C OMPARECENCIAS ANTE EL P LENO

any of the measured stabilizer eigenvalues are non-trivial, the resulting state is discarded. Fortunately, the process can be concatenated to an arbitrary number of levels, resulting in a state that is arbitrarily pure. Such purity is costly in terms of the resource of the noisy magic state ancillas. However, recent progress in more efficient encodings has been made [Bravyi and Haah,2012;Jones, 2013; Meier et al.,

2012].

Magic state injection by gate teleportation is one of the best-known techniques for implementing a transversal non-Clifford gate. However, as we have mentioned, codes exist that have a transversal non-Clifford gate but not a transversal Hadamard gate. In such cases the logical Hadamard can be realised using magic state injection or a technique called gauge fixing [Paetznick and Reichardt,2013]. Other techniques to achieve a universal fault-tolerant gate set exist, such as fault-tolerant ancilla preparation for the Toffoli gate [Shor, 1996], code concatenation [Jochym-O’Connor and Laflamme, 2014] and code conversion [Anderson et al.,2014], but these will not be discussed in this thesis.

We have now completed our brief introduction to fault-tolerant quantum comput- ing. We conclude this section by bringing to the reader’s attention the dominance of the qubit in all we have discussed thus far. This is a natural extension of the classical realm, where the bit likewise dominates. However, as in the classical domain, base 2 is not the only possible choice. In the next section we therefore introduce the notion of qudits, the quantum analogue of a base d classical bit.

1.5

Qudit Stabilizer Codes

Qubits have been studied as the basic building block for quantum computing and hence error correcting architectures because as 2-level systems they are the natural quantum analogue of classical bit. Qudits (d-level quantum systems) are the natural higher-dimensional generalisation of qubits, defined as a quantum superposition of d basis states. The primary advantage of qudit quantum error correcting codes is that the magic state distillation procedures are much more efficient [Campbell,2014;

Campbell et al.,2012]. There is also evidence for improved thresholds in some codes as the qudit Hilbert space dimension increases [Anwar et al., 2014], and we shall explore this more in Chapter 4.

1.5. Qudit Stabilizer Codes

Associated to each d-level quantum state is a Hilbert spaceHd. This is a complex

vector space with an inner product and norm, and is spanned by the computational basis states _{|αi where α ∈ Z}d. In the case of prime qudit dimensions, the d-element

cyclic group Zd ={0, . . . , d − 1} can be conveniently identified with addition over

integers modulo d. The fact it is a cyclic group means (d_{− a) ∈ Z}d≡ −a ∈ Zd.

The conventional single-qubit Pauli operators are generalised as:

X_da = X

j∈Zd

|j ⊕ ai hj| , Z_db = X

j∈Zd

ωbj_{|ji hj| ,} (1.16)

where ω = e2πi/d and the addition ‘_{⊕’ is taken to be modulo d throughout this} thesis. We shall drop the subscript d unless it is necessary for clarity. Notice that now Xd = Zd = I. Unlike qubit Pauli operators, these unitary operators are not Hermitian when d > 2, but they possess orthogonal eigenspaces with eigenvalues of the form ωj_{, for some j. Hence, we can still interpret them as corresponding to}

physical observables with measurement outputs labelled by their complex eigenvalues. The qudit Pauli operators obey the ω-commutation relation Xa_Zb _{= ω}−ab_Zb_Xa

for arbitrary a, b_{∈ Z}d. They generate the single-qudit Pauli group Pd =hX, Zi up

to a global phase of ωj _{for j ∈ Z}

d. The n-qudit Pauli group Pnd is the n-fold tensor

product of the single-qudit Pauli group P⊗n_d .

Similarly to the qubit case, the code space of a qudit stabilizer code is defined as the ‘+1’ eigenspace of an abelian subgroup S ∈ Pn_d, such that ωjI _{6∈ S for non-zero j.} The qudit syndrome measurements can have outcomes ωj _{∀j ∈ Z}

d, and we often

denote these outcomes simply by j, which we call the charge of the syndrome1_. In the next chapter we begin by introducing both qubit and qudit variants of a family of degenerate quantum error correcting codes that have useful properties for FTQC applications: the surface codes. In Chapter3we study some of the properties of the qubit version of the surface code. The decoders for such codes are nontrivial and in Chapter 4 we introduce a new decoder that performs well for both the qubit and qudit variants of the code. We conclude in Chapter 5 by generalising a second family of quantum error correcting codes, colour codes, from the qubit to the qudit domain.

1_{This comes from the anyon (quasi-particle) description of topological codes. We do not consider}

Chapter 2

Topological Codes

Topological codes are a class of stabilizer error correcting codes defined on lattices with useful properties. Consequently, they have become the leading candidates in the search for quantum error correcting codes to form part of a scheme to realise an experimentally implementable quantum computer. Their defining feature is that the encoded information is dependent not on local parameters but rather on the nature of the manifold in which the lattice is embedded. The information is hence associated with global degrees of freedom. Thus, each code family is parametrised by the lattice size, and increasing the lattice size increases the error protection offered by the code. Furthermore, the stabilizer generators are geometrically local. This offers two key benefits: firstly they are easily measured in an experimental setting, and secondly any noise introduced in measuring them remains local. By employing suitable recovery strategies that aim to minimise the chances of a local error being mapped to a global operation, the chances of failing to correct the errors are low as long as the code is operating below the error threshold. In this chapter we introduce several variants of one prominent example of a topological code: the surface code.

2.1. Introduction

2.1

Introduction

So far we have discussed protection against errors and fault-tolerant implementations of gates for universal quantum computation. In this chapter we introduce surface codes: a class of codes invented in 1997 by Alexei Kitaev. There are two overwhelming benefits of this class of codes over the code constructions by Calderbanke, Shor and Steane introduced in Chapter 1. First, the code distance scales very efficiently, for example as O(n1/2_{) in 2D, where n is the number of qubits comprising the code. The}

second benefit of these codes is that their check operators are geometrically local. Since physics acts locally this makes the check operators easier to realise, thus this property increases the possibility of experimentally implementing such a code.

The surface code [Bravyi and Kitaev,1998;Kitaev,1997b,2003] is one of a family of topological stabilizer codes, and is the basis for an approach to fault-tolerant quantum computing for which high thresholds have been reported [Fowler et al.,

2012a; Raussendorf and Harrington, 2007; Raussendorf et al., 2007; Wang et al.,

2003]. Surface codes are able to perform quantum error correction fault-tolerantly, that is, they are robust not only to physical errors on the individual qubits, but also to noise in the syndrome measurements.

The toric code [Dennis et al., 2002] is among the most extensively studied of this family of codes, revealing much insight into related topologically ordered systems. The great benefits that these codes offer is reflected in the degree of experimental interest, for example in superconducting qubits [Kelly et al., 2015]. A great deal of theoretical work has concentrated on calculating thresholds for various error models, for example models including lost qubits [Stace et al., 2009], and on the discovery and implementation of new classical decoding algorithms such as the soft-decision renormalisation group algorithm [Duclos-Cianci and Poulin,2010a,b]. The toric code performs well, with high thresholds for some commonly studied noise models.

In this chapter we shall outline the qubit toric code construction due to Kitaev and while discussing the planar variant often known as the surface code we shall include its generalisation to a qudit form. The introduction of the other widely studied family of topological codes, colour codes, is left until Chapter 5.