Low-rank optimal control of quantum devices

Abstract

We demonstrate that the control protocols of quantum information devices can be simulated by assuming a low-rank ansatz for the density matrix. The rationale underlying this assumption is that quantum information protocols, by design, operate in a regime of nearly pure quantum states. Within the low-rank assumption, the simulation of these protocols is considerably faster than solving the full Lindblad master equation. This advantage can be used to increase the accuracy of the simulation by avoiding uncontrolled approximations, and to streamline protocol optimization. We benchmark our approach on the optimization of the transmon qubit dispersive readout in a realistic transmon-resonator-filter model. With Hilbert space dimension $N = 2000$, assuming a rank as low as $M = 20$ we achieve a nearly 100-fold speedup compared to full master equation integration while accurately reproducing all relevant observables. By combining the low-rank approximation with a compact pulse parametrization and gradient-free optimization, we obtain state-of-the-art readout assignment errors $\varepsilon_a \approx 1.2 \times 10^{-3}$ for a 40 ns readout pulse schedule, while comfortably running on a laptop and not relying on the rotating-wave approximation. Our approach is broadly applicable to most quantum control protocols, including quantum gates, state preparation, and fast reset operations. This establishes low-rank methods as a general tool for optimal control across diverse quantum platforms.

Figures (7)

• Figure 1: (a) Schematic of the transmon readout model. The transmon (yellow) is capacitively coupled to a resonator (red), itself coupled to a Purcell filter (blue). The filter is coupled to the environment through a readout line that enables driving and introduces dissipation at a rate $\kappa$. (b) Illustrative example of driving pulses $\Omega_d(t)$. Both the square (dark blue) and stepwise (light blue) pulses are smoothed out using a logistic function, as detailed in Ref. SM. The thin lines show the stepwise pulse before smoothing.
• Figure 2: (a) The phase-space trajectories $\beta_{g/e} = \mathrm{tr}(\hat{\rho}_{g/e} \hat{f})$ for a readout time $\tau = 40$ ns and a square pulse profile with amplitude $|\Omega_d| / 2\pi = 150$ MHz. The full, low-rank with $M=20$, and RWA evolutions correspond to the full, dashed, and dotted lines, respectively. The red (blue) lines correspond to an initial ground (first excited) transmon state. The thin lines connecting red and blue trajectories highlight equal times on the two curves. (b) The photon occupation $n_{g/e} = \mathrm{tr}(\rho_{g/e} \hat{f}^\dagger f) = |\beta_{g/e}|^2$ in the filter mode and the relative errors between the full and low-rank (dashed), and full and RWA (dotted) results. The relative error between the full and approximate values, defined for a quantity $y$ as $\delta=|(y_{\mathrm{full}} - y_{\mathrm{LRA}})/y_{\mathrm{full}}|$ is much smaller for the LRA than the RWA. The Hilbert space truncations are $N_l = 100$, $N_u = 4$, and $N_t = 5$, resulting in $N = 2000$ states.
• Figure 3: (a) The optimal value of the loss function for various readout times $\tau$ between $40$ and $100$ ns. Inset: a typical variation of the loss function along the optimization schedule for $\tau = 40$ ns. (b) Real (solid) and imaginary (dotted) parts of the optimal pulse envelope (light blue) for $\tau=40~\mathrm{ns}$. The envelope of the square pulse (dark blue) is plotted for comparison. (c) Phase-space trajectories for the optimal readout schedule and $\tau=40~\mathrm{ns}$, for initial states $\ket{g}$ (red) and $\ket{e}$ (blue), computed with $N_t = 5$, $N_u = 4$ and $N_l = 100$. The full and RWA trajectories are taken from a simulation in a Hilbert space of increased dimension $N_t = 6$, $N_u = 6$ and $N_l = 100$. Inset: difference between $M = 20$ (dashed purple) and $M = 80$ (dashed blue) low-rank evolutions compared to the full result (blue) in the region of phase-space where the discrepancy is most pronounced.
• Figure S1: Time evolution of (a) the purity $\mathrm{tr}(\hat{\rho}^2)$ and (b) the control quantity $p_M/p_1$, as computed from integrating the full master equation Eq. \ref{['eq:hamiltonian']}, the LRA Eq. \ref{['eq:nosse']} with $M = 20$, and the RWA Eq.\ref{['eq:hamiltonian_rwa']} for the transmon readout. The solid, dashed, and dotted lines correspond to the full evolution, low-rank, and rotating-wave approximation evolutions, respectively. The red (blue) lines correspond to an initial transmon ground (first excited) state. The purity generally decreases in time as the system evolves towards a mixed state. The relative error between the full and approximate values $\delta$ is much smaller for the LRA than for the RWA. (b) The control quantity $p_M/p_1$ increases as the state becomes mixed, providing a practical indicator of the LRA’s accuracy.
• Figure S2: Transmon readout error and ionization for a square pulse compared to the LRA with rank $M=20$ and RWA. The full, low-rank, and RWA evolutions are plotted in grey, dashed orange, and dotted turquoise lines, respectively. (a) The assignment error $\epsilon_a(\tau)$ as a function of the readout time. The dashed black line corresponds to the long-$\tau$ limit $\epsilon_a \simeq \tau\gamma/2$. The relative error $\delta$, shown in the lower panel, demonstrates that the LRA captures the exact assignment error slightly better than the RWA, though the RWA still performs well due to the relatively weak drive strength. (b) The transmon ionization Eq. \ref{['eq:transmon_ionization']}. Again, the LRA is a better match to the full dynamics than the RWA, though the latter still manages to capture the transmon ionizations since the drive is relatively weak.