Single-shot read-out of a superconducting qubit using a Josephson parametric oscillator

Abstract

We propose and demonstrate a read-out 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 quantum-limited amplifier. We demonstrate proof-of-principle, single-shot read-out performance, and present an error budget indicating that this method can surpass the fidelity threshold required for quantum computing.

ARTICLE

Received 22 Oct 2015 | Accepted 24 Mar 2016 | Published 9 May 2016

Philip Krantz1, Andreas Bengtsson1, Michal Simoen1, Simon Gustavsson2, Vitaly Shumeiko1, W.D. Oliver2,3, C.M. Wilson4, Per Delsing1 & Jonas Bylander1

We propose and demonstrate a read-out 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 18515 photons in the resonator, and one with zero oscillation amplitude. This high contrast obviates a following quantum-limited amplier. We demonstrate proof-of-principle, single-shot read-out performance, and present an error budget indicating that this method can surpass the delity threshold required for quantum computing.

DOI: 10.1038/ncomms11417 OPEN

Single-shot read-out of a superconducting qubit using a Josephson parametric oscillator

1 Microtechnology and Nanoscience, Chalmers University of Technology, Kemivagen 9, SE-41296 Gothenburg, Sweden. 2 Research Laboratory of Electronics, Massachusetts Institute of Technology, Cambridge, Massachusetts 02139, USA. 3 MIT Lincoln Laboratory, 244 Wood Street, Lexington, Massachusetts 02420, USA. 4 Institute of Quantum Computing, University of Waterloo, Waterloo, Ontario, Canada N2L 3G1. Correspondence and requests for materials should be addressed to P.K. (email: mailto:[email protected]

Web End [email protected] ) or to J.B. (email: mailto:[email protected]

Web End [email protected] ).

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ARTICLE NATURE COMMUNICATIONS | DOI: 10.1038/ncomms11417

The read-out scheme for quantum bits of information (qubits) constitutes one essential component of a quantum information processor1. During the course of a quantum

algorithm, qubit-state 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 shotwithout averaging of the output signals of repeated measurements on identically prepared qubitswith delity exceeding approximately 99% (ref. 2).

The commonly used measurement scheme for a super-conducting qubit coupled with a linear microwave resonator does not, by itself, offer single-shot measurement performance. The qubit imparts a state-dependent (dispersive) frequency shift on the resonator, which can be determined by applying a probe signal and measuring the reected or transmitted signal, although only for weak probing, rendering an inadequate signal-to-noise ratio (SNR)3,4.

Researchers have addressed the problem of insufcient SNR in essentially two ways. One approach is to feed the weak output signal into a following, parametric linear amplier that adds only the minimum amount of noise allowed by quantum mechanics58. 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 contrast913.

In this paper, we propose and demonstrate a simplied read-out 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 proof-of-concept, single-shot read-out performance (SNR41). We obtain 81.5% qubit-state discrimination for a read-out time t600 ns; however, from the

error analysis, we infer a read-out delity of 98.71.2%, taking into account known and reparable errors due to qubit initialization and decoherence (17.21.2%). A realistically achievable qubit relaxation time, T1 50 ms, and a Purcell band-pass lter

would reduce these errors from 17.2 to o0.5%, as well as shorten the required read-out time to to100 ns. The remaining errors, which are due to the switching events in the oscillator(1.20.3%), can be eliminated by improving the data aquisition protocolsee Discussion and Supplementary Note 1. These qubit and detection improvements would bring the read-out delity to E99.5%.

Our read-out scheme relies on parametric pumping of a frequency-tunable 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 steady-state resonator eld corresponding to 18515 photons, whose output we can clearly distinguish from the nonoscillating state when followed by a commercial semiconductor amplier, eliminating the need for a quantum-limited amplier. Conceptually, this method can yield arbitrarily large contrast due to the parametric instability, and moreover, only requires a pump but no input signal.

This read-out scheme is well aligned with scalable, multi-qubit implementations. Parametric oscillators can be readily frequency-multiplexed14 and allow for a simplied experimental set-up (compared with conventional microwave reectometry) without a separate input port to the resonator or a following parametric

amplier, 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 ux-pumping line only, which further reduces the number of cables and interconnects.

