Gradient-based closed-loop quantum optimal control in a solid-state two-qubit system

Guanru Feng, Franklin H. Cho, Hemant Katiyar, Jun Li, Dawei Lu, Jonathan Baugh, and Raymond Laflamme

Phys. Rev. A 98, 052341 – Published 26 November 2018

Abstract

Quantum optimal control can play a crucial role in realizing a set of universal quantum logic gates with error rates below the threshold required for fault tolerance. Open-loop quantum optimal control relies on accurate modeling of the quantum system under control and does not scale efficiently with system size. These problems can be avoided in closed-loop quantum optimal control, which utilizes feedback from the system to improve control fidelity. In this paper, two gradient-based closed-loop quantum optimal control algorithms, the hybrid quantum-classical approach (HQCA) described by Li et al. [Phys. Rev. Lett. 118, 150503 (2017)] and the finite-difference (FD) method, are experimentally investigated and compared to the open-loop quantum optimal control utilizing the gradient ascent method. We employ a solid-state ensemble of coupled electron-nuclear spins serving as a two-qubit system. Specific single-qubit and two-qubit state preparation gates are optimized using the closed-loop and open-loop methods. The experimental results demonstrate the implemented closed-loop quantum control outperforms the open-loop control in our system. Furthermore, simulations reveal that HQCA is more robust than the FD method to gradient noise which originates from measurement noise in this experimental setting. On the other hand, the FD method is more robust to control field distortions coming from nonideal hardware.

Received 29 May 2018

DOI:https://doi.org/10.1103/PhysRevA.98.052341

Physics Subject Headings (PhySH)

Quantum control Quantum gates

Quantum Information, Science & Technology

Authors & Affiliations

Guanru Feng^1,2, Franklin H. Cho^1,2, Hemant Katiyar^1,2, Jun Li^1,2,3, Dawei Lu^1,2,3, Jonathan Baugh^1,2,4,*, and Raymond Laflamme^1,2,5,6,†

¹Institute for Quantum Computing, Waterloo, Ontario, Canada N2L 3G1
²Department of Physics and Astronomy, University of Waterloo, Waterloo, Ontario, Canada N2L 3G1
³Department of Physics and Shenzhen Institute for Quantum Science and Engineering, Southern University of Science and Technology, Shenzhen 518055, China
⁴Department of Chemistry, University of Waterloo, Waterloo, Ontario, Canada N2L 3G1
⁵Perimeter Institute for Theoretical Physics, Waterloo, Ontario, Canada N2J 2W9
⁶Canadian Institute for Advanced Research, Toronto, Ontario, Canada M5G 1Z8

^*baugh@uwaterloo.ca
^†laflamme@iqc.ca

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Vol. 98, Iss. 5 — November 2018

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Figure 1
Flow diagram of gradient-based closed-loop optimal control as applied to ESR. Arrows label the direction of information flow. Error sources are labeled using orange dashed-line circles. Both the HQCA and FD methods can compensate for the control errors caused by the uncertainty in the system Hamiltonian. Theoretically, the FD method finds the gradient $\partial F / \partial u$ , while HQCA finds the gradient $\partial F / \partial \tilde{u}$ . The pulse shape represented by $\tilde{u}$ is distorted by the hardware.
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Figure 2
(a) Energy-level diagram of the two-qubit system. Each level is designated by electron and nuclear spin quantum number, $m_{s}$ and $m_{I}$ , respectively. (b) Pulse sequence used for the reference (spin echo), gate $1 (ZI$ to $XI)$ , and gate $2 (ZI$ to $ZZ)$ measurement. For the reference and readout, 200-ns long square $π$ /2 and $π$ pulses with excitation bandwidth of 5 MHz were used throughout the experiments, providing selective excitation of one allowed transition without affecting the other. The echo is formed after $τ = 1 μ s$ from the $π$ pulse. Phase cycling was implemented in gate-2 measurement to remove possible signal contributions from the transversal polarizations, which we do not want to measure. (c) ESR spectra acquired by sweeping the magnetic field (blue, lower trace) and frequency (red, upper trace) using the reference spin-echo sequence. For the field swept spectrum, the microwave pulse frequency is fixed at the resonance frequency of the loop-gap resonator, determined from a separate microwave reflection (S11) measurement as shown in the inset. For the frequency-swept spectrum, $B_{0}$ is fixed at 3401 G (denoted by the arrow in the figure). Two strong signals at $3401 \pm 14$ G or $\pm 36$ MHz $(S_{L}$ and $S_{R})$ correspond to the resonance condition of the two allowed ESR transitions. The unequal intensities obtained in the frequency-swept spectrum are due to the frequency dependence of the spectrometer's transfer function.
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Figure 3
Process flow for optimizing closed-loop control. Starting with the pulse obtained from previous iteration (step 1; shown here is a square pulse as an example), two control qualities $F^{k, \pm x}$ are measured (step 2) to calculate the gradient in a particular direction of a basis denoted as $v^{k}$ (step 3). For the HQCA method, a $\pm π / 2$ rotation pulse around the $x$ axis $(R_{x} (\pm π / 2)$ ; see Appendix pp2) is inserted at the $k th$ segment. For the FD method, the pulse is perturbed by the addition or subtraction of a small amplitude vector proportional to the $k th$ basis element in the $x$ phase. Steps 2 and 3 are repeated for all elements of the basis, and the final gradient is obtained by summing all gradients (step 4). The pulse is updated by adding the final gradient in $x$ phase (step 5). A similar procedure is carried out to obtain the final gradient in the $y$ phase. More details of the procedure, including information of different basis sets used, are given in the main text.
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Figure 4
(a) $F_{ZZ}$ as a function of the number of iterations (HQCA method). Here, the fidelity measurement for each pulse shape, together with the reference signal measurements, was repeated five times for the first 13 iterations and 50 times for the last four iterations. Each individual measurement was an average of 16 000 phase-cycled scans. The error bars indicate one standard deviation of the measurement results obtained over the repetitions (5 or 50). (b) Fluctuation of normalized reference signal intensity (here ${\bar{S}}_{R})$ during 100 repetitions. Dashed lines indicate $\pm 0.03$ deviation from the normalized mean. (c) Example of pulse update. The solid yellow (light gray) curve represents the initial $x$ -phase shape. The solid orange (medium gray) curve is the experimentally measured gradient $g$ . The dashed brown (dark gray) curve is the updated pulse obtained by adding $c g$ .
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Figure 5
The transfer function used in open-loop pulse design and simulation. (a) The experimentally measured transfer function which has the bandwidth of $\sim 130$ MHz. (b) Obtained by artificial alteration of (a) to make a narrower bandwidth of $\sim 70$ MHz. The absolute, real, and imaginary values are in solid yellow (light gray), solid orange (medium gray), and dashed brown (dark gray) line, respectively. The transfer function is a complex function, and its amplitude and phase are used to adjust the amplitude and phase of our pulses.
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