Golden codes: quantum LDPC codes built from regular tessellations of hyperbolic 4-manifolds

12/22/2017
by   Vivien Londe, et al.
0

We adapt a construction of Guth and Lubotzky [arXiv:1310.5555] to obtain a family of quantum LDPC codes with non-vanishing rate and minimum distance scaling like n^0.2 where n is the number of physical qubits. Similarly as in [arXiv:1310.5555], our homological code family stems from tessellated hyperbolic 4-manifolds. The main novelty of this work is that we consider a regular tessellation consisting of hypercubes. We exploit this strong local structure to design and analyze an efficient decoding algorithm.

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1 Introduction

Building a large-scale quantum computer is certainly one of the main challenges faced by the physics community in the 21th century. This turns out to be a daunting task because of the extreme fragility of quantum information: any uncontrolled interaction between the qubits and the environment leads to decoherence and quickly causes any computation to fail. Fortunately, theoretical solutions exist under the form of quantum error correcting codes which allow one to encode logical qubits into a larger number of physical qubits, in such a way that logical information can be preserved and recovered despite potential errors occurring on the physical qubits [Sho95], [CS96].

Mathematically, a quantum code of dimension and length is a subspace of of dimension . A possible way to specify such a subspace is via a stabilizer group: an Abelian subgroup of the

-qubit Pauli group. In that case, the quantum code is defined as the common eigenspace of the stabilizers with eigenvalue

. Such a code is called a stabilizer code [Got97]. Among these, CSS codes, due to Calderbank, Shor and Steane, are those for which the stabilizer group admits a list of generators which are either products of -type Pauli operators, or of -type Pauli operators. A convenient way to find CSS codes, initiated by Kitaev [Kit03], is to consider tessellations of manifolds. In that case, the physical qubits are associated with -dimensional faces of the tessellation, the -type stabilizers are associated with -faces and the -type stabilizers with -faces. Such codes are called homological. The stabilizer group of a homological code is commutative as required because ()-faces, -faces and ()-faces form a chain complex. In other words, given an -face , and an -face , there is an even number of -faces incident to both and [BMD07], [Zém09], [BT16].

A major advantage of homological codes is that they are naturally of the low-density parity-check (LDPC) type, meaning that generators of the stabilizer group act nontrivially on a constant number of qubits and that each qubit is acted upon by a constant number of generators. This is of course especially interesting for potential implementations, but also at a more mathematical level since classical LDPC codes play a central role in classical coding theory. A second advantage of homological codes is that they can lead to simple and efficient decoding algorithms which directly exploit the local structure of the code on the manifold [DKLP02], [Har04], [DZ17], [DN17].

The parameters of homological codes can be derived from the properties of the underlying manifold: the length of the code is given by the number of -faces in the tessellation, the dimension is related to the rank of the homology group, and the minimum distance, that is the minimum weight of a nontrivial Pauli error, is related to the homological systole of the manifold, that is the minimal number of -faces forming an homologically nontrivial -cycle. Exploiting this connection with manifolds exhibiting systolic freedom, Freedman, Meyer and Luo [FML02] were able to construct the quantum LDPC codes with the best minimum distance presently known, achieving .

An important question is to understand what parameters can be achieved with LDPC codes. The toric code and the code of [FML02] display a large minimum distance but only encode a constant number of qubits, . If the manifold is Euclidean, strong constraints are known to apply on the code parameters: namely the parameters have to satisfy for some constant [BPT10]. For tessellations of 2-dimensional hyperbolic manifolds, Delfosse showed that [Del13]. In particular, these results show that one cannot get a good minimum distance for surface codes with constant rate.

In many cases, it is natural to consider constant-rate codes where : such codes for instance allow one to obtain quantum fault-tolerant computation with constant space overhead [Got14]. For a long time, it was believed that constant-rate homological codes could not have a large minimum distance, that is growing polynomially with their length [Zém09]. A recent breakthrough was the work of Guth and Lubotzky [GL14] who gave a construction of homological codes in hyperbolic 4-space that combine a constant rate with a minimum distance . It was later shown by Murillo that . Quickly after this result, Hastings proposed a decoding algorithm for such codes [Has16]. Unfortunately, the analysis of Hastings’ decoding algorithm is only valid when local brute-force decoding is performed at a scale that may not be computationally practical. In fact, it is difficult to precisely analyze the performance of Hastings’ decoder because the local structure of the codes of [GL14] is not completely explicit.

In this work, we give a variant of the construction of Guth and Lubotzky which admits a simple explicit local structure: it is based on a regular tessellation of the 4-dimensional hyperbolic space by means of hypercubes. We then exploit this local structure to design an efficient decoding algorithm which tries to locally shorten cycles. In Section 2, we give an overview of our approach compared to that of Guth and Lubotzky. In Section 3, we explain how to obtain a regular tessellation of hyperbolic -space with hypercubes. In Section 4, we detail how to quotient the space to get a compact manifold, which then yields the quantum code. We finally describe our local decoder and analyze its performances in Section 5.

