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20 | 20 | "id": "a3c9a6e8",
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21 | 21 | "metadata": {},
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22 | 22 | "outputs": [
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23 |
| - { |
24 |
| - "name": "stderr", |
25 |
| - "output_type": "stream", |
26 |
| - "text": [ |
27 |
| - "<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" |
28 |
| - ] |
29 |
| - }, |
30 | 23 | {
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31 | 24 | "name": "stdout",
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32 | 25 | "output_type": "stream",
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131 | 124 | "output_type": "stream",
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132 | 125 | "text": [
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133 | 126 | "Result: ATOSDevice, MYQLM_PYLINALG\n",
|
134 |
| - " Expectation value: -3.3170640012883936\n", |
| 127 | + " Expectation value: -3.317064001288396\n", |
135 | 128 | " Error/Variance: 0.0\n",
|
136 | 129 | "\n",
|
137 |
| - "-3.3170640012883936\n" |
| 130 | + "-3.317064001288396\n" |
138 | 131 | ]
|
139 | 132 | }
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140 | 133 | ],
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|
168 | 161 | "text": [
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169 | 162 | "BatchResult: 2 results\n",
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170 | 163 | "Result: IBMDevice, AER_SIMULATOR\n",
|
171 |
| - " Expectation value: -3.0685000000000002\n", |
172 |
| - " Error/Variance: 41.63205175\n", |
| 164 | + " Expectation value: -3.2504999999999997\n", |
| 165 | + " Error/Variance: 41.46215925\n", |
173 | 166 | "Result: ATOSDevice, MYQLM_PYLINALG\n",
|
174 |
| - " Expectation value: -3.05\n", |
175 |
| - " Error/Variance: 0.14420550306266672\n", |
| 167 | + " Expectation value: -3.2715\n", |
| 168 | + " Error/Variance: 0.1439368300002305\n", |
176 | 169 | "\n"
|
177 | 170 | ]
|
178 | 171 | }
|
|
181 | 174 | "results = run(circuit, [ATOSDevice.MYQLM_PYLINALG, IBMDevice.AER_SIMULATOR])\n",
|
182 | 175 | "print(results)"
|
183 | 176 | ]
|
| 177 | + }, |
| 178 | + { |
| 179 | + "cell_type": "markdown", |
| 180 | + "id": "057b9856", |
| 181 | + "metadata": {}, |
| 182 | + "source": [ |
| 183 | + "## Pauli String representation\n", |
| 184 | + "\n", |
| 185 | + "An observable can also be represented by a Pauli string. This section will demonstrate how to create and manipulate PauliString objects." |
| 186 | + ] |
| 187 | + }, |
| 188 | + { |
| 189 | + "cell_type": "code", |
| 190 | + "execution_count": 9, |
| 191 | + "id": "4ac9c7e0", |
| 192 | + "metadata": {}, |
| 193 | + "outputs": [], |
| 194 | + "source": [ |
| 195 | + "from mpqp.measures import I, X as Pauli_X, Y as Pauli_Y, X as Pauli_Z" |
| 196 | + ] |
| 197 | + }, |
| 198 | + { |
| 199 | + "cell_type": "markdown", |
| 200 | + "id": "aed703a9", |
| 201 | + "metadata": {}, |
| 202 | + "source": [ |
| 203 | + "> ⚠ **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", |
| 204 | + "> - **Rename Import:**\n", |
| 205 | + "> ```python \n", |
| 206 | + "> from mpqp.measures import X as Pauli_X \n", |
| 207 | + "> ```\n", |
| 208 | + "> - **Import Only Pauli String:**\n", |
| 209 | + "> ```python\n", |
| 210 | + "> from mpqp.measures import pauli_string \n", |
