We present and demonstrate a general three-step method for extracting the quantum efficiency of dispersive qubit readout in circuit QED. We use active depletion of post-measurement photons and optimal integration weight functions on two quadratures to maximize the signal-to-noise ratio of the non-steady-state homodyne measurement. We derive analytically and demonstrate experimentally that the method robustly extracts the quantum efficiency for arbitrary readout conditions in the linear regime. We use the proven method to optimally bias a Josephson traveling-wave parametric amplifier and to quantify different noise contributions in the readout amplification chain.

Many protocols in quantum information processing, like quantum error correction,^{1,2} require rapid interleaving of qubit gates and measurements. These measurements are ideally nondemolition, fast, and of high fidelity. In circuit QED,^{3–5} a leading platform for quantum computing, nondemolition readout is routinely achieved by off-resonantly coupling a qubit to a resonator. The qubit-state-dependent dispersive shift of the resonator frequency is inferred by measuring the resonator response to an interrogating pulse by homodyne detection. A key element setting the speed and fidelity of dispersive readout is the quantum efficiency,^{6} which quantifies how the signal-to-noise ratio is degraded with respect to the limit imposed by quantum vacuum fluctuations.

In recent years, the use of superconducting parametric amplifiers^{7–11} as the front end of the readout amplification chain has boosted the quantum efficiency towards unity, leading to readout infidelity on the order of one percent^{12,13} in individual qubits. Most recently, the development of traveling-wave parametric amplifiers^{14,15} (TWPAs) has extended the amplification bandwidth from tens of MHz to several GHz and with a sufficient dynamic range to readout tens of qubits. For characterization and optimization of the amplification chain, the ability to robustly determine the quantum efficiency is an important benchmark.

A common method for quantifying the quantum efficiency *η* that does not rely on calibrated noise sources compares the information obtained in a weak qubit measurement (expressed by the signal-to-noise-ratio, SNR) to the dephasing of the qubit (expressed by the decay of the off-diagonal elements of the qubit density matrix),^{16,17} $\eta =SNR24\beta m$, with $e\u2212\beta m=|\rho 01(T)||\rho 01(0)|$, where *T* is the measurement duration. Previous experimental work^{14,18–20} has been restricted to fast resonators driven under specific symmetry conditions such that information is contained in only one quadrature of the output field and in a steady state. To allow *in-situ* calibration of *η* in multi-qubit devices under development,^{21–25} a method is desirable that does not rely on either of these conditions.

In this letter, we present and demonstrate a general three-step method for extracting the quantum efficiency of linear dispersive readout in cQED. Our method disposes with previous requirements in both the dynamics and the phase space trajectory of the resonator field while requiring two easily met conditions: the depletion of resonator photons post measurement^{26,27} and the ability to perform weighted integration of both quadratures of the output field.^{28,29} We experimentally test the method on a qubit-resonator pair with a Josephson TWPA (JTWPA)^{14} at the front end of the amplification chain. To prove the generality of the method, we extract a consistent value of *η* for different readout drive frequencies and drive envelopes. Finally, we use the method to optimally bias the JTWPA and to quantify different noise contributions in the readout amplification chain.

We first derive the method, obtaining experimental boundary conditions. For a measurement in the linear dispersive regime of cQED, the internal field *α*(*t*) of the readout resonator, driven by a pulse with envelope *εf*(*t*) and detuned by Δ from the resonator center frequency, is described by^{16,30}

where *κ* is the resonator linewidth and 2*χ* is the dispersive shift. The upper (lower) sign has to be chosen for the qubit in the ground $|0\u27e9$ [excited $|1\u27e9$] state. We study the evolution of the SNR and the measurement-induced dephasing as a function of the drive amplitude *ε* while keeping *T* constant. We find that the SNR scales linearly, SNR = *aε*, and that coherence elements exhibit a Gaussian dependence, $|\rho 01(T,\epsilon )|=|\rho 01(T,0)|e\u2212\epsilon 22\sigma m2$, with *a* and *σ*_{m} being dependent on *κ*, *χ*, Δ, and *f*(*t*). Furthermore, we find (supplementary material)

provided that two conditions are met. The conditions are as follows: (i) optimal integration functions^{28,29} are used to optimally extract information from both quadratures and (ii) the intra-resonator field vanishes at the beginning and end, i.e., photons are depleted from the resonator post-measurement.

To meet these conditions, we introduce a three-step experimental method: first, tuneup active photon depletion (or depletion by waiting) and calibration of the optimal integration weights; second, obtaining the measurement-induced dephasing of the variable-strength weak measurement by including the pulse within a Ramsey sequence; and third, measuring the SNR of the variable-strength weak measurement from single-shot readout histograms.

