QuantumLegos

Documentation for QuantumLegos.

All contents:

Example

CheckMatrix and defining Lego

julia> using QuantumLegos

julia> stabilizers = pauliop.(["IIXXXX", "IIZZZZ", "ZIZZII", "IZZIZI", "IXXXII", "XIXIXI"])
6-element Vector{StaticArraysCore.SVector{6, QuantumLegos.PauliOps.SinglePauliOp}}:
 pauliop("IIXXXX")
 pauliop("IIZZZZ")
 pauliop("ZIZZII")
 pauliop("IZZIZI")
 pauliop("IXXXII")
 pauliop("XIXIXI")

julia> cmat = checkmatrix(stabilizers)
CheckMatrix with 6 generators, 6 legs:
  0  0  1  1  1  1 | 0  0  0  0  0  0
  0  0  0  0  0  0 | 0  0  1  1  1  1
  0  0  0  0  0  0 | 1  0  1  1  0  0
  0  0  0  0  0  0 | 0  1  1  0  1  0
  0  1  1  1  0  0 | 0  0  0  0  0  0
  1  0  1  0  1  0 | 0  0  0  0  0  0


julia> cmat.nlegs
6

julia> cmat.ngens
6

julia> cmat.cmat
6×12 Matrix{Bool}:
 0  0  1  1  1  1  0  0  0  0  0  0
 0  0  0  0  0  0  0  0  1  1  1  1
 0  0  0  0  0  0  1  0  1  1  0  0
 0  0  0  0  0  0  0  1  1  0  1  0
 0  1  1  1  0  0  0  0  0  0  0  0
 1  0  1  0  1  0  0  0  0  0  0  0

julia> # define lego

julia> lego = Lego(stabilizers)
Lego{6}(6, StaticArraysCore.SVector{6, QuantumLegos.PauliOps.SinglePauliOp}[pauliop("IIXXXX"), pauliop("IIZZZZ"), pauliop("ZIZZII"), pauliop("IZZIZI"), pauliop("IXXXII"), pauliop("XIXIXI")])

julia> lego.stabgens |> checkmatrix
CheckMatrix with 6 generators, 6 legs:
  0  0  1  1  1  1 | 0  0  0  0  0  0
  0  0  0  0  0  0 | 0  0  1  1  1  1
  0  0  0  0  0  0 | 1  0  1  1  0  0
  0  0  0  0  0  0 | 0  1  1  0  1  0
  0  1  1  1  0  0 | 0  0  0  0  0  0
  1  0  1  0  1  0 | 0  0  0  0  0  0

Defining and Updating State

julia> using QuantumLegos

julia> stabilizers = pauliop.(["IIXXXX", "IIZZZZ", "ZIZZII", "IZZIZI", "IXXXII", "XIXIXI"])
6-element Vector{StaticArraysCore.SVector{6, QuantumLegos.PauliOps.SinglePauliOp}}:
 pauliop("IIXXXX")
 pauliop("IIZZZZ")
 pauliop("ZIZZII")
 pauliop("IZZIZI")
 pauliop("IXXXII")
 pauliop("XIXIXI")

julia> lego = Lego(stabilizers)
Lego{6}(6, StaticArraysCore.SVector{6, QuantumLegos.PauliOps.SinglePauliOp}[pauliop("IIXXXX"), pauliop("IIZZZZ"), pauliop("ZIZZII"), pauliop("IZZIZI"), pauliop("IXXXII"), pauliop("XIXIXI")])

julia> # state with 1 lego, 0 leg

julia> st = State([lego, ], Tuple{LegoLeg, LegoLeg}[])
State(Lego[Lego{6}(6, StaticArraysCore.SVector{6, QuantumLegos.PauliOps.SinglePauliOp}[pauliop("IIXXXX"), pauliop("IIZZZZ"), pauliop("ZIZZII"), pauliop("IZZIZI"), pauliop("IXXXII"), pauliop("XIXIXI")])], Tuple{LegoLeg, LegoLeg}[], CheckMatrix(Bool[0 0 … 0 0; 0 0 … 1 1; … ; 0 1 … 0 0; 1 0 … 0 0], 6, 6))

