marcusps · marcusps · Jul 4, 2026 · Jul 5, 2026 · Jul 5, 2026 · Jul 17, 2026
diff --git a/paulimer/README.md b/paulimer/README.md
@@ -21,6 +21,9 @@ the building blocks for stabilizer quantum mechanics and quantum error correctio
 - **Clifford Unitaries**: Efficient representation enabling fast operations
   - [`CliffordUnitary`]: O(n²) Pauli conjugation via binary symplectic matrix
   - Supports all standard Clifford gates (H, S, CNOT, etc.)
+  - Decomposition into Clifford transvections (`π/4` Pauli exponents), including a
+    strict-minimum-length variant, via [`clifford_to_transvections`] and
+    [`clifford_to_transvections_minimal`]
 
 Based on algorithms from [arXiv:2309.08676](https://arxiv.org/abs/2309.08676).
 
@@ -171,6 +174,8 @@ Key documentation:
 - [`SparsePauli`](src/pauli/sparse.rs) - Sparse Pauli representation for large systems
 - [`PauliGroup`](src/pauli_group.rs) - Subgroup operations and stabilizer groups
 - [`CliffordUnitary`](src/clifford.rs) - Clifford gates and Pauli conjugation
+- [Transvection decomposition](src/clifford/transvection.rs) - Decomposing Cliffords into `π/4`
+  Pauli exponents (`clifford_to_transvections`, `clifford_to_transvections_minimal`)
 - [Trait documentation](src/lib.rs) - `Pauli`, `Clifford`, and other core traits
 
 ## Contributing

diff --git a/paulimer/bindings/python/README.md b/paulimer/bindings/python/README.md
@@ -22,6 +22,11 @@ print(q * q)  # Identity
 h = paulimer.CliffordUnitary.from_name("Hadamard", [0], qubit_count=1)
 print(h.image_of(paulimer.DensePauli("X")))  # Z
 
+# Decompose a Clifford into pi/4 Pauli exponents (Clifford transvections)
+cnot = paulimer.CliffordUnitary.from_name("ControlledX", [0, 1], qubit_count=2)
+factors = cnot.to_transvections_minimal()
+print(factors)  # minimal-length list of transvection Paulis reproducing the symplectic action
+
 # Stabilizer simulation
 sim = paulimer.OutcomeCompleteSimulation(2)
 sim.apply_unitary(paulimer.UnitaryOpcode.Hadamard, [0])
@@ -32,7 +37,8 @@ sim.measure(paulimer.SparsePauli("Z0"))
 ## Features
 
 - **DensePauli / SparsePauli** - Pauli operators with phase tracking and multiplication
-- **CliffordUnitary** - Clifford gates with conjugation and composition
+- **CliffordUnitary** - Clifford gates with conjugation, composition, and decomposition into `π/4`
+  Pauli exponents (`to_transvections`, `to_transvections_minimal`)
 - **PauliGroup** - Group operations including membership testing and factorization
 - **Stabilizer Simulation** - Noiseless (OutcomeComplete, OutcomeFree, OutcomeSpecific) and noisy (Faulty) modes
 

