QuantumCore

Hands-on lab · 2 steps

Pauli products and the Pauli decomposition

Multiply two Pauli matrices by hand to see XY=iZXY=iZ (and why the order matters), then expand a Hermitian 2×22\times2 matrix in the Pauli basis using aP=12Tr(PA)a_P=\tfrac12\mathrm{Tr}(PA).

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Step 1: Compute XY

Compute the matrix product XYXY of X=(0110)X=\begin{pmatrix}0&1\\1&0\end{pmatrix} and Y=(0−ii0)Y=\begin{pmatrix}0&-i\\i&0\end{pmatrix} (first XX, then YY, as written: row of XX times column of YY). Enter all four entries; type the imaginary unit as i. Entries are accepted within 10−310^{-3} in Frobenius distance.

Enter each entry as an expression, e.g. 1/sqrt(2), -i/2, exp(i*pi/4). Rows run top to bottom, columns left to right.

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Fill every entry with a valid expression.

Step 2: Decompose A in the Pauli basis

Write A=(21−i1+i3)=a0I+axX+ayY+azZA=\begin{pmatrix}2&1-i\\1+i&3\end{pmatrix}=a_0I+a_xX+a_yY+a_zZ and enter the four real coefficients a0,ax,ay,aza_0,a_x,a_y,a_z in this order. Use aP=12Tr(PA)a_P=\tfrac12\mathrm{Tr}(PA) for P∈{I,X,Y,Z}P\in\lbrace I,X,Y,Z\rbrace. Values are accepted within 10−310^{-3}.

Enter exact expressions or decimals, e.g. 1/sqrt(2), pi/3. Values must be real.

enter a value
enter a value
enter a value
enter a value
Fill every field with a valid expression.