Notebooks/ToyQubitization.ipynb

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    "# Qubitization methodology for simulating Coupled Classical Harmonic Oscillators using Classiq\n",
    "Authors: Viraj Dsouza, Cristina Radian, Kerem Yurtseven\n",
    "\n",
    "Date: 09.08.2024\n",
    "\n",
    "This work is designed under and for the final project of the Womanium Quantum+AI Program 2024\n",
    "\n",
    "Reference Paper: Exponential Quantum Speedup in Simulating Coupled Classical Oscillators, Babbush et.al, 2023 [1](https://arxiv.org/abs/2303.13012)\n",
    "\n",
    "We also refer the readers to [Clasiqdocs](https://github.com/Classiq/classiq-library/tree/9c43f05f3d498c8c72be7dcb3ecdaba85d9abd6e/tutorials/hamiltonian_simulation/hamiltonian_simulation_with_block_encoding) for the qubitization methodology and use this for the purpose of our work.\n",
    "___\n"
   ]
  },
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   "cell_type": "markdown",
   "metadata": {},
   "source": [
    " \n",
    "\n",
    "\n",
    "## Qubitization\n",
    "\n",
    "Our purpose in this notebook is to block encode the Hamiltonian evolution matrix such that:\n",
    "\\begin{equation*}\n",
    "    U = \\begin{pmatrix} \\text{exp}(iHt)/ \\lambda & * \\\\\n",
    "    * & *\n",
    "\\end{pmatrix}\n",
    "\\end{equation*}\n",
    "where $H$ is the system Hamiltonian and $\\lambda$ is the normalization factor and use that unitary matrix to create a random walk operator and simulate the behavior of the system at a given $t$.\n",
    "\\begin{equation*}\n",
    "    \\ket{\\psi(t)} = e^{-iHt} \\ket{\\psi(0)}\n",
    "\\end{equation*}\n",
    "\n",
    "In order to create a block encoding, we first imply LCU method to the system Hamiltonian to first form a block encoding of $H$.\n",
    "___"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Linear Combination of Unitaries\n",
    "\n",
    "Assume $H$ can be written in Pauli basis as:\n",
    "\\begin{equation*}\n",
    "    H = \\sum_{i=0}^{L-1} \\alpha_i U_i\n",
    "\\end{equation*}\n",
    "\n",
    "LCU refers to $(\\bar{\\alpha}, m, 0)$-block encoding of a matrix given as a sum of unitaries:\n",
    "$$\n",
    "U_{(\\bar{\\alpha},m,0)-H} =\\begin{pmatrix}\n",
    "H/\\bar{\\alpha} & * \\\\\n",
    "* & *\n",
    "\\end{pmatrix}, \\text{   for     } H = \\sum^{L-1}_{i=0} \\alpha_i U_i, \\,\\,\\,\\,\\, \\alpha_i\\geq 0\n",
    "$$\n",
    "with $\\bar{\\alpha}\\equiv\\sum^{L-1}_{i=0}\\alpha_i$ and $m= \\lceil\\log_2(L) \\rceil$ represents the numner of block qubits. \n",
    "\n",
    "\n",
    "More details on the LCU methodology can be found in classiq's [LCU tutorial](https://github.com/Classiq/classiq-library/blob/main/tutorials/linear_combination_of_unitaries/linear_combination_of_unitaries.ipynb)\n",
    "\n",
    "We can define the normalization as $ \\lambda = \\sum |\\alpha_i|$ and find the `PREPARE` and `SELECT` functions.\n",
    "___ "
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Quantum Walk-Operator and block encoding $e^{iHt}$\n",
    "To simulate the Hamiltonian, we can use the Szegedy Quantum Walk-Operator [2](https://ieeexplore.ieee.org/abstract/document/1366222).\n",
    "\n",
    "\n",
    "\n",
    "\n",
    "Given block-encoding of the Hamiltonian $H$, we can define the Szegedy quanutm walk-operator as: \n",
    "$$\n",
    "W\\equiv -\\Pi_{|0\\rangle_m} U_{(\\bar{\\alpha},m,0)-H},\n",
    "$$  \n",
    "where $\\Pi_{|0\\rangle_m}$ is a reflection operator about the block state. The powers of $W$ correspond to a $(1,m,0)$ block encoding of the Chebyshev polynomials of the matrix $H$:\n",
    "$$\n",
    "W^k = \\begin{pmatrix}\n",