We test the method on a cQED test chip containing seven transmon qubits with dedicated readout resonators, each coupled to one of the two feedlines (see supplementary material). We present data for one qubit-resonator pair but have verified the method with other pairs in this and other devices. The qubit is operated at its flux-insensitive point with a qubit frequency *f*_{q} = 5.070 GHz, where the measured energy relaxation and echo dephasing times are *T*_{1} = 15 *μ*s and *T*_{2,echo} = 26 *μ*s, respectively. The resonator has a low-power fundamental at $fr,|0\u27e9=7.852400\u2009GHz\u2009(fr,|1\u27e9=fr,|0\u27e9+\chi /\pi =7.852295\u2009GHz)$ for qubit in $|0\u27e9\u2009(|1\u27e9)$, with linewidth *κ*/2*π* = 1.4 MHz. The readout pulse generation and readout signal integration are performed by single-sideband mixing. Pulse-envelope generation, demodulation, and signal processing are performed using a Zurich Instruments UHFLI-QC with 2 AWG channels and 2 ADC channels running at 1.8 GSample/s with 14- and 12-bit resolution, respectively.

In the first step, we tune up the depletion steps and calibrate the optimal integration weights. We use a measurement ramp-up pulse of duration *τ*_{up} = 600 ns, followed by a photon-depletion counter pulse^{26,27} consisting of two steps of 200 ns duration each, for a total depletion time *τ*_{d} = 400 ns. To successfully deplete without relying on symmetries that are specific to a readout frequency at the midpoint between ground and excited state resonances (i.e., Δ = 0), we vary 4 parameters of the depletion steps (details are provided in the supplementary material). From the averaged transients of the finally obtained measurement pulse, we extract the optimal integration weights given by^{28,29} the difference between the averaged transients for $|0\u27e9$ and $|1\u27e9$ [Fig. 1(a)]. The success of the active depletion is evidenced by the nulling at the end of *τ*_{d}. In this initial example, we connect to previous work by setting Δ = 0, leaving all measurement information in one quadrature.

We next use the tuned readout to study its measurement-induced dephasing and SNR to finally extract *η*. We measure the dephasing by including the measurement-and-depletion pulse in a Ramsey sequence and varying its amplitude, *ε* [Figs. 1(b)]. By varying the azimuthal angle of the second qubit pulse, we allow distinguishing dephasing from deterministic phase shifts and extract $|\rho 01|$ from the amplitude of the fitted Ramsey fringes. The SNR at various *ε* is extracted from single-shot readout experiments, preparing the qubit in $|0\u27e9$ and $|1\u27e9$ [Figs. 1(c)]. We use double Gaussian fits in both cases, neglecting measurement results in the spurious Gaussians to reduce corruption by imperfect state preparation and residual qubit excitation and relaxation. As expected, as a function of $\epsilon ,\u2009|\rho 01|$ decreases following a Gaussian form and the SNR increases linearly [Fig. 1(d)]. The best fits to both dependencies give *η*_{e} = 0.165 ± 0.002. Note that both dephasing and SNR measurements include ramp-up, depletion, and an additional *τ*_{buffer} = 100 ns, making the total measurement window *T* = *τ*_{up} + *τ*_{d} + *τ*_{buffer} = 1100 ns.

We next demonstrate the generality of the method by extracting *η* as a function of the readout drive frequency. We repeat the method at fifteen readout drive detunings over a range of 2.8 MHz ∼ *κ*/*π* ∼ 14*χ*/*π* around Δ = 0 [Figs. 2(a) and 2(b)]. Furthermore, we compare the effect of using optimal weight functions versus square weight functions and the effect of using active versus passive photon depletion. The square weight functions correspond to a single point in phase space during *T*, with unit amplitude and an optimized phase maximizing SNR. We satisfy the zero-photon field condition by depleting the photons actively with *T* = 1100 ns (as in Fig. 1) or passively by waiting with *T* = 2100 ns. When information is extracted from both quadratures using optimal weight functions, we measure an average *η*_{e} = 0.167 with a standard deviation of 0.04. The extracted optimal integration functions in the time domain [Figs. 2(c) and 2(d)] show how the resonator returns to the vacuum for both active depletion and passive depletion. Square weight functions are not able to track the measurement dynamics in the time domain (even at Δ = 0), leading to a reduction in *η*_{e}. Figures 2(e) and 2(f) show the weight functions in phase space. The opening of the trajectories with detuning illustrates the rotating optimal measurement axis during the measurement and leads to a further reduction of the increase in *η*_{e} when square weight functions are used. The dynamics and the *η*_{e} dependence on Δ are excellently described by the linear model, which uses Eq. (1), the separately calibrated *κ* and *χ* [Fig. 2(a)], and *η* = 0.1670 (details are given in the supplementary material). Furthermore, the matching of the dynamics and depletion pulse parameters (see supplementary material) when using active photon depletion confirms the numerical optimization techniques.