julia> st.cmat.cmat
6×12 Matrix{Bool}:
 0  0  1  1  1  1  0  0  0  0  0  0
 0  0  0  0  0  0  0  0  1  1  1  1
 0  0  0  0  0  0  1  0  1  1  0  0
 0  0  0  0  0  0  0  1  1  0  1  0
 0  1  1  1  0  0  0  0  0  0  0  0
 1  0  1  0  1  0  0  0  0  0  0  0

julia> add_lego!(st, lego)
State(Lego[Lego{6}(6, StaticArraysCore.SVector{6, QuantumLegos.PauliOps.SinglePauliOp}[pauliop("IIXXXX"), pauliop("IIZZZZ"), pauliop("ZIZZII"), pauliop("IZZIZI"), pauliop("IXXXII"), pauliop("XIXIXI")]), Lego{6}(6, StaticArraysCore.SVector{6, QuantumLegos.PauliOps.SinglePauliOp}[pauliop("IIXXXX"), pauliop("IIZZZZ"), pauliop("ZIZZII"), pauliop("IZZIZI"), pauliop("IXXXII"), pauliop("XIXIXI")])], Tuple{LegoLeg, LegoLeg}[], CheckMatrix(Bool[0 0 … 0 0; 0 0 … 0 0; … ; 0 0 … 0 0; 0 0 … 0 0], 12, 12))

julia> st.cmat.cmat
12×24 Matrix{Bool}:
 0  0  1  1  1  1  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0
 0  0  0  0  0  0  0  0  0  0  0  0  0  0  1  1  1  1  0  0  0  0  0  0
 0  0  0  0  0  0  0  0  0  0  0  0  1  0  1  1  0  0  0  0  0  0  0  0
 0  0  0  0  0  0  0  0  0  0  0  0  0  1  1  0  1  0  0  0  0  0  0  0
 0  1  1  1  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0
 1  0  1  0  1  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0
 0  0  0  0  0  0  0  0  1  1  1  1  0  0  0  0  0  0  0  0  0  0  0  0
 0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  1  1  1  1
 0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  1  0  1  1  0  0
 0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  1  1  0  1  0
 0  0  0  0  0  0  0  1  1  1  0  0  0  0  0  0  0  0  0  0  0  0  0  0
 0  0  0  0  0  0  1  0  1  0  1  0  0  0  0  0  0  0  0  0  0  0  0  0

julia> # state with 2 legos, 0 leg

julia> st2 = State([lego, lego], Tuple{LegoLeg, LegoLeg}[])
State(Lego[Lego{6}(6, StaticArraysCore.SVector{6, QuantumLegos.PauliOps.SinglePauliOp}[pauliop("IIXXXX"), pauliop("IIZZZZ"), pauliop("ZIZZII"), pauliop("IZZIZI"), pauliop("IXXXII"), pauliop("XIXIXI")]), Lego{6}(6, StaticArraysCore.SVector{6, QuantumLegos.PauliOps.SinglePauliOp}[pauliop("IIXXXX"), pauliop("IIZZZZ"), pauliop("ZIZZII"), pauliop("IZZIZI"), pauliop("IXXXII"), pauliop("XIXIXI")])], Tuple{LegoLeg, LegoLeg}[], CheckMatrix(Bool[0 0 … 0 0; 0 0 … 0 0; … ; 0 0 … 0 0; 0 0 … 0 0], 12, 12))

julia> st == st2
true

2 Lego 1 edge state

julia> using QuantumLegos

julia> stabilizers = pauliop.(["IIXXXX", "IIZZZZ", "ZIZZII", "IZZIZI", "IXXXII", "XIXIXI"])
6-element Vector{StaticArraysCore.SVector{6, QuantumLegos.PauliOps.SinglePauliOp}}:
 pauliop("IIXXXX")
 pauliop("IIZZZZ")
 pauliop("ZIZZII")
 pauliop("IZZIZI")
 pauliop("IXXXII")
 pauliop("XIXIXI")

