Floquet-engineered fast SNAP gates in weakly coupled circuit-QED systems

TL;DR

The paper addresses the slow SNAP gate problem in ultra-high-coherence cavity–transmon systems caused by small bare dispersive shifts $\chi_0$. It introduces a Floquet-engineered protocol that dynamically boosts the dispersive shift to $\chi_d$ via sideband driving and implements SNAP gates in the resulting Floquet basis, achieving speeds exceeding the bare dispersive limit. By extending quantum optimal control to the Floquet SNAP framework, the authors demonstrate additional order-of-magnitude improvements in gate fidelity and duration, validated by Floquet–Markov open-system simulations. The approach enables high-fidelity, selective control of weakly coupled, high-coherence cavities and suggests robust, scalable control strategies for Floquet quantum systems, with potential extensions to multi-mode architectures and fault-tolerant schemes.

Abstract

Superconducting cavities with high quality factors, coupled to a fixed-frequency transmon, provide a state-of-the-art platform for quantum information storage and manipulation. The commonly used selective number-dependent arbitrary phase (SNAP) gate faces significant challenges in ultra-high-coherence cavities, where the weak dispersive shifts necessary for preserving high coherence typically result in prolonged gate times. Here, we propose a protocol to achieve high-fidelity SNAP gates that are orders of magnitude faster than the standard implementation, surpassing the speed limit set by the bare dispersive shift. We achieve this enhancement by dynamically amplifying the dispersive coupling via sideband interactions, followed by quantum optimal control on the Floquet-engineered system. We also present a unified perturbation theory that explains both the gate acceleration and the associated benign drive-induced decoherence, corroborated by Floquet-Markov simulations. These results pave the way for the experimental realization of high-fidelity, selective control of weakly coupled, high-coherence cavities, and expanding the scope of optimal control techniques to a broader class of Floquet quantum systems.

Figures (9)