To further test the robustness of the method to arbitrary pulse envelopes, we have used a measurement-and-depletion pulse envelope *f*(*t*) resembling a typical Dutch skyline. The pulse envelope outlines five canal houses, of which the first three ramp up the resonator and the latter two are used as the tunable depletion steps. Completing the three steps, we extract (see supplementary material) *η*_{e} = 0.167 ± 0.005, matching our previous value to within error.

We use the proven method to optimally bias the JTWPA and to quantify different noise contributions in the readout chain. To this end, we map *η*_{e} as a function of pump power and frequency, just below the dispersive feature of the JTWPA, finding the maximum *η*_{e} = 0.1670 at (*P*_{pump} = –71.0 dBm and *f*_{pump} = 8.13 GHz) [Figs. 3(a)–3(c)]. We next compare the obtained *η*_{e} at the optimal bias frequency to independent low-power measurements of the JTWPA gain *G*_{JTWPA}, and we find *G*_{JTWPA} = 21.6 dB at the optimal bias point. We fit this parametric plot with a three-stage model, with noise contributions before, in, and after the JTWPA, *η*(*G*_{JTWPA}) = *η*_{pre} × *η*_{JTWPAd} (*G*_{JTWPA}) × *η*_{post} (*G*_{JTWPA}). The parameter *η*_{pre} captures losses in the device and the microwave network between the device and the JTWPA and is therefore independent of *G*_{JTWPA}. The JTWPA has a distributed loss along the amplifying transmission line, which is modeled as an array of interleaved sections with quantum-limited amplification and sections with attenuation adding up to the total insertion loss of the JTWPA (as in Ref. 14). Finally, the post-JTWPA amplification chain is modeled with a fixed noise temperature, whose relative noise contribution diminishes as *G*_{JTWPA} is increased. The best fit [Figs. 3(d) and 3(e)] gives *η*_{pre} = 0.22, consistent with 50% photon loss due to symmetric coupling of the resonator to the feedline input and output, an attenuation of the microwave network between the device and the JTWPA of 2 dB and residual loss in the JTWPA of 27%. We fit a distributed insertion loss of the JTWPA of 4.6 dB, closely matching the separate calibration of 4.2 dB [Fig. 3(c)]. Finally, we fit a noise temperature of 2.6 K, close to the HEMT amplifier's factory specification of 2.2 K.

We identify room for improving *η*_{e} to ∼0.5 by implementing Purcell filters with asymmetric coupling^{20,31} (primarily to the output line) and decreasing the insertion loss in the microwave network, by optimizing the setup for shorter and superconducting cabling between the device and JTWPA.

In conclusion, we have presented and demonstrated a general three-step method for extracting the quantum efficiency of linear dispersive qubit readout in cQED. We have derived analytically and demonstrated experimentally that the method robustly extracts the quantum efficiency for arbitrary readout conditions in the linear regime. This method will be used as a tool for readout performance characterization and optimization.

See supplementary material for a description of the linear model, the derivation of Eq. (2), a description of the depletion tuneup, and additional figures.

We thank W. D. Oliver for providing the JTWPA, N. K. Langford for experimental contributions, M. A. Rol for software contributions, and C. Dickel and F. Luthi for discussions. This research was supported by the Office of the Director of National Intelligence (ODNI), Intelligence Advanced Research Projects Activity (IARPA), via the U.S. Army Research Office Grant No. W911NF-16-1-0071. Additional funding was provided by Intel Corporation and the ERC Synergy Grant QC-lab. The views and conclusions contained herein are those of the authors and should not be interpreted as necessarily representing the official policies or endorsements, either expressed or implied, of the ODNI, IARPA, or the U.S. Government. The U.S. Government is authorized to reproduce and distribute reprints for Governmental purposes notwithstanding any copyright annotation thereon.

## References

*η*= 1 corresponds to an ideal phase preserving amplification. Furthermore, we define the SNR as the average separation of the integrated homodyne voltage histograms as obtained from single-shot readout experiments for $|0\u27e9$ and $|1\u27e9$, divided by their average standard deviation (see supplementary material).