We formulate a time-optimal approach to adiabatic quantum computation (AQC). A corresponding natural Riemannian metric is also derived, through which AQC can be understood as the problem of finding a geodesic on the manifold of control parameters. This geometrization of AQC is demonstrated through two examples, where we show that it leads to improved performance of AQC, and sheds light on the roles of entanglement and curvature of the control manifold in algorithmic performance.
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 study the assisted adiabatic passage, and equivalently the transitionless quantum driving, as a quantum brachistochrone trajectory. The optimal Hamiltonian for given constraints is constructed from the quantum brachistochrone equation. We discuss how the adiabatic passage is realized as the solution of the equation. The formulation of the quantum brachistochrone is based on the principle of least action. We utilize it to discuss the stability of the adiabatic passage.
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…
We study some aspects of the Quantum Brachistochrone Problem. Physical realizability of the faster pseudo Hermitian version of the problem is also discussed. This analysis, applied to simple quantum gates, supports an informational interpretation of the problem that is quasi Hermitian invariant.
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…
In this brief comment we attempt to clarify the apparent discrepancy between the papers [1] and [2] on the quantum brachistochrone, namely whether it is possible to use a judicious mixture of Hermitian and non-Hermitian quantum mechanics to evade the standard lower limit on the time taken for evolution by a Hermitian Hamiltonian with given energy dispersion between two given states.
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 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.
Recently, the quantum brachistochrone problem is discussed in the literature by using non-Hermitian Hamilton operators of different type. Here, it is demonstrated that the passage time is tunable in realistic open quantum systems due to the biorthogonality of the eigenfunctions of the non-Hermitian Hamilton operator. As an example, the numerical results obtained by Bulgakov et al. for the transmission through microwave cavities of different shape are analyzed from the point of view of the brachistochrone problem. The passage time is shortened in the crossover from the weak-coupling to the stro…
Entanglement is closely related to some fundamental features of the dynamics of composite quantum systems: quantum entanglement enhances the "speed" of evolution of certain quantum states, as measured by the time required to reach an orthogonal state. The concept of "speed" of quantum evolution constitutes an important ingredient in any attempt to determine the fundamental limits that basic physical laws impose on how fast a physical system can process or transmit information. Here we explore the relationship between entanglement and the speed of quantum evolution in the context of the quantum…
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 quantum brachistochrone problem for the spin-1 system in a magnetic field of a constant absolute value. Such system gives us a possibility to examine in detail the statement of papers [A. Carlini {\it et al.}, Phys. Rev. Lett. {\bf 96}, 060503 (2006), D. C. Brody, D. W. Hook, J. Phys. A {\bf 39}, L167, (2006)] that {\it the state vectors realizing the evolution with the minimal time of passage evolve along the subspace spanned by the initial and final state vectors.} Using explicit example we show the existence of quantum brachistochrone with minimal possible time, but the state vecto…
We explore the connection between quantum brachistochrone (time-optimal) evolution of a three-qubit system and its residual entanglement called three-tangle. The result shows that the entanglement between two qubits is not required for some brachistochrone evolutions of a three-qubit system. However, the evolution between two distinct states cannot be implemented without its three-tangle, except for the trivial cases in which less than three qubits attend evolution. Although both the probability density function of the time-averaged three-tangle and that of the time-averaged squared concurrenc…
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…
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 present a general formalism based on the variational principle for finding the time-optimal quantum evolution of mixed states governed by a master equation, when the Hamiltonian and the Lindblad operators are subject to certain constraints. The problem reduces to solving first a fundamental equation (the {\it quantum brachistochrone}) for the Hamiltonian, which can be written down once the constraints are specified, and then solving the constraints and the master equation for the Lindblad and the density operators. As an application of our formalism, we study a simple one-qubit model where …
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 …
We study the quantum brachistochrone evolution for a system of two spins-$\frac{1}{2}$ described by an anisotropic Heisenberg Hamiltonian without $zx$, $zy$ interacting couplings in magnetic field directed along the z-axis. This Hamiltonian realizes quantum evolution in two subspaces spanned by $\|\uparrow\uparrow\rangle$, $\|\downarrow\downarrow\rangle$ and $\|\uparrow\downarrow\rangle$, $\|\downarrow\uparrow\rangle$ separately and allows to consider the brachistochrone problem on each subspace separately. Using the evolution operator for this Hamiltonian we generate quantum gates, namely an …
We address the question: Given an arbitrary initial state and a general physical interaction what is the minimum time for reaching a target entangled state? We show that the minimum time is inversely proportional to the quantum mechanical uncertainty in the non-local Hamiltonian. We find that the presence of initial entanglement helps to minimize the waiting time. Furthermore, we find that in a bi-local rotating frame the entangling capability is actually a geometric quantity. We give an universal bound for the time average of entanglement rate for general quantum systems. The time average of …
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 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.
