API Reference

This page provides a complete reference for all exported functions in FibonacciChain.jl.

Basis and Hamiltonian Functions

FibonacciChain.anyon_basis — Function

anyon_basis(model::AnyonModel; symmetry_block=nothing)
anyon_basis(anyon_type::AbstractAnyonType, ::Type{T}; pbc::Bool=true, symmetry_block=nothing) where {N, T <: BitStr{N}}

Generate basis states for an anyon chain model.

This function provides two interfaces: a high-level one that takes an AnyonModel object, and a low-level one that takes the model parameters directly.

Arguments

High-level interface

model::AnyonModel: An AnyonModel object containing the system parameters (anyon type, size, and boundary conditions).
symmetry_block: Optional. The topological charge sector to filter the basis. Not yet implemented.

Low-level interface

anyon_type::AbstractAnyonType: The type of anyon, e.g., FibonacciAnyon() or IsingAnyon().
T::Type: The BitStr type specifying the chain length N.
pbc::Bool=true: Specifies whether to use periodic (true) or open (false) boundary conditions.
symmetry_block: Optional. The topological charge sector.

Returns

Vector{T}: A sorted vector of the basis states.

Examples

julia> using FibonacciChain, BitBasis

julia> model = AnyonModel(FibonacciAnyon(), 4; pbc=true); basis = anyon_basis(model); length(basis)
7

julia> basis_fibo = anyon_basis(FibonacciAnyon(), BitStr{4, Int}; pbc=true); length(basis_fibo)
7

julia> model_ising = AnyonModel(IsingAnyon(), 4; pbc=true); basis_ising = anyon_basis(model_ising); length(basis_ising)
16

source

anyon_basis(AT::AbstractAnyonType, ::Type{T}, k::Int; symmetry_block=nothing) where {N, T <: BitStr{N}}
anyon_basis(model::AnyonModel, k::Int; symmetry_block=nothing)

Generate basis states in specific momentum sector k and topological sector symmetry_block.

Arguments

Low-level interface

AT::AbstractAnyonType: Anyon type instance (e.g., FibonacciAnyon(), IsingAnyon())
T::Type: BitStr type specifying chain length N
k::Int: Momentum sector (0 ≤ k ≤ N-1)
symmetry_block: Topological charge sector (optional)

High-level interface

model::AnyonModel: Anyon model containing system parameters
k::Int: Momentum sector (0 ≤ k ≤ N-1)
symmetry_block: Topological charge sector (optional)

Returns

Vector{T}: Basis states in momentum sector k
Dict{T, Vector{T}}: Representative mapping for translation equivalence classes

Examples

julia> using FibonacciChain, BitBasis

julia> model = AnyonModel(FibonacciAnyon(), 6; pbc=true);

julia> basisK, basis_dic = anyon_basis(model, 0);

julia> length(basisK) > 0
true

source

FibonacciChain.anyon_ham — Function

anyon_ham(model::AnyonModel; kwargs...)
anyon_ham(model::AnyonModel, ::Type{T}; kwargs...) where {N, T <: BitStr{N}}

Construct the Hamiltonian matrix for a 1D anyon chain.

Arguments

High-level interface

model::AnyonModel: An AnyonModel object specifying the anyon type, system size, boundary conditions, and interaction type.
kwargs...: Additional model-specific parameters, such as J and h for the Ising model.

Low-level usage (discouraged)

While not a direct low-level function, you can construct the model on the fly: anyon_ham(AnyonModel(IsingAnyon(), N), ...)

Returns

Matrix{Float64}: The Hamiltonian matrix constructed in the chosen basis.

Supported Models

Fibonacci Anyons: Supports :Antiferro and :Ferro interaction terms.
Ising Anyons: Transverse field Ising model.

Examples

julia> using FibonacciChain, BitBasis

julia> N = 4; model_fibo = AnyonModel(FibonacciAnyon(), N; pbc=true, measure_operator=:Antiferro);

julia> H_fibo = anyon_ham(model_fibo); size(H_fibo)
(7, 7)

julia> model_ising = AnyonModel(IsingAnyon(), N; pbc=true); H_ising = anyon_ham(model_ising; J=1.0, h=1.0); size(H_ising)
(16, 16)

source

anyon_ham(model::AnyonModel, ::Type{T}, k::Int; symmetry_block=nothing, kwargs...) where {N, T <: BitStr{N}}
anyon_ham(model::AnyonModel, k::Int; symmetry_block=nothing, kwargs...)

Construct Hamiltonian matrix in specific momentum sector for 1D anyon chain.

