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doc: combined notebook 3 and 6
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‎examples/notebooks/3_Expectation_value_of_observables.ipynb

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"id": "a3c9a6e8",
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"metadata": {},
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"outputs": [
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{
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"name": "stderr",
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"output_type": "stream",
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"text": [
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"<frozen importlib._bootstrap>:228: RuntimeWarning: scipy._lib.messagestream.MessageStream size changed, may indicate binary incompatibility. Expected 56 from C header, got 64 from PyObject\n"
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]
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},
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{
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"name": "stdout",
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"output_type": "stream",
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"output_type": "stream",
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"text": [
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"Result: ATOSDevice, MYQLM_PYLINALG\n",
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" Expectation value: -3.3170640012883936\n",
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" Expectation value: -3.317064001288396\n",
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" Error/Variance: 0.0\n",
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"\n",
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"-3.3170640012883936\n"
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"-3.317064001288396\n"
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]
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}
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],
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"text": [
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"BatchResult: 2 results\n",
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"Result: IBMDevice, AER_SIMULATOR\n",
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" Expectation value: -3.0685000000000002\n",
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" Error/Variance: 41.63205175\n",
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" Expectation value: -3.2504999999999997\n",
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" Error/Variance: 41.46215925\n",
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"Result: ATOSDevice, MYQLM_PYLINALG\n",
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" Expectation value: -3.05\n",
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" Error/Variance: 0.14420550306266672\n",
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" Expectation value: -3.2715\n",
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" Error/Variance: 0.1439368300002305\n",
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"\n"
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]
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}
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"results = run(circuit, [ATOSDevice.MYQLM_PYLINALG, IBMDevice.AER_SIMULATOR])\n",
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"print(results)"
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]
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},
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{
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"cell_type": "markdown",
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"id": "057b9856",
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"metadata": {},
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"source": [
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"## Pauli String representation\n",
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"\n",
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"An observable can also be represented by a Pauli string. This section will demonstrate how to create and manipulate PauliString objects."
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]
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},
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{
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"cell_type": "code",
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"execution_count": 9,
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"id": "4ac9c7e0",
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"metadata": {},
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"outputs": [],
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"source": [
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"from mpqp.measures import I, X as Pauli_X, Y as Pauli_Y, X as Pauli_Z"
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]
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},
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{
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"cell_type": "markdown",
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"id": "aed703a9",
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"metadata": {},
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"source": [
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"> ⚠ **pauli_string import**: pauli atoms are named I, X, Y, and Z. If you have conflicts with `mpqp.gates import X, Y, Z,` , you can:\n",
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"> - **Rename Import:**\n",
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"> ```python \n",
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"> from mpqp.measures import X as Pauli_X \n",
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"> ```\n",
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"> - **Import Only Pauli String:**\n",
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"> ```python\n",
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"> from mpqp.measures import pauli_string \n",
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"> pauli_string.X\n",
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"> ```\n",
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">"
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]
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},
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{
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"cell_type": "markdown",
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"id": "30d41d0f",
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"metadata": {},
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"source": [
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"A `PauliString` is based on the following hierarchy:\n",
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"- an *atom* is the most elemental building brick of pauli string, it it either\n",
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" `I`, `X`, `Y` or `Z`;\n",
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"- a *monomial* is a tensor product of *atoms* multiplied by a real coefficient,\n",
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" for instance `0.3 * I⊗Z⊗Y`. In MPQP, the tensor product is denoted as `@`, so\n",
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" the previous example would be expressed as `0.3 * I@Z@Y`;\n",
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"- a *string* is a sum of *monomials*, for instance `0.3 * I@Z@Y + X@X@X`.\n",
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"\n",
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"In practice, you never need to handle these types independently, you can express\n",
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"everything in term of expression based on the fundamental atoms. Let's see how."
