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Clifford induces symplectic map

Statement

For each $U \in C_{n}$ , conjugation induces an invertible $F_{2}$ -linear map $S_{U} : V_{n} \to V_{n}$ satisfying $ω (S_{U} a, S_{U} b) = ω (a, b)$ for all $a, b \in V_{n}$ . Thus $S_{U}$ is symplectic. The assignment respects multiplication: $S_{U V} = S_{U} \circ S_{V}$ for all $U, V \in C_{n}$ .

Facts & Assumptions

Given: $U$ is a unitary normalizer of the phase-aware Pauli group and $V_{n} = P_{n} / Z_{n}$ is identified with its binary labels.

[F1]

Binary Pauli quotient isomorphism: the label quotient is a group isomorphism, with scalar center as kernel.

[F2]

Pauli commutator is symplectic pairing: the commutator sign of two Pauli operators is $(- 1)^{ω (a, b)}$ .

[F3]

Clifford normalizer: $U$ and $U^{*}$ normalize $P_{n}$ .

Proof

1.1givenF1F3algebra

Conjugation sends each Pauli to a Pauli by [F3]. It fixes every scalar, so representatives in the same scalar coset have images in the same coset. Therefore it defines a map $S_{U}$ on $V_{n}$ under [F1].

2.1givenF1F3step 1.1algebra

For Pauli operators $P, Q$ , $U (P Q) U^{*} = (U P U^{*}) (U Q U^{*})$ , so the quotient map [F1] shows $S_{U} (a + b) = S_{U} (a) + S_{U} (b)$ . An additive map between $F_{2}$ vector spaces is linear because the only scalars are $0$ and $1$ . Conjugation by $U^{*}$ is its inverse, so $S_{U}$ is invertible. Applying the same identity successively gives $S_{U V} = S_{U} \circ S_{V}$ for Clifford unitaries $U, V$ .

3.1givenF2step 1.1algebra∎

The commutator identity from [F2] remains the same after conjugating both sides by $U$ , because conjugation preserves products and scalar signs. Applying [F2] to the image Paulis yields $(- 1)^{ω (S_{U} a, S_{U} b)} = (- 1)^{ω (a, b)}$ . The map $r \mapsto (- 1)^{r}$ is injective on $F_{2}$ , hence $ω (S_{U} a, S_{U} b) = ω (a, b)$ .

Depends on

Used by

Clifford to binary symplectic action is surjective Theorem

Dependency tree · two levels

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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 Preskill, Quantum Error Correction, Physics 229 Lecture Notes, Chapter 7 (standard reference, not scraped)