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.
https://www.thofma.com/Hecke.jl/stable/
Some relevant functions from Hecke:
- Relative automorphism generators from
Hecke.automorphism_groupQQ - Defining ray class fields:
rcf = ray_class_field((iseven(d) ? 2*d : d)*OK,infinite_places(K))forK = quadratic_field((d-3)*(d+1))[1]andOK = maximal_order(K). number_field(rcf)computes generatorsabsolute_automorphism_group(rcf)of the full automorphism group and now works for alld.- TO DO: check that we can construct the ring ray class fields.
MapClassGrp: {quotient of the class group} -> {ideals}MapRayClassGrp: {quotient of a ray class group} -> {ideals prime to the conductor}ClassFieldClassField_ppCyclic class field of prime-power degreeartin_map(rcf)gives map from the ideal group to the set of automorphisms ofnumber_field(rcf)i.e. from aFacElemMon{Hecke.NfAbsOrdIdlSet{AnticNumberField, nf_elem}}to aHecke.NfMorSet{NfRelNS{nf_elem}}complex_conjugation(F,infplace)extends the Artin map to infinite places, giving complex conjugation in the corresponding complex embeddings.automorphism_group(rcf)gives a map from aGrpGento the set of automorphisms ofnumber_field(rcf)fixing the base, andinv(rcf.quotientmap)works.
- Group actions
- 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$N$ .
Let
- An
$R$ -lattice is a finitely generated torsion-free$R$ -module. - A fractional
$R$ -ideal$I$ of$B$ is a full-rank$R$ -sublattice.-
Left order
$O_\ell(I) = \{ b \in B : bI \subset I \}$ -
Right order
$O_r(I) = \{ b \in B : bI \subset I \}$ -
$I$ is an$O_\ell(I)-O_r(I)$ -bimodule, a left fractional$O_\ell(I)$ ideal and a right fractional$O_r(I)$ -ideal. - There is a category whose objects are the orders and whose morphisms are the fractional ideals, with composition given by proper products (requiring matching left and right orders). Fractional ideals connect their left/right orders.
-
Left order
-
$O\simeq O'$ iff$bOb^{-1} = O'$ for some$b \in B^\times$ iff they are connected by a principal ideal$bO = O'b$ . So the isomorphism classes are the conjugacy classes. - An invertible ideal of
$B$ is a fractional ideal$I$ with an inverse$I^{-1}$ , satisfying$I^{-1}I = O_R(I)$ and therefore$I I^{-1} = O_\ell(I)$ .- The orders and the invertible ideals form a groupoid called the core of the above category.
- A normal ideal of
$B$ is a fractional$R$ -ideal with maximal left/right orders.- normal
$\Rightarrow$ invertible. - The maximal orders and normal ideals of
$B$ form the Brandt groupoid.
- normal
- An integral ideal of
$B$ is a normal$R$ -ideal$I$ with$I \subset O_\ell(I) \cap O_r(I)$ . - A maximal left ideal
$I$ of$B$ is a maximal left ideal of$O_\ell(I)$ . - Each integral ideal
$I$ of$B$ can be factored into a proper product$I = M_1 M_2 \cdots M_n$ of maximal left ideals$M_i$ of$B$ with$O_r(M_i) = O_\ell(M_{i+1})$ for$1 \leq i < n$ , where$n$ is the length of$O_\ell(I)/I$ .
We need the following basic operations to compute with maximal or Eichler quaternion orders (see https://arxiv.org/abs/0808.3833):
- Check isomorphism of fractional ideals:
is_isomorphic(Oscar)IsIsomorphic(Magma), reduces tois_principal/IsPrincipal- Indefinite case: check image in ray class group mod the infinite ramified primes of
$B$ . - Definite case: Solve shortest lattice vector problem
- Indefinite case: check image in ray class group mod the infinite ramified primes of
- Compute connecting fractional ideals
I(OO,O)such that left ideal is OO and right ideal is O.
The main difficult tasks are the following:
-
Compute representatives for the conjugacy classes = types = isomorphism classes of orders
-
ConjugacyClasses(O)in Magma
-
-
Compute representatives for the 2-sided ideal class group
- Extends the class group of the base maximal order by square roots of ramified primes
-
TwoSidedIdealClassGroup(O)in Magma
-
Combining the previous two (KV 2.10) computes representatives
[J*I(OO,O) : OO in ConjugacyClasses(O), J in TwoSidedIdealClassGroup(OO)]for the set$\mathrm{Pic}_\ell(O)$ of left-equivalence classes of invertible right$O$ -ideals.-
RightIdealClasses(O)in Magma - Cardinality is the class number
-
-
Hecke does one split prime. Can we use
PolyMake.jlfor totally positive and more general?
- 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)