Abstract
The stochastic evolution of quantum systems during measurement is arguably the most enigmatic feature of quantum mechanics. Measuring a quantum system typically steers it towards a classical state, destroying the coherence of an initial quantum superposition and the entanglement with other quantum systems. Remarkably, the measurement of a shared property between non-interacting quantum systems can generate entanglement, starting from an uncorrelated state. Of special interest in quantum computing is the parity measurement1, which projects the state of multiple qubits (quantum bits) to a state with an even or odd number of excited qubits. A parity meter must discern the two qubit-excitation parities with high fidelity while preserving coherence between same-parity states. Despite numerous proposals for atomic2, semiconducting1,3,4,5,6,7 and superconducting qubits8,9, realizing a parity meter that creates entanglement for both even and odd measurement results has remained an outstanding challenge. Here we perform a time-resolved, continuous parity measurement of two superconducting qubits using the cavity in a three-dimensional circuit quantum electrodynamics10,11 architecture and phase-sensitive parametric amplification12. Using postselection, we produce entanglement by parity measurement reaching 88 per cent fidelity to the closest Bell state. Incorporating the parity meter in a feedback-control loop, we transform the entanglement generation from probabilistic to fully deterministic, achieving 66 per cent fidelity to a target Bell state on demand. These realizations of a parity meter and a feedback-enabled deterministic measurement protocol provide key ingredients for active quantum error correction in the solid state13,14,15.
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Acknowledgements
We thank C. C. Bultink and H.-S. Ku for experimental assistance, and G. Haack, R. Hanson, G. Johansson, A. F. Kockum and L. Tornberg for discussions. We acknowledge funding from the Dutch Organization for Fundamental Research on Matter (FOM), the Netherlands Organization for Scientific Research (NWO, VIDI scheme), the EU FP7 integrated projects SOLID and SCALEQIT, and partial support from the DARPA QuEST programme.
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D.R. fabricated the device. D.R. and C.A.W. performed the measurements. D.R., C.A.W. and G.d.L. analysed the data. M.D., Ya.M.B. and L.D.C. provided theory support. M.J.T. and R.N.S. produced the feedback controller. K.W.L. designed the JPA. D.R., G.d.L. and L.D.C. wrote the manuscript with feedback from all authors. L.D.C. designed and supervised the project.
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Extended data figures and tables
Extended Data Figure 1 Spectroscopy of the two-qubit and cavity system.
The transition frequency of QB is tuned by applying magnetic flux through its SQUID loop with an external coil. QA (fA = 5.52 GHz) is a single-junction transmon and thus not tunable. QB was designed tunable to allow trimming of the dispersive-shift matching condition. However, the maximal frequency of QB (fB = 7.80 GHz) is still approximately 20 MHz lower than needed for a perfect match of dispersive shifts. Thus, we flux bias QB at this maximal frequency, which is also optimal for coherence. Inset, higher resolution spectroscopy of the avoided crossing of QB with the cavity fundamental mode (fr = 6.55 GHz), revealing a minimum splitting of 167 MHz.
Extended Data Figure 2 Detailed schematic of the experimental set-up.
Complete wiring of electronic components outside and inside the 3He/4He dilution refrigerator (Leiden Cryogenics CF-650). Readout and qubit-drive pulses, shaped by a Tektronix AWG5014 and two Tektronix AWG520, enter the cavity via a single transmission line. The cavity output is reflected by the JPA, which is biased by a superconducting coil and a strong pump tone, bending its resonance down to fP and providing parametric amplification12. The signal is further amplified at the 3 K stage (Caltech Cryo1-12, 0.06 dB noise figure) and at room temperature (two Miteq AFS3-04000800-10-ULN amplifiers, 0.8 dB noise figure). Demodulation to baseband is provided by a generator at fP, also used for readout and pump. Two phase shifters allow adjusting the relative phase between the three tones at fP. The demodulated signal is split into three separate arms after amplification by a Stanford Research Systems SR445A. One arm stabilizes the JPA flux bias via an ADwin-GOLD processor programmed as a PID controller31. In the second arm, the signal is filtered by a bias tee, amplified with a custom-built amplifier, and integrated and thresholded by the FPGA. The FPGA conditionally triggers a QA π pulse from an AWG520 (Fig. 4). The third arm connects to an AlazarTech ATS9870 digitizer for data storage and processing after a second SR445A amplification stage. Red colour highlights the key components of the feedback loop.
