Quantum_Gradients_with_Backpropagation.ipynb

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        "# Quantum Gardients with BackPropagation\n",
        "\n",
        "Any quantum device, whether a hardware device or a simulator can be trained using the **parameter-shift rule**\n",
        "\n",
        "-----\n",
        "-----\n",
        "\n",
        "### Parameter-shift rule\n",
        "\n",
        "The output of a variational circuit (i.e the expectation of an observable) can be written as **quantum function** $f(\\theta)$ parameterized by $\\theta = \\theta_1 , \\theta_2...$\n",
        "\n",
        "The partial derivative of $f(\\theta)$ can in mnay cases be expressed as a linear combination of other quantum functions. Importantly, these other quantum functions typically use the same circuit, differing only in the shift of the argument. This means that partial derivatives of a variational circuit can be computed by using the same variational circuit architecture.\n",
        "\n",
        "Recipes of how to get partial derivatives by evaluated parameter-shifted instances of a variational circuit are called parameter-shift rules, and have been first introduced to quantum machine learning in [Mitarai et al. (2018)](https://arxiv.org/abs/1803.00745), and extended in [Schuld et al. (2018)](https://arxiv.org/abs/1811.11184).\n",
        "\n",
        "\n",
        "\n",
        "![alt text](https://pennylane.ai/qml/_images/gradients2.png)\n",
        "\n",
        "\n",
        "Making a rough analogy to classically computable functions, this is similar to how the derivative of the function $f(x) = \\sin(x)$ is identical to $\\frac{1}{2} \\sin(x +\\frac{\\pi}{2}) - \\frac{1}{2} \\sin(x - \\frac{\\pi}{2})$.\n",
        "\n",
        "So the same underlying algorithm can be reused to compute both $\\sin(x)$ and its derivative by evaluating at $x \\pm \\frac{\\pi}{2}$. This inituition holds for many quantum functions of interest :  \n",
        "\n",
        "*the same circuit can be used to compute both the quantum function and the gradient of the quantum function.*"
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        "## A more technical explaination\n",
        "\n",
        "Quantum Circuits are specified by sequence of gates. The unitary transformation carried out by the circuit thus be broken down into a product of unitaries:\n",
        "\n",
        "$$U(x; \\theta) = U_N(\\theta_{N}) U_{N-1}(\\theta_{N-1}) \\cdots U_i(\\theta_i) \\cdots U_1(\\theta_1) U_0(x).$$\n",
        "\n",
        "Each of these gates is unitary, and therefore must have the form $U_{j}(\\gamma_j)=\\exp{(i\\gamma_j H_j)}$ where $H_j$ is the Hermitian operator which generates the gate and $\\gamma_j$ is the gate parameter.\n",
        "\n",
        "\n",
        "\n"
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        "## A single Parameterised gate\n",
        "\n",
        "Let us single out a single parameter $\\theta_i$ and its associated gate $U_i(\\theta_i)$, For simplicity we remove all gates except $U_i(\\theta)$ and $U_o(x)$ for the moment. In this case, we have a simplified quantum circuit function\n",
        "\n",
        "$$f(x; \\theta_i) = \\langle 0 | U_0^\\dagger(x)U_i^\\dagger(\\theta_i)\\hat{B}U_i(\\theta_i)U_0(x) | 0 \\rangle = \\langle x | U_i^\\dagger(\\theta_i)\\hat{B}U_i(\\theta_i) | x \\rangle.$$\n",
        "\n",
        "For convenience, we rewrite the unitary conjugation as a linear transformation $\\mathcal{M}_{\\theta_i}$ acting on the operator $\\hat{B}$\n",
        "\n",
        "$$U_i^\\dagger(\\theta_i)\\hat{B}U_i(\\theta_i) = \\mathcal{M}_{\\theta_i}(\\hat{B}).$$\n",
        "\n",
        "The transformation $\\mathcal{M}_{\\theta_i}$ , depends smoothly on the parameter $\\theta_i$, so this qunatum function will have a well-defined gardient:\n",
        "\n",
        "$$\\nabla_{\\theta_i}f(x; \\theta_i) = \\langle x | \\nabla_{\\theta_i}\\mathcal{M}_{\\theta_i}(\\hat{B}) | x \\rangle \\in \\mathbb{R}.$$\n",
        "\n",
        "The key insight is that we can, in many cases of interest, express this gradient as a linear combination of the same transformation $\\mathcal{M}$, but with different parameters. Namely,\n",
        "\n",
