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‎doc/pub/week12/html/week12-bs.html‎

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@@ -72,6 +72,22 @@
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None,
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'controlled-phase-rotation'),
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('Circuit Diagram', 2, None, 'circuit-diagram'),
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('Swap gates', 2, None, 'swap-gates'),
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('Swap gates and step-by-step mathematical derivation',
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2,
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None,
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'swap-gates-and-step-by-step-mathematical-derivation'),
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('Circuit‐level decomposition and implicit bit reversal',
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2,
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None,
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'circuit-level-decomposition-and-implicit-bit-reversal'),
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('Qubit ordering', 2, None, 'qubit-ordering'),
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('Change of basis order', 2, None, 'change-of-basis-order'),
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('The bit-reversal permutation and Swap gates',
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2,
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None,
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'the-bit-reversal-permutation-and-swap-gates'),
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('Can we avoid them?', 2, None, 'can-we-avoid-them'),
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('Python code with PennyLane',
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2,
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None,
@@ -213,6 +229,13 @@
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<!-- navigation toc: --> <li><a href="#qft-on-three-qubits" style="font-size: 80%;">QFT on three qubits</a></li>
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<!-- navigation toc: --> <li><a href="#controlled-phase-rotation" style="font-size: 80%;">Controlled Phase Rotation</a></li>
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<!-- navigation toc: --> <li><a href="#circuit-diagram" style="font-size: 80%;">Circuit Diagram</a></li>
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<!-- navigation toc: --> <li><a href="#swap-gates" style="font-size: 80%;">Swap gates</a></li>
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<!-- navigation toc: --> <li><a href="#swap-gates-and-step-by-step-mathematical-derivation" style="font-size: 80%;">Swap gates and step-by-step mathematical derivation</a></li>
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<!-- navigation toc: --> <li><a href="#circuit-level-decomposition-and-implicit-bit-reversal" style="font-size: 80%;">Circuit‐level decomposition and implicit bit reversal</a></li>
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<!-- navigation toc: --> <li><a href="#qubit-ordering" style="font-size: 80%;">Qubit ordering</a></li>
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<!-- navigation toc: --> <li><a href="#change-of-basis-order" style="font-size: 80%;">Change of basis order</a></li>
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<!-- navigation toc: --> <li><a href="#the-bit-reversal-permutation-and-swap-gates" style="font-size: 80%;">The bit-reversal permutation and Swap gates</a></li>
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<!-- navigation toc: --> <li><a href="#can-we-avoid-them" style="font-size: 80%;">Can we avoid them?</a></li>
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<!-- navigation toc: --> <li><a href="#python-code-with-pennylane" style="font-size: 80%;">Python code with PennyLane</a></li>
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<!-- navigation toc: --> <li><a href="#complexity" style="font-size: 80%;">Complexity</a></li>
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<!-- navigation toc: --> <li><a href="#inverse-qft-and-the-phase-estimation-algorithm" style="font-size: 80%;">Inverse QFT and the phase estimation algorithm</a></li>
@@ -464,6 +487,126 @@ <h2 id="circuit-diagram" class="anchor">Circuit Diagram </h2>
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</center>
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<br/><br/>
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<!-- !split -->
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<h2 id="swap-gates" class="anchor">Swap gates </h2>
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<p>In the circuit implementation of the \( n \)-qubit Quantum Fourier
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Transform (QFT), one finds at the end a cascade of Swap gates
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<em>crossing</em> qubit \( 1 \) with qubit \( n \), qubit \( 2 \) with qubit \( n-1 \), and
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so on. These are not arbitrary extra operations, but arise inevitably
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from the fact that the standard QFT&#8212;when written as a matrix&#8212;maps
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computational&#8208;basis states into <em>bit&#8208;reversed</em> amplitudes. The Swap
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gates simply undo that bit-reversal so that the output register has
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its qubits in the natural (most&#8208;significant&#8211;to&#8211;least&#8208;significant)
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order.
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</p>
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<!-- !split -->
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<h2 id="swap-gates-and-step-by-step-mathematical-derivation" class="anchor">Swap gates and step-by-step mathematical derivation </h2>
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<p>On an \( n \)-qubit register, label the computational basis states by integers</p>
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$$
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\vert x\rangle \quad,\quad x\in\{0,1,\dots,2^n-1\}\,.
