Abstract
We propose and demonstrate a readout technique for a superconducting qubit by dispersively coupling it with a Josephson parametric oscillator. We employ a tunable quarter wavelength superconducting resonator and modulate its resonant frequency at twice its value with an amplitude surpassing the threshold for parametric instability. We map the qubit states onto two distinct states of classical parametric oscillation: one oscillating state, with 185±15 photons in the resonator, and one with zero oscillation amplitude. This high contrast obviates a following quantumlimited amplifier. We demonstrate proofofprinciple, singleshot readout performance, and present an error budget indicating that this method can surpass the fidelity threshold required for quantum computing.
Introduction
The readout scheme for quantum bits of information (qubits) constitutes one essential component of a quantum information processor^{1}. During the course of a quantum algorithm, qubitstate errors need to be corrected; in many implementations, this is done by quantum error correction, where each operation is based on the outcomes of stabilizer measurements that indicate the qubit errors. The stabilizers must therefore be determined in a single shot—without averaging of the output signals of repeated measurements on identically prepared qubits—with fidelity exceeding approximately 99% (ref. 2).
The commonly used measurement scheme for a superconducting qubit coupled with a linear microwave resonator does not, by itself, offer singleshot measurement performance. The qubit imparts a statedependent (dispersive) frequency shift on the resonator, which can be determined by applying a probe signal and measuring the reflected or transmitted signal, although only for weak probing, rendering an inadequate signaltonoise ratio (SNR)^{3,4}.
Researchers have addressed the problem of insufficient SNR in essentially two ways. One approach is to feed the weak output signal into a following, parametric linear amplifier that adds only the minimum amount of noise allowed by quantum mechanics^{5,6,7,8}. Another approach is to insert a nonlinear element into the system and apply a strong drive tone, such that the resonator enters a bistable regime, hence enhancing the detection contrast^{9,10,11,12,13}.
In this paper, we propose and demonstrate a simplified readout technique in which a superconducting qubit is directly integrated into a Josephson parametric oscillator (JPO). We map the qubit states onto the ground and excited states of the oscillator, and demonstrate proofofconcept, singleshot readout performance (SNR>1). We obtain 81.5% qubitstate discrimination for a readout time ; however, from the error analysis, we infer a readout fidelity of 98.7±1.2%, taking into account known and reparable errors due to qubit initialization and decoherence (17.2±1.2%). A realistically achievable qubit relaxation time, T_{1}=50 μs, and a Purcell bandpass filter would reduce these errors from 17.2 to <0.5%, as well as shorten the required readout time to . The remaining errors, which are due to the switching events in the oscillator (1.2±0.3%), can be eliminated by improving the data aquisition protocol—see Discussion and Supplementary Note 1. These qubit and detection improvements would bring the readout fidelity to ≈99.5%.
Our readout scheme relies on parametric pumping of a frequencytunable resonator by modulation of its inductance. The pumping amplitude exceeds the threshold for parametric instability, the point above which the resonator oscillates spontaneously, even in the absence of an input probe signal. This instability threshold is controlled by the state of the qubit, whose ground and excited states correspond to the nonoscillating and oscillating states of the resonator, respectively. In our measurement, the oscillating state produces a steadystate resonator field corresponding to 185±15 photons, whose output we can clearly distinguish from the nonoscillating state when followed by a commercial semiconductor amplifier, eliminating the need for a quantumlimited amplifier. Conceptually, this method can yield arbitrarily large contrast due to the parametric instability, and moreover, only requires a pump but no input signal.
This readout scheme is well aligned with scalable, multiqubit implementations. Parametric oscillators can be readily frequencymultiplexed^{14} and allow for a simplified experimental setup (compared with conventional microwave reflectometry) without a separate input port to the resonator or a following parametric amplifier, and consequently, also without additional bulky microwave circulators that would normally route the input and parametric pumping tones. It is also possible to manipulate the qubit via the fluxpumping line only, which further reduces the number of cables and interconnects.