The PT-symmetric (PTS) quantum brachistochrone problem is reanalyzed as quantum system consisting of a non-Hermitian PTS component and a purely Hermitian component simultaneously. Interpreting this specific setup as subsystem of a larger Hermitian system, we find non-unitary operator equivalence classes (conjugacy classes) as natural ingredient which contain at least one Dirac-Hermitian representative. With the help of a geometric analysis the compatibility of the vanishing passage time solution of a PTS brachistochrone with the Anandan-Aharonov lower bound for passage times of Hermitian brach…
There are a number of tasks in quantum information science that exploit non-transitional adiabatic dynamics. Such a dynamics is bounded by the adiabatic theorem, which naturally imposes a speed limit in the evolution of quantum systems. Here, we investigate an approach for quantum state engineering exploiting a shortcut to the adiabatic evolution, which is based on rapid quenches in a continuous-time Hamiltonian evolution. In particular, this procedure is able to provide state preparation faster than the adiabatic brachistochrone. Remarkably, the evolution time in this approach is shown to be …
A non-Hermitian operator with a real spectrum and a complete set of eigenvectors may serve as the Hamiltonian operator for a unitary quantum system provided that one makes an appropriate choice for the defining inner product of the physical Hilbert state. We study the consequences of such a choice for the representation of states in terms of projection operators and the geometry of the state space. This allows for a careful treatment of the quantum Brachistochrone problem and shows that it is indeed impossible to achieve faster unitary evolutions using PT-symmetric or other non-Hermitian Hamil…
Recently Bender, Brody, Jones and Meister found that in the quantum brachistochrone problem the passage time needed for the evolution of certain initial states into specified final states can be made arbitrarily small, when the time-evolution operator is taken to be non-Hermitian but PT-symmetric. Here we demonstrate that such phenomena can also be obtained for non-Hermitian Hamiltonians for which PT-symmetry is completely broken, i.e. dissipative systems. We observe that the effect of a tunable passage time can be achieved by projecting between orthogonal eigenstates by means of a time-evolut…
The quantum mechanical brachistochrone system with PT-symmetric Hamiltonian is Naimark dilated and reinterpreted as subsystem of a Hermitian system in a higher-dimensional Hilbert space. This opens a way to a direct experimental implementation of the recently hypothesized PT-symmetric ultra-fast brachistochrone regime of [C. M. Bender et al, Phys. Rev. Lett. {\bf 98}, 040403 (2007)] in an entangled two-spin system.
We study suppression of nonadiabatic transitions during adiabatic generation of a cat-like state (a superposition of different size cat states) and a spin squeezed state in a bosonic Josephson junction. In order to minimize the adiabatic error, we use quantum adiabatic brachistochrone, which enables us to track a geometrically efficient path in parameter space under given conditions without requiring additional terms. For creation of a cat-like state, divergence of the quantum geometric tensor associated with gap closing at the critical point is avoided because of the parity conservation. The …
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）
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…
We introduce the concept of interpolation in quantum evolution and present a general framework to find the energy optimal Hamiltonian for a quantum system evolving among a given set of middle states using variational and geometric methods. A few special cases are carefully studied. The quantum brachistochrone problem is proved as a special case.
