Quantum Mechanics for Graphs and CW-complexes

Zhe Hu (UIUC)Michael Toriyama (UIUC)Boyan Xu (UIUC)Chengzheng Yu (UIUC)

IntroductionThe goal of this project is to develop a version of quantum mechanicsbased on a graph-theoretic analogue of the Schrodinger equation. To agraph we associate a partition function, a discretization of the Feynmanpath integral which we use to count special types of paths and computetopological invariants. We apply this framework to model heat diffusion inmetals and propagation of information in networks.Definitions

A quantum state Ψ is a complex-valued function on the vertices of a graphΓ. With respect to a labelling of the Γ, the Laplacian of Γ is

∆ = val−A

where A and val are adjacency and valency operators, respectively, of Γ.Note that

∆ = II∗

where I is the incidence operator with respect to a chosen orientation,which maps an edge to the difference of its endpoints. Thus ∆ is acomposition of boundary and coboundary operators, hence a discreteanalogue of the usual Laplacian on euclidean space.

Schrodinger equationThe quantum system on Γ is given by

∂

∂tΨt = −∆Ψt

Combinatorial InterpretationWe studied the graph Laplacian through its combinatorial properties. Bydefining a super-walk, one can understand the Laplacian and its higherpowers by connecting them to finite graphs via our theorem. A super-walkis an sequence v0,e0, v1, . . . , vk where vi are vertices, ei are edges, and eiis incident to vi−1 and vi (we allow vi−1 = vi). The length of a super-walk isthe number of edges. The sign of a superwalk is (−1)` where ` is thenumber of times vi−1 6= vi.

Theorem

(∆+Γ )k(i , j) =

∑γ,i→j ,k

(γ)

where γ, i → j , k is a super-walk γ that starts at vertex i, ends at vertex j,and has length k.

Applications: TopologyOne of the applictions of the Laplacian is in topology. One idea is that wecan use the Laplacian ∆ to calculate Betti numbers of graphs:b0=dim(ker(∆)) and b1=dim(coker(∆)). In a more general way, the firstnumber b0 gives the number of connected components and the second oneb1 gives the number of generating cycles in a graph. We can use those 2Betti numbers to distinguish following 3 graphs topologically.

1 2

3 1 2

34

1 2

34

Evolution of Quantum StatesThe main idea to discover how quantum states evolves along the time is tosolve the Schrodinger equation. The Schrodinger operator in QM is−∆ + u(x). Here, for convenience, we use u(x) = 0. The coefficient of theODE is the Laplacian matrix. We then use MATLAB to solve the ODE.

Applications: Discretized Heat EquationA crucial application of the graph Laplacian is to produce a discretizedanalogue of the heat equation.

∂u∂t

= α∇2u

To address this, we present in a simulation some crucial aspect of discretetime-evolution of energy states intrinsic to a set of atoms in a material.Specifically, a simulation of heat diffusing through a bulk structure of galliumwas run through Molecular Dynamics (MD), where a ”heat zone” wasinitially established on the bottom of the material before the heat wasallowed to diffuse through the material.

(Left) Time-evolution of the discrete heat equation acting in a gallium bulkstructure. The particles are colored based on the amount of kinetic energyit has. (Right) Schematic of heat diffusion for the first two timesteps of asimulation by applying the graph Laplacian.

The simulation illustrates a discrete-time progression of energy transferfrom an initially unstable state. Although the graph Laplacian was not usedto perform this simulation, similar results can be obtained by an iterativeapplication of the graph Laplacian to a given state.Although this is open to future work, from the perspective of computingapplications, the matrix form of the graph Laplacian can pave way to moretime-efficient simulation techniques for the evolution of states.Future DirectionsThus far our theory has been confined to graphs, i.e. 1-dimensional objects.In future work, we hope to generalize our results to CW-complexes ofhigher dimension.Our ongoing work develops Morse theory on graphs and Hodge theory withrespect to the supersymmetric Laplacian ∆. Drawing inspiration from ideasof Witten [1], we seek to prove discrete Morse inequalities usingsupersymmetry.References

[1] Witten, Edward. Supersymmetry and Morse theory. J. Differential Geom. 17 (1982), no. 4, 661–692.

Support for this project was provided by the Illinois Geometry Lab and the Department of Mathematics at the University of Illinois at Urbana-Champaign.

Acknowledgement goes to Professor Ivan Contreras and Sarah Loeb for their gracious supervision of this project.