We present a general framework for finding the time-optimal evolution and the optimal Hamiltonian for a quantum system with a given set of initial and final states. Our formulation is based on the variational principle and is analogous to that for the brachistochrone in classical mechanics. We reduce the problem to a formal equation for the Hamiltonian which depends on certain constraint functions specifying the range of available Hamiltonians. For some simple examples of the constraints, we explicitly find the optimal solutions.
Minimum-time quantum control protocols can be obtained from the quantum brachistochrone formalism [Carlini, Hosoya, Koike, and Okudaira, Phys. Rev. Lett. 96, 06053, (2006)]. We point out that the original treatment implicitly applied the variational calculus with fixed boundary conditions. We argue that the genuine quantum brachistochrone problem involves a variational problem with a movable endpoint, contrary to the classical brachistochrone problem. This formulation not only simplifies the derivation of the quantum brachistochrone equation but introduces an additional constraint at the endpo…
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 essentially any quantum control problem. So far this potential has not been realized, however, due to the inadequacy of conventional numerical methods to solve it. Here, using differential geometry, we reformulate the quantum brachistochrone curves as geodesics on the unitary group. With this identification we are able to obtain a numerical method that efficiently solves the brachistochrone prob…
We introduce an iterative method to search for time-optimal Hamiltonians that drive a quantum system between two arbitrary, and in general mixed, quantum states. The method is based on the idea of progressively improving the efficiency of an initial, randomly chosen, Hamiltonian, by reducing its components that do not actively contribute to driving the system. We show that our method converges rapidly even for large dimensional systems, and that its solutions saturate any attainable bound for the minimal time of evolution. We provide a rigorous geometric interpretation of the iterative method …
Projector Form of the Quantum Brachistochrone and Its Relation to Two-Boundary Quantum Algorithm Design
Graduate School of Information Sciences, Tohoku University（东北大学信息科学研究院）；Department of Physics, Institute of Science Tokyo（东京科学大学物理系）；Sigma-i Co., Ltd.（Sigma-i有限公司）
We propose a variational principle to compute a quantum adiabatic brachistochrone (QAB) for open systems. Using the notion of "adiabatic speed" based on the energy gaps, we derive a Lagrangian associated to the functional measuring the time spent to achieve adiabatic behavior, which in turn allows us to perform the optimization. The QAB is illustrated for non-unitary dynamics of STIRAP process, the Deutsch-Jozsa quantum computing algorithm and of a transmon qutrit. A numerical protocol is devised, which allows to compute the QAB for arbitrary quantum systems for which exact simulations can be …
We analytically investigate the role of entanglement in time-optimal state evolution as an appli- cation of the quantum brachistochrone, a general method for obtaining the optimal time-dependent Hamiltonian for reaching a target quantum state. As a model, we treat two qubits indirectly cou- pled through an intermediate qubit that is directly controllable, which represents a typical situation in quantum information processing. We find the time-optimal unitary evolution law and quantify residual entanglement by the two-tangle between the indirectly coupled qubits, for all possible sets of initia…
Transforming an initial quantum state into a target state through the fastest possible route---a quantum brachistochrone---is a fundamental challenge for many technologies based on quantum mechanics. Here, we demonstrate fast coherent transport of an atomic wave packet over a distance of 15 times its size---a paradigmatic case of quantum processes where the target state cannot be reached through a local transformation. Our measurements of the transport fidelity reveal the existence of a minimum duration---a quantum speed limit---for the coherent splitting and recombination of matter waves. We …
We experimentally demonstrate fast and high-fidelity geometric control of a quantum system with the most brachistochrone method on hybrid spin registers in diamond. Based on the time-optimal universal geometric control, single geometric gates with the fidelities over 99.2% on the spin state of nitrogen-vacancy center are realized with average durations shortened by 74.9%, comparing with conventional geometric method. The fidelity of the fast geometric two-qubit gate exceeds 96.5% on the hybrid spin registers. With these fast high-fidelity gates available, we implement quantum entanglement-enha…
In quantum control, geometrical operations could provide an extra layer of robustness against control errors. However, the conventional non-adiabatic holonomic quantum computation (NHQC) is limited by the fact that all of the operations require exactly the same amount of evolution time, even for a small-angle rotation. Furthermore, NHQC confines the driving part of the Hamiltonian to strictly cover a fixed pulse area, making it sensitive to control errors. Here we present an unconventional approach of NHQC, termed brachistochronic NHQC, for bypassing these limitations. Specifically, with B-NHQ…
Efficient control of qubits plays a key role in quantum information processing. In the current work, an alternative set of differential equations are derived for an optimal quantum control of single or multiple qubits with or without interaction. The new formulation enables a great reduction of the computation load by eliminating redundant complexity involved in previous formulations. A relaxation technique is designed for numerically detecting optimal paths involving entanglement. Interesting continuous symmetries are identified in the Lagrangian, which indicates the existence of physically e…
Using the quantum-brachistochrone formalism, we address the problem of finding the fastest possible (time-optimal) deterministic conversion between $W$ and Greenberger-Horne-Zeilinger (GHZ) states in a system of three identical and equidistant neutral atoms that are acted upon by four external laser pulses. Assuming that all four pulses are close to being resonant with the same internal (atomic) transition -- the one between the atomic ground state and a high-lying Rydberg state -- each atom can be treated as an effective two-level system ($gr$-type qubit). Starting from an effective system Ha…
We consider quantum brachistochrone evolution for a spin-$s$ system on rotational manifolds. Such manifolds are determined by the rotation of the eigenstates of the operator of projection of spin-$s$ on some direction. The Fubini-Study metrics of these manifolds are those of spheres with radii dependent on the value of the spin and on the value of the spin projection. The conditions for optimal evolution of the spin-$s$ system on rotational manifolds are obtained.