| 211 | + "> pauli_string.X\n", |
| 212 | + "> ```\n", |
| 213 | + ">" |
| 214 | + ] |
| 215 | + }, |
| 216 | + { |
| 217 | + "cell_type": "markdown", |
| 218 | + "id": "30d41d0f", |
| 219 | + "metadata": {}, |
| 220 | + "source": [ |
| 221 | + "A `PauliString` is based on the following hierarchy:\n", |
| 222 | + "- an *atom* is the most elemental building brick of pauli string, it it either\n", |
| 223 | + " `I`, `X`, `Y` or `Z`;\n", |
| 224 | + "- a *monomial* is a tensor product of *atoms* multiplied by a real coefficient,\n", |
| 225 | + " for instance `0.3 * I⊗Z⊗Y`. In MPQP, the tensor product is denoted as `@`, so\n", |
| 226 | + " the previous example would be expressed as `0.3 * I@Z@Y`;\n", |
| 227 | + "- a *string* is a sum of *monomials*, for instance `0.3 * I@Z@Y + X@X@X`.\n", |
| 228 | + "\n", |
| 229 | + "In practice, you never need to handle these types independently, you can express\n", |
| 230 | + "everything in term of expression based on the fundamental atoms. Let's see how." |
| 231 | + ] |
| 232 | + }, |
| 233 | + { |
| 234 | + "cell_type": "markdown", |
| 235 | + "id": "7d07ec4b", |
| 236 | + "metadata": {}, |
| 237 | + "source": [ |
| 238 | + "### Creating and Manipulating PauliString Objects\n", |
| 239 | + "\n", |
| 240 | + "Let's illustrate how to create and manipulate PauliString objects:" |
| 241 | + ] |
| 242 | + }, |
| 243 | + { |
| 244 | + "cell_type": "code", |
| 245 | + "execution_count": 10, |
| 246 | + "id": "bc5ed517", |
| 247 | + "metadata": {}, |
| 248 | + "outputs": [ |
| 249 | + { |
| 250 | + "name": "stdout", |
| 251 | + "output_type": "stream", |
| 252 | + "text": [ |
| 253 | + "ps_1=1*I@X + -3*X@Y\n" |
| 254 | + ] |
| 255 | + } |
| 256 | + ], |
| 257 | + "source": [ |
| 258 | + "ps_1 = I @ Pauli_Z - 3 * Pauli_X @ Pauli_Y\n", |
| 259 | + "print(f\"{ps_1=}\")" |
| 260 | + ] |
| 261 | + }, |
| 262 | + { |
| 263 | + "cell_type": "markdown", |
| 264 | + "id": "7824baf5", |
| 265 | + "metadata": {}, |
| 266 | + "source": [ |
| 267 | + "#### Simplifying and Rounding PauliStrings\n", |
| 268 | + "\n", |
| 269 | + "When dealing with PauliStrings, simplification and rounding are common operations:\n", |
| 270 | + "\n", |
| 271 | + "- **Simplify**: We combine like terms and eliminate those with zero coefficients;\n", |
| 272 | + "- **Round**: We can round the coefficients to a specified number of decimals (5 by default).\n", |
| 273 | + "\n", |
| 274 | + "> `str` on a `PauliString` will call both methods: `round()` and `simplify()`" |
| 275 | + ] |
| 276 | + }, |
| 277 | + { |
| 278 | + "cell_type": "code", |
| 279 | + "execution_count": 11, |
| 280 | + "id": "06784ef3", |
| 281 | + "metadata": {}, |
| 282 | + "outputs": [ |
| 283 | + { |
| 284 | + "name": "stdout", |
| 285 | + "output_type": "stream", |
| 286 | + "text": [ |
| 287 | + "ps_2 = 1*I@X + 2.555555555*Y@I + 1*X@X + -1*X@X\n", |
| 288 | + " = 1*I@X + 2.555555555*Y@I\n", |
| 289 | + " ~= 1*I@X + 2.6*Y@I + 1*X@X + -1*X@X\n", |
| 290 | + " ~= 1*I@X + 2.5556*Y@I\n" |
| 291 | + ] |
| 292 | + } |
| 293 | + ], |
| 294 | + "source": [ |
| 295 | + "ps_2 = I @ Pauli_Z + 2.555555555 * Pauli_Y @ I + Pauli_X @ Pauli_Z - Pauli_X @ Pauli_Z\n", |
| 296 | + "print(\"ps_2 =\",repr(ps_2))\n", |
| 297 | + "print(\" =\",repr(ps_2.simplify()))\n", |
| 298 | + "print(\" ~=\",repr(ps_2.round(1)))\n", |
| 299 | + "print(\" ~=\",ps_2)" |
| 300 | + ] |
| 301 | + }, |
| 302 | + { |
