{
"cells": [
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"cell_type": "markdown",
"id": "equivalence-0",
"metadata": {},
"source": [
"# Are two circuits the same?\n",
"\n",
"`equivalent` says whether two programs apply the same gates up to a global phase, and how it decided: by their stabilizer tableaus (Clifford circuits, at any width), on every basis state (up to 8 qubits), by a chain of checked rewrites, or on random product states (evidence, not proof)."
]
},
{
"cell_type": "code",
"execution_count": 1,
"id": "equivalence-1",
"metadata": {
"execution": {
"iopub.execute_input": "2026-11-11T12:23:54.515997Z",
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"shell.execute_reply": "2026-11-11T12:23:54.520519Z"
}
},
"outputs": [],
"source": [
"import wai_quantum as wq"
]
},
{
"cell_type": "code",
"execution_count": 2,
"id": "equivalence-2",
"metadata": {
"execution": {
"iopub.execute_input": "2026-11-11T12:23:54.521714Z",
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"name": "stdout",
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"text": [
"two Hadamards and nothing equivalent by tableau both are Clifford circuits with the same tableau: the same unitary up to a global phase, at any number of qubits\n",
"T twice and S equivalent by exact every one of the 2 basis states gives the same output up to one global phase\n",
"X and Z different by tableau both are Clifford circuits, and their tableaus differ\n",
"a CX, and H CZ H equivalent by tableau both are Clifford circuits with the same tableau: the same unitary up to a global phase, at any number of qubits\n",
"different widths different by shape one has 1 qubits and the other 2\n"
]
}
],
"source": [
"pairs = {\n",
" \"two Hadamards and nothing\": (\"qubit[1] q; h q[0]; h q[0];\", \"qubit[1] q;\"),\n",
" \"T twice and S\": (\"qubit[1] q; t q[0]; t q[0];\", \"qubit[1] q; s q[0];\"),\n",
" \"X and Z\": (\"qubit[1] q; x q[0];\", \"qubit[1] q; z q[0];\"),\n",
" \"a CX, and H CZ H\": (\n",
" \"qubit[2] q; cx q[0], q[1];\",\n",
" \"qubit[2] q; h q[1]; cz q[0], q[1]; h q[1];\",\n",
" ),\n",
" \"different widths\": (\"qubit[1] q; x q[0];\", \"qubit[2] q; x q[0];\"),\n",
"}\n",
"for name, (a, b) in pairs.items():\n",
" v = wq.equivalent(\"OPENQASM 3.0; \" + a, \"OPENQASM 3.0; \" + b)\n",
" print(f\"{name:<26} {v['verdict']:<11} by {v['how']:<8} {v['why']}\")"
]
},
{
"cell_type": "markdown",
"id": "equivalence-3",
"metadata": {},
"source": [
"## Improve a design, proved equal\n",
"\n",
"`improve` makes the critique's fixes and offers each variant only after the engine has checked it equivalent to the program."
]
},
{
"cell_type": "code",
"execution_count": 3,
"id": "equivalence-4",
"metadata": {
"execution": {
"iopub.execute_input": "2026-11-11T12:23:54.524950Z",
"iopub.status.busy": "2026-11-11T12:23:54.524890Z",
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"shell.execute_reply": "2026-11-11T12:23:54.529545Z"
}
},
"outputs": [
{
"name": "stdout",
"output_type": "stream",
"text": [
"changes: ['h on qubit 0 twice in a row (lines 3 and 4): together they do nothing', 'rz on qubit 1 twice in a row (lines 5 and 6): one rz with the angles added']\n",
"gates 5 -> 2 | every one of the 4 basis states gives the same output up to one global phase\n",
"OPENQASM 3.0;\n",
"qubit[2] q;\n",
"rz(0.25 + 0.5) q[1];\n",
"cx q[0], q[1];\n",
"\n"
]
}
],
"source": [
"src = '''OPENQASM 3.0;\n",
"qubit[2] q;\n",
"h q[0];\n",
"h q[0];\n",
"rz(0.25) q[1];\n",
"rz(0.5) q[1];\n",
"cx q[0], q[1];\n",
"'''\n",
"v = wq.improve(src)\n",
"best = v[\"candidates\"][0]\n",
"print(\"changes:\", best[\"changes\"])\n",
"print(\"gates\", v[\"base\"][\"gates\"], \"->\", best[\"measures\"][\"gates\"], \"|\", best[\"equivalence\"][\"why\"])\n",
"print(best[\"qasm\"])"
]
},
{
"cell_type": "code",
"execution_count": 4,
"id": "equivalence-5",
"metadata": {
"execution": {
"iopub.execute_input": "2026-11-11T12:23:54.530871Z",
"iopub.status.busy": "2026-11-11T12:23:54.530783Z",
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"shell.execute_reply": "2026-11-11T12:23:54.582841Z"
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""
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"Circuit(2 qubits, 7 operations in 4 columns)"
]
},
"execution_count": 4,
"metadata": {},
"output_type": "execute_result"
}
],
"source": [
"wq.Circuit(src)"
]
},
{
"cell_type": "code",
"execution_count": 5,
"id": "equivalence-6",
"metadata": {
"execution": {
"iopub.execute_input": "2026-11-11T12:23:54.584337Z",
"iopub.status.busy": "2026-11-11T12:23:54.584255Z",
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"shell.execute_reply": "2026-11-11T12:23:54.586867Z"
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""
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"text/plain": [
"Circuit(2 qubits, 4 operations in 3 columns)"
]
},
"execution_count": 5,
"metadata": {},
"output_type": "execute_result"
}
],
"source": [
"wq.Circuit(best[\"qasm\"])"
]
}
],
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