Crosstalk-Robust Quantum Gate Sets

• Crosstalk-robust gate sets are engineered quantum control primitives designed to suppress interference by optimizing pulse shapes, effective Hamiltonians, or compilation rules.
• They employ diverse methods tailored to hardware platforms—such as echo sequences in trapped ions, pairwise robust pulses in superconducting circuits, and neural-network-optimized controls in spin qubits.
• Experimental implementations demonstrate significant improvements in gate fidelity and error mitigation, highlighting the impact of platform-specific error modeling and scalable design strategies.

Crosstalk-robust gate sets are collections of quantum control primitives whose pulse shapes, effective Hamiltonians, or compilation rules are designed so that dominant crosstalk channels do not substantially degrade gate performance under realistic parallel operation. In the literature, the term covers several distinct constructions: pairwise-robust single-qubit pulse libraries on superconducting heavy-hex devices, condition-conditioned optimal-control pulses for tunable-coupler CZ gates, target-only echo sequences for trapped-ion Mølmer–Sørensen gates, coupling-matrix engineering that nulls target-neighbor couplings without knowing the leakage level, geometric and analytic single-qubit constructions for always-on-coupled spin systems, and scalable subsystem or tensor-network robust-control frameworks for large parallel circuits (Goldschmidt et al., 16 Mar 2026, Yu et al., 23 Mar 2026, Yang et al., 5 Jan 2026, Le et al., 4 Mar 2026).

1. Conceptual scope and error models

Crosstalk is not a single error mechanism. In fixed-frequency transmons, a standard model is always-on $ZZ$ coupling, with perturbative magnitude

$$\zeta^{(ij)}_{ZZ} = - J^2 \frac{2(\alpha_i + \alpha_j)}{(\alpha_i + \Delta_{ij})(\alpha_j - \Delta_{ij})},$$

which produces state-dependent phase accumulation during nominally single-qubit layers (Goldschmidt et al., 16 Mar 2026). In tunable-coupler superconducting gates, the error model expands to condition-dependent distortions within a gate instance: static or dynamically biased $ZZ$ interactions, classical cross-drive leakage, coupler-mediated parasitic $Z$-shifts, AC Stark shifts, amplitude-dependent detuning biases, narrowband spectral bleed-through, and leakage to non-computational transmon levels (Yu et al., 23 Mar 2026). In trapped ions, the leading coherent crosstalk during an MS gate appears as residual target-spectator entanglers of the form $X^{(i)} \hat{\sigma}_{\phi}^{(j)}$ (Fang et al., 2022). In neutral-atom CZ gates with local addressing, the dominant spectator error is a leakage-induced amplitude error $p(\epsilon)=\alpha \epsilon + O(\epsilon^2)$ produced by a reduced Rabi frequency $\sqrt{\epsilon}\Omega(t)$ on the spectator (Warttmann et al., 14 Jul 2025).

Semiconductor spin-qubit work emphasizes a different regime: off-resonant spectator rotation when the difference in resonance frequencies is not sufficiently large compared to the driving strength, together with always-on exchange and exchange-noise-induced coherent errors (Kanaar et al., 2023, Güngördü et al., 2019). Multimode bosonic processors coupled to a transmon ancilla exhibit ancilla-frequency detuning conditioned on spectator-mode occupation,

$$\Delta_a(\{n_j\})=\sum_j \left[\chi_j n_j + \frac{\chi'_j}{2} n_j(n_j-1)\right],$$

so a pulse calibrated at one ancilla frequency becomes off-resonant in the presence of populated spectator modes (You et al., 2024).

A recurrent misconception is that crosstalk is synonymous with classical pulse spillover. Simultaneous GST distinguishes crosstalk-free, context-dependent, and general two-qubit models, and shows that the relevant signature can be spectator-dependent local maps, residual entangling terms such as $Z\otimes Z$, or correlated SPAM. In the reported AQT and QSCOUT experiments, the context-dependent local model sufficed, while correlated SPAM was not required by the data (Rudinger et al., 2021). This suggests that “crosstalk-robust” does not denote a single mitigation protocol, but a design objective defined relative to a platform-specific error taxonomy.

