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@@ -190,7 +190,7 @@ <h4>March 12, 2025</h4>
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<center style="font-size:80%">
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<!-- copyright --> &copy; 1999-2024, Morten Hjorth-Jensen. Released under CC Attribution-NonCommercial 4.0 license
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<!-- copyright --> &copy; 1999-2025, Morten Hjorth-Jensen. Released under CC Attribution-NonCommercial 4.0 license
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</center>
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</section>
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@@ -201,14 +201,32 @@ <h2 id="plans-for-the-week-of-march-10-14-solving-quantum-mechanical-problems-wi
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<b></b>
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<p>
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<ol>
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<p><li> Project 1 and simulating one- and two-qubit systems with the VQE algorithm</li>
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<p><li> Introducing our final model, the Lipkin model, a two-qubit and a four-qubit system</li>
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<p><li> <a href="https://youtu.be/" target="_blank">Video of lecture to be added</a>
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<p><li> Discussion of our final model, the Lipkin model, a two-qubit and a four-qubit system
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<!-- o <a href="https://youtu.be/" target="_blank">Video of lecture to be added</a> -->
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<!-- o <a href="https://github.com/CompPhysics/QuantumComputingMachineLearning/blob/gh-pages/doc/HandWrittenNotes/2024/NotesMarch6.pdf" target="_blank">Whiteboard notes</a> --></li>
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</ol>
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</div>
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</section>
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<section>
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<h2 id="reading-recommendation">Reading recommendation </h2>
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<p>Hundt's chapter 3 and 4 contain many useful
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hints for computing properties of Pauli matrices. Similarly, section
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6.11 has many details of relevance for project 1.
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</p>
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</section>
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<section>
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<h2 id="more-background-material">More background material </h2>
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<p>For the Lipkin model, we recommend strongly the work of LaRose and collaborators, see
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<a href="https://journals.aps.org/prc/abstract/10.1103/PhysRevC.106.024319" target="_blank"><tt>https://journals.aps.org/prc/abstract/10.1103/PhysRevC.106.024319</tt></a>, the article is also available at <a href="https://github.com/CompPhysics/QuantumComputingMachineLearning/blob/gh-pages/doc/articles/PhysRevC.106.024319.pdf" target="_blank"><tt>https://github.com/CompPhysics/QuantumComputingMachineLearning/blob/gh-pages/doc/articles/PhysRevC.106.024319.pdf</tt></a>. See in particular section 3.
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</p>
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<p>For codes, feel free to be inspired and/or reuse the codes at <a href="https://github.com/CompPhysics/QuantumComputingMachineLearning/tree/gh-pages/doc/Programs/LipkinModel" target="_blank"><tt>https://github.com/CompPhysics/QuantumComputingMachineLearning/tree/gh-pages/doc/Programs/LipkinModel</tt></a>.</p>
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</section>
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<section>
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<h2 id="lipkin-model">Lipkin model </h2>
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@@ -1774,13 +1792,298 @@ <h2 id="quantum-circuit-rewriting-the-lipkin-model-in-terms-of-pauli-matrices">Q
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<p>&nbsp;<br>
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<p>It then describes how to one can create an arbitrary superposition of Dicke states, which we modify here to restrict ourselves to a Hamming weight of constant parity. The circuit to construct such a state (for the \( k=6 \) case, as an example) is discussed next week.</p>
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</section>
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<section>
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<h2 id="summarizing-the-lipkin-model">Summarizing the Lipkin model </h2>
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<p>The Lipkin model is rewritten in terms of the quasipin operators as</p>
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<p>&nbsp;<br>
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$$
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\begin{align*}
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\begin{split}
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H_0 &= \epsilon J_z, \\
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H_1 &= \frac{1}{2}V (J_+^2 + J_-^2), \\
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H_2 & = \frac{1}{2}W ( J_+J_- +J_{-}J_+ - N ),
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\end{split}
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\end{align*}
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$$
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<p>&nbsp;<br>
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</section>
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<section>
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<h2 id="quasispin-operators">Quasispin operators </h2>
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<p>These quasi spin operators obey the normal spin commutator relations</p>
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<p>&nbsp;<br>
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$$
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\begin{align*}
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&[J_z,J_{\pm}] = \pm J_\pm,
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&[J_+,J_-] = 2 J_z, \\
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&[J^2,J_{\pm}] = 0,
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&[J^2,J_z] = 0,
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\end{align*}
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$$
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<p>&nbsp;<br>
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<p>in addition to commuting with the number operator</p>
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<p>&nbsp;<br>
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$$
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\begin{align*}
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[N,J_z] = [N,J_{\pm}] = [N,J^2] = 0.
