Experimental integrated open-source Julia/Oscar/Nemo/Hecke/GAP/Polymake/ANTIC workflow for quantum computing research as alternative to Python/Magma/Sage/Pari. Heavily using Oscar.jl which in turn wraps GAP.jl for group theory, Polymake.jl for polyhedral geometry and Singular.jl for algebraic geometry and invariant theory. Oscar incorporates Hecke.jl for computational algebraic number theory and class field theory, which wraps ANTIC for fast number theory computations in C.
julia> using Oscar, QP
julia> ZN(3)
Integers modulo 3
julia> Z3
Integers modulo 3
julia> Z3[4 2]
[1 2]
julia> ket(Z3[0 0]) + ket(Z3[1 1]) + ket(Z3[2 2])
[1]
[0]
[0]
[0]
[1]
[0]
[0]
[0]
[1]
julia> heis(Z3[4 2])
[ 0 0 -z_3 - 1]
[z_3 0 0]
[ 0 1 0]
julia> weil_U(Z5[2 1; 0 3])
[-1 0 0 0 0]
[ 0 0 -z_5 0 0]
[ 0 0 0 0 z_5^3 + z_5^2 + z_5 + 1]
[ 0 z_5^3 + z_5^2 + z_5 + 1 0 0 0]
[ 0 0 0 -z_5 0]
julia> X = gen(matrix_polynomial_ring(QQ,8))
[X_{0,0} X_{0,1} X_{0,2} X_{0,3} X_{0,4} X_{0,5} X_{0,6} X_{0,7}]
[X_{1,0} X_{1,1} X_{1,2} X_{1,3} X_{1,4} X_{1,5} X_{1,6} X_{1,7}]
[X_{2,0} X_{2,1} X_{2,2} X_{2,3} X_{2,4} X_{2,5} X_{2,6} X_{2,7}]
[X_{3,0} X_{3,1} X_{3,2} X_{3,3} X_{3,4} X_{3,5} X_{3,6} X_{3,7}]
[X_{4,0} X_{4,1} X_{4,2} X_{4,3} X_{4,4} X_{4,5} X_{4,6} X_{4,7}]
[X_{5,0} X_{5,1} X_{5,2} X_{5,3} X_{5,4} X_{5,5} X_{5,6} X_{5,7}]
[X_{6,0} X_{6,1} X_{6,2} X_{6,3} X_{6,4} X_{6,5} X_{6,6} X_{6,7}]
[X_{7,0} X_{7,1} X_{7,2} X_{7,3} X_{7,4} X_{7,5} X_{7,6} X_{7,7}]
julia> tr(X^2)
X_{0,0}^2 + 2*X_{0,1}*X_{1,0} + 2*X_{0,2}*X_{2,0} + 2*X_{0,3}*X_{3,0} + 2*X_{0,4}*X_{4,0} + 2*X_{0,5}*X_{5,0} + 2*X_{0,6}*X_{6,0} + 2*X_{0,7}*X_{7,0} + X_{1,1}^2 + 2*X_{1,2}*X_{2,1} + 2*X_{1,3}*X_{3,1} + 2*X_{1,4}*X_{4,1} + 2*X_{1,5}*X_{5,1} + 2*X_{1,6}*X_{6,1} + 2*X_{1,7}*X_{7,1} + X_{2,2}^2 + 2*X_{2,3}*X_{3,2} + 2*X_{2,4}*X_{4,2} + 2*X_{2,5}*X_{5,2} + 2*X_{2,6}*X_{6,2} + 2*X_{2,7}*X_{7,2} + X_{3,3}^2 + 2*X_{3,4}*X_{4,3} + 2*X_{3,5}*X_{5,3} + 2*X_{3,6}*X_{6,3} + 2*X_{3,7}*X_{7,3} + X_{4,4}^2 + 2*X_{4,5}*X_{5,4} + 2*X_{4,6}*X_{6,4} + 2*X_{4,7}*X_{7,4} + X_{5,5}^2 + 2*X_{5,6}*X_{6,5} + 2*X_{5,7}*X_{7,5} + X_{6,6}^2 + 2*X_{6,7}*X_{7,6} + X_{7,7}^2
julia> partial_trace(X,[2,2,2],[1,3])
[X_{0,0} + X_{1,1} + X_{4,4} + X_{5,5} X_{0,2} + X_{1,3} + X_{4,6} + X_{5,7}]
[X_{2,0} + X_{3,1} + X_{6,4} + X_{7,5} X_{2,2} + X_{3,3} + X_{6,6} + X_{7,7}]
julia> reduced_operator(X,[2,2,2],[2])
[X_{0,0} + X_{1,1} + X_{4,4} + X_{5,5} X_{0,2} + X_{1,3} + X_{4,6} + X_{5,7}]
[X_{2,0} + X_{3,1} + X_{6,4} + X_{7,5} X_{2,2} + X_{3,3} + X_{6,6} + X_{7,7}]
julia> sic(3)
9-element Vector{AbstractAlgebra.Generic.MatSpaceElem{nf_elem}}:
[0 0 0; 0 1 -1; 0 -1 1]
[0 0 0; 0 1 -z_3; 0 z_3+1 1]
[0 0 0; 0 1 z_3+1; 0 -z_3 1]
[1 -1 0; -1 1 0; 0 0 0]
[1 -z_3 0; z_3+1 1 0; 0 0 0]
[1 z_3+1 0; -z_3 1 0; 0 0 0]
[1 0 -1; 0 0 0; -1 0 1]
[1 0 z_3+1; 0 0 0; -z_3 0 1]
[1 0 -z_3; 0 0 0; z_3+1 0 1]
julia> 2*fiducial("7b")[1,:]
