Basis expansion method with expand

src/ITensorTDVP.jl

@@ -1,5 +1,5 @@
 module ITensorTDVP
-export TimeDependentSum, dmrg_x, linsolve, tdvp, to_vec
+export TimeDependentSum, dmrg_x, expand, linsolve, tdvp, to_vec
 include("ITensorsExtensions.jl")
 using .ITensorsExtensions: to_vec
 include("applyexp.jl")
@@ -17,6 +17,7 @@ include("contract.jl")
 include("reducedconstantterm.jl")
 include("reducedlinearproblem.jl")
 include("linsolve.jl")
+include("expand.jl")
 using PackageExtensionCompat: @require_extensions
 function __init__()
   @require_extensions

src/expand.jl

@@ -0,0 +1,103 @@
+using ITensors:
+  ITensors,
+  Algorithm,
+  Index,
+  ITensor,
+  @Algorithm_str,
+  δ,
+  commonind,
+  dag,
+  denseblocks,
+  directsum,
+  hasqns,
+  prime,
+  scalartype,
+  uniqueinds
+using ITensors.ITensorMPS: MPO, MPS, apply, dim, linkind, maxlinkdim, orthogonalize
+using LinearAlgebra: normalize, svd, tr
+
+# Possible improvements:
+#  - Allow a maxdim argument to be passed to `expand`.
+#  - Current behavior is letting bond dimension get too
+#    big when used in imaginary time evolution.
+#  - Use (1-tau*operator)|state> to generate "Krylov" vectors
+#    instead of operator|state>. Is that needed?
+
+function expand(state, reference; alg, kwargs...)
+  return expand(Algorithm(alg), state, reference; kwargs...)
+end
+
+"""
+Given an MPS state and a collection of MPS references,
+returns an MPS which is equal to state
+(has fidelity 1.0 with state) but whose MPS basis
+is expanded to contain a portion of the basis of
+the references that is orthogonal to the MPS basis of state.
+"""
+function expand(
+  ::Algorithm"orthogonalize",
+  state::MPS,
+  references::Vector{MPS};
+  cutoff=10^2 * eps(real(scalartype(state))),
+)
+  n = length(state)
+  state = orthogonalize(state, n)
+  references = map(reference -> orthogonalize(reference, n), references)
+  s = siteinds(state)
+  for j in reverse(2:n)
+    # SVD state[j] to compute basisⱼ
+    linds = [s[j - 1]; linkinds(state, j - 1)]
+    _, λⱼ, basisⱼ = svd(state[j], linds; righttags="bψ_$j,Link")
+    rinds = uniqueinds(basisⱼ, λⱼ)
+    # Make projectorⱼ
+    idⱼ = prod(r -> denseblocks(δ(scalartype(state), r', dag(r))), rinds)
+    projectorⱼ = idⱼ - prime(basisⱼ, rinds) * dag(basisⱼ)
+    # Sum reference density matrices
+    ρⱼ = sum(reference -> prime(reference[j], rinds) * dag(reference[j]), references)
+    # TODO: Fix bug that `tr` isn't preserving the element type.
+    ρⱼ /= scalartype(state)(tr(ρⱼ))
+    # Apply projectorⱼ
+    ρⱼ_projected = apply(apply(projectorⱼ, ρⱼ), projectorⱼ)
+    expanded_basisⱼ = basisⱼ
+    if norm(ρⱼ_projected) > 10^3 * eps(real(scalartype(state)))
+      # Diagonalize projected density matrix ρⱼ_projected
+      # to compute reference_basisⱼ, which spans part of right basis
+      # of references which is orthogonal to right basis of state
+      dⱼ, reference_basisⱼ = eigen(
+        ρⱼ_projected; cutoff, ishermitian=true, righttags="bϕ_$j,Link"
+      )
+      state_indⱼ = only(commoninds(basisⱼ, λⱼ))
+      reference_indⱼ = only(commoninds(reference_basisⱼ, dⱼ))
+      expanded_basisⱼ, expanded_indⱼ = directsum(
+        basisⱼ => state_indⱼ, reference_basisⱼ => reference_indⱼ
+      )
+    end
+    # Shift ortho center one site left using dag(expanded_basisⱼ)
+    # and replace tensor at site j with expanded_basisⱼ
+    state[j - 1] = state[j - 1] * (state[j] * dag(expanded_basisⱼ))
+    state[j] = expanded_basisⱼ
+    for reference in references
+      reference[j - 1] = reference[j - 1] * (reference[j] * dag(expanded_basisⱼ))
+      reference[j] = expanded_basisⱼ
+    end
+  end
+  return state
+end
+
+function expand(
+  ::Algorithm"global_krylov",
+  state::MPS,
+  operator::MPO;
+  krylovdim=3,
+  cutoff=(√(eps(real(scalartype(state))))),
+)
+  # TODO: Try replacing this logic with `Base.accumulate`.
+  references = Vector{MPS}(undef, krylovdim - 1)
+  for k in 1:(krylovdim - 1)
+    previous_reference = get(references, k - 1, state)
+    references[k] = normalize(
+      apply(operator, previous_reference; maxdim=maxlinkdim(state) + 1)
+    )
+  end
+  return expand(state, references; alg="orthogonalize", cutoff)
+end

