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Hadamard test example #1867

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Hadamard test example #1867

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hadamard test
marwafar Jun 28, 2024
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hadamard test
marwafar Jun 28, 2024
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Merge branch 'main' into Hadamard-test
schweitzpgi Jul 10, 2024
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add space
marwafar Jul 17, 2024
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Merge branch 'main' into Hadamard-test
sacpis Jul 17, 2024
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bmhowe23 Jul 18, 2024
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Rename for capitalization consistency
bmhowe23 Jul 18, 2024
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Add to tutorials.rst (required to fix docs)
bmhowe23 Jul 18, 2024
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344 changes: 344 additions & 0 deletions docs/sphinx/examples/python/tutorials/hadamard_test.ipynb
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{
"cells": [
{
"cell_type": "markdown",
"metadata": {},
"source": [
"# Hadamard Test\n",
"\n",
"Consider the observable $O$ and two generic quantum states $\\ket{\\psi}$ and $\\ket{\\phi}$. We want to calculate the quantity\n",
"$$\n",
"\\bra{\\psi} O \\ket{\\phi}.\n",
"$$\n",
"where $O$ is a Pauli operator.\n",
"\n",
"First of all we shall prepare the states $\\ket{\\psi}$ and $\\ket{\\phi}$ using a quantum circuit for each of them. So we have\n",
"$$\n",
"\\ket{\\psi} = U_{\\psi}\\ket{0} \\qquad \\ket{\\phi} = U_{\\phi}\\ket{0}\n",
"$$\n",
"\n",
"Let's define an observable we want to use:\n",
"$$\n",
"O = X_1X_2\n",
"$$\n",
"\n",
"Now we can evaluate the matrix element using the following fact:\n",
"$$\n",
"\\bra{\\psi}O\\ket{\\phi} = \\bra{0}U_\\psi^\\dagger O U_\\phi\\ket{0}\n",
"$$\n",
"This is just an expectation value which can be solved with a simple Hadamard test. The probability to measure $0$ or $1$ in the ancilla qubit is\n",
"\n",
"$$\n",
"P(0) = \\frac{1}{2} \\left[ I + Re \\bra{\\psi} O \\ket{\\phi} \\right]\n",
"$$\n",
"\n",
"$$\n",
"P(1) = \\frac{1}{2} \\left[ I - Re \\bra{\\psi} O \\ket{\\phi} \\right]\n",
"$$\n",
"\n",
"The difference between the probability of $0$ and $1$ gives \n",
"\n",
"$$\n",
"P(0)-P(1) = Re \\bra{\\psi} O \\ket{\\phi}\n",
"$$"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"### A- Numerical result as a reference: "
]
},
{
"cell_type": "code",
"execution_count": 18,
"metadata": {},
"outputs": [
{
"name": "stdout",
"output_type": "stream",
"text": [
"Psi state: (0.707107,0)\n",
"(0,0)\n",
"(0.707107,0)\n",
"(0,0)\n",
"\n",
"Phi state: (0,0)\n",
"(1,0)\n",
"(0,0)\n",
"(0,0)\n",
"\n",
"hamiltonian: [[0.+0.j 0.+0.j 0.+0.j 1.+0.j]\n",
" [0.+0.j 0.+0.j 1.+0.j 0.+0.j]\n",
" [0.+0.j 1.+0.j 0.+0.j 0.+0.j]\n",
" [1.+0.j 0.+0.j 0.+0.j 0.+0.j]] \n",
"\n",
"Numerical expectation value: (0.7071067690849304+0j)\n"
]
}
],
"source": [
"import cudaq\n",
"import numpy as np\n",
"from functools import reduce\n",
"\n",
"cudaq.set_target('nvidia')\n",
"\n",
"qubit_num = 2\n",
"\n",
"@cudaq.kernel\n",
"def psi(num:int):\n",
" q = cudaq.qvector(num)\n",
" h(q[1])\n",
" \n",
"@cudaq.kernel\n",
"def phi(n:int):\n",
" q = cudaq.qvector(n)\n",
" x(q[0])\n",
"\n",
"psi_state = cudaq.get_state(psi, qubit_num)\n",
"print('Psi state: ', psi_state)\n",
"\n",
"phi_state=cudaq.get_state(phi, qubit_num)\n",
"print('Phi state: ', phi_state)\n",
"\n",
"ham=cudaq.spin.x(0) * cudaq.spin.x(1)\n",
"ham_matrix = ham.to_matrix()\n",
"print('hamiltonian: ', np.array(ham_matrix), '\\n')\n",
"\n",
"exp_val=reduce(np.dot,(np.array(psi_state).conj().T, ham_matrix, phi_state))\n",
"\n",
"print('Numerical expectation value: ', exp_val) "
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"### B- Using ``sample`` algorithmic primitive to sample the ancilla qubit and compute the expectation value."
