Frequency- and Amplitude-Modulated Gates for Universal Quantum Control

TL;DR

The paper addresses achieving high-fidelity universal quantum gates on fixed-frequency superconducting qubits using a framework of frequency- and amplitude-modulated microwave control. It introduces a Floquet-engineered extended Hilbert space approach that converts drive-frequency modulation into effective qubit interactions, enabling both adiabatic and nonadiabatic gates with a structured five-stage pulse protocol and FAQUAD-based optimization. Numerical simulations on transmon-like parameters demonstrate a universal gate set (X, Hadamard, phase, CZ) with errors below $0.1\%$ and gate times from tens to over a hundred nanoseconds, plus an always-on CZ variant that further reduces two-qubit times. The framework broadens the microwave-control toolbox for scalable quantum processors and is extendable to larger multi-qubit systems, with potential enhancements in robustness and hardware integration.

Abstract

Achieving high-fidelity single- and two-qubit gates is essential for executing arbitrary digital quantum algorithms and for building error-corrected quantum computers. We propose a theoretical framework for implementing quantum gates using frequency- and amplitude-modulated microwave control, which extends conventional amplitude modulation by introducing frequency modulation as an additional degree of control. Our approach operates on fixed-frequency qubits, converting the need for qubit frequency tunability into drive frequency modulation. Using Floquet theory, we analyze and design these drives for optimal fidelity within specified criteria. Our framework spans adiabatic to nonadiabatic gates within the Floquet framework, ensuring broad applicability across gate types and control schemes. Using typical transmon qubit parameters in numerical simulations, we demonstrate a universal gate set-including the X, Hadamard, phase, and CZ gates-with control error well below 0.1% and gate times of 25-40 ns for single-qubit operations and 125-135 ns for two-qubit operations. Furthermore, we show an always-on CZ gate tailored for driven qubits, which has gate times of 80-90 ns.

Figures (9)

• Figure 1: Overview of the gate protocol. (a) Qubit-coupler-qubit transmon-based circuit with two fixed-frequency qubits (QBa, QBb) and a flux-tunable coupler (CPLRc). Each qubit (coupler) has a microwave charge drive line. Proposed gates are implemented using microwave control only. (b) The staged gate protocol manifested in the $\omega_\text{inst}(t)-\Omega(t)$ parameter space. In stages , , we turn on/off the drive amplitude while keeping $\omega_\text{inst}(t)=\omega_0$. In stages , , we modulate the drive frequency while keeping $\Omega(t)=\Omega_1$. In stage , we hold at $\Omega(t)=\Omega_1, \omega_\text{inst}(t)=\omega_1$. (c) Mapping between the original Hilbert space $\mathcal{H}$ and the extended Hilbert space $\mathcal{K}$, where the example system is driven by a frequency-modulated microwave pulse. Stage : turn on the drive, mapping from $\mathcal{H}$ to $\mathcal{K}$. Stages , , : state evolution can be best analyzed and designed in $\mathcal{K}$ where the dynamics are simpler. Stage : turn off the drive, mapping from $\mathcal{K}$ back to $\mathcal{H}$. (d) High-level illustrations of adiabatic gates (purple trajectory, e.g., Z and CZ gates) and nonadiabatic gates (orange trajectory, e.g., X gate). (e)(f) Descriptive profiles of $\Omega(t)$ and $\omega_\text{inst}(t)$, and the segmentation with time parameters corresponding to the staged gate protocol.
• Figure 2: CZ gate. (a)(c) Quasienergy spectrum of the computational Floquet mode $\ket{0,ege}$ and non-computational Floquet mode $\ket{-1,gef}$ as a function of drive amplitude $\Omega(t)$ and instantaneous frequency $\omega_\mathrm{inst}(t)$. $\omega^*/2\pi=5.771$ GHz indicates where the closest point in quasienergy takes place, which also dictates the maximal allowed ZZ interaction $-9.61$ MHz. (b)(d) ZZ interaction as a function of drive amplitude $\Omega(t)$ and instantaneous frequency $\omega_\text{inst}(t)$. (e)(f) Phase accumulation $\phi$ and control error of CZ gates as a function of instantaneous frequency endpoint $\omega_1$. Purple star indicates the best CZ gate among the samples shown. (g)(h) The optimized control waveforms $\Omega(t)$ and $\omega_\text{inst}(t)$ for the best CZ gate.
• Figure 3: Always-on CZ gate. (a) Quasienergy spectrum of the computational Floquet mode $\ket{0,ege}$ coming into avoided crossing with non-computational Floquet mode $\ket{-2,ghe}$ as a function of instantaneous frequency $\omega_\mathrm{inst}(t)$. $\omega^*/2\pi=5.892$ GHz indicates where the closest point in quasienergy takes place, which also dictates the maximal allowed ZZ interaction $12.3$ MHz. (b) ZZ interaction as a function of instantaneous frequency $\omega_\mathrm{inst}(t)$. (c)(d) Phase accumulation and control error of always-on CZ gates as a function of instantaneous frequency endpoint $\omega_1$. Purple star indicates the best CZ gate among the samples shown. (e) The optimized control waveform $\omega_\text{inst}(t)$ for the best CZ gate.
• Figure 4: Z gate (a-d) and X gate (e-h). (a) Quasienergy spectrum of the Floquet modes $\{\ket{1,g},\ket{0,e},\ket{-1,f}\}$-subspace as a function of $\omega_\text{inst}(t)$ for the Z gate. The black curve indicates that the population stays in $\ket{1,g}$ during the Z gate if initialized in $\ket{g}$. (b)(c) Phase accumulation and control error of Z gates as a function of instantaneous frequency endpoint $\omega_1$. Purple star indicates the best Z gate among the samples shown. (d) The optimized control waveforms $\Omega(t)$ and $\omega_\text{inst}(t)$ for the best Z gate. (e) Quasienergy spectrum of the Floquet modes $\{\ket{1,g},\ket{0,e},\ket{-1,f}\}$-subspace as a function of $\omega_\text{inst}(t)$ for the X gate. The red curves (solid, dashed, dash-dot line) show the population transitions during the frequency modulation of the X gate if initialized in $\ket{g}$. Part I: the population remains in $\ket{1,g}$. Part II: the population transfers to $\ket{1,g}$, $\ket{0,e}$ and $\ket{-1,f}$. Part III: the population goes to $\ket{0,e}$. (f) Average overlap as a function of $t_\omega,t_h$ when $\omega_1/2\pi=5.24$ GHz at the end of the frequency modulation during the X gate. Red area in the middle indicates top candidates for potential high-fidelity X gates. (g) The X gate control error as a function of $t_a$ for one specific set of $t_\omega,t_h,\omega_1$ among the top candidates in (f). (h) The optimized control waveforms $\Omega(t)$ and $\omega_\text{inst}(t)$ for the best X gate.
• Figure 5: CZ gate. (a) Static ZZ interaction as a function of coupler frequency $\omega_c$. We choose $\omega_c/2\pi=5.431$ GHz to have a vanishing ZZ interaction. (b)(c) Quasienergy spectrum of the computational Floquet modes $\ket{0,ggg},\ket{0,gge},\ket{0,egg},\ket{0,ege}$ as a function of drive amplitude $\Omega(t)$ and instantaneous frequency $\omega_\text{inst}(t)$, respectively.