    "T_k(H) & * \\\\\n",
    "* & *\n",
    "\\end{pmatrix}=U_{(1,m,0)-T_k(H)},\n",
    "$$\n",
    "with $T_k$ being the k-th Chebyshev polynomial. Its useful to note that this property holds only for an **Hermitian block-encoding**. From the Jacobi -Anger expansion (given below) we can have an $\\epsilon$-approximation of $\\exp(iHt)\\approx \\sum^d_{i=0} \\beta_{i} T_{i}(H)$, for which we can perform the following encoding\n",
    "$$\n",
    "U_{(\\bar{\\beta},\\tilde{m},\\epsilon)-\\exp{(iHt)}} =\n",
    "\\begin{pmatrix}\n",
    "\\exp{(iHt)}/\\bar{\\beta} & * \\\\\n",
    "* & *\n",
    "\\end{pmatrix}=\n",
    "% \\begin{pmatrix}\n",
    "% \\frac{1}{\\bar{\\beta}}\\sum^d_{k=0} \\beta_{k} T_{k}(Ht) & * \\\\\n",
    "% * & *\n",
    "% \\end{pmatrix}=\n",
    "\\begin{pmatrix}\n",
    "\\frac{1}{\\bar{\\beta}}\\sum^d_{k=0} \\beta_{k} W^k & * \\\\\n",
    "* & *\n",
    "\\end{pmatrix}\n",
    "% \\begin{pmatrix}\n",
    "% \\frac{1}{\\bar{\\beta}}\\sum^d_{k=0} W^k & * \\\\\n",
    "% * & *\n",
    "% \\end{pmatrix},\n",
    "$$\n",
    "where $\\tilde{m}=m+\\lceil \\log_2(d+1) \\rceil$ \n",
    "\n",
    "We evaluate the coefficients $\\beta_{i}$ as the bessel coefficients ($J(t)$) of the Jacobi-anger expressions for sine and cosine as follows\n",
    "$$\n",
    "\\cos(Ht)= J_0(t) + 2\\sum^{d}_{k=1} (-1)^k J_{2k}(t) T_{2k}(H)\\\\\n",
    "\\sin(Ht)= 2\\sum^{d}_{k=0} (-1)^k J_{2k+1}(t) T_{2k+1}(H),\n",
    "$$\n",
    "___"
   ]
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"
    }
   },
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "##  System Configuration\n",
    "In this example, let exist two masses with $m_1 = 1$ and $m_2 = 1$. Also, assume spring constants are $\\kappa_{11} = \\kappa_{22} = 0$ and $\\kappa_{12} = 1$. Initial states are chosen as $\\vec{x}(t) = (0,1)^T$ and $\\vec{\\dot{x}}(t) = (0,0)^T$\n",
    "\n",
    "![Toy_Example.png](attachment:Toy_Example.png)\n",
    "___"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Pre-Processing\n",
    "\n",
    "Create necessary matrix and initial states using functions created in `ccho_helpers.py`"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 1,
   "metadata": {},
   "outputs": [],
   "source": [
    "## Import Libraries\n",
    "import numpy as np\n",
    "import matplotlib.pyplot as plt\n",
    "import scipy\n",
    "from ccho_helpers import *\n",
    "from classiq import * \n",
    "from classiq.execution import ExecutionPreferences, ClassiqBackendPreferences, IBMBackendPreferences, IBMBackendProvider, IonqBackendPreferences, GCPBackendPreferences"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 2,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "Initial Energy:  0.5\n",
      "B matrix:  [[ 0.+0.j  1.+0.j  0.+0.j]\n",
      " [ 0.+0.j -1.+0.j  0.+0.j]]\n",
      "Hamiltonian formed: [[-0.-0.j -0.-0.j -0.-0.j -0.-0.j -0.-0.j -1.-0.j -0.-0.j -0.-0.j]\n",
      " [-0.-0.j -0.-0.j -0.-0.j -0.-0.j -0.-0.j  1.-0.j -0.-0.j -0.-0.j]\n",
      " [-0.-0.j -0.-0.j -0.-0.j -0.-0.j -0.-0.j -0.-0.j -0.-0.j -0.-0.j]\n",
      " [-0.-0.j -0.-0.j -0.-0.j -0.-0.j -0.-0.j -0.-0.j -0.-0.j -0.-0.j]\n",
      " [-0.+0.j -0.+0.j -0.+0.j -0.+0.j -0.-0.j -0.-0.j -0.-0.j -0.-0.j]\n",
      " [-1.+0.j  1.+0.j -0.+0.j -0.+0.j -0.-0.j -0.-0.j -0.-0.j -0.-0.j]\n",
      " [-0.+0.j -0.+0.j -0.+0.j -0.+0.j -0.-0.j -0.-0.j -0.-0.j -0.-0.j]\n",
      " [-0.+0.j -0.+0.j -0.+0.j -0.+0.j -0.-0.j -0.-0.j -0.-0.j -0.-0.j]]\n",
      "Initial State: [[ 0.+0.j]\n",
      " [ 0.+0.j]\n",
      " [ 0.+0.j]\n",
      " [ 0.+0.j]\n",
      " [ 0.+0.j]\n",
      " [-0.-1.j]\n",
      " [ 0.+0.j]\n",
      " [ 0.+0.j]]\n"
     ]
    }
   ],
   "source": [