        "$$\\nabla_{\\theta_i}\\mathcal{M}_{\\theta_i}(\\hat{B}) = c[\\mathcal{M}_{\\theta_i + s}(\\hat{B}) - \\mathcal{M}_{\\theta_i - s}(\\hat{B})],$$\n",
        "\n",
        "where the multiplier $c$ and the shift $s$ are determined completely by the type of transformation $\\mathcal{M}$ and independent of the value of $\\theta_i$\n",
        "\n",
        "\n",
        "\n"
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        "## Multiple Parameterized Gates\n",
        "\n",
        "To complete the story, we now go back to the case where there are many gates in the circuit. We can absorb any gates applied before gate i into the initial state:\n",
        "\n",
        "$$|\\psi_{i-1}\\rangle = U_{i-1}(\\theta_{i-1}) \\cdots U_{1}(\\theta_{1})U_{0}(x)|0\\rangle$$\n",
        "\n",
        "Similarly, any gates applied after gate $i$ are combined with the observable $\\hat{B}$:\n",
        "\n",
        "$$\\hat{B}_{i+1} = U_{N}^\\dagger(\\theta_{N}) \\cdots U_{i+1}^\\dagger(\\theta_{i+1}) \\hat{B} U_{i+1}(\\theta_{i+1}) \\cdots U_{N}(\\theta_{N}) $$\n",
        "\n",
        "With this simplification, the quantum circuit function becomes\n",
        "\n",
        "$$ f(x; \\theta) = \\langle \\psi_{i-1} | U_i^\\dagger(\\theta_i) \\hat{B}_{i+1} U_i(\\theta_i) | \\psi_{i-1} \\rangle = \\langle \\psi_{i-1} | \\mathcal{M}_{\\theta_i} (\\hat{B}_{i+1}) | \\psi_{i-1} \\rangle,$$\n",
        "\n",
        "and its gradient is \n",
        "\n",
        "$$ \\nabla_{\\theta_i}f(x; \\theta) = \\langle \\psi_{i-1} | \\nabla_{\\theta_i}\\mathcal{M}_{\\theta_i} (\\hat{B}_{i+1}) | \\psi_{i-1} \\rangle.$$\n",
        "\n",
        "This gradient has the exact same form as the single-gate case, except we modify the state $|x⟩ → |\\psi_{i-1}⟩$ and the measurement operator $\\hat{B}\\rightarrow\\hat{B}_{i+1}$. In terms of the circuit, this means we can leave all other gates as they are, and only modify gate $U(\\theta_i)$ when we want to differentiate with respect to the paramter $\\theta_i$\n",
        "\n",
        "------\n",
        "------\n"
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        "## Pauli gate example\n",
        "Consider a quantum computer with parameterized gates of the form,\n",
        "\n",
        "$$ U_i(\\theta_i)=\\exp\\left(-i\\tfrac{\\theta_i}{2}\\hat{P}_i\\right),$$\n",
        "\n",
        "where $\\hat{P}_i=\\hat{P}_i^\\dagger$ is a Pauli operator.\n",
        "\n",
        "The gradient of this unitary is \n",
        "\n",
        "$$\\nabla_{\\theta_i}U_i(\\theta_i) = -\\tfrac{i}{2}\\hat{P}_i U_i(\\theta_i) = -\\tfrac{i}{2}U_i(\\theta_i)\\hat{P}_i .$$\n",
        "\n",
        "Substituting this into the quantum circuit function $f(x;\\theta)$, we get\n",
        "\n",
        "$$\\begin{align}\n",
        "    \\nabla_{\\theta_i}f(x; \\theta) = &\n",
        "    \\frac{i}{2}\\langle \\psi_{i-1} | U_i^\\dagger(\\theta_i) \\left( P_i \\hat{B}_{i+1} - \\hat{B}_{i+1} P_i \\right) U_i(\\theta_i)| \\psi_{i-1} \\rangle \\\\\n",
        "    = & \\frac{i}{2}\\langle \\psi_{i-1} | U_i^\\dagger(\\theta_i) \\left[P_i, \\hat{B}_{i+1}\\right]U_i(\\theta_i) | \\psi_{i-1} \\rangle,\n",
        "\\end{align}$$\n",
        "\n",
        "\n",
        "where $[X,Y] = XY-YX$ is the commutator.\n",
        "\n",
        "\n",
        "We now make use of the following mathematical identity for the commutators involving Pauli operators\n",
        "\n",
        "$$ \\left[ \\hat{P}_i, \\hat{B} \\right] = -i\\left(U_i^\\dagger\\left(\\tfrac{\\pi}{2}\\right)\\hat{B}U_i\\left(\\tfrac{\\pi}{2}\\right) - U_i^\\dagger\\left(-\\tfrac{\\pi}{2}\\right)\\hat{B}U_i\\left(-\\tfrac{\\pi}{2}\\right) \\right). $$\n",
        "\n",
        "Substituing this into the previous equation, we obtain the gradient expression\n",
        "\n",
        "$$ \\begin{align}\n",
        "    \\nabla_{\\theta_i}f(x; \\theta) = & \\hphantom{-} \\tfrac{1}{2} \\langle \\psi_{i-1} | U_i^\\dagger\\left(\\theta_i + \\tfrac{\\pi}{2} \\right) \\hat{B}_{i+1} U_i\\left(\\theta_i + \\tfrac{\\pi}{2} \\right) | \\psi_{i-1} \\rangle \\\\\n",
        "    & - \\tfrac{1}{2} \\langle \\psi_{i-1} | U_i^\\dagger\\left(\\theta_i - \\tfrac{\\pi}{2} \\right) \\hat{B}_{i+1} U_i\\left(\\theta_i - \\tfrac{\\pi}{2} \\right) | \\psi_{i-1} \\rangle.\n",
        "\\end{align}$$\n",
