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Shor and Steane ancillae 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 codes, however, Shor and Steane measurements have impractical time and space overhead, respectively. In this work, we widen the choice of ancilla codes by unifying Shor and Steane measurements into a single framework, called homomorphic measurements. For any Calderbank-Shor-Steane code with the appropriate ancilla code, one can avoid repetitive measurements or complicated ancilla state preparation procedures such as distillation, which overcomes the difficulties of both Shor and Steane methods. As an example, we utilize the theory of covering spaces to construct homomorphic measurement protocols for arbitrary X- or Z-type logical Pauli operators on surface codes in general, including the toric code and hyperbolic surface codes. Conventional surface code decoders, such as minimum-weight perfect matching, can be directly applied to our constructions.

We investigate topological order on fractal geometries embedded in n dimensions. We consider the n-dimensional lattice with holes at all length scales the corresponding fractal (Hausdorff) dimension of which is D_{H}=n−δ. In particular, we diagnose the existence of the topological order through the lens of quantum information and geometry, i.e., via its equivalence to a quantum error-correcting code with a macroscopic code distance or the presence of macroscopic systoles in systolic geometry. We first prove a no-go theorem that Z_{N} topological order cannot survive on any fractal embedded in two spatial dimensions and with D_{H}=2−δ. For fractal-lattice models embedded in three dimensions (3D) or higher spatial dimensions, Z_{N} topological order survives if the boundaries on the holes condense only loop or, more generally, k-dimensional membrane excitations (k≥2), thus predicting the existence of fractal topological quantum memories (at zero temperature) or topological codes that are embeddable in 3D. Moreover, for a class of models that contain only loop or membrane excitations and are hence self-correcting on an n-dimensional manifold, we prove that Z_{N} topological order survives on a large class of fractal geometries independent of the type of hole boundary and is hence extremely robust. We further construct fault-tolerant logical gates in the Z_{2} version of these fractal models, which we term fractal surface codes, using their connection to global and higher-form topological symmetries equivalent to sweeping the corresponding gapped domain walls. In particular, we discover a logical controlled-controlled-Z (ccz) gate corresponding to a global symmetry in a class of fractal codes embedded in 3D with Hausdorff dimension asymptotically approaching D_{H}=2+ϵ for arbitrarily small ϵ, which hence only requires a space overhead Ω(d^{2+ϵ}), where d is the code distance. This in turn leads to the surprising discovery of certain exotic gapped boundaries that only condense the combination of loop excitations and certain gapped domain walls. We further obtain logical C^{p}Z gates with p≤n−1 on fractal codes embedded in n dimensions. In particular, for the logical C^{n−1}Z in the nth level of the Clifford hierarchy, we can reduce the space overhead to Ω(d^{n−1+ϵ}). On the mathematical side, our findings in this paper also lead to the discovery of macroscopic relative systoles in a class of fractal geometries.