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‎doc/pub/week16/html/week16-bs.html‎

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'implementation-with-pennylane'),
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('Kullback-Leibler divergence',
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('Summary of steps in PennyLane implementation',
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'kullback-leibler-divergence')]}
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'summary-of-steps-in-pennylane-implementation')]}
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<body>
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<!-- navigation toc: --> <li><a href="#gibbs-sampling" style="font-size: 80%;">Gibbs sampling</a></li>
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<!-- navigation toc: --> <li><a href="#parameter-optimization-and-variational-techniques" style="font-size: 80%;">Parameter Optimization and Variational Techniques</a></li>
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<!-- navigation toc: --> <li><a href="#implementation-with-pennylane" style="font-size: 80%;">Implementation with PennyLane</a></li>
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<!-- navigation toc: --> <li><a href="#kullback-leibler-divergence" style="font-size: 80%;">Kullback-Leibler divergence</a></li>
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<!-- navigation toc: --> <li><a href="#summary-of-steps-in-pennylane-implementation" style="font-size: 80%;">Summary of steps in PennyLane implementation</a></li>
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</ul>
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@@ -786,36 +786,69 @@ <h2 id="parameter-optimization-and-variational-techniques" class="anchor">Parame
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<!-- !split -->
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<h2 id="implementation-with-pennylane" class="anchor">Implementation with PennyLane </h2>
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<p>As a concrete example, we outline how to implement an RQBM in
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PennyLane. We consider \( n_v \) visible and \( n_h \) hidden qubits. The ansatz
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can be, for instance, layers of parameterized single-qubit rotations
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and entangling gates that respect the bipartite structure. Below is
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illustrative code (in Python) using PennyLane&#8217;s default.qubit
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simulator.
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<p>Below we sketch a toy example: a two-qubit Quantum Circuit Born Machine
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(QCBM), which generates a probability distribution via a parameterized
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quantum circuit. While a QCBM is not exactly a QBM (it has no thermal
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state), it serves to show how to code a quantum generative model.
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</p>
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<!-- !bc pycod -->
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<!-- !ec -->
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<!-- code=python (!bc pycod) typeset with pygments style "default" -->
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<div class="cell border-box-sizing code_cell rendered">
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<div class="input">
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<div class="inner_cell">
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<div class="input_area">
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<div class="highlight" style="background: #f8f8f8">
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<pre style="line-height: 125%;"><span style="color: #008000; font-weight: bold">import</span> <span style="color: #0000FF; font-weight: bold">pennylane</span> <span style="color: #008000; font-weight: bold">as</span> <span style="color: #0000FF; font-weight: bold">qml</span>
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<span style="color: #008000; font-weight: bold">from</span> <span style="color: #0000FF; font-weight: bold">pennylane</span> <span style="color: #008000; font-weight: bold">import</span> numpy <span style="color: #008000; font-weight: bold">as</span> np
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<span style="color: #408080; font-style: italic"># Use a 2-qubit simulator</span>
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dev <span style="color: #666666">=</span> qml<span style="color: #666666">.</span>device(<span style="color: #BA2121">&#39;default.qubit&#39;</span>, wires<span style="color: #666666">=2</span>)
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<span style="color: #408080; font-style: italic"># Define a simple 2-qubit QCBM circuit</span>
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<span style="color: #AA22FF">@qml</span><span style="color: #666666">.</span>qnode(dev)
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<span style="color: #008000; font-weight: bold">def</span> <span style="color: #0000FF">circuit</span>(params):
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<span style="color: #408080; font-style: italic"># params = 4 angles</span>