diff --git a/paulimer/bindings/python/examples/clifford-transvection-decomposition.ipynb b/paulimer/bindings/python/examples/clifford-transvection-decomposition.ipynb
@@ -0,0 +1,300 @@
+{
+ "cells": [
+  {
+   "cell_type": "markdown",
+   "id": "a95d9b9a",
+   "metadata": {},
+   "source": [
+    "# Decomposing Cliffords into transvections (π/4 Pauli exponents)\n",
+    "\n",
+    "Every Clifford unitary can be written as an ordered product of **Clifford transvections** — the\n",
+    "`π/4` Pauli exponents $\\exp\\!\\big(i\\tfrac{\\pi}{4} P_v\\big)$. Conjugation by such an exponent acts on\n",
+    "Pauli operators as a **symplectic transvection**\n",
+    "\n",
+    "$$\n",
+    "x \\;\\mapsto\\; x + \\langle x, v\\rangle\\, v,\n",
+    "$$\n",
+    "\n",
+    "where $\\langle\\cdot,\\cdot\\rangle$ is the symplectic (commutation) form. `paulimer` exposes two\n",
+    "decompositions, following the transvection framework of\n",
+    "[arXiv:2102.11380](https://arxiv.org/abs/2102.11380) (Pllaha, Volanto & Tirkkonen,\n",
+    "*Decomposition of Clifford Gates*):\n",
+    "\n",
+    "- [`CliffordUnitary.to_transvections`](../paulimer.pyi) — a greedy reduction that always returns a\n",
+    "  **linear** number of factors ($O(n)$),\n",
+    "- [`CliffordUnitary.to_transvections_minimal`](../paulimer.pyi) — the **strict minimum** number of\n",
+    "  factors.\n",
+    "\n",
+    "Both reproduce the Clifford's **symplectic (conjugation) action** only; the Pauli-image signs and\n",
+    "the global phase are *not* preserved (the sign of a transvection does not change its symplectic\n",
+    "action)."
+   ]
+  },
+  {
+   "cell_type": "code",
+   "execution_count": 1,
+   "id": "9e2b876e",
+   "metadata": {
+    "execution": {
+     "iopub.execute_input": "2026-07-05T18:09:15.236725Z",
+     "iopub.status.busy": "2026-07-05T18:09:15.236584Z",
+     "iopub.status.idle": "2026-07-05T18:09:15.240675Z",
+     "shell.execute_reply": "2026-07-05T18:09:15.239473Z"
+    }
+   },
+   "outputs": [],
+   "source": [
+    "import paulimer\n",
+    "from paulimer import CliffordUnitary, SparsePauli, DensePauli"
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "id": "b42aee0d",
+   "metadata": {},
+   "source": [
+    "## A single transvection\n",
+    "\n",
+    "A `π/4` Pauli exponent *is* a Clifford transvection, so the simplest Cliffords decompose into a\n",
+    "single factor. The phase gate $S = \\exp(-i\\tfrac{\\pi}{4} Z)$ and the Hadamard are both single\n",
+    "transvections (recall the returned sign is irrelevant to the symplectic action):"
+   ]
+  },
+  {
+   "cell_type": "code",
+   "execution_count": 2,
+   "id": "9bbcb823",
+   "metadata": {
+    "execution": {
+     "iopub.execute_input": "2026-07-05T18:09:15.242261Z",
+     "iopub.status.busy": "2026-07-05T18:09:15.242209Z",
+     "iopub.status.idle": "2026-07-05T18:09:15.244600Z",
+     "shell.execute_reply": "2026-07-05T18:09:15.243912Z"
+    }
+   },
+   "outputs": [
+    {
+     "name": "stdout",
+     "output_type": "stream",
+     "text": [
+      "S        -> [Z]\n",
+      "Hadamard -> [-𝑖Y]\n"
+     ]
+    }
+   ],
+   "source": [
+    "s_gate = CliffordUnitary.from_name(\"SqrtZ\", [0], qubit_count=1)\n",
+    "hadamard = CliffordUnitary.from_name(\"Hadamard\", [0], qubit_count=1)\n",
+    "\n",
+    "print(\"S        ->\", s_gate.to_transvections_minimal())\n",
+    "print(\"Hadamard ->\", hadamard.to_transvections_minimal())"
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "id": "c78b1510",
+   "metadata": {},
+   "source": [
+    "## Rebuilding a Clifford and checking the symplectic action\n",
+    "\n",
+    "Applying the returned transvections in order with\n",
+    "[`left_mul_pauli_exp`](../paulimer.pyi) reconstructs the original **symplectic matrix**. We compare\n",
+    "`symplectic_matrix` (not the full signed tableau, since signs and global phase are not tracked by\n",