Arguments

model::AnyonModel: Anyon model containing system parameters
T::Type: BitStr type specifying chain length N (optional)
k::Int: Momentum sector (0 ≤ k ≤ N-1)
symmetry_block: Topological charge sector (optional)
kwargs...: Additional model parameters, e.g., J, h for Ising model

Returns

Matrix: Hamiltonian matrix in chosen momentum sector (ComplexF64, or real if k=0 or k=N/2)

Examples

julia> using FibonacciChain

julia> model = AnyonModel(FibonacciAnyon(), 6; pbc=true);

julia> H_k0 = anyon_ham(model, 0);

julia> size(H_k0)[1] > 0
true

source

anyon_ham(model::AnyonModel, sites::Vector{<:Index}; kwargs...)

Construct anyon chain Hamiltonian as Matrix Product Operator (MPO).

Arguments

model::AnyonModel: Anyon model containing system parameters
sites::Vector{<:Index}: ITensor site indices
kwargs...: Additional parameters (e.g., J, h for Ising model)

Returns

MPO: Hamiltonian as Matrix Product Operator
- Fibonacci: three-body interactions based on Fibonacci fusion rules
- Ising: transverse field Ising model with ZZ and X terms

source

FibonacciChain.anyon_ham_sparse — Function

anyon_ham_sparse(model::AnyonModel; kwargs...)

Construct the anyon chain Hamiltonian as a sparse matrix.

Arguments

model::AnyonModel: An AnyonModel object containing the system parameters.
kwargs...: Additional keyword arguments passed to actingHam, e.g., J, h for the Ising model.

Returns

SparseMatrixCSC{Float64, Int}: The sparse Hamiltonian matrix in the anyon basis.

Examples

julia> using FibonacciChain, BitBasis, SparseArrays, LinearAlgebra

julia> N = 6; model = AnyonModel(FibonacciAnyon(), N; pbc=true); H_sparse = anyon_ham_sparse(model);

julia> H_sparse isa SparseMatrixCSC
true

julia> ishermitian(H_sparse)
true

julia> H_dense = anyon_ham(model); norm(Matrix(H_sparse) - H_dense) < 1e-10
true

source

anyon_ham_sparse(model::AnyonModel, k::Int; symmetric_block=nothing, kwargs...)

Construct the Hamiltonian in a specific momentum and/or topological sector as a sparse matrix.

Arguments

model::AnyonModel: An AnyonModel object containing the system parameters.
k::Int: The momentum quantum number (0 ≤ k ≤ N-1).
symmetric_block: Optional. The topological charge sector (e.g., :trivial, :nontrivial).
kwargs...: Additional keyword arguments passed to actingHam, e.g., J, h.

Returns

SparseMatrixCSC: The sparse Hamiltonian in the specified symmetric sector. The element type will be ComplexF64 for general k and Float64 for k=0 or k=N/2.

source

FibonacciChain.mapst_sec2tot — Function

mapst_sec2tot(AT::AbstractAnyonType, ::Type{T}, state::Vector{ET}, k::Int; pbc::Bool=true) where {N, T <: BitStr{N}, ET}
mapst_sec2tot(model::AnyonModel, state::Vector{ET}, k::Int)

Map state in momentum sector to total Hilbert space.

Arguments

High-level interface

model::AnyonModel: An AnyonModel object containing system parameters
state::Vector{ET}: The state vector in the Hilbert space of the specified momentum sector
k::Int: The momentum sector index (0 ≤ k ≤ N-1)

Low-level interface

AT::AbstractAnyonType: The anyon type.
T::Type: The BitStr type specifying the chain length N.
state::Vector{ET}: The state vector in the momentum sector.
k::Int: The momentum sector index.
pbc::Bool=true: Specifies periodic boundary conditions (required for momentum sectors).

Returns

Vector{ET}: The state vector represented in the total Hilbert space basis.

Examples

julia> using FibonacciChain, LinearAlgebra

julia> model = AnyonModel(FibonacciAnyon(), 6; pbc=true); H_k0 = anyon_ham(model, 0);

julia> eigvecs_k0 = eigvecs(H_k0); kstate = eigvecs_k0[:, 1];

julia> total_state = mapst_sec2tot(model, kstate, 0); length(total_state) == length(anyon_basis(model))
true

source

Symmetry and Topological Operations

FibonacciChain.translation_matrix — Function

translation_matrix(model::AnyonModel)

Generate translation operator matrix for anyon basis states.

Arguments

model::AnyonModel: Anyon model containing system parameters (N, pbc, anyon_type)

Returns

Matrix{Float64}: Translation matrix mapping each basis state to its translated version

source

FibonacciChain.inversion_matrix — Function

inversion_matrix(model::AnyonModel)

Generate spatial inversion operator matrix for anyon basis states.

Arguments

model::AnyonModel: Anyon model containing system parameters (N, pbc, anyon_type)

Returns

Matrix{Float64}: Inversion matrix mapping each basis state to its spatially reflected version

source

FibonacciChain.braidingsqmap — Function

braidingsqmap(model::AnyonModel{FibonacciAnyon}, state::Vector{ET}, idx::Int) where {ET}

Apply braiding squared operation to quantum state at specified site.