| 303 | + "cell_type": "markdown", |
| 304 | + "id": "bb3a48ec", |
| 305 | + "metadata": {}, |
| 306 | + "source": [ |
| 307 | + "#### Arithmetic Operations\n", |
| 308 | + "\n", |
| 309 | + "We can perform various arithmetic operations on PauliString objects, including addition, subtraction, scalar multiplication, scalar division, and matrix multiplication:" |
| 310 | + ] |
| 311 | + }, |
| 312 | + { |
| 313 | + "cell_type": "code", |
| 314 | + "execution_count": 12, |
| 315 | + "id": "b9cadb08", |
| 316 | + "metadata": {}, |
| 317 | + "outputs": [ |
| 318 | + { |
| 319 | + "name": "stdout", |
| 320 | + "output_type": "stream", |
| 321 | + "text": [ |
| 322 | + "Addition:\n", |
| 323 | + "(1*I@X + -3*X@Y) + (1*I@X + 2.6*Y@I) = 2*I@X + -3*X@Y + 2.6*Y@I\n", |
| 324 | + "\n", |
| 325 | + "Subtraction:\n", |
| 326 | + "(1*I@X + -3*X@Y) - (1*I@X + 2.6*Y@I) = -3*X@Y + -2.6*Y@I\n", |
| 327 | + "\n", |
| 328 | + "Scalar product:\n", |
| 329 | + "2 * (1*I@X + -3*X@Y) = 2*I@X + -6*X@Y\n", |
| 330 | + "\n", |
| 331 | + "Scalar division:\n", |
| 332 | + "(1*I@X + 2.6*Y@I) / 3 ~= 0.3333*I@X + 0.8667*Y@I\n", |
| 333 | + "\n", |
| 334 | + "Tensor product:\n", |
| 335 | + "(1*I@X + -3*X@Y) @ Z = 1*I@X@X + -3*X@Y@X\n", |
| 336 | + "\n", |
| 337 | + "(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" |
| 338 | + ] |
| 339 | + } |
| 340 | + ], |
| 341 | + "source": [ |
| 342 | + "ps_2 = ps_2.round(1).simplify()\n", |
| 343 | + "print(f\"\"\"Addition:\n", |
| 344 | + "({ps_1}) + ({ps_2}) = {ps_1 + ps_2}\n", |
| 345 | + "\n", |
| 346 | + "Subtraction:\n", |
| 347 | + "({ps_1}) - ({ps_2}) = {ps_1 - ps_2}\n", |
| 348 | + "\n", |
| 349 | + "Scalar product:\n", |
| 350 | + "2 * ({ps_1}) = {2 * ps_1}\n", |
| 351 | + "\n", |
| 352 | + "Scalar division:\n", |
| 353 | + "({ps_2}) / 3 ~= {ps_2 / 3}\n", |
| 354 | + "\n", |
| 355 | + "Tensor product:\n", |
| 356 | + "({ps_1}) @ Z = {ps_1 @ Pauli_Z}\n", |
| 357 | + "\n", |
| 358 | + "({ps_1}) @ ({ps_2}) = {ps_1 @ ps_2}\"\"\")" |
| 359 | + ] |
| 360 | + }, |
| 361 | + { |
| 362 | + "cell_type": "markdown", |
| 363 | + "id": "59c34f41", |
| 364 | + "metadata": {}, |
| 365 | + "source": [ |
| 366 | + "### Create an Observable with a Pauli string" |
| 367 | + ] |
| 368 | + }, |
| 369 | + { |
| 370 | + "cell_type": "code", |
| 371 | + "execution_count": 13, |
| 372 | + "id": "5b0537a1", |
| 373 | + "metadata": {}, |
| 374 | + "outputs": [], |
| 375 | + "source": [ |
| 376 | + "obs1 = Observable(ps_1)" |
| 377 | + ] |
| 378 | + }, |
| 379 | + { |
| 380 | + "cell_type": "markdown", |
| 381 | + "id": "3b411669", |
| 382 | + "metadata": {}, |
| 383 | + "source": [ |
| 384 | + "Since there is an equivalence between definition from a Pauli string or from a\n", |
| 385 | + "Hermitian matrix, both these observable wan can also be expressed in term of the\n", |
| 386 | + "mean through which it was not defined (Pauli for matrix and vice versa):" |
| 387 | + ] |
| 388 | + }, |
| 389 | + { |
| 390 | + "cell_type": "code", |
| 391 | + "execution_count": 31, |
| 392 | + "id": "1732e099", |
| 393 | + "metadata": {}, |
| 394 | + "outputs": [ |
| 395 | + { |
| 396 | + "name": "stdout", |
| 397 | + "output_type": "stream", |
| 398 | + "text": [ |
| 399 | + "`obs` created with matrix:\n", |
| 400 | + "matrix:\n", |
| 401 | + "[[ 4.+0.j 2.+0.j 3.+0.j 8.+0.j]\n", |
| 402 | + " [ 2.+0.j -3.+0.j 1.+0.j 0.+0.j]\n", |
| 403 | + " [ 3.+0.j 1.+0.j -1.+0.j 5.+0.j]\n", |
