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Adiabatic quantum computation

Lukáš Charamza

Published Jan 1, 2014• 0 citations
OpenAlex

Abstract

In this thesis we summarize the principles of quantum computing. We specifically consider adiabatic quantum computing, whose principles are explained and shown on several examples. To explain the principle of adiabatic quantum computing we review the adiabatic theorem. We also outline possibility of using a particular Hamiltonian by Berry, which enables us to evolve system adiabatically in arbitrarily short time. In the final part of this thesis, we explain the concept of quantum phase transitions. We discuss a relationship between quantum phase transitions and adiabatic quantum computing and show that adiabatic quantum computing scales polynomially with the number of qubits only for quantum phase transitions of second or higher order. Powered by TCPDF (www.tcpdf.org)

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In this thesis we summarize the principles of quantum computing. We specifically consider adiabatic quantum computing, whose principles are explained and shown on several examples. To explain the principle of adiabatic quantum computing we review the adiabatic theorem. We also outline possibility of using a particular Hamiltonian by Berry, which enables us to evolve system adiabatically in arbitrarily short time. In the final part of this thesis, we explain the concept of quantum phase transitions. We discuss a relationship between quantum phase transitions and adiabatic quantum computing and show that adiabatic quantum computing scales polynomially with the number of qubits only for quantum phase transitions of second or higher order. Powered by TCPDF (www.tcpdf.org)
This Thesis focuses on different aspects of quantum computation theory: adiabatic quantum algorithms, decoherence during the adiabatic evolution and quantum simulators. After an overview on the area of quantum computation and setting up the formal ground for the rest of the Thesis we derive a general error estimate for adiabatic quantum computing. We demonstrate that the first-order correction, which has frequently been used as a condition for adiabatic quantum computation, does not yield a good estimate for the computational error. Therefore, a more general criterion is proposed, which includes higher-order corrections and shows that the computational error can be made exponentially small – which facilitates significantly shorter evolution times than the first-order estimate in certain situations. Based on this criterion and rather general arguments and assumptions, it can be demonstrated that a run-time of order of the inverse minimum energy gap is sufficient and necessary. Furthermore, exploiting the similarity between adiabatic quantum algorithms and quantum phase transitions, we study the impact of decoherence on the sweep through a second-order quantum phase transition for the prototypical example of the Ising chain in a transverse field and compare it to the adiabatic version of Grover’s search algorithm. It turns out that (in contrast to first-order transitions) the impact of decoherence caused by a weak coupling to a rather general environment increases with system size (i.e., number of spins/qubits), which might limit the scalability of the system. Finally, we propose the use of electron systems to construct laboratory systems based on present-day technology which reproduce and thereby simulate the quantum dynamics of the Ising model and the O(3) nonlinear sigma model.
Abstract In this chapter, we finally have our first look at quantum computing. In particular, we consider adiabatic quantum computing which is a paradigm tailored to the general problem of bipolar QUBO solving. Given our study up to this point, its basic principles will be easy to understand since Hopfield nets for problem solving and adiabatic quantum computing are closely related concepts. Indeed, we already know about tensor products, Hopfield energies and their representation in terms of Hamiltonians, annealing, the axioms of quantum mechanics, the Schrödinger equation, and quantum superposition. All these concepts will now be jointly put to use and brought together with yet another quantum mechanical principle, namely the adiabatic theorem. Based on two examples of simple QUBO problems, we demonstrate how adiabatic quantum computers can explore their exponentially large search spaces in parallel and thus achieve quadratic speedup over classical exhaustive search procedures.
This thesis explores two mathematical aspects of adiabatic quantum computation. Adiabatic quantum computation depends on the adiabatic theorem of quantum mechanics, and (a) we provide a rigorous formulation of the adiabatic theorem with explicit definitions of constants, and (b) we bound error in the adiabatic approximation under conditions of noise and experimental error. We apply the new results to a standard example of violation of the adiabatic approximation, and to a superconducting flux qubit. Further, adiabatic quantum computation requires large ground-state energy gaps throughout a Hamiltonian evolution if it is to solve problems in polynomial time. We identify a class of random Hamiltonians with non-nearest-neighbor interactions and a ground-state energy gap of O(1/n)\mathcal{O}(1/\sqrt{n})O(1/n​), where nnn is the number of qubits. We also identify two classes of Hamiltonians with non-nearest-neighbor interactions whose ground state can be found in polynomial time with adiabatic quantum computing. We then use the Jordan-Wigner transformation to derive equivalent results for Hamiltonians defined using Pauli operators.
The quantum adiabatic theorem is a widely applied result in quantum mechanics. However, recently, counterexamples were produced, indicating that the transition probabilities in some systems do not converge as predicted by the adiabatic theorem [K. P. Marzlin and B. C. Sanders, Phys. Rev. Lett. 93, 160408 (2004)]. In this paper, we examine a proof of the adiabatic theorem, paying specific attention to the validity of the assumptions made in the proof. We show that the convergence of the transition probabilities is path dependent when the change in the Hamiltonian is large. Finally, we consider the application of this reformulated form of the adiabatic theorem to the adiabatic evolution paradigm in quantum computation.
The quantum adiabatic theorem ensures that a slowly changing system, initially prepared in its ground state, will evolve to its final ground state with arbitrary precision. As a first result this thesis extends the original theorem to projection operators keeping the statement valid for Hamiltonians with degenerate ground spaces. Yet the main focus of this work lies in studying the efficiency of quantum circuit simulations by adabatic quantum computation. The standard Hamiltonian construction by Kitaev is based on a path graph reflecting the LLL computation steps and influencing the scaling of the necessary evolution time by its spectral gap of O(1L2)\mathcal{O}\left(\frac{1}{L^2}\right)O(L21​). Aspiring to an improved running time we generalize Kitaev's Hamiltonian to so-called standard graph Hamiltonians based on graph families with a different spectral gap. In this generalized construction the first two time derivatives of the Hamiltonian and the fraction of initial vertices appear as additional parameters of running time. In a first step the time derivatives can be proven to be constant. Expansion results from spectral graph theory however impose a trade-off between the spectral gap and the fraction of initial vertices as well as the fraction of final vertices which corresponds to the probability for obtaining the correct computational output. Graphs with spectral gap O(1Lk)\mathcal{O}\left(\frac{1}{L^k}\right)O(Lk1​), k<2k<2k<2, turn out to contradict very likely, graphs with k<1k<1k<1 even for sure at least one of the efficiency criterias for running time, output probability or Hamiltonian implementation. The above results may also be obstacles for a possible quantum PCP-theorem in complexity theory claiming the local Hamiltonian problem with constant gap to be QMA-complete since the very same Kitaev Hamiltonian is constructed in the QMA-hardness proof for the Local Hamiltonian problem.