We analyze the dispersive readout of superconducting charge and flux qubits as a quantum measurement process. The measurement oscillator frequency is considered much lower than the qubit frequency. This regime is interesting because large detuning allows for strong coupling between the measurement oscillator and the signal transmission line, thus allowing for fast readout. Due to the large detuning we may not use the rotating wave approximation in the oscillator-qubit coupling. Instead we start from an approximation where the qubit follows the oscillator adiabatically, and show that non-adiabatic corrections are small. We find analytic expressions for the measurement time, as well as for the back-action, both while measuring and in the off-state. The quantum efficiency is found to be unity within our approximation, both for charge and flux qubit readout.
We analyze the dispersive readout of superconducting charge and flux qubits as a quantum measurement process. The measurement oscillator frequency is considered much lower than the qubit frequency. This regime is interesting because large detuning allows for strong coupling between the measurement oscillator and the signal transmission line, thus allowing for fast readout. Due to the large detuning we may not use the rotating wave approximation in the oscillator-qubit coupling. Instead we start from an approximation where the qubit follows the oscillator adiabatically, and show that non-adiabatic corrections are small. We find analytic expressions for the measurement time, as well as for the back-action, both while measuring and in the off-state. The quantum efficiency is found to be unity within our approximation, both for charge and flux qubit readout.
Superconducting flux qubits have many advantages as a storage of quantum information, such as broad range tunability of frequency, small-size fabricability, and high controllability. In the flux qubit-oscillator, qubits are connected to SQUID resonators for the purpose of performing dispersive non-destructive readouts of qubit signals with high fidelity. In this work, we propose a theoretical model for analyzing quantum characteristics of a flux qubit-oscillator on the basis of quantum solutions obtained using a unitary transformation approach. The energy levels of the combined system (qubit + resonator) are analyzed in detail. Equally spaced each energy level of the resonator splits into two parts depending on qubit states. Besides, coupling of the qubit to the resonator brings about an additional modification in the split energy levels. So long as the coupling strength and the tunnel splitting are not zero but finite values, the energy-level splitting of the resonator does not disappear. We conclude that quantum nondemolition dispersive measurements of the qubit states are possible by inducing bifurcation of the resonator states through the coupling.
Dispersive readout, the current standard for measuring superconducting qubits, relies on relatively easy to engineer, static transverse coupling between a qubit and a detuned cavity mode. However, this method requires circuit design trade-offs between qubit-cavity coupling, cavity-feedline coupling, detuning, and the resultant decoherence introduced by the cavity mode. Additionally, the criteria for performing quantum nondemolition measurements, including reduced cavity photon numbers, further limits speed and fidelity. In this work, we introduce a circuit with pulse-tunable qubit-cavity and cavity-feedline couplings so that the entire qubit-measurement process can be fully optimized. Here, the core measurement process resembles a quantum gate operation between the qubit and the cavity, with the cavity acting like an ancilla, storing the qubit-state information. To fully explore time-dependent effects of the full qubit-cavity dynamics, we go beyond the rotating-wave, Jaynes-Cummings, or dispersive approximations, to develop a pulsed-measurement protocol. In addition, we propose a realistic circuit architecture that can provide a very fast, high-fidelity, QND measurement with near-unit efficiency.
We analyze the relaxation of a superconducting flux qubit during measurement. The qubit state is measured with a nonlinear oscillator driven across the threshold of bifurcation, acting as a switching dispersive detector. This readout scheme is of quantum nondemolition type. Two successive readouts are used to analyze the evolution of the qubit and the detector during the measurement. We introduce a simple transition rate model for characterizing the qubit relaxation and the detector switching process. Corrected for qubit relaxation the readout fidelity is at least 95%. Qubit relaxation strongly depends on the driving strength and the state of the oscillator.
