[WIP] Implicit derivatives for CTMRG

docs/src/assets/pepskit.bib

@@ -157,3 +157,13 @@ @misc{zhang_accelerating_2025
       primaryClass={cond-mat.str-el},
       url={https://arxiv.org/abs/2505.00494}, 
 }
+
+@misc{burgelman_implicit_2026,
+      title={Implicit differentiation of tensor network algorithms}, 
+      author={Lander Burgelman and Anna Francuz and Paul Brehmer and Lukas Devos and Jutho Haegeman and Frank Verstraete and Bram Vanhecke},
+      year={2026},
+      eprint={2607.15030},
+      archivePrefix={arXiv},
+      primaryClass={quant-ph},
+      url={https://arxiv.org/abs/2607.15030}, 
+}

src/Defaults.jl

@@ -85,6 +85,9 @@ Module containing default algorithm parameter values and arguments.
     - `:GeomSum` : Geometric sum approximation of the Neumann series of the inverse Jacobian, see [`PEPSKit.GeomSum`](@ref) for details
     - `:ManualIter` : Manual fixed-point iteration, see [`PEPSKit.ManualIter`](@ref) for details
 * `gradient_fixedpoint_solver_eager=$(Defaults.gradient_fixedpoint_solver_eager)` : Enables `:Arnoldi` solver algorithm to finish before the full Krylov dimension is reached.
+* `gradient_implicit_solver_alg=:$(Defaults.gradient_implicit_solver_alg)` : Default solver algorithm for the `ImplicitGradient` gradient algorithm.
+    - `:GMRES` : GMRES iterative linear solver, see [`KrylovKit.GMRES`](@extref) for details
+    - `:BiCGStab` : BiCGStab iterative linear solver, see [`KrylovKit.BiCGStab`](@extref) for details
 
 ## Optimization
 
@@ -151,6 +154,7 @@ const gradient_verbosity = -1
 const gradient_alg = :FixedPointGradient
 const gradient_fixedpoint_solver_alg = :Arnoldi # ∈ {:GMRES, :BiCGStab, :Arnoldi, :GeomSum, :ManualIter}
 const gradient_fixedpoint_solver_eager = true
+const gradient_implicit_solver_alg = :GMRES # ∈ {:GMRES, :BiCGStab}
 
 # Optimization
 const reuse_env = true

src/PEPSKit.jl

@@ -93,6 +93,7 @@ include("algorithms/contractions/ctmrg/renormalize_corner.jl")
 include("algorithms/contractions/ctmrg/renormalize_edge.jl")
 include("algorithms/contractions/ctmrg/contract_site.jl")
 include("algorithms/contractions/ctmrg/gaugefix.jl")
+include("algorithms/contractions/ctmrg/characteristic_equations.jl")
 
 include("algorithms/contractions/absorb_weight.jl")
 include("algorithms/contractions/transfer.jl")
@@ -140,7 +141,7 @@ include("algorithms/toolbox.jl")
 include("algorithms/correlator_adapters.jl")
 include("algorithms/correlators.jl")
 
-include("algorithms/optimization/fixed_point_differentiation.jl")
+include("algorithms/optimization/implicit_differentiation.jl")
 include("algorithms/optimization/peps_optimization.jl")
 
 include("algorithms/select_algorithm.jl")
@@ -161,7 +162,7 @@ export product_peps
 export reduced_densitymatrix, expectation_value, network_value, cost_function
 export correlator, correlation_length
 export leading_boundary
-export PEPSOptimize, FixedPointGradient, GeomSum, ManualIter
+export PEPSOptimize, FixedPointGradient, GeomSum, ManualIter, ImplicitGradient
 export fixedpoint
 
 export LocalTruncation