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Update week7.do.txt
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‎doc/src/week7/week7.do.txt‎

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@@ -451,15 +451,174 @@ represented by a Pauli-$\bm{Z}$ gate/matrix.
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!split
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===== Explicit expressions =====
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In order to perform our measurements we need the following operators $\bm{U}$
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===== Rewriting our strings of Pauli matrices =====
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As discussed last week and reviewed above, to perform a measurement,
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it is often useful to rewrite the string of Pauli matrices in a
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specific order or to simplify the expression.
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Consider a string of Pauli matrices of the form:
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!bt
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\begin{align*}
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\bm{Z}\otimes\bm{I}\hspace{1cm} & \bm{U}=\bm{I}\otimes\bm{I}\\
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\bm{I}\otimes\bm{Z}\hspace{1cm} & \bm{U}=\text{SWAP}\\
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\bm{Z}\otimes\bm{Z}\hspace{1cm} & \bm{U}=CX_{10}\\
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\bm{X}\otimes\bm{X}\hspace{1cm} & \bm{U}=CX_{10}(\bm{H}\otimes\bm{H})\\
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\end{align*}
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\[
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P = \sigma_{i_1} \otimes \sigma_{i_2} \otimes \cdots \otimes \sigma_{i_n},
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\]
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!et
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where each $\sigma_{i_k}$ is one of the Pauli matrices $\bm{X}$, $\bm{Y}$, $\bm{Z}$, or the identity matrix $\bm{I}$.
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!split
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===== Rewriting the string of matrices =====
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To rewrite this string for measurement purposes, follow these steps:
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!bblock Commute and reorder:
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Use the commutation relations of the Pauli matrices to reorder the string. The commutation relations are:
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!bt
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\[
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[\sigma_i, \sigma_j] = 2i \epsilon_{ijk} \sigma_k,
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\]
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where $\epsilon_{ijk}$ is the Levi-Civita symbol. This allows you to move certain Pauli matrices to the left or right in the string.
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!et
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!eblock
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!bblock Simplify using identities:
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Use the fact that $\sigma_i^2 = I$ and $\sigma_i \sigma_j = -\sigma_j \sigma_i$ for $i \neq j$ to simplify the expression. For example:
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!bt
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\[
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\sigma_x \sigma_y = -\sigma_y \sigma_x = i \sigma_z.
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\]
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!et
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!eblock
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!split
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===== More rewriting tricks =====
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!bblock Group terms:
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Group terms that are easier to measure together. For example, if you have a term like $\sigma_x \otimes \sigma_x$, you can measure both qubits in the $X$-basis simultaneously.
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!eblock
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!bblock Diagonalize if necessary:
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If the final expression is not diagonal, you may need to apply a unitary transformation to diagonalize it before measurement. For example,to measure $\sigma_x$, you can apply the Hadamard gate $H$ to transform it into $\sigma_z$:
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!bt
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\[
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H \sigma_x H = \sigma_z.
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\]
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!et
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!bblock
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!split
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===== Example =====
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A simple example is
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!bt
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\[
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\bm{P} = \bm{X} \bm{Y} \otimes bm{Z}.
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\]
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!et
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To rewrite this for measurement we use the commutation relations to reorder the terms if needed.
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Then we eimplify using identities:
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!bt
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\[
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\bm{X} \bm{Y} = i \bm{Z}.
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\]
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!et
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We obtain then
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!bt
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\[
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\bm{P} = (i \bm{Z}) \otimes \bm{Z} = i (\bm{Z} \otimes \bm{Z}).
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\]
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!et
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!split
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===== The $\bm{Z}\otimes \bm{I}$ term =====
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The explicit matrix is
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!bt
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\[
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\bm{Z}\otimes \bm{I} = \begin{bmatrix}
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1 & 0 & 0 & 0 \\
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0 & 1 & 0 & 0 \\
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0 & 0 & -1 & 0 \\
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0 & 0 & 0 & -1
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\end{bmatrix}.
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\]
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!et
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When we perform a measurement on the first qubit, we see that this
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matrix gives us the correct eigenvalues for the first qubit (but not
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for the second one). To see this multiply the above matrix with our
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computational basis states, that is the states $\vert 00\rangle =\vert 0\rangle
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\times \vert 0\rangle$, $\vert 01 \rangle$, $\vert 10\rangle$ and
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$\vert 11\rangle$.
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!split
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===== The specific eigenvalues =====
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Multiplying with the $\vert 00\rangle$ state we get
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!bt
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\[
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\bm{Z}\otimes \bm{I} = \begin{bmatrix}
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1 & 0 & 0 & 0 \\
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0 & 1 & 0 & 0 \\
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0 & 0 & -1 & 0 \\
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0 & 0 & 0 & -1
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\end{bmatrix}\begin{bmatrix} 1\\ 0 \\ 0 \\ 0\end{bmatrix}=\begin{bmatrix} 1\\ 0 \\ 0 \\ 0\end{bmatrix},
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\]
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!et
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and
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!bt
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\[
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\bm{Z}\otimes \bm{I} = \begin{bmatrix}
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1 & 0 & 0 & 0 \\
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0 & 1 & 0 & 0 \\
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0 & 0 & -1 & 0 \\
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0 & 0 & 0 & -1
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\end{bmatrix}\begin{bmatrix} 0\\ 0 \\ 1 \\ 0\end{bmatrix}=-1\begin{bmatrix} 0\\ 0 \\ 1 \\ 0\end{bmatrix}.
