Zero-ZZ Interaction Points in Superconducting Qubits

• Zero-ZZ interaction points are specific circuit settings in superconducting qubits where the cross-Kerr (ZZ) coupling cancels, eliminating state-dependent phase shifts.
• Analytical methods like level repulsion and diagrammatic perturbation theory enable precise tuning by balancing direct and indirect virtual transition pathways.
• Experimental protocols and state-assignment algorithms have demonstrated that achieving zero-ZZ conditions can boost two-qubit gate fidelities up to 99.9% while reducing crosstalk.

A zero-ZZ interaction point refers to a set of circuit and/or control parameters in a multi-qubit superconducting device at which the effective ZZ coupling term (also known as the cross-Kerr or two-qubit longitudinal interaction) vanishes. This suppresses the always-on conditional phase shifts that degrade quantum gate fidelity and crosstalk performance. The existence, classification, and exploitation of zero-ZZ points is a central design premise for high-fidelity, scalable superconducting quantum circuits.

1. Formal Definition and Physical Origin

The ZZ coupling, denoted ζ or χ_ZZ, quantifies the state-dependent frequency shift when two (typically transmon) qubits are capacitively or inductively coupled, with or without an intermediary coupler. In a dressed eigenbasis, the two-qubit effective Hamiltonian takes the form: $$H' = \frac{1}{2}\omega'_1 \sigma_z \otimes I + \frac{1}{2}\omega'_2 I \otimes \sigma_z + \frac{1}{2}\zeta \sigma_z \otimes \sigma_z$$ The ZZ rate is defined via four dressed energies: $\zeta = E'_{11} - E'_{10} - E'_{01} + E'_{00}$ In circuit QED architectures, ζ emerges from virtual transitions—second- and fourth-order—involving computational and noncomputational states (e.g. $|20\rangle$, $|02\rangle$). These contributions may add or cancel depending on device parameters. Importantly, zero-ZZ points may exist even at finite exchange coupling, enabling strong two-qubit gates without deleterious idling crosstalk.

2. Analytical Frameworks for ZZ Cancellation

2.1 Level-Repulsion (“Intuitive”) Picture

Each virtual process (e.g., $|11\rangle \leftrightarrow |02\rangle$) induces an avoided crossing/repulsion between many-body states, shifting their energies by $+\frac{g_{ij}^2}{\Delta_{ij}}$ (for $|i\rangle$) and $-\frac{g_{ij}^2}{\Delta_{ij}}$ (for $|j\rangle$). ZZ arises when the net repulsions on $|11\rangle$ vs $|10\rangle/|01\rangle$ are imbalanced. By adjusting detunings and couplings, repulsions can be engineered to cancel exactly, yielding ζ=0 (Fors et al., 27 Aug 2024).

2.2 Diagrammatic Perturbation Theory

Systematic expansion in small couplings $g_{ij}$ (Schrieffer–Wolff or Brillouin–Wigner) allows diagrammatic enumeration of processes up to fourth order: $$\zeta = \zeta_w + \zeta_e + \zeta_{nw} + \zeta_{ne} + \zeta_n + \zeta_3$$ where each $\zeta_{*}$ (w,e,nw,ne,n,3) is a distinct level-repulsion or loop process with explicit detuning and coupling dependence. The zero-ZZ condition is the algebraic equation: $$\zeta_w + \zeta_e + \zeta_{nw} + \zeta_{ne} + \zeta_n + \zeta_3 = 0$$ Explicitly, near zeros occur where: $$\frac{g_{12}^2}{\Delta_{12}} \approx \frac{(g_{13}g_{23})^2}{\Delta_{c2}^2 \Delta_{12}} \pm \cdots$$ allowing for compensation of direct and indirect ZZ pathways (Fors et al., 27 Aug 2024, Sung et al., 2020, Collodo et al., 2020).

2.3 State Assignment (Eigenstate Continuity) Approach

In parameter regimes where simple energy-label assignment fails (e.g., strong hybridization), the correct extraction of ζ requires algorithmic label assignment. A stable-matching (Gale–Shapley) algorithm assigns dressed eigenenergies to the bare computational labels by maximum overlap $|\langle \text{bare}|\text{dressed}\rangle|$, lending numerical stability and exactness across all regimes (Fors et al., 27 Aug 2024).

