Experimental Realization of a C-Shunt Flux-Qubit Coupler for Controlled-Z Gates between Transmons

AI summary

Overview

Research area: Superconducting quantum computing — specifically the hardware engineering of tunable couplers that mediate two-qubit entangling gates between transmon qubits.

Technical level: Intermediate. The paper assumes familiarity with transmon qubits, anharmonicity, ZZ crosstalk, and randomized benchmarking, but its central argument (a coupler with a positive anharmonicity spectrum gives extra design freedom) is accessible to anyone who knows the basics of circuit QED.

Scope (one sentence): The authors build and benchmark a two-transmon device in which a three-junction capacitively shunted (C-shunt) flux-qubit coupler sits between the data qubits, characterizing a low-interaction idle bias, a 45 ns nonadiabatic controlled-Z gate, the error budget, and the prospects for scaling the architecture.

What This Paper Is About

Superconducting processors need two contradictory things from the same pair of qubits: a strong, switchable interaction when an entangling gate is running, and essentially zero interaction when the qubits are idling, since leftover ZZ crosstalk corrupts whatever else is happening on the chip. Most tunable couplers are themselves transmons, whose nearly harmonic energy spectrum leaves little room to satisfy both requirements at once. This paper tests a different coupler element — a capacitively shunted flux qubit, which has a flux-tunable spectrum with large positive anharmonicity and needs no Josephson-junction-array superinductor — placed between two ordinary transmon data qubits, and asks whether it can deliver both a quiet idle point and a fast CZ gate.

Key Contributions

1. First experimental realization of a transmon–C-shunt–transmon (TCT) device. A three-junction C-shunt flux-qubit coupler, with a large shunt capacitance and a flux-tunable loop, capacitively couples two transmon data qubits (qubit-coupler couplings g₁c/2π ≃ 101 MHz and g₂c/2π ≃ 105 MHz, plus a residual direct qubit-qubit capacitance giving g₁₂/2π ≃ 7.5 MHz).

2. Demonstration of a genuinely low-interaction idle bias. Conditional Ramsey measurements show the residual ZZ interaction changing sign and being tunable from roughly −5.2 MHz to 10.5 MHz, with a zero crossing near Φ_ext/Φ₀ ≃ 0.46; swap oscillations confirm that the transverse exchange is strongly suppressed at that same bias.

3. A 45 ns nonadiabatic CZ gate with interleaved randomized benchmarking fidelity 99.23(15)%, obtained with simultaneous single-qubit X-gate fidelities above 99.7% that are statistically consistent with individual operation.

4. A quantified error budget plus a scalability analysis. Lindblad simulations attribute a significant part of the CZ error to the coupler's short energy-relaxation time, and numerical modeling of a 3 × 3 nearest-neighbor array and a six-qubit chain shows how the coupler's positive anharmonicity moves a two-excitation spectral collision away from the gate-control band and how gate-induced ZZ changes decay with distance along a chain.

Main Findings

• Device parameters: Data qubit transition frequencies f₁₀ = 5.334 GHz (Qubit 1) and 5.43 GHz (Qubit 2), coupler f₁₀ = 4.03 GHz; anharmonicities −256 MHz and −254 MHz for the data qubits versus α_c/2π ≃ 703 MHz (positive) for the coupler; energy-relaxation times T₁ = 17.50 μs, 25.17 μs, and 1.1 μs respectively. A fit to the measured spectra puts the minimum coupler frequency near 4.00 GHz.

• Tunable ZZ with a usable zero: The extracted ζ_ZZ/2π spans approximately −5.2 MHz to 10.5 MHz across coupler flux, crossing zero near Φ_ext/Φ₀ ≃ 0.46, which is adopted as the idle point where direct and coupler-mediated contributions to the conditional frequency shift cancel.

• Single-qubit gates are essentially crosstalk-free at idle: Simultaneous X-gate fidelities are 99.711(32)% (Qubit 1) and 99.826(56)% (Qubit 2); individual values are 99.759(26)% and 99.836(28)%. The simultaneous errors are larger by only 0.048 and 0.010 percentage points, neither resolved within statistical uncertainty.