    "## Creating the 2 masses system\n",
    "\n",
    "# Number of masses\n",
    "N =2**1\n",
    "\n",
    "# ass Matrix\n",
    "M = np.array([[1,0],[0,1]])\n",
    "\n",
    "# Spring Matrix\n",
    "K = np.array([[0,1],[1,0]])\n",
    "\n",
    "# Initial Conditions\n",
    "x_0 = np.zeros([N,1])\n",
    "xdot_0 = np.zeros([N,1])\n",
    "x_0[1] = 1\n",
    "\n",
    "# F Matrix\n",
    "F= create_matrix_F(K, N)\n",
    "\n",
    "# A Matrix\n",
    "A = create_matrix_A(M, F)\n",
    "\n",
    "# Transform coordinates\n",
    "y = coordinate_transformation(M, x_0, xdot_0)\n",
    "y_0 = y[\"y_0\"]\n",
    "ydot_0 = y[\"ydot_0\"]\n",
    "\n",
    "# B Matrix\n",
    "B = create_matrix_B(M, K, A, N)\n",
    "\n",
    "# Stacking B with zeros to get square matrix N^2 x N^2\n",
    "B_padded = padding_B(B, N)\n",
    "\n",
    "# Hamiltonian\n",
    "H = create_Hamiltonian(B_padded)\n",
    "\n",
    "# Initial State\n",
    "init_state = create_init_state(B_padded, y_0, ydot_0, N)\n",
    "E0_y = calculate_energy(y_0, ydot_0, M, K)\n",
    "\n",
    "# Normalize the Initial State\n",
    "normalization = normalize_init_state(init_state)\n",
    "normalized_init_state = normalization[\"normalized_init_state\"]\n",
    "norm = normalization[\"norm\"]\n",
    "\n",
    "# Print the results\n",
    "print(\"Initial Energy: \", E0_y)\n",
    "print(\"B matrix: \", B)\n",
    "print(\"Hamiltonian formed:\", H)\n",
    "print(\"Initial State:\", init_state)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Pauli Matrix Decomposition\n",
    "\n",
    "Create the Pauli List for the Hamiltonian. Pauli Decomposition is taken from [Classiq Library](https://github.com/Classiq/classiq-library/blob/main/algorithms/hhl/hhl/hhl.ipynb) thanks a lot to the authors!"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 3,
   "metadata": {},
   "outputs": [
    {
     "name": "stderr",
     "output_type": "stream",
     "text": [
      "100%|██████████| 64/64 [00:00<00:00, 32936.87it/s]"
     ]
    },
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "[('XII', (0.25+0j)), ('XIZ', (-0.25+0j)), ('XIX', (-0.25+0j)), ('XZI', (0.25+0j)), ('XZZ', (-0.25+0j)), ('XZX', (-0.25+0j)), ('YIY', (0.25+0j)), ('YZY', (0.25+0j))]\n"
     ]
    },
    {
     "name": "stderr",
     "output_type": "stream",
     "text": [
      "\n"
     ]
    }
   ],
   "source": [
    "## Pauli Decomposition and transforming it into Classiq syntax\n",
    "\n",
    "# Create the Pauli Matrix Decomposition of the Hamiltonian\n",
    "pauli_list = lcu_naive(H)\n",
    "\n",
    "# Transform Pauli Matrix Decomposition to Classiq compatible\n",
    "classiq_pauli_list = pauli_list_to_hamiltonian(pauli_list)\n",
    "\n",
    "print(pauli_list)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## LCU and Chebyshev Functions\n",
    "\n",
    "Create functions to find LCU probabilities and Chebyshev coefficients. These functions are taken from the Classiq Github [4](https://github.com/Classiq/classiq-library/blob/main/tutorials/hamiltonian_simulation/hamiltonian_simulation_with_block_encoding/hamiltonian_simulation_with_block_encoding.ipynb). Thanks a lot to the authors!"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 4,
   "metadata": {},
   "outputs": [],
   "source": [
    "def get_cheb_coef(epsilon, t):\n",
    "    \"\"\"\n",
    "        Definition:\n",
    "            Find the Chebyshev coefficients\n",
    "        Args:\n",
    "            epsilon (float): Sensitivity\n",
    "            t (float): Evolution time for the system\n",
    "        Outputs:\n",
    "            cos_coef (list): Coefficients of cosine function\n",