ResultsThe Josephson parametric oscillator. Our device consists of a quarter wavelength (l/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, LJ F; I f

F0=

2p cos pF=F0

I20 I2 f

p

, where I0 is the critical current and F0 is the ux quantum. This inductance can be controlled by the external magnetic ux through the SQUID loop, F(t) Fd.c.

Fa.c.(t), and by the superconducting phase difference across the JJs, f(t), via its currentphase relation, I(t) I0 sinf(t).

Time-varying modulations of F and fparametric pumpingaffect 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 ux, F(t), which can lead to frequency mixing as well as parametric effects such as noiseless amplication of a signal, frequency conversion and instabilities6,1519.

The state of the JPO has a rich dependence on several parameters, some of which was studied recently, both theoretically20,21 and experimentally7,17,19. The equation of motion for the intra-resonator electric eld amplitude, A, can be written as

i _

A EA dA a A

j j2A iGA

p

2G0

B t

: 1 Here E is proportional to the externally applied pump amplitude, Fa.c., which modulates the resonant frequency parametrically at close to twice its value, opE2or (degenerate pumping), and d op/2 or is the resonators detuning from

half of the pump frequency. The eld amplitude, A, and its complex conjugate, A , are slow variables in a frame rotating at op/2, and |A|2 is the equivalent number of photons in the resonator. The Dufng parameter, a, associated with a cubic eld nonlinearity, arises from the nonlinear Josephson inductance. The linear damping rate has two components, G G0 GR,

where G0/2p 1.02 MHz is the external damping rate, associated

with the photon decay through the coupling capacitor, and GR/2p 0.30 MHz is the internal loss rate. The equations

right-hand side represents the input probe signal, such that |B(t)|2 has units of photons per second. The output ow of photons per second, |C(t)|2, is given by C t

B t

i

2G0

p A.

For low pumping amplitude, below the parametric instability threshold, EoEth, this device works as a phase-sensitive parametric amplier (JPA) for an input B(t) at signal frequency os

op/2 (refs 6,1517,22). Note, however, that we keep B(t) 0

in the measurements reported here. For a pumping amplitude exceeding the threshold, E4Eth, spontaneous parametric oscillations set insee Fig. 1b and equation (10) in Methods. The resonator eld builds up exponentially in time, even in the absence of an input probe signal until it becomes limited by the Dufng and pump-induced nonlinearities and reaches a steady state17,19.

We connected a transmon qubit capacitively to the resonator23see Fig. 1a. The state of the JPO (oscillating or nonoscillating) can then be controlled by the qubit-state-dependent, dispersive frequency shift, w, which the qubit exerts

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NATURE COMMUNICATIONS | DOI: 10.1038/ncomms11417 ARTICLE

a

a

ADC Image

rej. filter

Digitizer

LO

I Q

5.4

Frequency (GHz)

5.6

300 K

5.2

5.0

48 GHz

HEMT

4.8

Parametric resonator

Transmon

2.8 K

F= 0.185[afii9843]

Resonator Qubit Pump

10 mK

4.6

60 dB

4.4

40 dB

40 dB

20 dB

4.2 1.0 0.8 0.6 0.4 0.2 0.0 0.2 0.4 0.6 0.8

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|A|2

|C |2

External d.c. flux (F/[afii9843])

b

5.25

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BPF

5.20

Parametric

resonator

d.c.

5.15

2g01/2[afii9843] = 92 MHz

Sample box

Qubit

5.10

5.05

b

4

2.52.01.51.00.5

Pump amplitude (/)

0.0

0

1

0.30 0.25 0.20 0.15

External d.c. flux (F/[afii9843])

Figure 2 | Combined resonator-qubit frequency spectra. (a) Qubit spectroscopy was used to map out the transmon spectrum (in red), whereas the resonator spectrum (in blue) was extracted using standard reectometry. The solid red and grey lines are ts. The dashed grey line, at resonator ux bias F 0.185p, indicates the bias point at which we later

demonstrate the read-out method. (b) Vacuum Rabi splitting around the ux bias point where the transmon frequency crosses that of the resonator, indicated by the grey box in a. The minimum frequency splitting yields a qubitresonator coupling g01/2p 46 MHz.