• CZ gate performance: Reference and interleaved RB fits give p_ref = 0.9570(11) and p_int = 0.9472(16), hence p_int/p_ref = 0.9897(21), r_CZ = 0.771(154)%, and F_CZ = 99.23(15)%.

• Coupler relaxation is a leading CZ error source: With data-qubit lifetimes fixed to measured values and a fixed optimized 45 ns pulse, relaxation-only simulations raise simulated CZ fidelity from 99.5131(43)% at T₁,c = 0.5 μs to 99.7984(17)% at 5 μs; values at 1.074 μs and 2 μs are 99.6822(26)% and 99.7507(21)%. At T₁,c = 1.074 μs the simulated CZ error is 0.3178(26)%, well below the measured 0.771(154)%, leaving room for control imperfections on top of relaxation.

• Single-qubit errors are only partly relaxation-limited: Using r ≃ t/(3T₁) for a 20 ns rotation gives relaxation errors of 0.038% and 0.026%, about 13% and 15% of the measured simultaneous gate errors of 0.289(32)% and 0.174(56)%.

• Positive anharmonicity relieves a frequency-collision constraint: The bare-state |020⟩–|101⟩ collision sits at ω_c^col = ω̄_q − α_c/2, so it is pushed downward. For the device-model value α_c/2π = 0.703 GHz the clearance from the 5.55–6.10 GHz gate-control band is at least 431.5 MHz (580 MHz for an assumed 1 GHz anharmonicity), whereas transmon-coupler reference values of −0.370 GHz (Xu et al.) and −0.140 GHz (Stehlik et al.) place the collision inside or only 10 MHz below the band.

• Spectator ZZ changes are localized: In a six-qubit chain (Q1–Q2–Q3–Q6–Q5–Q4) with only C₁₂ pulsed from 6.630 GHz to 5.860 GHz, |Δζ_ZZ|/2π is 3.97 MHz on the target link, 9.18 kHz on the adjacent link (0.231% of the target), 20.3 Hz at distance 2, and below 1 Hz at distances 3 and 4.

Methodology in Plain English

The team fabricated a three-element superconducting circuit: two standard transmon qubits with a small direct capacitance between them, and a central coupler made from a three-junction flux-qubit loop with a large shunting capacitor. The shunt capacitance suppresses charge noise while the loop keeps the coupler's frequency and anharmonicity tunable with an external magnetic flux, so the coupler's spectrum can be moved during an experiment.

They first mapped the device with pulsed spectroscopy, fitting the observed avoided crossings to a three-mode circuit model to extract frequencies, anharmonicities, and couplings. To find the idle point they used a conditional Ramsey experiment: apply a flux pulse to the coupler, drive Ramsey fringes on one qubit while preparing the other in |0⟩ or |1⟩, and read the conditional phase difference to get ζ_ZZ. To measure the usable interaction strength they prepared a single excitation on one qubit and watched swap oscillations as a function of interaction time, extracting |g_eff|. Both the theoretical picture and the numerics treat the direct qubit-qubit exchange and the coupler-mediated virtual exchange as two paths that can cancel.

Single-qubit gates were 20 ns Gaussian microwave pulses with DRAG quadrature correction, calibrated for amplitude, detuning, quadrature, and virtual-Z, and benchmarked with simultaneous and individual interleaved randomized benchmarking. The CZ gate was implemented as a fast, nonadiabatic flux excursion of the coupler away from the idle bias and back: the pulse is tuned so that the |11⟩ state transiently hybridizes with non-computational states, picks up a conditional phase of π, and returns to the computational subspace, after which virtual-Z rotations remove single-qubit phases. Gate fidelity came from interleaved RB with the standard d = 4 prefactor of 3/4.

For the error analysis they solved a Lindblad master equation on a three-mode Hamiltonian with four levels per mode, including counter-rotating terms, varying the coupler T₁ with the pulse held fixed, then ran simulated reference and interleaved RB with leakage handled in an absorbing-loss limit. Finally, they ran numerical studies of a hypothetical 3 × 3 array and a six-qubit chain to check where the |020⟩–|101⟩ collision falls relative to the gate-control band and how much a gate on one link disturbs its neighbors.

