Robust and efficient in situ quantum control

Christopher Ferrie and Osama Moussa

Phys. Rev. A 91, 052306 – Published 7 May 2015

Abstract

Precision control of quantum systems is the driving force for both quantum technology and the probing of physics at the quantum and nanoscale levels. We propose an implementation-independent method for in situ quantum control that leverages recent advances in the direct estimation of quantum gate fidelity. Our algorithm takes account of the stochasticity of the problem, is suitable for closed-loop control, and requires only a constant number of fidelity-estimating experiments per iteration independent of the dimension of the control space. It is efficient and robust to both statistical and technical noise.

Received 2 December 2014

DOI:https://doi.org/10.1103/PhysRevA.91.052306

Authors & Affiliations

Christopher Ferrie^1,2,* and Osama Moussa^3,†

¹Center for Quantum Information and Control, University of New Mexico, Albuquerque, New Mexico 87131-0001, USA
²Centre for Engineered Quantum Systems, School of Physics, University of Sydney, Sydney, New South Wales, Australia
³Institute for Quantum Computing and Department of Physics and Astronomy, University of Waterloo, Waterloo, Ontario N2L 3G1, Canada

^*csferrie@gmail.com
^†omoussa@gmail.com

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Vol. 91, Iss. 5 — May 2015

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Images

Figure 1
The median infidelity (1-fidelity) of our in situ algorithm and an ex situ algorithm as a function of the number of iterations. The problem is that of designing a random single-qubit gate and is described in more detail in Sec. 3. The in situ algorithm was afforded only $10^{3}$ experiments per fidelity estimate while the ex situ algorithm was given the ability to compute the fidelity to machine precision but saturates since the assumed Hamiltonian was incorrect by $| | ▵ H | | = 10^{- 2} | | H | |$ .
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Figure 2
For the single-qubit random gate design problem, plotted is the infidelity (1-fidelity) of the ACRONYM and NM algorithms as a function of the number of iterations for three different numbers of experiments per iteration: (Left to right) $10^{2}, 10^{3}$ , and $10^{4}$ . The thick lines are the median of the data and the shaded region around each is the interquartile range (the middle 50% of the data). The thin black lines are fits to the theoretical $O (k^{β})$ scaling. We see that NM is limited by the noise in the function call while ACRONYM is not.
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Figure 3
For the single-qubit random gate design problem, plotted is the infidelity (1-fidelity) of the ACRONYM as a function of $N$ , the number of experiments per fidelity estimate at $k = 10^{4}$ iterations. The dimension of the control space is $p = 10$ . The thick line is the median of the data and the shaded region is the interquartile range (the middle 50% of the data). This demonstrates, with the results of Fig. 2, that more experiments per iteration of the algorithm are not necessary to converge. At some point (here $\approx 10^{4}$ ), a constant number of experiments suffices.
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Figure 4
For the control problem of designing a controlled-not (c-not) gate, plotted is the infidelity (1-fidelity) of ACRONYM and NM as a function of the number of iterations for $N = 10^{3}$ and $N = 10^{4}$ experiments per iteration. The thick lines are the median of the data and the shaded region is the interquartile range (the middle 50% of the data). As shown for the single-qubit problem in the figures above, ACRONYM converges while NM does not.
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Figure 5
Back to the single-qubit problem, now with added with control noise. Plotted is the infidelity (1-fidelity) of the ACRONYM and NM algorithms as a function of the number of iterations. ACRONYM used $N = 10^{3}$ experiments per function evaluation, but NM was given exact fidelity values. Both methods were subjected to independent zero-mean Gaussian noise on each control component with standard deviation $10^{- 2}$ . Even with exactly fidelity values, NM is limited by the control noise. ACRONYM is robust to both statistical and control noise.
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