2. Superconducting implementations

In superconducting circuits, one line of work defines Crosstalk-Robust Gate Sets (CRGS) as co-designed single-qubit pulse libraries that suppress dominant pairwise $ZZ$ errors during parallel layers. The construction exploits locality: the device graph is vertex-colored, each logical gate in the native set $$\zeta^{(ij)}_{ZZ} = - J^2 \frac{2(\alpha_i + \alpha_j)}{(\alpha_i + \Delta_{ij})(\alpha_j - \Delta_{ij})},$$0 is realized by a small number of colored pulse shapes, and simultaneously applied adjacent pulses are made mutually “orthogonal” with respect to the relevant crosstalk susceptibility. On IBM heavy-hex devices, two colors suffice. The synthesis problem is formulated as constrained robust trajectory optimization with fidelity threshold $$\zeta^{(ij)}_{ZZ} = - J^2 \frac{2(\alpha_i + \alpha_j)}{(\alpha_i + \Delta_{ij})(\alpha_j - \Delta_{ij})},$$1, amplitude and curvature bounds, and a susceptibility functional that factorizes over edges (Goldschmidt et al., 16 Mar 2026).

The experimental motivation for CRGS is explicit. On IBM Brisbane, eight-qubit XY4 experiments showed visible oscillations at $$\zeta^{(ij)}_{ZZ} = - J^2 \frac{2(\alpha_i + \alpha_j)}{(\alpha_i + \Delta_{ij})(\alpha_j - \Delta_{ij})},$$2–$$\zeta^{(ij)}_{ZZ} = - J^2 \frac{2(\alpha_i + \alpha_j)}{(\alpha_i + \Delta_{ij})(\alpha_j - \Delta_{ij})},$$3 MHz for non-CRGS gates, while with CRGS those oscillations vanished and the decay rate improved from $$\zeta^{(ij)}_{ZZ} = - J^2 \frac{2(\alpha_i + \alpha_j)}{(\alpha_i + \Delta_{ij})(\alpha_j - \Delta_{ij})},$$4 MHz and $$\zeta^{(ij)}_{ZZ} = - J^2 \frac{2(\alpha_i + \alpha_j)}{(\alpha_i + \Delta_{ij})(\alpha_j - \Delta_{ij})},$$5 MHz to $$\zeta^{(ij)}_{ZZ} = - J^2 \frac{2(\alpha_i + \alpha_j)}{(\alpha_i + \Delta_{ij})(\alpha_j - \Delta_{ij})},$$6 MHz. In simultaneous four-qubit random Clifford identity circuits, the decay parameter improved from $$\zeta^{(ij)}_{ZZ} = - J^2 \frac{2(\alpha_i + \alpha_j)}{(\alpha_i + \Delta_{ij})(\alpha_j - \Delta_{ij})},$$7 to $$\zeta^{(ij)}_{ZZ} = - J^2 \frac{2(\alpha_i + \alpha_j)}{(\alpha_i + \Delta_{ij})(\alpha_j - \Delta_{ij})},$$8, and in a four-qubit TFIM simulation the median KL-divergence improvement over Gaussian pulses was approximately $$\zeta^{(ij)}_{ZZ} = - J^2 \frac{2(\alpha_i + \alpha_j)}{(\alpha_i + \Delta_{ij})(\alpha_j - \Delta_{ij})},$$9 (Goldschmidt et al., 16 Mar 2026). A plausible implication is that, in this setting, crosstalk robustness is not merely a gate-level property but a circuit-layer property tied to scheduling by color classes.

A second superconducting direction uses condition-conditioned optimal control. The Physics-Guided Neural Control framework introduces a normalized condition vector $ZZ$0 that encodes concurrent-drive settings and enters an affine first-order distortion map for $ZZ$1, cross-drive ratio, and detuning biases. Controls are generated by a Fourier-feature MLP with a hardware-aware flattop envelope and amplitude-bounding $ZZ$2 map; training differentiates through a Lindblad solver and aggregates cost over sampled crosstalk conditions. In a tunable-coupler CZ example with $ZZ$3 ns and $ZZ$4 steps, PGNC produced the tightest fidelity distributions and the smallest worst-case degradation under off-grid conditions, while remaining gate-agnostic in the sense that the same pipeline can target other single- or two-qubit operations by replacing $ZZ$5 and the relevant control generators (Yu et al., 23 Mar 2026).