Results
The Josephson parametric oscillator
Our device consists of a quarter wavelength (λ/4), superconducting coplanar waveguide resonator, shorted to ground in one end via two parallel Josephson tunnel junctions (JJs)—see Fig. 1a. The JJs form a superconducting quantum interference device (SQUID), which acts as a variable Josephson inductance, , where I_{0} is the critical current and Φ_{0} is the flux quantum. This inductance can be controlled by the external magnetic flux through the SQUID loop, Φ(t)=Φ_{d.c.}+Φ_{a.c.}(t), and by the superconducting phase difference across the JJs, φ(t), via its current—phase relation, I(t)=I_{0} sinφ(t).
Timevarying modulations of Φ and φ—parametric pumping—affect the resonator dynamics, albeit in rather different ways; moreover, the Josephson inductance is indeed both parametric and nonlinear. We explain these differences in the Discussion section below. The resonant frequency of the JPO is parametrically modulated via the magnetic flux, Φ(t), which can lead to frequency mixing as well as parametric effects such as noiseless amplification of a signal, frequency conversion and instabilities^{6,15,16,17,18,19}.
The state of the JPO has a rich dependence on several parameters, some of which was studied recently, both theoretically^{20,21} and experimentally^{7,17,19}. The equation of motion for the intraresonator electric field amplitude, A, can be written as
Here is proportional to the externally applied pump amplitude, Φ_{a.c.}, which modulates the resonant frequency parametrically at close to twice its value, ω_{p}≈2ω_{r} (degenerate pumping), and δ=ω_{p}/2−ω_{r} is the resonator’s detuning from half of the pump frequency. The field amplitude, A, and its complex conjugate, A*, are slow variables in a frame rotating at ω_{p}/2, and A^{2} is the equivalent number of photons in the resonator. The Duffing parameter, α, associated with a cubic field nonlinearity, arises from the nonlinear Josephson inductance. The linear damping rate has two components, Γ=Γ_{0}+Γ_{R}, where Γ_{0}/2π=1.02 MHz is the external damping rate, associated with the photon decay through the coupling capacitor, and Γ_{R}/2π=0.30 MHz is the internal loss rate. The equation’s righthand side represents the input probe signal, such that B(t)^{2} has units of photons per second. The output flow of photons per second, C(t)^{2}, is given by .
For low pumping amplitude, below the parametric instability threshold, , this device works as a phasesensitive parametric amplifier (JPA) for an input B(t) at signal frequency ω_{s}=ω_{p}/2 (refs 6, 15, 16, 17, 22). Note, however, that we keep B(t)=0 in the measurements reported here. For a pumping amplitude exceeding the threshold, , spontaneous parametric oscillations set in—see Fig. 1b and equation (10) in Methods. The resonator field builds up exponentially in time, even in the absence of an input probe signal until it becomes limited by the Duffing and pumpinduced nonlinearities and reaches a steady state^{17,19}.
We connected a transmon qubit capacitively to the resonator^{23}—see Fig. 1a. The state of the JPO (oscillating or nonoscillating) can then be controlled by the qubitstatedependent, dispersive frequency shift, χ, which the qubit exerts on the resonator^{24,25}. When the JPO is being pumped above the threshold for parametric oscillation, with amplitude and frequency detuning, δ, then a change of qubit state effectively pulls the resonator to a different value of the detuning, outside of the region of parametric oscillations—see Fig. 1b. We denote the qubitstatedependent detunings by δ^{0〉}=δ−χ and δ^{1〉}=δ+χ. The resulting mapping of the qubit state onto the average number of photons in the resonator provides us with a qubitstate readout mechanism, which we exploit in this work.
Characterization of qubit and JPO
The device and cryogenic experimental setup are depicted in Fig. 1a. The sample is thermally anchored to the mixing chamber of a dilution refrigerator with a base temperature of 10 mK. The parametric λ/4 resonator (in blue) is capacitively coupled with the transmission line (C_{c}=11.9 fF), yielding an external quality factor Q_{ext}=ω_{r}/2Γ_{0}=2555. A transmon qubit (in red) is also coupled near this end of the resonator.