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]
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},
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{
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"cell_type": "markdown",
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"id": "7d07ec4b",
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"metadata": {},
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"source": [
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"### Creating and Manipulating PauliString Objects\n",
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"\n",
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"Let's illustrate how to create and manipulate PauliString objects:"
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]
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},
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{
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"cell_type": "code",
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"execution_count": 10,
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"id": "bc5ed517",
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"metadata": {},
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"outputs": [
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{
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"name": "stdout",
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"output_type": "stream",
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"text": [
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"ps_1=1*I@X + -3*X@Y\n"
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]
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}
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],
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"source": [
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"ps_1 = I @ Pauli_Z - 3 * Pauli_X @ Pauli_Y\n",
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"print(f\"{ps_1=}\")"
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]
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},
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{
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"cell_type": "markdown",
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"id": "7824baf5",
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"metadata": {},
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"source": [
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"#### Simplifying and Rounding PauliStrings\n",
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"\n",
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"When dealing with PauliStrings, simplification and rounding are common operations:\n",
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"\n",
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"- **Simplify**: We combine like terms and eliminate those with zero coefficients;\n",
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"- **Round**: We can round the coefficients to a specified number of decimals (5 by default).\n",
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"\n",
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"> `str` on a `PauliString` will call both methods: `round()` and `simplify()`"
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]
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},
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{
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"cell_type": "code",
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"execution_count": 11,
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"id": "06784ef3",
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"metadata": {},
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"outputs": [
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{
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"name": "stdout",
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"output_type": "stream",
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"text": [
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"ps_2 = 1*I@X + 2.555555555*Y@I + 1*X@X + -1*X@X\n",
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" = 1*I@X + 2.555555555*Y@I\n",
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" ~= 1*I@X + 2.6*Y@I + 1*X@X + -1*X@X\n",
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" ~= 1*I@X + 2.5556*Y@I\n"
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]
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}
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],
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"source": [
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"ps_2 = I @ Pauli_Z + 2.555555555 * Pauli_Y @ I + Pauli_X @ Pauli_Z - Pauli_X @ Pauli_Z\n",
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"print(\"ps_2 =\",repr(ps_2))\n",
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"print(\" =\",repr(ps_2.simplify()))\n",
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"print(\" ~=\",repr(ps_2.round(1)))\n",
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"print(\" ~=\",ps_2)"
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]
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},
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{
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"cell_type": "markdown",
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"id": "bb3a48ec",
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"metadata": {},
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"source": [
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"#### Arithmetic Operations\n",
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"\n",
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"We can perform various arithmetic operations on PauliString objects, including addition, subtraction, scalar multiplication, scalar division, and matrix multiplication:"
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]
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},
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{
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"cell_type": "code",
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"execution_count": 12,
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"id": "b9cadb08",
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"metadata": {},
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"outputs": [
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{
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"name": "stdout",
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"output_type": "stream",
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"text": [
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"Addition:\n",
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"(1*I@X + -3*X@Y) + (1*I@X + 2.6*Y@I) = 2*I@X + -3*X@Y + 2.6*Y@I\n",
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"\n",
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"Subtraction:\n",
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"(1*I@X + -3*X@Y) - (1*I@X + 2.6*Y@I) = -3*X@Y + -2.6*Y@I\n",
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"\n",
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"Scalar product:\n",
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"2 * (1*I@X + -3*X@Y) = 2*I@X + -6*X@Y\n",
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"\n",
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"Scalar division:\n",
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"(1*I@X + 2.6*Y@I) / 3 ~= 0.3333*I@X + 0.8667*Y@I\n",
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"\n",
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"Tensor product:\n",
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"(1*I@X + -3*X@Y) @ Z = 1*I@X@X + -3*X@Y@X\n",
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"\n",
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"(1*I@X + -3*X@Y) @ (1*I@X + 2.6*Y@I) = 1*I@X@I@X + -3*I@X@X@Y + 2.6*Y@I@I@X + -7.8*Y@I@X@Y\n"
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]
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}
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],
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"source": [
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"ps_2 = ps_2.round(1).simplify()\n",
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"print(f\"\"\"Addition:\n",
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"({ps_1}) + ({ps_2}) = {ps_1 + ps_2}\n",
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"\n",
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"Subtraction:\n",
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"({ps_1}) - ({ps_2}) = {ps_1 - ps_2}\n",
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"\n",
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"Scalar product:\n",
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"2 * ({ps_1}) = {2 * ps_1}\n",
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"\n",
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"Scalar division:\n",
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"({ps_2}) / 3 ~= {ps_2 / 3}\n",
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"\n",
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"Tensor product:\n",
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"({ps_1}) @ Z = {ps_1 @ Pauli_Z}\n",
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"\n",
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"({ps_1}) @ ({ps_2}) = {ps_1 @ ps_2}\"\"\")"
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]
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},
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{
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"cell_type": "markdown",
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"id": "59c34f41",
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"metadata": {},
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"source": [
