Comprehensive explanation of ZZ coupling in superconducting qubits

TL;DR

This work tackles unwanted static ZZ couplings in superconducting qubits by developing a unified, multi-faceted framework. It combines a diagrammatic Schrieffer–Wolff perturbation theory, a Hamiltonian-graph representation, a level-repulsion–based intuitive picture, and a stable-marriage–based state-assignment algorithm to analyze ZZ in a three-transmon circuit with a flux-tunable coupler. The authors identify 24 energy-level configurations and reveal two main classes of zero ZZ regions (level-repulsion–type and three-loop–type) along with regions of strong ZZ near perturbative poles, providing both analytical and numerical predictions that guide device design. The framework not only clarifies the mechanisms behind ZZ coupling but also offers practical guidance for engineering weak ZZ for high-fidelity gates and exploiting strong ZZ for fast CZ/CPHASE operations, with clear pathways to extend to larger qubit networks and higher-order couplings.

Abstract

A major challenge for scaling up superconducting quantum computers is unwanted couplings between qubits, which lead to always-on ZZ couplings that impact gate fidelities by shifting energy levels conditional on qubit states. To tackle this challenge, we introduce analytical and numerical techniques, including a diagrammatic perturbation theory and a state-assignment algorithm. Together, these tools enable us to explain the emergence of ZZ coupling in three linked pictures, where each picture tells us more about the underlying mechanisms creating the ZZ coupling. These pictures generalize previous efforts, which focused on specific setups and a single mechanism. The deeper understanding that we provide of the mechanisms behind the ZZ coupling facilitate finding parameter regions of weak and strong ZZ coupling. We showcase our techniques for a system consisting of two fixed-frequency transmon qubits connected by a flux-tunable transmon coupler. There, we find three types of parameter regions with zero or near-zero ZZ coupling, all of which are accessible with current technology. We furthermore find regions of strong ZZ coupling nearby, which may be used to implement adiabatic controlled-phase gates and quantum simulations. Our framework is applicable to many types of qubits and opens up for the design of large-scale quantum computers with improved gate fidelities.

Figures (14)

• Figure 1: Circuit diagram for two fixed-frequency qubits (blue and green) coupled through both a direct capacitive coupling and a flux-tunable coupler (orange). The qubits and the coupler are implemented with transmon circuits, where the nonlinear inductances are the Josephson junctions (crossed boxes). Two Josephson junctions in a loop create a superconducting quantum interference device (SQUID) with an effective inductance tunable by the external magnetic flux through the loop. The direct capacitive coupling and the coupler together generate an effective ZZ coupling (red arrows) between the qubits.
• Figure 2: Hamiltonian graph representation of the effective Hamiltonian in Eqs. (\ref{['eq:effective_H0']}) and (\ref{['eq:effective_V']}). The effective Hamiltonian conserves the parity of a state's total excitation number $N_e$. As a result, the graph is decoupled into two subgraphs with states of even (left) and odd (right) total number of excitations. Excitation-conserving edges (black solid lines) connect states into excitation subgraphs. Non-excitation-conserving edges (orange solid lines) connect states between excitation subgraphs. The edges are undirected since the Hamiltonian is Hermitian. The five excitation subgraphs with the lowest excitation number are shown. To simplify the representation, we have removed loop edges and non-excitation-conserving edges that have contributions less than the approximation precision defined in Section \ref{['sec:analytical_predictions']} below. We only include the edge weights for the excitation-conserving edges in the three lowest excitation subgraphs; see the main text in Section \ref{['sec:Hamiltonian_graph']} for further details.
• Figure 3: The nine main level repulsions on the energies defining the ZZ coupling. (a) The energy spectrum of the two-level system of $\ket{010}$ and $\ket{100}$ as a function of the bare detuning. Here, we neglect effects from other couplings outside of the two-level system and position the energy levels around $E = 0$. The energy spectrum shows the avoided level crossing of the eigenenergies $E_\pm$ (black solid lines) which deviate from the bare energies (black dashed lines) as a result of (fictive) level repulsions. Similar avoided level crossings are present for the other eight level repulsions that are represented with black arrows towards the states associated with the energies. (b) The main level repulsions following the edges of the Hamiltonian graph in Fig. \ref{['fig:Hamiltonian_graph']}. The black dashed line highlights the level repulsions shown in detail in (a). We note that the level repulsion from the energy level of the $\ket{002}$ is mediated by the edges through the states $\ket{101}$ and $\ket{011}$. (c) The level-diagram representation of the Hamiltonian graph with level repulsions in (b). We show two examples of level configurations that have balanced level repulsions. The balanced configurations predict likely parameter regions with zero ZZ coupling. Note that the level repulsions in the first-excitation subgraph have an opposite contribution to the ZZ coupling.
• Figure 4: Predictions for the static ZZ coupling from the intuitive picture. The predictions are given in the parameter space of the shifted coupler frequency relative to the mean qubit frequency $\omega'_3 = \omega_3 - (\omega_1 + \omega_2)/2$ and the qubit detuning $\Delta_{12} = \omega_1 - \omega_2$. The parameter space is partitioned according to the 24 configurations, resulting in 24 bounded parameter regions for $\Delta_{12} > 0$. We note that the figure is symmetric under $\Delta_{12} \to -\Delta_{12}$ due to the freedom to label the qubits. The partition lines (solid black) are the borders between configurations; they represent the conditions for two or more energy levels permuting. We predict nine configurations for $\Delta_{12} > 0$ where it is likely to find a zero ZZ coupling. We note that we do not expect zero ZZ coupling everywhere in the white regions. The regions colored blue and orange are predicted to have negative and positive ZZ coupling, respectively. The dark orange region is predicted to yield the strongest ZZ coupling.
• Figure 5: One of the closed paths contributing to the third-order energy correction in Eq. (\ref{['eq:E3']}) for the initial state $\ket{100}$. The initial state is highlighted by making its vertex a diamond. The closed path is generated by the commutator $[S_1,[S_1,V]]$ acting on the initial state $\ket{100}$, and it visits all vertices in the first-excitation subgraph in Fig. \ref{['fig:Hamiltonian_graph']}. The sum of all possible paths similar to the one here forms the diagram in Eq. (\ref{['eq:tri']}).