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\end{align*}
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$$
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<p>&nbsp;<br>
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<p>Using these relations we can show that the Hamiltonian (a product of
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quasi spin operators and the number operator) also commutes with
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\( J^2 \), that is \( [H,J^2]=0 \). This means that \( H \) and \( J^2 \) have a shared
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eigenbasis, with \( J \) being a so-called <b>good</b> quantum number.
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</p>
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</section>
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<section>
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<h2 id="system-to-diagonalize-by-traditional-methods">System to diagonalize by traditional methods </h2>
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<p>Using spin-eigenstates as the Hamiltonian basis, we define states
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through the normal approach \( \vert J,J_z\rangle \) with \( J \) and $ J_z$ as spin
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and spin-projections, respectively. The states \( J_z = \pm J \) are the
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easiest to construct, corresponding to a single level being completely
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filled. States in between can then be found using the quasi-spin
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ladder operators following
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</p>
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<p>&nbsp;<br>
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$$
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J_\pm \vert J,J_z\rangle = \sqrt{J(J+1)-J_z (J_z \pm 1)} \vert J,J_z \pm 1\rangle.
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$$
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<p>&nbsp;<br>
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<p>Using this basis for the quasi-spin Hamiltonian, we can rewrite the explicit matrix \( H_{J_z,J'_z} = \langle J,J_z\vert H \vert J,J_z'\rangle \) can be constructed.</p>
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</section>
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<section>
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<h2 id="the-j-1-case">The \( J=1 \) case </h2>
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<p>For \( N=2 \) particles, we have the triplet \( J=1 \), with three possible projection \( J_z = 0, \pm 1 \). This gives the following matrix</p>
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<p>&nbsp;<br>
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$$
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H = \begin{bmatrix}
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-\epsilon & 0 & V \\
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0 & W & 0 \\
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V & 0 & \epsilon
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\end{bmatrix}.
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$$
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<p>&nbsp;<br>
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<p>By solving the eigenvalue problem, the ground state energy can be exactly found.</p>
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</section>
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<section>
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<h2 id="the-j-2-case">The \( J=2 \) case </h2>
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<p>Similarly, for the \( N=4 \) particles case we have \( J = 2 \) with five possible projections \( J_z = 0,\pm1,\pm2 \). This gives us the Hamiltonian matrix</p>
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<p>&nbsp;<br>
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$$
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H = \begin{bmatrix}
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-2\epsilon & 0 & \sqrt{6}V &0 &0 \\
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0 & -\epsilon + 3W & 0 & 3V & 0 \\
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\sqrt{6}V & 0 & 4W & 0 & \sqrt{6}V \\
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0 & 3V & 0 & \epsilon + 3W & 0 \\
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0 & 0 & \sqrt{6}V & 0 & 2\epsilon
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\end{bmatrix}.
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$$
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<p>&nbsp;<br>
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</section>
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<section>
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<h2 id="more-rewriting">More rewriting </h2>
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<p>Following the work of LaRose and collaborators, see
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<a href="https://journals.aps.org/prc/abstract/10.1103/PhysRevC.106.024319" target="_blank"><tt>https://journals.aps.org/prc/abstract/10.1103/PhysRevC.106.024319</tt></a>,
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we can rewrite the Lipkin Hamiltonian for efficient adaptations to
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quantum computing.
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</p>
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<p>We have</p>
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<p>&nbsp;<br>
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$$
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J_z = \sum_{i}^{N} j_{z}^{i} \hspace{20px} J_\pm = \sum_i^N j_{\pm}^{i} = \sum_i^N (j_{x}^{i}\pm ij_{y}^{i}),
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$$
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<p>&nbsp;<br>
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<p>with \( N \) being the number of particles. Additionally, since we have spin-\( 1/2 \) fermions, the mapping to Pauli matrices follow</p>
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<p>&nbsp;<br>
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$$
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j_{x}^{i} = \frac{1}{2}X_i,\hspace{20px}j_{y}^{i} = \frac{1}{2}Y_i,\hspace{20px}j_{z}^{i} = \frac{1}{2}Z_i.
