add linear algebra and documentation (#61)

* add linalg and docs

* remove space

* doc fix

* typo fix in docs

* Update docs/src/introduction.md

Co-authored-by: Stefan Krastanov <[email protected]>

* updating docs to support autogenerated API

* superop rearranging

* intro.md change

* update changelog and project.toml

---------

Co-authored-by: Stefan Krastanov <[email protected]>

CHANGELOG.md

@@ -1,9 +1,11 @@
 # News
 
-## v0.3.4 - dev
+## v0.3.4 - 2024-07-22
 
 - Added `tr` and `ptrace` functionalities.
 - New symbolic superoperator representations.
+- Added linear algebra operations `exp`, `vec`, `conj`, `transpose`.
+- Created a getting-started guide in docs.
 
 ## v0.3.3 - 2024-07-12
 

Project.toml

@@ -1,7 +1,7 @@
 name = "QuantumSymbolics"
 uuid = "efa7fd63-0460-4890-beb7-be1bbdfbaeae"
 authors = ["QuantumSymbolics.jl contributors"]
-version = "0.3.4-dev"
+version = "0.3.4"
 
 [deps]
 Latexify = "23fbe1c1-3f47-55db-b15f-69d7ec21a316"

docs/make.jl

@@ -4,8 +4,7 @@ push!(LOAD_PATH,"../src/")
 using Documenter
 using DocumenterCitations
 using QuantumSymbolics
-using QuantumOptics
-using QuantumClifford
+using QuantumInterface
 
 DocMeta.setdocmeta!(QuantumSymbolics, :DocTestSetup, :(using QuantumSymbolics, QuantumOptics, QuantumClifford); recursive=true)
 
@@ -25,8 +24,9 @@ function main()
     authors = "Stefan Krastanov",
     pages = [
         "QuantumSymbolics.jl" => "index.md",
-        "Qubit Basis Choice" => "qubit_basis.md",
+        "Getting Started with QuantumSymbolics.jl" => "introduction.md",
         "Express Functionality" => "express.md",
+        "Qubit Basis Choice" => "qubit_basis.md",
         "API" => "API.md",
     ]
     )

docs/src/index.md

@@ -202,4 +202,4 @@ express(MixedState(X1)/2+SProjector(Z1)/2, CliffordRepr())
 
 !!! warning "Stabilizer state expressions"
 
-    The state written as $\frac{|Z₁⟩⊗|Z₁⟩+|Z₂⟩⊗|Z₂⟩}{√2}$ is a well known stabilizer state, namely a Bell state. However, automatically expressing it as a stabilizer is a prohibitively expensive computational operation in general. We do not perform that computation automatically. If you want to ensure that states you define can be automatically converted to tableaux for Clifford simulations, avoid using summation of kets. On the other hand, in all of our Clifford Monte-Carlo simulations, `⊗` is fully supported, as well as [`SProjector`](@ref), [`MixedState`](@ref), [`StabilizerState`](@ref), and summation of density matrices.
+    The state written as $\frac{|Z₁⟩⊗|Z₁⟩+|Z₂⟩⊗|Z₂⟩}{√2}$ is a well known stabilizer state, namely a Bell state. However, automatically expressing it as a stabilizer is a prohibitively expensive computational operation in general. We do not perform that computation automatically. If you want to ensure that states you define can be automatically converted to tableaux for Clifford simulations, avoid using summation of kets. On the other hand, in all of our Clifford Monte-Carlo simulations, `⊗` is fully supported, as well as [`projector`](@ref), [`MixedState`](@ref), [`StabilizerState`](@ref), and summation of density matrices.