0

V Q(mV)

8 6 4 2 0

0

4 4

4

Pump detuning ([afii9829]|0/) VI (mV)

4

V Q(mV)

2.52.01.51.00.50.0

0

2[H9273]/

8 6 4 2 0Pump detuning ([afii9829]|0/) VI (mV)

0

4 4

4

Figure 1 | Experimental set-up and read-out mechanism. (a) Schematic of the cryogenic microwave reectometry set-up. The transmon qubit (red) is capacitively coupled with the coplanar waveguide parametric resonator (blue). The input and output ows of photons are denoted |B|2 and |C|2, respectively, whereas the number of photons in the resonator is denoted |A|2. The output signal is acquired using heterodyne detection of the amplied microwave signal. The components drawn in lighter grey are those that are rendered unnecessary by the JPO read-out method, thereby offering a simplied experimental set-up (see text). (b) Parametric oscillation regions for the qubit ground state |0i (solid blue line) and

excited state |1i (dashed blue line), respectively. These blue lines represent

the instability boundaries, EEth, where the number of steady-state

solutions to equation (1) changes. The two panels on the right are measured [I,Q]-quadrature voltage histograms of the device output for the pump bias point indicated by the circles, revealing two different oscillator states: outside of the region of parametric oscillations, the resonator is quiet (|A|2 0). Within the region, the resonator has two oscillating states

(|A|240), with a phase difference of p radians.

on the resonator24,25. When the JPO is being pumped above the threshold for parametric oscillation, with amplitude E and frequency detuning, d, then a change of qubit state effectively pulls the resonator to a different value of the detuning, outside of the region of parametric oscillationssee Fig. 1b. We denote the qubit-state-dependent detunings by d|0i d w and d|1i d w.

The resulting mapping of the qubit state onto the average number of photons in the resonator provides us with a qubit-state read-out mechanism, which we exploit in this work.

Characterization of qubit and JPO. The device and cryogenic experimental set-up 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 l/4 resonator (in blue) is capacitively coupled with the transmission line (Cc 11.9 fF), yielding an external quality factor

Qext or/2G0 2555. A transmon qubit (in red) is also coupled

near this end of the resonator.

The resonator output signal is amplied using a 48 GHz high-electron-mobility transistor amplier, with a noise temperature TN 2.2 K, followed by two room-temperature ampliers. We

detect the outgoing signal using heterodyne mixing. The signal is rst downconverted to a frequency (oRF oLO)/2p 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 rst characterize the transmon spectroscopicallysee Fig. 2afrom which we extract the Josephson and charging energies, EJ/2p 9.82 GHz and EC/2p 453 MHz, respectively.

From the vacuum Rabi splitting, we extract a qubit resonator

coupling rate g01/2p 46 MHzsee Fig. 2b.

Next, we t the frequency tuning curve of the resonator (with the qubit in the |0i-state) to the relationo 0j ir F

or F

g201=D F

; 2 where F pFd.c./F0 denotes the static ux bias, normalized to the

magnetic ux quantum. The effective dispersive shift due to the qubit is

w F

g201

D F

; 3

EC D F

EC

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a

which, in turn, depends on the qubitresonator detuning, D(F)

oa(F0) or(F0), with F0 F/8.88 0.58 representing the effective

magnetic ux of the transmon. Moreover, the qubit and resonator frequency spectra are well approximated by23,26

oa F0

8EJ cos F0

Time (s)

0.15 0.45

0.0 0.60

0.30

1

0

Qubit

Pump

j jEC q

EC; 4

or F

A q(a.u.)

A p(a.u.)

[afii9848][afii9843] = 52 ns

[afii9848]d = 20 ns

[afii9848]r

1

ol=4

1 g0= cos F

j j

; 5

where ol/4/2p 5.55 GHz is the bare resonant frequency (in

absence of the SQUID), and g0 LJ(F 0)/Lr 5.30.1% is the

inductive participation ratio between the SQUID (at zero ux) and the resonator. The solid grey and red lines in Fig. 2a are ts to equations (2) and (4), respectively.

