Quantum Brachistochrone Curves as Geodesics: Obtaining Accurate Minimum-Time Protocols for the Control of Quantum Systems

Abstract

Most methods of optimal control cannot obtain accurate time-optimal protocols. The quantum brachistochrone equation is an exception, and has the potential to provide accurate time-optimal protocols for a wide range of quantum control problems. So far, this potential has not been realized, however, due to the inadequacy of conventional numerical methods to solve it. Here we show that the quantum brachistochrone problem can be recast as that of finding geodesic paths in the space of unitary operators. We expect this brachistochrone-geodesic connection to have broad applications, as it opens up minimaltime control to the tools of geometry. As one such application, we use it to obtain a fast numerical method to solve the brachistochrone problem, and apply this method to two examples demonstrating its power. The ability to realize a prescribed evolution for a given physical quantum device is important in a range of applications. A powerful approach to this task is to vary the Hamiltonian of the device with time [1][2][3]. A prescription for a time-dependent Hamiltonian realizing a desired evolution is called a "control protocol," and a protocol that achieves this task in the minimal time is called "timeoptimal" [4,5]. Because the ever-present noise from the environment degrades quantum states over time, generating the fastest possible evolution is important in information processing [6][7][8], metrology [9][10][11], cooling [12,13], and experiments that probe quantum behavior [14][15][16][17]. Finding accurate time-optimal protocols is difficult because it is a two-objective optimization problem: one must minimize the error in the resulting evolution while simultaneously minimizing the time taken by the protocol (hereafter, the "protocol time"). Finding approximate protocols, on the other hand, is relatively easy: one can minimize a weighted sum of the two objectives [1], or search for protocols at a range of fixed times to locate a likely minimum time. But there is presently no practical way to refine these further. Analytical methods that use the Pontryagin maximum principle or the geometry of the unitary group are useful only for specific kinds of problems and constraints [5,[18][19][20][21][22][23][24][25][26]. In view of this, the quantum brachistochrone equation (QBE) was a significant development [27][28][29]. It has the potential to provide accurate time-optimal protocols under two physically relevant constraints: (i) the system has a finite energy bandwidth (the norm of the Hamiltonian is bounded); and (ii) the Hamiltonian is restricted to a subspace of Hermitian operators. Nevertheless, an obstacle remains that has prevented the QBE from becoming a practical tool: the QBE transforms the optimization problem into that of solving an ordinary differential equation (ODE) with boundary values, but there exists no numerical method that can solve such a boundary-value problem (BVP) efficiently in high dimensions. The available methods, namely "simple shooting," "multiple shooting," finite difference, and finite-el...

Zermelo navigation in the quantum brachistochrone

“…Other recent work [35] also discusses finding Finsler geodesics on SU (n) in the context of quantum optimal control. We feel that the relative simplicity of the EP equations, which hold on a vector space su(n), compared to the methods described in [35,45] that hold on SU (n), justify the usefulness of our approach. Furthermore, they avoid the need to ever determine a geodesic on SU (n) when all that is practically needed is the Hamiltonian that drivesÛ t along that geodesic.…”

“…This question has been discussed from many perspectives before, for example [1,11,20,25,30,31,36,41,46]. Notable recent works based on geometric methods are [11,12,45]. Our previous work [39] begins an investigation into the application of "Zermelo Navigation" to determining speed limits for implementing quantum gates in systems of the form eqn.…”

“…We predict that the main obstacle to this will be solving for the metric F in terms ofF . Recent work [45] contains a large appendix "Euler-Lagrange equation on SU (n)" discussing methods for finding geodesics on SU (n). Other recent work [35] also discusses finding Finsler geodesics on SU (n) in the context of quantum optimal control.…”

Fundamental Speed Limits to the Generation of Quantumness

“…The modern formulation of QSL for unitary processes takes into account an alternative expression as an upper bound for the speed of evolution, the mean energy of the system, that can replace the role of energy dispersion Δ E 1232021. A geometric interpretation provides an intuitive understanding of the QSL bound as a brachistochrone22 where the geodesic2324 set by the Fubini-Study metric in (projective) Hilbert space is travelled at the maximum speed of evolution achievable under a given Hamiltonian dynamics25262728. Time-optimal evolutions are often explored in the context of quantum control theory, where the existence of a QSL has been shown to limit the performance of algorithms aimed at identifying optimal driving protocols1112.…”

Time-optimal navigation through quantum wind

“…With the advances in the implementation of quantum technologies, the theoretical understanding of controlled quantum dynamics and, in particular, of their limits, is becoming increasingly important. One aspect of such limits that has been investigated extensively in the literature concerns the time-optimal manoeuvring of quantum states [1][2][3][4][5][6][7][8][9][10][11][12][13][14][15][16][17][18]. If the time-evolution is unconstrained (apart from a bound on the energy resource), then this amounts to finding the time-independent Hamiltonian that generates maximum speed of evolution.…”