Transitionless driving on adiabatic search algorithm

2004

Grover's algorithm is one of the most important quantum algorithms, which performs the task of searching an unsorted database without a priori probability. Recently the adiabatic evolution has been used to design and reproduce quantum algorithms, including Grover's algorithm. In this paper, we show that quantum search algorithm by adiabatic evolution has two properties that conventional quantum search algorithm doesn't have. Firstly, we show that in the initial state of the algorithm only the amplitude of the basis state corresponding to the solution affects the running time of the algorithm, while other amplitudes do not. Using this property, if we know a priori probability about the location of the solution before search, we can modify the adiabatic evolution to make the algorithm faster. Secondly, we show that by a factor for the initial and finial Hamiltonians we can reduce the running time of the algorithm arbitrarily. Especially, we can reduce the running time of adiabatic search algorithm to a constant time independent of the size of the database. The second property can be extended to other adiabatic algorithms.

New Journal of Physics, 2010

We consider the effect of two different environments on the performance of the quantum adiabatic search algorithm, a thermal bath at finite temperature, and a structured environment similar to the one encountered in systems coupled to the electromagnetic field that exists within a photonic crystal. While for all the parameter regimes explored here, the algorithm performance is worsened by the contact with a thermal environment, the picture appears to be different when considering a structured environment. In this case we show that, by tuning the environment parameters to certain regimes, the algorithm performance can actually be improved with respect to the closed system case. Additionally, the relevance of considering the dissipation rates as complex quantities is discussed in both cases. More particularly, we find that the imaginary part of the rates can not be neglected with the usual argument that it simply amounts to an energy shift, and in fact influences crucially the system dynamics.

Physical Review Letters, 2008

We study the effect of a thermal environment on adiabatic quantum computation using the Bloch-Redfield formalism. We show that in certain cases the environment can enhance the performance in two different ways: (i) by introducing a time scale for thermal mixing near the anticrossing that is smaller than the adiabatic time scale, and (ii) by relaxation after the anticrossing. The former can enhance the scaling of computation when the environment is superohmic, while the latter can only provide a prefactor enhancement. We apply our method to the case of adiabatic Grover search and show that performance better than classical is possible with a superohmic environment, with no a priori knowledge of the energy spectrum.

Physical Review A, 2005

This paper explores several aspects of the adiabatic quantum computation model. We first show a way that directly maps any arbitrary circuit in the standard quantum computing model to an adiabatic algorithm of the same depth. Specifically, we look for a smooth time-dependent Hamiltonian whose unique ground state slowly changes from the initial state of the circuit to its final state. Since this construction requires in general an n-local Hamiltonian, we will study whether approximation is possible using previous results on ground state entanglement and perturbation theory. Finally we will point out how the adiabatic model can be relaxed in various ways to allow for 2-local partially adiabatic algorithms as well as 2-local holonomic quantum algorithms.

2006

Quantum adiabatic computation is a novel paradigm for the design of quantum algorithms, which is usually used to find the minimum of a classical function. In this paper, we show that if the initial hamiltonian of a quantum adiabatic evolution with a interpolation path is too simple, the minimal gap between the ground state and the first excited state of this quantum adiabatic evolution is an inverse exponential distance. Thus quantum adiabatic evolutions of this kind can't be used to design efficient quantum algorithms. Similarly, we show that a quantum adiabatic evolution with a simple final hamiltonian also has a long running time, which suggests that some functions can't be minimized efficiently by any quantum adiabatic evolution with a interpolation path.

2003

We report on a detailed analysis of generalization of the local adiabatic search algorithm. Instead of evolving directly from an initial ground state Hamiltonian to a solution Hamiltonian a different evolution path is introduced and is shown that the time required to find an item in a database of size $N$ can be made to be independent of the size of the database by modifying the Hamiltonian used to evolve the system.