Arguments

model::AnyonModel{FibonacciAnyon}: Fibonacci anyon model containing system parameters (N, pbc)
state::Vector{ET}: Quantum state vector in anyon basis
idx::Int: Site index for braiding operation

Returns

Vector{ComplexF64}: Transformed state after braiding operation

source

FibonacciChain.topological_symmetry_basismap — Function

topological_symmetry_basismap(model::AnyonModel{FibonacciAnyon}, state::T) where {N, T <: BitStr{N}}

Compute topological symmetry coefficients for all basis states relative to given state.

Arguments

model::AnyonModel{FibonacciAnyon}: Fibonacci anyon model
state::T: Reference state configuration

Returns

Vector{Float64}: Coefficients for each basis state

Examples

julia> using FibonacciChain, BitBasis

julia> N = 4; T = BitStr{N, Int};

julia> model = AnyonModel(FibonacciAnyon(), N; pbc=true);

julia> state = T(0b0000);  # Vacuum state

julia> coeffs = topological_symmetry_basismap(model, state);

julia> length(coeffs) == length(anyon_basis(model))
true

source

FibonacciChain.Fsymmetry_coef — Function

Fsymmetry_coef(::FibonacciAnyon, ::Type{T}, state::T, base::T; pbc::Bool=true) where {N, T <: BitStr{N}}
Fsymmetry_coef(model::AnyonModel{FibonacciAnyon}, state::T, base::T)

Compute topological symmetry coefficient for state in given base configurations for Fibonacci anyon chain.

Arguments

Low-level interface

::FibonacciAnyon: Fibonacci anyon type instance
::Type{T}: BitStr type specifying chain length N
state::T: Target state configuration
base::T: Base state configuration
pbc::Bool=true: Periodic boundary conditions

High-level interface

model::AnyonModel{FibonacciAnyon}: Anyon model containing system parameters
state::T: Target state configuration
base::T: Base state configuration

Returns

Float64: Topological symmetry coefficient based on Fibonacci fusion rules

Examples

julia> using FibonacciChain, BitBasis

julia> N = 4; T = BitStr{N, Int};

julia> model = AnyonModel(FibonacciAnyon(), N; pbc=true);

julia> state = T(0b1010); ϕ = (1 + sqrt(5)) / 2;

julia> base = T(0b0101);

julia> coef = Fsymmetry_coef(model, state, base); coef ≈ 0.3819660112501051
true

source

Ladder System Functions

FibonacciChain.ladderChoi — Function

ladderChoi(model::AnyonModel{FibonacciAnyon}, p::Float64, state::Vector{ET}) where {ET}

Apply probabilistic braiding noise channel to density matrix state in vec form.

Arguments

model::AnyonModel{FibonacciAnyon}: Fibonacci anyon model containing system parameters (N, pbc)
p::Float64: Braiding probability (0 ≤ p ≤ 1)
state::Vector{ET}: Density matrix state in vectorized form

Returns

Vector{ET}: State after applying Choi map with braiding noise

Implements noise channel: ρ → (1-p)ρ + p B(ρ) where B is braiding operation.

source

FibonacciChain.ladderrdm — Function

ladderrdm(model::AnyonModel, subsystems::Vector{Int64}, state::Vector{ET}) where {ET}

Compute reduced density matrix for specified subsystem from vectorized density matrix.

Arguments

model::AnyonModel: Anyon model containing system parameters (N, pbc, anyon_type)
subsystems::Vector{Int64}: Indices of subsystem sites to trace out
state::Vector{ET}: Full density matrix state in vectorized form

Returns

Matrix{Float64}: Reduced density matrix of the subsystem

For ladder systems where both subsystems have equal dimensions.

source

FibonacciChain.ladderbraidingsqmap — Function

ladderbraidingsqmap(model::AnyonModel{FibonacciAnyon}, state::Vector{ET}, idx::Int) where {ET}

Apply braiding squared operation to density matrix state in vectorized form. Effectively, it describes the noise induced by inter-two layer chain braiding.

Arguments

model::AnyonModel{FibonacciAnyon}: Fibonacci anyon model containing system parameters (N, pbc)
state::Vector{ET}: Density matrix state in vectorized form
idx::Int: Site index for braiding operation

Returns

Vector{ET}: Transformed density matrix state after braiding

Operates on superposition states represented as vectorized density matrices.

