Robust quantum control against clock noises in multiqubit systems

High-precision manipulation of multiqubit quantum systems requires strictly clocked and synchronized multichannel control signals. However, practical arbitrary wave-form generators always suffer from random signal jitters and channel latencies that induce nonignorable state or gate operation errors. In this paper, we analyze the average gate error caused by clock noises, from which an estimation formula is derived for quantifying the control robustness against clock noises. This measure is then employed for finding robust controls via a homotopic optimization algorithm. We also introduce our recently proposed stochastic optimization algorithm, b-grape, for training robust controls via randomly generated clock-noise samples. Numerical simulations on a two-qubit example demonstrate that both algorithms can greatly improve the control robustness against clock noises. The homotopic algorithm converges much faster than the b-grape algorithm, but the latter can achieve higher control robustness.

Figure 1

Schematic diagram for clock noises in piecewise-constant control signals. The (channel) latency is referred to as the arrival delay time of the AWG signal and the timing error of each subpulse consists of latency and random jitter noises. The deviation of the actual signal with clock noise from the ideally clocked signal consists of small subpulses described by Eq. (7).

Figure 2

The learning curves of homotopic and b-grape algorithms under (a) latency and jitter noises and under (b) latency noise only. The average gate errors estimated in the homotopic algorithm and in the training process of the b-grape algorithm are both consistent with their testing performances (each intermediate control field is tested by averaging ${10}^4$ random clock-noise samples).

Figure 3

The two-qubit gate (controlled-not) error distribution under optimal controls obtained by grape, homotopic and b-grape algorithms, (a) under latency and jitter clock noises and (b) under latency noise only. The robust optimization can effectively push the entire error distribution to the high-precision regime.

Figure 4

The dependence of gate error on two latency variables under optimal controls obtained by grape, homotopic, and b-grape algorithms.

Figure 5

The optimized $y$-axis control fields on the second qubit obtained by grape (gray solid curve), homotopic (blue solid curve), and b-grape (red dash-dotted curve) algorithms, (a) under latency and jitter noises and (b) under only latency noise. The robust optimization against random jitter noises leads to much smoother wave forms shown in (a) than without jitter noises in (b).

Figure 6

The level sets of infidelity at 0.001 under the control fields optimized with grape (red dotted curve) and b-grape (black solid curve), whose areas are used to quantify the control robustness against uncertainty parameters $({\epsilon }_1,{\epsilon }_2)$ and $({\tau }_1,{\tau }_2)$ described in Eq. (21). (a), (b) Robust optimization with respect to only (${\tau }_1,{\tau }_2$); (c), (d) robust optimization with respect to only (${\epsilon }_1,{\epsilon }_2$); (e), (f) robust optimization with respect to both $({\tau }_1,{\tau }_2)$ and $({\epsilon }_1,{\epsilon }_2)$.