We determine the optimal parameters for a simple and efficient scheme of dispersive readout of a qubit. Depending on the qubit state (ground or excited), the resonance of a cavity is shifted either to the red or to the blue side. Qubit state is inferred by detecting the photon number transmitted through the cavity. It turns out that this kind of detection provides better measurement fidelity than the detection of the presence or absence of photons only. We show that radiating the cavity on either of the frequencies it shifts to results in a suboptimal measurement. The optimal frequency of the probe photons is determined, as well as the optimal ratio of the shift to the resonator leakage. It is shown that to maximize the fidelity of a long-lasting measurement, it is sufficient to use the parameters optimizing the signal-to-noise ratio in the photon count. One can reach 99% fidelity for a single-shot measurement in various physical realizations of the scheme.
We determine the optimal parameters for a simple and efficient scheme of dispersive readout of a qubit. Depending on the qubit state (ground or excited), the resonance of a cavity is shifted either to the red or to the blue side. Qubit state is inferred by detecting the photon number transmitted through the cavity. It turns out that this kind of detection provides better measurement fidelity than the detection of the presence or absence of photons only. We show that radiating the cavity on either of the frequencies it shifts to results in a suboptimal measurement. The optimal frequency of the probe photons is determined, as well as the optimal ratio of the shift to the resonator leakage. It is shown that to maximize the fidelity of a long-lasting measurement, it is sufficient to use the parameters optimizing the signal-to-noise ratio in the photon count. One can reach 99% fidelity for a single-shot measurement in various physical realizations of the scheme.
Fast and high-fidelity qubit measurement is essential for realizing quantum error correction, which is in turn a key ingredient to universal quantum computing. For electron spin qubits, fast readout is one of the significant road blocks toward error correction. Here we examine the dispersive readout of a single spin in a semiconductor double quantum dot coupled to a microwave resonator. We show that using displaced squeezed vacuum states for the probing photons can improve the qubit readout fidelity and speed. Under condition of proper phase matching, we find that a moderate, and only moderate, squeezing can enhance both the signal-to-noise ratio and the fidelity of the qubit-state readout, and the optimal readout time can be shortened to the sub-microsecond range with above 99% fidelity. These enhancements are achieved at low probing microwave intensity, ensuring non-demolition qubit measurement.
High fidelity single shot qubit state readout is essential for many quantum information processing protocols. In superconducting quantum circuit, the qubit state is usually determined by detecting the dispersive frequency shift of a microwave cavity from either transmission or reflection. We demonstrate the use of constructive interference between the transmitted and reflected signal to optimize the qubit state readout, with which we find a better resolved state discrimination and an improved qubit readout fidelity. As a simple and convenient approach, our scheme can be combined with other qubit readout methods based on the discrimination of cavity photon states to further improve the qubit state readout.
We study the dispersive readout of a qubit in the ultimate limit of a single-photon probe. The use of a single-photon probe avoids the errors due to nonorthogonality of coherent states. A photodetector is used in the scheme we consider. The dynamics of the system is studied using the Heisenberg-Langevin equations. We treat the counter-rotating terms in the Hamiltonian perturbatively, which leads to the Bloch-Siegert shift in the resonator frequency. It is shown how the shift can improve the readout. The theory of photon transport through the qubit and the resonator it couples to is provided while taking the effect of the counter-rotating terms into account. To calculate the readout contrast, we use two approaches. The first one neglects the qubit relaxation and allows us to derive a compact expression for the contrast. Also, we obtain simple estimates for the system parameters to maximize the contrast. The second approach accounts for the qubit relaxation, which allows us to further improve the contrast. We demonstrate that for a readout time of $1\phantom{\rule{4pt}{0ex}}\ensuremath{\mu}\mathrm{s}$, a contrast of more than 75% can be achieved for an ideal detector and single-photon source.