Examples

julia> using FibonacciChain, LinearAlgebra, BitBasis

julia> N = 4;

julia> model = AnyonModel(FibonacciAnyon(), N; pbc=true);

julia> # Create PBC Fibonacci anyon basis 
       basis = anyon_basis(model);

julia> l = length(basis);

julia> ρ_vec = zeros(ComplexF64, l^2);

julia> # Initialize as identity matrix (vectorized)
       for i in 1:l
           ρ_vec[(i-1)*l + i] = 1.0/l;
       end

julia> # Apply braiding at site 2
       ρ_braided = ladderbraidingsqmap(model, ρ_vec, 2);

julia> norm(ρ_braided) > 0  # Should be non-zero
true

source

FibonacciChain.laddertranslationmap — Function

laddertranslationmap(model::AnyonModel, state::Vector{ET}) where {ET}

Apply translation operator to vectorized density matrix state.

Arguments

model::AnyonModel: Anyon model containing system parameters (N, pbc, anyon_type)
state::Vector{ET}: Density matrix state in vectorized form

Returns

Vector{ET}: Translated density matrix state

Translates both bra and ket parts of the density matrix consistently.

source

Measurement Functions

FibonacciChain.measure_basismap — Function

measure_basismap(model::AnyonModel, τ::Float64, state::T, i::Int, sign::Bool) where {T}

Map single basis state under measurement operation at site i.

Arguments

model::AnyonModel: Anyon model containing system parameters (N, pbc, measure_operator)
τ::Float64: Measurement strength parameter
state::T: Input basis state
i::Int: Measurement site index (1 ≤ i ≤ N)
sign::Bool: Measurement outcome (false for +, true for -)

Returns

Tuple: Either (basis1, basis2, coeff1, coeff2) for superposition output or (basis, coeff) for single output

Maps individual basis states according to measurement protocols and fusion rules.

Examples

julia> using FibonacciChain, BitBasis

julia> N = 4; T = BitStr{N, Int};

julia> model = AnyonModel(FibonacciAnyon(), N; pbc=true, measure_operator=:Antiferro);

julia> state = T(0b0100);  # Single τ at site 2

julia> τ = 1.0;  # Measurement strength

julia> result = measure_basismap(model, τ, state, 2, false);

julia> length(result) ∈ [2, 4]  # Returns 2 or 4 elements depending on configuration
true

source

FibonacciChain.measuremap — Function

measuremap(model::AnyonModel, τ::Float64, state::Vector{ET}, idx::Int, sign::Bool)
measuremap(model::AnyonModel, ψ::MPS, sites, i::Int, τ::Float64, sign::Bool; 
           cutoff::Float64=1e-10, maxdim::Int=100)

Apply measurement operator to quantum state and return post-measurement state.

Methods

Vector version (from Measurement.jl)

Apply measurement to a state vector.

Arguments

model::AnyonModel: Anyon model containing system parameters
τ::Float64: Measurement strength parameter
state::Vector{ET}: Input quantum state vector
idx::Int: Measurement site index
sign::Bool: Measurement outcome (false for +, true for -)

Returns

Vector{ET}: Post-measurement quantum state (unnormalized)

MPS version (from MPSMeasurement.jl)

Apply measurement to an MPS state.

Arguments

model::AnyonModel: Anyon model containing system parameters
ψ::MPS: Input quantum state
sites: ITensor site indices
i::Int: Measurement site
τ::Float64: Measurement strength parameter
sign::Bool: Measurement outcome (false for +, true for -)
cutoff::Float64=1e-10: MPS truncation cutoff
maxdim::Int=100: Maximum bond dimension

Returns

MPS: Post-measurement quantum state (normalized)
Float64: Measurement probability

source

FibonacciChain.measurement_enumeration — Function

measurement_enumeration(model::AnyonModel, τ::Float64, initial_state::Vector{ET}, measurement_sites::Vector{Int}) where {ET}

Enumerate all trajectories of measurements on a given initial state.

Arguments

model::AnyonModel: Anyon model containing system parameters
τ::Float64: Measurement strength parameter
initial_state::Vector{ET}: Initial quantum state vector
measurement_sites::Vector{Int}: Sites to measure

Returns

Tuple{Vector{Vector{ET}}, Vector{Vector{Int64}}, Vector{Float64}}: (final_states, trajectories, probabilities)

Examples

julia> using FibonacciChain, LinearAlgebra

julia> model = AnyonModel(FibonacciAnyon(), 4; pbc=true, measure_operator=:Antiferro);

julia> basis = anyon_basis(model);

julia> state = normalize(ones(length(basis)));

julia> states, trajs, probs = measurement_enumeration(model, 1.0, state, [2]);

julia> length(states) == 2  # Two possible outcomes
true

source

FibonacciChain.measurement_tree_visualization — Function

measurement_tree_visualization(trajectories::Vector{Vector{Int64}}, probabilities::Vector{Float64})

Visualize a measurement tree given trajectories and their probabilities.

Arguments

trajectories::Vector{Vector{Int64}}: Measurement outcome sequences at each branch.
probabilities::Vector{Float64}: Corresponding probabilities for each trajectory.