]
},
{
"cell_type": "code",
"execution_count": 19,
"metadata": {},
"outputs": [
{
"name": "stdout",
"output_type": "stream",
"text": [
"{ 0:85356 1:14644 }\n",
"\n",
"Observable QC: 0.70712 + - 0.0015811092713661505\n",
"Numerical result 0.7071067690849304\n"
]
}
],
"source": [
"import cudaq\n",
"\n",
"cudaq.set_target('nvidia')\n",
"\n",
"@cudaq.kernel\n",
"def U_psi(q:cudaq.qview):\n",
" h(q[1])\n",
"\n",
"@cudaq.kernel\n",
"def U_phi(q:cudaq.qview):\n",
" x(q[0])\n",
"\n",
"@cudaq.kernel \n",
"def ham_cir(q:cudaq.qview):\n",
" x(q[0])\n",
" x(q[1])\n",
"\n",
"@cudaq.kernel\n",
"def kernel(n:int):\n",
" ancilla=cudaq.qubit()\n",
" q = cudaq.qvector(n)\n",
" h(ancilla)\n",
" cudaq.control(U_phi,ancilla,q)\n",
" cudaq.control(ham_cir,ancilla,q)\n",
" cudaq.control(U_psi,ancilla,q)\n",
" \n",
" h(ancilla)\n",
" \n",
" mz(ancilla)\n",
"\n",
"shots = 100000 \n",
"qubit_num=2\n",
"count = cudaq.sample(kernel, qubit_num, shots_count = shots) \n",
"print(count)\n",
"\n",
"mean_val = (count['0']-count['1']) / shots\n",
"error = np.sqrt(2* count['0'] * count['1'] / shots) / shots\n",
"print('Observable QC: ', mean_val,'+ -', error)\n",
"print('Numerical result', np.real(exp_val))\n",
" "
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"### C- Use multi-GPUs to compute the matrix elements"
]
},
{
"cell_type": "code",
"execution_count": 20,
"metadata": {},
"outputs": [
{
"name": "stdout",
"output_type": "stream",
"text": [
"Number of QPUs: 5\n",
"0\n",
"{ 0:63807 1:36193 }\n",
"\n",
"Observable QC: 0.27614 + - 0.0021491238917289066\n",
"1\n",
"{ 0:49929 1:50071 }\n",
"\n",
"Observable QC: -0.00142 + - 0.0022360657230949183\n",
"2\n",
"{ 0:50041 1:49959 }\n",
"\n",
"Observable QC: 0.00082 + - 0.0022360672257336093\n",
"3\n",
"{ 0:50276 1:49724 }\n",
"\n",
"Observable QC: 0.00552 + - 0.0022360339102974265\n"
]
}
],
"source": [
"import cudaq\n",
"\n",
"cudaq.set_target(\"nvidia-mqpu\")\n",
"\n",
"target = cudaq.get_target()\n",
"qpu_count = target.num_qpus()\n",
"print(\"Number of QPUs:\", qpu_count)\n",
"\n",
"@cudaq.kernel\n",
"def U_psi(q:cudaq.qview, theta:float):\n",
" ry(theta, q[1])\n",
"\n",
"@cudaq.kernel\n",
"def U_phi(q:cudaq.qview, theta: float):\n",
" rx(theta, q[0])\n",
"\n",
"@cudaq.kernel \n",
"def ham_cir(q:cudaq.qview):\n",
" x(q[0])\n",
" x(q[1])\n",
"\n",
"@cudaq.kernel\n",
"def kernel(n:int, angle:float, theta:float):\n",
" ancilla = cudaq.qubit()\n",
" q = cudaq.qvector(n)\n",
" h(ancilla)\n",
" cudaq.control(U_phi, ancilla, q, theta)\n",
" cudaq.control(ham_cir, ancilla, q)\n",
" cudaq.control(U_psi, ancilla, q, angle)\n",
