Bond dimension

Reference for changing the bond dimension of a state — expanding or truncating its virtual spaces — and for inspecting those virtual spaces directly. For a task-oriented walkthrough see Controlling bond dimension; the full, canonical docstrings for the whole package live in the Library index.

Interface

MPSKit.changebonds Function

julia

changebonds(ψ::AbstractMPS, H, alg, envs) -> ψ′, envs′
changebonds(ψ::AbstractMPS, alg) -> ψ′

Change the bond dimension of ψ using the algorithm alg, and return the new ψ and the new envs. For AbstractInfiniteMPS, changebonds returns new environments without modifying the one provided. changedbonds! can modify both the provided state and environments, depending on the algorithm. For FiniteMPS, changebonds also modifies the environments.

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MPSKit.changebonds! Function

julia

changebonds(ψ::AbstractMPS, H, alg, envs) -> ψ′, envs′
changebonds(ψ::AbstractMPS, alg) -> ψ′

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Expansion and truncation algorithms

MPSKit.OptimalExpand Type

julia

struct OptimalExpand{S} <: MPSKit.Algorithm

An algorithm that expands the given mps as described in Zauner-Stauber et al. Phys. Rev. B 97 (2018), by selecting the dominant contributions of a two-site updated MPS tensor, orthogonal to the original ψ.

The expansion is state-preserving: the added directions are connected through a zero block, so that the expanded state represents the same physical state as the original one (as required for e.g. TDVP).

Note

changebonds! is only defined for FiniteMPS, and modifies both the state and its environment.

Fields

alg_svd::Any: algorithm used for the singular value decomposition
trscheme::MatrixAlgebraKit.TruncationStrategy: algorithm used for truncating the expanded space

See also

Used as the algorithm argument of changebonds and changebonds!.

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MPSKit.RandExpand Type

julia

struct RandExpand{S} <: MPSKit.Algorithm

An algorithm that expands the bond dimension by adding random unitary vectors that are orthogonal to the existing state. This means that additional directions are added to AL and AR that are contained in the nullspace of both. Note that this is happens in parallel, and therefore the expansion will never go beyond the local two-site subspace.

The truncation strategy dictates the number of expanded states, by generating uniformly distributed weights for each state in the two-site space and truncating that.

Note

The environments are not used here, but changebonds! modifies both the state and environment so they remain consistent.

Fields

alg_svd::Any: algorithm used for the singular value decomposition
trscheme::MatrixAlgebraKit.TruncationStrategy: algorithm used for truncation of the expanded space

See also

Used as the algorithm argument of changebonds and changebonds!.

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MPSKit.SvdCut Type

julia

struct SvdCut{S} <: MPSKit.Algorithm

An algorithm that uses truncated SVD to change the bond dimension of a state or operator. This is achieved by a sweeping algorithm that locally performs (optimal) truncations in a gauged basis.

changedbonds! is only defined for FiniteMPS and FiniteMPO.

Fields

alg_svd::Any: algorithm used for the singular value decomposition
trscheme::MatrixAlgebraKit.TruncationStrategy: algorithm used for truncation of the gauge tensors

See also

Used as the algorithm argument of changebonds and changebonds!.

References

Parker et al. Phys. Rev. B 102 (2020)

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MPSKit.VUMPSSvdCut Type

julia

struct VUMPSSvdCut <: MPSKit.Algorithm

An algorithm that uses a two-site update step to change the bond dimension of a state.

Note

changebonds! is not defined.

Fields

alg_gauge::Any: algorithm used for gauging the InfiniteMPS
alg_eigsolve::Any: algorithm used for the eigenvalue solvers
alg_svd::Any: algorithm used for the singular value decomposition
trscheme::MatrixAlgebraKit.TruncationStrategy: algorithm used for truncation of the two-site update

See also

Used as the algorithm argument of changebonds.

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MPSKit.SketchedExpand Type

julia

struct SketchedExpand{S} <: MPSKit.Algorithm

An algorithm that expands the bond dimension like OptimalExpand — selecting the dominant directions of the projected two-site update orthogonal to the current state — but at single-site cost using the randomized "shrewd selection" of Controlled Bond Expansion (Gleis et al. Phys. Rev. Lett. 130, 246402 (2023)). A random sketch of the orthogonal complement is folded into the effective environment, collapsing the large bond before the two-site update is ever formed, and the dominant directions are read off a small singular value decomposition.

The state-preserving behaviour matches OptimalExpand.

Note

Only defined for FiniteMPS (through changebond!), so it can be used standalone or as the alg_expand strategy of DMRG. The reported ϵ_2site is a randomized estimate, and the folded application does not exploit JordanMPO sparsity.

Fields

alg_orth::Any: algorithm used to orthonormalize the sketched complement (passed as the alg of left_orth!/right_orth!); nothing selects QR without oversampling and an SVD-based decomposition otherwise
trscheme::MatrixAlgebraKit.TruncationStrategy: algorithm used for truncating the expanded space
oversampling::Int64: number of extra sketch columns drawn beyond the target rank (range-finder oversampling)

See also

Used as the algorithm argument of changebonds and changebonds!.

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Note

SketchedExpand is experimental: it uses randomized controlled bond expansion (CBE), so its reported error estimate is itself randomized, and it is only defined for FiniteMPS.

Inspecting the virtual spaces

The bond dimension of an MPS or MPO is the dimension of the virtual space living on a given bond. The accessors below return that VectorSpace, whose dim gives the numeric bond dimension.

MPSKit.left_virtualspace Function

julia

left_virtualspace(ψ::AbstractMPS, [pos=1:length(ψ)])

Return the virtual space of the bond to the left of sites pos.

Warning

In rare cases, the gauge tensor on the virtual space might not be square, and as a result it cannot always be guaranteed that right_virtualspace(ψ, i - 1) == left_virtualspace(ψ, i)

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MPSKit.right_virtualspace Function

julia

right_virtualspace(ψ::AbstractMPS, [pos=1:length(ψ)])

Return the virtual space of the bond to the right of site(s) pos.

Warning

In rare cases, the gauge tensor on the virtual space might not be square, and as a result it cannot always be guaranteed that right_virtualspace(ψ, i - 1) == left_virtualspace(ψ, i)

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MPSKit.physicalspace Function

julia

physicalspace(ψ::AbstractMPS, [pos=1:length(ψ)])

Return the physical space of the site tensor at site i.

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<!– Maintainer notes (docs-writer):

Symbols included: changebonds, changebonds!, OptimalExpand, RandExpand, SvdCut, VUMPSSvdCut, SketchedExpand, left_virtualspace, right_virtualspace, physicalspace.

Caveats / REVIEW items for the maintainer: (a) The 4-arg changebonds(ψ, H, alg, envs) default keyword/positional behavior for envs is inconsistent across algorithms: envs=nothing for RandExpand/SvdCut vs envs=environments(...) for OptimalExpand/SketchedExpand/VUMPSSvdCut. This is surfaced verbatim in the docstrings pulled in above; not restated here. (b) SvdCut applied to an InfiniteMPO carries an in-source TODO about generalizing to finite MPOs — worth confirming intended scope before promoting this further. (c) The singular changebond/changebond! functions exist and are documented in source but are NOT exported, so they are intentionally omitted from this reference page (kept consistent with canonical=false covering only exported, public-facing symbols). –>

Bond dimension ​

Interface ​

Expansion and truncation algorithms ​

Inspecting the virtual spaces ​

Bond dimension

Interface

Expansion and truncation algorithms

Inspecting the virtual spaces