Vecs2Pauli: software for converting between quantum-state vectors and the (Pauli) stabilizer formalism

Vecs2Pauli solves the Local-Pauli-Transformation task (and more)

Local-Pauli-Transformation Task: find all matrices of the form

α · P

₁ ⊗ P₂ ... ⊗ P_n which map one given vector

s⃗

to another vector

t⃗

, where

$n$ is the number of qubits
$s⃗$ are $t⃗$ are vectors of length $2$ ⁿ whose entries are complex numbers
$α$ a complex number
each $P$ _j for $j=1,2,...,n$ is a single-qubit Pauli matrix: \[ \text{I} = \begin{bmatrix} 1 & 0\\ 0 & 1 \end{bmatrix} , \qquad \text{X} = \begin{bmatrix} 0 & 1\\ 1 & 0 \end{bmatrix} , \qquad \text{Y} = \begin{bmatrix} 0 & -\mathrm{i}\\ \mathrm{i} & 0 \end{bmatrix} , \qquad \text{Z} = \begin{bmatrix} 1 & 0\\ 0 &-1 \end{bmatrix} . \]
the symbol " $⊗$ " is the Kronecker product.

By setting

t⃗

equal to

s⃗

, the solution to the task are precisely the (Pauli) stabilisers of

s⃗

. So Vecs2Pauli will find all (Pauli) stabilisers of a vector.

Example 1: take as source vector

s⃗ = (0.71, 0.71)

and as target vector the same:

t⃗ = (0.71, 0.71)

. Then Vecs2Pauli finds all Local Pauli matrices which map

s⃗

to itself, i.e. it finds the stabilisers of

s⃗

. These stabilisers are

I

and

X

, which can be written as

X

to the power zero and

X

to the power one. Hence,

X

is a generator for the group of stabilisers.

Example 2: take as source vector

s⃗ = (1, 0, 0, 1)

and as target

t⃗ = (0, 1, -1, 0)

(both are so-called (unnormalised) Bell states). Then

s⃗

is mapped to

t⃗

by precisely all Local Pauli matrices which can be written as

-i · Y ⊗ I

times

g

, where

g

is any operator that can be written as a multiplication of

X ⊗ X

and

Z ⊗ Z

(the stabilisers of

s⃗

).

Notation: in general, a state of

n

quantum bits can be represented as a vector consisting of

2

ⁿ complex numbers. Typically, these are written as a sum of terms of the form

c |x⟩

where

x

is a string of

n

bits and

c

the complex number which is found as the

x

-th position of the vector (so-called Dirac notation). For example, the vector

s⃗

from Example 1 is written

0.71 |0⟩ + 0.71 |1⟩

, and the vectors from Example 2 are

0|00⟩ + 1|01⟩ + 1|10⟩ + 0|00⟩ = |01⟩ + |10⟩

and

|01⟩ - |10⟩

Algorithm

The underlying algorithm of Vecs2Pauli which solves the Local-Pauli-Transformation task is based on our earlier work on a new data structure for simulation of quantum computing, combining decision diagrams and the stabilizer formalism. A full description of the algorithm will be made public soon.