LDPC-cat codes for low-overhead quantum computing in 2D

Abstract

Quantum low-density parity-check (qLDPC) codes are a promising construction for drastically reducing the overhead of fault-tolerant quantum computing (FTQC) architectures. However, all of the known hardware implementations of these codes require advanced technologies, such as long-range qubit connectivity, high-weight stabilizers, or multi-layered chip layouts. An alternative approach to reduce the hardware overhead of fault-tolerance is to use bosonic cat qubits where bit-flip errors are exponentially suppressed by design. In this work, we combine both approaches and propose an architecture based on cat qubits concatenated in classical LDPC codes correcting for phase-flips. We find that employing such phase-flip LDPC codes provides two major advantages. First, the hardware implementation of the code can be realised using short-range qubit interactions in 2D and low-weight stabilizers, which makes it readily compatible with current superconducting circuit technologies. Second, we demonstrate how to implement a fault-tolerant universal set of logical gates with a second layer of cat qubits while maintaining the local connectivity. We conduct a numerical brute force optimisation of these classical codes to find the ones with the best encoding rate for algorithmically relevant code distances. We discover that some of the best codes benefit from a cellular automaton structure. This allows us to define families of codes with high encoding rates and distances. Finally, we numerically assess the performance of our codes under circuit-level noise. Assuming a physical phase-flip error probability $\epsilon \approx 0.1\%$, our $[165+8\ell ,34+2\ell ,22]$ code family allows to encode $100$ logical qubits with a total logical error probability (including both logical phase-flip and bit-flip) per cycle and per logical qubit ${\epsilon }_L\le 10^{-8}$ on a $758$ cat qubit chip.

23 pages, 11 figures

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• Preserving phase coherence and linearity in cat qubits with exponential bit-flip suppression
• Fault-tolerant quantum computing with the parity code and noise-biased qubits
• Stabilization of cat-state manifolds using nonlinear reservoir engineering
• A cat qubit stabilization scheme using a voltage biased Josephson junction
• Unfolded distillation: very low-cost magic state preparation for biased-noise qubits
• Multimode rotationally symmetric bosonic codes from group-theoretic construction
• Exact and Efficient Stabilizer Simulation of Thermal-Relaxation Noise for Quantum Error Correction
• Quantum Error Correction with Superpositions of Squeezed Fock States
• Fault Tolerant Quantum Simulation via Symplectic Transvections
• Hierarchical Quantum Error Correction with Hypergraph Product Code and Rotated Surface Code
• Non-perturbative switching rates in bistable open quantum systems: from driven Kerr oscillators to dissipative cat qubits