Abstract
Quantum error correction is required for a practical quantum computer because of the fragile nature of quantum information. In quantum error correction, information is redundantly stored in a large quantum state space and one or more observables must be monitored to reveal the occurrence of an error, without disturbing the information encoded in an unknown quantum state. Such observables, typically multiquantumbit parities, must correspond to a special symmetry property inherent in the encoding scheme. Measurements of these observables, or error syndromes, must also be performed in a quantum nondemolition way (projecting without further perturbing the state) and more quickly than errors occur. Previously, quantum nondemolition measurements of quantum jumps between states of welldefined energy have been performed in systems such as trapped ions^{1,2,3}, electrons^{4}, cavity quantum electrodynamics^{5,6}, nitrogen–vacancy centres^{7,8,9} and superconducting quantum bits^{10,11}. So far, however, no fast and repeated monitoring of an error syndrome has been achieved. Here we track the quantum jumps of a possible error syndrome, namely the photon number parity of a microwave cavity, by mapping this property onto an ancilla quantum bit, whose only role is to facilitate quantum state manipulation and measurement. This quantity is just the error syndrome required in a recently proposed scheme for a hardwareefficient protected quantum memory using Schrödinger cat states (quantum superpositions of different coherent states of light) in a harmonic oscillator^{12}. We demonstrate the projective nature of this measurement onto a region of state space with welldefined parity by observing the collapse of a coherent state onto even or odd cat states. The measurement is fast compared with the cavity lifetime, has a high singleshot fidelity and has a 99.8 per cent probability per single measurement of leaving the parity unchanged. In combination with the deterministic encoding of quantum information in cat states realized earlier^{13,14}, the quantum nondemolition parity tracking that we demonstrate represents an important step towards implementing an active system that extends the lifetime of a quantum bit.
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Acknowledgements
We thank L. Jiang and S. M. Girvin for discussions. Facilities use was supported by the Yale Institute for Nanoscience and Quantum Engineering and the US NSF MRSEC DMR 1119826. This research was supported in part by the Office of the Director of National Intelligence (ODNI), Intelligence Advanced Research Projects Activity (IARPA), through the Army Research Office (W911NF0910369) and in part by the US Army Research Office (W911NF0910514). All statements of fact, opinion or conclusions contained herein are those of the authors and should not be construed as representing the official views or policies of IARPA, the ODNI or the US government. M.M. acknowledges partial support from the Agence National de Recherche under the project EPOQ2, ANR09JCJC0070. B.V. acknowledges partial support from NSF under the project PHY1309996.
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L.S. and A.P. performed the experiment and analysed the data. Z.L. and M.M. provided theoretical support. B.V. and G.K. provided further experimental contributions. K.M.S., A.N., M.H. and S.S. contributed to the doublepumped Josephson bifurcation amplifier under the supervision of M.H.D. J.B. and L.F. fabricated the device. R.J.S. designed and supervised the project. L.S., A.P., L.F. and R.J.S. wrote the manuscript with feedback from all authors.
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Extended data figures and tables
Extended Data Figure 1 Schematic of the measurement setup.
We use two separate lines to drive the readout and the storage cavity. Qubit state manipulations are realized through the readout cavity input line. The readout cavity output signal is first amplified by a JBA operating in a doublepumped mode, and the reflected signal then goes through three isolators in series before being further amplified by a HEMT at 4 K. The amplified signal is finally downconverted to 50 MHz and then digitized by a fast 1 GS dataacquisition card.
Extended Data Figure 2 Poisson distribution of photon numbers in the cavity.
Dotted colour lines are data for the first eight Fock states n = 0, 1, 2, …, 7 as functions of displacement amplitude α. The measurements are performed with a selective πpulse on each number splitting peak, and the resulting signal amplitude should be proportional to the corresponding number population. These oscillation amplitudes have been normalized to probabilities such that the sum of the amplitudes corresponding to n = 0 and n = 1 equals unity. Dashed lines are theoretical curves with a Poisson distribution , where the x axis has had a single scale factor adjusted to fit all these probabilities. The excellent agreement indicates good control of the coherent state in the cavity and also gives a good calibration of the cavity displacement amplitude. On the basis of the probability of n = 1 at α = 0, we find a background photon population of n_{th} = 0.02 in the cavity. Inset bottom panel: spectroscopy (left axis) of the number splitting peaks of the qubit when populating different photon numbers in the cavity. Inset top panel: difference between peak positions and a linear fit. The curvature necessitates a secondorder polynomial fit, resulting in a linear dispersive shift χ_{qs}/2π = 1.789 ± 0.002 MHz and a nonlinear dispersive shift .
Extended Data Figure 3 Ensembleaveraged free parity evolution of a coherent state.
The measurement protocol is shown in the inset. The single parity measurement gives a readout voltage that has been converted to parity through thresholding. All measured evolution curves saturate at the same value in the long time limit. This saturation level has been forced to 0.96 (because n_{th} = 0.02), represented by the dashed horizontal line. The solid lines are global fits, giving a time constant of τ_{0} = 55 µs.