docs/src/introduction.md

@@ -0,0 +1,251 @@
+# Getting Started with QuantumSymbolics.jl
+
+```@meta
+DocTestSetup = quote
+    using QuantumSymbolics
+end
+```
+
+QuantumSymbolics is designed for manipulation and numerical translation of symbolic quantum objects. This tutorial introduces basic features of the package.
+
+## Installation
+
+QuantumSymbolics.jl can be installed through the Julia package system in the standard way:
+
+```
+using Pkg
+Pkg.add("QuantumSymbolics")
+```
+
+## Literal Symbolic Quantum Objects
+
+Basic objects of type [`SBra`](@ref), [`SKet`](@ref), [`SOperator`](@ref), and [`SSuperOperator`](@ref) represent symbolic quantum objects with `name` and `basis` properties. Each type can be generated with a straightforward macro:
+
+```jldoctest
+julia> using QuantumSymbolics
+
+julia> @bra b # object of type SBra
+⟨b|
+
+julia> @ket k # object of type SKet
+|k⟩
+
+julia> @op A # object of type SOperator
+A
+
+julia> @superop S # object of type SSuperOperator
+S
+```
+
+By default, each of the above macros defines a symbolic quantum object in the spin-1/2 basis. One can simply choose a different basis, such as the Fock basis or a tensor product of several bases, by passing an object of type `Basis` to the second argument in the macro call:
+
+```jldoctest
+julia> @op B FockBasis(Inf, 0.0)
+B
+
+julia> basis(B)
+Fock(cutoff=Inf)
+
+julia> @op C SpinBasis(1//2)⊗SpinBasis(5//2)
+C
+
+julia> basis(C)
+[Spin(1/2) ⊗ Spin(5/2)]
+```
+Here, we extracted the basis of the defined symbolic operators using the `basis` function.
+
+Symbolic quantum objects with additional properties can be defined, such as a Hermitian operator, or the zero ket (i.e., a symbolic ket equivalent to the zero vector $\bm{0}$).
+
+## Basic Operations
+
+Expressions containing symbolic quantum objects can be built with a variety of functions. Let us consider the most fundamental operations: multiplication `*`, addition `+`, and the tensor product `⊗`. 
+
+We can multiply, for example, a ket by a scalar value, or apply an operator to a ket:
+
+```jldoctest
+julia> @ket k; @op A;
+
+julia> 2*k
+2|k⟩
+
+julia> A*k
+A|k⟩
+```
+
+Similar scaling procedures can be performed on bras and operators. Addition between symbolic objects is also available, for instance:
+
+```jldoctest
+julia> @op A₁; @op A₂;
+
+julia> A₁+A₂
+(A₁+A₂)
+
+julia> @bra b;
+
+julia> 2*b + 5*b
+7⟨b|
+```
+Built into the package are straightforward automatic simplification rules, as shown in the last example, where `2⟨b|+5⟨b|` evaluates to `7⟨b|`. 
+
+Tensor products of symbolic objects can be performed, with basis information transferred:
+
+```jldoctest
+julia> @ket k₁; @ket k₂;
+
+julia> tp = k₁⊗k₂
+|k₁⟩|k₂⟩
+
+julia> basis(tp)
+[Spin(1/2) ⊗ Spin(1/2)]
+```
+
+Inner and outer products of bras and kets can be generated:
+
+```jldoctest
+julia> @bra b; @ket k;
+
+julia> b*k
+⟨b||k⟩
+
+julia> k*b
+|k⟩⟨b|
+```
+
+More involved combinations of operations can be explored. Here are few other straightforward examples:
+
+```jldoctest
+julia> @bra b; @ket k; @op A; @op B;
+
+julia> 3*A*B*k
+3AB|k⟩
+
+julia> A⊗(k*b + B)
+(A⊗(B+|k⟩⟨b|))
+
+julia> A-A
+𝟎
+```
+In the last example, a zero operator, denoted `𝟎`, was returned by subtracting a symbolic operator from itself. Such an object is of the type [`SZeroOperator`](@ref), and similar objects [`SZeroBra`](@ref) and [`SZeroKet`](@ref) correspond to zero bras and zero kets, respectively.
+
+## Linear Algebra on Bras, Kets, and Operators
+
+QuantumSymbolics supports a wide variety of linear algebra on symbolic bras, kets, and operators. For instance, the commutator and anticommutator of two operators, can be generated:
+
+```jldoctest
+julia> @op A; @op B;
+
+julia> commutator(A, B)
+[A,B]
+
+julia> anticommutator(A, B)
+{A,B}
+
+julia> commutator(A, A)
+𝟎
+```
+Or, one can take the dagger of a quantum object with the [`dagger`](@ref) function:
+
+```jldoctest
+julia> @ket k; @op A; @op B;
+
+julia> dagger(A)
+A†
+
+julia> dagger(A*k)
+|k⟩†A†
+
+julia> dagger(A*B)
+B†A†
+```
+Below, we state all of the supported linear algebra operations on quantum objects:
+