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$$
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<p>The QFT is the linear map</p>
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$$
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\mathrm{QFT}n:\;\vert x \rangle \;\mapsto\;\frac{1}{\sqrt{2^n}}\sum{y=0}^{2^n-1}
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\exp{(2\pi i\,xy / 2^n)}\,\vert y\rangle.
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$$
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<p>Viewed as a \( 2^n\times2^n \) unitary matrix \( F_{2^n} \), its \( (y,x) \)&#8211;entry is</p>
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$$
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[F_{2^n}]_{y,x} \;=\;\frac{1}{\sqrt{2^n}}\,e^{2\pi i\,x y / 2^n}\,.
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$$
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<!-- !split -->
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<h2 id="circuit-level-decomposition-and-implicit-bit-reversal" class="anchor">Circuit&#8208;level decomposition and implicit bit reversal </h2>
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<p>A standard circuit for \( \mathrm{QFT}_n \) proceeds qubit by qubit (from
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most significant qubit down to least significant), applying:
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</p>
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<ol>
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<li> A Hadamard on qubit \( j \);</li>
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<li> Controlled-\( R_2,R_3,\dots,R_{n-j+1} \) phase gates from qubit \( k>j \) to qubit \( j \), where</li>
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</ol>
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$$
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R_m \;=\;\begin{pmatrix}1&0\\0&e^{2\pi i/2^m}\end{pmatrix}.
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$$
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<ol>
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<li> Move on to the next qubit \( j+1 \).</li>
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</ol>
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<!-- !split -->
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<h2 id="qubit-ordering" class="anchor">Qubit ordering </h2>
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<p>If you label the input qubits as
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\( \vert x_{n-1}\rangle\otimes\vert x_{n-2}\rangle\otimes\cdots\otimes\vert x_0\rangle \) (with
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\( x_{n-1} \) the most&#8208;significant bit), then the order in which
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phase&#8208;gates tie those bits ends up producing output amplitudes in the
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reverse bit&#8208;order, that is
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</p>
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$$
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\mathrm{QFT}n \;=\; (\text{controlled‐phase and Hadamard layers})\;\times\; \underbrace{\bigl(\mathrm{bit\!-\!reversal}\bigr)}{P},
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$$
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<p>where \( P \) is the permutation matrix that sends</p>
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$$
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\vert y_{n-1}y_{n-2}\dots y_0\rangle \longmapsto \vert y_{0}y_{1}\dots y_{n-1}\rangle.
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$$
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<!-- !split -->
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<h2 id="change-of-basis-order" class="anchor">Change of basis order </h2>
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<p>Concretely, one finds for each basis vector</p>
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$$
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(\text{circuit without swaps})\;\vert x_{n-1}\dots x_0\rangle\;=\;\frac{1}{\sqrt{2^n}}
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\sum_{y} \exp{(2\pi i\,x\,y/2^n)}\;\vert y_{\text{reversed}}\rangle,
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$$
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<p>where \( y_{\text{reversed}} = \sum_j y_j,2^{n-1-j} \) instead of the natural \( \sum_j y_j,2^j \).</p>
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<!-- !split -->
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<h2 id="the-bit-reversal-permutation-and-swap-gates" class="anchor">The bit-reversal permutation and Swap gates </h2>
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<p>The bit-reversal permutation \( P \) is exactly the unitary that swaps
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qubit \( j \) with qubit \( (n-1-j) \) for all
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\( j=0,\dots,\lfloor(n-1)/2\rfloor \). In matrix form,
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</p>
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$$
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\Pi \;=\;\prod_{j=0}^{\lfloor(n-1)/2\rfloor}\;\mathrm{SWAP}{j,\;n-1-j},
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$$
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<p>where each \( \mathrm{SWAP}{a,b} \) acts as</p>
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$$
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\mathrm{SWAP}:\;\vert u\rangle_a\!\otimes\!\vert v\rangle_b \;\longmapsto\;\vert v\rangle_a\!\otimes\!\vert u\rangle_b.
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$$
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<p>Since the circuit we built implements \( F{2^n},\Pi \) rather than
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\( F{2^n} \), appending those swaps at the end yields exactly the matrix
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\( F{2^n} \) in the natural qubit ordering.
