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arXiv:2107.02343 · Physical Review Applied

Accurate Methods for the Analysis of Strong-Drive Effects in Parametric Gates

Alexandru Petrescu, Camille Le Calonnec, Catherine Leroux, Agustín Di Paolo, Pranav Mundada, Sara Sussman, Andrei Vrajitoarea, Andrew Houck, Alexandre Blais

Published Apr 3, 2023• 39 citations
arXivHTMLPDF

Abstract

The ability to perform fast, high-fidelity entangling gates is a requirement for a viable quantum processor. In practice, achieving fast gates often comes with the penalty of strong-drive effects that are not captured by the rotating-wave approximation. These effects can be analyzed in simulations of the gate protocol, but those are computationally costly and often hide the physics at play. Here, we show how to efficiently extract gate parameters by directly solving a Floquet eigenproblem using exact numerics and a perturbative analytical approach. As an example application of this toolkit, we study the space of parametric gates generated between two fixed-frequency transmon qubits connected by a parametrically driven coupler. Our analytical treatment, based on time-dependent Schrieffer-Wolff perturbation theory, yields closed-form expressions for gate frequencies and spurious interactions, and is valid for strong drives. From these calculations, we identify optimal regimes of operation for different types of gates including $i$swap, controlled-Z, and cnot. These analytical results are supplemented by numerical Floquet computations from which we directly extract drive-dependent gate parameters. This approach has a considerable computational advantage over full simulations of time evolutions. More generally, our combined analytical and numerical strategy allows us to characterize two-qubit gates involving parametrically driven interactions, and can be applied to gate optimization and cross-talk mitigation such as the cancelation of unwanted $ZZ$ interactions in multiqubit architectures.