Single-shot qubit read-out. We now demonstrate our method for reading out the qubit with the JPO. We choose a static ux bias point F 0.185p for the resonator SQUID, corresponding to

a resonant frequency o 0j ir=2p5:218 GHz and qubit transition

frequency oa/2p 4.885 GHzsee dashed grey line in Fig. 2a.

Consequently, the qubitresonator detuning is D/2p 334

MHz, and the effective dispersive shift is 2w/2p 7.258 MHz.

We measured a Purcell-limited qubit relaxation time, T1 4.240.21 ms, and Ramsey free-induction decay time

T 21:66 0:32 mssee Methods, Supplementary Fig. 1 and

Supplementary Table 1.

To operate the parametric oscillator as a high-delity qubit read-out 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 |0i in the quiet

state (the empty resonator) and the excited state |1i in the

populated state of the resonator. Figure 3a shows the pulse sequence for qubit manipulation and read-out, and Fig. 3b shows the resulting output from the JPO, operated with the pump settings d|0i/G 5.34, E=G 3:56.

The populated oscillator in Fig. 3b contains 18515 photons. We obtained this estimate from a comparison between the probe-amplitude dependence of the resonant frequency and the expected photon number dependence of the Dufng shiftsee Methods and Supplementary Fig. 3. This number of photons should be compared with |A|2 2003 photons, which is the

solution to equation (1) in the steady state _

A0 .

1

0

1

0.5 0.0 0.5 1.0 1.5 2.0 2.5 3.0Time (s)

Time

b

200

50

Photon number, |A|2

150

1

|A|2

[afii9848]s

100

104 Avg.

0

0

0.5 0.0 0.5 1.0 1.5 2.0 2.5 3.0Time (s)

Figure 3 | Qubit read-out by the Josephson parametric oscillator.(a) Pulse sequence: the qubit p-pulse (in red), with Gaussian edges and a plateau of duration tp52 ns, is followed by a short delay, td20 ns, before

the pump is turned on at time t 0. (b) The solid blue and red traces show

the inferred photon number, |A|2, in the resonator, with and without a prior p-pulse on the qubit, respectively. Note that the resonator latches, once it has entered into the oscillating state, and remains there even if the qubit relaxes. The traces are the result of 104 averages of the raw data; the inset shows a single instance of the raw data on the same time axis as the main plot. Before the sampling window of width ts300 ns, a delay tr300 ns is

added to avoid recording the transient oscillator response. The hatched region around the average photon number represents our uncertainty, originating from the amplier gain calibration.

To achieve such clear qubit-state discrimination as in Fig. 3b, we needed to make a judicious choice of ux bias point, F, to mitigate the effects of two nonlinear shifts of the resonant frequency19. The Dufng shift dominates when F-p/2, whereas a pump-induced frequency shift dominates when F-0. These shifts can move the resonator away from the proper pump condition, thereby effectively restricting the output powersee Methods and Supplementary Fig. 2.

Moreover, the qubit resonator detuning should be in the

dispersive regime D g01

, in which the qubit state controls the

resonant frequency of the resonator. Yet it must yield a sufciently large dispersive shift, w4G (equation 3), to produce clearly distinguishable output levels, corresponding to the |0i and

|1i states. For our chosen ux bias point, we identify the optimal

pump settings by mapping out the parametric oscillation region as a function of pump frequency and amplitudesee Fig. 4a,b.

An interesting feature is present within the left half of Fig. 4a,b (where the populated resonator encodes |1i). Here when the qubit

is initially in the |1i 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 tri-stable oscillation state17,21,

associated with red detuning of the above-threshold region for the |0i state. When the qubit relaxes, there occurs an instantaneous

shift of the pseudopotential for the amplitude A, from bistable (with two p-shifted, nite-amplitude states; see Fig. 1b)

to tri-stable (with one additional zero-amplitude state). The elds initial condition at the time of this shift, Aa0, 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 d; E

-planesee Fig. 4c.