The resonator output signal is amplified using a 4–8 GHz highelectronmobility transistor amplifier, with a noise temperature T_{N}=2.2 K, followed by two roomtemperature amplifiers. We detect the outgoing signal using heterodyne mixing. The signal is first downconverted to a frequency (ω_{RF}−ω_{LO})/2π=187.5 MHz; then, the [I,Q]quadrature voltages are sampled at 250 MS s^{−1}, before they are digitally downsampled at a rate of 20 MS s^{−1}.
We first characterize the transmon spectroscopically—see Fig. 2a—from which we extract the Josephson and charging energies, E_{J}/2π=9.82 GHz and E_{C}/2π=453 MHz, respectively. From the vacuum Rabi splitting, we extract a qubit−resonator coupling rate g_{01}/2π=46 MHz—see Fig. 2b.
Next, we fit the frequency tuning curve of the resonator (with the qubit in the 0〉state) to the relation
where F=πΦ_{d.c.}/Φ_{0} denotes the static flux bias, normalized to the magnetic flux quantum. The effective dispersive shift due to the qubit is
which, in turn, depends on the qubit–resonator detuning, Δ(F)=ω_{a}(F′)−ω_{r}(F′), with F′=F/8.88+0.58 representing the effective magnetic flux of the transmon. Moreover, the qubit and resonator frequency spectra are well approximated by^{23,26}
where ω_{λ/4}/2π=5.55 GHz is the bare resonant frequency (in absence of the SQUID), and γ_{0}=L_{J}(F=0)/L_{r}=5.3±0.1% is the inductive participation ratio between the SQUID (at zero flux) and the resonator. The solid grey and red lines in Fig. 2a are fits to equations (2) and (4), respectively.
Singleshot qubit readout
We now demonstrate our method for reading out the qubit with the JPO. We choose a static flux bias point F=0.185π for the resonator SQUID, corresponding to a resonant frequency and qubit transition frequency ω_{a}/2π=4.885 GHz—see dashed grey line in Fig. 2a. Consequently, the qubit–resonator detuning is Δ/2π=−334 MHz, and the effective dispersive shift is 2χ/2π=−7.258 MHz. We measured a Purcelllimited qubit relaxation time, T_{1}=4.24±0.21 μs, and Ramsey freeinduction decay time —see Methods, Supplementary Fig. 1 and Supplementary Table 1.
To operate the parametric oscillator as a highfidelity qubit readout device, we must be able to map the states of the qubit onto different states of the oscillator, which we must then clearly distinguish. We encode the qubit ground state 0〉 in the ‘quiet’ state (the empty resonator) and the excited state 1〉 in the ‘populated’ state of the resonator. Figure 3a shows the pulse sequence for qubit manipulation and readout, and Fig. 3b shows the resulting output from the JPO, operated with the pump settings δ^{0〉}/Γ=−5.34, .
The populated oscillator in Fig. 3b contains 185±15 photons. We obtained this estimate from a comparison between the probeamplitude dependence of the resonant frequency and the expected photon number dependence of the Duffing shift—see Methods and Supplementary Fig. 3. This number of photons should be compared with A^{2}=200±3 photons, which is the solution to equation (1) in the steady state .
To achieve such clear qubitstate discrimination as in Fig. 3b, we needed to make a judicious choice of flux bias point, F, to mitigate the effects of two nonlinear shifts of the resonant frequency^{19}. The Duffing shift dominates when F→±π/2, whereas a pumpinduced frequency shift dominates when F→0. These shifts can move the resonator away from the proper pump condition, thereby effectively restricting the output power—see Methods and Supplementary Fig. 2.
