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Mapping the optimal route between two quantum states


A central feature of quantum mechanics is that a measurement result is intrinsically probabilistic. Consequently, continuously monitoring a quantum system will randomly perturb its natural unitary evolution. The ability to control a quantum system in the presence of these fluctuations is of increasing importance in quantum information processing and finds application in fields ranging from nuclear magnetic resonance1 to chemical synthesis2. A detailed understanding of this stochastic evolution is essential for the development of optimized control methods. Here we reconstruct the individual quantum trajectories3,4,5 of a superconducting circuit that evolves under the competing influences of continuous weak measurement and Rabi drive. By tracking individual trajectories that evolve between any chosen initial and final states, we can deduce the most probable path through quantum state space. These pre- and post-selected quantum trajectories also reveal the optimal detector signal in the form of a smooth, time-continuous function that connects the desired boundary conditions. Our investigation reveals the rich interplay between measurement dynamics, typically associated with wavefunction collapse, and unitary evolution of the quantum state as described by the Schrödinger equation. These results and the underlying theory6, based on a principle of least action, reveal the optimal route from initial to final states, and may inform new quantum control methods for state steering and information processing.

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Figure 1: Set-up.
Figure 2: Quantum trajectories of the quantum state on the Bloch sphere are plotted against time.
Figure 3: Greyscale histograms of quantum trajectories in the undriven case.
Figure 4: Greyscale histograms of quantum trajectories in the driven case.

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We thank A. N. Korotkov, S. G. Rajeev, N. Roch and D. Toyli for discussions. This research was supported in part by the Army Research Office, Office of Naval Research and the Office of the Director of National Intelligence (ODNI), Intelligence Advanced Research Projects Activity (IARPA), through the Army Research Office. 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. A.N.J. acknowledges support from NSF grant no. DMR-0844899 (CAREER).

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Authors and Affiliations



S.J.W. and K.W.M. performed the experiment and analysed the experimental data. J.D. and A.C. wrote the trajectory simulation code. A.C., J.D. and A.N.J. contributed the theory. All work was carried out under the supervision of I.S. All authors contributed to writing the manuscript.

Corresponding author

Correspondence to K. W. Murch.

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The authors declare no competing financial interests.

Extended data figures and tables

Extended Data Figure 1 Experimental schematic.

The weak measurement tone is always on. The projective readout tone is pulsed. The amplitude and phase of the signal displacement tone are adjusted to displace the measurement signals back to the origin of the X1X2 plane, which allows the LJPA to perform in the linear regime.

Extended Data Figure 2 Paths slightly varied from the optimal solution.

a, Overplotted x and z coordinates of 11 trajectories slightly varied from an optimized solution with boundary conditions (xIzI) = (0.88, 0), (xFzFTF) = (−0.683, −0.227, 0.464 µs) and the Rabi drive Ω/2π = 1.08 MHz. b, The corresponding conjugate variables px and pz. c, d, Plots of the unnormalized probability versus changes of the constant δ1 in the px differential equation (c) and the unnormalized probability versus changes of the constant δ2 in the pz differential equation (d). In this case, the optimized solution gives a maximum value of the path probability density.

Extended Data Figure 3 Optimal time between starting and destination states.

The probability density functions P(zF|zI = 0) plotted as functions of time T (solid curves) along with experimental data (dotted curves) with τ = 1.25 µs. The red, green and blue curves are the distribution functions P(zF = 0.2|zI = 0), P(zF = 0.4|zI = 0) and P(zF = 0.6|zI = 0), respectively. The optimized times Topt for the three cases are shown as the vertical black dashed lines with the labels T0.2, T0.4 and T0.6.

Extended Data Figure 4 Greyscale histograms of ensemble and post-selected trajectories for different Rabi frequencies and measurement strengths.

a, Ensemble and post-selected trajectories for Ω/2π = 1.08 MHz and τ = 1.25 µs. The post-selections for times {t1 = 464 ns, t2 = 944 ns, t3 = 1.424 µs} are (xFzF) = {(−0.78, −0.5), (0.7, −0.5), (−0.73, −0.5)} with a post selection window of ±0.08. b, Trajectories for Ω/2π = 1.08 MHz and τ = 315 ns with (xFzF) = {(−0.69, −0.5), (0.5, −0.5), (−0.73, −0.5)}. c, Trajectories for Ω/2π = 0.58 MHz and τ = 315 ns with (xFzF) = {(−0.35, −0.5), (−0.5, −0.5), (−0.56, −0.5)}. Note that all the trajectories use the same value of zF. The values of xF were chosen to give a large number of trajectories in the post-selected ensemble.

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Weber, S., Chantasri, A., Dressel, J. et al. Mapping the optimal route between two quantum states. Nature 511, 570–573 (2014).

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