We choose the pump operation point d|0i/G 5.34,

E=G 3:56, 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 in-phase (VI) and quadrature (VQ) voltage measurements from 105 read-out cycles, with each measurement being the mean quadrature voltage within the sampling time ts (blue window in Fig. 3). We project each of the 2D histograms onto its real axis, and thus construct 1D histograms of the VI componentsee Fig. 5c. We can then extract a SNR, SNR jm 1

j i m 0

j ij= s 1

j i s 0

j i

3:39, where m and s denote the mean value and s.d., respectively, of the Gaussians used to t the histograms. The peak separation of the histograms gives a condence level of 99.998% for the read-out

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a c

5.0

1.0

2.52.01.51.00.5

2.52.01.51.00.5

0

4.5

0.9

4.0

0.8

3.5

0.7

Pump strength (/)

8 6 4 2 0 2

8 6 4 2 0 2

Output power (a.u.)

Pump strength (/)

2.5

State discrimination

3.0

0.6

0.5

b

1

2[H9273]/

2.0

0.4

1.5

0.3

1.0

0.2

0.5

0.1

0.0 6 4 2 0 2

0.0

Pump detuning ([afii9829]|0/) Pump detuning ([afii9829]|0/)

Figure 4 | Parametric oscillations and state discrimination. Output eld of the resonator when the qubit is in its (a) ground state |0i and (b) excited

state |1i. (c) Contour plot of the state discrimination within the two parametric oscillation regions. The black circle in the left region, located at

d|0i/G 5.34, E=G 3:56, represents the bias point used throughout the rest of the analysis. The state discrimination in this point is 81.5%.

delity. The peak appearing in the centre of the blue trace arises mainly from qubit relaxation before and during the read-out. 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 delity 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, Vth. From

these sums, we obtain the S-curves of the probability to nd the qubit in its ground state as a function of the voltage threshold valuesee Fig. 5d. We dene the delity of the measurement as the maximum separation between the two S-curves.

DiscussionTo evaluate the delity of the read-out itself, as compared with the delity loss associated with qubit errors, we now present an error budget. From the histograms in Fig. 5c, we can account for81.5% 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 105 read-out cycles as in the measured histograms. The simulation results are binned in the same way as the measurements, using the Gaussian ts as boundaries, and taking into account the following statistics: rst, qubit relaxation and preparation errors; second, thermal population of the qubit; third, spurious switching events by p-radians of the oscillator phase during read-out (yielding a reduced sampled voltage); and fourth, peak separation error due to the limited SNR.

We nd that the main contribution to the loss of delity is due to qubit relaxation before and during the read-out. From the measured relaxation time, T1 4.240.21 ms, we obtain a delity

loss of 11.60.5%. However, this error can be reduced substantially (to o0.5%) by introducing a Purcell band-pass lter2729 at the output of the JPO; since the qubit is detuned from the JPO, this decreases its relaxation into the 50-O transmission line. Such a lter would allow us to increase the resonator damping rate, G0, substantially reducing the read-out time without compromising T1. 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 qubitresonator coupling, g01, and detuning, D, need to be chosen accordingly to render a sufciently large dispersive frequency shift.

From the simulation, we further attribute 4.50.3% to qubit preparation errors. Another 1.10.4% can be explained from

thermal population of the qubit; the effective qubit temperature is Tq 453 mK. By adding these delity loss contributions due to

the qubit to the measured state discrimination, we can account for 81.5% 11.60.5% 4.50.3% 1.10.4% 98.71.2%.

There are also errors introduced by the parametric oscillator itself: switchings between the p-shifted oscillating states reduce the overall measured voltage. We performed a separate control measurement that yielded 2.40.5% switching probability, which translates into a maximal delity loss of half of that, 1.20.25%.

a b

4 0

2

2

4

0

V Q(mV)

V Q(mV)

2

2

4

4

0

4 4

2

2

|1

0

4 4

2

2

VI (mV)

VI (mV)

VI (mV)

c d

8

1.0

0

# of counts (103 )

0.8

6 Vth

V

V

V

V

Vth

0.6

81.5%

4

1

P | 0

0.4

2

0.2

0

0 4 4

2

2

0.0 0 1 2 3 4

Threshold, Vth (mV)