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"### Create an Observable with a Pauli string"
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]
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},
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{
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"cell_type": "code",
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"execution_count": 13,
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"id": "5b0537a1",
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"metadata": {},
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"outputs": [],
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"source": [
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"obs1 = Observable(ps_1)"
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]
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},
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{
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"cell_type": "markdown",
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"id": "3b411669",
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"metadata": {},
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"source": [
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"Since there is an equivalence between definition from a Pauli string or from a\n",
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"Hermitian matrix, both these observable wan can also be expressed in term of the\n",
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"mean through which it was not defined (Pauli for matrix and vice versa):"
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]
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},
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{
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"cell_type": "code",
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"execution_count": 31,
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"id": "1732e099",
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"metadata": {},
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"outputs": [
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{
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"name": "stdout",
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"output_type": "stream",
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"text": [
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"`obs` created with matrix:\n",
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"matrix:\n",
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"[[ 4.+0.j 2.+0.j 3.+0.j 8.+0.j]\n",
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" [ 2.+0.j -3.+0.j 1.+0.j 0.+0.j]\n",
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" [ 3.+0.j 1.+0.j -1.+0.j 5.+0.j]\n",
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" [ 8.+0.j 0.+0.j 5.+0.j 2.+0.j]]\n",
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"Pauli string:\n",
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"0.5*I@I + 3.5*I@X + 1*I@Z + 1.5*X@I + 4.5*X@X + 1.5*X@Z + -3.5*Y@Y + -1.5*Z@X + 2.5*Z@Z\n",
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"\n",
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"\n",
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"`obs1` created with Pauli string:\n",
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"Pauli string:\n",
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"1*I@X + -3*X@Y\n",
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"matrix:\n",
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"[[0.+0.j 1.+0.j 0.+0.j 0.-3.j]\n",
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" [1.+0.j 0.+0.j 0.+3.j 0.+0.j]\n",
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" [0.+0.j 0.-3.j 0.+0.j 1.+0.j]\n",
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" [0.+3.j 0.+0.j 1.+0.j 0.+0.j]]\n"
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]
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}
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],
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"source": [
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"print(\"`obs` created with matrix:\")\n",
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"print(\"matrix:\")\n",
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"print(obs.matrix)\n",
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"print(\"Pauli string:\")\n",
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"print(obs.pauli_string)\n",
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"\n",
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"\n",
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"print(\"\\n\\n`obs1` created with Pauli string:\")\n",
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"print(\"Pauli string:\")\n",
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"print(obs1.pauli_string)\n",
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"print(\"matrix:\")\n",
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"print(obs1.matrix)"
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]
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},
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{
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"cell_type": "markdown",
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"id": "a63b436e",
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"metadata": {},
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"source": [
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"### ExpectationMeasure\n",
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"\n",
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"Next, we measure the expectation value of the observable by adding it to the circuit."
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]
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},
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{
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"cell_type": "code",
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"execution_count": 24,
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"id": "892f0b6e",
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"metadata": {},
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"outputs": [],
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"source": [
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"circuit = circuit.without_measurements()\n",
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"circuit.add(ExpectationMeasure([0, 1], observable=obs1, shots=1000))"
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]
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},
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{
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"cell_type": "code",
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"execution_count": 25,
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"id": "745aa999",
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"metadata": {},
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"outputs": [
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{
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"name": "stdout",
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"output_type": "stream",
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"text": [
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"BatchResult: 3 results\n",
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"Result: IBMDevice, AER_SIMULATOR\n",
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" Expectation value: 0.08399999999999999\n",
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" Error/Variance: 9.98776\n",
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"Result: ATOSDevice, MYQLM_PYLINALG\n",
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" Expectation value: 0.010000000000000009\n",
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" Error/Variance: 0.10000486474653607\n",
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"Result: ATOSDevice, MYQLM_CLINALG\n",
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" Expectation value: -0.017999999999999933\n",
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" Error/Variance: 0.10001671531970716\n",
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"\n"
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]
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}
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],
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"source": [
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"results = run(\n",
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" circuit,\n",
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" [\n",
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" ATOSDevice.MYQLM_PYLINALG,\n",
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" IBMDevice.AER_SIMULATOR,\n",
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" ATOSDevice.MYQLM_CLINALG,\n",
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" ],\n",
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")\n",
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"print(results)"
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]
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}
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],
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"metadata": {
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"name": "python",
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"nbconvert_exporter": "python",
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"pygments_lexer": "ipython3",
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"version": "3.9.12"
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"version": "3.9.5"
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}
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},
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"nbformat": 4,

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