scholarly journals The performance of the quantum adiabatic algorithm on spike Hamiltonians

2017 ◽  
Vol 15 (02) ◽  
pp. 1750011 ◽  
Author(s):  
Linghang Kong ◽  
Elizabeth Crosson
Keyword(s):  
Simulated Annealing ◽  
Lower Bound ◽  
Ground State ◽  
Recursion Relation ◽  
Spectral Gap ◽  
Hamming Weight ◽  
Wkb Method ◽  
Adiabatic Evolution ◽  
Comparison Argument ◽  
Avoided Crossings

Spike Hamiltonians arise from optimization instances for which the adiabatic algorithm provably out performs classical simulated annealing. In this work, we study the efficiency of the adiabatic algorithm for solving the “the Hamming weight with a spike” problem by analyzing the scaling of the spectral gap at the critical point for various sizes of the barrier. Our main result is a rigorous lower bound on the minimum spectral gap for the adiabatic evolution when the bit-symmetric cost function has a thin but polynomially high barrier, which is based on a comparison argument and an improved variational ansatz for the ground state. We also adapt the discrete WKB method for the case of abruptly changing potentials and compare it with the predictions of the spin coherent instanton method which was previously used by Farhi, Goldstone and Gutmann. Finally, our improved ansatz for the ground state leads to a method for predicting the location of avoided crossings in the excited energy states of the thin spike Hamiltonian, and we use a recursion relation to understand the ordering of some of these avoided crossings as a step towards analyzing the previously observed diabatic cascade phenomenon.

scholarly journals Spectral Gap of the Largest Eigenvalue of the Normalized Graph Laplacian

2021 ◽  
Author(s):  
Jürgen Jost ◽  
Raffaella Mulas ◽  
Florentin Münch
Keyword(s):  
Lower Bound ◽  
Bipartite Graph ◽  
Spectral Gap ◽  
Complete Bipartite Graph ◽  
Graph Laplacian ◽  
New Method ◽  
Single Edge ◽  
Largest Eigenvalue ◽  
Complement Graph ◽  
The Largest Eigenvalue

AbstractWe offer a new method for proving that the maxima eigenvalue of the normalized graph Laplacian of a graph with n vertices is at least $$\frac{n+1}{n-1}$$ n + 1 n - 1 provided the graph is not complete and that equality is attained if and only if the complement graph is a single edge or a complete bipartite graph with both parts of size $$\frac{n-1}{2}$$ n - 1 2 . With the same method, we also prove a new lower bound to the largest eigenvalue in terms of the minimum vertex degree, provided this is at most $$\frac{n-1}{2}$$ n - 1 2 .

scholarly journals A lower bound for the spectral gap of the conjugate Kac process with 3 interacting particles

2021 ◽  
Vol 0 (0) ◽  
pp. 0
Author(s):  
Luís Simão Ferreira
Keyword(s):  
Monte Carlo ◽  
Lower Bound ◽  
Monte Carlo Simulations ◽  
Entropy Production ◽  
Quantitative Estimate ◽  
Final Section ◽  
Spectral Gap ◽  
Empirical Estimate ◽  
Interacting Particles

<p style='text-indent:20px;'>In this paper, we proceed as suggested in the final section of [<xref ref-type="bibr" rid="b2">2</xref>] and prove a lower bound for the spectral gap of the conjugate Kac process with 3 interacting particles. This bound turns out to be around <inline-formula><tex-math id="M1">\begin{document}$ 0.02 $\end{document}</tex-math></inline-formula>, which is already physically meaningful, and we perform Monte Carlo simulations to provide a better empirical estimate for this value via entropy production inequalities. This finishes a complete quantitative estimate of the spectral gap of the Kac process.</p>

scholarly journals Engineering Reflective Metasurfaces with Ising Hamiltonian and Quantum Annealing

2021 ◽  
Author(s):  
Zhen Peng ◽  
Charles Ross ◽  
Qi Jian Lim ◽  
Gabriele Gradoni
Keyword(s):  
Cross Section ◽  
Mobile Networks ◽  
Ground State ◽  
Energy Minimization ◽  
Phase Modulation ◽  
Optimization Problem ◽  
Ground State Solution ◽  
Quantum Annealing ◽  
Adiabatic Evolution ◽  
Annealing Algorithms

<div><div><div><p>We present a novel and flexible method to optimize the phase response of reflective metasurfaces towards a desired scattering profile. The scattering power is expressed as a spin-chain Hamiltonian using the radar cross section formalism. For metasurfaces reflecting an oblique plane wave, an Ising Hamiltonian is obtained. Thereby, the problem of achieving the scattering profile is recast into finding the ground-state solution of the associated Ising Hamiltonian. To rapidly explore the configuration states, we encode the Ising coefficients with quantum annealing algorithms, taking advantage of the fact that the adiabatic evolution efficiently performs energy minimization in the Ising model. Finally, the optimization problem is solved on the D-Wave 2048-qubit quantum adiabatic optimizer machine for binary phase as well as quadriphase reflective metasurfaces. Even though the work is focused on the phase modulation of metasurfaces, we believe this approach paves the way to fast optimization of reconfigurable intelligent surfaces that are mod- ulated in both amplitude and phase for multi-beam generation in and beyond 5G/6G mobile networks.</p></div></div></div>