A third approach treats crosstalk suppression and gate speed as a geometric-control problem. In tunable-coupler superconducting two-qubit gates, residual $ZZ$6 decreases as the coupler frequency increases, but the effective entangling rate $ZZ$7 also decreases, producing a speed–crosstalk trade-off. The proposed solution introduces additional parametric degrees of freedom—drive phase $ZZ$8, small detuning $ZZ$9, and trajectory shaping on the $Z$0 Bloch sphere—to realize nonadiabatic geometric $Z$1 gates. Numerically, the optimal trajectory family was found near $Z$2, and under realistic decoherence the best operating point occurred near $Z$3 MHz (Deng et al., 10 Apr 2026). This suggests that geometric path design can serve as an alternative to brute-force spectral isolation.

3. Trapped-ion and neutral-atom constructions

In trapped ions, one major result is that coherent crosstalk during individually addressed MS gates can be canceled by target-only echoing. If the unwanted Hamiltonian contains bilinear terms $Z$4, a midpoint $Z$5 echo flips $Z$6 while leaving the desired $Z$7 interaction invariant, so the first-order Magnus term for crosstalk vanishes. This yields two practical constructions: a collective-gate echo that wraps blocks of $Z$8 gates, and a per-gate embedded echo $Z$9 (Fang et al., 2022).

The experimental performance is precise. In a 5-ion chain, target-only local suppression achieved Bell-state fidelity per gate $X^{(i)} \hat{\sigma}_{\phi}^{(j)}$0 for collective-gate echo and $X^{(i)} \hat{\sigma}_{\phi}^{(j)}$1 for per-gate echo. Spectator population after 21 consecutive gates dropped from $X^{(i)} \hat{\sigma}_{\phi}^{(j)}$2 without suppression to $X^{(i)} \hat{\sigma}_{\phi}^{(j)}$3 with suppression. After crosstalk cancellation, residual error was dominated by motional dephasing rather than coherent target-spectator coupling (Fang et al., 2022). The significance is methodological: the robust gate set is compiler-friendly because only target qubits are pulsed for mitigation.

A more structural ion-trap strategy is coupling-matrix engineering. Here the effective XX coupling matrix $X^{(i)} \hat{\sigma}_{\phi}^{(j)}$4 is synthesized so that $X^{(i)} \hat{\sigma}_{\phi}^{(j)}$5 for all target-neighbor pairs while $X^{(i)} \hat{\sigma}_{\phi}^{(j)}$6 remains large. Writing the crosstalk vectors as columns of $X^{(i)} \hat{\sigma}_{\phi}^{(j)}$7, the design condition is

$X^{(i)} \hat{\sigma}_{\phi}^{(j)}$8

Because $X^{(i)} \hat{\sigma}_{\phi}^{(j)}$9, enforcing $p(\epsilon)=\alpha \epsilon + O(\epsilon^2)$0 makes the gate insensitive to the unknown leakage level $p(\epsilon)=\alpha \epsilon + O(\epsilon^2)$1. The paper gives both a linearized loop-superposition synthesis and a full quadratic AM segmented optimization, and demonstrates in a QSCOUT three-ion experiment that the engineered composite gate remained insensitive to target-neighbor parity up to $p(\epsilon)=\alpha \epsilon + O(\epsilon^2)$2 (Kashyap et al., 28 Aug 2025).

