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).
We present a family of quantum error-correcting codes that support a universal set of transversal logic gates using only local operations on a two-dimensional array of physical qubits. The construction is a subsystem version of color codes where gauge fixing through local measurements dynamically determines which gates are transversal. Although the operations are local, the underlying code is not topological in structure, which is how the construction circumvents no-go constraints imposed by the Bravyi-K\"onig and Pastawski-Yoshida theorems. We provide strong evidence that the encoding has no error threshold in the conventional sense, though it is still possible to have logical gates with error probability much lower than that of physical gates.
Recently, S. Bravyi and R. K\"onig [Phys. Rev. Lett. 110, 170503 (2013)] 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 show that, for local stabilizer codes in
D spatial dimensions, locality-preserving gates are restricted to a set of unitary gates known as the
Dth level of the Clifford hierarchy. In this paper, we explore this idea further by providing several extensions and applications of their characterization to qubit stabilizer and subsystem codes. First, we present a no-go theorem for self-correcting quantum memory. 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 non-Clifford gates do not admit such implementations in Haah's cubic code and Michnicki's welded code. Second, we prove that the code distance of a
D-dimensional local stabilizer code with a nontrivial locality-preserving
mth-level Clifford logical gate is upper bounded by $O({L}^{D+1\ensuremath{-}m})$. For codes with non-Clifford gates (
m>2), this improves the previous best bound by S. Bravyi and B. Terhal [New. J. Phys. 11, 043029 (2009)]. Topological color codes, introduced by H. Bombin and M. A. Martin-Delgado [Phys. Rev. Lett. 97, 180501 (2006); Phys. Rev. Lett. 98, 160502 (2007); Phys. Rev. B 75, 075103 (2007)], saturate the bound for
m=D. Third, we prove that the qubit erasure threshold for codes with a nontrivial transversal
mth-level Clifford logical gate is upper bounded by
1/m. This implies that 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. Fourth, we extend the result of Bravyi and K\"onig to subsystem codes. Unlike stabilizer codes, the so-called union lemma does not apply to subsystem codes. This problem is avoided by assuming the presence of an error threshold in a subsystem code, and a conclusion analogous to that of Bravyi and K\"onig is recovered.
Topological color codes defined by the 4.8.8 semiregular lattice feature geometrically local check operators and admit transversal implementation of the entire Clifford group, making them promising candidates for fault-tolerant quantum computation. Recently, several efficient algorithms for decoding the syndrome of color codes were proposed. Here, we modify one of these algorithms to account for errors affecting the syndrome, applying it to the family of triangular 4.8.8 color codes encoding one logical qubit. For a three-dimensional bit-flip channel, we report a threshold error rate of 0.0208(1), compared with 0.0305(4) previously reported for an integer-program-based decoding algorithm. When we account for circuit details, this threshold is reduced to 0.00143(1) per gate, compared with 0.00672(1) per gate for the surface code under an identical noise model.
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.
Two-level quantum systems, qubits, are not the only basis for quantum computation. Advantages exist in using qudits,
d-level quantum systems, as the basic carrier of quantum information. We show that color codes, a class of topological quantum codes with remarkable transversality properties, can be generalized to the qudit paradigm. In recent developments it was found that in three spatial dimensions a qubit color code can support a transversal non-Clifford gate and that in higher spatial dimensions additional non-Clifford gates can be found, saturating Bravyi and K\"onig's bound [S. Bravyi and R. K\"onig, Phys. Rev. Lett. 111, 170502 (2013)]. Furthermore, by using gauge fixing techniques, an effective set of Clifford gates can be achieved, removing the need for state distillation. We show that the qudit color code can support the qudit analogs of these gates and also show that in higher spatial dimensions a color code can support a phase gate from higher levels of the Clifford hierarchy that can be proven to saturate Bravyi and K\"onig's bound in all but a finite number of special cases. The methodology used is a generalization of Bravyi and Haah's method of triorthogonal matrices [S. Bravyi and J. Haah, Phys. Rev. A 86, 052329 (2012)], which may be of independent interest. For completeness, we show explicitly that the qudit color codes generalize to gauge color codes and share many of the favorable properties of their qubit counterparts.