To use quantum systems for technological applications one first needs to preserve their coherence for macroscopic time scales, even at finite temperature. Quantum error correction has made it possible to actively correct errors that affect a quantum memory. An attractive scenario is the construction of passive storage of quantum information with minimal active support. Indeed, passive protection is the basis of robust and scalable classical technology, physically realized in the form of the transistor and the ferromagnetic hard disk. The discovery of an analogous quantum system is a challenging open problem, plagued with a variety of no-go theorems. Several approaches have been devised to overcome these theorems by taking advantage of their loopholes. The state-of-the-art developments in this field are reviewed in an informative and pedagogical way. The main principles of self-correcting quantum memories are given and several milestone examples from the literature of two-, three-and higher-dimensional quantum memories are analyzed.

The surface code is currently the leading proposal to achieve fault-tolerant quantum computation. Among its strengths are the plethora of known ways in which fault-tolerant Clifford operations can be performed, namely, by deforming the topology of the surface, by the fusion and splitting of codes and even by braiding engineered Majorana modes using twist defects. Here we present a unified framework to describe these methods, which can be used to better compare different schemes, and to facilitate the design of hybrid schemes. Our unification includes the identification of twist defects with the corners of the planar code. This identification enables us to perform single-qubit Clifford gates by exchanging the corners of the planar code via code deformation. We analyse ways in which different schemes can be combined, and propose a new logical encoding. We also show how all of the Clifford gates can be implemented with the planar code without loss of distance using code deformations, thus offering an attractive alternative to ancilla-mediated schemes to complete the Clifford group with lattice surgery. arXiv:1609.04673v5 [quant-ph]

Performing large calculations with a quantum computer will likely require a fault-tolerant architecture based on quantum error-correcting codes. The challenge is to design practical quantum error-correcting codes that perform well against realistic noise using modest resources. Here we show that a variant of the surface code—the XZZX code—offers remarkable performance for fault-tolerant quantum computation. The error threshold of this code matches what can be achieved with random codes (hashing) for every single-qubit Pauli noise channel; it is the first explicit code shown to have this universal property. We present numerical evidence that the threshold even exceeds this hashing bound for an experimentally relevant range of noise parameters. Focusing on the common situation where qubit dephasing is the dominant noise, we show that this code has a practical, high-performance decoder and surpasses all previously known thresholds in the realistic setting where syndrome measurements are unreliable. We go on to demonstrate the favourable sub-threshold resource scaling that can be obtained by specialising a code to exploit structure in the noise. We show that it is possible to maintain all of these advantages when we perform fault-tolerant quantum computation.

Qudit toric codes are a natural higher-dimensional generalization of the wellstudied qubit toric code. However, standard methods for error correction of the qubit toric code are not applicable to them. Novel decoders are needed. In this paper we introduce two renormalization group decoders for qudit codes and analyse their error correction thresholds and efficiency. The first decoder is a generalization of a 'hard-decisions' decoder due to Bravyi and Haah (arXiv:1112.3252). We modify this decoder to overcome a percolation effect which limits its threshold performance for many-level quantum systems. The second decoder is a generalization of a 'soft-decisions' decoder due to Poulin and Duclos-Cianci (2010 Phys. Rev. Lett. 104 050504), with a small cell size to optimize the efficiency of implementation in the high dimensional case. In each case, we estimate thresholds for the uncorrelated bit-flip error model and provide a comparative analysis of the performance of both these approaches to error correction of qudit toric codes.

The constituent parts of a quantum computer are inherently vulnerable to errors. To this end, we have developed quantum error-correcting codes to protect quantum information from noise. However, discovering codes that are capable of a universal set of computational operations with the minimal cost in quantum resources remains an important and ongoing challenge. One proposal of significant recent interest is the gauge color code. Notably, this code may offer a reduced resource cost over other well-studied fault-tolerant architectures by using a new method, known as gauge fixing, for performing the non-Clifford operations that are essential for universal quantum computation. Here we examine the gauge color code when it is subject to noise. Specifically, we make use of single-shot error correction to develop a simple decoding algorithm for the gauge color code, and we numerically analyse its performance. Remarkably, we find threshold error rates comparable to those of other leading proposals. Our results thus provide the first steps of a comparative study between the gauge color code and other promising computational architectures.

The color code is both an interesting example of an exactly solvable topologically ordered phase of matter and also among the most promising candidate models to realize faulttolerant quantum computation with minimal resource overhead. The contributions of this work are threefold. First of all, we build upon the abstract theory of boundaries and domain walls of topological phases of matter to comprehensively catalog the objects as they are realizable in color codes. Together with our classification we also provide lattice representations of these objects which include three new types of boundaries as well as a generating set for all 72 color code twist defects. Our work thus provides an explicit toy model that will help to better understand the abstract theory of domain walls. Secondly, building upon the established framework, we discover a number of interesting new applications of the cataloged objects for devising quantum information protocols. These include improved methods for performing quantum computations by code deformation, a new four-qubit error-detecting code, as well as families of new quantum errorcorrecting codes we call stellated color codes, which encode logical qubits at the same distance as the next best color code, but using approximately half the number of physical qubits. To the best of our knowledge, our new topological codes have the highest encoding rate of local stabilizer codes with boundedweight stabilizers in two dimensions. Finally, we show how the boundaries and twist defects of the color code are represented by multiple copies of other phases. Indeed, in addition to the well studied comparison between the color code and two copies of the surface code, we also compare the color code to two copies of the three-fermion model. In particular, we find that this analogy offers a very clear lens through which we can view the symmetries of the color code which gives rise to its multitude of domain walls.

