Homomorphic Logical Measurements

Shor and Steane ancilla are two well-known methods for fault-tolerant logical measurements,

which are successful on small codes and their concatenations. On large quantum low-density-parity-

check (LDPC) codes, however, Shor and Steane measurements have impractical time and space over-

head respectively. In this work, we widen the choice of ancilla codes by unifying Shor and Steane

measurements into a single framework, called homormophic measurements. As an example, we utilize

the theory of covering spaces to construct homomorphic measurement protocols for arbitrary X- or

Z-type logical Pauli measurements on surface codes in general, including the toric code and hyper-

bolic surface codes. These protocols do not require repetitive measurements or complicated ancilla

state preparation procedures such as distillation, which overcomes the difficulties of both Shor and

Steane methods. In contrast to other existing methods such as lattice surgery, conventional surface

code decoders can be directly applied to our constructions. The conceptual idea of our framework can

be extended to other quantum LDPC codes, which presents a new direction of fault-tolerant logical

operation design.