Spectral properties of a quantum circuit are efficiently read out by monitoring the resonance frequency shift it induces in a microwave resonator coupled to it. When the two systems are strongly detuned, theory attributes the shift to an effective resonator capacitance or inductance that depends on the quantum circuit state. At small detuning, the shift arises from the exchange of virtual photons, as described by the Jaynes-Cummings model. Here we present a theory bridging these two limits and illustrate, with several examples, its necessity for a general description of quantum circuits readout. Circuit Quantum Electrodynamics (cQED) is at the heart of most advanced superconducting quantum technologies. Different types of superconducting qubits can be strongly coupled to microwave resonators thus achieving regimes and phenomenena which cannot be reached within the realm of quantum optics [1]. More recently, strong coupling between microwave resonators and a variety of other quantum systems not necessarily involving superconductors has been achieved [2], extending further the realm of cQED. In all these applications the measurement of the qubit or the hybrid device state is achieved by monitoring the resonator properties. Theoretically , two regimes have been approached using disconnected descriptions [3]: the dispersive regime, where the qubit-resonator detuning is larger than the coupling strength yet small enough to allow the exchange of virtual photons, and the adiabatic regime, where the de-tuning is sufficiently large for virtual processes to be strongly suppressed. The dispersive regime, which describes level repulsion between those of the quantum circuit and of the resonator, is typically dealt with using a Jaynes-Cummings Hamiltonian within different levels of approximation [3-10]. In contrast, the adiabatic regime accounts for the renormalization of the resonator capaci-tance/inductance by the effective capacitance of the circuit , including its "quantum capacitance" [11, 12], or its effective inductance [13, 14], which modifies the resonator frequency [15, 16]. However, there is no actual border between these two regimes which could justify a separate treatment, as illustrated by recent experiments on hybrid cQED setups [17] that reveal features of both regimes for the same device. This situation claims for a unified description of quantum circuits readout, going beyond the standard Jaynes-Cummings model, which could be applied to different types of devices over a large range of parameters. * Corresponding author : a.l.yeyati@uam.es In the present Letter we derive a general expression for the resonator frequency shift when coupled to a generic quantum circuit. This expression naturally interpolates between the adiabatic and the dispersive regimes, thus allowing to clarify their origin from the same coupling Hamiltonian. In addition our formalism is not restricted to the usual two-level approximations but any multilevel situation can be described on the same footing. We illustrate the importance of the different terms in our expression by analyzing well-known models like a short single channel superconducting weak link hosting Andreev states, the RF-SQUID and the Cooper pair box. Resonator-quantum circuit coupling.-The system we consider comprises a resonant circuit and a quantum circuit coupled through phase or charge fluctuations as depicted in Figs. 1(a), 2(a) and in the inset of Fig. 3. The resonant circuit is represented as a lumped-element LC resonator with bare resonance frequency f r = ω r /2π, with ω r = 1/ √ L r C r. Introducing the photon annihilation (creation) operators a (a †), it can be described by the Hamiltonian H r = ω r a † a. On the other hand, the quantum circuit Hamiltonian,Ĥ qc (x), depends on a dimensionless control parameter x, corresponding to an excess charge on a capacitor or a flux through a loop. We denote by |Φ i (x) the eigenstates of the uncoupled quantum circuit,Ĥ qc (x)|Φ i (x) = E i (x)|Φ i (x). Flux (charge) fluctuations in the resonator lead to x → x 0 +x r , wherex r = λ(s a + s * a †) with a coupling constant λ, depending on a coupling scheme [19], and s = 1 (−i). We assume λ 1 in accordance with experiments. The resonator-quantum circuit coupling HamiltonianĤ c is obtained by expandingĤ qc (x 0 +x r) up to second order inx rĤ c (x 0) =x rĤ qc (x 0) +x 2 r 2Ĥ qc (x 0), (1) where the prime stands for the derivative with respect to x. The Hamiltonian describing resonator, quantum
The performance of a wide range of quantum computing algorithms and protocols depends critically on the fidelity and speed of the employed qubit readout. Examples include gate sequences benefiting from midcircuit real-time measurement-based feedback, such as qubit initialization, entanglement generation, teleportation, and, perhaps most importantly, quantum error correction. A prominent and widely used readout approach is based on the dispersive interaction of a superconducting qubit strongly coupled to a large-bandwidth readout resonator, frequently combined with a dedicated or shared Purcell filter protecting qubits from decay. By dynamically reducing the qubit-resonator detuning and thus increasing the dispersive shift, we demonstrate a beyond-state-of-the-art two-state-readout error of only 0.25% in 100-ns integration time. Maintaining low-readout-drive strength, we nearly quadruple the signal-to-noise ratio of the readout by doubling the readout-mode line width, which we quantify by considering the hybridization of the readout resonator and its dedicated Purcell filter. We find excellent agreement between our experimental data and our theoretical model. The presented results are expected to further boost the performance of new and existing algorithms and protocols critically depending on high-fidelity fast midcircuit measurements. Published by the American Physical Society 2024
We analyze a single-shot readout for superconducting qubits via the controlled catch, dispersion, and release of a microwave field. A tunable coupler is used to decouple the microwave resonator from the transmission line during the dispersive qubit-resonator interaction, thus circumventing damping from the Purcell effect. We show that, if the qubit frequency tuning is sufficiently adiabatic, a fast high-fidelity qubit readout is possible, even in the strongly nonlinear dispersive regime. Interestingly, the Jaynes-Cummings nonlinearity leads to the quadrature squeezing of the resonator field below the standard quantum limit, resulting in a significant decrease of the measurement error.