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arXiv:2408.15402

Comprehensive explanation of ZZ coupling in superconducting qubits

Simon Pettersson Fors, Jorge Fernández-Pendás, Anton Frisk Kockum

Published Aug 27, 2024• 0 citations
arXivHTMLPDF

Abstract

A major challenge for scaling up superconducting quantum computers is unwanted couplings between qubits, which lead to always-on ZZ couplings that impact gate fidelities by shifting energy levels conditional on qubit states. To tackle this challenge, we introduce analytical and numerical techniques, including a diagrammatic perturbation theory and a state-assignment algorithm. Together, these tools enable us to explain the emergence of ZZ coupling in three linked pictures, where each picture tells us more about the underlying mechanisms creating the ZZ coupling. These pictures generalize previous efforts, which focused on specific setups and a single mechanism. The deeper understanding that we provide of the mechanisms behind the ZZ coupling facilitate finding parameter regions of weak and strong ZZ coupling. We showcase our techniques for a system consisting of two fixed-frequency transmon qubits connected by a flux-tunable transmon coupler. There, we find three types of parameter regions with zero or near-zero ZZ coupling, all of which are accessible with current technology. We furthermore find regions of strong ZZ coupling nearby, which may be used to implement adiabatic controlled-phase gates and quantum simulations. Our framework is applicable to many types of qubits and opens up for the design of large-scale quantum computers with improved gate fidelities.

DOI
10.48550/arxiv.2408.15402

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A major challenge for scaling up superconducting quantum computers is unwanted couplings between qubits, which lead to always-on ZZ couplings that impact gate fidelities by shifting energy levels conditional on qubit states. To tackle this challenge, we introduce analytical and numerical techniques, including a diagrammatic perturbation theory and a state-assignment algorithm. Together, these tools enable us to explain the emergence of ZZ coupling in three linked pictures, where each picture tells us more about the underlying mechanisms creating the ZZ coupling. These pictures generalize previous efforts, which focused on specific setups and a single mechanism. The deeper understanding that we provide of the mechanisms behind the ZZ coupling facilitate finding parameter regions of weak and strong ZZ coupling. We showcase our techniques for a system consisting of two fixed-frequency transmon qubits connected by a flux-tunable transmon coupler. There, we find three types of parameter regions with zero or near-zero ZZ coupling, all of which are accessible with current technology. We furthermore find regions of strong ZZ coupling nearby, which may be used to implement adiabatic controlled-phase gates and quantum simulations. Our framework is applicable to many types of qubits and opens up for the design of large-scale quantum computers with improved gate fidelities.
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This Master thesis investigates ZZ and ZZZ couplings in a system of three superconducting transmon qubits. Such couplings represent energy shifts arising from qubit-qubit interactions, and they are important for the performance of quantum computers. Adopting an analytical method based on the Schrieffer-Wolff transformation, we extend it with new insight and compute energy corrections associated with ZZZ and ZZ couplings.The thesis explores a known parameter region, where one ZZ coupling can be eliminated, to find if the ZZZ coupling can be made zero simultaneously. Lastly, an attempt is made to connect ZZZ and ZZ couplings with a simple linear relationship.
Generating high-fidelity, tunable entanglement between qubits is crucial for realizing gate-based quantum computation. In superconducting circuits, tunable interactions are often implemented using flux-tunable qubits or coupling elements, adding control complexity and noise sources. Here, we realize a tunable ZZ interaction between two transmon qubits with fixed frequencies and fixed coupling, induced by driving both transmons off resonantly. We show tunable coupling over 1 order of magnitude larger than the static coupling, and change the sign of the interaction, enabling cancellation of the idle coupling. Further, this interaction is amenable to large quantum processors: the drive frequency can be flexibly chosen to avoid spurious transitions, and because both transmons are driven, it is resilient to microwave cross talk. We apply this interaction to implement a controlled phase (CZ) gate with a gate fidelity of 99.43(1)% as measured by cycle benchmarking, and we find the fidelity is limited by incoherent errors.
To reduce circuit depth when executing Quantum algorithms, it is necessary to maximize qubit connectivity on a near-term quantum processor. While addressing this, we also need to ensure high gate fidelity, suppression of unwanted ZZ cross-talk, a compact layout footprint, and minimal control hardware complexity to support scalability. In current superconducting quantum chips, fixed coupling is used as it is easier to scale, but it is limited by unwanted static ZZ interaction during single qubit operations, which degrades system performance. To overcome these challenges, we have introduced a first-of-its-kind hybrid tunable-coupling architecture that connects four fixed-frequency transmon qubits using a single coupler. This hybrid coupler uses off-resonant Stark drives to tune ZZ strength between qubit pairs. Experimentally backed simulation results indicate that our proposed hybrid design maximizes the qubit connectivity while reducing control overhead. This design achieves a near 20% reduction in circuit depth compared to IBM's Heavy-Hexagonal layout, showing its potential for scalability.
Overcoming the issue of qubit-frequency fluctuations is essential to realize stable and practical quantum computing with solid-state qubits. Static ZZ interaction, which causes a frequency shift of a qubit depending on the state of neighboring qubits, is one of the major obstacles to integrating fixed-frequency transmon qubits. Here we propose and experimentally demonstrate ZZ-interaction-free single-qubit-gate operations on a superconducting transmon qubit by utilizing a semi-analytically optimized pulse based on a perturbative analysis. The gate is designed to be robust against slow qubit-frequency fluctuations. The robustness of the optimized gate spans a few MHz, which is sufficient for suppressing the adverse effects of the ZZ interaction. Our result paves the way for an efficient approach to overcoming the issue of ZZ interaction without any additional hardware overhead.
In the companion paper (Part I), we developed an optimal-control framework to suppress intrinsic and decoherence-induced errors in controlled-Z (CZ) gates between directly coupled, frequency-tunable transmons. Here, we extend this framework to incorporate uncertainties in coupling strength, anharmonicities, and zero-flux qubit frequencies, enabling the design of high-fidelity gates robust against noise and parameter uncertainties. We classify these uncertainties by whether they perturb the off-diagonal coupling strengths or the diagonal detunings of the Hamiltonian, and we identify distinct suppression mechanisms for each class. Coupling errors are mitigated through dynamical decoupling induced by strong detuning modulation. For a 30-ns gate, this strategy reduces the coupling error to 6.21×10−76.21\times10^{-7}6.21×10−7 and tolerates coupling-strength uncertainties up to 2π×42π\times42π×4 MHz while keeping total gate infidelity below 10−310^{-3}10−3. Detuning errors are minimized near a symmetric-detuning operating point that balances the two dominant leakage-mediated interactions. A 24-ns gate operated near this point tolerates individual uncertainties of up to 2π×52π\times52π×5 MHz in each relevant diagonal parameter under the same infidelity threshold. We further optimize gate performance under realistic conditions where intrinsic errors, decoherence, and parameter uncertainties coexist across six representative stages of progressively improving hardware performance. The resulting stage-specific controls achieve lower infidelity and greater robustness. Together with the companion study (Part I), this framework offers a practical approach to balance speed, fidelity, and robustness while reducing the need for exhaustive characterization and frequent recalibration in large-scale superconducting quantum processors.