For Rydberg CZ gates with local addressing, the robust construction is a midpoint phase-hop echo. Splitting a shaped single-pulse CZ into two symmetric halves and applying a phase jump $p(\epsilon)=\alpha \epsilon + O(\epsilon^2)$3 cancels the first-order spectator excitation amplitude, promoting the leading spectator error from $p(\epsilon)=\alpha \epsilon + O(\epsilon^2)$4 to $p(\epsilon)=\alpha \epsilon + O(\epsilon^2)$5. For the shaped pulses studied, the worst-case spectator infidelity obeyed $p(\epsilon)=\alpha \epsilon + O(\epsilon^2)$6, and numerical simulations showed approximately two orders of magnitude fidelity improvement over the single-pulse baseline across a broad parameter range (Warttmann et al., 14 Jul 2025). The same work notes a non-robust window near $p(\epsilon)=\alpha \epsilon + O(\epsilon^2)$7, which directly limits the geometry and spectator layout compatible with the robust gate set.

4. Semiconductor spin-qubit and fixed-coupling approaches

One analytic route to crosstalk-robust single-qubit gate sets in always-on-coupled systems maps the driven dynamics to curves on a sphere. For a driven block Hamiltonian $p(\epsilon)=\alpha \epsilon + O(\epsilon^2)$8, the pulse waveform is the geodesic curvature of a trajectory $p(\epsilon)=\alpha \epsilon + O(\epsilon^2)$9, and crosstalk suppression is obtained when the trajectory is a closed loop with zero enclosed area,

$\sqrt{\epsilon}\Omega(t)$0

This guarantees that the same $\sqrt{\epsilon}\Omega(t)$1 is implemented for $\sqrt{\epsilon}\Omega(t)$2 without assuming small couplings. The resulting practical gate set consists of robust $\sqrt{\epsilon}\Omega(t)$3 pulses and virtual $\sqrt{\epsilon}\Omega(t)$4 rotations (Zeng et al., 16 Mar 2025).

A closely related fixed-Ising program exploits two-color scheduling and a decomposition into commuting $\sqrt{\epsilon}\Omega(t)$5 blocks. Driving only one color class at a time makes the sitewise Hamiltonians commute, and each driven site further decomposes into $\sqrt{\epsilon}\Omega(t)$6 commuting blocks labeled by neighbor-$\sqrt{\epsilon}\Omega(t)$7-parity. Shaped pulses are then optimized simultaneously across all distinct blocks for $\sqrt{\epsilon}\Omega(t)$8-rotations, $\sqrt{\epsilon}\Omega(t)$9, and shaped identity operations. The reported $$\Delta_a(\{n_j\})=\sum_j \left[\chi_j n_j + \frac{\chi'_j}{2} n_j(n_j-1)\right],$$0-robust pulses achieved trace infidelity $$\Delta_a(\{n_j\})=\sum_j \left[\chi_j n_j + \frac{\chi'_j}{2} n_j(n_j-1)\right],$$1 for identity, $$\Delta_a(\{n_j\})=\sum_j \left[\chi_j n_j + \frac{\chi'_j}{2} n_j(n_j-1)\right],$$2, and $$\Delta_a(\{n_j\})=\sum_j \left[\chi_j n_j + \frac{\chi'_j}{2} n_j(n_j-1)\right],$$3, while leakage-robust transmon $$\Delta_a(\{n_j\})=\sum_j \left[\chi_j n_j + \frac{\chi'_j}{2} n_j(n_j-1)\right],$$4 pulses achieved infidelity $$\Delta_a(\{n_j\})=\sum_j \left[\chi_j n_j + \frac{\chi'_j}{2} n_j(n_j-1)\right],$$5 at $$\Delta_a(\{n_j\})=\sum_j \left[\chi_j n_j + \frac{\chi'_j}{2} n_j(n_j-1)\right],$$6 MHz, $$\Delta_a(\{n_j\})=\sum_j \left[\chi_j n_j + \frac{\chi'_j}{2} n_j(n_j-1)\right],$$7 MHz, and $$\Delta_a(\{n_j\})=\sum_j \left[\chi_j n_j + \frac{\chi'_j}{2} n_j(n_j-1)\right],$$8 MHz (Kanaar et al., 2023). In this literature, a crosstalk-robust gate set is explicitly universal: single-qubit rotations plus a nontrivial $$\Delta_a(\{n_j\})=\sum_j \left[\chi_j n_j + \frac{\chi'_j}{2} n_j(n_j-1)\right],$$9 entangler, with virtual $Z\otimes Z$0 gates completing the compilation basis.