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qml<span style="color: #666666">.</span>RX(params[<span style="color: #666666">0</span>], wires<span style="color: #666666">=0</span>)
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qml<span style="color: #666666">.</span>RY(params[<span style="color: #666666">1</span>], wires<span style="color: #666666">=1</span>)
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qml<span style="color: #666666">.</span>CNOT(wires<span style="color: #666666">=</span>[<span style="color: #666666">0</span>, <span style="color: #666666">1</span>])
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qml<span style="color: #666666">.</span>RX(params[<span style="color: #666666">2</span>], wires<span style="color: #666666">=0</span>)
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qml<span style="color: #666666">.</span>RY(params[<span style="color: #666666">3</span>], wires<span style="color: #666666">=1</span>)
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<span style="color: #008000; font-weight: bold">return</span> qml<span style="color: #666666">.</span>probs(wires<span style="color: #666666">=</span>[<span style="color: #666666">0</span>,<span style="color: #666666">1</span>]) <span style="color: #408080; font-style: italic"># return probabilities of |00&gt;,|01&gt;,|10&gt;,|11&gt;</span>
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<span style="color: #408080; font-style: italic"># Initialize random parameters</span>
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params <span style="color: #666666">=</span> np<span style="color: #666666">.</span>random<span style="color: #666666">.</span>randn(<span style="color: #666666">4</span>, requires_grad<span style="color: #666666">=</span><span style="color: #008000; font-weight: bold">True</span>)
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<span style="color: #408080; font-style: italic"># Get the output distribution from the circuit</span>
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probs <span style="color: #666666">=</span> circuit(params)
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<span style="color: #008000">print</span>(<span style="color: #BA2121">&quot;Probabilities:&quot;</span>, probs)
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</pre>
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</div>
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</div>
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</div>
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</div>
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<div class="output_wrapper">
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<div class="output">
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<div class="output_area">
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<div class="output_subarea output_stream output_stdout output_text">
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</div>
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</div>
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</div>
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</div>
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<p>This circuit takes a parameter vector params of length \( n_v+n_h \) and
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returns the probabilities \( p_{\Theta}(v) \) of measuring each visible
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bitstring \( v \). Notice we measure only the visible wires (the
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wires=list(range(n$\_$v)) in qml.probs marginalizes out the hidden
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qubit).
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<p>In this code, we create a two-qubit variational circuit and return the
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probabilities of each basis state. One could train params to match a
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target distribution by defining a cost function between
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probs and the data distribution, and using PennyLane&#8217;s automatic
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differentiation to update params. Though not a true QBM (no
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Hamiltonian or Gibbs state), this demonstrates how one might build and
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train a small quantum generative model in PennyLane.
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</p>
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<!-- !split -->
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<h2 id="kullback-leibler-divergence" class="anchor">Kullback-Leibler divergence </h2>
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<p>Next, we train this model to match a target dataset distribution.
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Suppose our data has distribution target = \( [p(00), p(01), p(10),
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p(11)] \). We can define the (classical) loss as the Kullback-Leibler
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divergence \( D_{\mathrm{KL}}(p_{\mathrm{data}}\vert\vert q_\Theta) \) or simply the
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negative log-likelihood. Then we update params by gradient descent.
