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Two anticommuting toric logical Pauli pairs

Statement

The representatives ${\overset{ˉ}{Z}}_{1}, {\overset{ˉ}{Z}}_{2}, {\overset{ˉ}{X}}_{1}, {\overset{ˉ}{X}}_{2}$ of Noncontractible toric logical loops obey ${\overset{ˉ}{Z}}_{i} {\overset{ˉ}{X}}_{j} = (- 1)^{δ_{i j}} {\overset{ˉ}{X}}_{j} {\overset{ˉ}{Z}}_{i}$ and commute within each Z pair and each X pair. Their four classes form a basis of the logical Pauli quotient of the two-qubit toric code.

Facts & Assumptions

Given: The four specified normalizer representatives and the two-qubit toric stabilizer code.

[F1]

Noncontractible toric logical loops: ${\overset{ˉ}{Z}}_{1}$ uses the horizontal primal row $h_{i, 0}$ , ${\overset{ˉ}{Z}}_{2}$ the vertical primal column $v_{0, j}$ , ${\overset{ˉ}{X}}_{1}$ uses $h_{0, j}$ , and ${\overset{ˉ}{X}}_{2}$ uses $v_{i, 0}$ .

[F2]

Pauli commutator is symplectic pairing: the commutator sign of two Pauli strings is $(- 1)^{ω}$ , where $ω$ is their binary symplectic pairing; for pure X/Z supports it is the overlap parity.

[F3]

Stabilizer syndromes and logical Pauli operators: logical Paulis are classes in $N (S) / (Z_{n} S)$ , with scalar phases and stabilizers identified.

[F4]

The periodic toric code encodes two qubits: the code encodes two qubits.

[F5]

Dimension of the logical Pauli quotient: an $n$ -qubit stabilizer code encoding $k$ qubits has a binary logical Pauli quotient of dimension $2 k$ .

[F6]

A single primal-dual crossing anticommutes: primal Z and dual X strings with exactly one shared physical edge anticommute.

Proof

1.1givenF1F2F6algebra

By [F1], ${\overset{ˉ}{Z}}_{1}$ and ${\overset{ˉ}{X}}_{1}$ share only $h_{0, 0}$ , while ${\overset{ˉ}{Z}}_{2}$ and ${\overset{ˉ}{X}}_{2}$ share only $v_{0, 0}$ . [F6] gives anticommutation for these equal-index pairs. The cross pairs $({\overset{ˉ}{Z}}_{1}, {\overset{ˉ}{X}}_{2})$ and $({\overset{ˉ}{Z}}_{2}, {\overset{ˉ}{X}}_{1})$ have disjoint physical-edge supports, so [F2] gives commutation for unequal indices. Two Z strings commute with one another and two X strings commute with one another because their factors have the same Pauli type.

2.1F3step 1.1algebra

Suppose a product ${\overset{ˉ}{Z}}_{1}^{a} {\overset{ˉ}{Z}}_{2}^{b} {\overset{ˉ}{X}}_{1}^{c} {\overset{ˉ}{X}}_{2}^{d}$ , with $a, b, c, d \in F_{2}$ , represented the identity class in [F3]. It would then commute with every normalizer representative. Commuting with ${\overset{ˉ}{X}}_{1}, {\overset{ˉ}{X}}_{2}, {\overset{ˉ}{Z}}_{1}, {\overset{ˉ}{Z}}_{2}$ in turn and using step 1.1 forces $a = b = c = d = 0$ . Hence the four classes are independent.

3.1F4F5step 2.1algebra∎

By [F4] the code has $k = 2$ , and [F5] gives logical quotient dimension $2 k = 4$ . The four independent classes from step 2.1 therefore form a basis of that quotient.

Depends on

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Sources

Alexei Kitaev, Fault-tolerant quantum computation by anyons (standard reference, not scraped)
Eric Dennis, Alexei Kitaev, Andrew Landahl, and John Preskill, Topological quantum memory (standard reference, not scraped)