Ansatz

Tenet.Lattice Type

Lattice

A lattice is a graph where the vertices are Sites and the edges are virtual bonds. It is used for representing the topology of a Ansatz Tensor Network. It fulfills the AbstractGraph interface.

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Tenet.Lattice Method

Lattice(::Val{:chain}, n; periodic=false)

Create a chain lattice with n sites.

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Tenet.Lattice Method

Lattice(::Val{:lieb}, nrows, ncols)

Create a Lieb lattice with nrows cell rows and ncols cell columns.

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Tenet.Lattice Method

Lattice(::Val{:rectangular}, nrows, ncols; periodic=false)

Create a rectangular lattice with nrows rows and ncols columns.

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Graphs.edges Method

Graphs.edges(::Lattice)

Return the edges of the lattice; i.e. pairs of Lanes.

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Graphs.has_edge Method

Graphs.has_edge(lattice::Lattice, edge)
Graphs.has_edge(lattice::Lattice, a::Lane, b::Lane)

Return true if the lattice has the given edge.

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Graphs.has_vertex Method

Graphs.has_vertex(lattice::Lattice, lane::AbstractLane)

Return true if the lattice has the given Lane.

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Graphs.ne Method

Graphs.ne(::Lattice)

Return the number of edges in the lattice.

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Graphs.neighbors Method

Graphs.neighbors(lattice::Lattice, lane::AbstractLane)

Return the neighbors Lanes of the given Lane.

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Graphs.nv Method

Graphs.nv(::Lattice)

Return the number of vertices; i.e. Lanes, in the lattice.

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Graphs.vertices Method

Graphs.vertices(::Lattice)

Return the vertices of the lattice; i.e. the list of Lanes.

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Tenet.AbstractAnsatz Type

AbstractAnsatz

Abstract type for Ansatz-derived types. Its subtypes must implement conversion or extraction to the underlying Ansatz.

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Tenet.Ansatz Type

Ansatz

AbstractQuantum Tensor Network together with a Lattice for connectivity information between Lanes.

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Tenet.Boundary Type

Boundary

Abstract type representing the boundary condition trait of a AbstractAnsatz Tensor Network.

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Tenet.Canonical Type

Canonical

Form trait representing a AbstractAnsatz Tensor Network in canonical form or Vidal gauge.

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Tenet.Form Type

Form

Abstract type representing the canonical form trait of a AbstractAnsatz Tensor Network.

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Tenet.MixedCanonical Type

MixedCanonical

Form trait representing a AbstractAnsatz Tensor Network in mixed-canonical form.

• The orthogonality center is a Lane or a vector of Lanes. The tensors to the left of the orthogonality center are left-canonical and the tensors to the right are right-canonical.

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Tenet.NonCanonical Type

NonCanonical

Form trait representing a AbstractAnsatz Tensor Network in non-canonical form.

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Tenet.Open Type

Open

Boundary trait representing an open boundary condition.

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Tenet.Periodic Type

Periodic

Boundary trait representing a periodic boundary condition.

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Tenet.Quantum Method

Quantum(tn::AbstractAnsatz)

Return the underlying Quantum Tensor Network of an AbstractAnsatz.

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Base.truncate Method

truncate(tn::AbstractAnsatz, bond; threshold = nothing, maxdim = nothing)

Like truncate!, but returns a new Tensor Network instead of modifying the original one.

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EinExprs.inds Method

inds(tn::AbstractAnsatz; bond)

Return the index of the virtual bond between two AbstractLanes in a AbstractAnsatz Tensor Network.

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Graphs.has_edge Method

has_edge(tn::AbstractAnsatz, a::AbstractLane, b::AbstractLane)

Check whether there is an edge between two AbstractLanes in the Lattice of the AbstractAnsatz Tensor Network.

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Graphs.neighbors Method

neighbors(tn::AbstractAnsatz, lane::AbstractLane)

Return the neighboring sites of a given AbstractLane in the Lattice of the AbstractAnsatz Tensor Network.

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LinearAlgebra.normalize Method

normalize!(ψ::AbstractAnsatz, at)

Normalize the state at a given Site or bond in a AbstractAnsatz Tensor Network.

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Tenet.canonize! Function

canonize!(tn::AbstractAnsatz)

Transform an AbstractAnsatz Tensor Network into the canonical form (aka Vidal gauge); i.e. the singular values matrix Λᵢ between each tensor Γᵢ₋₁ and Γᵢ.

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Tenet.canonize Method

canonize(tn::AbstractAnsatz)

Like canonize!, but returns a new Tensor Network instead of modifying the original one.

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Tenet.contract! Method

contract!(tn::AbstractAnsatz; bond)

Contract the virtual bond between two AbstractLanes in a AbstractAnsatz Tensor Network.

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Tenet.evolve! Method

evolve!(ψ::AbstractAnsatz, gate; threshold = nothing, maxdim = nothing, normalize = false)

Evolve (through time) a AbstractAnsatz Tensor Network with a gate operator.

Note

Currently only the "Simple Update" algorithm is implemented.

Arguments

• ψ: Tensor Network representing the state.

• gate: The gate operator to evolve the state with.

Keyword Arguments

• threshold: The threshold to truncate the bond dimension.

• maxdim: The maximum bond dimension to keep.