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</p>
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<!-- !split -->
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<h2 id="can-we-avoid-them" class="anchor">Can we avoid them? </h2>
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<p>One might ask: <em>Couldn&#8217;t we just re&#8208;label wires?</em> In principle yes,
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but in a physical device the wiring corresponds to real qubit lines.
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If you want the logical output bit \( y_{n-1} \) on the same physical wire
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as the input \( x_{n-1} \), you must physically swap the states. Hence
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the Swap gates are unavoidable whenever you demand that qubit indices
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retain their original semantic meaning.
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</p>
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<p>In summary, the Swap gates at the end of the QFT circuit correct the
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inherent bit-reversal that arises from the order in which Hadamard and
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controlled-phase gates entangle the qubits. Without them, the output
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register would hold the Fourier&#8208;transformed amplitudes in
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reverse&#8208;index order.
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</p>
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<!-- !split -->
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<h2 id="python-code-with-pennylane" class="anchor">Python code with PennyLane </h2>
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<p>We now implement a full QFT circuit manually for three qubits using PennyLane.</p>

‎doc/pub/week12/html/week12-reveal.html‎

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@@ -399,6 +399,152 @@ <h2 id="circuit-diagram">Circuit Diagram </h2>
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<br/><br/>
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</section>
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<section>
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<h2 id="swap-gates">Swap gates </h2>
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<p>In the circuit implementation of the \( n \)-qubit Quantum Fourier
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Transform (QFT), one finds at the end a cascade of Swap gates
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<em>crossing</em> qubit \( 1 \) with qubit \( n \), qubit \( 2 \) with qubit \( n-1 \), and
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so on. These are not arbitrary extra operations, but arise inevitably
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from the fact that the standard QFT&#8212;when written as a matrix&#8212;maps
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computational&#8208;basis states into <em>bit&#8208;reversed</em> amplitudes. The Swap
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gates simply undo that bit-reversal so that the output register has
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its qubits in the natural (most&#8208;significant&#8211;to&#8211;least&#8208;significant)
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order.
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</p>
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</section>
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<section>
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<h2 id="swap-gates-and-step-by-step-mathematical-derivation">Swap gates and step-by-step mathematical derivation </h2>
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<p>On an \( n \)-qubit register, label the computational basis states by integers</p>
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<p>&nbsp;<br>
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$$
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\vert x\rangle \quad,\quad x\in\{0,1,\dots,2^n-1\}\,.
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$$
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<p>&nbsp;<br>
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<p>The QFT is the linear map</p>
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<p>&nbsp;<br>
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$$
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\mathrm{QFT}n:\;\vert x \rangle \;\mapsto\;\frac{1}{\sqrt{2^n}}\sum{y=0}^{2^n-1}
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\exp{(2\pi i\,xy / 2^n)}\,\vert y\rangle.
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$$
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<p>&nbsp;<br>
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<p>Viewed as a \( 2^n\times2^n \) unitary matrix \( F_{2^n} \), its \( (y,x) \)&#8211;entry is</p>
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<p>&nbsp;<br>
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$$
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[F_{2^n}]_{y,x} \;=\;\frac{1}{\sqrt{2^n}}\,e^{2\pi i\,x y / 2^n}\,.
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$$
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<p>&nbsp;<br>
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</section>
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<section>
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<h2 id="circuit-level-decomposition-and-implicit-bit-reversal">Circuit&#8208;level decomposition and implicit bit reversal </h2>
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<p>A standard circuit for \( \mathrm{QFT}_n \) proceeds qubit by qubit (from
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most significant qubit down to least significant), applying:
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</p>
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<ol>
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<p><li> A Hadamard on qubit \( j \);</li>
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<p><li> Controlled-\( R_2,R_3,\dots,R_{n-j+1} \) phase gates from qubit \( k>j \) to qubit \( j \), where</li>
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</ol>
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<p>
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<p>&nbsp;<br>
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$$
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R_m \;=\;\begin{pmatrix}1&0\\0&e^{2\pi i/2^m}\end{pmatrix}.