    "            sin_coef (list): Coefficients of sine function\n",
    "    \"\"\"\n",
    "    poly_degree = int(\n",
    "        np.ceil(\n",
    "            t\n",
    "            + np.log(epsilon ** (-1)) / np.log(np.exp(1) + np.log(epsilon ** (-1)) / t)\n",
    "        )\n",
    "    )\n",
    "    cos_coef = [scipy.special.jv(0, t)] + [\n",
    "        2 * scipy.special.jv(2 * k, t) * (-1) ** k for k in range(1, poly_degree // 2 + 1)\n",
    "    ]\n",
    "    sin_coef = [\n",
    "        -2 * scipy.special.jv(2 * k - 1, t) * (-1) ** k for k in range(1, poly_degree // 2 + 1)\n",
    "    ]\n",
    "    return cos_coef, sin_coef"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "#### **Block Encoding the Hamiltonian (LCU):**\n",
    "\n",
    "##### Preparing the normalized coefficients for the PREPARE operation\n",
    "\n",
    "The PREPARE operation prepares a quantum state that corresponds to the probabilities $|\\alpha_i| /\\sum|\\alpha_i|$. We define a function that gets the list of  coefficients of the Hamiltonian and returns the probabilities to be loaded as part of the PREPARE operation. The block size is also determined based on the number of terms in the Hamiltonian."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 5,
   "metadata": {},
   "outputs": [],
   "source": [
    "def get_normalized_lcu_coef(lcu_coef):\n",
    "    \"\"\"\n",
    "        Definition:\n",
    "            Find the LCU probabilities, normaliziation factor, and size of the probabilities\n",
    "        Args:\n",
    "            lcu_coef (list): List of coefficients for Pauli matrices\n",
    "        Outputs:\n",
    "            normalization_factor (float): 1-norm of the coefficients\n",
    "            coef_size (int): Number of Qubits to create the probabilities\n",
    "            prepare_prob(list): Probabilities of normalized LCU coefficients\n",
    "    \"\"\"\n",
    "    \n",
    "    lcu_coef = [abs(c) for c in lcu_coef]\n",
    "    normalization_factor = sum(lcu_coef)\n",
    "    prepare_prob = [c / normalization_factor for c in lcu_coef]\n",
    "    coef_size = int(np.ceil(np.log2(len(prepare_prob))))\n",
    "    prepare_prob += [0] * (2**coef_size - len(prepare_prob))\n",
    "\n",
    "    return normalization_factor, coef_size, prepare_prob"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "#### Defining a function that applies the $(PREPARE) SELECT (PREPARE^{\\dagger})$ operation\n",
    "\n",
    "The SELECT operation acts on the desired target state controlled by the quantum state of the block qubits. This constitution ends our construction of the unitary block Hamiltonian. "
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 6,
   "metadata": {},
   "outputs": [],
   "source": [
    "@qfunc\n",
    "def apply_pauli_term(pauli_string: PauliTerm, state: QArray[QBit]):\n",
    "    \"\"\"\n",
    "    Definition:\n",
    "        Apply the Pauli Matrix to the state\n",
    "    Args:\n",
    "        pauli_string (PauliTerm): Pauli Matrix sequence\n",
    "        state (QArray[QBit]): State to apply pauli matrices\n",
    "    \"\"\"\n",
    "    repeat(\n",
    "        count=state.len,\n",
    "        iteration=lambda index: switch(\n",
    "            pauli_string.pauli[index],\n",
    "            [\n",
    "                lambda: IDENTITY(state[pauli_string.pauli.len - index - 1]),\n",
    "                lambda: X(state[pauli_string.pauli.len - index - 1]),\n",
    "                lambda: Y(state[pauli_string.pauli.len - index - 1]),\n",
    "                lambda: Z(state[pauli_string.pauli.len - index - 1]),\n",
    "            ],\n",
    "        ),\n",
    "    )\n",
    "\n",