Silicon spin-qubit work extends this program to nonperturbative pulse synthesis under appreciable off-resonant spectator drive. A physics-informed neural network was used to generate robust $Z\otimes Z$1, $Z\otimes Z$2, and iToffoli-equivalent pulses in a three-qubit Heisenberg model with crosstalk and charge noise. The robust pulses maintained infidelity $Z\otimes Z$3 for average quasistatic voltage fluctuations up to a few mV instead of tenths of mV for non-robust pulses; specifically, the robust $Z\otimes Z$4 remained below $Z\otimes Z$5 up to $Z\otimes Z$6 mV, while the robust iToffoli remained below $Z\otimes Z$7 up to $Z\otimes Z$8 mV average fluctuations (Kanaar et al., 2023). A related neural-network optimization for silicon CZ gates showed that, even when simple analytical control fields were ineffective because of appreciable crosstalk, optimization could maintain order-of-magnitude improvement over a simple cosine pulse (Kanaar et al., 2022).

Another silicon route treats crosstalk compensation as a synchronization problem. In linear spin-qubit arrays, the off-resonant spectator Rabi rate $Z\otimes Z$9 is synchronized so that $ZZ$0, forcing the spectator to undergo a full rotation and return to its initial state. Under realistic parameters, the paper identifies nearly crosstalk-free CNOT operation at $ZZ$1 MHz, $ZZ$2 MHz, and $ZZ$3 ns (Heinz et al., 2021). This suggests that some “gate-set” constructions are best understood as synchronized parameter manifolds rather than as fixed pulse templates.

5. Scalable robust control beyond few-qubit models

For large parallel circuits, one strategy is to reduce the full-system problem to constant-sized subsystem pairs. If $ZZ$4 in the absence of crosstalk, then the first-order deviation caused by $ZZ$5 factorizes, and robustness is achieved by enforcing

$ZZ$6

This subsystem-optimized robust-control framework produced analytic pulses for parallel single-qubit $ZZ$7 gates and numerical pulses for parallel multi-qubit gates, with validation on NV centers, a 12-spin NMR processor, and superconducting arrays up to 200 qubits. The reported scaling reduced noise growth from exponential to linear for parallel single-qubit gates and achieved an order-of-magnitude reduction for parallel multi-qubit gates (Yang et al., 5 Jan 2026).

A complementary many-body strategy uses tensor-network robust control. Here the full Hamiltonian on a one-dimensional chain includes local $ZZ$8 drives, tunable nearest-neighbor $ZZ$9 couplings, and unknown static Heisenberg-type parasitic couplings

$$\zeta^{(ij)}_{ZZ} = - J^2 \frac{2(\alpha_i + \alpha_j)}{(\alpha_i + \Delta_{ij})(\alpha_j - \Delta_{ij})},$$00

with $$\zeta^{(ij)}_{ZZ} = - J^2 \frac{2(\alpha_i + \alpha_j)}{(\alpha_i + \Delta_{ij})(\alpha_j - \Delta_{ij})},$$01. MPO/MPS simulations combined with GRAPE and ensemble averaging produced robust parallel $$\zeta^{(ij)}_{ZZ} = - J^2 \frac{2(\alpha_i + \alpha_j)}{(\alpha_i + \Delta_{ij})(\alpha_j - \Delta_{ij})},$$02 and parallel CNOT gates on chains of 50 qubits. At $$\zeta^{(ij)}_{ZZ} = - J^2 \frac{2(\alpha_i + \alpha_j)}{(\alpha_i + \Delta_{ij})(\alpha_j - \Delta_{ij})},$$03 crosstalk, the reported many-body infidelities were $$\zeta^{(ij)}_{ZZ} = - J^2 \frac{2(\alpha_i + \alpha_j)}{(\alpha_i + \Delta_{ij})(\alpha_j - \Delta_{ij})},$$04 for parallel $$\zeta^{(ij)}_{ZZ} = - J^2 \frac{2(\alpha_i + \alpha_j)}{(\alpha_i + \Delta_{ij})(\alpha_j - \Delta_{ij})},$$05 and $$\zeta^{(ij)}_{ZZ} = - J^2 \frac{2(\alpha_i + \alpha_j)}{(\alpha_i + \Delta_{ij})(\alpha_j - \Delta_{ij})},$$06 for parallel CNOT, while non-robust controls approached $$\zeta^{(ij)}_{ZZ} = - J^2 \frac{2(\alpha_i + \alpha_j)}{(\alpha_i + \Delta_{ij})(\alpha_j - \Delta_{ij})},$$07 at $$\zeta^{(ij)}_{ZZ} = - J^2 \frac{2(\alpha_i + \alpha_j)}{(\alpha_i + \Delta_{ij})(\alpha_j - \Delta_{ij})},$$08 (Le et al., 4 Mar 2026). This work extends the gate-set idea from local spectator suppression to explicit many-body robustification.

