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Physical review. B./Physical review. B

Longitudinal (curvature) couplings of an N -level qudit to a superconducting resonator at the adiabatic limit and beyond

Rusko Ruskov, Charles Tahan

Published Jun 14, 2024• 8 citations
Publisher

Abstract

Understanding the coupling of superconducting (SC) wires to solid state qubits is vital to the design of quantum computers. To date, there has not been a complete theory applicable to all parameter regimes, from the traditional dispersive regime of circuit QED to the resonance regime to the far-off resonance situation in the adiabatic regime. This work presents that theory in the context of modulated longitudinal coupling, which promises to enhance the readout and coupling speed and fidelity through a resonator of solid-state qubit operations.

DOI
10.1103/physrevb.109.245303

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Understanding the coupling of superconducting (SC) wires to solid state qubits is vital to the design of quantum computers. To date, there has not been a complete theory applicable to all parameter regimes, from the traditional dispersive regime of circuit QED to the resonance regime to the far-off resonance situation in the adiabatic regime. This work presents that theory in the context of modulated longitudinal coupling, which promises to enhance the readout and coupling speed and fidelity through a resonator of solid-state qubit operations.
Longitudinal (curvature) couplings of an NNN-level qudit to a superconducting resonator at the adiabatic limit and beyond
Understanding how and to what magnitude solid-state qubits couple to metallic wires is crucial to the design of quantum systems such as quantum computers. Here, we investigate the coupling between a multi-level system, or qudit, and a superconducting (SC) resonator's electromagnetic field, focusing on the interaction involving both the transition and diagonal dipole moments of the qudit. Specifically, we explore the effective dynamical (time-dependent) longitudinal coupling that arises when a solid-state qudit is adiabatically modulated at small gate frequencies and amplitudes, in addition to a static dispersive interaction with the SC resonator. For the first time, we derive Hamiltonians describing the longitudinal multi-level interactions in a general dispersive regime, encompassing both dynamical longitudinal and dispersive interactions. These Hamiltonians smoothly transition between their adiabatic values, where the couplings of the n-th level are proportional to the level's energy curvature concerning a qudit gate voltage, and the substantially larger dispersive values, which occur due to a resonant form factor. We provide several examples illustrating the transition from adiabatic to dispersive coupling in different qubit systems, including the charge (1e DQD) qubit, the transmon, the double quantum dot singlet-triplet qubit, and the triple quantum dot exchange-only qubit. In some of these qubits, higher energy levels play a critical role, particularly when their qubit's dipole moment is minimal or zero. For an experimentally relevant scenario involving a spin-charge qubit with magnetic field gradient coupled capacitively to a SC resonator, we showcase the potential of these interactions. They enable close-to-quantum-limited quantum non-demolition (QND) measurements and remote geometric phase gates, demonstrating their practical utility in quantum information processing.
We devise a Floquet theory of longitudinal and dispersive readout in circuit quantum electrodynamics (cQED). By studying qubits coupled to cavity photons and driven at the resonance frequency of the cavity ${\ensuremath{\omega}}_{\mathrm{r}}$, we establish a universal connection between the qubit ac Stark shift and the longitudinal and dispersive coupling to photons. We find that the longitudinal coupling ${g}_{\ensuremath{\parallel}}$ is controlled by the slope of the ac Stark shift as function of the driving strength Aq{A}_{\mathrm{q}}Aq​, while the dispersive shift $\ensuremath{\chi}$ depends on its curvature. The two quantities become proportional to each other in the weak drive limit (${A}_{\mathrm{q}}\ensuremath{\rightarrow}0$). Our approach unifies the adiabatic limit (${\ensuremath{\omega}}_{\mathrm{r}}\ensuremath{\rightarrow}0$)---where ${g}_{\ensuremath{\parallel}}$ is generated by the static spectrum curvature (or quantum capacitance)---with the diabatic limit, where ${\ensuremath{\omega}}_{\mathrm{r}}$ is large and the static spectrum plays no role. We derive analytical results supported by exact numerical simulations. We apply them to superconducting and spin-hybrid cQED systems, showcasing the flexibility of faster-than-dispersive longitudinal readout.
Despite the significant progress in superconducting quantum computation over the past years, quantum state measurement still lags nearly an order of magnitude behind quantum gate operations in speed and fidelity. The main challenge is that the strong coupling and readout signal used to probe the quantum state may also introduce additional channels which may cause qubit state transitions. Here, we design a novel architecture to implement the long-sought longitudinal interaction scheme between qubits and resonators. This architecture not only provides genuine longitudinal interaction by eliminating residual transversal couplings, but also introduces proper nonlinearity to the resonator that can further minimize decay error and measurement-induced excitation error. Combined with the multilevel readout protocol, we achieved a measurement fidelity of 99.8% in 202 ns without requiring any first-stage amplification. This highlights its potential as a highly promising architecture for superconducting qubit measurement.
Existing schemes for semiconductor spin qubit readout involve either spin-to-charge conversion or electric dipole coupling to a superconducting resonator. The former requires destructive readout while the latter suffers from enhanced qubit dephasing, limiting the potential performance of quantum measurement: an open challenge in semiconductor quantum computing. Here we propose a coupling to a superconducting resonator based on qubit energy curvature versus gate voltage, which enables quantum nondemolition (QND) readout while the qubit resides in its full sweet spot and with zero dipole coupling (offering resilience to charge noise). Readout strengths of tens to hundred megahertz (MHz) can be reached, at least ten times larger than in the previously proposed schemes. We show how this curvature interaction generates two strictly QND readout approaches: (i) via a ``dispersive-like'' (quantum capacitance) spin-dependent resonator energy shift, similar in implementation to circuit-QED dispersive readout, but avoiding the Purcell effect; and (ii) with the addition of qubit gate voltage modulation, a longitudinal readout which can be selectively switched on, for a chosen n-qubit subsystem. Neither scheme requires the qubit and resonator to be close in frequency.
Superconducting circuits consisting of a few low-anharmonic transmons coupled to readout and bus resonators can perform basic quantum computations. Since the number of qubits in such circuits is limited to not more than a few tens, the qubits can be designed to operate within the dispersive regime where frequency detunings are much stronger than coupling strengths. However, scaling up the number of qubits will bring the circuit out of the regime, and this invalidates current theories. We develop a formalism that allows to consistently diagonalize the superconducting circuit Hamiltonian beyond the dispersive regime. This will allow to study qubit-qubit interaction unperturbatively, therefore, our formalism remains valid and accurate at small or even negligible frequency detuning; thus, our formalism serves as a theoretical ground for designing qubit characteristics for scaling up the number of qubits in superconducting circuits. We study the most important circuits with single- and two-qubit gates, i.e., a single transmon coupled to a resonator and two transmons sharing a bus resonator. Surprisingly, our formalism allows to determine the circuit characteristics, such as dressed frequencies and Kerr couplings, in closed-form formulas that not only reproduce perturbative results, but also extrapolate beyond the dispersive regime and can ultimately reproduce (and even modify) the Jaynes-Cumming results at resonant frequencies.