Moreover, the qubit−resonator detuning should be in the dispersive regime , in which the qubit state controls the resonant frequency of the resonator. Yet it must yield a sufficiently large dispersive shift, χ>Γ (equation 3), to produce clearly distinguishable output levels, corresponding to the 0〉 and 1〉 states. For our chosen flux bias point, we identify the optimal pump settings by mapping out the parametric oscillation region as a function of pump frequency and amplitude—see Fig. 4a,b.
An interesting feature is present within the left half of Fig. 4a,b (where the populated resonator encodes 1〉). Here when the qubit is initially in the 1〉 state, the resonator latches into its oscillating state for as long as the pump is kept on, and does not transition into its quiet state when the qubit relaxes, as one might have expected. This latching is shown by the blue trace in Fig. 3b. We attribute it to the existence of a tristable oscillation state^{17,21}, associated with red detuning of the abovethreshold region for the 0〉 state. When the qubit relaxes, there occurs an instantaneous shift of the pseudopotential for the amplitude A, from bistable (with two πshifted, finiteamplitude states; see Fig. 1b) to tristable (with one additional zeroamplitude state). The field’s initial condition at the time of this shift, A≠0, causes the resonator to maintain its oscillating state. A separate study of this latching feature will be reported elsewhere.
We evaluate the obtainable state discrimination by collecting quadrature voltage histograms at every point within the two regions of parametric oscillations in the plane—see Fig. 4c. We choose the pump operation point δ^{0〉}/Γ=−5.34, , indicated by the black circle, and show the characterization in detail in Fig. 5. In this point, the state discrimination has reached a plateau around 81.5%. Each histogram in Fig. 5a,b contains inphase (V_{I}) and quadrature (V_{Q}) voltage measurements from 10^{5} readout cycles, with each measurement being the mean quadrature voltage within the sampling time (blue window in Fig. 3). We project each of the 2D histograms onto its real axis, and thus construct 1D histograms of the V_{I} component—see Fig. 5c. We can then extract a SNR, , where μ and σ denote the mean value and s.d., respectively, of the Gaussians used to fit the histograms. The peak separation of the histograms gives a confidence level of 99.998% for the readout fidelity. The peak appearing in the centre of the blue trace arises mainly from qubit relaxation before and during the readout. We analyse this and other contributions in the next section, as well as in Supplementary Note 1 and Supplementary Fig. 4.
To extract the measurement fidelity from the histograms, we plot the cumulative distribution function of each of the two traces in Fig. 5c, by summing up the histogram counts symmetrically from the centre and outward, using a voltage threshold, V_{th}. From these sums, we obtain the Scurves of the probability to find the qubit in its ground state as a function of the voltage threshold value—see Fig. 5d. We define the fidelity of the measurement as the maximum separation between the two Scurves.
Discussion
To evaluate the fidelity of the readout itself, as compared with the fidelity loss associated with qubit errors, we now present an error budget. From the histograms in Fig. 5c, we can account for 81.5% of the population, thus missing 18.5%. To understand the remaining contributions, we run a Monte Carlo simulation of the qubit population, consisting of the same number of 10^{5} readout cycles as in the measured histograms. The simulation results are binned in the same way as the measurements, using the Gaussian fits as boundaries, and taking into account the following statistics: first, qubit relaxation and preparation errors; second, thermal population of the qubit; third, spurious switching events by πradians of the oscillator phase during readout (yielding a reduced sampled voltage); and fourth, peak separation error due to the limited SNR.
We find that the main contribution to the loss of fidelity is due to qubit relaxation before and during the readout. From the measured relaxation time, T_{1}=4.24±0.21 μs, we obtain a fidelity loss of 11.6±0.5%. However, this error can be reduced substantially (to <0.5%) by introducing a Purcell bandpass filter^{27,28,29} at the output of the JPO; since the qubit is detuned from the JPO, this decreases its relaxation into the 50Ω transmission line. Such a filter would allow us to increase the resonator damping rate, Γ_{0}, substantially reducing the readout time without compromising T_{1}. This is shown in Supplementary Note 2 and Supplementary Table 2. Note, however, that an increased resonator damping rate yields an increased width of the parametric oscillation region: consequently, the qubit–resonator coupling, g_{01}, and detuning, Δ, need to be chosen accordingly to render a sufficiently large dispersive frequency shift.
