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arXiv:quant-ph/0606194 · Physical Review A

Adiabatic computation: A toy model

Pedro Ribeiro, Rémy Mosseri

Published Oct 26, 2006• 5 citations
arXivHTMLPDF

Abstract

We discuss a toy model for adiabatic quantum computation which displays some phenomenological properties expected in more realistic implementations. This model has two free parameters: the adiabatic evolution parameter $s$ and the $\ensuremath{\alpha}$ parameter, which emulates many-variable constraints in the classical computational problem. The proposed model presents, in the $s\text{\ensuremath{-}}\ensuremath{\alpha}$ plane, a line of first-order quantum phase transition that ends at a second-order point. The relation between computation complexity and the occurrence of quantum phase transitions is discussed. We analyze the behavior of the ground and first excited states near the quantum phase transition, the gap, and the entanglement content of the ground state.

DOI
10.1103/physreva.74.042333

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We discuss a toy model for adiabatic quantum computation which displays some phenomenological properties expected in more realistic implementations. This model has two free parameters: the adiabatic evolution parameter sss and the \ensuremathα\ensuremath{\alpha}\ensuremathα parameter, which emulates many-variable constraints in the classical computational problem. The proposed model presents, in the s\ensuremath-\ensuremathαs\text{\ensuremath{-}}\ensuremath{\alpha}s\ensuremath-\ensuremathα plane, a line of first-order quantum phase transition that ends at a second-order point. The relation between computation complexity and the occurrence of quantum phase transitions is discussed. We analyze the behavior of the ground and first excited states near the quantum phase transition, the gap, and the entanglement content of the ground state.
We analyze the ground-state entanglement in a quantum adiabatic evolution algorithm designed to solve the NPNPNP-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.
Many physically interesting models show a quantum phase transition when a single parameter is varied through a critical point, where the ground state and the first excited state become degenerate. When this parameter appears as a coupling constant, these models can be understood as straight-line interpolations between different Hamiltonians HI{H}_{\mathrm{I}}HI​ and HF{H}_{\mathrm{F}}HF​. For finite-size realizations however, there will usually be a finite energy gap between ground and first excited state. By slowly changing the coupling constant through the point with the minimum energy gap one thereby has an adiabatic algorithm that prepares the ground state of HF{H}_{\mathrm{F}}HF​ from the ground state of HI{H}_{\mathrm{I}}HI​. The adiabatic theorem implies that in order to obtain a good preparation fidelity the runtime $\ensuremath{\tau}$ should scale with the inverse energy gap and thereby also with the system size. In addition, for open quantum systems not only nonadiabatic but also thermal excitations are likely to occur. It is shown that---using only local Hamiltonians---for the one-dimensional quantum Ising model and the cluster model in a transverse field the conventional straight-line path can be replaced by a series of straight-line interpolations, along which the fundamental energy gap is always greater than a constant independent on the system size. The results are of interest for adiabatic quantum computation since strong similarities between adiabatic quantum algorithms and quantum phase transitions exist.
We investigate the connection between local minima in the problem Hamiltonian and first-order quantum phase transitions during adiabatic quantum computation. We demonstrate how some properties of the local minima can lead to an extremely small gap that is exponentially sensitive to the Hamiltonian parameters. Using perturbation expansion, we derive an analytical formula that cannot only predict the behavior of the gap, but also provide insight on how to controllably vary the gap size by changing the parameters. We show agreement with numerical calculations for a weighted maximum independent set problem instance.
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.
Adiabatic quantum computation is based on the adiabatic evolution of quantum systems. We analyze a particular class of quantum adiabatic evolutions where either the initial or final Hamiltonian is a one-dimensional projector Hamiltonian on the corresponding ground state. The minimum-energy gap, which governs the time required for a successful evolution, is shown to be proportional to the overlap of the ground states of the initial and final Hamiltonians. We show that such evolutions exhibit a rapid crossover as the ground state changes abruptly near the transition point where the energy gap is minimum. Furthermore, a faster evolution can be obtained by performing a partial adiabatic evolution within a narrow interval around the transition point. These results generalize and quantify earlier works.
In the context of adiabatic quantum computation (AQC), it has been argued that first-order quantum phase transitions (QPTs) due to localization phenomena cause AQC to fail by exponentially decreasing the minimal spectral gap of the Hamiltonian along the annealing path as a function of the qubit number. The vanishing of the spectral gap is often linked to the localization of the ground state in a local minimum, requiring the system to tunnel into the global minimum at a later stage of the annealing. Recent methods have been proposed to avoid this phenomenon by carefully designing the involved Hamiltonians. However, it remains a challenge to formulate a comprehensive theory of the effect of the various parameters and the conditions under which QPTs make the AQC algorithm fail. Equipped with concepts from graph theory, in this work we link graph quantities associated with the Hamiltonians along the annealing path with the occurrence of QPTs. These links allow us to derive bounds on the location of the minimal spectral gap along the annealing path, augmenting the toolbox for the analysis of strategies to improve the runtime of AQC algorithms.