Adiabatic Quantum Computation and Simulation

Quantum pattern recognition algorithm for two-qubit systems has been implemented by quantum adiabatic evolution. We will estimate required running time for this algorithm by means of an analytical solution of time-dependent Hamiltonian since the time complexity of adiabatic quantum evolution is a limitation on the quantum computing. These results can be useful for experimental implementation. Adiabatic quantum computation (AQC) [1,2] is a model of quantum computing designed to solve optimization problem [3,4] and then as an alternative method for standard circuit model of quantum computing. [5] By applying the AQC, several quantum algorithms such as Deutsch, [6] Detutsch-joza, [7] Bernstein-Vazirani [8] and Grover [9] are redesigned and the required running time for adiabatic evolution has been estimated. Quantum pattern recognition for two-qubit systems is another algorithm that has been designed by AQC. [10] In Ref. [10], a non-training quantum neural network (QNN) has been presented by the combination of classical Hopfield network and the AQC method for pattern recognition and associative memory. Moreover, an experimental proof of their model for the two-qubit system has been implemented by liquid state NMR. In this Letter, we estimate a lower bound of required computation time for the two-qubit quantum pattern recognition algorithm. [10] For this purpose, we use the adiabatic theorem condition analytically and discuss the runtime of the quantum system. This estimation may be useful in experimental implementation in the different quantum systems. Let us consider a quantum system which is described by a quantum state |ψ t to solve a problem instance. Suppose that H 0 and H T are initial and final Hamiltonians of the system. Suppose that |ψ 0 , the ground state of H 0 , is the initial state of the system and |ψ T , the ground state of H T is the final state of the system that encodes the solution of the problem. The evolution of the system can be derived according to a path Hamiltonian H(t) = (1 − s)H 0 + sH T , (1) where s = s(T) is a time-dependent parameter with s(0) = 0 and s(T) = 1, with T being the total running time of the evolution. Suppose that |E 0 (t) and |E 1 (t) are the ground and first excited states of the Hamiltonian at time t, corresponding to eigen-values E 0 (t) and E 1 (t), respectively. The adiabatic theorem [11,12] guarantees that if the system evolves slowly enough, at time T the system would be close to E 0 (T) with a high probability

Communications in Theoretical Physics, 2013

In this paper, we study two different nonlinear interpolating paths in adiabatic evolution algorithms for solving a particular class of quantum search problems where both the initial and final Hamiltonian are one-dimensional projector Hamiltonians on the corresponding ground state. If the overlap between the initial state and final state of the quantum system is not equal to zero, both of these models can provide a constant time speedup over the usual adiabatic algorithms by increasing some another corresponding "complexity". But when the initial state has a zero overlap with the solution state in the problem, the second model leads to an infinite time complexity of the algorithm for whatever interpolating functions being applied while the first one can still provide a constant running time. However, inspired by a related reference, a variant of the first model can be constructed which also fails for the problem when the overlap is exactly equal to zero if we want to make up the "intrinsic" fault of the second model-an increase in energy. Two concrete theorems are given to serve as explanations why neither of these two models can improve the usual adiabatic evolution algorithms for the phenomenon above. These just tell us what should be noted when using certain nonlinear evolution paths in adiabatic quantum algorithms for some special kind of problems.

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.

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

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