From the simulation, we further attribute 4.5±0.3% to qubit preparation errors. Another 1.1±0.4% can be explained from thermal population of the qubit; the effective qubit temperature is T_{q}=45±3 mK. By adding these fidelity loss contributions due to the qubit to the measured state discrimination, we can account for 81.5%+11.6±0.5%+4.5±0.3%+1.1±0.4%=98.7±1.2%.
There are also errors introduced by the parametric oscillator itself: switchings between the πshifted oscillating states reduce the overall measured voltage. We performed a separate control measurement that yielded 2.4±0.5% switching probability, which translates into a maximal fidelity loss of half of that, 1.2±0.25%. The switching rate of the parametric oscillator depends on many parameters, including damping rates and bias points; this error can therefore, with careful engineering, be decreased even further. We could, however, eliminate the effect of phaseswitching events using a rectifying detection scheme, for example, a diode or a fieldprogrammable gate array, tracking the absolute value of the output field instead of its amplitude.
The last and smallest contribution to the fidelity loss is the peak separation error, which accounts for the intrinsic overlap between the histograms. However, this contribution is <0.002% for our SNR of 3.39, and can therefore be neglected. For details on the error budget analysis, see Supplementary Note 1 and Supplementary Fig. 4.
By combining the abovementioned improvements (reduced qubit relaxation rate, optimized qubit manipulations and cooling, enhanced resonator output coupling, and rectifying data acquisition), the readout fidelity could realistically reach ≈99.5%, limited only by the qubit relaxation.
Finally, we demonstrate that the relaxation time of our qubit is not measurably afflicted by the pump—see Methods and Supplementary Fig. 1. Our measurement scheme is, in principle, quantum nondemolition, see Supplementary Note 3; however, a proper experimental and theoretical assessment of the backaction is outside the scope of this work.
Table 1 puts our results in the context of previous work on parametric and nonlinear Josephson amplification and detection circuits.
A fluxpumped, parametric phaselocked oscillator was used as a following amplifier, also enabling sensitive qubit readout^{7}. In our work, the qubit was directly coupled with the JPO, which simplifies the experimental setup by reducing the number of microwave components needed. Also, with a pumping amplitude below the parametric instability threshold, the fluxpumped JPA has been used to readout one qubit^{6}, as well as multiple qubits coupled with the same bus resonator^{28}.
There is another way of operating our device: instead of pumping the flux at ω_{p}≈2ω_{r}, we can apply an alternating pump current (, B(t≠0)), now at a frequency close to resonance, ω_{p}≈ω_{r}, and thereby directly modulate the phase difference, φ. Both methods can provide linear parametric gain on reflection of a detuned signal (ω_{s}≠ω_{p}/2 and ω_{s}≠ω_{p}, respectively). The fluxpumped JPA has a very wide frequency separation between pump tone and signal, because ω_{s}≈ω_{r}≈ω_{p}/2, which is a practical advantage since it makes the resonator’s entire instantaneous bandwidth available for amplification with no need to suppress or filter out the pump tone. Moreover, the λ/4 resonator has no mode in the vicinity of ω_{p} that the pump might otherwise populate.
We emphasize that there are indeed two different physical mechanisms in play, since flux and current pumping address orthogonal variables in the sense that and , where and denote the gaugeinvariant phase differences across the two parallel JJs. This distinction is also evident in equation (1). The parametric fluxpumping term, , modulates the resonant frequency; it couples the resonator field amplitude and its complex conjugate, which can provide quadrature squeezing of an input signal and enables phasesensitive parametric amplification; and for stronger modulation, there is a parametric instability threshold into the JPO regime—see Fig. 1b.