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$$
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<p>&nbsp;<br>
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<ol>
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<p><li> Move on to the next qubit \( j+1 \).</li>
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</ol>
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</section>
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<section>
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<h2 id="qubit-ordering">Qubit ordering </h2>
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<p>If you label the input qubits as
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\( \vert x_{n-1}\rangle\otimes\vert x_{n-2}\rangle\otimes\cdots\otimes\vert x_0\rangle \) (with
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\( x_{n-1} \) the most&#8208;significant bit), then the order in which
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phase&#8208;gates tie those bits ends up producing output amplitudes in the
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reverse bit&#8208;order, that is
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</p>
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<p>&nbsp;<br>
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$$
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\mathrm{QFT}n \;=\; (\text{controlled‐phase and Hadamard layers})\;\times\; \underbrace{\bigl(\mathrm{bit\!-\!reversal}\bigr)}{P},
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$$
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<p>&nbsp;<br>
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<p>where \( P \) is the permutation matrix that sends</p>
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<p>&nbsp;<br>
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$$
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\vert y_{n-1}y_{n-2}\dots y_0\rangle \longmapsto \vert y_{0}y_{1}\dots y_{n-1}\rangle.
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$$
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<p>&nbsp;<br>
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</section>
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<section>
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<h2 id="change-of-basis-order">Change of basis order </h2>
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<p>Concretely, one finds for each basis vector</p>
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<p>&nbsp;<br>
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$$
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(\text{circuit without swaps})\;\vert x_{n-1}\dots x_0\rangle\;=\;\frac{1}{\sqrt{2^n}}
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\sum_{y} \exp{(2\pi i\,x\,y/2^n)}\;\vert y_{\text{reversed}}\rangle,
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$$
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<p>&nbsp;<br>
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<p>where \( y_{\text{reversed}} = \sum_j y_j,2^{n-1-j} \) instead of the natural \( \sum_j y_j,2^j \).</p>
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</section>
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<section>
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<h2 id="the-bit-reversal-permutation-and-swap-gates">The bit-reversal permutation and Swap gates </h2>
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<p>The bit-reversal permutation \( P \) is exactly the unitary that swaps
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qubit \( j \) with qubit \( (n-1-j) \) for all
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\( j=0,\dots,\lfloor(n-1)/2\rfloor \). In matrix form,
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</p>
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<p>&nbsp;<br>
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$$
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\Pi \;=\;\prod_{j=0}^{\lfloor(n-1)/2\rfloor}\;\mathrm{SWAP}{j,\;n-1-j},
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$$
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<p>&nbsp;<br>
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<p>where each \( \mathrm{SWAP}{a,b} \) acts as</p>
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<p>&nbsp;<br>
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$$
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\mathrm{SWAP}:\;\vert u\rangle_a\!\otimes\!\vert v\rangle_b \;\longmapsto\;\vert v\rangle_a\!\otimes\!\vert u\rangle_b.
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$$
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<p>&nbsp;<br>
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<p>Since the circuit we built implements \( F{2^n},\Pi \) rather than
524+
\( F{2^n} \), appending those swaps at the end yields exactly the matrix
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\( F{2^n} \) in the natural qubit ordering.
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</p>
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</section>
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<section>
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<h2 id="can-we-avoid-them">Can we avoid them? </h2>
531+
532+
<p>One might ask: <em>Couldn&#8217;t we just re&#8208;label wires?</em> In principle yes,
533+
but in a physical device the wiring corresponds to real qubit lines.
534+
If you want the logical output bit \( y_{n-1} \) on the same physical wire
535+
as the input \( x_{n-1} \), you must physically swap the states. Hence
536+
the Swap gates are unavoidable whenever you demand that qubit indices
537+
retain their original semantic meaning.
538+
</p>
539+
540+
<p>In summary, the Swap gates at the end of the QFT circuit correct the
541+
inherent bit-reversal that arises from the order in which Hadamard and
542+
controlled-phase gates entangle the qubits. Without them, the output
543+
register would hold the Fourier&#8208;transformed amplitudes in
544+
reverse&#8208;index order.
545+
</p>
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</section>
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<section>
403549
<h2 id="python-code-with-pennylane">Python code with PennyLane </h2>
404550
<p>We now implement a full QFT circuit manually for three qubits using PennyLane.</p>

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