Physical Review A, 2004

We analyze the ground state entanglement in a quantum adiabatic evolution algorithm designed to solve the NP-complete Exact Cover problem. The entropy of entanglement seems to obey linear and universal scaling at the point where the energy gap becomes small, suggesting that the system passes near a quantum phase transition. Such a large scaling of entanglement suggests that the effective connectivity of the system diverges as the number of qubits goes to infinity and that this algorithm cannot be efficiently simulated by classical means. On the other hand, entanglement in Grover's algorithm is bounded by a constant. PACS numbers: 03.67.-a, 03.65.Ud, 03.67.Hk

Physical Review A, 2008

We propose a strategy to achieve the Grover search algorithm by adiabatic passage in a very efficient way. An adiabatic process can be characterized by the instantaneous eigenvalues of the pertaining Hamiltonian, some of which form a gap. The key to the efficiency is based on the use of parallel eigenvalues. This allows us to obtain non-adiabatic losses which are exponentially small, independently of the number of items in the database in which the search is performed.

Enthusiast's perspective: We analyze the effectiveness of AQC for a small rank problem Hamiltonian HF with the arbitrary initial Hamiltonian HI . We prove that for the generic HI the running time cannot be smaller than O( √ N ), where N is a dimension of the Hilbert space. We also construct an explicit HI for which the running time is indeed O( √ N ). Our algorithm can be used to solve the unstructured search problem with the unknown number of marked items.

We show that by a suitable choice of a time dependent Hamiltonian, Deutsch's algorithm can be implemented by an adiabatic quantum computer. We extend our analysis to the Deutsch-Jozsa problem and estimate the required running time for both global and local adiabatic evolutions. Quantum computation and quantum information theory has attracted a great deal of attention in recent times. Inherently quantum mechanical systems can in principle be used to implement a wide variety of computational algorithms wth enhanced efficiency [1–3]. The principle of superposition in quantum mechanics, according to which a system can be in a linearly superposed state of more than one eigenstate, is the key to this increased efficiency. One of the first algorithms that was first proposed in this context is Deutsch's algorithm [4]. In this, one would like to determine whether a function f : {0, 1} → {0, 1} is constant or balanced, i.e. whether f (0) = f (1) or f (0) = f (1) using a quantum computer. The four possible outcomes of f are: f (0) = f (1) = 0 (constant) f (0) = f (1) = 1 (constant) f (0) = 0 , f (1) = 1 (balanced) f (0) = 1 , f (1) = 0 (balanced) Ordinarily, one has to determine both f (0) and f (1) to infer the nature of the function, since the knowledge of one does not shed light on the value of the other. However, it was shown that by applying a certain sequence of unitary operators ('gates') on a given initial quantum mechanical state, and then making just one measurement on the final state, the nature of the function f can be determined [4]. Recently, a new framework of quantum computation has been proposed, in which the series of gates referred to above is entirely replaced by a Hamiltonian which changes continuously with time. The Hamiltonian is so chosen that the state of the system is its ground state at all times (although the ground state itself is time dependent), and the system slowly evolves to a desired final state [5]. Several applications of this have been considered [6]. Using this framework, it was shown that Grover's search algorithm can be efficiently implemented [7]. In this paper, we show that Deutsch's algorithm can be implemented as well, by choosing a suitable initial state and a Hamiltonian which evolves that state. Then a single measurement of the final state suffices to determine whether the function f is constant or balanced. Finally, we show that the results can be extended to the Deutsch-Jozsa algorithm involving n-qubits. Let us begin with a 2-level system, e.g. a spin 1/2 particle, with the basis kets {|0, |1}. We define the 'initial' and 'final' Hamiltonians H 0 , H 1 respectively as: H 0 = I − |ψ 0 ψ 0 | (1) H 1 = I − |ψ 1 ψ 1 | (2) (3) where the initial and final state vectors are given respectively by: |ψ 0 = 1 √ 2 (|0 + |1) (4) |ψ 1 = α|0 + β|1 (5) * Electronic address: saurya,randy,gabor@theory.uwinnipeg.ca