+    "this decomposition).\n",
+    "\n",
+    "The minimal factor count is either $r$ or $r+1$, where the **residue rank**\n",
+    "\n",
+    "$$\n",
+    "r \\;=\\; 2n - \\dim \\operatorname{Fix}(F)\n",
+    "$$\n",
+    "\n",
+    "is the codimension of the space of Pauli operators fixed under conjugation. In `paulimer`,\n",
+    "$\\dim\\operatorname{Fix}(F)$ is the size of the Clifford's centralizer."
+   ]
+  },
+  {
+   "cell_type": "code",
+   "execution_count": 3,
+   "id": "ed66171a",
+   "metadata": {
+    "execution": {
+     "iopub.execute_input": "2026-07-05T18:09:15.246310Z",
+     "iopub.status.busy": "2026-07-05T18:09:15.246264Z",
+     "iopub.status.idle": "2026-07-05T18:09:15.248826Z",
+     "shell.execute_reply": "2026-07-05T18:09:15.248499Z"
+    }
+   },
+   "outputs": [
+    {
+     "name": "stdout",
+     "output_type": "stream",
+     "text": [
+      "residue rank r      = 1\n",
+      "number of factors   = 1\n",
+      "symplectic action ok: True\n"
+     ]
+    }
+   ],
+   "source": [
+    "def residue_rank(clifford):\n",
+    "    return 2 * clifford.qubit_count - len(clifford.centralizer())\n",
+    "\n",
+    "\n",
+    "def rebuild(factors, qubit_count):\n",
+    "    rebuilt = CliffordUnitary.identity(qubit_count)\n",
+    "    for pauli in factors:\n",
+    "        rebuilt.left_mul_pauli_exp(pauli)\n",
+    "    return rebuilt\n",
+    "\n",
+    "\n",
+    "factors = s_gate.to_transvections_minimal()\n",
+    "rebuilt = rebuild(factors, s_gate.qubit_count)\n",
+    "print(\"residue rank r      =\", residue_rank(s_gate))\n",
+    "print(\"number of factors   =\", len(factors))\n",
+    "print(\"symplectic action ok:\", rebuilt.symplectic_matrix == s_gate.symplectic_matrix)"
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "id": "eab0ecb9",
+   "metadata": {},
+   "source": [
+    "## Greedy versus minimal, and the $r+1$ case\n",
+    "\n",
+    "For many Cliffords the greedy and minimal decompositions agree, but not always. The CNOT gate has\n",
+    "residue rank $r = 2$ yet needs $r + 1 = 3$ transvections: its symplectic action is *hyperbolic*\n",
+    "($\\langle v, vF\\rangle = 0$ for all $v$), which forces one extra factor."
+   ]
+  },
+  {
+   "cell_type": "code",
+   "execution_count": 4,
+   "id": "00096f40",
+   "metadata": {
+    "execution": {
+     "iopub.execute_input": "2026-07-05T18:09:15.250074Z",
+     "iopub.status.busy": "2026-07-05T18:09:15.249935Z",
+     "iopub.status.idle": "2026-07-05T18:09:15.252141Z",
+     "shell.execute_reply": "2026-07-05T18:09:15.251741Z"
+    }
+   },
+   "outputs": [
+    {
+     "name": "stdout",
+     "output_type": "stream",
+     "text": [
+      "residue rank r = 2\n",
+      "greedy  : [Z, ZX, IX]  ( 3 factors )\n",
+      "minimal : [Z, ZX, IX]  ( 3 factors )\n",
+      "symplectic action ok: True\n"
+     ]
+    }
+   ],
+   "source": [
+    "cnot = CliffordUnitary.from_name(\"ControlledX\", [0, 1], qubit_count=2)\n",
+    "\n",
+    "greedy = cnot.to_transvections()\n",
+    "minimal = cnot.to_transvections_minimal()\n",
+    "print(\"residue rank r =\", residue_rank(cnot))\n",
+    "print(\"greedy  :\", greedy, \" (\", len(greedy), \"factors )\")\n",
+    "print(\"minimal :\", minimal, \" (\", len(minimal), \"factors )\")\n",
+    "print(\"symplectic action ok:\", rebuild(minimal, 2).symplectic_matrix == cnot.symplectic_matrix)"
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "id": "6d7dfd52",
+   "metadata": {},
+   "source": [
+    "## A subtle case: non-hyperbolic maps that still need $r+1$\n",
+    "\n",
+    "The 2021 paper claims that *every* non-hyperbolic Clifford decomposes into exactly $r$ transvections.\n",
+    "That is **not correct over $\\mathbb{F}_2$**: some non-hyperbolic maps still require $r + 1$. The\n",