Behavior

Normalizes probabilities to sum to 1.
Prints levels from root to leaves with indentation representing depth.

source

FibonacciChain.transfer_matrix — Function

transfer_matrix(model::AnyonModel, τ::Float64; sign::Bool=true)

Construct the transfer matrix for two measurement layers.

Arguments

model::AnyonModel: Anyon model containing system parameters
τ::Float64: Measurement strength parameter
sign::Bool=true: Measurement outcome sign

Returns

Matrix{Float64}: Transfer matrix between two measurement layers

source

FibonacciChain.boundary_evolution — Function

boundary_evolution(model::AnyonModel, state::Vector{T}, measure_config::MeasureConfig, 
                   sample::Union{Nothing, BitVector}=nothing; layer_idx::Int=1)
boundary_evolution(model::AnyonModel, sites, state::MPS, sample::BitVector, 
                   measure_config::MeasureConfig; 
                   cutoff::Float64=1e-12, maxdim::Int=1000, layer_idx::Int=1)

Evolve a quantum state under a single layer of boundary measurements with a given trajectory.

Methods

Vector version (from Measurement.jl)

Evolve a state vector under boundary measurements.

Arguments

model::AnyonModel: The anyon model specifying system parameters.
state::Vector{T}: The initial state vector.
measure_config::MeasureConfig: Configuration containing measurement strength τ and mode.
sample::Union{Nothing, BitVector}=nothing: The measurement outcomes for the layer.
layer_idx::Int=1: The layer index for measurement configuration.

Returns

Measurement_outcome_boundary: A struct containing state, sample, and free_energy.

MPS version (from MPSMeasurement.jl)

Evolve an MPS state under boundary measurements.

Arguments

model::AnyonModel: The anyon model specifying system parameters.
sites: The ITensor site indices.
state::MPS: The initial MPS state.
sample::BitVector: The measurement outcomes for the layer.
measure_config::MeasureConfig: Configuration containing measurement strength τ and mode (:sample or :Born).
cutoff::Float64=1e-12: MPS truncation cutoff.
maxdim::Int=1000: Maximum bond dimension for MPS operations.
layer_idx::Int=1: The layer index for measurement configuration.

Returns

Measurement_outcome_mps_boundary: A struct containing:
- state::MPS: The evolved MPS state.
- samples::BitVector: The measurement outcomes for the layer.
- free_energy::Float64: The free energy associated with the measurement layer.

source

FibonacciChain.bulk_evolution — Function

bulk_evolution(model::AnyonModel, state::Vector{ET}, measure_config::MeasureConfig,
               samples::Union{Nothing,BitMatrix}=nothing)
bulk_evolution(model::AnyonModel, sites, state::MPS, measure_config::MeasureConfig,
               samples::Union{Nothing,BitMatrix}=nothing;
               cutoff::Float64=1e-10, maxdim::Int=100)

Perform bulk measurement evolution from t₁ to t₂ on quantum state.

Methods

Vector version (from Measurement.jl)

Evolve a state vector under bulk measurements.

Arguments

model::AnyonModel: Anyon model containing system parameters
state::Vector{ET}: Initial quantum state vector
measure_config::MeasureConfig: Configuration struct containing τ, t₁, t₂, mode, rng, etc.
samples::Union{Nothing,BitMatrix}=nothing: Predefined measurement sequences for :sample mode

Returns

Measurement_outcome_bulk: A struct containing:
- states::Vector{Vector{ET}}: Intermediate states at each time step
- samples::BitMatrix: Measurement outcome sequences
- free_energys::Vector{Float64}: Free energy for each layer

MPS version (from MPSMeasurement.jl)

Evolve an MPS state under bulk measurements.

Arguments

model::AnyonModel: Anyon model containing system parameters
sites: ITensor site indices
state::MPS: Initial MPS quantum state
measure_config::MeasureConfig: Configuration struct containing τ, t₁, t₂, mode, rng, etc.
samples::Union{Nothing,BitMatrix}=nothing: Predefined measurement sequences for :sample mode
cutoff::Float64=1e-10: MPS truncation cutoff
maxdim::Int=100: Maximum bond dimension

Returns

Measurement_outcome_bulk: A struct containing:
- states::Vector{MPS}: Intermediate states at each time step
- samples::BitMatrix: Measurement outcome sequences
- free_energy::Vector{Float64}: Free energy for each layer

Notes

In :Born mode, samples are generated probabilistically via Born rule
In :sample mode, samples must be provided as input
(2N+1) layers of measurements correspond to N time steps of evolution

source

MPS Functions

FibonacciChain.measurement_operator_mps — Function

measurement_operator_mps(model::AnyonModel, sites::Vector{<:Index}, i::Int, τ::Float64, sign::Bool)

Create local measurement operator at site i as ITensor.