| 404 | + " [ 8.+0.j 0.+0.j 5.+0.j 2.+0.j]]\n", |
| 405 | + "Pauli string:\n", |
| 406 | + "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", |
| 407 | + "\n", |
| 408 | + "\n", |
| 409 | + "`obs1` created with Pauli string:\n", |
| 410 | + "Pauli string:\n", |
| 411 | + "1*I@X + -3*X@Y\n", |
| 412 | + "matrix:\n", |
| 413 | + "[[0.+0.j 1.+0.j 0.+0.j 0.-3.j]\n", |
| 414 | + " [1.+0.j 0.+0.j 0.+3.j 0.+0.j]\n", |
| 415 | + " [0.+0.j 0.-3.j 0.+0.j 1.+0.j]\n", |
| 416 | + " [0.+3.j 0.+0.j 1.+0.j 0.+0.j]]\n" |
| 417 | + ] |
| 418 | + } |
| 419 | + ], |
| 420 | + "source": [ |
| 421 | + "print(\"`obs` created with matrix:\")\n", |
| 422 | + "print(\"matrix:\")\n", |
| 423 | + "print(obs.matrix)\n", |
| 424 | + "print(\"Pauli string:\")\n", |
| 425 | + "print(obs.pauli_string)\n", |
| 426 | + "\n", |
| 427 | + "\n", |
| 428 | + "print(\"\\n\\n`obs1` created with Pauli string:\")\n", |
| 429 | + "print(\"Pauli string:\")\n", |
| 430 | + "print(obs1.pauli_string)\n", |
| 431 | + "print(\"matrix:\")\n", |
| 432 | + "print(obs1.matrix)" |
| 433 | + ] |
| 434 | + }, |
| 435 | + { |
| 436 | + "cell_type": "markdown", |
| 437 | + "id": "a63b436e", |
| 438 | + "metadata": {}, |
| 439 | + "source": [ |
| 440 | + "### ExpectationMeasure\n", |
| 441 | + "\n", |
| 442 | + "Next, we measure the expectation value of the observable by adding it to the circuit." |
| 443 | + ] |
| 444 | + }, |
| 445 | + { |
| 446 | + "cell_type": "code", |
| 447 | + "execution_count": 24, |
| 448 | + "id": "892f0b6e", |
| 449 | + "metadata": {}, |
| 450 | + "outputs": [], |
| 451 | + "source": [ |
| 452 | + "circuit = circuit.without_measurements()\n", |
| 453 | + "circuit.add(ExpectationMeasure([0, 1], observable=obs1, shots=1000))" |
| 454 | + ] |
| 455 | + }, |
| 456 | + { |
| 457 | + "cell_type": "code", |
| 458 | + "execution_count": 25, |
| 459 | + "id": "745aa999", |
| 460 | + "metadata": {}, |
| 461 | + "outputs": [ |
| 462 | + { |
| 463 | + "name": "stdout", |
| 464 | + "output_type": "stream", |
| 465 | + "text": [ |
| 466 | + "BatchResult: 3 results\n", |
| 467 | + "Result: IBMDevice, AER_SIMULATOR\n", |
| 468 | + " Expectation value: 0.08399999999999999\n", |
| 469 | + " Error/Variance: 9.98776\n", |
| 470 | + "Result: ATOSDevice, MYQLM_PYLINALG\n", |
| 471 | + " Expectation value: 0.010000000000000009\n", |
| 472 | + " Error/Variance: 0.10000486474653607\n", |
| 473 | + "Result: ATOSDevice, MYQLM_CLINALG\n", |
| 474 | + " Expectation value: -0.017999999999999933\n", |
| 475 | + " Error/Variance: 0.10001671531970716\n", |
| 476 | + "\n" |
| 477 | + ] |
| 478 | + } |
| 479 | + ], |
| 480 | + "source": [ |
| 481 | + "results = run(\n", |
| 482 | + " circuit,\n", |
| 483 | + " [\n", |
| 484 | + " ATOSDevice.MYQLM_PYLINALG,\n", |
| 485 | + " IBMDevice.AER_SIMULATOR,\n", |
| 486 | + " ATOSDevice.MYQLM_CLINALG,\n", |
| 487 | + " ],\n", |
| 488 | + ")\n", |
| 489 | + "print(results)" |
| 490 | + ] |
184 | 491 | }
|
185 | 492 | ],
|
186 | 493 | "metadata": {
|
|
199 | 506 | "name": "python",
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200 | 507 | "nbconvert_exporter": "python",
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201 | 508 | "pygments_lexer": "ipython3",
|
202 |
| - "version": "3.9.12" |
| 509 | + "version": "3.9.5" |
203 | 510 | }
|
204 | 511 | },
|
205 | 512 | "nbformat": 4,
|
|
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