+    "smallest example already occurs on two qubits — the symplectic action built below (a product of the\n",
+    "transvections $X_0, X_1, X_0X_1, Z_0$) has residue rank $r = 3$, is non-hyperbolic, yet needs $4$\n",
+    "transvections. `to_transvections_minimal` returns the correct minimum. See\n",
+    "[`docs/transvection-minimality-correction.md`](../../../docs/transvection-minimality-correction.md)\n",
+    "for the full analysis and a machine-checked proof."
+   ]
+  },
+  {
+   "cell_type": "code",
+   "execution_count": 5,
+   "id": "3a68cc03",
+   "metadata": {
+    "execution": {
+     "iopub.execute_input": "2026-07-05T18:09:15.253271Z",
+     "iopub.status.busy": "2026-07-05T18:09:15.253223Z",
+     "iopub.status.idle": "2026-07-05T18:09:15.256499Z",
+     "shell.execute_reply": "2026-07-05T18:09:15.255166Z"
+    }
+   },
+   "outputs": [
+    {
+     "name": "stdout",
+     "output_type": "stream",
+     "text": [
+      "residue rank r      = 3\n",
+      "minimal factors     = [IX, XX, -𝑖Y, X] ( 4 factors )\n",
+      "needs r + 1          : True\n",
+      "symplectic action ok: True\n"
+     ]
+    }
+   ],
+   "source": [
+    "example = CliffordUnitary.identity(2)\n",
+    "for pauli in [\"X0\", \"X1\", \"X0 X1\", \"Z0\"]:\n",
+    "    example.left_mul_pauli_exp(SparsePauli(pauli))\n",
+    "\n",
+    "minimal = example.to_transvections_minimal()\n",
+    "r = residue_rank(example)\n",
+    "print(\"residue rank r      =\", r)\n",
+    "print(\"minimal factors     =\", minimal, \"(\", len(minimal), \"factors )\")\n",
+    "print(\"needs r + 1          :\", len(minimal) == r + 1)\n",
+    "print(\"symplectic action ok:\", rebuild(minimal, 2).symplectic_matrix == example.symplectic_matrix)"
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "id": "992a2da1",
+   "metadata": {},
+   "source": [
+    "## Summary\n",
+    "\n",
+    "- Clifford transvections are `π/4` Pauli exponents; `to_transvections` /\n",
+    "  `to_transvections_minimal` decompose any Clifford into them, reproducing its symplectic action\n",
+    "  with $O(n)$ factors.\n",
+    "- The minimal count is $r$ or $r + 1$, where $r = 2n - \\dim\\operatorname{Fix}(F)$.\n",
+    "- Only the symplectic action is reproduced — Pauli-image signs and the global phase are not.\n",
+    "\n",
+    "### References\n",
+    "\n",
+    "- T. Pllaha, K. Volanto, O. Tirkkonen, *Decomposition of Clifford Gates*, GLOBECOM 2021,\n",
+    "  [arXiv:2102.11380](https://arxiv.org/abs/2102.11380).\n",
+    "- [`docs/transvection-minimality-correction.md`](../../../docs/transvection-minimality-correction.md)\n",
+    "  — a correction to the paper's minimality claim, with a verified counterexample."
+   ]
+  }
+ ],
+ "metadata": {
+  "kernelspec": {
+   "display_name": "paulimer",
+   "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": "3.12.13"
+  }
+ },
+ "nbformat": 4,
+ "nbformat_minor": 5
+}
diff --git a/paulimer/bindings/python/paulimer.pyi b/paulimer/bindings/python/paulimer.pyi
@@ -543,6 +543,18 @@ class CliffordUnitary:
         """
         ...
 
+    def to_transvections_minimal(self) -> list[SparsePauli]:
+        """Decompose into a *minimal* ordered product of Clifford transvections (pi/4 Pauli exponents).
+
+        Returns Pauli operators ``[P_1, ..., P_k]`` such that applying ``exp(i pi/4 P_1)``, then
+        ``exp(i pi/4 P_2)``, ..., then ``exp(i pi/4 P_k)`` reproduces this Clifford's symplectic
+        (conjugation) action, with ``k`` the minimal transvection count (``r`` or ``r + 1``, where
+        ``r`` is the rank of the residue matrix). Pauli-image signs and the global phase are not
+        reproduced; :meth:`to_transvections` is the linear-time greedy variant, which may use more
+        factors.
+        """
+        ...
+
     def centralizer(self) -> list[SparsePauli]:
         """Generators of the centralizer: Paulis fixed up to sign under conjugation."""
         ...