Analysis of Frequency Collisions in Parametrically Modulated Superconducting Circuits

TL;DR

This work addresses spectral crowding in parametric modulation of superconducting circuits by developing a Floquet-based numerical framework complemented with analytical models for both qubit-modulated and coupler-modulated interactions. It maps the full landscape of parasitic sideband transitions, derives physics-informed constraints, and couples these with an SMT/OMT optimization workflow to identify safe operating points for multi-qubit architectures. Key contributions include explicit expressions for the effective couplings $g_{\text{eff}}^{(n)}$, detunings, collision-angle maps, dynamic ZZ analysis, and a practical two-stage optimization for frequency allocation applicable to qubit-qubit and qubit-coupler-qubit systems, validated against full dynamical simulations. The framework enables co-design of device parameters and control protocols to suppress crosstalk and frequency-collision errors, paving the way for scalable, high-fidelity parametric operations and analog quantum simulation on large superconducting processors.

Abstract

Superconducting circuits are a leading platform for scalable quantum computing, where parametric modulation is a widely used technique for implementing high-fidelity multi-qubit operations. A critical challenge, however, is that this modulation can induce a dense landscape of parasitic couplings, leading to detrimental frequency collisions that constrain processor performance. In this work, we develop a comprehensive numerical framework, grounded in Floquet theory, to systematically analyze and mitigate these collisions. Our approach integrates this numerical analysis with newly derived analytical models for both qubit-modulated and coupler-modulated schemes, allowing us to characterize the complete map of parasitic sideband interactions and their distinct error budgets. This analysis forms the basis of a constraint-based optimization methodology designed to identify parameter configurations that satisfy the derived physical constraints, thereby avoiding detrimental parasitic interactions. We illustrate the utility of this framework with applications to analog quantum simulation and gate design. Our work provides a predictive tool for co-engineering device parameters and control protocols, enabling the systematic suppression of crosstalk and paving the way for large-scale, high-performance quantum processors.

Figures (11)