7-element Vector{Hecke.RelNonSimpleNumFieldElem{AbsSimpleNumFieldElem}}:
2
-_$1 - sqrt(2) - 1
-_$1 - sqrt(2) - 1
_$1 - sqrt(2) - 1
-_$1 - sqrt(2) - 1
_$1 - sqrt(2) - 1
_$1 - sqrt(2) - 1
julia> SicData(5)
Constructing number field
Finding complex conjugation
SicData(5, 12, 12, 1, Polynomial ring in 5×5 variables X_{⋅,⋅} over Rational field, Real quadratic field defined by x^2 - 3, InfPlc[Infinite place corresponding to (Complex embedding corresponding to -1.73 of real quadratic field), Infinite place corresponding to (Complex embedding corresponding to 1.73 of real quadratic field)], Maximal order of Real quadratic field defined by x^2 - 3
with basis AbsSimpleNumFieldElem[1, sqrt(3)], sqrt(3) + 2, Order of Real quadratic field defined by x^2 - 3
with Z-basis AbsSimpleNumFieldOrderElem[1, -sqrt(3) + 6], Order of Real quadratic field defined by x^2 - 3
with Z-basis AbsSimpleNumFieldOrderElem[1, -sqrt(3) + 2], -sqrt(3) + 2, Class field defined mod (<5, 5>, InfPlc{AbsSimpleNumField, AbsSimpleNumFieldEmbedding}[Infinite place corresponding to (Complex embedding corresponding to -1.73 of real quadratic field), Infinite place corresponding to (Complex embedding corresponding to 1.73 of real quadratic field)]) of structure Z/2 x Z/8, Non-simple number field of degree 16 over real quadratic field, Map: non-simple number field -> non-simple number field, #undef, #undef, #undef)
Some julia tips can be found here.
Supported in part by the NSERC Discovery under Grant No. RGPIN-2018-04742, the NSERC/EU project FoQaCiA under Grant No. ALLRP-569582-21, the Institute for Quantum Computing and the Perimeter Institute for Theoretical Physics.
- Rings of polynomial functions on matrices over number fields
- Explicit computing in bases (vector, operator, etc)
- Useful matrices and tensor products
- Multiqudit bases parameterized by elements of finite cyclic rings (
nmodzzModRingElem), of modules over them (nmod_matzzModMatrix) or of finite abelian groups (FinGenAbGroupElem) - Single-qudit generalized Paulis in all dimensions
- Qudit generalized Clifford = Weil representation in prime dimensions
- Projective linear groups
- Computing properties of SIC-POVMs
SicData(d)(orSicData(d,build_nf=false)to skip building the number field)fiducial(d)andsic(d)ford=2,3,3,4,5,7as well asfiducial("7a"),fiducial("7b)etc.- Class field thery in Hecke
- Group actions in GAP/Oscar
- Arithmetic of quantum circuits
- Characteristic and Wigner functions
- Generalize Weil to composite
$N$ - Sort out even case
-
Fix this Oscar issue to implement$\mathrm{SL}(2,\mathbb{Z}/N)$ for all$N$ .
- Stabilizer/graph states
- Parametric models / exponential families
- Solving pentagon and hexagon equations
- Ross-Selinger and extensions
- Schur transform
- Abstract framework for modeling quantum systems
- Implementation-agnostic (native julia, arb, FLINT, ..., TensorFlow)