Quantum interference device for controlled two-qubit operations

The paper addresses how to realize high-fidelity, programmable entangling operations essential for universal quantum computing, where fault-tolerant thresholds motivate two-qubit gate fidelities above ~0.99. While single-qubit gates routinely exceed this threshold, achieving similar performance for two-qubit gates remains challenging across platforms. Prior work has demonstrated high-fidelity two-qubit gates in trapped ions and in superconducting architectures (e.g., CZ and cross-resonance), with other gates approaching but often below 0.99. The authors propose and analyze a four-qubit quantum interference device—the diamond gate—in which two control qubits determine which two-qubit operation is enacted on two target qubits. Implemented natively in a diamond-shaped network of capacitively coupled superconducting transmons, the device realizes four distinct controlled two-qubit operations (two entangling swap/phase gates, a parity-selective phase gate, and an idle identity). Analytical treatment via Floquet/Magnus expansion and numerical Lindblad simulations indicate the gate can achieve ~0.99 fidelity in under 100 ns with state-of-the-art parameters. The study also examines leakage to higher transmon levels and presents a passive mitigation strategy via engineered crosstalk.

The introduction surveys two-qubit gate performance across platforms. In trapped ions, Mølmer–Sørensen-type gates achieved fidelities ≥0.993 and have since improved. Silicon quantum dots have demonstrated controlled-rotation gates with ~0.98 fidelity. In superconducting qubits, CZ and cross-resonance gates have surpassed 0.99 fidelity, while iSWAP/νiSWAP, bSWAP, RIP, and parametric CZ gates typically yield fidelities in the 0.9 range. Architectures include fixed/frequency-tunable transmons coupled directly or via resonators/tunable couplers. Related theoretical works consider 2D transmon arrays and spin networks with exchange interactions, though with different coupling schemes or encodings. This context motivates seeking native multi-qubit operations that can simplify compilation and enhance connectivity.

The system comprises four qubits: two targets (T1, T2) and two controls (C1, C2) in a diamond geometry with exchange-type couplings. In the qubit model, the Hamiltonian has a noninteracting term setting target/control frequencies Ω±Δ and interaction terms Jc (control-control) and J (cross couplings between target and control qubits). Moving to a rotating frame and assuming |Ω| ≫ |J|,|Jc| (RWA), the effective time-dependent Hamiltonian contains static Jc terms and drive-like terms oscillating at ±Δ. Using Floquet theory and a first-order Magnus expansion in J/Δ (valid for |Δ| ≫ |J|,|Jc|), the authors derive an effective time-independent Hamiltonian over one period, enabling analytic expressions for the controlled target operations. The four control basis states {|00⟩c, |11⟩c, |Ψ+⟩c, |Ψ−⟩c} each catalyze a distinct two-qubit unitary on the targets at gate time τg = πΔ/(4J^2), with the control state ideally unaltered. Nonzero Jc facilitates preparing control Bell states but induces small mixing among triplet control states during operation, leading to infidelities ∝ (2J/Δ)^2. The authors detail an extensible architecture connecting multiple diamond modules in a 2D lattice, detuning A/B plaquettes to run gates in parallel and tuning pairs into resonance for inter-plaquette swaps. To quantify performance under decoherence and imperfections, they solve a Lindblad master equation with collapse operators for dephasing and relaxation on each qubit (rate γ), using QuTiP. Average gate fidelity is computed for the full four-qubit gate and for each controlled two-qubit gate by restricting initial control states appropriately and accounting for leakage. Parameter sweeps examine dependence on Δ, J, and Jc, and robustness to crosstalk Jr, coupling asymmetries, imperfect control-state preparation, and decoherence. Beyond the two-level approximation, they include the second excited state of each transmon (qutrit model) to analyze leakage processes such as |11⟩c ↔ {|02⟩c, |20⟩c} mixing and target-to-target leakage via higher levels that bypass control during idle and parity-controlled operations. They propose engineering a small direct target-target crosstalk Jr to destructively interfere with these leakage pathways and derive an approximate optimal Jr that cancels the unwanted processes.