• normalize: Whether to normalize the state after truncation.

Notes

• The gate must act on neighboring sites according to the Lattice of the Tensor Network.

• The gate must have the same number of inputs and outputs.

• Currently only the "Simple Update" algorithm is used and the gate must be a 1-site or 2-site operator.

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Tenet.expect Method

expect(ψ::AbstractAnsatz, observable)

Compute the expectation value of an observable on a AbstractAnsatz Tensor Network.

Arguments

• ψ: Tensor Network representing the state. It must be a state (i.e. socket returns State(dual=false)).

• observable: The observable to compute the expectation value. If a Vector or Tuple of observables is provided, the sum of the expectation values is returned.

Keyword Arguments

• bra: The bra state. It must be a dual state (i.e. socket returns State(dual=true)). Defaults to the adjoint of ψ.

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Tenet.form Function

form(tn::AbstractAnsatz)

Return the canonical form of the AbstractAnsatz Tensor Network.

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Tenet.isisometry Function

isisometry(tn::AbstractAnsatz, lane; dir, kwargs...)

Check if the tensor at a given Lane in a AbstractAnsatz Tensor Network is an isometry. The dir keyword argument specifies the direction of the isometry to check.

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Tenet.lattice Method

lattice(tn::AbstractAnsatz)

Return the Lattice of the AbstractAnsatz Tensor Network.

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Tenet.mixed_canonize! Function

mixed_canonize!(tn::AbstractAnsatz, orthog_center)

Transform an AbstractAnsatz Tensor Network into the mixed-canonical form, that is, for i < orthog_center the tensors are left-canonical and for i >= orthog_center the tensors are right-canonical, and in the orthog_center there is a tensor with the Schmidt coefficients in it.

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Tenet.mixed_canonize Method

mixed_canonize(tn::AbstractAnsatz, orthog_center)

Like mixed_canonize!, but returns a new Tensor Network instead of modifying the original one.

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Tenet.overlap Method

overlap(a::AbstractAnsatz, b::AbstractAnsatz)

Compute the overlap between two AbstractAnsatz Tensor Networks.

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Tenet.simple_update! Method

simple_update!(ψ::AbstractAnsatz, gate; threshold = nothing, maxdim = nothing, kwargs...)

Update a AbstractAnsatz Tensor Network with a gate operator using the "Simple Update" algorithm. kwargs are passed to the truncate! method in the case of a multi-site gate.

Warning

Currently only 1-site and 2-site gates are supported.

Arguments

• ψ: Tensor Network representing the state.

• gate: The gate operator to update the state with.

Keyword Arguments

• threshold: The threshold to truncate the bond dimension.

• maxdim: The maximum bond dimension to keep.

• normalize: Whether to normalize the state after truncation.

Notes

• If both threshold and maxdim are provided, maxdim is used.

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Tenet.tensors Method

tensors(tn::AbstractAnsatz; bond)

Return the Tensor in a virtual bond between two AbstractLanes in a AbstractAnsatz Tensor Network.

Notes

• If the AbstractAnsatz Tensor Network is in the canonical form, Tenet stores the Schmidt coefficients of the bond in a vector connected to the bond hyperedge between the two sites and the vector.

• If the bond contains no Schmidt coefficients, this method will throw a MissingSchmidtCoefficientsException.

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Tenet.truncate! Method

truncate!(::Canonical, tn::AbstractAnsatz, bond; canonize=true, kwargs...)

Truncate the dimension of the virtual bond of a Canonical Tensor Network by keeping the maxdim largest Schmidt coefficients or those larger than threshold, and then canonizes the Tensor Network if canonize is true.

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Tenet.truncate! Method

truncate!(::NonCanonical, tn::AbstractAnsatz, bond; threshold, maxdim, compute_local_svd=true)

Truncate the dimension of the virtual bond of a NonCanonical Tensor Network by contracting the bond, performing an SVD and keeping the maxdim largest singular values or those larger than threshold.

Arguments

• tn: The AbstractAnsatz Tensor Network.

• bond: The bond to truncate.

Keyword Arguments

• threshold: The threshold to truncate the bond dimension.

• maxdim: The maximum bond dimension to keep.

• compute_local_svd: Whether to compute the local SVD of the bond. If true, it will contract the bond and perform a SVD to get the local singular values. Defaults to true.

• normalize: Whether to normalize the state at the bond after truncation. Defaults to false.

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Tenet.truncate! Method

truncate!(tn::AbstractAnsatz, bond; threshold = nothing, maxdim = nothing)

Truncate the dimension of the virtual bond``of an [Ansatz](@ref) Tensor Network. Dispatches to the appropriate method based on the [form`](@ref) of the Tensor Network:

• If the Tensor Network is in the MixedCanonical form, the bond is truncated by moving the orthogonality center to the bond and keeping the maxdim largest Schmidt coefficients or those larger than threshold.

• If the Tensor Network is in the Canonical form, the bond is truncated by keeping the maxdim largest Schmidt coefficients or those larger than threshold, and then recanonizing the Tensor Network.

• If the Tensor Network is in the NonCanonical form, the bond is truncated by contracting the bond, performing an SVD and keeping the maxdim largest singular values or those larger than threshold.

Notes

• Either threshold or maxdim must be provided. If both are provided, maxdim is used.

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