Abstract

Fault-tolerant logical entangling gates are essential for scalable quantum computing, but are limited by the error rates and overheads of physical two-qubit gates and measurements. To address this limitation, we introduce phantom codes-quantum error-correcting codes that realize entangling gates between all logical qubits in a code block purely through relabelling of physical qubits during compilation, yielding perfect fidelity with no spatial or temporal overhead. We present a systematic study of such codes. First, we identify phantom codes using complementary numerical and analytical approaches. We exhaustively enumerate all 2.71×10^10 inequivalent CSS codes up to n=14 and identify additional instances up to n=21 via SAT-based methods. We then construct higher-distance phantom-code families using quantum Reed-Muller codes and the binarization of qudit codes. Across all identified codes, we characterize other supported fault-tolerant logical Clifford and non-Clifford operations. Second, through end-to-end noisy simulations with state preparation, full QEC cycles, and realistic physical error rates, we demonstrate scalable advantages of phantom codes over the surface code across multiple tasks. We observe a one-to-two order-of-magnitude reduction in logical infidelity at comparable qubit overhead for GHZ-state preparation and Trotterized many-body simulation tasks, given a modest preselection acceptance rate. Our work establishes phantom codes as a viable architectural route to fault-tolerant quantum computation with scalable benefits for workloads with dense local entangling structure, and introduces general tools for systematically exploring the broader landscape of quantum error-correcting codes.

Personal information

Shayan Majidy is a Banting Postdoctoral Fellow at Harvard University, working with Misha Lukin and Michael Gullans. He completed his PhD at the University of Waterloo as a Vanier Scholar, supervised by Raymond Laflamme and Nicole Yunger Halpern. During his PhD, he co-wrote the textbook Building Quantum Computers as lead author, published by Cambridge University Press. His PhD work was recognized with top honours, including the WB Pearson Medal, John Brodie Memorial Award, and the IQC Achievement Award.

Reference

https://arxiv.org/abs/2601.20927