Unitary Error Basis

ianmcloughlin.github.io linkedin github

A set of $n^2$ orthonormal $n \times n$ matrices with respect to the inner product $\langle A, B \rangle = \text{Tr}(A^\dagger B)/n$ where $\text{Tr}(M)$ is the trace of the matrix $M$ and $M^\dagger$ is its conjugate transpose.

Example: Pauli Matrices

$\left\{\begin{bmatrix} 1 & 0 \\ 0 & 1 \\ \end{bmatrix}, \; \begin{bmatrix} 0 & 1 \\ 1 & 0 \\ \end{bmatrix}, \ \begin{bmatrix} 1 & 0 \\ 0 & -1 \\ \end{bmatrix}, \ \begin{bmatrix} 0 & -i \\ i & 0 \\ \end{bmatrix}\right\}$

Conjugate Transpose

$A^\dagger = A^\text{H} = A^\dagger = \bar{A}^\textsf{T}$

$\begin{bmatrix} i & 2+i \\ 1 & 3-2i \\ \end{bmatrix}^\dagger = \begin{bmatrix} -i & 1 \\ 2-i & 3+2i \\ \end{bmatrix}$

Trace

$A = [a_{ij}]$

$\text{Tr}(A) = \sum_i a_{ii}$

$\text{Tr}(A \otimes B) = \text{Tr}(A) \text{Tr}(B)$

$\text{Tr}(A + B) = \text{Tr}(A) + \text{Tr}(B)$

$\text{Tr}(cA) = c\text{Tr}(A)$

$\text{Tr}(A) = \sum_i \lambda_i$

$\text{Tr}(A) = c\text{Tr}(A)$

Inner Product

$A, B$: $n \times n$ matrices.

$\langle A, B \rangle = \text{Tr}(A^\dagger B) / n$

Orthogonality

$A = \begin{bmatrix} 0 & -i \\ i & 0 \\ \end{bmatrix}\qquad B = \begin{bmatrix} 1 & 0 \\ 0 & -1 \\ \end{bmatrix}$

$A^\dagger = \begin{bmatrix} 0 & -i \\ i & 0 \\ \end{bmatrix}$

$A^\dagger B = \begin{bmatrix} 0 & -i \\ i & 0 \\ \end{bmatrix}\begin{bmatrix} 1 & 0 \\ 0 & -1 \\ \end{bmatrix}=\begin{bmatrix} 0 & i \\ i & 0 \\ \end{bmatrix}$

$\textsf{tr}(A^\dagger B) = 0 + 0 = 0$

$d = 2$

$\textsf{tr}(A^\dagger B)/2 = 0/2 = 0$

Normality

$A = \begin{bmatrix} 0 & -i \\ i & 0 \\ \end{bmatrix} \qquad A^\dagger = \begin{bmatrix} 0 & -i \\ i & 0 \\ \end{bmatrix}$

$A^\dagger A = \begin{bmatrix} 0 & -i \\ i & 0 \\ \end{bmatrix}\begin{bmatrix} 0 & -i \\ i & 0 \\ \end{bmatrix}=\begin{bmatrix} -i^2 & 0 \\ 0 & -i^2 \\ \end{bmatrix}$

$\textsf{tr}(A^\dagger A) = -i^2 + -i^2 = 2$

$d = 2$

$\textsf{tr}(A^\dagger B)/2 = 2/2 = 1$

Equivalence

Unitary matrix: $U^\dagger U = I$.

$\mathcal{U}(d)$: group of all $d \times d$ unitary matrices.

$\mathcal{E}, \mathcal{E}’$: $d \times d$ Unitary Error Bases.

$\mathcal{E} \equiv \mathcal{E}’ \Leftrightarrow \exists A, B \in \mathcal{U}(d), f:\mathcal{E} \rightarrow \mathbb{C} : \mathcal{E}’ = \{f(E)AEB \mid E \in \mathcal{E}\}$

Pauli Basis is Unique

Up to equivalence, the Pauli basis is unique in dimension 2.

Let $\mathcal{E} = \{E_1, E_2, E_3, E_4 \}$ be a $2 \times 2$ unitary error basis.

Then $E_1^\dagger \mathcal{E} = \{I, E_1^\dagger E_2, E_1^\dagger E_3, E_1^\dagger E_4 \}$ is also a unitary error basis.

Characterization

$U$
$d \times d$ unitary matrix.
$E_i(U)$
$\sqrt{d} (U_{i,nd+m})_{n,m=0,\ldots,d-1}$

$U = \begin{bmatrix} 0 & -i \\ i & 0 \\ \end{bmatrix}$

$E_0(U) = \sqrt{2} \begin{bmatrix} U_{0,0} & U_{0,1} \\ U_{0,2} & U_{0,3} \\ \end{bmatrix}$

TODO: Fix above

Nice Unitary Error Bases

https://people.engr.tamu.edu/andreas-klappenecker/ueb/uebdef.html

$G$: finite group.

