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Hyper Spin-glass model and Partition function (#55)
* hyper spin glass * new hyper spin glass model doc * fix doc make
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| # # Hyper-Spin-glass problem | ||
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| # !!! note | ||
| # It is highly recommended to read the [Independent set problem](@ref) chapter before reading this one. | ||
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| # ## Problem definition | ||
| # The hyper-spin-glass problem is a natural generalization of the spin-glass problem from simple graph to hypergraph. | ||
| # A hyper-spin-glass problem of hypergraph ``H = (V, E)`` can be characterized by the following energy function | ||
| # ```math | ||
| # E = \sum_{c \in E} w_{c} \prod_{v\in c} S_v | ||
| # ``` | ||
| # where ``S_v \in \{-1, 1\}``, ``w_c`` is coupling strength associated with hyperedge ``c``. | ||
| # In the program, we use boolean variable ``s_v = \frac{1-S_v}{2}`` to represent a spin configuration. | ||
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| using GenericTensorNetworks | ||
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| # In the following, we are going to defined an spin glass problem for the following hypergraph. | ||
| num_vertices = 15 | ||
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| hyperedges = [[1,3,4,6,7], [4,7,8,12], [2,5,9,11,13], | ||
| [1,2,14,15], [3,6,10,12,14], [8,14,15], | ||
| [1,2,6,11], [1,2,4,6,8,12]] | ||
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| weights = [-1, 1, -1, 1, -1, 1, -1, 1]; | ||
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| # The energy function is | ||
| # ```math | ||
| # \begin{align*} | ||
| # E = &-s_1s_3s_4s_6s_7 + s_4s_7s_8s_{12} - s_2s_5s_9s_{11}s_{13} +\\ | ||
| # &s_1s_2s_{14}s_{15} - s_3s_6s_{10}s_{12}s_{14} + s_8s_{14}s_{15} +\\ | ||
| # &s_1s_2s_6s_{11} - s_1s_2s_4s_6s_8s_{12} | ||
| # \end{align*} | ||
| # ``` | ||
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| # ## Generic tensor network representation | ||
| # We define an anti-ferromagnetic spin glass problem as | ||
| problem = HyperSpinGlass(num_vertices, hyperedges; weights); | ||
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| # ### Theory (can skip) | ||
| # Let ``H = (V, E)`` be a hypergraph. The tensor network for the partition function of the energy model for ``H`` can be defined as the following tiple of (alphabet of labels, input tensors, output labels). | ||
| # ```math | ||
| # \begin{cases} | ||
| # \Lambda &= \{s_v \mid v \in V\}\\ | ||
| # \mathcal{T} &= \{B^{(c)}_{s_{N(c, 1),N(c, 2),\ldots,N(c, d(c))}} \mid c \in E\} \cup \{W^{(v)}_{s_v} \mid v \in V\}\\ | ||
| # \sigma_o &= \varepsilon | ||
| # \end{cases} | ||
| # ``` | ||
| # where ``s_v \in \{0, 1\}`` is the boolean degreen associated to vertex ``v``, | ||
| # ``N(c, k)`` is the ``k``th vertex of hyperedge ``c``, and ``d(c)`` is the degree of ``c``. | ||
| # The edge tensor ``B^{(c)}`` is defined as | ||
| # ```math | ||
| # B^{(c)} = \begin{cases} | ||
| # x^{w_c} & (\sum_{v\in c} s_v) \;is\; even, \\ | ||
| # x^{-w_c} & otherwise. | ||
| # \end{cases} | ||
| # ``` | ||
| # and the vertex tensor ``W^{(v)}`` (used to carry labels) is defined as | ||
| # ```math | ||
| # W^{(v)} = \left(\begin{matrix}1_v\\ 1_v\end{matrix}\right) | ||
| # ``` | ||
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| # ## Solving properties | ||
| # ### Minimum and maximum energies | ||
| # Its ground state energy is -8. | ||
| Emin = solve(problem, SizeMin())[] | ||
| # While the state correspond to the highest energy has the ferromagnetic order. | ||
| Emax = solve(problem, SizeMax())[] | ||
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| # In this example, the spin configurations can be choosen to make all hyperedges having even or odd spin parity. | ||
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| # ### Counting properties | ||
| # ##### partition function and graph polynomial | ||
| # The graph polynomial defined for the hyper-spin-glass problem is a Laurent polynomial | ||
| # ```math | ||
| # Z(G, w, x) = \sum_{k=E_{\rm min}}^{E_{\rm max}} c_k x^k, | ||
| # ``` | ||
| # where ``E_{\rm min}`` and ``E_{\rm max}`` are minimum and maximum energies, | ||
| # ``c_k`` is the number of spin configurations with energy ``k``. | ||
| # Let the inverse temperature ``\beta = 2``, the partition function is | ||
| β = 2.0 | ||
| Z = solve(problem, PartitionFunction(β))[] | ||
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| # The infinite temperature partition function is the counting of all feasible configurations | ||