    "\n",
    "@qfunc\n",
    "def lcu_paulis(pauli_terms_list: CArray[PauliTerm], signs: CArray[CInt], probs: CArray[CReal], block: QNum, state: QArray[QBit]):\n",
    "    \"\"\"\n",
    "        Definition:\n",
    "            Apply all the Pauli sequences in LCU\n",
    "        Args:\n",
    "            pauli_terms_list (CArray[PauliTerm]): Pauli Matrix Decomposition of the Hamiltonian\n",
    "            signs (CArray[CInt]): Signs of PMD coefficients\n",
    "            probs (CArray[CReal]): Probabilities for LCU\n",
    "            block (QNum): PREPARE variables\n",
    "            state (QArray[QBit]): State to apply LCU\n",
    "    \"\"\"\n",
    "\n",
    "    within_apply(\n",
    "        lambda: inplace_prepare_state(probs, 0.0, block),\n",
    "        lambda: repeat(\n",
    "            count=pauli_terms_list.len,\n",
    "            iteration=lambda i: control(\n",
    "                block == i, lambda: (\n",
    "                                U(0, 0, 0, np.pi / 2 * signs[i], state[0]),\n",
    "                                apply_pauli_term(pauli_terms_list[i], state)\n",
    "                ),\n",
    "            ),\n",
    "        )\n",
    "    )"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 7,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "Normalization factor:  2.0\n",
      "LCU size:  3\n",
      "Probabilities:  [0.125, 0.125, 0.125, 0.125, 0.125, 0.125, 0.125, 0.125]\n",
      "Signs:  [0.0, 3.0, 2.0, 1.0, 2.0, 3.0, 0.0, 1.0]\n"
     ]
    }
   ],
   "source": [
    "## Find the LCU parameters for the Hamiltonian\n",
    "\n",
    "pauli_coeff = [p.coefficient for p in classiq_pauli_list]\n",
    "signs_pauli = np.sign(pauli_coeff).tolist()\n",
    "generalized_signs_pauli = [\n",
    "    (1 - signs_pauli[s]) + (s) % 2 for s in range(len(signs_pauli))\n",
    "]\n",
    "normalization_ham, lcu_size_ham, prepare_probs_ham = get_normalized_lcu_coef(pauli_coeff)\n",
    "\n",
    "print(\"Normalization factor: \", normalization_ham)\n",
    "print(\"LCU size: \", lcu_size_ham)\n",
    "print(\"Probabilities: \", prepare_probs_ham)\n",
    "print(\"Signs: \", generalized_signs_pauli)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Hamiltonian Simulation using Block Encoding $e^{iHt}$: (Qubitization method)\n",
    "Define necessary functions for Quantum Walk-Operators and Qubitization and simulate the Hamiltonian"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 8,
   "metadata": {},
   "outputs": [],
   "source": [
    "@qfunc\n",
    "def my_walk_operator(block: QArray[QBit], state: QArray[QBit]) -> None:\n",
    "    \"\"\"\n",
    "        Definition:\n",
    "            Applies the Szegedy Quantum Walk Operator\n",
    "        Args:\n",
    "            block (QArray[QBit]): PREPARE variables\n",
    "            state (QArray[QBit]): State to apply walk operator\n",
    "    \"\"\"\n",
    "    lcu_paulis(classiq_pauli_list, generalized_signs_pauli, prepare_probs_ham, block, state)\n",
    "    reflect_about_zero(block)\n",
    "    RY(2 * np.pi, block[0])"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "##### Evaluating the Chebyshev co-efficients\n",
    "\n",
    "We calculate the coefficients $\\beta_i$ for approximating the sine and cosine functions. We want to approximate $e^{iHt}$, but our Hamiltonian is encoded with a normalization factor $\\bar{\\alpha}$ that comes from the LCU block encoding. Therefore, we rescale the time by this factor. \n",
    "\n",
