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‎doc/pub/week15/html/week15-bs.html‎

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@@ -2034,7 +2034,7 @@ <h2 id="setting-up-a-vqc" class="anchor">Setting up a VQC </h2>
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</p>
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$$
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\vert \Psi(\mathbf{x};\boldsymbol\Theta)\rangle = W(\boldsymbol\Theta),U(\mathbf{x}),|0\rangle^{\otimes n}.
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\vert \Psi(\mathbf{x};\boldsymbol\Theta)\rangle = W(\boldsymbol\Theta)U(\mathbf{x})|0\rangle^{\otimes n}.
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$$
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<p>For instance, one common ansatz is the hardware-efficient circuit:
@@ -2052,7 +2052,7 @@ <h2 id="outputs" class="anchor">Outputs </h2>
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given by the expectation values:
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</p>
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$$
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f_k(\mathbf{x};\boldsymbol\Theta) ;=; \langle \Psi(\mathbf{x};\boldsymbol\Theta) | \hat B_k | \Psi(\mathbf{x};\boldsymbol\Theta)\rangle.
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f_k(\mathbf{x};\boldsymbol\Theta) = \langle \Psi(\mathbf{x};\boldsymbol\Theta) | \hat B_k | \Psi(\mathbf{x};\boldsymbol\Theta)\rangle.
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$$
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<p>Equivalently, with</p>
@@ -2062,7 +2062,7 @@ <h2 id="outputs" class="anchor">Outputs </h2>
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<p>one has</p>
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$$
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f_k(\mathbf{x};\boldsymbol\Theta) = \langle 0|U(\mathbf{x})^\dagger W(\boldsymbol\Theta)^\dagger ,\hat B_k, W(\boldsymbol\Theta) U(\mathbf{x}),|0\rangle.
2065+
f_k(\mathbf{x};\boldsymbol\Theta) = \langle 0|U(\mathbf{x})^\dagger W(\boldsymbol\Theta)^\dagger\hat B_k W(\boldsymbol\Theta) U(\mathbf{x})|0\rangle.
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$$
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<p>Commonly \( \hat B \) is a Pauli operator (e.g. \( Z \) on one qubit). In practice one runs many shots on quantum hardware or simulates this circuit classically to estimate \( \langle \hat B_k\rangle \) .</p>
@@ -2091,15 +2091,15 @@ <h2 id="mathematical-example" class="anchor">Mathematical example </h2>
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<p>and a variational layer is</p>
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$$
2094-
V(\boldsymbol\Theta)=R_y(\Theta_1)\otimes R_y(\Theta_2),\mathrm{CNOT}(0,1),
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V(\boldsymbol\Theta)=R_y(\Theta_1)\otimes R_y(\Theta_2)\mathrm{CNOT}(0,1),
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$$
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<p>(apply \( R_y \) on each qubit then entangle). After
2098-
applying \( W(\boldsymbol\Theta)=V(\boldsymbol\Theta) \) to \( |0,0\rangle \),
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applying \( W(\boldsymbol\Theta)=V(\boldsymbol\Theta) \) to \( |00\rangle \),
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we measure \( \hat B=Z\otimes I \) on qubit 0. The output is
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</p>
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$$
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f(\mathbf{x};\boldsymbol\Theta) = \langle 0,0|,U(\mathbf{x})^\dagger,V(\boldsymbol\Theta)^\dagger, (Z\otimes I), V(\boldsymbol\Theta),U(\mathbf{x}),|0,0\rangle.
2102+
f(\mathbf{x};\boldsymbol\Theta) = \langle 00|U(\mathbf{x})^\dagger V(\boldsymbol\Theta)^\dagger (Z\otimes I)V(\boldsymbol\Theta)U(\mathbf{x})|00\rangle.
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$$
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<p>This \( f(x;\Theta) \) is then compared to the target in a cost function for optimization.</p>