Current pumping by an input B(t), on the other hand, corresponds to an external force that directly contributes to the intraresonator field A and drives its nonlinear term αA^{2}. For zero detuning, ω_{s}=ω_{p}, this is the driven Duffing oscillator that has no gain (it offers no phasesensitive amplification); for stronger driving there occurs, a dynamical bifurcation but no internal instability or parametric oscillations.
Current pumping with a moderate amplitude is used for linear amplification with the JPA^{30,31}, which enabled, for example, the observation of quantum jumps in a qubit^{5}. Current modulation is also used in the latching detection scheme of the Josephson bifurcation amplifier^{9,10,14,32,33}. There, a higheramplitude input strongly drives the Duffing nonlinearity near its bifurcation point; the two qubit states can then be mapped onto two different resonator output field amplitudes. The Josephson bifurcation amplifier was used for quantum nondemolition measurement of a qubit, and in a lumpedelement resonator^{11}, in which a qubitstatesensitive autoresonance was observed in response to a frequencychirped current drive. Yet another method is to couple the qubit with a linear resonator, which inherits a crossKerr nonlinearity from the qubit; current pumping of the resonator can then yield a strong output signal that depends on the qubit state^{12,13}.
In conclusion, we have introduced a singleshot readout technique for superconducting qubits—the JPO readout. We demonstrated proofofprinciple operation, obtaining a barestate discrimination of 81.5%. After correcting for known and reparable errors, this translates into an inferred readout fidelity of 98.7±1.2%, which by implementing a rectifying detection scheme can be further increased by 1.2±0.3%. With foreseeable improvements and optimization, this device would be an attractive candidate for implementing multiqubit readout in the context of scalable error correction schemes. This fidelity and the readout time are both amenable to optimization.
Our system integrates a parametric readout mechanism into the resonator to which the qubit is coupled, substantially reducing the number of components needed to perform singleshot readout in a circuit quantum electrodynamics architecture. Advantages offered by this readout technique include the potential for multiplexing and scalability with no need for signalprobe inputs, additional microwave circulators, or separate parametric amplifiers. As opposed to other integrated readout devices, our pump frequency is far outside of the resonator band and can thus easily be spectrally separated from other transition frequencies in the system.
Note added in proof: During the preparation of this manuscript, a new class of broadband, Josephson parametric amplifier, the Josephson traveling wave parametric amplifier (JTWPA), was developed and published^{35}.
Methods
Device fabrication
We fabricated our device on sapphire, using niobium for the waveguides and the transmon paddles, and shadowevaporated aluminum for the Josephson junctions. To reduce the surface roughness before processing, the 2′′ c plane sapphire wafer was preannealed at 1,100 °C for 10 h in an atmosphere of N_{2}:O_{2}, 4:1, ramping the temperature by 5 °C min^{−1}. The annealed wafer was then sputtered with 80 nm of Nb in a nearultrahigh vacuum magnetron sputter. The first patterning of the sample consists of a photolithography step to define alignment marks and bond pads, deposited using electronbeam evaporation of 3 nm Ti and 80 nm Au. Next, the resonator, the transmon islands, and the pump line were defined in the Nb layer using a standard electronbeam lithography process at 100 keV, and etched using inductively coupled plasma reactive ion etching in NF_{3} gas.
The Al/AlO_{x}/Al Josephson junctions forming the SQUIDs, used to terminate the resonator and for connecting the transmon islands, were then defined in a second electronbeam step. After exposure, the 2′′wafer was diced into separate chips, using the exposed electronbeam resist as a protective resist. Before the first evaporation step, the surfaces of the Nb films where cleaned using in situ Arion milling inside of the Plassys evaporator. However, due to the substantially different regimes of critical currents, I_{0}, required for the Josephson junction of the transmons and the parametric resonator, two sequential evaporations and oxidations were performed within the same vacuum cycle by rotating a planetary aperture mounted inside the evaporator loadlock, effectively shielding one half of the sample at the time. Finally, a postdeposition ashing step was performed to clean the surfaces from organic residues.
