Mathieu Control of the Effective Coupling in Superconducting Qubits

TL;DR

This work addresses the trade-off between strong, tunable couplings and subspace leakage in superconducting qubits by introducing Mathieu control: a non-resonant two-photon drive on the quadratic potential creates a selective nonlinear frequency shift that renormalizes interactions without populating noncomputational states. The authors derive the effective dispersive Hamiltonian in the rotating frame, showing that the induced $ ( abla^2) $-type term enables continuous tuning of the ZZ coupling $ J_{zz} $ and demonstrate zero-coupling points for independent single-qubit operations. They further show that a two-qubit system can realize a tunable ZZ interaction with high gate fidelities and minimal leakage, and extend the scheme to a five-qubit chain to programmable simulate the XXZ Heisenberg model, including phase-transition signatures. Overall, Mathieu control provides a unifying, scalable approach for high-fidelity quantum logic and programmable quantum simulation, with potential applicability to other platforms featuring tunable oscillators.

Abstract

A common challenge in superconducting quantum circuits is the trade-off between strong coupling and computational subspace integrity. We present Mathieu control, which uses a non-resonant two-photon drive to create a selective nonlinear frequency shift. This shift modifies interactions while preserving qubit states, enabling continuous tuning of the ZZ coupling, including full suppression, and integrating single- and two-qubit gates with low leakage. For a qubit-coupler-qubit device, it allows independent ZZ control, facilitating a programmable Heisenberg (XXZ) Hamiltonian. Extended to a five-qubit chain, the system can be reconfigured to simulate dynamics of quantum magnetic phases. Mathieu control thus provides a framework for high-fidelity quantum logic and programmable simulation.

Figures (2)

• Figure 1: (a) Schematic of the two directly coupled transmon qubits and the applied two-photon flux drive $\hat{H}_d(t)$ on Qubit 2. The corresponding energy level diagram illustrates the off-resonant coupling (red arrow) inducing level repulsion between states $|02\rangle$ and $|04\rangle$. (b) The effective ZZ coupling strength $J_{zz}$ as a function of the Mathieu control drive amplitude $\epsilon$ at a fixed drive frequency $\omega_d$, calculated by exact diagonalization of the full Hamiltonian. The vertical dashed line marks the operating point $\epsilon_0$ where $J_{zz}=0$. (c) Schematic of the control pulse sequence for implementing single-qubit and two-qubit gates. The sequence executes an XI gate, followed by an IX gate, and then a CZ gate. The waveforms for the XY control lines (top two panels) and the Mathieu (Z) control lines (bottom two panels) of both qubits are shown. (d) Process matrices ($\chi$ matrices) in the Pauli transfer representation for the simulated XI gate (left panel) and IX gate (right panel). The basis is the two-qubit Pauli basis $\{I, \sigma_x, \sigma_y, \sigma_z\}^{\otimes 2}$. Color scale represents the absolute value $|\chi_{mn}|$ on a logarithmic scale. (e) Process matrices ($\chi$ matrices) for the simulated simultaneous two-qubit X gate (XX, left panel) and the controlled-phase (CZ) gate (right panel). Display conventions are identical to (d).
• Figure 2: (a) Schematic of an extensible and programmable Heisenberg interaction chain realized with an alternating arrangement of qubits and tunable couplers. Green arrows indicate the static capacitive coupling strengths between adjacent elements. Red arrows indicate the application of Mathieu control pulses exclusively to the couplers' SQUID loops, while the qubits remain idle. (b) Static tuning of the effective qubit-qubit couplings $J_{xx}$ and $J_{zz}$ in a single qubit-coupler-qubit unit as a function of the coupler frequency $\omega_c$. Working points A ($\omega_c = 4.55\,\text{GHz}$) and B ($\omega_c = 4.67\,\text{GHz}$) are marked. (c) Two-dimensional map of the effective $J_{zz}$ coupling strength for the system biased at static working point A under the applied two-photon Mathieu drive. The coupling is plotted as a function of the drive amplitude $\epsilon$ and frequency $\omega_d$. Contour lines are spaced at intervals of $2\,\text{MHz}$. (d) Corresponding two-dimensional map of the effective $J_{zz}$ coupling for the system biased at static working point B. (e) Normalized staggered $\sigma^z$ correlation function $\mathcal{C}^z_{\text{stag}}(t)$ as a function of time for three representative values of the anisotropy parameter $\Delta = J_{zz}/2J_{xx}$: $3.55$, $0.00$, and $-1.86$. The solid lines show the evolution in the five-qubit chain under Mathieu control. Dashed lines show an exact five-qubit Heisenberg chain. (f) Fitted parameters $t_0$ (right axes) and $n$ (left axes) obtained by fitting the early-time relaxation of $\mathcal{C}^z_{\text{stag}}(t)$ to the function $\exp[-(t/t_0)^n]$. The solid lines correspond to the Mathieu-controlled chain at different efficient $\Delta$ values. Light dashed lines show the corresponding fits for the exact five-qubit Heisenberg chain.