QuDirac.jl

QuDirac.jl is a Julia library for using Dirac notation to perform quantum mechanics computations.

Documentation for the current release version (v0.1) can be found here.

Installation

To install QuDirac.jl, you should have a working build of Julia v0.3. Then, you can grab QuDirac.jl via the package manager:

julia> Pkg.add("QuDirac")

Features

These are toy examples for demoing features. See below for more involved examples.

Ket, Bra, and Operator types

julia> bell = d" 1/√2 * (| 0,0 > + | 1,1 >) "
Ket{KroneckerDelta,2,Float64} with 2 state(s):
  0.7071067811865475 | 0,0 ⟩
  0.7071067811865475 | 1,1 ⟩

julia> bell'
Bra{KroneckerDelta,2,Float64} with 2 state(s):
  0.7071067811865475 ⟨ 0,0 |
  0.7071067811865475 ⟨ 1,1 |

julia> ptrace(bell * bell', 1)
OpSum{KroneckerDelta,1,Float64} with 2 operator(s):
  0.4999999999999999 | 0 ⟩⟨ 0 |
  0.4999999999999999 | 1 ⟩⟨ 1 |

Support for undefined inner products

# tells QuDirac to use the rule for undefined inner products
julia> default_inner(UndefinedInner())
INFO: QuDirac's default inner product type is currently UndefinedInner()

julia> d" < 0,0 | *  (| 0,0 > + | 1,1 >)/√2 "
((⟨ 0,0 | 0,0 ⟩ + ⟨ 0,0 | 1,1 ⟩) / 1.4142135623730951)

julia> s = d" (e^( < 1,2 | 3,4 > ) + < 5,6 | 7,8 > * im)^4 "
(((exp(⟨ 1,2 | 3,4 ⟩)) + (⟨ 5,6 | 7,8 ⟩ * im))^4)

julia> inner_eval((b, k) -> sum(k) - sum(b), s)
8.600194553751864e6 + 2.5900995362955774e6im

Custom inner product rules

julia> immutable MyInner <: AbstractInner end

julia> QuDirac.inner_rule(::MyInner, ktlabel, brlabel) = sqrt(ktlabel[1]+brlabel[1])
inner_rule (generic function with 3 methods)

julia> default_inner(MyInner())
INFO: QuDirac's default inner product type is currently MyInner()

# eval ⟨ π | e ⟩ with MyInner rule -> sqrt(π + e)
julia> d" < π | e > "
2.420717761749361

Functional operator construction

# define a₂ on Ket
julia> @def_op " a₂ | x, y, z > = √y * | x, y - 1, z > "
a₂ (generic function with 1 method)

# define a₂ on Bra
julia> @def_op " < x, y, z | a₂ = √(y + 1) * < x, y + 1, z | "
a₂ (generic function with 2 methods)

julia> d" a₂ * | 3,5,5 > "
Ket{KroneckerDelta,3,Float64} with 1 state(s):
  2.23606797749979 | 3,4,5 ⟩

julia> d" a₂' * | 3,4,5 > "
Ket{KroneckerDelta,3,Float64} with 1 state(s):
  2.23606797749979 | 3,5,5 ⟩

julia> d" < 3,4,5 | * a₂ * | 3,5,5 > "
2.23606797749979

# Hadamard operator
julia> @rep_op " H | n > = 1/√2 * ( | 0 > + (-1)^n *| 1 > ) " 0:1;

julia> H
OpSum{KroneckerDelta,1,Float64} with 4 operator(s):
  0.7071067811865475 | 1 ⟩⟨ 0 |
  0.7071067811865475 | 0 ⟩⟨ 0 |
  0.7071067811865475 | 0 ⟩⟨ 1 |
  -0.7071067811865475 | 1 ⟩⟨ 1 |

...and other stuff

Implementation of common operations like partial trace (ptrace) and partial transpose (ptranspose)
Treat states and operators as map-like data structures, enabling label-based analysis for spectroscopy purposes
xsubspace allows easy selection of excitation subspaces of states and operators
permute and switch allows generic permutation of factor labels for states
filter/filter! for the filtering out component states/operators via predicate functions
Arbitrary mapping functions (map/maplabels/mapcoeffs) for applying functions to labels and coefficients

Examples

There are currently two example files, qho.jl and randwalk.jl. The former implements methods for plotting quantum harmonic oscillator wave functions using Plotly. The latter is a simple implementation of a quantum random walk.

To run the examples, one can do the following (using qho.jl as an example):

julia> cd(Pkg.dir("QuDirac"))
julia> include("examples/qho.jl")