Arguments

model::AnyonModel: Anyon model containing system parameters
sites::Vector{<:Index}: ITensor site indices
i::Int: Measurement site
τ::Float64: Measurement strength parameter
sign::Bool: Measurement outcome (false for +, true for -)

Returns

ITensor: Local measurement operator incorporating neighboring site correlations

source

FibonacciChain.initial_mps — Function

anyon_mps_gst(model::AnyonModel; sweep_times=5, maxdim=5, cutoff=1e-10, outputlevel=1)

Find ground state of anyon chain Hamiltonian using DMRG.

Arguments

model::AnyonModel: Anyon model containing system parameters (N, pbc, anyon_type)
sweep_times=5: Number of DMRG sweeps
maxdim=5: Maximum bond dimension
cutoff=1e-10: Truncation cutoff
outputlevel=1: Verbosity level

Returns

MPS: Ground state as Matrix Product State
Float64: Ground state energy

Examples

julia> using FibonacciChain, ITensorMPS, ITensors

julia> model = AnyonModel(FibonacciAnyon(), 8; pbc=true);

julia> ψ_gs, E0 = anyon_mps_gst(model, maxdim=10, outputlevel=0);

julia> ψ_gs isa MPS
true

julia> E0 < 0 && imag(E0) ≈ 0
true

source

FibonacciChain.mps_measurement_enumeration — Function

mps_measurement_enumeration(model::AnyonModel, ψ::MPS, sites::Vector{<:Index}, measurement_sites::Vector{Int}, 
                            τ::Float64; cutoff::Float64=1e-10, maxdim::Int=100)

Enumerate all possible measurement trajectories on MPS state.

Arguments

model::AnyonModel: Anyon model containing system parameters
ψ::MPS: Input quantum state
sites: ITensor site indices
measurement_sites::Vector{Int}: Sites to measure
τ::Float64: Measurement strength parameter
cutoff::Float64=1e-10: MPS truncation cutoff
maxdim::Int=100: Maximum bond dimension

Returns

Vector{MPS}: Final states for each trajectory
Vector{Vector{Bool}}: Measurement trajectories
Vector{Float64}: Probabilities for each trajectory

source

Reference Qubit Functions

FibonacciChain.add_reference_qubits — Function

add_reference_qubits(model::AnyonModel, ψ::MPS, sites::Vector{<:Index}, site_idx::Int=1;
                     k_new::Int=1, verbose::Bool=false) -> Tuple{MPS, Vector{<:Index}}

Add reference qubits to an MPS state at specified site using copy method for correlation measurements.

Principle

Reference qubits are ancillary qubits used to probe spatial and temporal correlations in monitored quantum systems. The key idea is to create a maximally entangled state between the reference qubit and a system qubit at a specific site, enabling measurement of correlations through the reference qubit's reduced density matrix.

Copy Method

Creates a controlled-copy (CNOT-like) entanglement:

If the system qubit at site_idx is in state |0⟩, the reference qubit becomes |0⟩
If the system qubit at site_idx is in state |1⟩, the reference qubit becomes |1⟩

Mathematically, for a state |ψ⟩ = α|0⟩ + β|1⟩ at site_idx:

    |ψ⟩ → α|00⟩ + β|11⟩  (reference ⊗ system)

This preserves the original state's structure while creating classical correlation.

For reset method (measure system qubit first, then create Bell pair), use add_reference_qubits_reset instead.

Arguments

model::AnyonModel: Anyon model containing system parameters
ψ::MPS: Input MPS quantum state
sites: ITensor site indices
site_idx::Int=1: Site index to entangle with reference qubit (1 ≤ site_idx ≤ N)
k_new::Int=1: Number of reference qubits to add (must be 0 or 1)
verbose::Bool=false: Enable verbose output for debugging

Returns

MPS: New MPS with reference qubit added at the leftmost position
Vector{Index}: Updated site indices including the new reference qubit

Notes

Each system qubit can only be entangled with one reference qubit
To add multiple reference qubits at different sites, call this function multiple times
The reference qubit is inserted at position 1 (leftmost) of the MPS
Original MPS is not modified (returns a new copy)

Example

model = AnyonModel(FibonacciAnyon(), 6)
ψ, sites = anyon_mps_gst(model)
ψ_ref, sites_ref = add_reference_qubits(model, ψ, sites, 3)  # Add ref qubit at site 3

See also: add_reference_qubits

source

add_reference_qubits_reset(model::AnyonModel, state::Vector{ET}, site_idx::Int64; 
                            k_new::Int=1, verbose::Bool=false) -> Tuple{Float64, Float64, Vector{ET}, Vector{ET}}

Add reference qubits using reset method: first measures/resets the system qubit, then creates a Bell pair.

Principle

Project the system qubit onto |0⟩ or |1⟩ basis (strong measurement)
Create a Bell pair between the reset qubit and reference qubit
Returns both branches with their probabilities

This method is useful when you want to prepare a known initial state before adding the reference qubit.