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 present a systematic method for dealing with time dependent quantum dynamics, based on the quantum brachistochrone and matrix mechanics. We derive the explicit time dependence of the Hamiltonian operator for a number of constrained finite systems from this formalism. Once this has been achieved we go on to calculate the wavevector as a function of time, in order to demonstrate the use of matrix methods with respect to several concrete examples. Interesting results are derived for elliptic curves and periodic orbits on higher dimensional non-commutative geometries.
Bellman's optimality principle has been of enormous importance in the development of whole branches of applied mathematics, computer science, optimal control theory, economics, decision making, and classical physics. Examples are numerous: dynamic programming, Markov chains, stochastic dynamics, calculus of variations, and the brachistochrone problem. Here we show that Bellman's optimality principle is violated in a teleportation problem on a quantum network. This implies that finding the optimal fidelity route for teleporting a quantum state between two distant nodes on a quantum network with…
Recently, Bender et al. have considered the quantum brachistochrone problem for the non-Hermitian $\cal PT$-symmetric quantum system and have shown that the optimal time evolution required to transform a given initial state $|ψ_i\rangle$ into a specific final state $|ψ_f\rangle$ can be made arbitrarily small. Additionally, it has been shown that finding the shortest possible time requires only the solution of the two-dimensional problem for the quantum system governed by the effective Hamiltonian acting in the subspace spanned by $|ψ_i\rangle$ and $|ψ_f\rangle$. In this paper, we study a simil…
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…
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…
We show that the quantum Fisher information attained in an adiabatic approach to critical quantum metrology cannot lead to the Heisenberg limit of precision and therefore regular quantum metrology under optimal settings is always superior. Furthermore, we argue that even though shortcuts to adiabaticity can arbitrarily decrease the time of preparing critical ground states, they cannot be used to achieve or overcome the Heisenberg limit for quantum parameter estimation in adiabatic critical quantum metrology. As case studies, we explore the application of counter-diabatic driving to the Landau-…
We present a rigorous proof that quantum circuit algorithm can be transformed into quantum adiabatic algorithm with the exact same time complexity. This means that from a quantum circuit algorithm of $L$ gates we can construct a quantum adiabatic algorithm with time complexity of $O(L)$. Additionally, our construction shows that one may exponentially speed up some quantum adiabatic algorithms by properly choosing an evolution path.
Quantum adiabatic evolutions find a broad range of applications in quantum physics and quantum technologies. The traditional form of the quantum adiabatic theorem limits the speed of adiabatic evolution by the minimum energy gaps of the system Hamiltonian. Here, we experimentally show using a nitrogen-vacancy center in diamond that, even in the presence of vanishing energy gaps, quantum adiabatic evolution is possible. This verifies a recently derived necessary and sufficient quantum adiabatic theorem and offers paths to overcome the conventionally assumed constraints on adiabatic methods. By …
A major challenge facing adiabatic quantum computing is that algorithm design and error correction can be difficult for adiabatic quantum computing. Recent work has considered addressing his challenge by using coherently controlled adiabatic evolutions in the place of classically controlled evolution. An important question remains: what is the relative power of controlled adiabatic evolution to traditional adiabatic evolutions? We address this by showing that coherent control and measurement provides a way to average different adiabatic evolutions in ways that cause their diabatic errors to ca…
Numerous sufficient conditions for adiabaticity of the evolution of a driven quantum system have been known for quite a long time. In contrast, necessary adiabatic conditions are scarce. A practicable necessary condition well-suited for many-body systems has been proven recently in [Phys. Rev. Lett. 119, 200401 (2017)]. Here we tailor this condition for estimating run times of quantum adiabatic algorithms. As an illustration, the condition is applied to the adiabatic algorithm for searching in an unstructured database (adiabatic Grover search algorithm). We find that thus obtained lower bound …
The unitary operator corresponding to the classical canonical transformation that connects a general closed system to an open system under adiabatic conditions is found. The quantum invariant operator of the adiabatic open system is derived from the unitary transformation of the quantum Hamiltonian of the closed system. On the basis of these results, we investigate the evolution of the general quantum adiabatic system and construct a revised adiabatic theorem. The adiabatic theorem developed here exactly reduces to the well-known Berry adiabatic theorem when the control parameter