• Figure 1: Accelerating SNAP gates by boosting the dispersive shift through sideband engineering. (a) Spectrum of a standard ancilla drive in a SNAP gate. Each peak is centered at $\omega_\text{q} + n\chi$ with linewidth constrained by $\Delta\omega_n \ll |\chi|$. (b) Bloch-sphere trajectory of the ancilla under a drive conditioned on the cavity state $|n\rangle$. The enclosed geometric phase $2\theta_n$ imparts a relative phase $\theta_n$ onto the Fock state $|n\rangle$. (c) Any arbitrary unitary $\hat{U}$ for a single cavity mode can be decomposed into a sequence of displacement operations $\hat{D}(\alpha)$ interleaved with SNAP gates $\prod_n \hat{S}_n(\theta_n)$. (d) Energy-level diagram of the system. The blue and red arrows denote the sideband transitions $|g,n+1\rangle \leftrightarrow |f,n\rangle$ and $|e,n+1\rangle \leftrightarrow |h,n\rangle$, with amplitudes $\Omega_{g,e}$ and detunings $\Delta_{g,e}$. The resulting dispersive shifts induced by the sideband interactions are labeled $\delta\chi_{g,e}$. The inset (in the rotating frame of the drive frequency) illustrates that the sideband transition arises from a Raman-like process mediated by the Jaynes– Cummings interaction and a single-photon transmon transition. (e) Ancilla frequency shift $\delta\omega_\text{q}$ as a function of drive frequency for different cavity photon numbers. The inset zooms in near the $|e,n+1\rangle \leftrightarrow |h,n\rangle$ sideband resonance, showing both the bare dispersive shift $\chi_0$ and the driven dispersive shift $\chi_\text{d}$ at the selected drive frequency (red dashed line). Bottom panel shows the overlap between Floquet and static eigenstates. Blue and red dots correspond to $_\text{S}\langle e,1|h,0\rangle_\text{F}$ and $_\text{S}\langle g,1|f,0\rangle_\text{F}$, respectively, with fits to Eq. \ref{['eq:fit']} shown as transparent curves. The linewidth of each peak reflects the sideband coupling strength. (f) Schematic illustration of SNAP gate acceleration. The SNAP pulse duration is shortened by applying a background sideband drive that enhances the dispersive shift. Parameters: $\omega_\text{q}/2\pi = 6.4$ GHz, $\omega_\text{c}/2\pi = 4.5$ GHz, $\alpha/2\pi = -230$ MHz, $\chi_0/2\pi = 0.14$ MHz, $\epsilon/2\pi = 0.8$ GHz.
• Figure 2: Implementing the SNAP gate in the Floquet basis. (a) Floquet matrix elements of the ancilla ladder operator $\hat{q}$, maximized over the Floquet band index $k$. Red boxes highlight elements that are not present in the static basis. (b) The same quantities as in (a), obtained through a perturbative treatment of the sideband interactions. (c) Pulse sequence for Floquet SNAP (not to scale). The gray background represents the sideband drive, consisting of a ramp-up, flat-top, and ramp-down segment. The Gaussian SNAP pulse is shown in blue. (d) Population dynamics of various ancilla states, where solid and transparent curves correspond to eigenstates in the Floquet basis and dressed basis, respectively. The inset provides a zoomed-in view of the ramp-up region, illustrating the smooth transition from the static to Floquet eigenstate. (e) Trajectory of the ancilla state evolution on the Bloch sphere within the Floquet basis. Parameters: $\omega_\text{q}/2\pi=6.4$ GHz, $\omega_\text{c}/2\pi=4.5$ GHz, $\alpha/2\pi=-230$ MHz, $\chi_0/2\pi=0.14$ MHz, $\epsilon/2\pi = 0.8$ GHz, $\omega_\text{d}/2\pi = 7.56$ GHz, $t_\text{r} = 10$ ns, $t_\text{g} = 10$ µs.
• Figure 3: Optimal control pulses for high-fidelity SNAP gates in the Floquet basis. (a) $I$ and $Q$ quadratures of an optimized pulse implementing the $\exp(i\pi|0\rangle\langle0|)$ SNAP gate in 1500 ns. (b) Phase accumulation dynamics for various Fock states, demonstrating the desired $\pi$-phase difference between the target $|0\rangle$ state and all other states. (c) Ancilla trajectories on the Bloch sphere for selected cavity Fock states, illustrating the photon-number-dependent evolution induced by the optimized pulse. (d) SNAP gate infidelity versus gate duration. Results for standard SNAP (green), Floquet SNAP (red), and QOC Floquet SNAP (blue) are shown for comparison. Dashed vertical lines mark characteristic timescales associated with the undriven dispersive shift and driven dispersive shift. (e) Pulse sequence for Fock state $|1\rangle$ preparation (not to scale). The gray background represents the sideband drive, consisting of ramp-up, flat-top, and ramp-down segments. The QOC SNAP pulse is shown in blue, sandwiched between two cavity displacement pulses depicted in red. (f) Wigner function of the Fock $|1\rangle$ state, prepared using 1500-ns pulses obtained from different methods. As a reference, we also show the state $|1\rangle_\text{approx}$, constructed from a SNAP-gate decomposition and free from coherent errors. Fidelities with respect to the Fock $|1\rangle$ state, along with the $|1\rangle_\text{approx}$ state (in parentheses), are provided. Parameters: $\omega_\text{q}/2\pi=6.4$ GHz, $\omega_\text{c}/2\pi=4.5$ GHz, $\alpha/2\pi=-230$ MHz, $\chi_0/2\pi=0.14$ MHz, $\epsilon/2\pi = 0.8$ GHz, $\omega_\text{d}/2\pi = 7.56$ GHz, $t_\text{r} = 10$ ns, $t_\text{d} = 72$ ns, $t_\text{g}=1500$ ns.
• Figure 4: Infidelity of Fock $|1\rangle$ state preparation in the presence of decoherence. Infidelities obtained from standard SNAP (green), Floquet SNAP (red), and QOC Floquet SNAP (blue) gates are shown as functions of the gate duration. Solid curves represent infidelities with respect to the state from SNAP-gate decomposition $|1\rangle_\text{approx}$, while dashed transparent curves indicate infidelities relative to the target Fock state $|1\rangle$. Stars highlight the minimum infidelity for each gate implementation. The gray dashed line indicates the intrinsic decomposition error, serving as a baseline for comparison. Parameters: $\omega_\text{q}/2\pi=6.4$ GHz, $\omega_\text{c}/2\pi=4.5$ GHz, $\alpha/2\pi=-230$ MHz, $\chi_0/2\pi=0.14$ MHz, $\epsilon/2\pi = 0.8$ GHz, $\omega_\text{d}/2\pi = 7.56$ GHz, $t_\text{r} = 10$ ns, $t_\text{d} = 72$ ns, $t_\text{g}=1500$ ns, $\gamma_\text{q}=(300\,\text{µs})^{-1}$, $\gamma_{\phi,\text{q}}=(500\,\text{µs})^{-1}$, $\gamma_\text{c}=(30\,\text{ms})^{-1}$.
• Figure 5: Decoherence dynamics with a sideband drive. (a) Evolution of incoherent transitions when the system is initialized in the Floquet eigenstate $|e,1\rangle$. The population dynamics is explained by the transition diagram (inset), which highlights key states and the incoherent transitions between them. (b) Dominant decoherence rates as functions of the sideband drive frequency. The blue curve represents the primary decay channel, arising from sideband dressing of the ancilla decay, while the red curve describes the leading dephasing mechanism due to photon shot noise, which results from elevated ancilla temperatures caused by dressed dephasing. Solid and dashed curves correspond to results obtained from solving the Floquet– Markov equation and from the analytic expression, respectively. The gray curve shows the driven dispersive shift as a function of the drive frequency, included as a reference, with the drive frequency used in (a) marked by a vertical line. Parameters: $\omega_\text{q}/2\pi=6.4$ GHz, $\omega_\text{c}/2\pi=4.5$ GHz, $\alpha/2\pi=-230$ MHz, $\chi_0/2\pi=0.14$ MHz, $\epsilon/2\pi = 0.8$ GHz, $\omega_\text{d}/2\pi = 7.56$ GHz, $\gamma_\text{q}=(300\,\text{µs})^{-1}$, $\gamma_{\phi,\text{q}}=(500\,\text{µs})^{-1}$, $\gamma_\text{c}=(30\,\text{ms})^{-1}$.