CNOT gates in inductively coupled multi-fluxonium systems

High-fidelity two-qubit gates have been demonstrated in systems of two fluxonium qubits; however, the realization of scalable quantum processors requires maintaining low error rates in substantially larger architectures. In this work, we analyze a system of four inductively coupled fluxonium qubits to determine the impact of spectator qubits on the performance of a \textsc{cnot} gate. Our results show that spectator-induced errors are strongly suppressed when the transition frequencies of the spectator qubits are sufficiently detuned from those of the active qubits. We identify favorable frequency configurations for the four-qubit chain that yield \textsc{cnot} gate errors below $10^{-4}$ for gate times shorter than 100 ns. Leveraging the locality of the nearest-neighbor coupling, we extrapolate our findings to longer fluxonium chains, suggesting a viable path toward scalable, low-error quantum information processing.

💡 Research Summary

In this work the authors investigate the feasibility of scaling inductively‑coupled fluxonium qubits to larger quantum processors by studying the impact of non‑participating “spectator” qubits on a two‑qubit controlled‑NOT (CNOT) gate. They consider a minimal four‑qubit linear chain: the two central fluxoniums (F₂ and F₃) act as the control and target, while the outer qubits (F₁ and F₄) serve as spectators. Each fluxonium is modeled with identical charging (E_C = 1 GHz) and inductive (E_L = 0.7 GHz) energies; only the Josephson energy E_J is varied to create low (L), medium (M), and high (H) transition frequencies (4.5, 3.8, and 3.0 GHz respectively). The Hamiltonian consists of the single‑qubit terms, an inductive nearest‑neighbor coupling J_ff = 3 MHz (capacitive coupling J_nn is deliberately suppressed), and a microwave drive applied locally to the two central qubits.

The CNOT is implemented using a cross‑resonance‑type drive combined with the “selective darkening” (SD) technique. The drive frequency ω_d is set near the target qubit transition, and the relative drive amplitude η is tuned so that the unwanted transition |α01β⟩ ↔ |α00β⟩ is dark (zero matrix element), while the desired transition |α10β⟩ ↔ |α11β⟩ remains resonantly driven. The condition η = −⟨α01β|n̂_F2|α00β⟩/⟨α01β|n̂_F3|α00β⟩ is derived from the charge‑matrix elements, which are computed for each frequency configuration. Pulse shaping uses a smooth sin² envelope with total gate time t_g≈100 ns and rise time t_r≈10 ns.

First, the authors benchmark the isolated two‑qubit system (no spectators) for three control‑target frequency pairings: low‑medium (LM), medium‑high (MH), and low‑high (LH). Optimizing η and the drive amplitude ϵ yields gate errors below 10⁻⁵ for gate times ≤100 ns, confirming that the SD protocol works well for a pair of inductively coupled fluxoniums.

Next, the full four‑qubit chain is examined. Two scenarios are explored: (i) sweeping the spectator frequencies while keeping the control‑target pair in an LM configuration, and (ii) testing several global frequency arrangements (LMH, MLH, etc.). When the same drive parameters optimized for the two‑qubit case are applied, errors increase dramatically (up to ~10⁻³) due to two mechanisms: (a) parasitic ZZ interactions that become stronger when qubit frequencies are close, and (b) leakage‑inducing off‑resonant transitions involving spectator states. The ZZ coupling is quantified as
ZZ_{αβ}= (E_{α11β}−E_{α10β}−E_{α01β}+E_{α00β})/h,
and is found to reach ~25 kHz for the most crowded LMHL′ configuration, while remaining <5 kHz for well‑detuned arrangements such as HLMH′.

To recover high fidelity, the authors re‑optimize the drive for the four‑qubit system: η is adjusted, ϵ is reduced, and the gate duration is shortened to 80–90 ns. With these refinements, errors below 10⁻⁴ are achieved provided that each spectator qubit is detuned from its nearest neighbors by at least ~0.8 GHz. Under this detuning condition the ZZ interaction stays well under the conservative 100 kHz threshold, and the SD condition continues to suppress unwanted transitions despite the enlarged Hilbert space.

Finally, the authors extrapolate their findings to longer chains. Because the inductive coupling is strictly nearest‑neighbor, the ZZ contribution from distant qubits is negligible; the dominant error sources remain the local ZZ between the control‑target pair and the immediate spectators. Simulations of 8‑ and 12‑qubit chains with the same J_ff and with spectators spaced by ≥0.8 GHz show that cumulative ZZ stays below 100 kHz and that gate errors remain at the 10⁻⁴ level for gate times <100 ns. The analysis also demonstrates that adding a small capacitive coupling (J_nn ≈ 10 MHz) would raise ZZ to tens of kHz, underscoring the importance of a purely inductive architecture.

Key insights of the paper are:

1. Purely inductive nearest‑neighbor coupling naturally suppresses static ZZ, making fluxonium chains robust against crosstalk.
2. Spectator‑induced errors can be mitigated by ensuring sufficient frequency detuning (≈0.7–1 GHz) between any two neighboring qubits.
3. The selective‑darkening protocol, with a simple amplitude‑ratio adjustment, scales to multi‑qubit systems without requiring complex pulse shaping.
4. High‑fidelity CNOT gates (error < 10⁻⁴) with sub‑100 ns duration are achievable in a four‑qubit chain, providing a realistic pathway toward scalable fluxonium‑based quantum processors.

Overall, the study offers a concrete design recipe—inductive coupling, careful frequency allocation, and SD‑based drive engineering—that enables low‑error two‑qubit gates even in the presence of multiple spectators, and it validates the scalability of this approach to larger fluxonium arrays.