Two-qubit sweet spots for capacitively coupled exchange-only spin qubits

Semiconductor quantum dots are a promising platform for quantum computing due to their scalability, coherence, and compatibility with existing microelectronics. High-fidelity gate operations have been achieved in various quantum dot architectures, including triple quantum dots (TQDs). Exchange-only (EO) qubits, encoded in the decoherence-free subspace of three electron spins, offer fast, all-electrical control via exchange interactions. However, implementing high-fidelity two-qubit gates remains a challenge for universal and fault-tolerant quantum computation. While exchange coupling is fast, it's short-ranged and susceptible to leakage. Capacitive coupling, arising from electrostatic Coulomb interaction, offers longer-range interaction, less stringent addressability requirements, and reduced leakage. This research focuses on capacitively coupled EO qubits, aiming to develop exact gate sequences for CPHASE and CNOT gates and identify sweet spots for improved noise resilience. Previous work has identified single-qubit sweet spots (1QSS) for AEON and RX qubits, but 2QSS for capacitive coupling remain unexplored. This study aims to fill this knowledge gap by theoretically demonstrating the existence of 2QSS and evaluating their impact on two-qubit gate fidelities and times under realistic charge noise.

Several studies have explored two-qubit gates in semiconductor quantum dot systems. Exchange-based gates, while fast, often require multiple steps and are prone to leakage. Hybrid approaches, such as spin-shuttling and circuit QED, have been proposed to address these limitations. Capacitive coupling presents an alternative with advantages such as longer-range interaction and reduced leakage. Previous research on EO qubits demonstrated the existence of 1QSS for both AEON and RX implementations, offering protection against single-qubit parameter fluctuations. The concept of sweet spots has also been explored in the context of exchange-coupled qubits, but a comprehensive study of 2QSS for capacitively coupled EO qubits was lacking. This work builds upon these previous efforts, extending the investigation to two-qubit gates and exploring the potential for enhancing gate fidelity through the identification and utilization of 2QSS.

The study uses a model of two capacitively coupled EO qubits arranged linearly in a TQD array. Each TQD is described by a Hubbard Hamiltonian, incorporating nearest-neighbor tunneling, detuning, and Coulomb energy terms. The perturbative limit (tunneling energy much smaller than Coulomb energy) is assumed, allowing the focus to be on the (1,1,1) charge state (one electron per dot) while considering small admixtures from doubly occupied states. Effective single-qubit Hamiltonians are derived using a Schrieffer-Wolff transformation. Two-qubit capacitive coupling is modeled through inter-dot Coulomb interactions between the TQDs, represented by a diagonal Hamiltonian in the computational basis. Analytical expressions for Coulomb integrals are obtained using tri-quadratic confinement potentials. The effect of charge noise is incorporated by simulating random fluctuations in tunneling and detuning parameters with a 1/f power spectral density. Exact gate sequences for CPHASE and CNOT gates are derived using Makhlin invariants. The non-local interaction time is determined analytically. The average gate fidelity under noise is calculated using a numerical simulation involving averaging over multiple noise realizations and compared with an approximate analytical expression obtained using a cumulant expansion. The existence of 2QSS is investigated by analyzing the partial derivatives of the interaction terms with respect to TQD parameters. Analytical expressions for the loci of 2QSS are derived and used to identify optimal working points for the two-qubit gates. Fidelities are calculated at various operating points including single and multiple parameter 2QSS and compared with the AEON and RX operating points. The numerical simulations include the generation of 1/f noise using an algorithm adapted from existing literature.

The research demonstrates the existence of multiple two-qubit sweet spots (2QSS) in the parameter space of capacitively coupled exchange-only qubits. Specifically, 2QSS were found for the middle detuning (εm) and the left/right tunnel couplings (τl, τr) of each qubit. Analytical expressions for the loci of these 2QSS were derived. Exact gate sequences for CPHASE and CNOT gates were reported, revealing that the non-local interaction time is determined by the energy difference between doubly-occupied charge admixture configurations in the two TQDs. This energy difference is interpreted as a dipole-dipole interaction between the TQDs. Calculations of two-qubit gate fidelities under 1/f noise showed that operating at 2QSS significantly improves fidelity. The study found that gate times are faster near charge occupation boundaries, suggesting a trade-off between speed and noise resilience. Comparisons between the AEON and RX qubits showed that, for comparable gate times, operating at the AEON 1QSS provides better fidelity. However, using the fastest gate time for RX yielded slightly better fidelity. A particular tunnel coupling ratio allows the εm 2QSS to overlap with the εA = 0 1QSS, enabling operation at both simultaneously. Double 2QSS (εm and τl) were also identified, leading to further improvements in fidelity. Analytical and numerical infidelity calculations showed good agreement except at the double 2QSS where the analytical expression overestimates infidelity due to significantly longer gate times. Infidelity was found to be less sensitive to detuning noise than tunneling noise. A global fidelity optimum was identified, lying on the εm 2QSS line; this point achieves fault-tolerance thresholds (infidelity <10^-4) when tunneling noise is negligible. However, the presence of tunneling noise shifts this optimum.

The findings address the central research question by demonstrating the existence and utility of 2QSS for improving the fidelity of capacitively coupled two-qubit gates in EO qubits. The identification of multiple 2QSS provides flexibility in choosing optimal operating points based on specific noise characteristics and experimental constraints. The theoretical framework, including the analytical expressions for gate times and fidelities, and the noise model provide valuable tools for designing and optimizing two-qubit gates in these systems. The observed trade-off between gate speed and noise resilience highlights the importance of considering both factors when selecting operating points. The results provide guidance for experimental implementations, particularly in tuning tunnel coupling ratios to achieve optimal performance. The near-fault-tolerant fidelities achieved at certain sweet spot intersections demonstrate significant progress towards practical quantum computation. However, further optimization is necessary to address situations with dominant tunneling noise.

This paper demonstrates the existence of multiple two-qubit sweet spots (2QSS) for capacitively coupled exchange-only qubits, providing a pathway to significantly improve the fidelity of two-qubit gates. The analytical expressions for the loci of 2QSS and the identified optimal operating points are valuable tools for experimental implementations. The results suggest that operating at the intersection of 1QSS and 2QSS provides the best fidelity, but the stringent noise requirements highlight the necessity of further improvements in qubit coherence. Future research could focus on investigating the impact of correlated noise, exploring advanced control techniques for single-qubit gates, and developing more sophisticated models of the quantum dot system.

The study uses a simplified model of the quantum dot system and noise. The 1/f noise model, while common, might not fully capture the complexity of real-world noise processes. The analytical expression for average fidelity is based on a perturbative approach, limiting its accuracy at high noise levels. The assumption of uncorrelated noise might not be perfectly realistic in experimental settings. The study focuses on charge noise, ignoring other potential sources of decoherence, such as phonon noise. The analysis of global fidelity assumes independence of detuning and tunneling noise; in practice, there might be correlations between the two noise types.