Bosonic processors provide a distinct large-Hilbert-space example. In multimode cavity processors, QOC-designed ancilla pulses robust to detuning $$\zeta^{(ij)}_{ZZ} = - J^2 \frac{2(\alpha_i + \alpha_j)}{(\alpha_i + \Delta_{ij})(\alpha_j - \Delta_{ij})},$$09 MHz are embedded into the echoed conditional displacement protocol. In a representative Fock-state preparation task with spectator mode occupancy $$\zeta^{(ij)}_{ZZ} = - J^2 \frac{2(\alpha_i + \alpha_j)}{(\alpha_i + \Delta_{ij})(\alpha_j - \Delta_{ij})},$$10, DRAG yielded infidelity $$\zeta^{(ij)}_{ZZ} = - J^2 \frac{2(\alpha_i + \alpha_j)}{(\alpha_i + \Delta_{ij})(\alpha_j - \Delta_{ij})},$$11 for $$\zeta^{(ij)}_{ZZ} = - J^2 \frac{2(\alpha_i + \alpha_j)}{(\alpha_i + \Delta_{ij})(\alpha_j - \Delta_{ij})},$$12, whereas QOC reduced it below $$\zeta^{(ij)}_{ZZ} = - J^2 \frac{2(\alpha_i + \alpha_j)}{(\alpha_i + \Delta_{ij})(\alpha_j - \Delta_{ij})},$$13. For two-mode Bell-cat generation, the robust ancilla pulses reduced infidelity by factors $$\zeta^{(ij)}_{ZZ} = - J^2 \frac{2(\alpha_i + \alpha_j)}{(\alpha_i + \Delta_{ij})(\alpha_j - \Delta_{ij})},$$14 to $$\zeta^{(ij)}_{ZZ} = - J^2 \frac{2(\alpha_i + \alpha_j)}{(\alpha_i + \Delta_{ij})(\alpha_j - \Delta_{ij})},$$15, depending on pulse duration and decoherence model (You et al., 2024). In this context, a crosstalk-robust gate set is a hybrid library of robust ancilla $$\zeta^{(ij)}_{ZZ} = - J^2 \frac{2(\alpha_i + \alpha_j)}{(\alpha_i + \Delta_{ij})(\alpha_j - \Delta_{ij})},$$16, $$\zeta^{(ij)}_{ZZ} = - J^2 \frac{2(\alpha_i + \alpha_j)}{(\alpha_i + \Delta_{ij})(\alpha_j - \Delta_{ij})},$$17, virtual $$\zeta^{(ij)}_{ZZ} = - J^2 \frac{2(\alpha_i + \alpha_j)}{(\alpha_i + \Delta_{ij})(\alpha_j - \Delta_{ij})},$$18, ECD primitives, and virtual cavity-phase corrections.

