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Clifford to binary symplectic action is surjective

Statement

For every $n \geq 1$ , write $Sp (V_{n}, ω)$ for the group of invertible $F_{2}$ -linear maps $T : V_{n} \to V_{n}$ satisfying $ω (T a, T b) = ω (a, b)$ for all $a, b$ . For every $T \in Sp (V_{n}, ω)$ there is a Clifford circuit $U$ made from Hadamard gates $H$ , phase gates $S, S^{*}$ , CNOT, and wire swaps whose induced action on binary Pauli labels is $T$ . Since the phase gate satisfies $S^{*} = S^{3}$ , this group is also generated by $H$ , $S$ , CNOT, and swaps. Hence the conjugation homomorphism from the Clifford normalizer to $Sp (V_{n}, ω)$ is onto. The proof uses finite transvection reduction and exact mod- $4$ phase updates at every gate.

Facts & Assumptions

Given: the finite-dimensional binary symplectic space $V_{n}$ , the Clifford normalizer, and the stated transvection generation and lifting lemmas.

[F1]

Symplectic transvections generate the binary symplectic group: every symplectic map is a finite product of maps $τ_{v} (w) = w + ω (w, v) v$ .

[F2]

Clifford circuits lift binary symplectic transvections: each such transvection has a Clifford circuit lift built from $H, S, S^{*}$ , CNOT, and swaps, with its mod- $4$ phases computable gate by gate.

[F3]

Clifford normalizer: the Clifford normalizer is closed under products.

[F4]

Binary Pauli quotient isomorphism: binary labels identify the scalar quotient of the Pauli group.

[F5]

Clifford induces symplectic map: each Clifford has its induced symplectic map, and $S_{U V} = S_{U} \circ S_{V}$ .

Proof

1.1givenF1choose

Let $T \in Sp (V_{n}, ω)$ . By [F1], choose a finite factorization $T = τ_{v_{m}} \dots τ_{v_{1}}$ , omitting zero vectors if desired. This factorization is obtained by the finite induction in [F1], so no choice axiom is used.

2.1givenF2F3F4F5step 1.1constructalgebra∎

For each factor, take a Clifford circuit $U_{j}$ from [F2] whose binary action is $τ_{v_{j}}$ . The finite product $U = U_{m} \dots U_{1}$ lies in $C_{n}$ by [F3]. By [F5], its induced symplectic action is $S_{U} = S_{U_{m}} \dots S_{U_{1}} = τ_{v_{m}} \dots τ_{v_{1}} = T$ . The quotient identification [F4] ensures these are exactly the binary labels of conjugated Paulis. Every constituent gate updates its phase coordinate modulo $4$ by the formulas in [F2], so the product is an exact Clifford lift as well as a binary one.

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Sources

M. Hein et al., Entanglement in Graph States and Its Applications (standard reference, not scraped)
Ronald de Wolf, Quantum Computing: Lecture Notes (standard reference, not scraped)
John Watrous, Understanding Quantum Information and Computation, Lesson 14 (standard reference, not scraped)