Fault-tolerant logical gates in quantum error-correcting codes∗

Recently, Bravyi and König have shown that there is a trade-off between fault-tolerantly implementable logical gates and geometric locality of stabilizer codes. They consider locality-preserving operations which are implemented by a constant-depth geometrically-local circuit and are thus fault-tolerant by construction. In particular, they shown that, for local stabilizer codes in D spatial dimensions, locality preserving gates are restricted to a set of unitary gates known as the D-th level of the Clifford hierarchy. In this paper, we elaborate this idea and provide several extensions and applications of their characterization in various directions. First, we present a new no-go theorem for self-correcting quantum memory [17]. Namely, we prove that a three-dimensional stabilizer Hamiltonian with a locality-preserving implementation of a non-Clifford gate cannot have a macroscopic energy barrier. This result implies that in Haah’s Cubic code [18] and Michnicki’s [19] welded code non-Clifford gates do not admit such an implementation. Second, we prove that the code distance of a D-dimensional local stabilizer code with non-trivial locality-preserving m-th level Clifford logical gate is upper bounded by O(LD+1−m). For codes with non-Clifford gates (m > 2), this improves the previous best bound by Bravyi and Terhal. Bombin and Martin-Delgado’s topological color codes saturate the bound for m = D. Third we prove that a qubit loss threshold of codes with non-trivial transversal m-th level Clifford logical gate is upper bounded by 1/m. As such, no family of fault-tolerant codes with transversal gates in increasing level of the Clifford hierarchy may exist. This result applies to arbitrary stabilizer and subsystem codes, and is not restricted to geometrically-local codes. Finally, we extend the result of Bravyi and König to subsystem codes. A technical difficulty is that, unlike stabilizer codes, the so-called union lemma does not apply to subsystem codes. This problem is avoided by assuming the presence of error threshold in a subsystem code, and the same conclusion as Bravyi-König is recovered. Quantum error-correcting codes constitute an indispensable ingredient in the roadmap to faulttolerant quantum computation as they offer the framework of enabling imperfect quantum gates and resources to implement arbitrarily reliable quantum computation [2, 3]. An essential feature for such codes is to admit a fault-tolerant implementation of a universal gate-set where physical errors should propagate in a benign and controlled manner. A paragon for fault-tolerant implementation of logical gates is provided by transversal unitary operations, i.e. single qubit rotations acting independently on each physical qubit. However, Eastin and Knill have proved that the set of transversal gates constitutes a finite group, and hence is not universal for quantum computation [4], suggesting a tension between computational power and fault-tolerance. Recently, Bravyi and König have further sharpened this tension for topological stabilizer codes supported on a lattice with geometrically local generators [5]. By extending their consideration to logical gates implemented by constant depth local quantum circuits as feasible proxy, they have shown that, in D spatial dimensions, fault-tolerantly implementable logical gates ∗This extended abstract summarizes [1].