3. Classification of Zero-ZZ Regimes

Comprehensive mapping of parameter space reveals three nondegenerate solution branches, experimentally realized in transmon–coupler–transmon architectures:

Regime	Δ<sub\>12</sub> Range	ζ Cancellation Mechanism	Parameter Tuning Lever
I	$|\Delta_{12}| > \max(|\alpha_1|,|\alpha_2|)$	Second-excited coupler (e.g. $|002\rangle$)	$\omega_c'$ at $\pm|\alpha|$
II	$|\Delta_{12}| < |\alpha|$ (straddling)	$|11\rangle \leftrightarrow |20^\pm\rangle$ cancellation	Deeper $\omega_c'$
III	(within/near straddling band at large $\omega_c'$)	Loop diagram ($\propto g_{12}g_{13}g_{23}$, 4th order)	$g_{12}$ design, high $\omega_c'$

Regimes I/II achieve zero-ZZ mainly with second-order processes and are robust against moderate flux noise. Regime III depends more sensitively on direct coupling $g_{12}$ but allows for tuning at both positive and negative $g_{12}$. Numerical ZZ landscapes confirm the sharpness and robustness of these contours (Fors et al., 27 Aug 2024).

4. Experimental Realization and Tuning

4.1 Extraction Protocols

ZZ rates are measured via conditioned Ramsey sequences: a Ramsey fringe is performed on one qubit with the other prepared in $|0\rangle$ or $|1\rangle$; the difference in extracted frequency yields ζ. Dynamic and idling errors are benchmarked via Clifford randomized benchmarking, correlators $C_{ZZ}(t)$, and phase accumulation protocols (Fors et al., 27 Aug 2024, Sung et al., 2020).

4.2 Device Examples

• Two fixed-frequency transmons with a flux-tunable coupler: three zero-ZZ regions are directly observed as a function of detuning and coupler flux bias.
• Devices with a hybrid CSFQ–transmon pair (opposite sign anharmonicity): ZZ can be statically canceled without tunable coupler (Ku et al., 2020).
• All-transmon arrays with multi-mode couplers achieve zero-ZZ at non-zero exchange (Zhao et al., 2020).

Robust operational points are chosen not just at $\zeta=0$ but where $d\zeta/d\omega_c$ is minimized (to reduce sensitivity to low-frequency flux noise and parameter drift).

5. Algorithmic and Circuit Design Techniques

5.1 State Assignment Algorithm

The stable-matching protocol for assigning computational labels in dressed eigenstates—critical in the presence of strong hybridization or energy-level collision—operates as follows:

1. Construct an $N\times N$ overlap matrix $$O_{I,i}=|\langle\text{bare }i|\text{dressed }I\rangle|$$.
2. Dressed states ("Alices") propose to bare states ("Bobs") by descending overlap.
3. If a bare state is unmatched, accept; if matched but prefers new proposal, reassign; repeat until stable matching is achieved.

This procedure guarantees robust extraction of ζ independent of proximity to perturbative or nonperturbative boundaries (Fors et al., 27 Aug 2024).

5.2 Practical Device Design Guidelines

• Ensure $|\alpha_{1,2}|$ exceeds expected detuning spreads to enable regime II; use large negative $\alpha_c$ to separate $|002\rangle$ energy, facilitating tuning.
• For regime III operation, introduce (and control) direct $g_{12}$ via circuit topology (floating ground, adjustable shunt capacitance, etc.).
• Engineer maximum $\omega_c$ tuning range such that all three zero-ZZ regimes are accessible.
• Implement closed-loop in situ calibration via the state-assignment algorithm for continual drift correction or array initialization.

Arrays can be organized such that nearest-neighbor pairs are sufficiently detuned to avoid collision with undesired zero-ZZ regions, supporting modular, collision-free two-qubit gate operation.

6. Scalability, Performance Metrics, and Limitations

Zero-ZZ operation directly improves both single- and two-qubit gate fidelities by suppressing conditional phase errors and crosstalk. Gate errors at zero-ZZ points are limited primarily by coherence (T₁, T₂), with measured idling errors scaling as $\epsilon \sim \tau/(T_1)$, achieving the decoherence floor. Reported fidelities of 99.8–99.9% for two-qubit gates at these operating points are consistent with this principle (Fors et al., 27 Aug 2024, Sung et al., 2020).

The required drive (in microwave-activated architectures) is weak and non-intrusive, with dynamic Stark shifts easily balanced to exactly nullify ζ (Ni et al., 2021). Hardware overhead is linear in number of coupler links, and the technique is apparatus-agnostic (compatible with both fixed-frequency and tunable couplers).

Sensitivity to flux noise, design parameter disorder, and shared coupler/cancellation tone frequencies poses current scaling challenges. For large-scale arrays, parameter crowding and cross-talk from cancellation tones demand careful architecture and control resource allocation.

7. Broader Implications and Future Directions

Precise understanding and classification of zero-ZZ interaction points underpin the route to large-scale, error-corrected quantum computing. The ability to engineer, predict, and maintain these points in complex, multi-qubit topologies will remain a core requirement for crosstalk suppression, high-fidelity entangling operation, and the management of frequency crowding and parasitic interactions in future architectures.

The general framework—level repulsion, diagrammatic analysis, and robust state assignment—extends beyond the specific case of transmon qubits, being in principle applicable to any circuit QED platform where virtual transitions and multi-path interference can mediate both desired and undesired two-qubit couplings (Fors et al., 27 Aug 2024).