Single-photon detectors are ubiquitous and integral components of photonic quantum cryptography, communication, and computation. Many applications, however, require not only detecting the presence of any photons, but distinguishing the number present with a single shot. Here, we implement a single-shot, high-fidelity photon number-resolving detector of up to 15 microwave photons in a cavity-qubit circuit QED platform. This detector functions by measuring a series of generalized parity operators, which make up the bits in the binary decomposition of the photon number. Our protocol consists of successive, independent measurements of each bit by entangling the ancilla with the cavity, then reading out and resetting the ancilla. Photon loss and ancilla readout errors can flip one or more bits, causing nontrivial errors in the outcome, but these errors have a traceable form, which can be captured in a simple hidden Markov model. Relying on the independence of each bit measurement, we mitigate biases in ensembles of measurements, showing good agreement with the predictions of the model. The mitigation improves the average total variation distance error of Fock states from 13.5--1.1. We also show that the mitigation is efficiently scalable to an M-mode system provided that the errors are independent and sufficiently small. Our work motivates the development of new algorithms that utilize single-shot, high-fidelity PNR detectors.
When a two-level system – a qubit – is used to probe a larger system, it naturally leads to answering a single yes-no question about the system state. Identifying what is the state of a system thus comes down to ask a series of binary questions iteratively to refine our knowledge. However, this approach leads to long measurement times for large systems, such as a resonator containing a large number of photons. In this thesis, we propose a new approach which enables us to make a measurement in a time, which is independent of the system size. This new measurement uses the qubit as an encoder of information about the system state into the many propagating modes of a transmission line. Assuming an ideal detector, we show that photon counting can then be implemented in a fixed time whatever the number of photons. We demonstrate the practicality of this approach by counting the number of photons in a microwave resonator coupled dispersively to a single superconducting qubit. We observe the qubit fluorescence dependence on the resonator photon number when the qubit is driven by a microwave monochromatic tone. Using the backaction of this dispersive measurement and post-selection, we evidence the photon counting ability of the measurement. The dephasing rate between two Fock states induced by the photon number measurement is measured and compared to theory. The latter allows us the study the non-linear dependence of the dephasing rate on the microwave drive amplitude.In a second time, the qubit fluorescence is probed using a frequency comb. Multiplexed heterodyne detections are simultaneously performed at each comb frequency and allow us to measure the photon number in the microwave resonator. This multiplexed measurement benefits from the recent bandwidth improvements of near quantum limited amplifiers. The limited cavity lifetime and detector efficiency prevented us from reaching single shot readout of the photon number in this proof-of-principle experiment. However, unlike in sequential measurement schemes, a single run of our experiment does provide, in parallel, partial information about the occupancy of each Fock state. Besides, we manage to observe the multiplexed measurement backaction on the resonator using direct Wigner tomography, which allowed us to measure the decoherence rate of the resonator induced by the measurement. We evidence an optimal qubit drive amplitude for information extraction, which matches the expected dynamics of a qubit under a multifrequency drive.
Fast, accurate, and low-leakage qubit measurement is a key requirement for quantum error correction. Here, we demonstrate measurement of a superconducting transmon qubit with a total duration of 97(1) ns, defined as the time from the start of the measurement pulse until the measurement-induced error on a subsequent π-pulse operation falls below 10−4. By combining a large state-averaged resonator decay rate of κeff/2π = 30.8 MHz with a dispersive shift close to the optimal SNR-per-photon condition, we achieve an assignment error of 0.17(1)% using a 58-ns measurement pulse, with residual readout photons depleting passively in tens of nanoseconds without an active depletion pulse. Using a repeated-measurement sequence together with a leakage-sensitive measurement, we benchmark the measurement-induced state transitions, finding a per-measurement leakage rate of 2.7(2)×10−5, only twice the background rate and two orders of magnitude below the measurement-induced relaxation rate, which dominates the assignment error. Floquet simulations indicate that the multiphoton resonances present at the operating point are weakly coupled and traversed diabatically, without causing leakage. These results demonstrate that a large resonator decay rate, combined with a dispersive shift close to the optimal SNR-per-photon condition, can enable fast, high-fidelity, low-leakage dispersive readout at small qubit-resonator detuning.