" \n",
" h(ancilla)\n",
" \n",
" mz(ancilla)\n",
" \n",
"shots = 100000 \n",
"angle = [0.0, 1.5,3.14,0.7]\n",
"theta = [0.6, 1.2 ,2.2 ,3.0]\n",
"qubit_num = 2\n",
"\n",
"result = []\n",
"for i in range(4): \n",
" count = cudaq.sample_async(kernel, qubit_num, angle[i], theta[i], shots_count = shots, qpu_id = i) \n",
" result.append(count) \n",
"\n",
"mean_val = np.zeros(len(angle))\n",
"i = 0\n",
"for count in result:\n",
" print(i)\n",
" i_result = count.get()\n",
" print(i_result)\n",
" mean_val[i] = (i_result['0'] - i_result['1']) / shots\n",
" error = np.sqrt(2 * i_result['0'] * i_result['1'] / shots) / shots\n",
" print('Observable QC: ', mean_val[i],'+ -', error)\n",
" i += 1\n",
" "
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"#### Diagonalize the matrix using for example Numpy or CuPy. In this example, since we are having 2x2 matrix, we use numpy."
]
},
{
"cell_type": "code",
"execution_count": 21,
"metadata": {},
"outputs": [
{
"name": "stdout",
"output_type": "stream",
"text": [
"[[ 0.27614 -0.00142]\n",
" [ 0.00082 0.00552]]\n",
"Eigen values: \n",
"[0.00551752 0.27614248]\n",
"Eigenvector: \n",
"[[ 0.00303004 -0.99999541]\n",
" [-0.99999541 -0.00303004]]\n"
]
}
],
"source": [
"import numpy as np\n",
"\n",
"my_mat = np.zeros((2,2),dtype=float)\n",
"m = 0\n",
"for k in range(2):\n",
" for j in range(2):\n",
" my_mat[k,j] = mean_val[m]\n",
" m += 1 \n",
"\n",
"print(my_mat)\n",
"\n",
"E,V = np.linalg.eigh(my_mat)\n",
"\n",
"print('Eigen values: ')\n",
"print(E)\n",
"\n",
"print('Eigenvector: ')\n",
"print(V)"
]
}
],
"metadata": {
"kernelspec": {
"display_name": "Python 3",
"language": "python",
"name": "python3"
},
"language_info": {
"codemirror_mode": {
"name": "ipython",
"version": 3
},
"file_extension": ".py",
"mimetype": "text/x-python",
"name": "python",
"nbconvert_exporter": "python",
"pygments_lexer": "ipython3",
"version": "3.10.12"
}
},
"nbformat": 4,
"nbformat_minor": 2
}
1 change: 1 addition & 0 deletions docs/sphinx/using/tutorials.rst
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Expand Up @@ -11,6 +11,7 @@ Tutorials that give an in depth view of CUDA-Q and its applications in Python.
/examples/python/tutorials/cost_minimization.ipynb
/examples/python/tutorials/vqe.ipynb
/examples/python/tutorials/qaoa.ipynb
/examples/python/tutorials/hadamard_test.ipynb
/examples/python/tutorials/hybrid_qnns.ipynb
/examples/python/tutorials/noisy_simulations.ipynb
/examples/python/tutorials/readout_error_mitigation.ipynb
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