    "Given the coefficients, we calculate the probabilities for the LCU prepare variable. These must be positive numbers. Moreover, for the odd terms, which corresponds to the sine function, we should add a factor of $i$. Those factors should be added to the unitary operations $W^k$. To take this into account, for each term we define a \"generalized sign\" $\\sigma_k$ such that $\\beta_k = e^{\\frac{\\pi}{2}\\sigma_k i} |\\beta_k|$ with $\\sigma_k \\in \\{0,1,2,3\\}$ that corresponds to the factors $1,i,-1,-i$ respectively. "
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 9,
   "metadata": {},
   "outputs": [],
   "source": [
    "## Finc Chebyshev coefficients and take phases into account\n",
    "\n",
    "time = 0.5\n",
    "\n",
    "normalized_time = normalization_ham * time\n",
    "\n",
    "cos_coef, sin_coef = get_cheb_coef(0.0001, normalized_time)\n",
    "\n",
    "combined_sin_cos_coef = []\n",
    "for k in range(len(cos_coef) - 1):\n",
    "    combined_sin_cos_coef.append(cos_coef[k])\n",
    "    combined_sin_cos_coef.append(sin_coef[k])\n",
    "combined_sin_cos_coef.append(cos_coef[-1])\n",
    "if len(sin_coef) == len(cos_coef):\n",
    "    combined_sin_cos_coef.append(sin_coef[-1])\n",
    "\n",
    "signs_cheb_coef = np.sign(combined_sin_cos_coef).tolist()\n",
    "generalized_signs = [\n",
    "    (1 - signs_cheb_coef[s]) + (s) % 2 for s in range(len(signs_cheb_coef))\n",
    "]\n",
    "positive_cheb_lcu_coef = np.abs(combined_sin_cos_coef)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 10,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "Normalization factor:  1.9191858138051914\n",
      "LCU size:  3\n",
      "Probabilities:  [0.3987095366450216, 0.4585804903095238, 0.11974190732900435, 0.02038713900649352, 0.0025809267099567547, 0, 0, 0]\n"
     ]
    }
   ],
   "source": [
    "## Find the LCU parameters for the exponentiation\n",
    "\n",
    "normalization_exp, lcu_size_exp, prepare_probs_exp = get_normalized_lcu_coef(positive_cheb_lcu_coef)\n",
    "\n",
    "print(\"Normalization factor: \", normalization_exp)\n",
    "print(\"LCU size: \", lcu_size_exp)\n",
    "print(\"Probabilities: \", prepare_probs_exp)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 11,
   "metadata": {},
   "outputs": [],
   "source": [
    "## LCU function for Chebyshev coefficients\n",
    "\n",
    "@qfunc\n",
    "def lcu_cheb(coef: CArray[CReal], generalized_signs: CArray[CInt], walk_operator: QCallable[QNum, QArray], walk_block: QNum, state: QArray, cheb_block: QNum):\n",
    "    \"\"\"\n",
    "        Definition:\n",
    "            Apply all the Chebyshev coefficients in LCU\n",
    "        Args:\n",
    "            coef (CArray[CReal]): Chebyshev Coefficients\n",
    "            generalized_signs (CArray[CInt]): Signs of Chebyshev coefficients\n",
    "            walk_operator (QCallable[QNum, QArray]): Quantum Walk-Operator\n",
    "            walk_block (QNum): PREPARE variables for Quantum Walk\n",
    "            state (QArray[QBit]): State to apply LCU\n",
    "            cheb_block (QNum): PREPARE variables for Chebyshev coefficients\n",
    "    \"\"\"\n",
    "\n",
    "    within_apply(\n",
    "        lambda: inplace_prepare_state(coef, 0.0, cheb_block),\n",
    "        lambda: repeat(\n",
    "            generalized_signs.len,\n",
    "            lambda k: control(\n",
    "                cheb_block == k,\n",
    "                lambda: (\n",
    "                    U(0, 0, 0, np.pi / 2 * generalized_signs[k], state[0]),\n",
    "                    power(k, lambda: walk_operator(walk_block, state)),\n",
    "                ),\n",
    "            ),\n",
    "        ),\n",
    "    )"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 12,
   "metadata": {},
   "outputs": [],
   "source": [
    "## Main Evolution\n",