$|G| = n^2, n \in \mathbb{N}$.

Nice error basis

Image of $D: G \rightarrow \mathcal{U}(n)$ where:

Generators

$X$: set of generators for $G$.

$H$: matrix group generated by $D(X) = \{ D(x) | x \in X \}$.

$Z(H) = \{ x | x \in H, \forall h \in H : x h = hx \}$.

Traversal: set of coset representatives of cosets of a subgroup.

Nice error basis is constructed as a traversal of $Z(H)$ in $H$.

Example

https://people.engr.tamu.edu/andreas-klappenecker/ueb/uebex.html

Klein Four-Group

Generators

$x = \begin{bmatrix} 1 & 0 \\ 0 & -1 \end{bmatrix}$

$y = \begin{bmatrix} -1 & 0 \\ 0 & 1 \end{bmatrix}$

Group

$x = \begin{bmatrix} 1 & 0 \\ 0 & -1 \end{bmatrix}$

$x^2 = \begin{bmatrix} 1 & 0 \\ 0 & -1 \end{bmatrix}\begin{bmatrix} 1 & 0 \\ 0 & -1 \end{bmatrix} = \begin{bmatrix} 1 & 0 \\ 0 & 1 \end{bmatrix} = I$

$y = \begin{bmatrix} -1 & 0 \\ 0 & 1 \end{bmatrix}$

$y^2 = \begin{bmatrix} -1 & 0 \\ 0 & 1 \end{bmatrix}\begin{bmatrix} -1 & 0 \\ 0 & 1 \end{bmatrix} = \begin{bmatrix} 1 & 0 \\ 0 & 1 \end{bmatrix} = I$

$xy = \begin{bmatrix} 1 & 0 \\ 0 & -1 \end{bmatrix}\begin{bmatrix} -1 & 0 \\ 0 & 1 \end{bmatrix} = \begin{bmatrix} -1 & 0 \\ 0 & -1 \end{bmatrix}$

$xyx = \begin{bmatrix} -1 & 0 \\ 0 & -1 \end{bmatrix}\begin{bmatrix} 1 & 0 \\ 0 & -1 \end{bmatrix} = \begin{bmatrix} -1 & 0 \\ 0 & 1 \end{bmatrix} = y$

$(xy)^2 = y^2 = I$

$xyy = x(y^2) = xI = x$

$yx = \begin{bmatrix} -1 & 0 \\ 0 & 1 \end{bmatrix}\begin{bmatrix} 1 & 0 \\ 0 & -1 \end{bmatrix} = \begin{bmatrix} -1 & 0 \\ 0 & -1 \end{bmatrix} = xy$

$G = \{ I, x, y, xy \}$

$G = \langle x, y \; | \; x^2, y^2, (xy)^2 \rangle$

https://en.wikipedia.org/wiki/Klein_four-group

Centre

$Z(G) = \{ z \in G \; | \; \forall g \in G : gz = zg \}$

$ xy \neq yx $

$ \Rightarrow Z(G) = \{ I, x, y, xy \} $

Group of Order 8

Generators

$x = \begin{bmatrix} 1 & 0 \\ 0 & -1 \end{bmatrix}$

$y = \begin{bmatrix} 0 & 1 \\ 1 & 0 \end{bmatrix}$

Group

$x = \begin{bmatrix} 1 & 0 \\ 0 & -1 \end{bmatrix}$

$x^2 = \begin{bmatrix} 1 & 0 \\ 0 & -1 \end{bmatrix}\begin{bmatrix} 1 & 0 \\ 0 & -1 \end{bmatrix} = \begin{bmatrix} 1 & 0 \\ 0 & 1 \end{bmatrix} = I$

$y = \begin{bmatrix} 0 & 1 \\ 1 & 0 \end{bmatrix}$

$y^2 = \begin{bmatrix} 0 & 1 \\ 1 & 0 \end{bmatrix}\begin{bmatrix} 0 & 1 \\ 1 & 0 \end{bmatrix} = \begin{bmatrix} 1 & 0 \\ 0 & 1 \end{bmatrix} = x^2$

$xy = \begin{bmatrix} 1 & 0 \\ 0 & -1 \end{bmatrix}\begin{bmatrix} 0 & 1 \\ 1 & 0 \end{bmatrix} = \begin{bmatrix} 0 & 1 \\ -1 & 0 \end{bmatrix}$

$xyx = \begin{bmatrix} 0 & 1 \\ -1 & 0 \end{bmatrix}\begin{bmatrix} 1 & 0 \\ 0 & -1 \end{bmatrix} = \begin{bmatrix} 0 & -1 \\ -1 & 0 \end{bmatrix} = -y$