Arguments

model::AnyonModel: Anyon model containing system size N, boundary conditions, etc.
state::Vector{ET}: Quantum state vector (may already contain k_old reference qubits)
site_idx::Int64: Site index for reference qubit insertion (1 ≤ site_idx ≤ N)
k_new::Int=1: Number of new reference qubits to add (must be 0 or 1)
verbose::Bool=false: Enable verbose output for debugging

Returns

Tuple{Float64, Float64, Vector{ET}, Vector{ET}}: (prob₀, prob₁, stateafter0, stateafter1) for both measurement branches

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FibonacciChain.reference_measuremap — Function

reference_measuremap(model::AnyonModel, τ::Float64, state::Vector{ET}, idx::Int, sign::Bool;
                     extended_basis::Vector{newT}, k_old::Int64=1)
reference_measuremap(model::AnyonModel, ::Type{T}, ::Type{pretype}, τ::Float64, state::Vector{ET},
                     idx::Int, sign::Bool; extended_basis::Vector{newT})

Apply a measurement operator to a quantum state with reference qubits.

This function extends measuremap to handle states that include reference qubits, preserving the reference qubit subspace while applying measurements to the system.

Arguments

model::AnyonModel: Anyon model containing system parameters
τ::Float64: Measurement strength parameter
state::Vector{ET}: Quantum state vector with reference qubits
idx::Int: Site index for measurement
sign::Bool: Measurement outcome (false=0, true=1)
extended_basis::Vector{newT}: Extended basis including reference qubits
k_old::Int64=1: Number of reference qubits in the state

Returns

Vector{ET}: The post-measurement state (unnormalized)

Notes

The extended basis must be constructed via build_extended_basis before calling this function.

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FibonacciChain.reference_boundary_evolution — Function

reference_boundary_evolution(model::AnyonModel, state::Vector{T}, measure_config::MeasureConfig,
                             sample::Union{Nothing, BitVector}=nothing; layer_idx::Int64=1)

Perform a single measurement layer evolution on a state with reference qubits.

Arguments

model::AnyonModel: Anyon model containing system parameters
state::Vector{T}: Quantum state vector with reference qubits
measure_config::MeasureConfig: Configuration containing τ, mode, rng, etc.
sample::Union{Nothing, BitVector}=nothing: Predefined measurement outcomes for :sample mode
layer_idx::Int64=1: Layer index (1-based) to determine measurement pattern

Returns

Measurement_outcome_boundary: A struct containing:
- state::Vector{T}: Post-measurement state
- sample::BitVector: Measurement outcomes for this layer
- free_energy::Float64: Free energy contribution from this layer

Notes

In :Born mode, samples are generated probabilistically via Born rule
In :sample mode, sample must be provided as input

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FibonacciChain.reference_bulk_evolution — Function

reference_bulk_evolution(model::AnyonModel, state::Vector{T}, measure_config::MeasureConfig,
                         samples::Union{Nothing,BitMatrix}=nothing) where{T}

Perform bulk measurement evolution from t₁ to t₂ on quantum state with reference qubits.

Arguments

model::AnyonModel: Anyon model containing system parameters
state::Vector{T}: Initial quantum state vector with reference qubits
measure_config::MeasureConfig: Configuration struct containing τ, t₁, t₂, mode, rng, etc.
samples::Union{Nothing,BitMatrix}=nothing: Predefined measurement sequences for :sample mode

Returns

Measurement_outcome_bulk: A struct containing:
- states::Vector{Vector{T}}: Intermediate states at each time step
- samples::BitMatrix: Measurement outcome sequences
- free_energys::Vector{Float64}: Free energy for each layer

Notes

In :Born mode, samples are generated probabilistically via Born rule
In :sample mode, samples must be provided as input
The state should already contain reference qubits added via add_reference_qubits

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FibonacciChain.reference_rdm — Function

reference_rdm(model::AnyonModel, subsystems::Vector{Int64}, state::Vector{ET}; traceref::Bool=true)

Compute the reduced density matrix for a state with reference qubits.

Arguments

model::AnyonModel: Anyon model containing system parameters
subsystems::Vector{Int64}: Indices of system sites to keep (not trace out)
state::Vector{ET}: Quantum state vector with reference qubits
traceref::Bool=true: If true, trace out reference qubits and keep system subsystems; if false, trace out system and keep reference qubits

Returns

Matrix{ET}: The reduced density matrix

Notes

Subsystem indices count from the right of the binary string representation
The number of reference qubits k_old is automatically deduced from the state length

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FibonacciChain.reference_evolution — Function

reference_evolution(N::Int, τ::Float64, forward::Vector{ET}, sample::Matrix{Bool}, 
                    x₂::Int, t₁, t₂; x₁::Int=1, rng=MersenneTwister(), pbc=true, 
                    anyon_type::Symbol=:Fibo, verbose=false, mode::Symbol=:sample)
reference_evolution(model::AnyonModel, sites, forward::Vector{MPS}, sample::Matrix{Bool},
                    x₂::Int, t₁, t₂, measure_config::MeasureConfig; x₁::Int=1, verbose=false)

Compute temporal/spatial correlation between two spacetime points using cached forward evolution.

