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.
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.
Fault-tolerant quantum computation allows quantum computations to be carried out while resisting unwanted noise. Several error-correcting codes have been developed to achieve this task, but none alone are capable of transversal universal quantum computation. This universality is highly desired and often achieved using techniques such as code concatenation, code switching, magic state distillation, or pieceable fault tolerance. Existing constructions can incur substantial overhead or rely on properties specific to particular codes. This work proposes implementing logical Clifford and T gates through novel, ancilla-mediated, pieceable interactions to construct a universal fault-tolerant quantum gate set. Our construction is deterministic, does not modify the underlying data codes or registers, and is generic over all data stabilizer codes. Thus, any single code becomes capable of universal quantum computation by leveraging helper codes in ancilla registers and mid-circuit measurements. Furthermore, since these logical gates are stabilizer code-generic, these implementations enable communication between heterogeneous stabilizer codes.
We study the use of triorthogonal codes for universal fault-tolerant quantum computation and propose two methods to circumvent the Eastin-Knill theorem, which prohibits any single quantum error-correcting code from supporting both universality and a transversal gate set. We show that our methods reduce the resource overhead compared with existing fault-tolerant protocols. We develop a simple fault-tolerant implementation of the logical Hadamard gate for triorthogonal codes by exploiting the fact that they have transversal controlled-Z (CZ) gates, resulting in a circuit with reduced overhead. We also introduce a procedure for generating a symmetric Calderbank-Shor-Steane code paired with a triorthogonal code, which allows CNOT and CZ gate transversality across the pair of codes. In addition, we present logical state teleportation circuits that transfer encoded states between the two codes, allowing all logical operations to be performed transversally. Our methods can be integrated into the Steane error correction framework without incurring additional resource cost. Finally, using the 15-qubit code as an example, we demonstrate that our protocols significantly reduce the gate overhead compared with other existing methods. These results highlight the potential of combining distinct code structures to achieve low-overhead, universal fault-tolerant quantum computation.
It is an oft cited fact that no quantum code can support a set of fault-tolerant logical gates that is both universal and transversal. This no-go theorem is generally responsible for the interest in alternative universality constructions including magic state distillation. Widely overlooked, however, is the possibility of non-transversal, yet still fault-tolerant, gates that work directly on small quantum codes. Here we demonstrate precisely the existence of such gates. In particular, we show how the limits of non-transversality can be overcome by performing rounds of intermediate error-correction to create logical gates on stabilizer codes that use no ancillas other than those required for syndrome measurement. Moreover, the logical gates we construct, the most prominent examples being Toffoli and controlled-controlled-Z, often complete universal gate sets on their codes. We detail such universal constructions for the smallest quantum codes, the 5-qubit and 7-qubit codes, and then proceed to generalize the approach. One remarkable result of this generalization is that any nondegenerate stabilizer code with a complete set of fault-tolerant single-qubit Clifford gates has a universal set of fault-tolerant gates. Another is the interaction of logical qubits across different stabilizer codes, which, for instance, implies a broadly applicable method of code switching.
It is an oft-cited fact that no quantum code can support a set of fault-tolerant logical gates that is both universal and transversal.This no-go theorem is generally responsible for the interest in alternative universality constructions including magic state distillation.Widely overlooked, however, is the possibility of nontransversal, yet still fault-tolerant, gates that work directly on small quantum codes.Here, we demonstrate precisely the existence of such gates.In particular, we show how the limits of nontransversality can be overcome by performing rounds of intermediate error correction to create logical gates on stabilizer codes that use no ancillas other than those required for syndrome measurement.Moreover, the logical gates we construct, the most prominent examples being Toffoli and controlled-controlled-Z, often complete universal gate sets on their codes.We detail such universal constructions for the smallest quantum codes, the 5-qubit and 7-qubit codes, and then proceed to generalize the approach.One remarkable result of this generalization is that any nondegenerate stabilizer code with a complete set of fault-tolerant single-qubit Clifford gates has a universal set of fault-tolerant gates.Another is the interaction of logical qubits across different stabilizer codes, which, for instance, implies a broadly applicable method of code switching.
Using transversal gates is a straightforward and efficient technique for fault-tolerant quantum computing. Since transversal gates alone cannot be computationally universal, they must be combined with other approaches such as magic state distillation, code switching, or code concatenation to achieve universality. In this paper we propose an alternative approach for universal fault-tolerant quantum computing, mainly based on the code concatenation approach proposed in [T. Jochym-O'Connor and R. Laflamme, Phys. Rev. Lett. 112, 010505 (2014)], but in a nonuniform fashion. The proposed approach is described based on nonuniform concatenation of the 7-qubit Steane code with the 15-qubit Reed-Muller code, as well as the 5-qubit code with the 15-qubit Reed-Muller code, which lead to two 49-qubit and 47-qubit codes, respectively. These codes can correct any arbitrary single physical error with the ability to perform a universal set of fault-tolerant gates, without using magic state distillation.
As there is no quantum error correction code with universal set of transversal gates, several approaches have been proposed which, in combination of transversal gates, make universal fault-tolerant quantum computation possible. Magic state distillation, code switching, code concatenation and pieceable fault-tolerance are well-known examples of such approaches. However, the overhead of these approaches is one of the main bottlenecks for large-scale quantum computation. In this paper, a hybrid approach is proposed which combines the code concatenation technique with the other mentioned approaches. The proposed approach outperforms code concatenation in terms of both number of qubits and error threshold and also significantly reduces the resource overhead of code switching, magic state distillation and pieceable fault-tolerance at the cost of reducing the effective distance of the concatenated code for implementing non-transversal gates.