‎doc/pub/week15/html/week15-reveal.html‎

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@@ -1885,7 +1885,7 @@ <h2 id="setting-up-a-vqc">Setting up a VQC </h2>
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<p>&nbsp;<br>
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$$
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\vert \Psi(\mathbf{x};\boldsymbol\Theta)\rangle = W(\boldsymbol\Theta),U(\mathbf{x}),|0\rangle^{\otimes n}.
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\vert \Psi(\mathbf{x};\boldsymbol\Theta)\rangle = W(\boldsymbol\Theta)U(\mathbf{x})|0\rangle^{\otimes n}.
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$$
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<p>&nbsp;<br>
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@@ -1906,7 +1906,7 @@ <h2 id="outputs">Outputs </h2>
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</p>
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<p>&nbsp;<br>
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$$
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f_k(\mathbf{x};\boldsymbol\Theta) ;=; \langle \Psi(\mathbf{x};\boldsymbol\Theta) | \hat B_k | \Psi(\mathbf{x};\boldsymbol\Theta)\rangle.
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f_k(\mathbf{x};\boldsymbol\Theta) = \langle \Psi(\mathbf{x};\boldsymbol\Theta) | \hat B_k | \Psi(\mathbf{x};\boldsymbol\Theta)\rangle.
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$$
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<p>&nbsp;<br>
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@@ -1920,7 +1920,7 @@ <h2 id="outputs">Outputs </h2>
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<p>one has</p>
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<p>&nbsp;<br>
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$$
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f_k(\mathbf{x};\boldsymbol\Theta) = \langle 0|U(\mathbf{x})^\dagger W(\boldsymbol\Theta)^\dagger ,\hat B_k, W(\boldsymbol\Theta) U(\mathbf{x}),|0\rangle.
1923+
f_k(\mathbf{x};\boldsymbol\Theta) = \langle 0|U(\mathbf{x})^\dagger W(\boldsymbol\Theta)^\dagger\hat B_k W(\boldsymbol\Theta) U(\mathbf{x})|0\rangle.
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$$
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<p>&nbsp;<br>
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@@ -1955,17 +1955,17 @@ <h2 id="mathematical-example">Mathematical example </h2>
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<p>and a variational layer is</p>
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<p>&nbsp;<br>
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$$
1958-
V(\boldsymbol\Theta)=R_y(\Theta_1)\otimes R_y(\Theta_2),\mathrm{CNOT}(0,1),
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V(\boldsymbol\Theta)=R_y(\Theta_1)\otimes R_y(\Theta_2)\mathrm{CNOT}(0,1),
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$$
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<p>&nbsp;<br>
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<p>(apply \( R_y \) on each qubit then entangle). After
1963-
applying \( W(\boldsymbol\Theta)=V(\boldsymbol\Theta) \) to \( |0,0\rangle \),
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applying \( W(\boldsymbol\Theta)=V(\boldsymbol\Theta) \) to \( |00\rangle \),
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we measure \( \hat B=Z\otimes I \) on qubit 0. The output is
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</p>
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<p>&nbsp;<br>
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$$
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f(\mathbf{x};\boldsymbol\Theta) = \langle 0,0|,U(\mathbf{x})^\dagger,V(\boldsymbol\Theta)^\dagger, (Z\otimes I), V(\boldsymbol\Theta),U(\mathbf{x}),|0,0\rangle.
1968+
f(\mathbf{x};\boldsymbol\Theta) = \langle 00|U(\mathbf{x})^\dagger V(\boldsymbol\Theta)^\dagger (Z\otimes I)V(\boldsymbol\Theta)U(\mathbf{x})|00\rangle.
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$$
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<p>&nbsp;<br>
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‎doc/pub/week15/html/week15-solarized.html‎

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@@ -1933,7 +1933,7 @@ <h2 id="setting-up-a-vqc">Setting up a VQC </h2>
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</p>
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$$
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\vert \Psi(\mathbf{x};\boldsymbol\Theta)\rangle = W(\boldsymbol\Theta),U(\mathbf{x}),|0\rangle^{\otimes n}.
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\vert \Psi(\mathbf{x};\boldsymbol\Theta)\rangle = W(\boldsymbol\Theta)U(\mathbf{x})|0\rangle^{\otimes n}.
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$$
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<p>For instance, one common ansatz is the hardware-efficient circuit:
@@ -1951,7 +1951,7 @@ <h2 id="outputs">Outputs </h2>
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given by the expectation values:
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</p>
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$$
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f_k(\mathbf{x};\boldsymbol\Theta) ;=; \langle \Psi(\mathbf{x};\boldsymbol\Theta) | \hat B_k | \Psi(\mathbf{x};\boldsymbol\Theta)\rangle.
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f_k(\mathbf{x};\boldsymbol\Theta) = \langle \Psi(\mathbf{x};\boldsymbol\Theta) | \hat B_k | \Psi(\mathbf{x};\boldsymbol\Theta)\rangle.
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$$
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<p>Equivalently, with</p>
@@ -1961,7 +1961,7 @@ <h2 id="outputs">Outputs </h2>
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<p>one has</p>
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$$
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f_k(\mathbf{x};\boldsymbol\Theta) = \langle 0|U(\mathbf{x})^\dagger W(\boldsymbol\Theta)^\dagger ,\hat B_k, W(\boldsymbol\Theta) U(\mathbf{x}),|0\rangle.
1964+
f_k(\mathbf{x};\boldsymbol\Theta) = \langle 0|U(\mathbf{x})^\dagger W(\boldsymbol\Theta)^\dagger\hat B_k W(\boldsymbol\Theta) U(\mathbf{x})|0\rangle.
19651965
$$
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19671967
<p>Commonly \( \hat B \) is a Pauli operator (e.g. \( Z \) on one qubit). In practice one runs many shots on quantum hardware or simulates this circuit classically to estimate \( \langle \hat B_k\rangle \) .</p>
@@ -1990,15 +1990,15 @@ <h2 id="mathematical-example">Mathematical example </h2>
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19911991
<p>and a variational layer is</p>
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$$
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V(\boldsymbol\Theta)=R_y(\Theta_1)\otimes R_y(\Theta_2),\mathrm{CNOT}(0,1),
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V(\boldsymbol\Theta)=R_y(\Theta_1)\otimes R_y(\Theta_2)\mathrm{CNOT}(0,1),
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$$
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19961996
<p>(apply \( R_y \) on each qubit then entangle). After
1997-
applying \( W(\boldsymbol\Theta)=V(\boldsymbol\Theta) \) to \( |0,0\rangle \),
1997+
applying \( W(\boldsymbol\Theta)=V(\boldsymbol\Theta) \) to \( |00\rangle \),
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we measure \( \hat B=Z\otimes I \) on qubit 0. The output is
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</p>
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$$
2001-
f(\mathbf{x};\boldsymbol\Theta) = \langle 0,0|,U(\mathbf{x})^\dagger,V(\boldsymbol\Theta)^\dagger, (Z\otimes I), V(\boldsymbol\Theta),U(\mathbf{x}),|0,0\rangle.
2001+
f(\mathbf{x};\boldsymbol\Theta) = \langle 00|U(\mathbf{x})^\dagger V(\boldsymbol\Theta)^\dagger (Z\otimes I)V(\boldsymbol\Theta)U(\mathbf{x})|00\rangle.
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$$
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<p>This \( f(x;\Theta) \) is then compared to the target in a cost function for optimization.</p>