Figure 5 | Quadrature voltage histograms of the parametric oscillator output collected after digital sampling. The pump bias point was d|0i/G 5.34, E=G 3:56. In (a), the qubit was in its ground state;

in (b), a p-pulse was applied before the read-out pulse. (c) 1D histograms of the in-phase voltage component, VI, from the quadrature histograms in a and b. The black and white solid lines are Gaussian ts, from which we extracted a signal-to-noise ratio of 3.39. (d) Cumulative distribution functions, corresponding to the |0i and |1i states, obtained by sweeping a

threshold voltage, Vth, from the centre of the two histograms (VI 0). The

maximum separation between the two S-curves yields a state discrimination of 81.5%.

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Table 1 | Overview of different modes of operation for the various Josephson amplication and detection schemes.

Device E Bs Bp # Modes Reference JPO(*) 4Eth 0 0 1 This work

JPA tEth a0 0 1 6

JPA tEth a0 0 Multimode 34

PPLO 4Eth a0 a0 1 7

JPA 0 a0 a0 1 5

JBA(*) 0 0 a0 1 10

JPC 0 a0 a0 2 8

JBA, Josephson bifurcation amplier; JPO, Josephson parametric oscillator; PPLO, parametric phase-locked oscillatorThe variables refer to equation (1), where E denotes the ux-pumping amplitude (at o E2o ), and B and B denote a.c. signal and pump amplitudes, respectively (at o Eo ). The two read-out methods marked with an asterisk (*) have the qubit directly integrated with the detector, whereas the other devices are used as following ampliers.

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 phase-switching events using a rectifying detection scheme, for example, a diode or a eld-programmable gate array, tracking the absolute value of the output eld instead of its amplitude.

The last and smallest contribution to the delity loss is the peak separation error, which accounts for the intrinsic overlap between the histograms. However, this contribution is o0.002% 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 above-mentioned improvements (reduced qubit relaxation rate, optimized qubit manipulations and cooling, enhanced resonator output coupling, and rectifying data acquisition), the read-out delity could realistically reach E99.5%, limited only by the qubit relaxation.

Finally, we demonstrate that the relaxation time of our qubit is not measurably aficted by the pumpsee 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 back-action is outside the scope of this work.

Table 1 puts our results in the context of previous work on parametric and nonlinear Josephson amplication and detection circuits.

A ux-pumped, parametric phase-locked oscillator was used as a following amplier, also enabling sensitive qubit read-out7. In our work, the qubit was directly coupled with the JPO, which simplies the experimental set-up by reducing the number of microwave components needed. Also, with a pumping amplitude below the parametric instability threshold, the ux-pumped JPA has been used to read-out one qubit6, as well as multiple qubits coupled with the same bus resonator28.

There is another way of operating our device: instead of pumping the ux at opE2or, we can apply an alternating pump current (E0, B(ta0)), now at a frequency close to resonance,

opEor, and thereby directly modulate the phase difference, f. Both methods can provide linear parametric gain on reection of a detuned signal (osaop/2 and osaop, respectively). The ux-pumped JPA has a very wide frequency separation between pump tone and signal, because osEorEop/2, which is a practical advantage since it makes the resonators entire instantaneous bandwidth available for amplication with no need to suppress or lter out the pump tone. Moreover, the l/4 resonator has no mode in the vicinity of op that the pump might otherwise populate.

We emphasize that there are indeed two different physical mechanisms in play, since ux and current pumping address orthogonal variables in the sense that F j1 j2

F0=2p and

f j1 j2

=2, where j1 and j2 denote the gauge-invariant

phase differences across the two parallel JJs. This distinction is also evident in equation (1). The parametric ux-pumping term, EA , modulates the resonant frequency; it couples the resonator eld amplitude and its complex conjugate, which can provide quadrature squeezing of an input signal and enables phase-sensitive parametric amplication; and for stronger modulation, there is a parametric instability threshold into the JPO regimesee Fig. 1b.

