The conventional method of qubit measurements in circuit QED is employing the dispersive regime of qubit-cavity coupling, which results in an approximated scheme of quantum nondemolition (QND) readout. This scheme becomes problematic in the case of strong coupling and/or strong measurement drive, owing to the so-called Purcell effect. A recent proposal by virtue of longitudinal coupling suggests a new scheme to realize fast, high-fidelity, and ideal QND readout of qubit state. The aim of the present work is twofold: (i) In parallel to what has been done in the past years for the dispersive readout, we carry out the gradual partial-collapse theory for this recent scheme, in terms of both the quantum trajectory equation and quantum Bayesian approaches. The partial-collapse weak measurement theory is useful for such as the measurement-based feedback control and other quantum applications. (ii) In the physical aspect, we construct the joint qubit-plus-cavity entangled state under continuous measurement and present a comprehensive analysis for the quantum efficiency, qubit-state purity, and signal-to-noise ratio in the output currents. The combination of the joint state and the quantum Bayesian rule provides a generalized scheme of cavity reset associated with the longitudinal coupling, which can restore the qubit to a quantum pure state from entanglement with the cavity states, and thus benefits the successive partial-collapse measurements after qubit rotations.
The conventional method of qubit measurements in circuit QED is employing the dispersive regime of qubit-cavity coupling, which results in an approximated scheme of quantum nondemolition (QND) readout. This scheme becomes problematic in the case of strong coupling and/or strong measurement drive, owing to the so-called Purcell effect. A recent proposal by virtue of longitudinal coupling suggests a new scheme to realize fast, high-fidelity, and ideal QND readout of qubit state. The aim of the present work is twofold: (i) In parallel to what has been done in the past years for the dispersive readout, we carry out the gradual partial-collapse theory for this recent scheme, in terms of both the quantum trajectory equation and quantum Bayesian approaches. The partial-collapse weak measurement theory is useful for such as the measurement-based feedback control and other quantum applications. (ii) In the physical aspect, we construct the joint qubit-plus-cavity entangled state under continuous measurement and present a comprehensive analysis for the quantum efficiency, qubit-state purity, and signal-to-noise ratio in the output currents. The combination of the joint state and the quantum Bayesian rule provides a generalized scheme of cavity reset associated with the longitudinal coupling, which can restore the qubit to a quantum pure state from entanglement with the cavity states, and thus benefits the successive partial-collapse measurements after qubit rotations.
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.
Compared with the quantum trajectory equation (QTE), the quantum Bayesian approach has the advantage of being more efficient to infer a quantum state under monitoring, based on the integrated output of measurements. For weak measurement of qubits in circuit quantum electrodynamics (cQED), properly accounting for the measurement backaction effects within the Bayesian framework is an important problem of current interest. Elegant work towards this task was carried out by Korotkov in 'bad-cavity' and weak-response limits (Korotkov 2011 Quantum Bayesian approach to circuit QED measurement (arXiv: 1111.4016 )). In the present work, based on insights from the cavity-field states (dynamics) and the help of an effective QTE, we generalize the results of Korotkov to more general system parameters. The obtained Bayesian rule is in full agreement with Korotkovʼs result in limiting cases and as well holds satisfactory accuracy in non-limiting cases in comparison with the QTE simulations. We expect the proposed Bayesian rule to be useful for future cQED measurement and control experiments.
Superconducting electrical circuits can be used to study the physics of cavity quantum electrodynamics (QED) in new regimes, therefore realizing circuit QED. For quantum-information processing and quantum optics, an interesting regime of circuit QED is the dispersive regime, where the detuning between the qubit transition frequency and the resonator frequency is much larger than the interaction strength. In this paper, we investigate how nonlinear corrections to the dispersive regime affect the measurement process. We find that in the presence of pure qubit dephasing, photon population of the resonator used for the measurement of the qubit act as an effective heat bath, inducing incoherent relaxation and excitation of the qubit. Measurement thus induces both dephasing and mixing of the qubit, something that can reduce the quantum nondemolition aspect of the readout. Using quantum trajectory theory, we show that this heat bath can induce quantum jumps in the qubit state. Nonlinear effects can also reduce the achievable signal-to-noise ratio of a homodyne measurement of the voltage.
Realizing fault-tolerant quantum computing relies on efficient quantum error correction. Bosonic cat-state qubits offer a highly promising, hardware-efficient approach by intrinsically suppressing bit-flip errors. However, fully unlocking their potential demands fast and high-fidelity quantum nondemolition (QND) measurements. Here, we propose a high-fidelity QND readout protocol for cat-state qubits in a Kerr-nonlinear resonator utilizing an effective longitudinal interaction. By deriving analytical expressions and performing numerical simulations, we demonstrate that longitudinal readout achieves faster signal-to-noise ratio (SNR) growth and shorter measurement times than conventional dispersive readout. Furthermore, we show that injecting squeezed input states can exponentially enhance the SNR, while time-dependent coupling further reduces the measurement time. This scheme provides a robust pathway toward rapid and nondestructive measurement of cat qubits in near-term quantum processors. Reliable quantum computers require error-resistant components like cat-state qubits, along with fast, non-destructive ways to readout them. Here, the authors propose a longitudinal readout protocol for these qubits that achieves faster and higher-fidelity measurements than conventional methods.
We introduce and analyze a dispersive qubit readout scheme where two-mode squeezing is generated directly in the measurement cavities. The resulting suppression of noise enables fast, high-fidelity readout of naturally weakly coupled qubits, and the possibility to protect strongly coupled qubits from decoherence by weakening their coupling. Unlike other approaches exploiting squeezing, our setup avoids the difficult task of transporting and injecting with high fidelity an externally generated squeezed state. Our setup is also surprisingly robust against unwanted non-QND backaction effects, as interference naturally suppresses Purcell decay: the system acts as its own Purcell filter. Our setup is compatible with the experimental state-of-the-art in circuit QED systems, but the basic idea could also be realized in other systems.
The solid-state superconducting circuit-QED (quantum electrodynamics) system is a promising candidate for quantum computing and quantum information processing, which serves also as an ideal platform for quantum measurement and quantum control studies. In this context, a large number of cavity photons may be involved in the quantum dynamics and will degrade the simulation efficiency. To avoid this difficulty, it is helpful to eliminate the degrees of freedom of the cavity photons, and obtain an effective master-equation description which contains only the qubit states. In this work, we examine two such schemes, the adiabatic elimination (AE) and the more recently proposed polaron transformation (PT) approaches, by comparing their results with exact numerical simulations. We find that in the absence of qubit-flip, which is a specific quantum nondemolition (QND) measurement, the PT scheme is superior to the AE method. Actually, in this case the PT scheme catches the measurement dynamics exactly. However, in the presence of qubit-flip such as for qubit oscillation measurement, the PT scheme is no longer better than the AE approach. We conclude that both schemes, in the weak measurement regime, can work almost equally well. This corresponds to strong cavity damping or weak coupling between the qubit and cavity photons. Out of this regime, unfortunately, one has to include the cavity photons into numerical simulations and more advanced methods/techniques are waiting for their exploration in this field.