We consider the evolution of a quantum simple harmonic oscillator in a general Gaussian state under simultaneous time-continuous weak position and momentum measurements.We deduce the stochastic evolution equations for position and momentum expectation values and the covariance-matrix elements from the characteristic function of the system.By generalizing the Chantasri-Dressel-Jordan (CDJ) formalism (Chantasri et al., 2013(Chantasri et al., , 2015) ) to this continuous-variable system, we construct its stochastic Hamiltonian and action.Action extremization gives us the equations for the most likely readout paths and quantum trajectories.For steady states of the covariance-matrix elements, the analytical solutions for these most likely paths are obtained.Using the CDJ formalism, we calculate final-state probability densities exactly starting from any initial state.We also demonstrate the agreement between the optimal-path solutions and the averages of simulated clustered stochastic trajectories.Our results provide insights into the time dependence of the mechanical energy of the system during the measurement process, motivating their importance for quantum measurement-engine or refrigerator experiments.Although quantum mechanics fundamentally limits our ability to extract information via simultaneous measurements of noncommuting observables, insights into the dynamics of the system during such a process can lead to a deeper understanding of the physics of measurement.
We generalize and extend the stochastic path-integral formalism and action principle for continuous quantum measurement introduced in Chantasri, Dressel, and Jordan [Phys. Rev. A 88, 042110 (2013)], where the optimal dynamics, such as the most likely paths, is obtained by extremizing the action of the stochastic path integral. In this work, we apply exact functional methods as well as develop a perturbative approach to investigate the statistical behavior of continuous quantum measurement. Examples are given for the qubit case. For qubit measurement with zero-qubit Hamiltonian, we find analytic solutions for average trajectories and their variances while conditioning on fixed initial and final states. For qubit measurement with unitary evolution, we use the perturbation method to compute expectation values, variances, and multi-time-correlation functions of qubit trajectories in the short-time regime. Moreover, we consider continuous qubit measurement with feedback control, using the action principle to investigate the global dynamics of its most likely paths, and finding that in an ideal case, qubit state stabilization at any desired pure state is possible with linear feedback. We also illustrate the power of the functional method by computing correlation functions for the qubit trajectories with a feedback loop to stabilize the qubit Rabi frequency.
A stochastic procedure is developed which allows one to express Pontryagin's maximum principle for a dissipative quantum system solely in terms of stochastic wave functions. Time-optimal controls can be efficiently computed without computing the density matrix. Specifically, the proper dynamical update rules are presented for the stochastic costate variables introduced by Pontryagin's maximum principle, and restrictions on the form of the terminal cost function are discussed. The proposed procedure is confirmed by comparing the results to those obtained from optimal control on Lindbladian dynamics. Numerically, the proposed formalism becomes time and memory efficient for large systems, and it can be generalized to describe non-Markovian dynamics.
Quantum metrology comprises a set of techniques and protocols that utilize quantum features for parameter estimation which can in principle outperform any procedure based on classical physics. We formulate the quantum metrology in terms of an optimal control problem and apply Pontryagin's maximum principle to determine the optimal protocol that maximizes the quantum Fisher information for a given evolution time. As the quantum Fisher information involves a derivative with respect to the parameter which one wants to estimate, we devise an augmented dynamical system that explicitly includes gradients of the quantum Fisher information. The necessary conditions derived from Pontryagin's maximum principle are used to quantify the quality of the numerical solution. The proposed formalism is generalized to problems with control constraints, and can also be used to maximize the classical Fisher information for a chosen measurement.
Optimization is one of the key applications of quantum computing where a quantum speedup has been an eagerly anticipated outcome. A promising approach to optimization using quantum dynamics is to consider a linear combination $s(t)B+[1\ensuremath{-}s(t)]C$ of two noncommuting Hamiltonians
B and
C, where
C encodes the solution to the optimization problem in its ground state,
B is a Hamiltonian whose ground state is easy to prepare, and $s(t)\ensuremath{\in}[0,1]$ is the bounded ``switching schedule'' or ``path,'' with $t\ensuremath{\in}[0,{t}_{f}]$. This approach encompasses two of the most widely studied quantum-optimization algorithms: quantum annealing [QA; continuous
s(t)] and the quantum approximate optimization algorithm [QAOA; piecewise constant
s(t)]. While it is notoriously difficult to prove a quantum advantage for either algorithm, it is possible to compare and contrast them by finding the optimal
s(t). Here we provide a rigorous analysis of this quantum optimal control problem, entirely within the geometric framework of Pontryagin's maximum principle of optimal control theory. We extend earlier results, derived in a purely closed-system setting, to open systems. This is the natural setting for experimental realizations of QA and QAOA. In the closed-system setting it was shown that the optimal solution is a ``bang-anneal-bang'' schedule, with the bangs characterized by
s(t)=0 and
s(t)=1 in finite subintervals of
[0,tf], in particular,
s(0)=0 and
s(tf)=1, in contrast to the standard prescription
s(0)=1 and
s(tf)=0 of QA. As an example, we prove that for a single spin-
1/2, the optimal solution in the closed-system setting is the bang-bang schedule, switching midway from $s\ensuremath{\equiv}0$ to $s\ensuremath{\equiv}1$. For finite-dimensional environments and without any approximations we identify sufficient conditions ensuring that either the bang-anneal, anneal-bang, or bang-anneal-bang schedules are optimal, and recover the optimality of
s(0)=0 and
s(tf)=1. However, for infinite-dimensional environments and a system described by an adiabatic Redfield master equation we do not recover the bang-type optimal solution. In fact we can only identify conditions under which
s(tf)=1, and even this result is not recovered in the fully Markovian limit, suggesting that the pure anneal-type schedule is optimal. Our open-system results have implications for the use of experimental quantum-information processors, which are by necessity noisy, and suggest that in this practical sense the optimal schedules for quantum optimization are likely to be continuous.
In this paper, we develop the theory of the quantum Pontryagin principle for continuous measurements and feedback. The analysis is carried out under the assumption of compatible events in the output channel. The plant is a quantum system, which generally is in a mixed state, coupled to a continuous measurement channel. The Pontryagin maximum principle is derived in both the Schrödinger picture and Heisenberg picture, in particular, in statistical moment coordinates. To avoid solving stochastic equations, we derive a linear-quadratic-Gaussian scheme, which is more suitable for control purposes. Finally, we use the quantum harmonic oscillator as a concrete example to illustrate the performance of the controller.