        "\n",
        "Finally, we can rewrite this in terms of quantum function:\n",
        "\n",
        "$$ \\nabla_{\\theta}f(x; \\theta) = \\tfrac{1}{2}\\left[ f(x; \\theta + \\tfrac{\\pi}{2}) - f(x; \\theta - \\tfrac{\\pi}{2}) \\right].$$\n",
        "\n",
        "\n",
        "\n"
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        "------\n",
        "------\n",
        "\n",
        "## Gaussina gate example\n",
        "\n",
        "For a quantum device with continuous-valued operators, such as photonic quantum computers, it is convenient to employ the Heisenberg picture i.e to track how the gates $U_i(\\theta_i)$ transform the final measurement operator $\\hat{B}$\n",
        "\n",
        "As an example,we consider the [Squeezing gate](https://en.wikipedia.org/wiki/Squeeze_operator). In the Heisenberg picture, the Squeezing gate causes the quadrature operators $\\hat{x}$ and $\\hat{p}$ to become rescaled:\n",
        "\n",
        "$$\\begin{align}\n",
        "    \\mathcal{M}^S_r(\\hat{x}) = & S^\\dagger(r)\\hat{x}S(r) \\\\\n",
        "                                = & e^{-r}\\hat{x}\n",
        "\\end{align}$$\n",
        "\n",
        "and \n",
        "\n",
        "$$\\begin{align}\n",
        "    \\mathcal{M}^S_r(\\hat{p}) = & S^\\dagger(r)\\hat{p}S(r) \\\\\n",
        "                                = & e^{r}\\hat{p}.\n",
        "\\end{align} $$\n",
        "\n",
        "\n",
        "-----\n",
        "\n",
        "Expressing this in matrix notation, we have\n",
        "\n",
        "$$ \\begin{align}\n",
        "    \\begin{bmatrix}\n",
        "        \\hat{x} \\\\\n",
        "        \\hat{p}\n",
        "    \\end{bmatrix}\n",
        "    \\rightarrow\n",
        "    \\begin{bmatrix}\n",
        "       e^{-r} & 0 \\\\\n",
        "       0      & e^r\n",
        "    \\end{bmatrix}\n",
        "    \\begin{bmatrix}\n",
        "        \\hat{x} \\\\\n",
        "        \\hat{p}\n",
        "    \\end{bmatrix}.\n",
        "\\end{align}$$\n",
        "\n",
        "The gradient of this transformation can easily be found:\n",
        "\n",
        "$$\\begin{align}\n",
        "    \\nabla_r\n",
        "    \\begin{bmatrix}\n",
        "        e^{-r} & 0 \\\\\n",
        "        0 & e^r\n",
        "    \\end{bmatrix}\n",
        "    =\n",
        "    \\begin{bmatrix}\n",
        "        -e^{-r} & 0 \\\\\n",
        "        0 & e^r\n",
        "    \\end{bmatrix}.\n",
        "\\end{align} $$\n",
        "\n",
        "We notice that this can be rewritten as a linear combination of the squeeze operations:\n",
        "\n",
        "$$ \\begin{align}\n",
        "    \\begin{bmatrix}\n",
        "        -e^{-r} & 0 \\\\\n",
        "        0 & e^r\n",
        "    \\end{bmatrix}\n",
        "    =\n",
        "    \\frac{1}{2\\sinh(s)}\n",
        "    \\left(\n",
        "    \\begin{bmatrix}\n",
        "        e^{-(r+s)} & 0 \\\\\n",
        "        0 & e^{r+s}\n",
        "    \\end{bmatrix}\n",
        "    -\n",
        "    \\begin{bmatrix}\n",
        "        e^{-(r-s)} & 0 \\\\\n",
        "        0 & e^{r-s}\n",
        "    \\end{bmatrix}\n",
        "    \\right),\n",
        "\\end{align}$$\n",
        "\n",
        "where $s$ is an arbitrary nonzero shift.\n",
        "\n",
        "-----\n",
        " As before, assume that an input $y$ has already been embedded into a quantum state $|y\\rangle = U_0(y)|0\\rangle$ before we apply the squeeze gate. If we measure the $\\hat{x}$ operator, we will have the following quantum circuit function:\n",
        "\n",
        " $$f(y;r) = \\langle y | \\mathcal{M}^S_r (\\hat{x}) | y \\rangle. $$\n",
        "\n",
        "\n",
        "\n",
        "Finally, its gradient can be expressed as:\n",
        "\n",
        "\n",
        "$$\\begin{align}\n",
        "    \\nabla_r f(y;r) = &  \\frac{1}{2\\sinh(s)} \\left[\n",
        "                         \\langle y | \\mathcal{M}^S_{r+s} (\\hat{x}) | y \\rangle\n",
        "                        -\\langle y | \\mathcal{M}^S_{r-s} (\\hat{x}) | y \\rangle \\right] \\\\\n",
        "                    = & \\frac{1}{2\\sinh(s)}\\left[f(y; r+s) - f(y; r-s)\\right].\n",
        "\\end{align} $$"
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