Extended Data Figure 4 Effectiveness of the pulse.
Blue and red data (bottom axis) are ensembleaveraged qubit readouts after consecutively (with no wait time) applying (, ) and (, ), respectively, as functions of different introduced into the cavity. The curvature for comes from the finite bandwidth of the pulses in the frequency domain. Green curve (top axis) is a time Rabi trace for an amplitude comparison with no initial cavity displacement.
Extended Data Figure 5 Qubit readout properties.
a, Histogram of qubit readout for the parity protocol used in repeated singleshot traces in Fig. 3. The phase between the JBA readout and the pump has been adjusted such that g〉, e〉 and f〉 states can be distinguished with optimal spacings. Thresholds between g〉 and e〉, and between e〉 and f〉, have been chosen to digitize the readout signal to +1, −1 and 0 for g〉, e〉 and f〉, respectively. Note that we assign a zero to the f〉 states to indicate a ‘failed’ measurement with no useful information about the parity. b–d, Illustrations of pulse sequences (not to scale) producing the readout error matrix with the storage cavity left in vacuum. The g〉 state (b) is prepared through postselection of an initial qubit measurement M_{1}, whereas e〉 (c) and f〉 (d) are prepared by properly pulsing the selected g〉 state. A histogram of the second measurement, M_{2}, gives the qubit readout properties. e, Qubit readout properties for qubit initially in g〉, e〉 and f〉, respectively.
Extended Data Figure 6 Parity readout properties and Wigner tomography.
a, Parity readout property for given even and odd parity states for the protocol ( as the second qubit pulse) used in the singleshot traces in Fig. 3 (). b, Protocol to measure parity readout fidelity. An initial qubit measurement allows a postselection of the g〉 state of the qubit, followed by six consecutive parity measurements. The pulse sequence of each parity measurement is shown in P_{1} in c. are determined by postselecting the cases with the first five consecutive identical parity results, which give the photon state parity with good confidence, and then constructing a histogram for the sixth parity measurement. c, Illustration of pulse sequence (not to scale) for producing the cat states and the Wigner tomography shown in Fig. 2. The protocol starts with a postselection of the g〉 state of the qubit through an initial qubit measurement M_{1}. A parity measurement is performed immediately after a storage cavity displacement α, followed by Wigner tomography with varying displacements β. A 280 ns waiting time after each measurement has been chosen to ensure that the readout cavity is in the vacuum state. The qubit pulses have a Gaussian envelope truncated to 4σ = 8 ns, and the displacement pulses on the storage cavity are 10 ns square pulses. The dashed enclosures represent the pulse sequences for parity measurement. d, Error budgets for Wigner tomography fidelity. e, Error budgets for the parity readout fidelities with as the second qubit pulse.
Extended Data Figure 7 Schematic of the quantum filter.
At time t, the density matrix of the photon state is ρ(C_{t}), which depends on all previous correlations. At t + dt, only considering the decoherence of the cavity, the expected density matrix from free evolution becomes . The additional information C_{t}_{+dt} acquired from the parity measurement at t + dt changes the knowledge of the parity of the photon state according to equation (1).
Extended Data Figure 8 Effectiveness and response time of the quantum filter.
a, Ensembleaveraged parity dynamics obtained directly from the correlation of qubit states between neighbouring parity measurements. The data set is the same as that shown in Fig. 4. Solid lines are predictions based on equation (2), in excellent agreement with the measured data. The offset of the averaged parity at t = 0 comes from the asymmetry between the parity readout fidelities of the even and odd states. The fact that the saturated parity value in the long time limit is much lower than that in Fig. 4 is additional proof of the effectiveness of the quantum filter. b, Effectiveness of the quantum filter. Blue (raw) and red (filtered) curves are the same as those shown in Fig. 3e. The green curve is the direct correlation of qubit states between neighbouring parity measurements. The red curve is clearly much smoother and can reject the brief changes in the green curve. c, Response time of the quantum filter applied to typical photon jump events. The blue curve is the raw data from a repeated parity measurement. The red curve is the corresponding parity estimator based on the quantum filter. Green and cyan curves are fits to tanh functions of the parity estimator at the transitions down and up, respectively, giving a transition time constant of less than 1 µs. However, the response time of the filter to make a transition between −0.9 and +0.9 is τ_{f} ≈ 2 µs.
Extended Data Figure 9 Histograms of the number of jumps extracted from the parity estimator during 500 µs repeated parity measurements for an initial even or odd cat state by postselection.
a, b, α = 2.0; c, d, α = 1.4; e, f, α = 1.0. Solid lines are numerical simulations including the background thermal excitation and finite response time of the quantum filter. In the simulation, we use a coherent state as the initial state without distinguishing the parity. The good agreement between data and simulation demonstrates that the repeated parity measurement can track the error syndromes faithfully.
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Sun, L., Petrenko, A., Leghtas, Z. et al. Tracking photon jumps with repeated quantum nondemolition parity measurements. Nature 511, 444–448 (2014). https://doi.org/10.1038/nature13436
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DOI: https://doi.org/10.1038/nature13436
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