- The diamond gate implements four controlled two-qubit operations on the targets determined by the control state: two distinct entangling swap/phase gates (for control |00⟩c and |11⟩c), a parity-selective phase gate (for |Ψ+⟩c), and an identity/idle gate (for |Ψ−⟩c). Gate time scales as τg = πΔ/(4J^2). - Numerical Lindblad simulations with state-of-the-art parameters (γ = 0.01 MHz, coherence ~100 μs) confirm high-fidelity, fast operation and match the analytic gate-time prediction within a few percent. - Parameter set 1 (Jc/2π = 20 MHz, J/2π = 65 MHz, Δ/2π = 2 GHz): predicted tg = 59.2 ns; simulated tg = 59.3 ns. Gate fidelities at tg: F00 = 0.9943, F11 = 0.9931, F+ = 0.9881, F− = 0.9968; overall diamond gate F = 0.9923. - Parameter set 2 (Jc/2π = 20 MHz, J/2π = 45 MHz, Δ/2π = 0.5 GHz): predicted tg = 30.9 ns; simulated tg = 31.5 ns. Gate fidelities at tg: F00 = 0.9662, F11 = 0.9668, F+ = 0.9348, F− = 0.9983; overall F = 0.9637. This illustrates a speed–fidelity trade-off. - Fidelity trends: The idle gate (|Ψ−⟩c) is limited primarily by decoherence and remains highest. Gates controlled by |00⟩c and |11⟩c behave similarly; |Ψ+⟩c is most susceptible to control-state mixing. Fidelity generally improves with larger Δ or smaller J at the expense of longer gate times. - Robustness: The architecture tolerates sizable asymmetries in target–control couplings (random deviations δJ) with minor fidelity degradation. Gate fidelity decreases approximately linearly with control-state preparation infidelity and with decoherence rate γ; even at γ = 0.05 MHz (T ~ 20 μs), overall fidelity ≈ 0.98 for the 59 ns gate. - Crosstalk: Direct target–target coupling Jr is intrinsically harmful (fidelity degrades roughly with Jr^2) because it bypasses control. However, when including the transmon second excited state, a small engineered Jr can be beneficial: by tuning Jr to an optimal value (example: Jr,opt/2π ≈ −3.66 MHz) the unwanted leakage pathways that bypass control destructively interfere, restoring selective control of swaps (e.g., preserving the |11⟩c-controlled swap while suppressing leakage during idle/parity operations). This cancellation can increase the gate time for the preserved swap (e.g., to ~220 ns in the illustrated case).

The work demonstrates that a simple exchange-coupled four-qubit network can natively realize a four-way controlled two-qubit gate via quantum interference, addressing the need for compact, high-fidelity entangling operations that can simplify circuit synthesis. The approach provides multiple useful two-qubit primitives (entangling swap/phase, parity-phase, idle) conditioned on the control state, enabling algorithmic flexibility and potential reductions in compilation overhead. Analytical Floquet/Magnus treatment yields closed-form insight and design rules (e.g., τg scaling, Δ and J dependence), while open-system simulations validate performance with realistic decoherence and imperfections, balancing speed and fidelity. The device integrates naturally into an extensible 2D layout of diamond plaquettes, allowing parallel execution and inter-plaquette connectivity via resonance tuning. Importantly, accounting for higher transmon levels reveals critical leakage channels that compromise control selectivity; the proposed passive mitigation by crosstalk engineering leverages destructive interference to suppress these processes, preserving the intended controlled behavior. Overall, the findings substantiate the feasibility of the diamond gate as a building block for highly connected superconducting quantum processors.

The authors propose and analyze a quantum interference device—the diamond gate—comprised of four exchange-coupled qubits that implements four distinct controlled two-qubit operations on a pair of targets. With superconducting transmon implementations and state-of-the-art parameters, the gate operates in tens of nanoseconds with fidelities around 0.99, subject to a speed–fidelity trade-off. Extending the model to include the second excited state reveals leakage pathways that can bypass control; a passive mitigation strategy based on engineering a small target–target crosstalk cancels this leakage via destructive interference, at the cost of modifying one controlled operation into a phase gate and increasing the gate time for another. The device can tile into a 2D architecture supporting parallel diamond gates interleaved with selective two-qubit swaps and single-qubit rotations. Future directions include incorporating active microwave control to further suppress leakage or optimize phases, exploring qubits with larger anharmonicity, and adapting the concept to other platforms (e.g., ultracold atoms or ions). Algorithmic studies could identify workloads particularly well-suited to the diamond-plaquette architecture.

Performance depends on detuning and coupling ratios; nonzero control–control coupling Jc, while helpful for Bell-state preparation, induces mixing among triplet control states and limits fidelity by ~O((2J/Δ)^2), creating a speed–fidelity trade-off. Gate fidelity is bounded by decoherence; longer gates increase susceptibility. In transmon implementations, small anharmonicity enables leakage to higher levels, causing control-bypassing processes that degrade idle and parity operations; mitigating this via engineered crosstalk alters some gate behaviors and can lengthen gate times. Direct target–target crosstalk is generally harmful if not tuned appropriately. The scheme requires high-fidelity preparation of specific control states (including Bell states), and fabrication/parameter variations can introduce asymmetries, though simulations indicate robustness within realistic ranges.