src/sweep_update.jl

@@ -1,6 +1,14 @@
 using ITensors: ITensors, uniqueinds
 using ITensors.ITensorMPS:
-  ITensorMPS, MPS, isortho, orthocenter, orthogonalize!, position!, replacebond!, set_nsite!
+  ITensorMPS,
+  MPS,
+  isortho,
+  noiseterm,
+  orthocenter,
+  orthogonalize!,
+  position!,
+  replacebond!,
+  set_nsite!
 using LinearAlgebra: norm, normalize!, svd
 using Printf: @printf
 

test/test_expand_basis.jl

@@ -0,0 +1,91 @@
+@eval module $(gensym())
+using ITensors: scalartype
+using ITensors.ITensorMPS: OpSum, MPO, MPS, inner, linkdims, maxlinkdim, randomMPS, siteinds
+using ITensorTDVP: dmrg, expand, tdvp
+using LinearAlgebra: normalize
+using Test: @test, @testset
+const elts = (Float32, Float64, Complex{Float32}, Complex{Float64})
+@testset "expand (eltype=$elt)" for elt in elts
+  @testset "expand (alg=\"orthogonalize\", conserve_qns=$conserve_qns, eltype=$elt)" for conserve_qns in
+                                                                                         (
+    false, true
+  )
+    n = 6
+    s = siteinds("S=1/2", n; conserve_qns)
+    state = randomMPS(elt, s, j -> isodd(j) ? "↑" : "↓"; linkdims=4)
+    reference = randomMPS(elt, s, j -> isodd(j) ? "↑" : "↓"; linkdims=2)
+    state_expanded = expand(state, [reference]; alg="orthogonalize")
+    @test scalartype(state_expanded) === elt
+    @test inner(state_expanded, state) ≈ inner(state, state)
+    @test inner(state_expanded, reference) ≈ inner(state, reference)
+  end
+  @testset "expand (alg=\"global_krylov\", conserve_qns=$conserve_qns, eltype=$elt)" for conserve_qns in
+                                                                                               (
+    false, true
+  )
+    n = 10
+    s = siteinds("S=1/2", n; conserve_qns)
+    opsum = OpSum()
+    for j in 1:(n - 1)
+      opsum += 0.5, "S+", j, "S-", j + 1
+      opsum += 0.5, "S-", j, "S+", j + 1
+      opsum += "Sz", j, "Sz", j + 1
+    end
+    operator = MPO(elt, opsum, s)
+    state = MPS(elt, s, j -> isodd(j) ? "↑" : "↓")
+    state_expanded = expand(state, operator; alg="global_krylov")
+    @test scalartype(state_expanded) === elt
+    @test maxlinkdim(state_expanded) > 1
+    @test inner(state_expanded, state) ≈ inner(state, state)
+  end
+  @testset "Decoupled ladder (alg=\"global_krylov\", eltype=$elt)" begin
+    nx = 10
+    ny = 2
+    n = nx * ny
+    s = siteinds("S=1/2", n)
+    opsum = OpSum()
+    for j in 1:2:(n - 2)
+      opsum += 1 / 2, "S+", j, "S-", j + 2
+      opsum += 1 / 2, "S-", j, "S+", j + 2
+      opsum += "Sz", j, "Sz", j + 2
+    end
+    for j in 2:2:(n - 2)
+      opsum += 1 / 2, "S+", j, "S-", j + 2
+      opsum += 1 / 2, "S-", j, "S+", j + 2
+      opsum += "Sz", j, "Sz", j + 2
+    end
+    operator = MPO(elt, opsum, s)
+    init = randomMPS(elt, s; linkdims=30)
+    reference_energy, reference_state = dmrg(
+      operator,
+      init;
+      nsweeps=15,
+      maxdim=[10, 10, 20, 20, 40, 80, 100],
+      cutoff=(√(eps(real(elt)))),
+      noise=(√(eps(real(elt)))),
+    )
+    state = randomMPS(elt, s)
+    nexpansions = 10
+    tau = elt(0.5)
+    for step in 1:nexpansions
+      # TODO: Use `∜` instead of `fourthroot` in Julia 1.10 and above.
+      state = expand(
+        state, operator; alg="global_krylov", krylovdim=3, cutoff=fourthroot(eps(real(elt)))
+      )
+      state = tdvp(
+        operator,
+        -4tau,
+        state;
+        nsteps=4,
+        cutoff=1e-5,
+        updater_kwargs=(; tol=1e-3, krylovdim=5),
+      )
+      state = normalize(state)
+    end
+    @test scalartype(state) === elt
+    # TODO: Use `∜` instead of `fourthroot` in Julia 1.10 and above.
+    @test inner(state', operator, state) ≈ reference_energy rtol =
+      2 * fourthroot(eps(real(elt)))
+  end
+end
+end

test/test_exports.jl

@@ -4,7 +4,7 @@ using Test: @test, @testset
 @testset "Test exports" begin
   @test issetequal(
     names(ITensorTDVP),
-    [:ITensorTDVP, :TimeDependentSum, :dmrg_x, :linsolve, :tdvp, :to_vec],
+    [:ITensorTDVP, :TimeDependentSum, :dmrg_x, :expand_basis, :linsolve, :tdvp, :to_vec],
   )
 end
 end