6. Characterization, certification, and compilation

Robust gate sets require characterization methods that resolve context dependence rather than averaging it away. Simultaneous GST does this by fitting crosstalk-free, context-dependent, and general two-qubit models to the same parallel-gate data and comparing them by log-likelihood ratios and evidence ratios. In the AQT transmon experiment, spectator compensation reduced the average diamond-distance estimate from $$\zeta^{(ij)}_{ZZ} = - J^2 \frac{2(\alpha_i + \alpha_j)}{(\alpha_i + \Delta_{ij})(\alpha_j - \Delta_{ij})},$$19 to $$\zeta^{(ij)}_{ZZ} = - J^2 \frac{2(\alpha_i + \alpha_j)}{(\alpha_i + \Delta_{ij})(\alpha_j - \Delta_{ij})},$$20, and the context-dependent model remained sufficient relative to the general model. In QSCOUT, large context-dependent equatorial errors were identified, including rotation-angle variation up to $$\zeta^{(ij)}_{ZZ} = - J^2 \frac{2(\alpha_i + \alpha_j)}{(\alpha_i + \Delta_{ij})(\alpha_j - \Delta_{ij})},$$21 mrad on one active gate, again without requiring a fully entangling model (Rudinger et al., 2021). These results are important because they show that robust gate-set design is inseparable from model discrimination.

Randomized mirror circuits supply a scalable complementary diagnostic. They benchmark universal and continuously parameterized gate sets by fitting $$\zeta^{(ij)}_{ZZ} = - J^2 \frac{2(\alpha_i + \alpha_j)}{(\alpha_i + \Delta_{ij})(\alpha_j - \Delta_{ij})},$$22 and converting the decay constant to an average error per layer. On a 27-qubit IBM processor, the measured 27-qubit layer error was $$\zeta^{(ij)}_{ZZ} = - J^2 \frac{2(\alpha_i + \alpha_j)}{(\alpha_i + \Delta_{ij})(\alpha_j - \Delta_{ij})},$$23, and comparison to crosstalk-free predictions from 1- and 2-qubit data led to the conclusion that crosstalk contributes approximately $$\zeta^{(ij)}_{ZZ} = - J^2 \frac{2(\alpha_i + \alpha_j)}{(\alpha_i + \Delta_{ij})(\alpha_j - \Delta_{ij})},$$24 of the total error per gate in random many-qubit circuit layers (Hines et al., 2022). This provides a circuit-scale criterion for whether a purportedly robust gate set remains robust when embedded into large random layers.

Compilation can also be part of the gate-set construction. CAMEL pre-calibrates local $$\zeta^{(ij)}_{ZZ} = - J^2 \frac{2(\alpha_i + \alpha_j)}{(\alpha_i + \Delta_{ij})(\alpha_j - \Delta_{ij})},$$25 windows on frequency-tunable superconducting chips, uses compensation pulses on spectator couplers to minimize the in-gate swap channel $$\zeta^{(ij)}_{ZZ} = - J^2 \frac{2(\alpha_i + \alpha_j)}{(\alpha_i + \Delta_{ij})(\alpha_j - \Delta_{ij})},$$26, and then chooses the largest crosstalk-free parallel set in each layer by a maximum-independent-set computation on the crosstalk graph. A $$\zeta^{(ij)}_{ZZ} = - J^2 \frac{2(\alpha_i + \alpha_j)}{(\alpha_i + \Delta_{ij})(\alpha_j - \Delta_{ij})},$$27 window is reported as a practical sweet spot because it already outperforms static-frequency-aware compilation in XEB while keeping calibration cost feasible (Lu et al., 2023). This suggests that, in some architectures, “crosstalk-robust gate set” is partly a property of the compiler’s admissible parallel bundles.

Across these literatures, the main unresolved issues are consistent. First-order or pairwise models often dominate current experiments, but higher-order terms, long-range couplings, drift, and model mismatch remain open. Many methods assume locality, bounded spectator activity, or moderate $$\zeta^{(ij)}_{ZZ} = - J^2 \frac{2(\alpha_i + \alpha_j)}{(\alpha_i + \Delta_{ij})(\alpha_j - \Delta_{ij})},$$28, and several explicitly note the need for periodic re-identification or hardware-in-the-loop recalibration (Yu et al., 23 Mar 2026, Yang et al., 5 Jan 2026, Le et al., 4 Mar 2026). A plausible implication is that the long-term form of crosstalk-robust gate sets will be hybrid: analytically constrained where symmetry permits, numerically conditioned where hardware is nonlinear, and compiler-aware where parallel scheduling changes the operating point.