‎doc/pub/week15/html/week15.html‎

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@@ -2010,7 +2010,7 @@ <h2 id="setting-up-a-vqc">Setting up a VQC </h2>
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</p>
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$$
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\vert \Psi(\mathbf{x};\boldsymbol\Theta)\rangle = W(\boldsymbol\Theta),U(\mathbf{x}),|0\rangle^{\otimes n}.
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\vert \Psi(\mathbf{x};\boldsymbol\Theta)\rangle = W(\boldsymbol\Theta)U(\mathbf{x})|0\rangle^{\otimes n}.
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$$
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<p>For instance, one common ansatz is the hardware-efficient circuit:
@@ -2028,7 +2028,7 @@ <h2 id="outputs">Outputs </h2>
20282028
given by the expectation values:
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</p>
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$$
2031-
f_k(\mathbf{x};\boldsymbol\Theta) ;=; \langle \Psi(\mathbf{x};\boldsymbol\Theta) | \hat B_k | \Psi(\mathbf{x};\boldsymbol\Theta)\rangle.
2031+
f_k(\mathbf{x};\boldsymbol\Theta) = \langle \Psi(\mathbf{x};\boldsymbol\Theta) | \hat B_k | \Psi(\mathbf{x};\boldsymbol\Theta)\rangle.
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$$
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<p>Equivalently, with</p>
@@ -2038,7 +2038,7 @@ <h2 id="outputs">Outputs </h2>
20382038

20392039
<p>one has</p>
20402040
$$
2041-
f_k(\mathbf{x};\boldsymbol\Theta) = \langle 0|U(\mathbf{x})^\dagger W(\boldsymbol\Theta)^\dagger ,\hat B_k, W(\boldsymbol\Theta) U(\mathbf{x}),|0\rangle.
2041+
f_k(\mathbf{x};\boldsymbol\Theta) = \langle 0|U(\mathbf{x})^\dagger W(\boldsymbol\Theta)^\dagger\hat B_k W(\boldsymbol\Theta) U(\mathbf{x})|0\rangle.
20422042
$$
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20442044
<p>Commonly \( \hat B \) is a Pauli operator (e.g. \( Z \) on one qubit). In practice one runs many shots on quantum hardware or simulates this circuit classically to estimate \( \langle \hat B_k\rangle \) .</p>
@@ -2067,15 +2067,15 @@ <h2 id="mathematical-example">Mathematical example </h2>
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20682068
<p>and a variational layer is</p>
20692069
$$
2070-
V(\boldsymbol\Theta)=R_y(\Theta_1)\otimes R_y(\Theta_2),\mathrm{CNOT}(0,1),
2070+
V(\boldsymbol\Theta)=R_y(\Theta_1)\otimes R_y(\Theta_2)\mathrm{CNOT}(0,1),
20712071
$$
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20732073
<p>(apply \( R_y \) on each qubit then entangle). After
2074-
applying \( W(\boldsymbol\Theta)=V(\boldsymbol\Theta) \) to \( |0,0\rangle \),
2074+
applying \( W(\boldsymbol\Theta)=V(\boldsymbol\Theta) \) to \( |00\rangle \),
20752075
we measure \( \hat B=Z\otimes I \) on qubit 0. The output is
20762076
</p>
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$$
2078-
f(\mathbf{x};\boldsymbol\Theta) = \langle 0,0|,U(\mathbf{x})^\dagger,V(\boldsymbol\Theta)^\dagger, (Z\otimes I), V(\boldsymbol\Theta),U(\mathbf{x}),|0,0\rangle.
2078+
f(\mathbf{x};\boldsymbol\Theta) = \langle 00|U(\mathbf{x})^\dagger V(\boldsymbol\Theta)^\dagger (Z\otimes I)V(\boldsymbol\Theta)U(\mathbf{x})|00\rangle.
20792079
$$
20802080

20812081
<p>This \( f(x;\Theta) \) is then compared to the target in a cost function for optimization.</p>

‎doc/pub/week15/ipynb/ipynb-week15-src.tar.gz‎

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