{
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"cell_type": "markdown",
"id": "5f216002",
"metadata": {},
"source": [
"# Textbook QPE\n",
"\n",
"We introduce the Quantum Phase Estimation algorithm and show how to compute the ground state energy of a Hamiltonian $H$. We consider a small system where the exponentiation of the Hamiltonian can be performed exactly to get the exact time evolution operator $U(t) = \\exp(-iHt)$.\n",
"\n",
"First, let us briefly introduce the algorithm. For a more detailed introduction, we refer the reader to the famous book by Michael A. Nielsen and Isaac L. Chuang on *Quantum Computation and Quantum Information*, or to the [Quantum Phase Estimation algorithm Wikipedia page](https://en.wikipedia.org/wiki/Quantum_phase_estimation_algorithm).\n",
"\n",
"Consider a unitary operator $U$ and an eigenstate $\\ket{u}$ of $U$: $U \\ket{u} = e^{i \\theta} \\ket{u}$. We want to measure $\\theta$ with $m$-bits precision.\n",
"\n",
"The QPE circuit contains two registers: a physical register with $n$ qubits and a phase register with $m$ qubits.\n",
"\n",
"![](\"./figures/qpe.png\")
\n",
"\n",
"1. The physical register starts in state $\\ket{\\psi}$, where $\\ket{\\psi}$ is an estimate of $\\ket{u}$ with fidelity $\\Omega = \\vert \\langle \\psi \\vert u \\rangle \\vert^2$.\n",
"2. The phase register is initially in state $\\ket{0}$.\n",
"3. The circuit starts with a Hadamard wall to put the phase register into a superposition state.\n",
"4. Then we *encode* the phase into the phase register via a sequence of controlled powers of $U$:\n",
"\n",
" $U^{2^k}, k=0,1,...,m-1$ is applied to the physical register, conditioned on the $k$-th phase qubit.\n",
"5. Finally to *decode* the phase, we apply the inverse Quantum Fourier Transform (QFT) on the phase register.\n",
"6. We measure the phase register and find a $m$-bits approximation to $\\theta$ with probability $\\propto \\Omega$ (at least $4\\Omega/\\pi^2$, see below).\n",
"7. After the measure, the physical register has been projected onto $\\ket{u}$.\n",
"\n",
"The notebook is organised as follows:\n",
"In the first section, we illustrate and detail the different parts of the algorithm on a small 1D Heisenberg Hamiltonian.\n",
"In the second section, we study the precision and success probability of the algorithm in more detail.\n",
"Finally, in the third section we focus on the influence of the initial overlap $\\Omega$."
]
},
{
"cell_type": "code",
"execution_count": 1,
"id": "609e0811",
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"execution": {
"iopub.execute_input": "2026-04-03T04:03:44.074156Z",
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"shell.execute_reply": "2026-04-03T04:03:46.061995Z"
}
},
"outputs": [],
"source": [
"import time\n",
"\n",
"import matplotlib.pyplot as plt\n",
"import numpy as np\n",
"from IPython.display import display\n",
"from quimb.tensor import MatrixProductState\n",
"from tqdm import notebook as tqdm\n",
"\n",
"import qpe_toolbox.estimation as qpe\n",
"from qpe_toolbox import EXACT\n",
"from qpe_toolbox.circuit import make_circ\n",
"from qpe_toolbox.hamiltonian import do_dmrg, heisenberg_hamiltonian"
]
},
{
"cell_type": "code",
"execution_count": 2,
"id": "3a7d252a",
"metadata": {
"execution": {
"iopub.execute_input": "2026-04-03T04:03:46.064919Z",
"iopub.status.busy": "2026-04-03T04:03:46.064583Z",
"iopub.status.idle": "2026-04-03T04:03:46.067763Z",
"shell.execute_reply": "2026-04-03T04:03:46.067086Z"
}
},
"outputs": [],
"source": [
"plt.rcParams.update({\"font.size\": 12})"
]
},
{
"cell_type": "markdown",
"id": "4d4995b0",
"metadata": {},
"source": [
"## Quantum phase estimation\n",
"### Example : 1D Heisenberg Hamiltonian\n",
"\n",
"First let us define a simple Hamiltonian which we can\n",
"- diagonalize exactly\n",
"- encode as a quantum circuit\n",
"\n",
"Consider the nearest-neighbor Heisenberg Hamiltonian on a 1D chain wih open boundary conditions\n",
"\n",
"$$ H = J \\sum_{k=0}^{L-1} \\vec{S}_k \\cdot \\vec{S}_{k+1} $$\n",
"where $\\vec{S}_k = \\vec{\\sigma}_k /2 \\;$ are the spin-1/2 generators of SU(2) acting on site $k$, with $\\vec{\\sigma} = (\\sigma^x, \\sigma^y, \\sigma^z)$ the Pauli matrices.\n",
"\n",