julia> lego = Lego(stabilizers)
Lego{6}(6, StaticArraysCore.SVector{6, QuantumLegos.PauliOps.SinglePauliOp}[pauliop("IIXXXX"), pauliop("IIZZZZ"), pauliop("ZIZZII"), pauliop("IZZIZI"), pauliop("IXXXII"), pauliop("XIXIXI")])

julia> state = State([lego, lego], edge.([((1, 3), (2, 3))]))
State(Lego[Lego{6}(6, StaticArraysCore.SVector{6, QuantumLegos.PauliOps.SinglePauliOp}[pauliop("IIXXXX"), pauliop("IIZZZZ"), pauliop("ZIZZII"), pauliop("IZZIZI"), pauliop("IXXXII"), pauliop("XIXIXI")]), Lego{6}(6, StaticArraysCore.SVector{6, QuantumLegos.PauliOps.SinglePauliOp}[pauliop("IIXXXX"), pauliop("IIZZZZ"), pauliop("ZIZZII"), pauliop("IZZIZI"), pauliop("IXXXII"), pauliop("XIXIXI")])], Tuple{LegoLeg, LegoLeg}[(LegoLeg(1, 3), LegoLeg(2, 3))], CheckMatrix(Bool[1 0 … 0 0; 0 1 … 0 0; … ; 0 0 … 1 1; 0 0 … 0 1], 10, 10))

julia> state.cmat
CheckMatrix with 10 generators, 10 legs:
  1  0  1  0  1  0  0  0  0  0 | 0  0  0  0  0  0  0  0  0  0
  0  1  0  1  1  0  0  0  0  0 | 0  0  0  0  0  0  0  0  0  0
  0  0  1  1  1  0  0  1  1  1 | 0  0  0  0  0  0  0  0  0  0
  0  0  0  0  0  1  0  1  0  1 | 0  0  0  0  0  0  0  0  0  0
  0  0  0  0  0  0  1  0  1  1 | 0  0  0  0  0  0  0  0  0  0
  0  0  0  0  0  0  0  0  0  0 | 1  0  0  1  1  0  0  0  0  0
  0  0  0  0  0  0  0  0  0  0 | 0  1  1  0  1  0  0  0  0  0
  0  0  0  0  0  0  0  0  0  0 | 0  0  1  1  1  0  0  1  1  1
  0  0  0  0  0  0  0  0  0  0 | 0  0  0  0  0  1  0  0  1  1
  0  0  0  0  0  0  0  0  0  0 | 0  0  0  0  0  0  1  1  0  1


julia> pg = state.cmat |> generators |> GeneratedPauliGroup
GeneratedPauliGroup{10}(StaticArraysCore.SVector{10, QuantumLegos.PauliOps.SinglePauliOp}[pauliop("XIXIXIIIII"), pauliop("IXIXXIIIII"), pauliop("IIXXXIIXXX"), pauliop("IIIIIXIXIX"), pauliop("IIIIIIXIXX"), pauliop("ZIIZZIIIII"), pauliop("IZZIZIIIII"), pauliop("IIZZZIIZZZ"), pauliop("IIIIIZIIZZ"), pauliop("IIIIIIZZIZ")], IterTools.Subsets{Vector{StaticArraysCore.SVector{N, QuantumLegos.PauliOps.SinglePauliOp} where N}}(StaticArraysCore.SVector{N, QuantumLegos.PauliOps.SinglePauliOp} where N[pauliop("XIXIXIIIII"), pauliop("IXIXXIIIII"), pauliop("IIXXXIIXXX"), pauliop("IIIIIXIXIX"), pauliop("IIIIIIXIXX"), pauliop("ZIIZZIIIII"), pauliop("IZZIZIIIII"), pauliop("IIZZZIIZZZ"), pauliop("IIIIIZIIZZ"), pauliop("IIIIIIZZIZ")]))

julia> pauliop("XIIXIXIIXI") in pg
true

Internal(how it works)

Notes on Overall flow

Details on ^[1]

state is translated to a single check matrix
- the size is ≤ $N \times 2N$ where $N$ is maximum number of lego logs.
any contraction can be performed on this single check matrix
if the check matrix can be represented as direct sum of matrices with $k N$ columns where $k ∈ ℕ$, then they are not contracted

Construction of State

Construction of State is completed by calling State constructor recursively.

