Arda Aydin

We study relationships between permutation-invariant, bosonic Fock-state, and spin codes, which arise in different physical systems, but exhibit close mathematical affinity. We show that, starting with classical ell-1 codes, it is possible to construct qudit permutationally invariant (PI) codes of arbitrary dimension, spin codes, and Fock state codes, called collectively SU(q) codes. To maintain control of the code parameters in this transition, we rely on a classic result from convex geometry known as Tverberg's theorem. Constructing ell-1 codes based on combinatorial patterns called Sidon sets and utilizing their Tverberg partitions, we obtain new families of SU(q) codes with distance that scales almost linearly with the code length N. This improves upon the existing designs for all the three code families and yields a conceptually new framework for constructing spin codes. We further present explicit constructions of SU(2) codes with shorter length or lower total spin/excitation than the known codes with similar parameters, new bosonic codes with exotic Gaussian gates, as well as examples of some short codes with distance larger than the known constructions.

Quantum low-density parity-check (qLDPC) codes are a class of quantum error-correction (QEC) codes with low-weight parity-checks that each require only a few two-qubit gates to implement. In recent years, qLDPC codes have gained popularity, as concrete code constructions have been found that outperform the surface code, and correspondingly performant practical decoders have been built. An outstanding challenge, however, remains that their Tanner graphs are not 2D-local thereby necessitating entangling gates to operate on distant qubits on a 2D device. Trapped-ion and neutral-atom qubits possess the ability to move qubits around when necessary – i.e. “flying qubits” – obviating the need for long-range gates. Here we report on an explicit layout that leverages flying qubits, that is very low-overhead for many families of cyclic codes (including the most promising qLDPC instances found to-date). Crucially, our layout eschews more complicated qubit permutations, and instead favours the cyclic shift a simple re-ordering of qubits along a loop that can be realized in depth 1 even on current generation devices. This contrasts significantly with layouts on fixed qubits that depend on a large number of long-range (and more error-prone) hardware couplers for long-distance gates. We also report on two competitive new sets of cyclic qLDPC codes that we constructed. The first is a set of Bivariate-Bicycle (BB) codes with lower weight parity-checks and higher minimum distance while maintaining the same length and encoding rate as comparable BB codes in. Second, we also constructed new Hypergraph Product (HGP) codes, that significantly outperform previously state-of-the-art HGP instances that were optimized by machine-learning methods. Both sets of new codes are efficiently implementable with our cyclic layout with syndrome circuits of fixed depth, made up of alternating layers of parallel gates and only a very small number of cyclic shifts. Combining competitive new qLDPC codes alongside a simple layout implementable on existing hardware, our work suggest a concrete and practical path towards a fault-tolerant quantum computer.