Letter | Published:

Mapping the optimal route between two quantum states

Nature volume 511, pages 570573 (31 July 2014) | Download Citation


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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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).

Author information


  1. Quantum Nanoelectronics Laboratory, Department of Physics, University of California, Berkeley, California 94720, USA

    • S. J. Weber
    •  & I. Siddiqi
  2. Department of Physics and Astronomy and Center for Coherence and Quantum Optics, University of Rochester, Rochester, New York 14627, USA

    • A. Chantasri
    •  & A. N. Jordan
  3. Department of Electrical Engineering, University of California, Riverside, California 92521, USA

    • J. Dressel
  4. Institute for Quantum Studies, Chapman University, University Drive, Orange, California 92866, USA

    • A. N. Jordan
  5. Department of Physics, Washington University, St Louis, Missouri 63130, USA

    • K. W. Murch


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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.

Competing interests

The authors declare no competing financial interests.

Corresponding author

Correspondence to K. W. Murch.

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