 May 2015

DOI:https://doi.org/10.1103/PhysRevA.92.022325

Authors & Affiliations

P. Bažant^1,*, H. Frydrych², G. Alber², and I. Jex¹

¹Department of Physics, FNSPE, Czech Technical University in Prague, Břehová 7, 115 19 Praha 1, Staré Město, Czech Republic
²Institut für Angewandte Physik, Technische Universität Darmstadt, D-64289 Darmstadt, Germany

^*pbazant@gmail.com

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Issue

Vol. 92, Iss. 2 — August 2015

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Images

Figure 1
Performance of self-stabilizing CDD with a five-qubit bath, pairwise qubit coupling strength $\sim 1, Δ t = 0.0002$ , and $‖λ_{c} q‖ \sim 0.1$ . The performance is measured as $1 - F_{avg}$ , the average infidelity. The parameter $n$ denotes the concatenation level of CDD, with larger $n$ resulting in longer sequences, which in turn take more time.
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Figure 2
Comparison of periodic versions of orthogonal array decoupling (OAD), SS1-OAD, SS2-OAD, and SS3-OAD in terms of the effective interaction strength approximated as $G_{E} = \frac{\sqrt{1 - F_{e}}}{T_{c}}$ , which quantifies the rapidity of the quadratic fidelity decay associated with periodic decoupling. The quantity $G_{E}$ was evaluated for $n = 9$ qubits, after one completed cycle of duration $T_{c}$ , and is plotted as a function of the pulse rate $1 / Δ t$ . The plots were calculated for a single random realization of the system Hamiltonian $H$ and the nonideal controls. For each decoupling scheme, 25 control sequences were generated, and their performance was averaged. Other realizations of $H$ and control nonidealities of similar magnitude led to very similar plots, so they are not shown. The control nonidealities were realized as noise added to control rotation vectors with standard deviation $Δ v_{i} = 0.01$ . The quantity $G_{E}$ is a good performance measure only for not too large values of $1 - F_{e}$ , so only points with $1 - F < 0.2$ are shown. The advantage of self-stabilization is obvious, but higher SS orders do not further improve performance in this case, they only widen the range of applicable values of $Δ t$ . The figure is consistent with (21).
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Figure 3
Comparison of randomized versions of OAD, SS1-OAD, SS2-OAD, and SS3-OAD, in terms of thefidelity decay rate approximated as $\frac{1 - F}{T_{c}}$ after a single cycle, which quantifies the rapidity of the linear fidelity decay associated with ideally randomized decoupling. The performance is averaged over 25 sequence realizations. All parameters are the same as in Fig. 2. We do not have a proof that our randomization procedure achieves linear decay, but simulations (see Fig. 4) are consistent with linear fidelity decay for sequences up to 200 cycles long, justifying the use of $\frac{1 - F}{T_{c}}$ after a single cycle as a practical performance predictor.
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Figure 4
Time evolutions of the error probability $1 - F$ for no decoupling, randomized OAD for $Δ t = 0.01$ , and randomized SS1-OAD for $Δ t = 0.0001$ . The values of $Δ t$ were chosen to maximize the performance of each scheme as predicted in Fig. 3. All parameters are the same as in Fig. 2. The evolution of error for SS1-OAD is in good agreement with the evolution predicted by extrapolation of fidelity after the first cycle, suggesting that the randomization procedure used for SS1-OAD is effective in keeping the fidelity decay linear for at least 200 cycles. The ability of self-stabilization to improve decoupling performance in the presence of systematic pulse errors is remarkable.
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