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

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.


In-Phase
Supplementary Table 4. Summary of the error budget analysis, starting out from the state discrimination of 81.5%.

SUPPLEMENTARY NOTE 1: FIDELITY ERROR BUDGET
In this note, we present an analysis of how we can distinguish between the different sources of the reduction in measurement fidelity. We calculate the fidelity from the maximum separation of the S-curves of the histogram data, representing the qubit in the ground and excited states -see Fig. 5(d). To quantify the missing fidelity, we here present an error budget where we discuss five different sources of fidelity loss. In our system, there are losses that are due to the qubit and those that are due to the parametric oscillator. The different loss contributions, discussed in this section, are presented in Supplementary Table 2.

(i) Qubit relaxation and preparation errors
The largest contribution to the reduction of fidelity is associated with the qubit relaxation time, T 1 = 4.24 ± 0.21 µs. The time scales for the readout are τ π = 52 ns (π pulse), τ d = 20 ns (delay before readout), τ r = 300 ns (oscillator rise time), and τ s = 300 ns (sampling time of the parametric oscillator). This means that the pre-sampling time (including the π-pulse) equals τ pre = τ π + τ d + τ r = 372 ns, which, together with the relaxation during the readout yields a fidelity loss of 1 − exp (−(τ pre + τ s /2)/T 1 ) = 11.6 ± 0.5 %. Here, we count only half of the sampling time, τ s , since a qubit decay event occuring during the second half of the sampling time still ends up on the right side of the voltage threshold, V th .
Next, to rule out dark or missed counts due to spurious oscillation events as well as non-triggered oscillations, we performed two separate measurements on the JPO (without involving the qubit) in which statistics of raw data indicated the absence of both these events. The Monte Carlo simulation outcome yields an upper limit for our qubit preparation error of 4.5±0.3 %, thus summing up to 11.6±0.5 % + 4.5±0.3 % = 16.1 ± 0.8 %.
(ii) Thermal population of the qubit Already from Fig. 5(c) in the main text, we see that the thermal population of the qubit is small. However, from the ground-state histograms in the logaritmic plot, we can make an estimate of the qubit temperature using the Boltzmann distribution function, 1/ exp[ ω a /(k B T q )], with a temperature T q = 45±3 mK, and assuming that the qubit can only populate its two lowest energy levels. By summing up the counts outside of a voltage threshold of V th = ±1.8 mV, we arrive at a 0.55 ± 0.2 % probability of finding the qubit in its excited state. However, since this will act on both the ground and excited states, it renders a doubled fidelity loss contribution of 1.1±0.4 %.

(iii) Switching events during the readout cycle
Each readout count is obtained, in the digitizer, by averaging the downconverted voltage during the sampling window of τ s = 300 ns; see Fig. 3. There is a certain probability that a π-phase switching event of the parametric oscillator occurs during this time. Such switching events cause smearing of the histograms towards the center, which is most pronounced for the blue histogram in Fig. 4 (since it consists mainly of oscillating counts). To investigate this source of error, we analyzed in more detail the raw data for 10 3 readout cycles. After disregarding non-oscillating traces (which represent events of qubit relaxation prior to the readout), we found that 2.4±0.5 % of the oscillating traces contained switching events. The error bar on the extracted switching rate come from assuming a binominal distribution of the n = 1000 measurements, and a switching probability p = 2.4 %, yielding a standard deviation, σ/n = np(1 − p)/n ≈ 0.5 %. This translates to a fidelity loss of 1.2±0.25%, since half of the switching events give rise to the correct measurement outcome regardless of the switching event due to the fact that these still end up on the correct side of the threshold. This error can, however, be eliminated by implementing a rectifying detection scheme, using for instance an FPGA -making π and −π indistinguishable.
(iv) Peak separation To ensure that the peak separation between the histograms is sufficient to fully distinguish between the readout events, we did a full region map of the fidelity as a function of pump frequency and power, see Fig. 4(b). In addition, this map serves as a guide for optimizing the readout contrast. The indicated spot shows the bias point δ |0 /Γ = −5.34, /Γ = 3.56, used throughout this error-budget analysis.
This bias point is positioned on a plateau of fairly constant fidelities. As we move down in pump strength, the fidelity is reduced. This can be understood from the reduction of the signal-to-noise ratio, which results in an overlap between the Gaussian histograms. The same is true for the opposite encoding, found within the right region, where the populated resonator encodes qubit state |0 . There, the SNR is insufficient for full state discrimination. At the bias point used, the pump is sufficiently strong for a negligible probability of non-oscillations, given that the qubit has been properly initialized. We can extract this overlap by fitting two Gaussians to both oscillating states, and one Gaussian to the center peak. From these fits, we extract a peak separation error of < 0.002%.

