Given a quantum error correcting code, an important task is to find encoded operations that can be implemented efficiently and fault-tolerantly. In this Letter we focus on topological stabilizer codes and encoded unitary gates that can be implemented by a constant-depth quantum circuit. Such gates have a certain degree of protection since propagation of errors in a constant-depth circuit is limited by a constant size light cone. For the 2D geometry we show that constant-depth circuits can only implement a finite group of encoded gates known as the Clifford group. This implies that topological protection must be "turned off" for at least some steps in the computation in order to achieve universality. For the 3D geometry we show that an encoded gate U is implementable by a constant-depth circuit only if U P U † is in the Clifford group for any Pauli operator P . This class of gates includes some non-Clifford gates such as the π/8 rotation. Our classification applies to any stabilizer code with geometrically local stabilizers and sufficiently large code distance. Quantum error correcting codes play a central role in all proposed schemes for fault-tolerant quantum computation. By repeatedly measuring error syndromes and applying corresponding correction operations, encoded states can be stored reliably for extended periods of time. Furthermore, some codes permit a fault-tolerant implementation of a computationally universal set of operations on encoded states [1] . Topological codes such as the surface code family [2-5] are arguably closest to what can currently be achieved in experiments [6] . The unifying feature of all topological codes is the geometric locality of their check operators. The physical qubits of a D-dimensional topological code can be laid out on a regular lattice embedded in R D such that the support of any check operator has diameter O(1). The locality ensures that the syndrome readout requires only short-range quantum gates and that each qubit participates only in a few gates. In addition, any topological code has a macroscopic distance: a non-trivial operation on encoded states cannot be implemented by acting on fewer than d qubits, where d can be made arbitrarily large by increasing the lattice size. The subclass of topological stabilizer codes (TSCs) has an additional convenient feature: the parity check operators are tensor products of single-qubit Pauli operators. This simple structure allows one to understand properties of stabilizer codes in much more depth [7] . The subclass of TSCs includes the toric and the surface codes [2, 3] , the color codes [8] , as well as the surface codes with twists [9] or punctured holes [5] . Examples of topological codes which are not TSC are the quantum double models [2] and the Turaev-Viro codes [10] . In this Letter we show that the simple structure of TSCs comes at a price: the set of gates implementable in a fault-tolerant manner is rather restricted for any such code. To formulate this more precisely, let us say that an encoded gate is topologically protected if it can be realized by applying a constant-depth quantum circuit on the physical qubits. Here we only consider circuits with geometrically local gates. This definition is motivated by the fact that constant-depth circuits are inherently faulttolerant: a fault in any single gate can affect at most O(1) qubits and a pre-existing error can spread to at most O(1) qubits. Topologically protected gates can therefore be executed using noisy hardware without introducing too many errors [11] . To state our main result let us fix the number of logical qubits k. For any j ≥ 1 define a set of encoded gates P j , j ≥ 1, such that P 1 is the group of k-qubit Pauli operators, and P j is a set of all k-qubit unitary operators U such that U P 1 U † ⊆ P j−1 , where j ≥ 2. In particular, P 2 is the so-called Clifford group, that is, the group generated by the Hadamard gate H = (X + Z)/ √ 2, the CNOT gate, and the π/4 rotation K = exp (iπZ/4). The set P 3 includes some non-Clifford gates such as the π/8 rotation √ K. Note that P j is not a group unless j = 1, 2. Ignoring overall phase factors, P j is a finite set [12] . The sets P j were originally introduced by Gottesman and Chuang [13] who proposed a fault-tolerant implementation of any gate in P j through a recursive application of the gate teleportation method [14] . Surprisingly, the sets P j also naturally arise in the context of topological codes. Our main result is the following. Theorem 1. Suppose a unitary operator U implementable by a constant-depth quantum circuit preserves the codespace C of a topological stabilizer code on a Ddimensional lattice, D ≥ 2. Then the restriction of U onto C implements an encoded gate from the set P D . Note that the restriction of U onto the codespace C can be viewed as a k-qubit operator only with respect to some basis of C. Any stabilizer code has a basis such that all encoded Pauli operators are products of physical Pauli operators [7] . We shall always implicitly assume that such a basis is chosen. Theorem 1 also holds for any depth-h quantum circuit U with gates of range r such that ξ, hr d 1/D , where d is the code distance and ξ is the maximum range of the parity check operators.
More recently, REF placed restrictions on the more general class of topologically protected logical gates in topological stabilizer codes, which include transversal gates as an optimal subset.
Classification of topologically protected gates for local stabilizer codes
Phys. Rev. Lett. 110, 170503 (2013)
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