    "\n",
    "# Add i phase for the last half\n",
    "@qfunc\n",
    "def init_state_phase(state: QNum):\n",
    "    \"\"\"\n",
    "        Definition:\n",
    "            Imply -pi/2 phase to the last half (msb qubit) of the QNum state in order to add -i factor\n",
    "        Args:\n",
    "            state (QNum): Initial state with bare amplitudes\n",
    "        Outputs:\n",
    "            state (QNum): Initial state with phase\n",
    "    \"\"\"\n",
    "\n",
    "    state_in_qubit = QArray(\"state_in_qubit\")\n",
    "    msb = QArray(\"msb\", QBit)\n",
    "    \n",
    "    size = np.log2(init_state.size)\n",
    "    allocate(size, msb)\n",
    "\n",
    "    bind(state, state_in_qubit)\n",
    "    repeat(state_in_qubit.len, lambda i: CX(state_in_qubit[i], msb[i]))\n",
    "    control(msb[size-1], lambda: PHASE(np.pi/2, state_in_qubit[size-1]))\n",
    "    bind(state_in_qubit, state)\n",
    "\n",
    "# Main Function\n",
    "@qfunc\n",
    "def main(ham_block: Output[QNum], state: Output[QNum], exp_block: Output[QNum]):\n",
    "    \"\"\"\n",
    "        Definition:\n",
    "            Main function to make evolution\n",
    "\n",
    "        Outputs:\n",
    "            ham_block (Output[QNum]): PREPARE variables for Hamiltonian\n",
    "            state (Output[QNum]): Resulting state at the t = evolution_coefficient\n",
    "            exp_block (Output[QNum]): PREPARE variables for exponantioation\n",
    "    \"\"\"\n",
    "\n",
    "    allocate(lcu_size_exp, exp_block)\n",
    "    allocate(lcu_size_ham, ham_block)\n",
    "    prepare_amplitudes(list(normalized_init_state), 0.0, state)\n",
    "    init_state_phase(state)\n",
    "\n",
    "    lcu_cheb(\n",
    "        prepare_probs_exp,\n",
    "        generalized_signs,\n",
    "        lambda x, y: my_walk_operator(x, y),\n",
    "        ham_block,\n",
    "        state,\n",
    "        exp_block,\n",
    "    )\n",
    "\n",
    "\n",
    "qmod = create_model(main)\n",
    "write_qmod(qmod, \"Simulation with Qubitization\", decimal_precision=16)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 14,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "Opening: https://platform.classiq.io/circuit/d015725d-04d8-4de6-9545-d7ffe0317e91?version=0.43.3\n"
     ]
    }
   ],
   "source": [
    "# Create model and synthesis. Use \"simulator_statevector\" to take phases into account\n",
    "backend_preferences = ClassiqBackendPreferences(backend_name=\"simulator_statevector\")\n",
    "model_pref = set_execution_preferences(qmod, ExecutionPreferences(num_shots=1, backend_preferences=backend_preferences))\n",
    "qprog = synthesize(model_pref)\n",
    "show(qprog)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 15,
   "metadata": {},
   "outputs": [],
   "source": [
    "# Execute the job\n",
    "job = execute(qprog)\n",
    "results = job.result()"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Post-Processing and Comparision"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 16,
   "metadata": {},
   "outputs": [],
   "source": [
    "def get_projected_state_vector(execution_result, measured_var: str, projections: dict) -> np.ndarray:\n",
    "    \"\"\"\n",
    "    Definition:\n",
    "        This function returns a reduced statevector from execution results.\n",
    "        measured var: the name of the reduced variable\n",
    "        projections: on which values of the other variables to project, e.g., {\"ind\": 1}\n",
    "    \"\"\"\n",
    "    \n",
    "    projected_size = len(execution_result[0].value.output_qubits_map[measured_var])\n",
    "    proj_statevector = np.zeros(2**projected_size).astype(complex)\n",