Defects in topologically ordered models have interesting properties that are reminiscent of the anyonic excitations of the models themselves. For example, dislocations in the toric code model are known as twists and possess properties that are analogous to Ising anyons. We strengthen this analogy by using the topological entanglement entropy as a diagnostic tool to identify properties of both defects and excitations in the toric code. Specifically, we show, through explicit calculation, that the toric code model including twists and dyon excitations has the same quantum dimensions, the same total quantum dimension, and the same fusion rules as an Ising anyon model.A fascinating class of many-body quantum systems are those that exhibit topological order [1]. Such systems are characterized by a gapped ground state manifold, with a degeneracy that depends on the boundary conditions, and have anyonic quasi-particle excitations. The degeneracy is robust to local perturbations, and therefore such systems are promising candidates for storing and manipulating quantum information [2][3][4][5].The structure of topologically-ordered systems can be further enriched by the use of domain walls or defects, across which quasi-particles transform nontrivially. We will be most interested in studying these defects in explicit lattice models [6][7][8][9][10], and in particular the model introduced by Bombin [6]. Essentially the same defects have also been studied in Chern-Simons theories [11][12][13]. In both of these settings, it has been shown that such defects can be viewed as having anyonic-like properties that are not associated with the underlying model [6,9,[11][12][13]. More specifically, in a particular two-dimensional topologically ordered spin lattice model known as the toric code, such a lattice dislocation leads to interesting behaviour: the points where dislocations terminate, known as twists, interact with the anyons of the toric code to reproduce properties, such as fusion rules, of nonabelian (specifically, Ising) anyons [6]. This is particularly surprising, as all excitations of the toric code are abelian anyons.In this paper, we further interrogate the analogy between twists and Ising anyons by using topological entanglement entropy (TEE) [14,15] as a diagnostic. Specifically, for a toric code model containing twists, we use the TEE to determine: (i) the total quantum dimension of the lattice with twists; (ii) the quantum dimensions of the objects (quasi-particles and defects) on the lattice, and (iii) the quantum dimensions of all of their fusion products, allowing us to reconstruct the fusion rules for twists and excitations. Our results coincide precisely with those of the Ising anyon model, lending further support to the analogy between the latter and the toric code with twists. We note that TEE is a particularly useful quantity in the context of the toric code, as both the parent Hamiltonian and the modified one with twists are described within the stabilizer formalism and so allow the TEE to be calculated e...

Noise in quantum computing is countered with quantum error correction. Achieving optimal performance will require tailoring codes and decoding algorithms to account for features of realistic noise, such as the common situation where the noise is biased towards dephasing. Here we introduce an efficient high-threshold decoder for a noise-tailored surface code based on minimum-weight perfect matching. The decoder exploits the symmetries of its syndrome under the action of biased noise and generalises to the fault-tolerant regime where measurements are unreliable. Using this decoder, we obtain fault-tolerant thresholds in excess of 6% for a phenomenological noise model in the limit where dephasing dominates. These gains persist even for modest noise biases: we find a threshold of ∼ 5% in an experimentally relevant regime where dephasing errors occur at a rate one hundred times greater than bit-flip errors.The surface code [1,2] is among the most promising quantum error-correcting codes to realise the first generation of scalable quantum computers [3][4][5]. This is due to its two-dimensional layout and low-weight stabilizers that help give it its high threshold [2,6,7], and its universal set of fault-tolerant logical gates [2,[8][9][10][11]. Ongoing experimental work [12][13][14][15] is steadily improving the surface code error rates. Concurrent work on improved decoding algorithms [6,7,[16][17][18] is leading to higher thresholds and lower logical failure rates, reducing the exquisite control demanded of experimentalists to realise such a system.Identifying the best decoder for the surface code depends critically on the noise model. Minimum-weight perfect matching (MWPM) [19,20] is near-optimal in the case of a bit-flip error model [2] and for a phenomenological error model with unreliable measurements [6]; see [21,22]. More recently, attention has turned to tailoring the decoder to perform under more realistic types of noise, such as depolarising noise [16,18,23,24] and correlated errors [25][26][27]. Of particular note is noise that is biased towards dephasing: a common feature of many architectures. With biased noise and reliable measurements, it is known that the surface code can be tailored to accentuate commonly occurring errors and that an appropriate decoder will give substantially increased thresholds [28,29]. However, these high thresholds were obtained using decoders with no known efficient implementation in the realistic setting where measurements are unreliable.Here we propose an efficient decoder for the surface code that is tailored to correct for local noise biased towards dephasing, demonstrating exceptional fault-tolerant thresholds. Our decoder uses the MWPM algorithm together with a recent technique to exploit symmetries of a given quantum errorcorrecting code [30]. Rather than using the symmetries of the code, we generalize this idea and use the symmetries of the entire system. Specifically, we exploit the symmetries of the syndrome with respect to its incident error model. Applied to pure depha...