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$$
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<p>&nbsp;<br>
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<p>This means that we require \( N \) qubits to calculate properties of a \( N \) particle system, if no more symmetry reductions are considred.</p>
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</section>
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<section>
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<h2 id="lipkin-hamiltonian-for-quantum-computing">Lipkin Hamiltonian for quantum computing </h2>
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<p>We rewrote our Hamiltonian as</p>
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<p>&nbsp;<br>
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$$
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\begin{align*}
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\begin{split}
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H_0 &= \frac{\epsilon}{2}\sum_{p}Z_p, \\
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H_1 &= \frac{1}{2}V \sum_{p < q}( X_p X_q - Y_p Y_q), \\
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H_2 &= \frac{1}{2}W \sum_{p < q} \left( X_p X_q + Y_p Y_q \right).
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\end{split}
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\end{align*}
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$$
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<p>&nbsp;<br>
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<p>For two particles coupling to spin \( J=1 \) we have</p>
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<p>&nbsp;<br>
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$$
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H^{N=2} = \frac{\epsilon}{2}\left(Z_1 + Z_2\right) + \frac{W+V}{2} X_1 X_2 - \frac{W-V}{2} Y_1 Y_2.
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$$
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<p>&nbsp;<br>
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</section>
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<section>
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<h2 id="for-four-fermions">For four fermions </h2>
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<p>Our Hamiltonian for \( N=4 \) is</p>
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<p>&nbsp;<br>
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$$
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\begin{align*}
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\begin{split}
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H^{N=4} =& \frac{\epsilon}{2}\left(Z_1 + Z_2 + Z_3 + Z_4\right) + \frac{W-V}{2}\Big( X_1 X_2 +\\
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& X_1 X_3 + X_1 X_4 + X_2 X_3 + X_3 X_4 + X_3 X_4
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\Big) \\
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&+ \frac{W+V}{2}\Big( Y_1 Y_2 + Y_1 Y_3 + Y_1 Y_4 + Y_2 Y_3 \\
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& + Y_3 Y_4 + Y_3 Y_4\Big).
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\end{split}
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\end{align*}
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$$
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<p>&nbsp;<br>
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<p>Note that well that a term like \( Z_1 \) reads \( Z_1\otimes I_2 \otimes I_3 \otimes I_4 \) where the subscript points to qubit \( i \) and all matrices are \( 2\times 2 \) matrices.</p>
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</section>
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<section>
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<h2 id="leaving-out-the-w-term">Leaving out the \( W \) term </h2>
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<p>If we set \( W = 0 \) we can simplify the Hamiltonian. For this we note that if we only include the \( H_0 \) and \( H_1 \) terms, spins differing by \( \pm2 \) are the only possible non-diagonal couplings. Since \( H_0 \) is diagonal and single particle energies are degenerate, it does not break any symmetry.
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Instead of the \( 2J+1 \) spin projections, we simply have \( J+1 \) relevant states. From <a href="https://journals.aps.org/prc/abstract/10.1103/PhysRevC.106.024319" target="_blank"><tt>https://journals.aps.org/prc/abstract/10.1103/PhysRevC.106.024319</tt></a>,
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the two-qubit Hamiltonian for \( N=4 \) can be written as
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</p>
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<p>&nbsp;<br>
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$$
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H_{W=0}^{N=4} = \epsilon(Z_1 + Z_2) + \frac{\sqrt{6}}{2}V (X_1 + X_2 + Z_1 X_0 - X_1 Z_0).
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$$
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<p>&nbsp;<br>
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<p>This is computationally more efficient since we only need half of the qubits.</p>
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</section>
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<section>
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<h2 id="more-details">More details </h2>
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<p>Here we give some additional details behind the rewriting of the Lipkin Hamiltonian in term of Pauli matrices and the identity matrix.
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For the one-body term we have
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</p>
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<p>&nbsp;<br>
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$$
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\begin{align*}
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H_0 = \epsilon J_z = \epsilon \sum_{p} j_{z}^{p} = \frac{\epsilon}{2} \sum_p Z_p
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\end{align*}
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$$
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<p>&nbsp;<br>
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<p>Moving on to the \( H_1 \) term, we need to expand the square of the \( J_\pm \) operators. Starting with \( J_+ \) we find</p>
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<p>&nbsp;<br>
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$$
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\begin{align*}
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J_+^2 &= (\sum_p j_{x}^{p} + \imath j_{y}^{p})^2 = \sum_{pq}(j_{x}^{p} + \imath j_{y}^{p})(j_{x}^{q} + \imath j_{y}^{q}) \\
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&= \sum_{pq} \left(j_{x}^{p}j_{x}^{q} - j_{y}^{p}j_{y}^{q} + \imath j_{y}^{p}j_{x}^{q} + \imath j_{x}^{p}j_{y}^{q}\right).