+- commutator of two operators: [`commutator`](@ref),
+- anticommutator of two operators: [`anticommutator`](@ref),
+- complex conjugate: [`conj`](@ref),
+- transpose: [`transpose`](@ref),
+- projection of a ket: [`projector`](@ref),
+- adjoint or dagger: [`dagger`](@ref),
+- trace: [`tr`](@ref),
+- partial trace: [`ptrace`](@ref),
+- inverse of an operator: [`inv`](@ref),
+- exponential of an operator: [`exp`](@ref),
+- vectorization of an operator: [`vec`](@ref).
+
+## Simplifying Expressions
+
+For predefined objects such as the Pauli operators [`X`](@ref), [`Y`](@ref), and [`Z`](@ref), additional simplification can be performed with the [`qsimplify`](@ref) function. Take the following example:
+
+```jldoctest
+julia> qsimplify(X*Z)
+(0 - 1im)Y
+```
+
+Here, we have the relation $XZ = -iY$, so calling [`qsimplify`](@ref) on the expression `X*Z` will rewrite the expression as `-im*Y`.
+
+Note that simplification rewriters used in QuantumSymbolics are built from the interface of [`SymbolicUtils.jl`](https://github.com/JuliaSymbolics/SymbolicUtils.jl). By default, when called on an expression, [`qsimplify`](@ref) will iterate through every defined simplification rule in the QuantumSymbolics package until the expression can no longer be simplified. 
+
+Now, suppose we only want to use a specific subset of rules. For instance, say we wish to simplify commutators, but not anticommutators. Then, we can pass the keyword argument `rewriter=qsimplify_commutator` to [`qsimplify`](@ref), as done in the following example:
+
+```jldoctest
+julia> qsimplify(commutator(X, Y), rewriter=qsimplify_commutator)
+(0 + 2im)Z
+
+julia> qsimplify(anticommutator(X, Y), rewriter=qsimplify_commutator)
+{X,Y}
+```
+As shown above, we apply [`qsimplify`](@ref) to two expressions: `commutator(X, Y)` and `anticommutator(X, Y)`. We specify that only commutator rules will be applied, thus the first expression is rewritten to `(0 + 2im)Z` while the second expression is simply returned. This feature can greatly reduce the time it takes for an expression to be simplified.
+
+Below, we state all of the simplification rule subsets that can be passed to [`qsimplify`](@ref):
+
+- `qsimplify_pauli` for Pauli multiplication,
+- `qsimplify_commutator` for commutators of Pauli operators,
+- `qsimplify_anticommutator` for anticommutators of Pauli operators.
+
+## Expanding Expressions
+
+Symbolic expressions containing quantum objects can be expanded with the [`qexpand`](@ref) function. We demonstrate this capability with the following examples.
+
+```jldoctest
+julia> @op A; @op B; @op C;
+
+julia> qexpand(A⊗(B+C))
+((A⊗B)+(A⊗C))
+
+julia> qexpand((B+C)*A)
+(BA+CA)
+
+julia> @ket k₁; @ket k₂; @ket k₃;
+
+julia> qexpand(k₁⊗(k₂+k₃))
+(|k₁⟩|k₂⟩+|k₁⟩|k₃⟩)
+
+julia> qexpand((A*B)*(k₁+k₂))
+(AB|k₁⟩+AB|k₂⟩)
+```
+
+## Numerical Translation of Symbolic Objects
+
+Symbolic expressions containing predefined objects can be converted to numerical representations with [`express`](@ref). Numerics packages supported by this translation capability are [`QuantumOptics.jl`](https://github.com/qojulia/QuantumOptics.jl) and [`QuantumClifford.jl`](https://github.com/QuantumSavory/QuantumClifford.jl/).
+
+By default, [`express`](@ref) converts an object to the quantum optics state vector representation. For instance, we can represent the exponential of the Pauli operator [`X`](@ref) numerically as follows:
+
+```jldoctest
+julia> using QuantumOptics
+
+julia> express(exp(X))
+Operator(dim=2x2)
+  basis: Spin(1/2)sparse([1, 2, 1, 2], [1, 1, 2, 2], ComplexF64[1.5430806327160496 + 0.0im, 1.1752011684303352 + 0.0im, 1.1752011684303352 + 0.0im, 1.5430806327160496 + 0.0im], 2, 2)
+```
+
+To convert to the Clifford representation, an instance of `CliffordRepr` must be passed to [`express`](@ref). For instance, we can represent the projection of the basis state [`X1`](@ref) of the Pauli operator [`X`](@ref) as follows:
+
+```jldoctest
+julia> using QuantumClifford
+
+julia> express(projector(X1), CliffordRepr())
+𝒟ℯ𝓈𝓉𝒶𝒷
++ Z
+𝒮𝓉𝒶𝒷
++ X
+```
+For more details on using [`express`](@ref), refer to the [express functionality page](@ref express).

