Abstract
Stabilizer codes are a simple and successful class of quantum error-correcting codes. Yet this success comes in spite of some harsh limitations on the ability of these codes to fault-tolerantly compute. Here we introduce a new metric for these codes, the disjointness, which, roughly speaking, is the number of mostly non-overlapping representatives of any given non-trivial logical Pauli operator. We use the disjointness to prove that transversal gates on error-detecting stabilizer codes are necessarily in a finite level of the Clifford hierarchy. We also apply our techniques to topological code families to find similar bounds on the level of the hierarchy attainable by constant depth circuits, regardless of their geometric locality. For instance, we can show that symmetric 2D surface codes cannot have non-local constant depth circuits for non-Clifford gates.
Highlights
Quantum error-correcting codes form the foundation of scalable quantum computing [1,2,3]
Similar no-go theorems limiting logical operators to be in a finite level of the Clifford hierarchy were derived for transversal single-qubit gates and two-qubit diagonal gates on stabilizer codes [8], as well as for constant-depth, local circuits on stabilizer and subsystem topological codes [9,10]
We address several related questions regarding transversal and constant-depth logical operators on stabilizer codes using a new quantity called the disjointness of the code
Summary
Quantum error-correcting codes form the foundation of scalable quantum computing [1,2,3]. Similar no-go theorems limiting logical operators to be in a finite level of the Clifford hierarchy were derived for transversal single-qubit gates and two-qubit diagonal gates on stabilizer codes [8], as well as for constant-depth, local circuits on stabilizer and subsystem topological codes [9,10]. The latter result has an important implication—one cannot achieve a universal gate set with constant-depth local circuits on two-dimensional (2D) topological codes such as those in Refs. Asymmetry of logical operators appears in our bounds as a necessary condition for possessing constant-depth circuits for non-Clifford gates, such as those on 3D color and toric codes [16,17] and on asymmetric 2D Bacon-Shor codes [18]
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