"We take $J=1$ in the following, such that all energies are expressed in units of $J$.\n",
"\n",
"#### 1. Hamiltonian definition, circuit initialization"
]
},
{
"cell_type": "markdown",
"id": "9c0098a9",
"metadata": {},
"source": [
"Let us define the Hamiltonian and perform exact diagonalization"
]
},
{
"cell_type": "code",
"execution_count": 3,
"id": "90d3489c",
"metadata": {
"execution": {
"iopub.execute_input": "2026-04-03T04:03:46.069334Z",
"iopub.status.busy": "2026-04-03T04:03:46.069164Z",
"iopub.status.idle": "2026-04-03T04:03:46.074003Z",
"shell.execute_reply": "2026-04-03T04:03:46.073271Z"
}
},
"outputs": [
{
"name": "stdout",
"output_type": "stream",
"text": [
"E_ED : -0.7500\n"
]
}
],
"source": [
"n_qubits = 2\n",
"h_spin = heisenberg_hamiltonian(n_qubits)\n",
"\n",
"# Get matrix\n",
"hamilt_matrix = h_spin.to_dense()\n",
"\n",
"# Diagonalize hamiltonian\n",
"eigvals, eigvecs = np.linalg.eigh(hamilt_matrix)\n",
"# Ground state\n",
"E0 = eigvals[0]\n",
"psi0 = eigvecs[:, 0]\n",
"print(f\"E_ED : {E0:.4f}\")"
]
},
{
"cell_type": "code",
"execution_count": 4,
"id": "29374ff3",
"metadata": {
"execution": {
"iopub.execute_input": "2026-04-03T04:03:46.075354Z",
"iopub.status.busy": "2026-04-03T04:03:46.075166Z",
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"shell.execute_reply": "2026-04-03T04:03:46.105516Z"
}
},
"outputs": [
{
"name": "stdout",
"output_type": "stream",
"text": [
"E_DMRG : -0.7500\n"
]
}
],
"source": [
"# Ground state MPS\n",
"E0_dmrg, psi0_mps = do_dmrg(h_spin)\n",
"print(f\"E_DMRG : {E0_dmrg:.4f}\")"
]
},
{
"cell_type": "code",
"execution_count": 5,
"id": "b6703215",
"metadata": {
"execution": {
"iopub.execute_input": "2026-04-03T04:03:46.107505Z",
"iopub.status.busy": "2026-04-03T04:03:46.107359Z",
"iopub.status.idle": "2026-04-03T04:03:46.123720Z",
"shell.execute_reply": "2026-04-03T04:03:46.122875Z"
}
},
"outputs": [
{
"name": "stdout",
"output_type": "stream",
"text": [
"1 - ||^2 = 4.441e-16\n"
]
}
],
"source": [
"F = abs(psi0_mps.H @ MatrixProductState.from_dense(psi0)) ** 2\n",
"print(f\"1 - ||^2 = {abs(1 - F):.4g}\")"
]
},
{
"cell_type": "markdown",
"id": "36418b27",
"metadata": {},
"source": [
"We now initialize the QPE circuit with a data register containing $|\\psi_0\\rangle$, and a phase register with $m=4$ phase qubits; then measure the energy from the circuit"
]
},
{
"cell_type": "code",
"execution_count": 6,
"id": "2a6fda3f",
"metadata": {
"execution": {
"iopub.execute_input": "2026-04-03T04:03:46.125068Z",
"iopub.status.busy": "2026-04-03T04:03:46.124925Z",
"iopub.status.idle": "2026-04-03T04:03:46.168068Z",
"shell.execute_reply": "2026-04-03T04:03:46.167283Z"
}
},
"outputs": [
{
"name": "stdout",
"output_type": "stream",
"text": [
"measure H = -0.7500\n"
]
}
],
"source": [
"n_phase_bits = 4\n",
"\n",
"psi_target = psi0_mps\n",
"initial_circ = make_circ(n_phase_bits, psi_target)\n",
"\n",
"data_reg = list(range(n_phase_bits, n_phase_bits + n_qubits))\n",
"print(\n",
" f\"measure H = {initial_circ.local_expectation(hamilt_matrix, where=data_reg):.4f}\"\n",
")"
]
},
{
"cell_type": "markdown",
"id": "d927ca91",
"metadata": {},
"source": [
"#### First stage of Quantum Phase Estimation Algorithm\n",
"\n",
"See e.g. Nielsen and Chuang.\n",
"- First, initialize the phase register with a \"Hadamard wall\"\n",
"- Then build the operator $U = \\exp(-i H t)$ for a given evolution time $t$ and apply a sequence of gates ctrl-$U^k$ on the qubit-register conditioned on the $k$-th phase qubit.\n",
"\n",
" Since $|\\psi_0 \\rangle$ is an eigenstate of $H$, we have $U |\\psi_0 \\rangle = \\exp(-i2\\pi \\theta) |\\psi_0 \\rangle$ with $0 \\leq \\theta \\leq 1$ ($U$ is unitary by hermiticity of $H$).\n",
"\n",
" The state of the phase register is then\n",
"\n",
"$$ \\frac{1}{\\sqrt{2^m}} \\sum_{q=0}^{2^m-1} e^{i2\\pi \\theta q} |q \\rangle ,$$\n",
"(the data register stays in the state $|\\psi_0\\rangle$)"
]
},
{
"cell_type": "code",
"execution_count": 7,
"id": "1a4b851c",
"metadata": {
"execution": {
"iopub.execute_input": "2026-04-03T04:03:46.169637Z",
"iopub.status.busy": "2026-04-03T04:03:46.169465Z",
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"shell.execute_reply": "2026-04-03T04:03:46.201596Z"
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