Current pumping by an input B(t), on the other hand, corresponds to an external force that directly contributes to the intra-resonator eld A and drives its nonlinear term a|A|2. For zero detuning, os op, this is the driven Dufng oscillator that

has no gain (it offers no phase-sensitive amplication); 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 amplication with the JPA30,31, which enabled, for example, the observation of quantum jumps in a qubit5. Current modulation is also used in the latching detection scheme of the Josephson bifurcation amplier9,10,14,32,33. There, a higher-amplitude input strongly drives the Dufng nonlinearity near its bifurcation point; the two qubit states can then be mapped onto two different resonator output eld amplitudes. The Josephson bifurcation amplier was used for quantum nondemolition measurement of a qubit, and in a lumped-element resonator11, in which a qubit-state-sensitive autoresonance was observed in response to a frequency-chirped current drive. Yet another method is to couple the qubit with a linear resonator, which inherits a cross-Kerr nonlinearity from the qubit; current pumping of the resonator can then yield a strong output signal that depends on the qubit state12,13.

In conclusion, we have introduced a single-shot read-out technique for superconducting qubitsthe JPO read-out. We demonstrated proof-of-principle operation, obtaining a bare-state discrimination of 81.5%. After correcting for known and reparable errors, this translates into an inferred read-out delity of 98.71.2%, which by implementing a rectifying detection scheme can be further increased by 1.20.3%. With foreseeable improvements and optimization, this device would be an attractive candidate for implementing multi-qubit read-out in the context of scalable error correction schemes. This delity and the read-out time are both amenable to optimization.

Our system integrates a parametric read-out mechanism into the resonator to which the qubit is coupled, substantially reducing the number of components needed to perform single-shot read-out in a circuit quantum electrodynamics architecture. Advantages offered by this read-out technique include the potential for multiplexing and scalability with no need for signal-probe inputs, additional microwave circulators, or separate parametric ampliers. As opposed to other integrated read-out 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 broad-band, Josephson parametric amplier, the Josephson traveling wave parametric amplier (JTWPA), was developed and published35.

Methods

Device fabrication. We fabricated our device on sapphire, using niobium for the waveguides and the transmon paddles, and shadow-evaporated aluminum for the Josephson junctions. To reduce the surface roughness before processing, the 200 c

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NATURE COMMUNICATIONS | DOI: 10.1038/ncomms11417 ARTICLE

plane sapphire wafer was pre-annealed at 1,100 C for 10 h in an atmosphere of N2:O2, 4:1, ramping the temperature by 5 C min 1. The annealed wafer was then sputtered with 80 nm of Nb in a near-ultrahigh vacuum magnetron sputter.

The rst patterning of the sample consists of a photolithography step to dene alignment marks and bond pads, deposited using electron-beam evaporation of3 nm Ti and 80 nm Au. Next, the resonator, the transmon islands, and the pump line were dened in the Nb layer using a standard electron-beam lithography process at 100 keV, and etched using inductively coupled plasma reactive ion etching in NF3 gas.

The Al/AlOx/Al Josephson junctions forming the SQUIDs, used to terminate the resonator and for connecting the transmon islands, were then dened in a second electron-beam step. After exposure, the 200-wafer was diced into separate chips, using the exposed electron-beam resist as a protective resist. Before the rst evaporation step, the surfaces of the Nb lms where cleaned using in situ Ar-ion milling inside of the Plassys evaporator. However, due to the substantially different regimes of critical currents, I0, 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 load-lock, effectively shielding one half of the sample at the time. Finally, a post-deposition ashing step was performed to clean the surfaces from organic residues.

Finding the parametric oscillation threshold. It is hard to experimentally nd the parametric oscillation threshold with good precision, when only considering the parametric oscillation region, Fig. 4a, whose observed shape gets smearedby the amplied vacuum noise. In this section, we present an alternative method using a weak probe signal: we probe the parametrically amplied response as we sweep the pump amplitude across the instability threshold.

We apply a probe signal on resonance, oso 0j ir, while applying a detuned

pump signal, such that (op 2os)/2p 100 kHz. The signal then undergoes

degenerate, phase-preserving parametric amplication (red trace in Supplementary Fig. 5), while the parametric oscillations are cancelled out since we measure the average amplitude of the eld. The parametric amplication has maximum gain just at the threshold. We plot the magnitude of the reected signal as a function of the pump power (at the generator), yielding an oscillation thresholdPth 10.8 dBm, as indicated by the dashed red line. As a comparison, we

measure the output power of parametric oscillation, for op 2or 0 and

B(t) 0see the blue trace.