src/QSymbolicsBase/QSymbolicsBase.jl

@@ -6,7 +6,7 @@ using TermInterface
 import TermInterface: isexpr,head,iscall,children,operation,arguments,metadata,maketerm
 
 using LinearAlgebra
-import LinearAlgebra: eigvecs,ishermitian,inv
+import LinearAlgebra: eigvecs,ishermitian,conj,transpose,inv,exp,vec,tr
 
 import QuantumInterface:
     apply!,
@@ -23,7 +23,7 @@ export SymQObj,QObj,
        apply!,
        express,
        tensor,⊗,
-       dagger,projector,commutator,anticommutator,tr,ptrace,
+       dagger,projector,commutator,anticommutator,conj,transpose,inv,exp,vec,tr,ptrace,
        I,X,Y,Z,σˣ,σʸ,σᶻ,Pm,Pp,σ₋,σ₊,
        H,CNOT,CPHASE,XCX,XCY,XCZ,YCX,YCY,YCZ,ZCX,ZCY,ZCZ,
        X1,X2,Y1,Y2,Z1,Z2,X₁,X₂,Y₁,Y₂,Z₁,Z₂,L0,L1,Lp,Lm,Lpi,Lmi,L₀,L₁,L₊,L₋,L₊ᵢ,L₋ᵢ,
@@ -35,8 +35,8 @@ export SymQObj,QObj,
        SScaled,SScaledBra,SScaledOperator,SScaledKet,
        STensorBra,STensorKet,STensorOperator,
        SZeroBra,SZeroKet,SZeroOperator,
+       SConjugate,STranspose,SProjector,SDagger,SInvOperator,SExpOperator,SVec,STrace,SPartialTrace,
        MixedState,IdentityOp,
-       SProjector,SDagger,STrace,SPartialTrace,SInvOperator,
        SApplyKet,SApplyBra,SMulOperator,SSuperOpApply,SCommutator,SAnticommutator,SBraKet,SOuterKetBra,
        HGate,XGate,YGate,ZGate,CPHASEGate,CNOTGate,
        XBasisState,YBasisState,ZBasisState,