Finding the parametric oscillation threshold
It is hard to experimentally find the parametric oscillation threshold with good precision, when only considering the parametric oscillation region, Fig. 4a, whose observed shape gets smeared by the amplified vacuum noise. In this section, we present an alternative method using a weak probe signal: we probe the parametrically amplified response as we sweep the pump amplitude across the instability threshold.
We apply a probe signal on resonance, , while applying a detuned pump signal, such that (ω_{p}−2ω_{s})/2π=100 kHz. The signal then undergoes degenerate, phasepreserving parametric amplification (red trace in Supplementary Fig. 5), while the parametric oscillations are cancelled out since we measure the average amplitude of the field. The parametric amplification has maximum gain just at the threshold. We plot the magnitude of the reflected signal as a function of the pump power (at the generator), yielding an oscillation threshold P_{th}=−10.8 dBm, as indicated by the dashed red line. As a comparison, we measure the output power of parametric oscillation, for ω_{p}−2ω_{r}=0 and B(t)=0—see the blue trace.
Limits of the parametric oscillation amplitude
As briefly discussed in the main text, there are two nonlinear effects that move the resonator away from its pump condition, by means of their associated frequency shifts^{19},
The Duffing shift dominates near flux bias F=±π/2; the Duffing parameter is approximated as
where Z_{0}=50 Ω is the resonator’s characteristic impedance and R_{K}=h/e^{2} is the quantum resistance.
The pumpinduced frequency shift dominates near F=0; it is approximated as
The resonator’s frequency tuning versus F, equation (5) in the main text, is shown in Supplementary Fig. 3a, for the parameters of our device, and equations (7) and (8) are plotted in Supplementary Fig. 3b. This figure illustrates that it is essential to bias the system far enough away from the limiting points, F=0 or π/2, such that neither frequency shift pulls the resonator too far from its pump condition, thereby severely limiting the attainable output power.
The steadystate solution of equation (1) in the main text yields an analytic expression for the expected number of photons within the region of parametric oscillations,
which, for our analysed bias point, amounts to 200±3 photons in the resonator. From this number, we obtain a Duffing shift −αA^{2}/2π≈−5.4±0.3 MHz (for α/2π=27±1.5 kHz per photon) and a pumpinduced frequency shift (for β=(7.5±0.1) × 10^{−3}).
The parameter β has the effect of skewing the parametric oscillation region, yielding an expression for the thresholds plotted in Fig. 1b,
Calibration of attenuation and gain via the Duffing nonlinearity
In this section, we present how we calibrated the gain of the amplifier chain, using the photonnumberdependent frequency shift of the Duffing oscillator, −αA^{2}, which we recall from the previous section. The frequency of the resonator as a function of input probe power takes the following form,
where ω_{r}(0) denotes the resonant frequency with zero photons in the resonator, Γ_{0} and Γ are the external and total loss rates, respectively, and α is the Duffing frequency shift per photon—recall equation (7). Using equation (11), we can fit the extracted resonant frequencies as a function of input probe power at different flux bias points, F, with the attenuation, Att, as the only fitting parameter (since α can be extracted separately by fitting ω_{λ/4} and γ_{0}—recall equation (5)). This is shown in Supplementary Fig. 2, where the data for five different flux bias points are fitted to attenuations presented in Supplementary Table 3. From these values, we obtain an average attenuation, 〈Att〉=127.5 dB, which can be compared with the installed 120 dB, indicating that we have a cable loss of 7.5 dB at the measurement frequency.
Moreover, from the same measurement, we can also obtain an estimate for the gain of the amplifier chain by assuming that all the signal gets reflected when it is far off resonance with the resonator, that is, reflection coefficient S_{11}^{2}=1. Then, the gain is obtained from the relation
For the five gain estimates presented in Supplementary Table 3, we obtain a gain of G=81.0±0.37 dB, at our given bias point. The error bars for this gain estimation has two origins: ±0.17 dB from the residual of the linear fit to the gain values presented in Supplementary Table 3, and another ±0.2 dB from the gain drift over time, which can be compared with our 91 dB of installed amplification.