SUPPLEMENTARY NOTE 2: REDUCTION OF FIDELITY LOSS AND READOUT TIME USING A PURCELL FILTER
From the error budget presented in Supplementary Note 1, we conclude that the largest contribution to the loss of readout fidelity is associated with qubit relaxation prior to and during readout. Fortunately, this error can be substantially suppressed by implementing a bandpass (Purcell) filter at the JPO output [1,3,4]. Here we present an estimate of the expected performance.
A Purcell filter would improve our readout fidelity in two ways: (i) it would increase the Purcell limitation on the qubit relaxation time, and (ii), it would allow for an increase of the resonator external damping rate. The latter both reduces the resonator ring-up time, τ r , and increases the signal-to-noise ratio, allowing us to reduce the sampling time, τ s . In this Note, we elaborate on the details of these readout improvements.
(i) Increase of the Purcell limitation Let us first look at three different values of the external damping rate, Γ 0 /2π, and how an added bandpass filter, centered around ω r , with a quality factor Q f = 200, changes the qubit relaxation time. The Purcell-limited relaxation time is given by which for our resonator and bias point results in T 1 = 4.11 µs -see the first row in Supplementary Table 2. By adding a filter of bandwidth ω r /Q f , the effective damping of the qubit into the 50-Ω load decreases without compromising the readout time of the resonator, modifying Eq. (1) into In Supplementary Table 2, we make a theoretical comparison of the fidelity loss due to T 1 relaxation, for external quality factors Q ext = 2554, 1000, and 500. The calculations use a shorter measurement time (see section (ii) here below), but otherwise the same parameters as in our experiment: resonator frequency, ω r /2π = 5.212 GHz, qubit transition frequency, ω a /2π = 4.885 GHz, qubit-resonator coupling, g 01 /2π = 46 MHz, and qubit-resonator detuning, ∆/2π = −334 MHz. We do the calculations with (a) and without (b) an added Purcell filter, and then for a greater qubit-resonator detuning (c), and finally (d) for a realistically achievable transmon relaxation time T1 = 50 us (limited by other channels of decoherence). The conclusion is that the Purcell filter substantially suppresses the fidelity loss due to relaxation, well below 1 %, see the last row in the table.
(ii) Decrease of the external quality factor The enhanced resonator damping rate would, for a linear resonator, yield a reduced ring-up time τ r = 2Q ext /ω r . The nonlinearity of the JPO leads to saturation of the intra-resonator field, at a time scale related to Q ext . By numerically solving the time-dependent Eq. (1) in the main text, we achieve agreement with the measurements, allowing us to estimate τ r for various values of Q ext . We note that τ s can realistically be decreased by 4 times -see Supplementary  Table 2.
Moreover, the reduced Q-value enables a reduction in the sampling time while maintaining a sufficient SNR. Recalling the steady-state solution for the equation of motion (Eq. (9) from the Methods section), we see that for a constant normalized pump strength, /Γ, and zero pump-detuning, δ/Γ = 0, the intra-resonator field amplitude, A, scales with the square root of the damping rate, The output signal amplitude (in absence of an input probe field) can be expressed in terms of A, For an overcoupled resonator, Γ ≈ Γ 0 and Q ≈ Q ext , Eq. (4) can be written in terms of Q ext , The detected signal increases inversely proportionally to the external Q-value. To keep the same signal-to-noise ratio, while decreasing Q ext , we can thus afford to reduce the measurement time substantially. Reducing the external Q-value by a factor 5 to 500 yields a sampling time τ s = 300/25 = 12 ns. Using a π-pulse of duration τ π = 10 ns, the pre-sampling time becomes τ pre = τ π + τ d + τ r = 10 + 0 + 51 ns, and the total measurement time is reduced to τ < 100 ns, which is comparable with state-of-the-art readout schemes based on following parametric amplifiers. Consequently, the loss of fidelity due to qubit relaxation becomes, see the last row of Supplementary Table 2.
The readout time could be reduced even further by optimising the nonlinearity, α, and the pump amplitude, , and by using an even lower-noise amplifier [5]. We can increase the dispersive shift, χ, to > 50 MHz by using the qubit straddling regime [6], where the qubit-resonator detuning is such that 0 < ∆ < E c .