Kerdock Codes Determine Unitary 2-Designs

Abstract

The non-linear binary Kerdock codes are known to be Gray images of certain extended cyclic codes of length N = 2 m over Z4. We show that exponentiating these Z4-valued codewords by ı √ −1 produces stabilizer states, that are quantum states obtained using only Clifford unitaries. These states are also the common eigenvectors of commuting Hermitian matrices forming maximal commutative subgroups (MCS) of the Pauli group. We use this quantum description to simplify the derivation of the classical weight distribution of Kerdock codes. Next, we organize the stabilizer states to form N + 1 mutually unbiased bases and prove that automorphisms of the Kerdock code permute their corresponding MCS, thereby forming a subgroup of the Clifford group. When represented as symplectic matrices, this subgroup is isomorphic to the projective special linear group PSL(2, N). We show that this automorphism group acts transitively on the Pauli matrices, which implies that the ensemble is Pauli mixing and hence forms a unitary 2-design. The Kerdock design described here was originally discovered by Cleve et al. (2016), but the connection to classical codes is new which simplifies its description and translation to circuits significantly. Sampling from the design is straightforward, the translation to circuits uses only Clifford gates, and the process does not require ancillary qubits. Finally, we also develop algorithms for optimizing the synthesis of unitary 2-designs on encoded qubits, i.e., to construct logical unitary 2-designs. Software implementations are available at https://github.com/nrenga/symplectic-arxiv18a, which we use to provide empirical gate complexities for up to 16 qubits.

Logical Clifford Synthesis for Stabilizer Codes

“…Recently, we have used the LCS algorithm to translate the unitary 2-design we constructed from classical Kerdock codes into a logical unitary 2-design [28], and in general, any design consisting of only Clifford elements can be transformed into a logical design using our algorithm. An implementation of the design is available on github.…”