$xyxy = \begin{bmatrix} 0 & 1 \\ -1 & 0 \end{bmatrix}\begin{bmatrix} 0 & 1 \\ -1 & 0 \end{bmatrix} = \begin{bmatrix} -1 & 0 \\ 0 & -1 \end{bmatrix} = -I$

$xyxyx = \begin{bmatrix} -1 & 0 \\ 0 & -1 \end{bmatrix}\begin{bmatrix} 1 & 0 \\ 0 & -1 \end{bmatrix} = \begin{bmatrix} -1 & 0 \\ 0 & 1 \end{bmatrix} = -x$

$xyxyxy = \begin{bmatrix} -1 & 0 \\ 0 & 1 \end{bmatrix}\begin{bmatrix} 0 & 1 \\ 1 & 0 \end{bmatrix} = \begin{bmatrix} 0 & -1 \\ 1 & 0 \end{bmatrix} = -xy$

$xyxyxyx = \begin{bmatrix} 0 & -1 \\ 1 & 0 \end{bmatrix}\begin{bmatrix} 1 & 0 \\ 0 & -1 \end{bmatrix} = \begin{bmatrix} 0 & 1 \\ 1 & 0 \end{bmatrix} = y$

$yx = \begin{bmatrix} 0 & 1 \\ 1 & 0 \end{bmatrix}\begin{bmatrix} 1 & 0 \\ 0 & -1 \end{bmatrix} = \begin{bmatrix} 0 & -1 \\ 1 & 0 \end{bmatrix} = -xy$

$yxy = (-xy)y = -x(y^2) = -x$

$G = \{ I, x, y, xy, -I, -x, -y, -xy \}$

$G = \langle x, y \; | \; x^2, y^2, (xy)^4 \rangle$

Table

$ \begin{array}{c|cccccccc} & I & x & y & xy & -I & -x & -y & -xy \\ \hline I & I & x & y & xy & -I & -x & -y & -xy \\ x & x & I & xy & y & -x & -I & -xy & -y \\ y & y & -xy & I & -x & -y & xy & -I & x \\ xy & xy & -y & x & -I & -xy & y & -x & I \\ -I & -I & -x & -y & -xy & I & x & y & xy \\ -x & -x & -I & -xy & -y & x & I & xy & y \\ -y & -y & xy & -I & x & y & -xy & I & -x \\ -xy & -xy & y & -x & I & xy & -y & x & -I \\ \end{array} $

Isomorphism

$I \leftrightarrow e$

$xy \leftrightarrow b$

$\Rightarrow -I \leftrightarrow b^2$

$\Rightarrow -xy \leftrightarrow b^3$

$\Rightarrow b^4 \leftrightarrow e $

$x \leftrightarrow a$

$\Rightarrow a^2 = e$

$\Rightarrow ab \leftrightarrow x^2y = y$

$\Rightarrow y \leftrightarrow ab$

$ba\leftrightarrow xyx = -y$

$ab^3 \leftrightarrow x(-xy) = -x^2y = -y \leftrightarrow ba$

$\Rightarrow -y \leftrightarrow ab^3 = ba$

$\Rightarrow -x \leftrightarrow ab^2$

$ \begin{array}{c|cccccccc} & e & b & b^2 & b^3 & a & ab & ab^2 & ab^3 \\ \hline e & e & b & b^2 & b^3 & a & ab & ab^2 & ab^3 \\ b & b & e & b^3 & b^2 & ab^3 & a & ab & ab^2 \\ b^2 & b^2 & b^3 & e & b & ab^2 & ab^3 & a & ab \\ b^3 & b^3 & b^2 & b & e & ab & ab^2 & ab^3 & a \\ a & a & ab & ab^2 & ab^3 & e & b & b^2 & b^3 \\ ab & ab & ab^2 & ab^3 & a & b^3 & e & b & b^2 \\ ab^2 & ab^2 & ab^3 & a & ab & b^2 & b^3 & e & b \\ ab^3 & ab^3 & a & ab & ab^2 & b & b^2 & b^3 & e \\ \end{array} $

https://groupprops.subwiki.org/wiki/Dihedral_group:D8

Centre

$Z(G) = \{ z \in G \; | \; \forall g \in G : gz = zg \}$

$ ab \neq ba $

$ \Rightarrow Z(G) = \{ e, b^2 \} $

Traversal

https://proofwiki.org/wiki/Definition:Transversal_(Group_Theory)

A subset $S \subseteq G$ where every (left/right) coset contains exactly one element of $S$.

Nice Error Basis

Traversal of centre of $H$.

Centre

$\{ I, -I \}$

Cosets

$\{ I, -I \}$ $\{ x, -x \}$ $\{ y, -y \}$ $\{ xy, -xy \}$

Traversal

$\{ I, x, y, xy \}$

Knill

Group Representations, Error Bases and Quantum Codes
Emanuel Knill
https://www.osti.gov/biblio/378680