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\]
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!et
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We don't get the correct eigenvalues if we perform the measurement on the second qubits!
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!split
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===== The $\bm{I}\otimes \bm{Z}$ term =====
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Our strategy is to rewrite all the strings of Pauli operators via
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specific unitary transformations in order to obtain a final operator
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$\bm{P}=\bm{Z}_1\bm{I}_2\dots\bm{I}_N$, where the subscripts indicate
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the various qubits. We can then perform the measurement on the first qubit only.
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We can rewrite $\bm{I}\otimes \bm{Z}$ via the SWAP gate
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!bt
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\[
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\text{SWAP} = \begin{bmatrix}
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1 & 0 & 0 & 0 \\
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0 & 0 & 1 & 0 \\
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0 & 1 & 0 & 0 \\
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0 & 0 & 0 & 1
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\end{bmatrix}.
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\]
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!et
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We have then that
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!bt
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\[
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\bm{P}=\text{SWAP}^{\dagger}(\bm{I}\otimes\bm{Z})\text{SWAP}=\bm{Z}\otimes\bm{I}.
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\]
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!et
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We note that the original $\bm{I}\otimes \bm{Z}$ does not give the
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correct eigenvalues when measured on the first qubit. Try this as an exercise.
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!split
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===== The $\bm{Z}\otimes \bm{Z}$ term =====
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This term gives the correct eigenvalue when operating on the first
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qubit. In principle thus we don't need to rewrite string of operators.
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However, let us rewrite it via a unitary transformation in
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order to have $\bm{P}=\bm{Z}\otimes\bm{I}$. To do so, consider the
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transformation
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!bt
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\[
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\bm{P}= CX_{10}^{\dagger}(\bm{Z}\otimes\bm{Z})CX_{10},
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\]
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!et
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where we have
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!bt
@@ -473,11 +632,78 @@ where we have
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\]
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!et
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!split
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===== Performing the transformation =====
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Performing the operation gives
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!bt
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\[
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\bm{P}= CX_{10}^{\dagger}(\bm{Z}\otimes\bm{Z})CX_{10}=\begin{bmatrix}
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1 & 0 & 0 & 0 \\
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0 & 1 & 0 & 0 \\
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0 & 0 & -1 & 0 \\
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0 & 0 & 0 & -1
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\end{bmatrix}=\bm{Z}\otimes\bm{I},
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\]
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!et
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which allows us to transform the original tensor product
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$\bm{Z}\otimes \bm{Z}$ into a matrix where we perform the measurement
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in the basis of the first qubit.
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To see this, act with $\bm{P}$ on the states $\vert 00\rangle =\vert
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0\rangle \times \vert 0\rangle$, $\vert 01 \rangle$, $\vert 10\rangle$
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and $\vert 11\rangle$.
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!split
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===== More terms =====
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Let us look at the $\bm{X}\otimes \bm{X}$.
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This term can be rewritten as
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!bt
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\[
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\bm{P}= (CX_{10}(\bm{H}\otimes\bm{H}))^{\dagger}(\bm{X}\otimes\bm{X})(CX_{10}(\bm{H}\otimes\bm{H})),
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\]
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!et
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which results in
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!bt
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\[
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\begin{bmatrix}
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1 & 0 & 0 & 0 \\
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0 & 0 & 0 & 1 \\
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0 & 0 & 1 & 0 \\
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0 & 1 & 0 & 0
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\end{bmatrix}.
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\]
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!et
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!split
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===== Explicit expressions =====
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In order to perform our measurements for our simple two-qubit Hamiltonian
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we need the following unitary transformations $\bm{U}$
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!bt
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\begin{align*}
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\bm{Z}\otimes\bm{I}\hspace{1cm} & \bm{U}=\bm{I}\otimes\bm{I}\\
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\bm{I}\otimes\bm{Z}\hspace{1cm} & \bm{U}=\text{SWAP}\\
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\bm{Z}\otimes\bm{Z}\hspace{1cm} & \bm{U}=CX_{10}\\
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\bm{X}\otimes\bm{X}\hspace{1cm} & \bm{U}=CX_{10}(\bm{H}\otimes\bm{H})\\
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\end{align*}
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!et
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!split
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===== More complete list and derivations of expressions for strings of operators =====
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For a two qubit system we list here the possible transformations
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!bt
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\begin{align*}
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\bm{Z}\otimes\bm{I}\hspace{1cm} & \bm{U}=\bm{I}\otimes\bm{I}\\
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\bm{I}\otimes\bm{Z}\hspace{1cm} & \bm{U}=\text{SWAP}\\
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\bm{Z}\otimes\bm{Z}\hspace{1cm} & \bm{U}=CX_{10}\\
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\bm{X}\otimes\bm{X}\hspace{1cm} & \bm{U}=CX_{10}(\bm{H}\otimes\bm{H})\\
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\end{align*}
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!et
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!split
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===== Lipkin model =====

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