Simulated Annealing for Finding TSP Lower Bound

2017 ◽  
pp. 45-55
Author(s):  
Łukasz Strąk ◽  
Wojciech Wieczorek ◽  
Arkadiusz Nowakowski
Keyword(s):  
Simulated Annealing ◽  
Lower Bound

Spectral gap of a weighted 3-simplicial complex

2021 ◽  
pp. 2150084
Author(s):  
Khalid Hatim ◽  
Azeddine Baalal
Keyword(s):  
Lower Bound ◽  
Simplicial Complex ◽  
Upper Bound ◽  
Spectral Gap ◽  
Simplicial Complexes ◽  
Cheeger Constant ◽  
Nonzero Eigenvalue ◽  
New Framework

In this paper, we construct a new framework that’s we call the weighted [Formula: see text]-simplicial complex and we define its spectral gap. An upper bound for our spectral gap is given by generalizing the Cheeger constant. The lower bound for our spectral gap is obtained from the first nonzero eigenvalue of the Laplacian acting on the functions of certain weighted [Formula: see text]-simplicial complexes.

On the General Class of Models of Adiabatic Evolution

2016 ◽  
Vol 23 (03) ◽  
pp. 1650016 ◽  
Author(s):  
Jie Sun ◽  
Songfeng Lu ◽  
Fang Liu
Keyword(s):  
Ground State ◽  
General Class ◽  
Time Complexity ◽  
State Energy ◽  
Search Problem ◽  
Quantum Search ◽  
Final States ◽  
Adiabatic Evolution ◽  
Speed Up ◽  
Effective Use

The general class of models of adiabatic evolution was proposed to speed up the usual adiabatic computation in the case of quantum search problem. It was shown [8] that, by temporarily increasing the ground state energy of a time-dependent Hamiltonian to a suitable quantity, the quantum computation can perform the calculation in time complexity O(1). But it is also known that if the overlap between the initial and final states of the system is zero, then the computation based on the generalized models of adiabatic evolution can break down completely. In this paper, we find another severe limitation for this class of adiabatic evolution-based algorithms, which should be taken into account in applications. That is, it is still possible that this kind of evolution designed to deal with the quantum search problem fails completely if the interpolating paths in the system Hamiltonian are chosen inappropriately, while the usual adiabatic evolutions can do the same job relatively effectively. This implies that it is not always recommendable to use nonlinear paths in adiabatic computation. On the contrary, the usual simple adiabatic evolution may be sufficient for effective use.

Simulated Annealing for Ground State Energy of Ionized Donor Bound Excitons in Semiconductors

2004 ◽  
Vol 42 (5) ◽  
pp. 779-784 ◽  
Author(s):  
Yan Hai-Qing ◽  
Tang Chen ◽  
Liu Ming ◽  
Zhang Hao ◽  
Zhang Gui-Min
Keyword(s):  
Simulated Annealing ◽  
Ground State Energy ◽  
Ground State ◽  
State Energy

scholarly journals Asymptotics of the spectral gap with applications to the theory of simulated annealing

1989 ◽  
Vol 83 (2) ◽  
pp. 333-347 ◽  
Author(s):  
Richard A Holley ◽  
Shigeo Kusuoka ◽  
Daniel W Stroock
Keyword(s):  
Simulated Annealing ◽  
Spectral Gap

Lower bound on the time complexity of local adiabatic evolution

2006 ◽  
Vol 74 (5) ◽  
Author(s):  
Zhenghao Chen ◽  
Pang Wei Koh ◽  
Yan Zhao
Keyword(s):  
Lower Bound ◽  
Time Complexity ◽  
Adiabatic Evolution

Energy bounds for the spiked harmonic oscillator

1995 ◽  
Vol 73 (7-8) ◽  
pp. 493-496 ◽  
Author(s):  
Richard L. Hall ◽  
Nasser Saad
Keyword(s):  
Lower Bound ◽  
Harmonic Oscillator ◽  
Ground State Energy ◽  
Upper Bound ◽  
Ground State ◽  
Trial Function ◽  
State Energy ◽  
Parameter Range ◽  
Complementary Energy ◽  
Energy Bounds

A three-parameter variational trial function is used to determine an upper bound to the ground-state energy of the spiked harmonic-oscillator Hamiltonian [Formula: see text]. The entire parameter range λ > 0 and α ≥ 1 is treated in a single elementary formulation. The method of potential envelopes is also employed to derive a complementary energy lower bound formula valid for all the discrete eigenvalues.

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