Extended Data Figure 3 Readout configuration for parity measurement and state tomography.
a, Histograms for the computational basis for the parity measurement MP (τP = 300 ns, 
), as in Fig. 3a. At this measurement power, states within each parity subspace are largely indistinguishable (see also Fig. 1b and Extended Data Fig. 7). For an ideal parity measurement, βA = βB = 0. We extract βA = 0.0146 mV, βB = −0.123 mV, βBA = −6.25 mV and β0 = 7.46 mV. b, Histograms for the tomography measurement (integration time 850 ns, 
). At this power, the cavity response is nonlinear (critical photon number33 ncrit ≈ 60), causing the resonance for |10〉 to bend towards lower frequency. As the resonance for |01〉 is instead power-independent, this effect discriminates |01〉 from the other states. This gives the joint readout the sensitivity to single and two-qubit terms required to perform state tomography22. Averaging of raw tomography measurements yields βA = −8.10 mV, βB = 9.10 mV, βBA = −12.8 mV and β0 = 17.1 mV. Digitizing the single shots with threshold VD = 32 mV gives βA = 0.424, βB = −0.360, βBA = 0.379 and β0 = 0.540.
Extended Data Figure 4 Temporal evolution of two-qubit superposition state with and without continuous parity measurement.
Comparison of the unconditioned two-qubit evolution during parity measurement (filled symbols, same data as in Fig. 2b) and during a delay of the same duration τP (open symbols). In the latter case, the decay of |ρij,kl| is solely due to intrinsic qubit decoherence. Evidently, measurement-induced dephasing dominates over intrinsic qubit dephasing. For reference, we estimate that entanglement (
) would be achieved in the deterministic scheme provided the net qubit dephasing rate 
, under the experimental conditions of Fig. 4.
Extended Data Figure 5 Two-qubit evolution under continuous parity measurement.
Unconditioned and conditioned state tomography of the final two-qubit state similar to Figs 2 and 3, but at more values of τP and using the threshold Vth optimizing parity readout fidelity (
). Middle row, unconditioned evolution. For τP = 0, there is only a 10 ns buffer between state preparation and tomography, instead of the 350 ns used in Figs 2, 3, 4 and all other τP values here. The uniformity of |ρij,kl| for τP = 0 (<4% relative difference) attests to the preparation fidelity of the initial maximal superposition state. Top row, evolution conditioned on Vint > Vth (MP = −1); bottom row, evolution conditioned on Vint < Vth (MP = +1).
Extended Data Figure 6 Two-qubit unconditioned evolution and conditioned concurrence for different measurement strengths.
a–f, Experiment as in Figs 2 and 3 with measurement strength corresponding to 
(a, d), 1.4 ± 0.1 (b, e) and 3.9 ± 0.1 (c, f). The best-fit frequency mismatch 
(see also Extended Data Fig. 3) is 182 ± 32 kHz (a, d), 220 ± 18 kHz (b, e) and 275 ± 7 kHz (c, f). Concurrence is calculated after postselection on Vint < Vth (MP = +1) or Vint > Vth (MP = −1).
Extended Data Figure 7 Cumulative histograms of parity measurements.
The four computational states are subjected to a parity measurement with τP = 300 ns, 
, as in Fig. 3a. At the optimal threshold Vth (dashed line), the average errors in determining the parity are εe = 0.13, εo = 0.11, yielding a parity measurement fidelity of FP = 1 − εe − εo = 0.76 (corrected for residual qubit excitations, see Methods). In a similar manner, we define the distinguishability within each parity subspace as the fidelity of the measurement discriminating between those states, yielding 0.03 for the even subspace and 0.02 for the odd.
Extended Data Figure 8 Frequency-dependent coherence times of QB.
Energy relaxation times 
(filled circles) and 
(square) below the fundamental cavity resonance are consistent with the single-mode Purcell effect34 and a coupling strength g/π = 167 MHz at the QB-cavity avoided crossing, as extracted from spectroscopy (Extended Data Fig. 1). We attribute the lower 
above the fundamental resonance to the effect of higher cavity modes. Pure dephasing times 
(open circles) are in excellent agreement with the first-order approximation for flux noise35 with spectral density Sf (ω) = A2/|f| and best-fit A = (1.9 ± 0.1) × 10−5Φ0 (dashed line), with Φ0 the flux quantum.
Extended Data Figure 9 Pulse timing and measurement integration windows.
Extended view (not to scale) of the pulse sequence used in Figs 2, 3, 4, showing also the integration windows used for parity measurement (τint for signal and τoff for offset) and tomographic joint readout. All specified time intervals are expressed in ns. Qubit control is performed with DRAG pulses36 with Gaussian envelopes on the main quadrature (σ = 6 ns, 4σ total duration) and derivative-of-Gaussian envelopes of optimized amplitude on the other. Single-qubit pulses are applied sequentially (QB first), with 10 ns buffer between them. The tomography measurement pulse is 1 µs long, and the homodyne response integrated for 850 ns starting after the first 100 ns.
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Ristè, D., Dukalski, M., Watson, C. et al. Deterministic entanglement of superconducting qubits by parity measurement and feedback. Nature 502, 350–354 (2013). https://doi.org/10.1038/nature12513
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DOI: https://doi.org/10.1038/nature12513
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