This function avoids recomputing the forward evolution up to time t₁ by using the cached forward states.

Methods

Vector version (from ReferenceProbe.jl)

Compute correlation using state vectors.

Arguments

N::Int: System size
τ::Float64: Measurement strength parameter
forward::Vector{ET}: Cached forward state evolution trajectory
sample::Matrix{Bool}: Measurement sample configuration
x₂::Int: Site index for second reference qubit insertion
t₁::Int: First time slice index
t₂::Int: Second time slice index (must be >= t₁)
x₁::Int=1: Spatial site index for first reference qubit
rng::MersenneTwister=MersenneTwister(): Random number generator
pbc::Bool=true: Periodic boundary conditions
anyon_type::Symbol=:Fibo: Anyon type (:Fibo or :Ising)
verbose::Bool=false: Enable verbose output
mode::Symbol=:sample: Evolution mode (:sample or :Born)

Returns

Reference qubit added states between specified spacetime slices.

MPS version (from MPSMeasurement.jl)

Compute correlation using MPS states.

Arguments

model::AnyonModel: Anyon model containing system parameters
sites: ITensor site indices
forward::Vector{MPS}: Cached forward state evolution trajectory from a previous bulk_evolution run.
sample::Matrix{Bool}: The full measurement sample configuration for the entire evolution.
x₂::Int: The site index for the second reference qubit insertion.
t₁::Int: The first time slice index for correlation.
t₂::Int: Second time slice index (must be >= t₁)
measure_config::MeasureConfig: Configuration containing τ, rng, mode, etc.
x₁::Int=1: Site index for first reference qubit
verbose::Bool=false: Enable verbose output for debugging.

Returns

Vector{MPS}: A vector of MPS states for the specified spacetime slices.
BitMatrix: The measurement samples used for the evolution.
Vector{Float64}: The free energy calculated for each layer.

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reference_evolution(model::AnyonModel, forward::Vector{ET}, measure_config::MeasureConfig, 
                    sample::Union{BitMatrix, Nothing}=nothing) -> Measurement_outcome_bulk

Compute temporal/spatial correlation between two spacetime points using cached forward evolution.

This function avoids recomputing the forward evolution up to time t₁ by using the cached forward states, then adds reference qubits and continues evolution to measure correlations.

Spacetime Diagram

    | ---->|
    forward
          Ref1                    x₁ = 1            
           |
           |
          Ref2 --> Ref3               x₂ = N/2
    | ----> |______| -----> |
    1      t₁   t₂=t₁+δt   Δt
    | --------------------->|
    sample

Arguments

model::AnyonModel: Anyon model containing system parameters (N, pbc, anyon_type, etc.)
forward::Vector{ET}: Cached forward state evolution trajectory from a previous bulk_evolution run
measure_config::MeasureConfig: Configuration struct containing:
- τ::Float64: Measurement strength parameter
- t₁::Int: First time slice index for correlation
- t₂::Int: Second time slice index (must be >= t₁)
- x₁::Int: Site index for first reference qubit
- x₂::Int: Site index for second reference qubit (must be >= x₁)
- mode::Symbol: Evolution mode (:sample or :Born)
- rng::MersenneTwister: Random number generator (for :Born mode)
- verbose::Bool: Enable verbose output
sample::Union{BitMatrix, Nothing}=nothing: Full measurement sample configuration for the entire evolution

Returns

Measurement_outcome_bulk: A struct containing:
- states::Vector{ET}: State evolution trajectory with reference qubits
- samples::BitMatrix: Measurement samples used for the evolution
- free_energys::Vector{Float64}: Free energy calculated for each layer

Correlation Types

Based on δt = t₂ - t₁ and δx = |x₂ - x₁|:

δt > 0 && δx > 0: 3-point correlation (both spatial and temporal), uses 3 reference qubits
δt == 0: Pure 2-point spatial correlation at fixed time, uses 2 reference qubits
δx == 0: Pure 2-point temporal correlation at fixed site, uses 2 reference qubits

Notes

The forward states should come from a previous bulk_evolution call
Reference qubits are added via add_reference_qubits at the specified spacetime points
In :sample mode, sample must be provided; in :Born mode, samples are generated via Born rule

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API Reference

Basis and Hamiltonian Functions

Reduced Density Matrices

Entanglement and Observables

Symmetry and Topological Operations

Ladder System Functions

Measurement Functions

MPS Functions

Reference Qubit Functions