| solve(problem, PartitionFunction(0.0))[] | ||
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| # Let ``x = e^\beta``, it corresponds to the partition function of a spin glass at temperature ``\beta^{-1}``. | ||
| poly = solve(problem, GraphPolynomial())[] | ||
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| # ### Configuration properties | ||
| # ##### finding a ground state | ||
| ground_state = solve(problem, SingleConfigMin())[].c.data | ||
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| Emin_verify = hyperspinglass_energy(hyperedges, ground_state; weights) | ||
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| # You should see a consistent result as above `Emin`. |
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| """ | ||
| $(TYPEDEF) | ||
| The [hyper-spin-glass](https://queracomputing.github.io/GenericTensorNetworks.jl/dev/generated/HyperSpinGlass/) problem is a generalization of the spin-glass problem to hypergraphs. | ||
| Positional arguments | ||
| ------------------------------- | ||
| * `n` is the number of spins. | ||
| * `cliques` is a vector of cliques, each being a vector of vertices (integers). | ||
| Keyword arguments | ||
| ------------------------------- | ||
| * `weights` are associated with the cliques. | ||
| * `optimizer` and `simplifier` are for tensor network optimization, check [`optimize_code`](@ref) for details. | ||
| * `fixedvertices` is a dict to specify the values of degree of freedoms, where a value can be `0` (in one side of the cut) or `1` (in the other side of the cut). | ||
| * `openvertices` is a tuple of labels to specify the output tensor. Theses degree of freedoms will not be contracted. | ||
| """ | ||
| struct HyperSpinGlass{CT<:AbstractEinsum,WT<:Union{NoWeight, Vector}} <: GraphProblem | ||
| code::CT | ||
| n::Int | ||
| cliques::Vector{Vector{Int}} | ||
| weights::WT | ||
| fixedvertices::Dict{Int,Int} | ||
| end | ||
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| """ | ||
| $(TYPEDSIGNATURES) | ||
| """ | ||
| function HyperSpinGlass(n::Int, cliques::AbstractVector; weights=NoWeight(), openvertices=(), fixedvertices=Dict{Int,Int}(), optimizer=GreedyMethod(), simplifier=nothing) | ||
| clqs = collect(collect.(cliques)) | ||
| @assert weights isa NoWeight || length(weights) == length(clqs) | ||
| rawcode = EinCode([clqs..., [[i] for i=1:n]...], collect(Int, openvertices)) # labels for edge tensors | ||
| HyperSpinGlass(_optimize_code(rawcode, uniformsize_fix(rawcode, 2, fixedvertices), optimizer, simplifier), n, clqs, weights, Dict{Int,Int}(fixedvertices)) | ||
| end | ||
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| flavors(::Type{<:HyperSpinGlass}) = [0, 1] | ||
| # first `ne` indices are for edge weights, last `n` indices are for vertex weights. | ||
| terms(gp::HyperSpinGlass) = gp.cliques | ||
| labels(gp::HyperSpinGlass) = collect(1:gp.n) | ||
| fixedvertices(gp::HyperSpinGlass) = gp.fixedvertices | ||
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| # weights interface | ||
| get_weights(c::HyperSpinGlass) = c.weights | ||
| get_weights(gp::HyperSpinGlass, i::Int) = [-gp.weights[i], gp.weights[i]] | ||
| chweights(c::HyperSpinGlass, weights) = HyperSpinGlass(c.code, c.cliques, weights, c.fixedvertices) | ||
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| function generate_tensors(x::T, gp::HyperSpinGlass) where T | ||
| ixs = getixsv(gp.code) | ||
| l = length(gp.cliques) | ||
| tensors = [ | ||
| Array{T}[clique_tensor(length(gp.cliques[i]), _pow.(Ref(x), get_weights(gp, i))...) for i=1:l]..., | ||
| add_labels!(Array{T}[[one(T), one(T)] for i in labels(gp)], ixs[l+1:end], labels(gp))... | ||
| ] | ||
| return select_dims(tensors, ixs, fixedvertices(gp)) | ||
| end | ||
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| function clique_tensor(rank, a::T, b::T) where T | ||
| res = zeros(T, fill(2, rank)...) | ||
| for i=0:(1<<rank-1) | ||
| res[i+1] = (count_ones(i) % 2) == 1 ? a : b | ||
| end | ||
| return res | ||
| end | ||
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| """ | ||
| $(TYPEDSIGNATURES) | ||
| Compute the energy for spin configuration `config` (an iterator). | ||
| """ | ||
| function hyperspinglass_energy(cliques, config; weights=NoWeight())::Real | ||
| size = zero(eltype(weights)) | ||
| s = 1 .- 2 .* Int.(config) # 0 -> spin 1, 1 -> spin -1 | ||
| for (i, spins) in enumerate(cliques) | ||
| size += prod(s[spins]) * weights[i] | ||
| end | ||
| return size | ||
| end |
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