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.PACS numbers: 03.67.Lx, 03.65.Ca, 02.30.Xx, 02.30.Yy In quantum mechanics one can change a given state to another by applying a suitable Hamiltonian on the system. In many situations, e.g. quantum computation, it is desirable to know the pathway in the shortest time.In this paper we consider the problem of finding the time-optimal path for the evolution of a pure quantum state and the optimal driving Hamiltonian. Recently, a growing number of works related to this topic have appeared. For instance, Alvarez and Gómez [1] showed that the quantum state in Grover's algorithm [2], known as the optimal quantum search algorithm [3], actually follows a geodesic curve derived from the Fubini-Study metric in the projective space. Khaneja et al. [4] and Zhang et al.[5], using a Cartan decomposition scheme for unitary operations, discussed the time optimal way to realize a two-qubit universal unitary gate under the condition that one-qubit operations can be performed in an arbitrarily short time. On the other hand, Tanimura et al. [6] gave an adiabatic solution to the optimal control problem in holonomic quantum computation, in which a desired unitary gate is generated as the holonomy corresponding to the minimal length loop in the space of control parameters for the Hamiltonian. Schulte-Herbrüggen et al.[7] exploited the differential geometry of the projective unitary group to give the tightest known upper bounds on the actual time complexity of some basic modules of quantum algorithms. More recently, Nielsen [8] introduced a lower bound on the size of the quantum circuit necessary to realize a given unitary operator based on the geodesic distance, with a suitable metric, between the unitary and the identity operators. However, a general method for generating the time optimal Hamiltonian evolution of quantum states was not known until now.Here we are going to study this problem by exploiting the analogy with the so-called brachistochrone problem in classical mechanics and the elementary properties * Electronic address: carlini@th.phys.titech.ac.jp † Electronic address: ahosoya@th.phys.titech.ac.jp ‡ Electronic address: koike@phys.keio.ac.jp § Electronic address: okudaira@th.phys.titech.ac.jp of quantum mechanics. In ordinary quantum mechanics the initial state and the Hamiltonian of a physical system are given and one has to find the final state using the Schrödinger equation. In our work we generalize this framework so as to optimize a certain cost functional with respect to the Hamiltonian a...

Extending our previous work on time optimal quantum state evolution [A. Carlini, A. Hosoya, T. Koike and Y. Okudaira, Phys. Rev. Lett. 96, 060503 (2006)], we formulate a variational principle for finding the time optimal realization of a target unitary operation, when the available Hamiltonians are subject to certain constraints dictated either by experimental or by theoretical conditions. Since the time optimal unitary evolutions do not depend on the input quantum state this is of more direct relevance to quantum computation. We explicitly illustrate our method by considering the case of a two-qubit system self-interacting via an anisotropic Heisenberg Hamiltonian and by deriving the time optimal unitary evolution for three examples of target quantum gates, namely the swap of qubits, the quantum Fourier transform and the entangler gate. We also briefly discuss the case in which certain unitary operations take negligible time.

We consider the role of the detector size in the imaging of confocal systems. We introduce a criterion for detector size to give true confocal operation and show that the lateral resolution is considerably more sensitive to detector size than is the depth-discrimination or optical-sectioning property.

We consider a two-level quantum system prepared in an arbitrary initial state and relaxing to a steady state due to the action of a Markovian dissipative channel. We study how optimal control can be used for speeding up or slowing down the relaxation towards the fixed point of the dynamics. We analytically derive the optimal relaxation times for different quantum channels in the ideal ansatz of unconstrained quantum control (a magnetic field of infinite strength). We also analyze the situation in which the control Hamiltonian is bounded by a finite threshold. As byproducts of our analysis we find that: (i) if the qubit is initially in a thermal state hotter than the environmental bath, quantum control cannot speed up its natural cooling rate; (ii) if the qubit is initially in a thermal state colder than the bath, it can reach the fixed point of the dynamics in finite time if a strong control field is applied; (iii) in the presence of unconstrained quantum control it is possible to keep the evolved state indefinitely and arbitrarily close to special initial states which are far away from the fixed points of the dynamics.

Suppose that three kinds of quantum systems are given in some unknown states $\ket f^{\otimes N}$, $\ket{g_1}^{\otimes K}$, and $\ket{g_2}^{\otimes K}$, and we want to decide which \textit{template} state $\ket{g_1}$ or $\ket{g_2}$, each representing the feature of the pattern class ${\cal C}_1$ or ${\cal C}_2$, respectively, is closest to the input \textit{feature} state $\ket f$. This is an extension of the pattern matching problem into the quantum domain. Assuming that these states are known a priori to belong to a certain parametric family of pure qubit systems, we derive two kinds of matching strategies. The first is a semiclassical strategy which is obtained by the natural extension of conventional matching strategies and consists of a two-stage procedure: identification (estimation) of the unknown template states to design the classifier (\textit{learning} process to train the classifier) and classification of the input system into the appropriate pattern class based on the estimated results. The other is a fully quantum strategy without any intermediate measurement which we might call as the {\it universal quantum matching machine}. We present the Bayes optimal solutions for both strategies in the case of K=1, showing that there certainly exists a fully quantum matching procedure which is strictly superior to the straightforward semiclassical extension of the conventional matching strategy based on the learning process.Comment: 11 pages, RevTeX, 3 figure

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 may be reduced to solving first a fundamental equation, which can be written down once the constraints are specified, for the Hamiltonian 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 the optimal Lindblad operators can be simulated by a tunable coupling with an ancillary qubit.

We expand on the idea that the spacetime signature should be treated as a dynamical degree of freedom in quantum field theory. It has been argued that the probability distribution for the signature, induced by massless free fields, is peaked at the Lorentzian value uniquely in D=4 dimensions. This argument is reviewed, and certain consistency constraints on the generalized signature (i.e., the tangent-space metric q,b(x)=diag[eiO'"',l,l,l]) are derived. It is shown that only one dynamical "Wick angle" O(x) can be introduced in the generalized signature, and the magnitude of fluctuations away from the Lorentzian signature 68=a-8 is estimated to be of order ( I , / R ) 3 , where l p is the Planck length, and R is the length scale of the Universe. For massless fields, the case of D=2 dimensions and the case of supersymmetry are degenerate, in the sense that no signature is preferred. Mass effects lift this degeneracy, and we show that a dynamical origin of the Lorentzian signature is also possible for (broken) supersymmetry theories in D=6 dimensions, in addition to the more general nonsupersymmetric case in D = 4 dimensions.

We consider the quantum analogue of the pattern matching problem, which consists of classifying a given unknown system according to certain predefined pattern classes. We address the problem of quantum template matching in which each pattern class Ci is represented by a known quantum stateĝi called a template state, and our task is to find a template which optimally matches a given unknown quantum statef . We set up a precise formulation of this problem in terms of the optimal strategy for an associated quantum Bayesian inference problem. We then investigate various examples of quantum template matching for qubit systems, considering the effect of allowing a finite number of copies of the input statef . We compare quantum optimal matching strategies and semiclassical strategies and demonstrate an entanglement assisted enhancement of performance in the general quantum optimal strategy.