Construct State without edge. Just adding legos. Checkmatrix is just a direct sum of each lego's checkmatrix
Concatenate each edges. During this operation, self tracing of checkmatrix is evaluated.

Each constructor calls action function (which is a map from State to State). Therefore, action functions can be used both for direct construction of State and action application to State during the game.

API

QuantumLegos.Lego — Type

Quantum lego with N legs.

Fields

nlegs::Int64: number of legs, equals N
stabgens::SVector{N, PauliOp{N}}: stabilizer generators. vector of PauliOp

Constructor

Lego([nlegs::Integer], stabgens::AbstractVector{PauliOp{N}})

Constructor for Lego. nlegs is optional (default is length of the first stabilizer generator).

Example

julia> stabgens = pauliop.(["II", "XX"])
2-element Vector{StaticArraysCore.SVector{2, QuantumLegos.PauliOps.SinglePauliOp}}:
 pauliop("II")
 pauliop("XX")

julia> Lego(stabgens)
Lego{2}(2, StaticArraysCore.SVector{2, QuantumLegos.PauliOps.SinglePauliOp}[pauliop("II"), pauliop("XX")])

source

QuantumLegos.LegoLeg — Type

mutable struct State

To be used in State.

Fields

lego_id::Int64: index in legos in State
edge_id::Int64: index in Lego in legos in State. No validation check included in LegoLeg.

Example

julia> x = LegoLeg.([(2, 1), (1, 1), (1, 0)])
3-element Vector{LegoLeg}:
 LegoLeg(2, 1)
 LegoLeg(1, 1)
 LegoLeg(1, 0)

julia> sort(x)
3-element Vector{LegoLeg}:
 LegoLeg(1, 0)
 LegoLeg(1, 1)
 LegoLeg(2, 1)

source

QuantumLegos.State — Type

mutable struct State

State (in p.4)

Fields

legos: Vector{Lego}
edges: Vector of ((lego_i, leg_n), (lego_j, leg_m)). Each element is sorted (i.e. lego_i < lego_j or lego_i == lego_j && leg_n < leg_m). This feature is used in is_connected_to_firstlego.
cmat::CheckMatrix: CheckMatrix

Constructor

State(legos::Vector{Lego{N}}, edges::Vector{Tuple{LegoLeg, LegoLeg}})

Methods with

add_lego!
add_edge!

Example

TODO

source

QuantumLegos._naive_distance — Method

Calculate distance. Use minimum distance of all generated normalizers.

source

QuantumLegos.add_edge! — Method

add_edge!(state::State, leg_1::LegoLeg, leg_2::LegoLeg)::State

Add a new edge between leg_1 and leg_2, updating state.

source

QuantumLegos.add_lego! — Method

add_lego!(state::State, lego::Lego) -> State

Add a new lego, updating state.

source

QuantumLegos.cmat_index — Method

cmat_index(state::State, leg::LegoLeg)::Int64

Get column index corresponds to leg in check matrix of state. If given leg is already connected, it throws ArgumentError. If given lego_id of leg is out of state.legos, throws ArgumentError.

source

QuantumLegos.distance — Method

distance(state::State) -> Int

Calculate code distance when the first leg of state is assigned as logical.

source

QuantumLegos.edge — Function

Helper function to create Tuple{LegoLeg, LegoLeg} to represent edge.

source

QuantumLegos.edge — Method

edge(t::Tuple{T, T, T, T}) where {T <: Integer}

source

QuantumLegos.edge — Method

edge(t::Tuple{T, T}) where {T <: Tuple{Integer, Integer}}

source

QuantumLegos.edge — Method

edge(x::T, y::T, z::T, w::T) where {T <: Integer}

source

QuantumLegos.is_connected_to_firstlego — Method

is_connected_to_firstlego(state::State)::BitVector

Returns vector which stores whether each lego is connected to the first lego.

source

1 C. Cao and B. Lackey, ‘Approximate Bacon-Shor code and holography’, J. High Energ. Phys., vol. 2021, no. 5, p. 127, May 2021, doi: 10.1007/JHEP05(2021)127.