DOI
10.1103/physrevapplied.19.044003

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The ability to perform fast, high-fidelity entangling gates is a requirement for a viable quantum processor. In practice, achieving fast gates often comes with the penalty of strong-drive effects that are not captured by the rotating-wave approximation. These effects can be analyzed in simulations of the gate protocol, but those are computationally costly and often hide the physics at play. Here, we show how to efficiently extract gate parameters by directly solving a Floquet eigenproblem using exact numerics and a perturbative analytical approach. As an example application of this toolkit, we study the space of parametric gates generated between two fixed-frequency transmon qubits connected by a parametrically driven coupler. Our analytical treatment, based on time-dependent Schrieffer-Wolff perturbation theory, yields closed-form expressions for gate frequencies and spurious interactions, and is valid for strong drives. From these calculations, we identify optimal regimes of operation for different types of gates including iiiswap, controlled-Z, and cnot. These analytical results are supplemented by numerical Floquet computations from which we directly extract drive-dependent gate parameters. This approach has a considerable computational advantage over full simulations of time evolutions. More generally, our combined analytical and numerical strategy allows us to characterize two-qubit gates involving parametrically driven interactions, and can be applied to gate optimization and cross-talk mitigation such as the cancelation of unwanted ZZZZZZ interactions in multiqubit architectures.
Achieving fast gates and high-fidelity readout in superconducting quantum circuits often benefits from the use of large amplitude microwave drives. As the strength increases, unwanted effects such as excitation out of the logical states also become more prevalent. In transmon qubits, coherent controls relying on sub-resonant drives are fundamentally limited by the eventual appearance of strong hybridization in the spectrum. Understanding how this threshold depends on intrinsic device parameters is vital for the design and operation of circuits utilizing the transmon as a source of nonlinearity. In this work, we present an experimental study of 14 transmons with a wide range of anharmonicities, and use comparison with Floquet branch analysis and semiclassical simulations to establish a straightforward relationship between the critical value of the effective drive amplitude ηmax⁡η_{\max}ηmax​ and the ratio γ=ωq/αγ=ω_q/αγ=ωq​/α, where ωqω_qωq​ is the transmon frequency and −α-α−α is the anharmonicity. For the case where the transmon is used as a parametrically driven coupler, these results further allow us to estimate maximum parametric interaction rates, which informs the circuit design process for optimizing gate rates and fidelities.
A current bottleneck for quantum computation is the realization of high-fidelity two-qubit quantum operations between two or more quantum bits in arrays of coupled qubits. Gates based on parametrically driven tunable couplers offer a convenient method to entangle multiple qubits by selectively activating different interaction terms in the effective Hamiltonian. Here, we theoretically and experimentally study a superconducting qubit setup with two transmon qubits connected via a capacitively coupled tunable bus. We develop a time-dependent Schrieffer-Wolff transformation and derive analytic expressions for exchange-interaction gates swapping excitations between the qubits (iswap) and for two-photon gates creating and annihilating simultaneous two-qubit excitations (bswap). We find that the bswap gate is generally slower than the more commonly used iswap gate, but features favorable scalability properties with less severe frequency-crowding effects, which typically degrade the fidelity in multiqubit setups. Our theoretical results are backed by experimental measurements as well as exact numerical simulations including the effects of higher transmon levels and dissipation.
We consider the dynamics of a two-level system (qubit) driven by strong and short resonant pulses in the framework of Floquet theory. First we derive analytical expressions for the quasienergies and Floquet states of the driven system. If the pulse amplitude varies very slowly, the system adiabatically follows the instantaneous Floquet states, which acquire dynamical phases that depend on the evolution of the quasienergies over time. The difference between the phases acquired by the two Floquet states corresponds to a qubit state rotation, generalizing the notion of Rabi oscillations to the case of large driving amplitudes. If the pulse amplitude changes very fast, the evolution is nonadiabatic, with transitions taking place between the Floquet states. We quantify and analyze the nonadiabatic transitions during the pulse by employing adiabatic perturbation theory and exact numerical simulations. We find that, for certain combinations of pulse rise and fall times and maximum driving amplitude, a destructive interference effect leads to a remarkably strong suppression of transitions between the Floquet states. This effect provides the basis of a quantum control protocol, which we name Floquet interference efficient suppression of transitions in the adiabatic basis (FIESTA), that can be used to design ultrafast high-fidelity single-qubit quantum gates.
The controls enacting logical operations on quantum systems are described by time-dependent Hamiltonians that often include rapid oscillations. In order to accurately capture the resulting time dynamics in numerical simulations, a very small integration time step is required, which can severely impact the simulation run time. Here, we introduce a semianalytic method based on the Dyson expansion that allows us to time-evolve driven quantum systems much faster than standard numerical integrators. This solver, which we name Dysolve, efficiently captures the effect of the highly oscillatory terms in the system Hamiltonian, significantly reducing the simulation's run time as well as its sensitivity to the time-step size. Furthermore, this solver provides the exact derivative of the time-evolution operator with respect to the drive amplitudes. This key feature allows for optimal control in the limit of strong drives and goes beyond common pulse-optimization approaches that rely on rotating-wave approximations. As an illustration of our method, we show results of the optimization of a two-qubit gate using transmon qubits in the circuit QED architecture.
Parametric two-qubit gates using tunable couplers are a promising approach to scalable superconducting quantum processors. However, conventional theories formulated in the idle eigenbasis break down in the large-amplitude regime required for fast gate operations. Here, we show that parametric gates are more naturally and accurately described as coherent rotations between instantaneous eigenstates rather than idle eigenstates. This approach may appear counterintuitive, since instantaneous eigenbases are usually associated with adiabatic dynamics. However, they also form a natural moving frame for driven dynamics, where the non-Abelian Berry connection governs transitions and provides a compact description of the exchange and effective ZZZZZZ couplings. We show that this theory achieves quantitative agreement with exact numerical simulations across representative tunable-coupler architectures, including double-transmon, capacitively shunted double-transmon, and single-transmon couplers. The description remains accurate even in regimes where idle-frame descriptions fail, while substantially reducing the reliance on computationally demanding exact simulations. These results establish an instantaneous-frame theory of parametric gates beyond conventional idle-frame models and provide a new perspective on driven quantum dynamics.
Superconducting circuits are a leading platform for scalable quantum computing, where parametric modulation is a widely used technique for implementing high-fidelity multi-qubit operations. A critical challenge, however, is that this modulation can induce a dense landscape of parasitic couplings, leading to detrimental frequency collisions that constrain processor performance. In this work, we develop a comprehensive numerical framework, grounded in Floquet theory, to systematically analyze and mitigate these collisions. Our approach integrates this numerical analysis with newly derived analytical models for both qubit-modulated and coupler-modulated schemes, allowing us to characterize the complete map of parasitic sideband interactions and their distinct error budgets. This analysis forms the basis of a constraint-based optimization methodology designed to identify parameter configurations that satisfy the derived physical constraints, thereby avoiding detrimental parasitic interactions. We illustrate the utility of this framework with applications to analog quantum simulation and gate design. Our work provides a predictive tool for co-engineering device parameters and control protocols, enabling the systematic suppression of crosstalk and paving the way for large-scale, high-performance quantum processors.