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\end{align*}
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$$
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<p>&nbsp;<br>
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</section>
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<section>
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<h2 id="other-manipulations">Other manipulations </h2>
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<p>Similarly, the \( J_-^2 \) term yields</p>
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<p>&nbsp;<br>
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$$
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\begin{align*}
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J_-^2 = \sum_{pq} (j_{x}^{p}j_{x}^{q} - j_{y}^{p}j_{y}^{q} - \imath j_{y}^{p}j_{x}^{q} - \imath j_{x}^{p}j_{y}^{q}).
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\end{align*}
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$$
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<p>&nbsp;<br>
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<p>Rewriting we get</p>
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<p>&nbsp;<br>
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$$
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\begin{align*}
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J_+^2 + J_-^2 &= 2\sum_{pq} j_{x}^{p}j_{x}^{q} - j_{z}^{p}j_{z}^{q} \\
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&= 2 \sum_p (j_{x}^{p})^2 - (j_{y}^{p})^2 + 4\sum_{p > q}( j_{x}^{p}j_{x}^{q} - j_{y}^{p}j_{y}^{q}) \\
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&= \frac{1}{2} \sum_p (X_p^2 - Y_p^2) + \sum_{p > q} (X_p X_q - Y_p Y_q)
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\end{align*}
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$$
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<p>&nbsp;<br>
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</section>
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<section>
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<h2 id="final-expression-for-h-1-and-h-2">Final expression for \( H_1 \) and \( H_2 \) </h2>
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<p>The diagonal term will cancel out, since the Pauli matrices are involutory, resulting in</p>
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<p>&nbsp;<br>
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$$
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H_1 = \frac{1}{2}V \sum_{p > q}( X_p X_q - Y_p Y_q).
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$$
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<p>&nbsp;<br>
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<p>Following the same procedure as above, we have finally</p>
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<p>&nbsp;<br>
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$$
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H_2 = \frac{1}{2}W \sum_{p < q} (X_p X_q + Y_p Y_q).
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$$
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<p>&nbsp;<br>
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</section>
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<section>
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<h2 id="background-material-on-vqe-and-transformations-into-a-computational-basis">Background material on VQE and transformations into a computational basis </h2>
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<p>The review of Tilly et al., see <a href="https://discovery.ucl.ac.uk/id/eprint/10157639/1/1-s2.0-S0370157322003118-main.pdf" target="_blank"><tt>https://discovery.ucl.ac.uk/id/eprint/10157639/1/1-s2.0-S0370157322003118-main.pdf</tt></a> (Physics Reports <b>986</b>, 1 (2022)), gives examples on how to arrive at the equations (see section 5.2) listed in the tables here. The article gives also an excellent overview of the VQE method.</p>
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<p>See also Hundt section 6.11.</p>
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</section>
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<section>
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<h2 id="transforming-pauli-and-identity-matrices">Transforming Pauli and identity matrices </h2>
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<br/><br/>
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<center>
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<p><img src="figures/table1.png" width="700" align="bottom"></p>
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</center>
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<br/><br/>
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</section>
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<section>
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<h2 id="transforming-pauli-and-identity-matrices-four-particle-case">Transforming Pauli and identity matrices, four particle case </h2>
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<br/><br/>
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<center>
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<p><img src="figures/table2.png" width="700" align="bottom"></p>
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</center>
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<br/><br/>
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</section>
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<section>
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<h2 id="plans-for-the-week-march-17-21">Plans for the week March 17-21 </h2>
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<div class="alert alert-block alert-block alert-text-normal">
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<b>Plans for the week March 11-15</b>
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<b></b>
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<p>
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<ol>
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<p><li> Discussion of the project only and how to implement the VQE for the simpler matrix problems</li>
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<p><li> Solving the Lipkin model with VQE</li>
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<p><li> Discussion of the project only and work on finalizing the project</li>
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</ol>
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</div>
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</section>

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