Engineering, Control, and Longitudinal Readout of Floquet Qubits

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Engineering, Control, and Longitudinal Readout of Floquet Qubits

Anthony Gandon, Camille Le Calonnec, Ross Shillito, Alexandru Petrescu, and Alexandre Blais

Phys. Rev. Applied 17, 064006 – Published 2 June 2022

Abstract

Properties of time-periodic Hamiltonians can be exploited to increase the dephasing time of qubits and to design protected one- and two-qubit gates. Recently, Huang et al. [Phys. Rev. Applied 15, 034065 (2021)] have shown that Floquet states offer a manifold of working points with dynamical protection larger than the few, usual, static sweet spots. Here, we show how Floquet theory, often used on systems with a single drive tone, can be used to describe approaches to robustly control Floquet qubits in the presence of multiple commensurate drive tones. Using this formulation, we introduce a longitudinal readout protocol to measure the Floquet qubit without the need of first adiabatically mapping the Floquet states back to the static qubit states, resulting in a significant speedup in the measurement time of the Floquet qubit. The analytical approach developed here can be applied to any Hamiltonian involving a small number of distinct drive tones, typical in the study of standard parametric gates for qubits outside of the rotating-wave approximation.

Received 27 September 2021
Revised 25 April 2022
Accepted 28 April 2022

DOI:https://doi.org/10.1103/PhysRevApplied.17.064006

Physics Subject Headings (PhySH)

Optoelectronics Quantum gates Quantum information with solid state qubits Quantum protocols Quantum simulation

Floquet systems Superconducting qubits

Microwave techniques

Quantum Information, Science & TechnologyCondensed Matter, Materials & Applied Physics

Authors & Affiliations

Anthony Gandon^1,2,*, Camille Le Calonnec ^1,3, Ross Shillito¹, Alexandru Petrescu¹, and Alexandre Blais ^1,4

¹Institut Quantique and Département de Physique, Université de Sherbrooke, Sherbrooke, Québec J1K 2R1, Canada
²Mines ParisTech, PSL Research University, 75006 Paris, France
³University of Strasbourg and CNRS, CESQ and ISIS (UMR 7006), 67000 Strasbourg, France
⁴Canadian Institute for Advanced Research, Toronto, Ontario M5G1M1, Canada

^*anthony.gandon@mines-paristech.fr

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Vol. 17, Iss. 6 — June 2022

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Images

Figure 1
(a) Population of the Floquet modes $| ϕ_{0} (t) ⟩$ and $| ϕ_{1} (t) ⟩$ in the numerically evolved state $| ψ (t) ⟩$ as a function of time under the Hamiltonian Eq. (5) for the parameters found in the main text. (b) Typical drive amplitude as a function of time. The first drive $ε_{d 1}$ (green line) is used to generate the Floquet qubit states, while the second drive $ε_{d 2}$ (blue line) drives Rabi oscillations between these levels. (c) Four quasiphase spectra for different numerators $p = 1, 2, 3, 4$ in the ratio $ω_{d 2} / ω_{d 1} = p / q$ . The colored dots are obtained from diagonalization of the propagator associated with Eq. (5) after a common period $2 π / ω_{GCD}$ and by sweeping the values of $q$ . Because of the folded space, several crossings are observed. However, a unique anticrossing corresponding to the resonance of the second drive with the Floquet qubit is observed (vertical line). The solid black lines are obtained from Eq. (6).
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Figure 2
(a) Pointer state separation $D (t)$ as a function of time for longitudinal Floquet readout (solid blue line), dispersive readout without state mapping (solid green line), and dispersive readout with the necessary state mapping with a ramp time of $T_{map} = 30 ns$ (dashed green line). The dashed blue line corresponds to the hypothetical situation where longitudinal readout is preceded by a mapping stage. This information is included only for comparison. (b) Ratio of the pointer state separation $D (t)$ and steady-state separation $D (\infty) = 0.47 \approx \tilde{g} / κ = 0.5$ as obtained from numerical integration under Eq. (7) as a function of time and for different ratios $Δ / ε_{d 1}$ . As expected, for small $Δ / ε_{d 1}$ the pointer states follow the ideal longitudinal dynamics expected from Eq. (10). (c) Pointer state separation $D (t)$ over the steady-state separation $D (\infty)$ as found from numerical simulation of the system dynamics under the Hamiltonian of Eq. (15). Here, $D (\infty) = 0.55$ is numerically evaluated at long times and at small $Δ / ε_{d 1}$ , corresponding to the average value of the bottom-right corner in panel (c). As in panel (b), the pointer state dynamics follow the expected behavior for small with $Δ / ε_{d 1}$ . The parameters used in panel (c) are ${ω_{a}, ω_{b}, ω_{c}} / 2 π = {8.2, 5.2, 7.78} GHz$ , ${α_{b} / 2, α_{c} / 2} / 2 π = {- 0.17, 0.4} GHz$ , ${g_{a}, g_{b}} / 2 π = {0.2, 0.2} GHz$ , ${\tilde{ϵ}}_{d 1} / 2 π = 0.35 GHz$ , and $κ / 2 π = 0.05 GHz$ .
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Figure 3
Possible realization of the longitudinal Floquet readout. A driven transmon qubit (green) is coupled to a readout cavity (blue) via a flux-modulated coupler (gray).
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Figure 4
Ramp profiles for (a) adiabatic and (b) instantaneous preparation pulses, as well as illustrative paths on the Bloch sphere. To facilitate the comparison between (a) and (b), both pulses are illustrated over the total time $T_{tot}$ of the adiabatic pulse, with padding added before and after the sudden pulse. (c) Initialization fidelity versus the ratio $Δ / ε_{d 1}$ and ramp time $T_{ramp}$ . The different areas correspond to sectors where an initialization fidelity higher than 99% (plain), 99.9% (hatched) and 99.99% (dotted) can be obtained in the adiabatic limit (green) and the instantaneous (blue) regimes.
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Figure 5
Quasiphase spectrum for numerators $1 \leq p_{k} \leq 15$ . The local minima of the avoided crossing can be observed on the first subplots and are characterized by a size $p \times ε_{d 2}$ . As this quantity goes beyond $π / 2$ , the local minimum becomes a local maximum in the folded space. The gray areas are used to visualize the triplet ${ϕ_{p_{k}} [q^{'} - 1], ϕ_{p_{k}} [q^{'}], ϕ_{p_{k}} [q^{'} + 1]}$ corresponding to local extrema with a nonempty intersection with the gray areas in the above subplots.
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Figure 6
(a) Average wall time per data point for each numerator $p$ using the QuTiP solver (green) and Dysolve (blue). Dashed lines are linear fits to the numerical results in solid lines. (b) Comparison of the precision for the two numerical solvers and for the approximate analytical solution obtained using the RWA. The metric is the squared sum of the quasiphase difference for all data points corresponding to one numerator $p$ . We find a good agreement between the two solvers (orange) and we also highlight the poor precision of the analytical result based on the RWA (blue and red), even for a TLS system.
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