    "    for sample in execution_result[0].value.parsed_state_vector:\n",
    "        if all(sample.state[key] == projections[key] for key in projections.keys()):\n",
    "            proj_statevector[int(sample.state[measured_var])] += sample.amplitude\n",
    "    return proj_statevector"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 17,
   "metadata": {},
   "outputs": [],
   "source": [
    "## Get the states where blocks are in 0 state\n",
    "\n",
    "state_result = get_projected_state_vector(\n",
    "    results, \"state\", {\"exp_block\": 0.0, \"ham_block\": 0.0}\n",
    ")"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 18,
   "metadata": {},
   "outputs": [],
   "source": [
    "## Renormalize the final result with the norm of the initial state\n",
    "final_state_result = norm * state_result"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 19,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "Expected State: [[-0.2393529+0.j        ]\n",
      " [ 0.2393529+0.j        ]\n",
      " [ 0.       +0.j        ]\n",
      " [ 0.       +0.j        ]\n",
      " [ 0.       +0.j        ]\n",
      " [ 0.       -0.39612871j]\n",
      " [ 0.       +0.j        ]\n",
      " [ 0.       +0.j        ]]\n",
      "Resulting State: [-1.62132687e-01+3.54829846e-17j  1.62132687e-01-1.83459399e-15j\n",
      "  3.80342924e-17-5.92531917e-17j  5.66967202e-17+3.52190715e-17j\n",
      " -1.61220529e-15-3.33056534e-15j -3.23352879e-01-2.38682564e-01j\n",
      "  5.33793253e-17+7.24025475e-17j  7.85852506e-17+2.81873394e-17j]\n"
     ]
    }
   ],
   "source": [
    "## Compare with classical results\n",
    "expected_state = (1 / normalization_exp * scipy.linalg.expm(1j * H * time) @ init_state)\n",
    "\n",
    "relative_phase = np.angle(expected_state[0] / state_result[0])\n",
    "\n",
    "final_state_result = final_state_result * np.exp(1j * relative_phase)\n",
    "\n",
    "print(\"Expected State:\", expected_state)\n",
    "print(\"Resulting State:\", final_state_result)\n"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 20,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "Overlap: 0.890038098309116\n"
     ]
    }
   ],
   "source": [
    "## Find the overlap between two result\n",
    "\n",
    "overlap = np.abs(np.vdot(state_result, expected_state)) * normalization_exp/ np.linalg.norm(state_result)\n",
    "print(\"Overlap:\", overlap)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Discussion and Results\n",
    "\n",
    "##### This work shows that Qubitization and Quantum Walk-Operators are important tools in Hamiltonian evolutions. We see a 89% overlap between classical and quantum results and this can be imrpoved by decreasing the epsilon for Chebyshev coefficients. Moreover the codes are general and can be used for simulating for larger values of $N=2**n$, but the process might be slow. \n",
    "\n",
    "##### Note that simulating for **larger times** doesn't seem to work well, because as evolution time increases the depth of the quauntum circuit increases exponentially, due to the power involved in the walk operator. This takes extremely long time to synthesize and thus we chose a small 't' to get the methodology working for benchmarking purposes. But there is a trade-off here, lower 't' also implies lower degree in the polynomial expansion, which is an issue for accuracy. Hence we had to find a sweet spot, where accuracy is not too compromised and the citcuit depth is reasonable for synthesis purposes. \n"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "___\n",
    "___"
   ]
  }
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