Majorization in quantum adiabatic algorithms (original) (raw)
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From quantum circuits to adiabatic algorithms
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.
Quantum adiabatic algorithm for factorization and its experimental implementation
We propose an adiabatic quantum algorithm capable of factorizing numbers, using fewer qubits than Shor's algorithm. We implement the algorithm in a NMR quantum information processor and experimentally factorize the number 21. In the range that our classical computer could simulate, the quantum adiabatic algorithm works well, providing evidence that the running time of this algorithm scales polynomially with the problem size.
Computing Research Repository, 2010
Two recent preprints [B. Altshuler, H. Krovi, and J. Roland, "Quantum adiabatic optimization fails for random instances of NP-complete problems", arXiv:0908.2782 and "Anderson localization casts clouds over adiabatic quantum optimization", arXiv:0912.0746] argue that random 4th order perturbative corrections to the energies of local minima of random instances of NP-complete problem lead to avoided crossings that cause the failure of quantum adiabatic algorithm (due to exponentially small gap) close to the end, for very small transverse field that scales as an inverse power of instance size N . The theoretical portion of this work does not to take into account the exponential degeneracy of the ground and excited states at zero field. A corrected analysis shows that unlike those in the middle of the spectrum, avoided crossings at the edge would require high [O(1)] transverse fields, at which point the perturbation theory may become divergent due to quantum phase transition. This effect manifests itself only in large instances [exp(0.02N ) ≫ 1], which might be the reason it had not been observed in the authors' numerical work. While we dispute the proposed mechanism of failure of quantum adiabatic algorithm, we cannot draw any conclusions on its ultimate complexity.
Adiabatic Quantum Computation is Equivalent to Standard Quantum Computation
SIAM Journal on Computing, 2007
Adiabatic quantum computation has recently attracted attention in the physics and computer science communities, but its computational power was unknown. We describe an efficient adiabatic simulation of any given quantum algorithm, which implies that the adiabatic computation model and the conventional quantum computation model are polynomially equivalent. Our result can be extended to the physically realistic setting of particles arranged on a two-dimensional grid with nearest neighbor interactions. The equivalence between the models provides a new vantage point from which to tackle the central issues in quantum computation, namely designing new quantum algorithms and constructing fault tolerant quantum computers. In particular, by translating the main open questions in the area of quantum algorithms to the language of spectral gaps of sparse matrices, the result makes these questions accessible to a wider scientific audience, acquainted with mathematical physics, expander theory and rapidly mixing Markov chains.
Finite temperature quantum algorithm and majorization
2008
It is often believed that quantum entanglement plays an important role in the speed-up of quantum algorithms. In addition, a few research groups have found that Majorization behavior may also play an important role in some quantum algorithms. In some of our previous work we showed that for a simple spin 1/2 system, consisting of two or three qubits, the value of a Groverian entanglement (a rather useful measure of entanglement) varies inversely with the temperature. In practical terms this means that more iterations of the Grover's algorithm may be needed when a quantum computer is working at finite temperature. That is, the performance of a quantum algorithm suffers due to temperature-dependent changes on the density matrix of the system. Most recently, we have been interested in the behavior of Majorization for the same types of quantum system, and we are trying to determine the relationship between Groverian entanglement and Majorization at finite temperature. As Majorization entails the probability distribution arising out of the evolving quantum state from the probabilities of the final outcomes, our study will reveal how Majorization affects the evolution of Grover's algorithm at finite temperature.
Adiabatic Quantum Computation and Deutsch's Algorithm
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
A Factorisation Algorithm in Adiabatic Quantum Computation
Journal of Physics Communications, 2019
The problem of factorising positive integer N into two integer factors x and y is first reformu-lated as an optimisation problem over the positive integer domain of either of the Diophantine polynomials Q N (x, y) = N 2 (N − xy) 2 + x(x − y) 2 or R N (x, y) = N 2 (N − xy) 2 + (x − y) 2 + x, of each of which the optimal solution is unique with x ≤ √ N ≤ y, and x = 1 if and only if N is prime. An algorithm in the context of Adiabatic Quantum Computation is then proposed for the general factorisation problem.
Adiabatic quantum computation and quantum phase transitions
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
On Models of Nonlinear Evolution Paths in Adiabatic Quantum Algorithms
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.