We analyse the optimal times for implementing unitary quantum gates in a constrained finite dimensional controlled quantum system. The family of constraints studied is that the permitted set of (time dependent) Hamiltonians is the unit ball of a norm induced by an inner product on su(n). We also consider a generalisation of this to arbitrary norms. We construct a Randers metric, by applying a theorem of Shen on Zermelo navigation, the geodesics of which are the time optimal trajectories compatible with the prescribed constraint. We determine all geodesics and the corresponding time optimal Ham…
Fermionic systems differ from bosonic ones in several ways, in particular that the time-reversal operator $T$ is odd, $T^2=-1$. For $PT$-symmetric bosonic systems, the no-signaling principle and the quantum brachistochrone problem have been studied to some degree, both of them controversially. In this paper, we apply the basic methods proposed for bosonic systems to {\it fermionic} two- and four-dimensional $PT$-symmetric Hamiltonians, and obtain several surprising results: We find - in contrast to the bosonic case - that the no-signaling principle is upheld for two-dimensional fermionic Hamil…
We discuss applications of shortcuts to adiabaticity (STA) to adiabatic quantum computation. After reviewing the fundamental properties and the present status of STA from the author's personal point of view, we apply the method to the adiabatic algorithm of the Grover's problem. We discuss two possible implementations of STA for the adiabatic quantum computations: the method of quantum adiabatic brachistochrone and the Lewis-Riesenfeld invariant-based inverse engineering.
We study a deformation of the counterdiabatic-driving Hamiltonian as a systematic strategy for an adiabatic control of quantum states. Using a unitary transformation, we design a convenient form of the driver Hamiltonian. We apply the method to a particle in a confining potential and discrete systems to find explicit forms of the Hamiltonian and discuss the general properties. The method is derived by using the quantum brachistochrone equation, which shows the existence of a nontrivial dynamical invariant in the deformed system.
Geometry and quantum brachistochrone analysis of multiple entangled spin-1/2 particles under all-range Ising interaction
Chouaïb Doukkali University（舒艾卜杜卡利大学）；Mohammed V University in Rabat（拉巴特穆罕默德五世大学）；Ibn Zohr University（伊本·佐赫尔大学）；Princess Nourah bint Abdulrahman University（公主努拉·宾特·阿卜杜勒拉赫曼大学）；Prince Sultan University（苏丹王子大学）；Menoufia University（门尤菲亚大学）
Comprehending the connections between the geometric, topological, and dynamical structures of integrable quantum systems with quantum phenomena exploitable in quantum information tasks, such as quantum entanglement, is a major problem in geometric information science. In this work we investigate these issues in a physical system of $\mathcal{N}$ interacting spin-$\mathtt{s}$ under long-range Ising model. We discover the relevant dynamics, identify the corresponding quantum phase space and we derive the associated Fubini-Study metric. Through the application of the Gauss-Bonnet theorem and the …
We present a general protocol to control closed quantum systems that is based on minimising the adiabatic action. Using tools based on the geometry of quantum evolutions through the quantum adiabatic brachistochrone, we show that high fidelity control of the ground state of critical quantum systems can be achieved and requires only a reasonable approximation for the ground state spectral gap. We demonstrate our results for three widely applicable systems: the Landau-Zener, Ising, and fully connected spin models.
An analysis of the motion of a relativistic electron under a linear constraint in four dimensions is presented. Interesting results are given that show that the state of the electron is well defined under the formalism of time optimal quantum state control. We establish compact mechanisms for achieving time dependent unitary evolution, and present new calculation methods for time-ordered exponential operators. A powerful modification of the brachistochrone technique is presented that allows solution of a class of problems via matrix decomposition of the Hamiltonian. These techniques allow us t…
Quantum control of systems plays important roles in modern science and technology. The ultimate goal of quantum control is to achieve high fidelity universal control in the time-optimal way. Although high fidelity universal control has been reported in various quantum systems, experimental implementation of time-optimal universal control remains elusive. Here we report the experimental realization of time-optimal universal control of spin qubits in diamond. By generalizing a recent method for solving quantum brachistochrone equations [X. Wang, et al., Phys. Rev. Lett. 114, 170501 (2015)], we o…
Quantum batteries are miniature energy storage devices and play a very important role in quantum thermodynamics. In recent years, quantum batteries have been extensively studied, but limited in theoretical level. Here we report the experimental realization of a quantum battery based on superconducting qubits. Our model explores dark and bright states to achieve stable and powerful charging processes, respectively. Our scheme makes use of the quantum adiabatic brachistochrone, which allows us to speed up the {battery ergotropy injection. Due to the inherent interaction of the system with its su…
In this paper, we briefly discuss the methodology for simulating a quantum computer which performs Shor's algorithm on a 7-qubit system to factorise 15. Using this simulation and the overlooked quantum brachistochrone method, we devised a Monte Carlo algorithm to calculate the expected time a theoretical quantum computer could perform this calculation under the same energy conditions as current working quantum computers. We found that, experimentally, a nuclear magnetic resonance quantum computer would take $1.59 \pm 0.04$ s to perform our simulated computation, whereas the expected optimal ti…