Why This Matters

Impact on research. Nearly harmonic transmon couplers force a compromise between exchange cancellation (needed for a quiet idle) and low ZZ (needed for accurate simultaneous operation), and they constrain frequency allocation because the coupler's own two-excitation states can collide with the gate band. Demonstrating that a C-shunt flux-qubit coupler supplies a positive, flux-tunable anharmonicity in a device that still uses ordinary transmons as data qubits gives the field a new design axis without changing the control and readout stack that transmon processors already rely on.

Real-world applications (each follows from the demonstrated capabilities):

• Fixed-frequency transmon processors with tunable couplers, where idle-time ZZ crosstalk is the dominant limit on parallel circuit execution.
• Modular or lattice architectures, where the six-qubit chain simulation's result — gate-induced ZZ falling from 3.97 MHz on the target link to 9.18 kHz next door — indicates that a gate on one link need not disturb distant qubits.
• Frequency-allocation planning for larger chips, since a positive coupler anharmonicity relaxes the constraint that forces careful placement of coupler modes relative to data-qubit transitions.
• Cryogenic control hardware design, since the reported setup uses RFSoC-based direct digital synthesis in the second Nyquist zone, removing the conventional IQ-mixing chain on the XY lines.

Industry relevance. The paper's error budget is directly actionable for foundry-style process development: the dominant identified CZ limitation is the coupler's 1.1 μs energy-relaxation time, and the simulations show that improving it to 5 μs would push simulated CZ fidelity to almost 99.8% with everything else unchanged — a fabrication target rather than a control-software target.

Future Directions

• Improve coupler coherence. The measured coupler T₁ of roughly 1.1 μs is far below the data-qubit values of 17.50 μs and 25.17 μs; design and fabrication changes aimed at raising it are the most direct route the authors identify to higher CZ fidelity.

• Close the gap between simulated and measured CZ error. The simulated relaxation-only error at the measured coupler lifetime (0.3178(26)%) is well below the measured 0.771(154)%, so leakage, residual conditional-phase error, and flux-pulse distortion remain unaccounted for and need targeted calibration.

• Measure relaxation at the gate bias. The authors note that gate-bias relaxation measurements, leakage diagnostics, and refined pulse calibration are the practical tools needed to verify where the remaining error actually lives.

• Validate the scalability predictions on a larger array. The 3 × 3 lattice and six-qubit chain results are numerical; the frequency-allocation strategy and the spectator-interaction suppression still need experimental confirmation in multi-qubit TCT networks, where parameter spread will matter.

Target Audience

Experimental superconducting-qubit researchers and hardware engineers who work on coupler design, gate calibration, or frequency allocation for multi-qubit chips. It is also well suited to graduate students already comfortable with transmon physics and randomized benchmarking who want a concrete worked example of how a new coupler element is characterized from spectroscopy through gate benchmarking to a numerical error budget. Readers without a circuit-QED background will find the device and measurement descriptions intelligible but will need supplementary reading for the perturbative ZZ analysis and the Lindblad modeling.

Authors’ abstract

Fast entangling gates and low-interaction idling require distinct operating conditions in superconducting quantum processors. We experimentally realize a transmon-C-shunt-transmon (TCT) device in which a flux-tunable, three-junction capacitively shunted flux-qubit coupler connects two transmon data qubits. The C-shunt coupler provides a flux-tunable, positively anharmonic spectrum for tailoring the interactions between the data qubits. Conditional Ramsey and swap measurements identify an idle region with near-zero residual ZZ interaction strength and suppressed transverse exchange. At this bias, simultaneous single-qubit interleaved randomized benchmarking yields X-gate fldelities above 99.7%, consistent with individual operation. A 45 ns nonadiabatic controlled-Z (CZ) gate achieves an interleaved randomized benchmarking fldelity estimate of 99.23(15)%. Numerical simulations identify coupler energy relaxation as an important limitation on CZ-gate fldelity and indicate that extending the coupler lifetime can improve gate performance. These results demonstrate the experimental feasibility of C-shunt flux-qubit couplers for combining low-interaction idling with fast entangling operations between transmon qubits.