Error Correction 5

Abstract: To quantify the accuracy of logical gates in approximate quantum error correction, we introduce the {\em composable logical gate error}. This quantity accounts for both deviation from the target gate and leakage out of the code space. It is subadditive under gate composition, enabling simple circuit analysis, and can be bounded using matrix elements of physical unitaries between (approximate) logical basis states. As a case study, we study the composable logical gate error of linear optics implementations of Paulis and Cliffords in approximate Gottesman-Kitaev-Preskill (GKP) codes. We find that the logical gate error for implementations of Pauli gates depends linearly on the squeezing parameter. This means that their accuracy increases monotonically with the amount of squeezing. In contrast, implementations of some Clifford gates retain a constant logical gate error even in the limit of infinite squeezing. This highlights that results derived for ideal GKP codes do not always translate to physically realistic approximate codes. We propose a way of sidestepping this no-go result in hybrid qubit-oscillator systems with Gaussian, multi-qubit, and qubit-controlled Gaussian unitaries. We propose implementations of logical gates using two oscillators and three qubits, whose logical gate error is bounded by a linear function of the squeezing parameter and scales polynomially with the number of encoded qubits.

Abstract: 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.

Abstract: Efficient and high-performance quantum error correction is essential for achieving fault-tolerant quantum computing. Low-depth random circuits offer a promising approach to identifying effective and practical encoding strategies. In this work, we rigorously prove through information-theoretic analysis that one-dimensional logarithmic-depth random Clifford encoding circuits can achieve high quantum error correction performance. We demonstrate that these random codes typically exhibit good approximate quantum error correction capability by proving that their encoding rate achieves the hashing bound for Pauli noise and the channel capacity for erasure errors. We show that the error correction inaccuracy decays once a threshold of logarithmic depth is exceeded, resulting in negligible recovery errors. This threshold is shown to be lower than that of the simple separate block encoding, and the decay rate is higher. We further establish that these codes are optimal by proving that logarithmic depth is necessary to maintain a constant encoding rate and high error correction performance. To prove our results, we propose new decoupling theorems for one-dimensional low-depth circuits. These results also imply strong decoupling and rapid thermalization properties in low-depth random circuits and have potential applications in quantum information science and physics.