Limits of the parametric oscillation amplitude. As briey 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 shifts19,

Do a A

j j2 bG E=G

2: 6 The Dufng shift dominates near ux bias F p/2; the Dufng parameter is

approximated as

a F

p2ol=4Z0

RK

10 P Att 30=10

or 0

; 11

where or(0) denotes the resonant frequency with zero photons in the resonator, G0 and G are the external and total loss rates, respectively, and a is the Dufng frequency shift per photonrecall equation (7). Using equation (11), we can t the extracted resonant frequencies as a function of input probe power at different ux bias points, F, with the attenuation, Att, as the only tting parameter (since a can be extracted separately by tting ol/4 and g0recall equation (5)). This is shown in

Supplementary Fig. 2, where the data for ve different ux bias points are tted to attenuations presented in Supplementary Table 3. From these values, we obtain an average attenuation, hAtti 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 amplier chain by assuming that all the signal gets reected when it is far off resonance with the resonator, that is, reection coefcient |S11|2 1. Then,

the gain is obtained from the relation

G S11

j j2 Att: 12 For the ve gain estimates presented in Supplementary Table 3, we obtain a gain of G 81.00.37 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 t 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 amplication.

Calibration of the resonator photon number. From the obtained calibration of the gain of our amplier 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,

A

j j2

Ps Pn2 G0=2p o 0j ir10

3

a0 g0

cos F

g0 cos F

3

G=10 ; 13

where Ps and Pn denote our signal and noise power levels, respectively. We demonstrate this for Fig. 3b, where the resonator is probed at a frequencyo 0j ir=2p5:212 GHz. The external damping rate is G0/2p 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/SNR2 16.11.3.

Quantum coherence and read-out nondestructiveness. To study how the parametric pump strength affects the qubits relaxation time, we here present coherence measurements for the transmon. First, we calibrate a qubit pulse duration corresponding to a p-pulse, using a Rabi measurement, where the pulse duration time is swept, for a xed pulse amplitude. From the t in Supplementary Fig. 1a, a pulse length of tp52 ns was obtained, and the Rabi decay time was

Trabi 2.530.15 ms. The histograms corresponding to the rst 0.5 ms 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 T1 measurements for different pump amplitudes E=G, and compare these with traditional reection readout, where we apply a weak resonant probe signal, but no pump B t

6 0; E0

; 7 where Z0 50 O is the resonators characteristic impedance and RK h/e2 is the

quantum resistance.The pump-induced frequency shift dominates near F 0; it is approximated as

b F

G ol=4g0

cos3 F

sin2 F

b0 cos3 F

sin2 F

: 8 The resonators 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 gure illustrates that it is essential to bias the system far enough away from the limiting points, F 0 or p/2,

such that neither frequency shift pulls the resonator too far from its pump condition, thereby severely limiting the attainable output power.

The steady-state solution of equation (1) in the main text yields an analytic expression for the expected number of photons within the region of parametric oscillations,

A

j j2

G a

.

The ts to the relaxation times suggest that our read-out is not any more destructive to the quantum state of the transmon than the traditional read-out technique is. We note, however, that our extracted relaxation time is limited by the Purcell effect, yielding T1E[2G0(g01/D)2] 1 4.11 ms. Also see Supplementary

Note 3.

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Acknowledgements

We would like to thank Jared Cole, Gran 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 #FA8721-05-C-0002. Opinions, interpretations,

conclusions and recommendations are those of the author and are not necessarily

endorsed by the United States Government.

Author contributions

P.K., A.B., M.S., C.M.W., P.D. and J.B. designed the experimental set-up. 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 co-authors.

Additional information

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How to cite this article: Krantz, P. et al. Single-shot read-out of a superconducting

qubit using a Josephson parametric oscillator. Nat. Commun. 7:11417

doi: 10.1038/ncomms11417 (2016).

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