Transversal gates play an important role in the theory of fault-tolerant quantum computation due to their simplicity and robustness to noise. By definition, transversal operators do not couple physical subsystems within the same code block. Consequently, such operators do not spread errors within code blocks and are, therefore, fault tolerant. Nonetheless, other methods of ensuring fault tolerance are required, as it is invariably the case that some encoded gates cannot be implemented transversally. This observation has led to a long-standing conjecture that transversal encoded gate sets cannot be universal. Here we show that the ability of a quantum code to detect an arbitrary error on any single physical subsystem is incompatible with the existence of a universal, transversal encoded gate set for the code.
Transversal gates play an important role in the theory of fault-tolerant quantum computation due to their simplicity and robustness to noise. By definition, transversal operators do not couple physical subsystems within the same code block. Consequently, such operators do not spread errors within code blocks and are, therefore, fault tolerant. Nonetheless, other methods of ensuring fault tolerance are required, as it is invariably the case that some encoded gates cannot be implemented transversally. This observation has led to a long-standing conjecture that transversal encoded gate sets cannot be universal. Here we show that the ability of a quantum code to detect an arbitrary error on any single physical subsystem is incompatible with the existence of a universal, transversal encoded gate set for the code.
Logic gates can be performed on data encoded in quantum code blocks such that errors introduced by faulty gates can be corrected. The important class of transversal gates acts bitwise between corresponding qubits of code blocks and thus limits error propagation. If any quantum gate could be implemented using transversal gates, the set would be universal. We study the structure ofGF(4)-additive quantum codes and prove that no universal set of transversal logic gates exists for these codes. This result is in stark contrast with the classical case, where universal transversal gate sets exist, and strongly supports the idea that additional quantum techniques, based, for example, on quantum teleportation or magic state distillation, are necessary to achieve universal fault-tolerant quantum computation on additive codes.
A long-standing open problem in fault-tolerant quantum computation has been to find a universal set of transversal gates. As shown by Zeng et al. (e-print arXiv:0706.1382), such a set does not exist for binary stabilizer codes. Here we generalize our work to show that for subsystem stabilizer codes in d-dimensional Hilbert space, such a universal set of transversal gates cannot exist for even one encoded qudit, for any dimension d, prime or nonprime. This result strongly supports the idea that other primitives, such as quantum teleportation, are necessary for universal fault-tolerant quantum computation, and may be an important factor for fault-tolerance noise thresholds.
In fault-tolerant quantum computation [1], quantum information is protected from noise by encoding it in somewhat non-local degrees of freedom, thus distributing it among many smaller subsystems, typically qubits. This makes sense under the physically relevant assumption that interactions with the environment have a local nature. The implementation of gates, consequently, must be also as local as possible to preserve the structure of noise. This is naturally achieved with transversal gates, i.e. unitary operators that transform encoded states by acting separately on suitable subsystems, in the simplest case independently on each qubit. Unfortunately, no code admits a universal transversal set of gates [2]. Topological quantum error correcting codes [3] introduce a richer notion of locality by considering the spatial location of the physical qubits, which are assumed to be arranged on a lattice. They come in families parametrized with a lattice size, for a fixed spatial dimension. Their defining features are (i) that the measurements needed to recover information about errors only involve a few neighbouring qubits and (ii) that no encoded information can be recovered without access to a number of physical qubits comparable to the system size. Rather than sticking to the above definition of transversality, for topological codes it is natural to consider instead quantum circuits of fixed depth with geometrically local gates [4, 5]. Remarkably, it has been recently shown that spatial dimension imposes constraints on the transversal gates that are allowed [5]. In particular, this is true in the case of topological stabilizer codes, where it has been found that transversal gates on D-dimensional codes have to belong to certain sets PD. A first main result of this work [6] is that codes that saturate these bounds can be constructed for every D, in the sense that they admit at least a transversal gate in PD (not belonging to PD−1). In particular, the result applies to color codes [7, 8, 9, 10], where the single-qubit gate RD := ( 1 0 0 e2πi/2 D ) is transversal (in the more strict non-topological sense).
Building fault-tolerant quantum computers is a current priority for the quantum computing industry and academic counterparts. We present a quantum error correction code encoding a single logical qubit into n=L2 physical qubits with distance d=L as a subsystem code, and n−O(n) logical qubits with distance d=3 as a standard stabiliser code. The logical single-qubit gates in the subsystem code are transversal. Under independent X,Z noise we find a threshold of 1.4%±0.1% for odd distance and 1.8%±0.2% for even distance for the subsystem variant. The code detects every 1- and 2-qubit error, and additionally every error of weight 3 and higher (constrained by the distance) that is not a product of the 3-qubit check operations. These operations act on the gauge qubits leaving the subsystem code space invariant. The code can be implemented by qubits in a triangular configuration with the plaquettes as 3-qubit check operations.
Fault-tolerant quantum computation is a technique that is necessary to build a scalable quantum computer from noisy physical building blocks. Key for the implementation of fault-tolerant computations is the ability to perform a universal set of quantum gates that act on the code space of an underlying quantum code. To implement such a universal gate set fault-tolerantly is an expensive task in terms of physical operations, and any possible shortcut to save operations is potentially beneficial and might lead to a reduction in overhead for fault-tolerant computations. We show how the automorphism group of a quantum code can be used to implement some operators on the encoded quantum states in a fault-tolerant way by merely permuting the physical qubits. We derive conditions that a code has to satisfy in order to have a large group of operations that can be implemented transversally when combining transversal CNOT with automorphisms. We give several examples for quantum codes with large groups, including codes with parameters [8, 3, 3], [15, 7, 3], [22, 8, 4], and [31, 11, 5].
Transversal implementations of encoded unitary gates are highly desirable for fault-tolerant quantum computation. Though transversal gates alone cannot be computationally universal, they can be combined with specially distilled resource states in order to achieve universality. We show that "triorthogonal" stabilizer codes, introduced for state distillation by Bravyi and Haah [Phys. Rev. A 86, 052329 (2012)], admit transversal implementation of the controlled-controlled-Z gate. We then construct a universal set of fault-tolerant gates without state distillation by using only transversal controlled-controlled-Z, transversal Hadamard, and fault-tolerant error correction. We also adapt the distillation procedure of Bravyi and Haah to Toffoli gates, improving on existing Toffoli distillation schemes.