PauliOperators

Simple package providing bitstring representations of tensor products of Pauli operators for efficient manipulations. In this representation, the Pauli string operator is represented as two binary strings, one for x and one for z.

The format used for an arbitrary Pauli is as follows:

$$ \begin{align} P_n =& i^\theta Z^{z_1} X^{x_1} ⊗ Z^{z_2} X^{x_2} ⊗ ⋯ ⊗ Z^{z_N} X^{x_N} \ =& i^\theta \bigotimes_j Z^{z_j} X^{x_j} \end{align} $$ where the $z$ and $x$ strings are encoded as the bitwise representation of an integer. Currently, we use Int128 integers so that have access to relatively large pauli strings, i.e., up to 128 qubits. Using only $X$ and $Z$ operators gives access to $Y$ (or rather $iY$) by simply setting both $x_i$ and $z_i$ to 1.

Types

This package provides the following types:

• FixedPhasePauli, which contains only the base operator string. Here, N is the number of qubits (currently maxed at 128).

struct FixedPhasePauli{N} <: AbstractPauli{N}
    z::Int128
    x::Int128
end

• Pauli, which includes both the base operator string, as well as a phase, $\theta$. This allows us to multiply and such, while keep track of the phases.

struct Pauli{N} <: AbstractPauli{N}
    θ::UInt8
    pauli::FixedPhasePauli{N}
end

• ScaledPauli. This allows us to describe a Pauli scaled by an arbitrary Float. We may want to parameterize the type here in the future.

struct ScaledPauli{N} <: AbstractPauli{N}
    coeff::ComplexF64
    pauli::FixedPhasePauli{N}
end

• PauliSum which provides a way to define a sum of ScaledPauli's. While there are multiple ways one might choose to do this, here, we use a dictionary, where the FixedPhasePauli is the key, and the value is the coefficient.

struct PauliSum{N}  
    ops::Dict{FixedPhasePauli{N},ComplexF64}
end

• ScaledPauliVector is simply an alias to a vector of ScaledPauli's. This also represents a sum of ScaledPauli's.

ScaledPauliVector{N} = Vector{ScaledPauli{N}}

• KetBitString is a simple bitstring representation of a computational basis state. This is provided to give access to computing things like expectation values and just general wavefunctions. p

"""
An occupation number vector, up to 128 qubits
"""
struct KetBitString{N} 
    v::Int128
end

Functions

The following functions are overloaded to allow the objects to interact intuitively:

• *, ⊗, ⊕, +

Addition needs a bit more specification, since, unlike *, a sum of Pauli's is not a Pauli. As such, we simply choose to return a PauliSum type when adding two AbstractPauli's.