coq-quantum / CoqQ

Compiling the development

We depend on the following external libraries:

  "coq"                      {>= "8.13" & < "8.16~" }
  "dune"                     {>= "3.2"              }
  "coq-mathcomp-ssreflect"   {>= "1.14" & < "1.15~" }
  "coq-mathcomp-algebra"     {>= "1.14" & < "1.15~" }
  "coq-mathcomp-fingroup"    {>= "1.14" & < "1.15~" }
  "coq-mathcomp-analysis"    { = "0.5.2"            }
  "coq-mathcomp-real-closed" {>= "1.1"  & < "1.2~"  }
  "coq-mathcomp-finmap"      {>= "1.5"  & < "1.6~"  }
  "coq-elpi"                 {>= "1.13" & < "1.14~" }

They can be installed in a local opam switch by executing:

opam switch create \
    --yes \
    --deps-only \
    --packages=coq-mathcomp-analysis=0.5.2,coq-mathcomp-fingroup=1.14.0 \
    --repositories=default=https://opam.ocaml.org,coq-released=https://coq.inria.fr/opam/released \
    .

Then, you can compile the development by just typing make (or opam config exec -- make if you used a local opam switch to install the dependencies).

Axioms present in the develoment

Our development is made assuming the informative excluded middle and functional extensionality. The axioms are not explicitly stated in our development but inherited from mathcomp analysis.

Comments for each file

xvector.v : extra of mathcomp/algebra/vector.v

mcextra.v : extra of mathcomp and mathcomp-real_closed

setdec.v : a prove-by-reflection tactic for efficient automated reasoning about set theory goals based on the tableau decision procedure in [Anisimov 2015].

mxpred.v : predicate for matrix and their hierarchy theory.

cpo.v : module for complete partial order.

orthomodular.v : module for orthomodular lattice

hermitian.v : define the Hermitian space and its instance chsType -- hermitian type with a orthonormal canonical basis. define and prove correct the Gram–Schmidt process that allows the orthonormalization a set of vectors w.r.t. an inner product.

prodvect.v : variant of dependent finite function.

tensor.v : define the tensor product over a family of Hermitian space based on their bases. define multi-linear maps. prove that the tensor produce of Hermitian/chsType is still a Hermitian/chsType with inner product consistent with each components. For a given L, define Hilbert space for any subsystem S $\subseteq$ L.

mxtopology.v : topology of complex number and matrix/finite vector space over complex number. modules for vector norm, vector order, finite dimensional normed module type (equipped with a vector order). prove the Bolzano-Weierstrass theorem, the equivalence of vector norms, the monotone convergence theorem for vector space w.r.t. arbitrary vector order with closed condition.

mxnorm.v : define matrix norm includes l1norm/l2norm (entry-wise 1/2-norm), i2norm (induced 2-norm), trnorm (trace/nuclear norm/schatten 1 norm), fbnorm (Frobenius norm/schatten 2 norm). provide singular value decomposition. define Lowner order and show density matrix forms a cpo w.r.t. Lowner order.

lfundef.v : define the tensor product and outer product of vectors, tensor product and general composition of linear functions.

quantum.v : define most of the basic concept of quantum mechanics based on linear function representation (lfun). concepts includes: hermitian/positive-semidefinite/density/observable/projection/unitary linear operators, super-operators and its topology/tensor product, (partial) orthonormal basis, normalized state, trace-nonincreasing / trace-preserving (quantum measurement) maps, completely-positive super-operators (CP, via choi matrix theory), quantum operation (QO), quantum channel (QC). basic constructs of super-operator (initialization, unitary transformation, if and while, dual super-operator) and their canonical structure to CP/QO/QC. define the cylindrical extension (lifting to a larger space).

hspace.v : subspace theory based on projection representation; i.e., the theory of projection lattice.

dirac.v : labelled Dirac notation, defined as a non-dependent type and have linear algebraic structure. using canonical structures to trace the domain and codomain of a labelled Dirac notation.

inhabited.v : define inhabited finite type (ihbFinType), Hilbert space associated to a ihbFinType, tensor product of state/operator in/on associated Hilbert space (for pair, tuple, finite function and dependent finite function)

qwhile.v : define abstract syntax of qwhile as inductive type, semantics, valid judgments, quantum variable, concrete syntax. more rewrite rules for labelled Dirac notation with quantum variable/data type. Validity is a mild extension of [D’Hondt and Panangaden 2006, Ying 2011].

qtype.v : utility of quantum data type; includes common 1/2-qubit gates, multiplexer, quantum Fourier bases/transformation, (phase) oracle (i.e., quantum access to a classical function) etc.

qhl.v : formalize quantum Hoare logic; includes rules for basic constructs and structure rules, frame rules and parallel rules, together with several useful rules such as (R.Inner) that has never been proposed before. Most of the rules are inspired from [Ying 2011, 2019] [Ying et al . 2018, 2022] and adopted with minor changes to allow all linear functions as assertions rather than restricted to quantum predicates (effect).

example.v : a representative set of case studies, including HHL algorithm for solving linear equations [Harrow et al. 2009], Grover search algorithm [Grover 1996], quantum phase estimation (QPE) and the hidden subgroup problem (HSP) algorithm [Kitaev 1995; Lomont 2004], together with the circuit implementation of parallel Hadamard, reverse circuit, quantum Fourier transformation (QFT) and Bravyi-Gosset-Konig’s algorithm for hidden linear function (HLF) problem [Bravyi et al . 2018].