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PennyLane&#8217;s automatic differentiation can compute gradients via the
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parameter-shift rule, but we show an explicit parameter-shift
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computation for demonstration:
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<p>To make it more QBM-like, one could encode the classical visible units
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as fixed inputs. For example, to compute the energy of a QBM
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Hamiltonian for a given visible bitstring, one might do:
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</p>
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<div class="inner_cell">
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<div class="input_area">
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<div class="highlight" style="background: #f8f8f8">
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<pre style="line-height: 125%;">Initialize parameters <span style="color: #AA22FF; font-weight: bold">and</span> perform a simple gradient descent
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<pre style="line-height: 125%;"><span style="color: #408080; font-style: italic"># Example: compute expectation of a simple Hamiltonian for a given state |v&gt;</span>
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H <span style="color: #666666">=</span> <span style="color: #666666">1.5</span> <span style="color: #666666">*</span> qml<span style="color: #666666">.</span>PauliZ(<span style="color: #666666">0</span>) <span style="color: #666666">+</span> <span style="color: #666666">0.7</span> <span style="color: #666666">*</span> qml<span style="color: #666666">.</span>PauliZ(<span style="color: #666666">1</span>) <span style="color: #666666">+</span> <span style="color: #666666">0.9</span> <span style="color: #666666">*</span> qml<span style="color: #666666">.</span>PauliZ(<span style="color: #666666">0</span>)<span style="color: #AA22FF">@qml</span><span style="color: #666666">.</span>PauliZ(<span style="color: #666666">1</span>)
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<span style="color: #AA22FF">@qml</span><span style="color: #666666">.</span>qnode(dev)
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<span style="color: #008000; font-weight: bold">def</span> <span style="color: #0000FF">energy_of_state</span>():
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<span style="color: #408080; font-style: italic"># Prepare visible qubits in state |v&gt; = |01&gt;, say</span>
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qml<span style="color: #666666">.</span>PauliX(wires<span style="color: #666666">=1</span>) <span style="color: #408080; font-style: italic"># flips qubit 1 to |1&gt;</span>
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<span style="color: #408080; font-style: italic"># No hidden qubits in this simple example</span>
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<span style="color: #408080; font-style: italic"># Return expectation of H in this state</span>
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<span style="color: #008000; font-weight: bold">return</span> qml<span style="color: #666666">.</span>expval(H)
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<span style="color: #008000">print</span>(<span style="color: #BA2121">&quot;Energy of |01&gt; state:&quot;</span>, energy_of_state())
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</pre>
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</div>
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</div>
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<p>This code illustrates the training loop. At each epoch we evaluate
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the loss, compute gradients (via two forward passes per parameter),
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and update the parameters. In practice one can use qml.grad for
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automatic gradients, and more sophisticated optimizers (Adam, natural
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gradient, etc.). The above shows that PennyLane can seamlessly
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integrate quantum circuit definitions with classical training logic.
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<p>Here, we used a two-qubit Hamiltonian
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\( H=1.5\sigma_z^0+0.7\sigma_z^1+0.9\sigma_z^0\sigma_z^1 \) and prepared
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the state \( \vert v\rangle=\vert 01\rangle \). The <b>expval(H)</b> call returns
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</p>
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$$
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\langle 01 \vert H \vert 01\rangle = -1.5 + 0.7 - 0.9 = -1.7,
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$$
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<p>(up to sign
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conventions). This shows how PennyLane can compute energies of basis
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states, which is a key step in evaluating the QBM energy and
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probability of data states.
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</p>
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<p>In practice, training a QBM in PennyLane would involve preparing
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quantum states corresponding to the Gibbs distribution. One could use
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functions like qml.qaoa.cost.Hamiltonian or custom circuits to
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approximate \( \exp{-\beta H} \), and then use PennyLane&#8217;s optimizers. Also,
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PennyLane can interface with PyTorch or TensorFlow, enabling hybrid
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optimization of quantum circuits with classical parameters (e.g. the
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Hamiltonian weights).
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</p>
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<!-- !split -->
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<h2 id="summary-of-steps-in-pennylane-implementation" class="anchor">Summary of steps in PennyLane implementation </h2>
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<ol>
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<li> Define a quantum device (qml.device) with enough wires for visibles+hidden.</li>
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</ol>
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<p>0 Construct a parametric quantum circuit (ansatz) that depends on trainable parameters (e.g. angles in rotations and entangling gates).</p>
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<ol>
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<li> If modeling an RQBM, one might clamp visible qubits (preparing them according to training data) and apply gates only to hidden qubits.</li>
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<li> Measure relevant observables: one can return probabilities (probs) or expectation values (expval) of Pauli operators, depending on the chosen loss function.</li>
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<li> Define a cost function, such as the negative log-likelihood or a distance between output and target distribution.</li>
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</ol>
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<p>Use an optimizer (e.g. gradient descent, Adam) with PennyLane&#8217;s gradient calculations to update parameters.</p>
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