Calibration of the resonator photon number
From the obtained calibration of the gain of our amplifier chain, G, we can now calculate the conversion factor between our measured power on the digitizer and the number of photons in the resonator, using the following relation,
where P_{s} and P_{n} denote our signal and noise power levels, respectively. We demonstrate this for Fig. 3b, where the resonator is probed at a frequency . The external damping rate is Γ_{0}/2π=1.02 MHz, and we calculate the background power level from the end of the trace (when the pump is off). From the obtained SNR, the number of added noise photons can be estimated accordingly, A^{2}/SNR^{2}=16.1±1.3.
Quantum coherence and readout nondestructiveness
To study how the parametric pump strength affects the qubit’s relaxation time, we here present coherence measurements for the transmon. First, we calibrate a qubit pulse duration corresponding to a πpulse, using a Rabi measurement, where the pulse duration time is swept, for a fixed pulse amplitude. From the fit in Supplementary Fig. 1a, a pulse length of was obtained, and the Rabi decay time was T_{rabi}=2.53±0.15 μs. The histograms corresponding to the first 0.5 μs are plotted in Supplementary Fig. 1b, using the same projective technique as for the histograms in Fig. 5c in the main text. Finally, we perform a set of T_{1} measurements for different pump amplitudes , and compare these with traditional reflection readout, where we apply a weak resonant probe signal, but no pump . The fits to the relaxation times suggest that our readout is not any more destructive to the quantum state of the transmon than the traditional readout technique is. We note, however, that our extracted relaxation time is limited by the Purcell effect, yielding T_{1}≈[2Γ_{0}(g_{01}/Δ)^{2}]^{−1}=4.11 μs. Also see Supplementary Note 3.
Additional information
How to cite this article: Krantz, P. et al. Singleshot readout of a superconducting qubit using a Josephson parametric oscillator. Nat. Commun. 7:11417 doi: 10.1038/ncomms11417 (2016).
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Acknowledgements
We would like to thank Jared Cole, Göran Johansson and Baladitya Suri for fruitful discussions. All devices were fabricated in the Nanofabrication Laboratory at MC2, Chalmers. Support came from the Wallenberg foundation, the European Research Council (ERC), the Royal Swedish Academy of Sciences (KVA), the European project ScaleQIT, STINT and Marie Curie CIG. The MIT and Lincoln Laboratory portions of this work were sponsored by the Assistant Secretary of Defense for Research & Engineering under Air Force Contract #FA872105C0002. Opinions, interpretations, conclusions and recommendations are those of the author and are not necessarily endorsed by the United States Government.
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Affiliations
Microtechnology and Nanoscience, Chalmers University of Technology, Kemivägen 9, SE41296 Gothenburg, Sweden
 Philip Krantz
 , Andreas Bengtsson
 , Michaël Simoen
 , Vitaly Shumeiko
 , Per Delsing
 & Jonas Bylander
Research Laboratory of Electronics, Massachusetts Institute of Technology, Cambridge, Massachusetts 02139, USA
 Simon Gustavsson
 & W. D. Oliver
MIT Lincoln Laboratory, 244 Wood Street, Lexington, Massachusetts 02420, USA
 W. D. Oliver
Institute of Quantum Computing, University of Waterloo, Waterloo, Ontario, Canada N2L 3G1
 C. M. Wilson
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Contributions
P.K., A.B., M.S., C.M.W., P.D. and J.B. designed the experimental setup. P.K. modelled and fabricated the device. P.K., A.B., S.G., W.D.O. and J.B. carried out the measurements. V.S. gave input on theoretical matters. P.K., P.D. and J.B. wrote the manuscript with input from all coauthors.
Competing interests
The authors declare no competing financial interests.
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Correspondence to Philip Krantz or Jonas Bylander.
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Supplementary Figures 15, Supplementary Tables 14, Supplementary Notes 13 and Supplementary References
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