• Figure 1: Two-qubit systems for qubit-modulated and coupler-modulated parametric interactions. (a) Schematic of two capacitively coupled transmon qubits with the static coupling strength $J$. A fixed-frequency qubit ($Q_2$) is coupled to a tunable transmon ($Q_1$). A parametric flux pulse is applied to $Q_1$ to modulate its frequency and induce a parametric interaction. (b) Energy level diagram of the two-qubit system. Solid and dashed arrows indicate co-rotating and counter-rotating transitions, respectively. The system is labeled using excitation manifolds $\mathcal{M}_{0,1,2,3,4}$ and the subsequent demos focus on the single-excitation manifold $\mathcal{M}_1$, i.e., the $\{|10\rangle,| 01\rangle\}$ subspace. (c) Schematic of the circuit architecture, featuring two transmon qubits ($Q_1, Q_2$) capacitively coupled to a central tunable transmon-type coupler ($C$) with static strengths $J_{1c}$ and $J_{2c}$, respectively. A direct static coupling of strength $J_{12}$ also exists between the two qubits. A parametric flux pulse is applied to the coupler to mediate parametric interactions. (d) Energy level diagram showing the lowest three excitation manifolds of the system: the ground state manifold $\mathcal{M}_0$, the single-excitation manifold $\mathcal{M}_1$, and the double-excitation manifold $\mathcal{M}_2$. Black solid arrows represent the strong, static qubit-coupler strengths (with strengths $\propto J_{1c},J_{2c}$), while the blue dotted arrow indicates the weak direct qubit-qubit strengths (with strength $\propto J_{12}$). Counter-rotating terms are omitted for clarity.
• Figure 2: Illustration for Floquet and Schrödinger equation (SE) methods of qubit-modulated parametric coupling in a qubit-qubit system. (a) Chevron pattern showing time-dependent population oscillations between states $|10\rangle$ and $|01\rangle$ as a function of the modulation frequency, $\omega_{p}$. The data is obtained from the SE method. (b) Schematic of the corresponding anticrossing between the two Floquet states $|\psi_{1,2}\rangle$. The quasienergy splitting between $|\psi_{1,2}\rangle$ varies as the modulation frequency $\omega_p$ increases, and the minimum quasienergy splitting, which occurs at the resonance point, is equal to the parametric coupling strength, $2g$. (c) Comparison of the generalized Rabi frequency extracted from the SE method in (a) (circles) and the Floquet quasienergy splitting $\Delta\epsilon$ in (b) (solid line). The excellent agreement validates the Floquet method. The vertex of the parabola corresponds to the on-resonance effective coupling, $2g$. (d) The collision angle, $\theta$, as a function of modulation frequency $\omega_{p}$, derived from the coupling strengths and detunings shown in (c). The collision angle is equal to $\pi/2$ at the above resonance point, and the collision angle is close to zero as $\omega_p$ moves away from the resonance point (the strong-dispersive condition).
• Figure 3: Comparison of parametric coupling strengths for different modulation schemes. (a) Parametric coupling strengths for the zeroth, first, and second harmonic orders of the $|10\rangle \leftrightarrow |01\rangle$ transition in a qubit-modulated qubit-qubit system, plotted as a function of the normalized modulation amplitude $\epsilon_p/\omega_p$. Black solid lines, red dashed lines, and gold markers correspond to results from Floquet theory, an analytical model given in Eq. \ref{['eq:qubit_strength']}, and the SE method, respectively. (b) Parametric coupling strengths for the first and second harmonic orders of the $|100\rangle \leftrightarrow |001\rangle$ transition in a coupler-modulated qubit-coupler-qubit system, as a function of the modulation amplitude $\epsilon_p$. The analytical model (dashed red lines) is derived from Eq. \ref{['eq:coupler_strength']}, with the summation truncated at $n=3$ for the first order and $n=4$ for the second. The adiabatic model (dashed black lines) is derived from Eq. \ref{['eq:coupler_strength']}, with the summation truncated at $n=1$ for the first order and $n=2$ for the second. The qubit and coupler parameters for numerical simulation and analytic expressions are listed in Table \ref{['tab:paras_qq']} for (a) and Table \ref{['tab:paras_qcq']} for (b).
• Figure 4: Maximum collision angle landscape for qubit-modulated interactions in a qubit-qubit system. The plot shows the simulated maximum collision angles, $\max \theta$, as a function of the modulation frequency, $\omega_p$, in the range of $100$ to $400$ MHz from Floquet theory. Each red branch represents a specific sideband transition, with its prominence determined by the coupling strength and detuning. The markers (circle, star, and triangle) indicate operating points for three target first-order interactions — $|01\rangle\leftrightarrow|10\rangle$, $|11\rangle\leftrightarrow|02\rangle$, and $|11\rangle\leftrightarrow|20\rangle$, respectively—chosen at modulation amplitudes that maximize their coupling strengths (i.e., $\epsilon_{p}/\omega_p=1.84$). For illustration, the second- (black dashed line) and third-order (gray dashed line) sidebands for the $|11\rangle\leftrightarrow|20\rangle$ transition are explicitly highlighted. The qubit parameters used for the simulation are listed in Table \ref{['tab:paras_qq']}.
• Figure 5: Population error analysis and numerical validation for qubit-modulated coupling in the qubit-qubit system. (a)-(c) Calculated population errors for three different target first-order interactions: (a) $|01\rangle\leftrightarrow|10\rangle$, (b) $|11\rangle\leftrightarrow|02\rangle$, and (c) $|11\rangle\leftrightarrow|20\rangle$. The operating points are chosen to maximize the respective coupling strengths, as marked in Fig. \ref{['fig:thetatf']}. The bars show the contributions from parasitic co-rotating transitions---$|01\rangle\leftrightarrow|10\rangle$ (blue), $|11\rangle\leftrightarrow|02\rangle$ (gold), and $|11\rangle\leftrightarrow|20\rangle$ (red)---and the total counter-rotating errors (teal). Solid bars represent the calculated error ($P_e$), while dashed borders indicate the upper bound ($P_e^{\text{bound}}$). The results confirm that co-rotating terms are the dominant source of error with negligible counter-rotating errors. (d)-(f) Error breakdown by harmonic order, $n$, for the corresponding target interactions in (a-c). The colored bars identify the source transition for each harmonic's error contribution. The dominant errors originate from low-order harmonics, which have stronger couplings. The harmonic corresponding to the target interaction is omitted from each plot (e.g., $n=-1$ is omitted in (d)). (g)-(i) Validation of the error model via direct dynamical simulation. The top panels show the time-domain micromotion of a non-target state's population over $0.5~\mu\text{s}$ ($100,000$ points), while the bottom panels show the corresponding Fourier transform at the range $[0, 500]$ MHz. Plotted are the populations of (g) $|11\rangle$ (blue line) when targeting $|01\rangle\leftrightarrow|10\rangle$, (h) $|01\rangle$ (gold line) when targeting $|11\rangle\leftrightarrow|02\rangle$, and (i) $|01\rangle$ (red line) when targeting $|11\rangle\leftrightarrow|20\rangle$. The peaks in the Fourier spectrum, which represent the frequencies of the population micromotion, align perfectly with the theoretically predicted frequencies of parasitic sideband transitions listed in Table \ref{['tab:TF']} (marked with blue circles, gold stars, and red triangles). The numbers in boxes label the maximum Fourier amplitudes from non-target states, which correspond to the errors in (d-f). This confirms that the population errors originate from these unwanted off-resonant couplings. The blue, gold, and red dashed lines in the bottom panels represent the modulation frequency of these three target interactions, respectively. All dynamical simulations were performed in QuTiP Lambert2024 using the vern9 solver with tolerances of $\text{atol} = \text{rtol} = 10^{-12}$.