2 A variant of Guth and Lubotzky’s construction based on a regular tessellation of hyperbolic space

The family of manifolds considered in [GL14] is a family of 4-dimensional hyperbolic coverings. The tessellations can be obtained by pulling back the natural tessellation of the base space. Each covering equipped with its natural tessellation gives rise to a quantum error correcting code. Unfortunately the fundamental polytope of this natural tessellation is not regular. In particular, it is nontrivial to obtain the local structure of the tessellation, and therefore an explicit description of the code generators. While this did not prevent Hastings from designing a decoding algorithm for these codes [Has16], simulating its performance for the codes of [GL14] appears quite impractical. (Note, however, that Hastings’ decoder was recently implemented for the 4-dimensional toric code, in Euclidean space [BDMT16].)

It is useful to see the 4-dimensional homological quantum error correcting codes that Guth and Lubotzky and we construct as generalisations of the 2-dimensional toric code. Let us therefore give the arithmetic manifold viewpoint on the toric code. We consider the ordinary tessellation of the Euclidean plane by unit squares such that vertices have integer coordinates. The translation group of Euclidean plane is . We denote by the subgroup of this translation group. Elements of stabilize the ordinary tessellation of Euclidean plane. Let be an ideal of , with a positive integer and define to be . The quotient of the Euclidean plane by is a torus, that naturally inherits the tessellation by unit squares from the Euclidean plane. The constructions of [GL14] and of the present work are generalisations of the 2-dimensional Euclidean toric code in a 4-dimensional hyperbolic setting. To help the reader make analogies with the toric code, we introduced in this paragraph notations similar to the notations used in the sequel.

We now summarise the construction of Guth and Lubotzky and explain the advantages of our approach. In [GL14], the construction is based on tessellations of hyperbolic 4-space. To each code corresponds a manifold equipped with a tessellation. The base space is constructed by considering the action of a cocompact discrete group of isometries on hyperbolic 4-space : . To each finite index subgroup of corresponds a covering of given by . It is natural to tessellate with a single 4-face and to tessellate with a number of 4-faces equal to the index of in . All 4-faces are isomorphic to the first one. Unfortunately the 4-face is not regular in [GL14], which makes the local description of the quantum code rather complicated. To obtain a similar construction with a regular 4-face, we reverse the process: we start with a convenient regular 4-face and then build a corresponding discrete group of isometries .

For its symmetries and because it tessellates hyperbolic 4-space, we choose the 4-dimensional hypercube as our targeted regular 4-face. We embed it in hyperbolic 4-space and scale it according to the tessellation of hyperbolic 4-space (see Section 3.2.1 for a definition of Schäfli symbols). The group is generated by the direct isometries of hyperbolic 4-space sending opposing faces of the hypercube to each other with no rotation. The tessellating 4-face we obtain is a hypercube by construction.

The tricky part of the construction is to define finite index subgroups of our discrete group of isometries in a way similar to [GL14]. Indeed, arithmeticity of subgroups plays a central role in lower bounding the minimum distance of the corresponding error correcting codes. To achieve this goal, we rely on arithmetic structures defined over the number field . Replacing by this number field, by its ring of integers and ideals by ideals of this ring of integers where is the golden ratio (giving its name to our construction), it is possible to define principal congruence subgroups

such that the corresponding family of error correcting codes satisfies the same asymptotic estimates as in

[GL14]. We therefore obtain a family of codes with a regular local structure, a non-vanishing rate and a minimum distance lower bounded by , where is the number of physical qubits.

We take advantage of the regular local structure to design an efficient decoding algorithm. This algorithm is highly local since it decreases the syndrome at the scale of a single 4-face. In particular, our algorithm is much more local and explicit than Hastings’ decoder [Has16]. We prove that syndromes associated with errors of weight below the injectivity radius of the manifold always contain a pattern that can be locally shortened so as to decrease the weight of the syndrome. In other words, the algorithm simply consists in examining the neighborhood of the syndrome and acting on qubits to decrease the syndrome weight. We show that arbitrary errors of size

are corrected by this algorithm, which in turn implies that random errors will be corrected with high probability if the error rate is small enough. These results are similar to those of Hastings’ decoder, but with the advantage of an entirely explicit algorithm with precise bounds on its performances.


3 Hyperbolic 4-space and its regular tessellation by hypercubes

In this section, we first introduce the minimal background on hyperbolic 4-space and regular tessellations. We then focus on the tessellation of hyperbolic 4-space by 4-dimensional hypercubes on which our quantum code construction is based.

3.1 Hyperbolic space

We use the hyperboloid model to describe 4-dimensional hyperbolic space. As a set, 4-dimensional hyperbolic space is identified with and . It is endowed with a Riemannian metric such as to make it a space of constant negative sectional curvature. Its isometry group is , the identity component of the special indefinite orthogonal group.
The reader is referred to [Rat06] for a comprehensive introduction to hyperbolic geometry. To give some intuition about hyperbolic space we will merely give the perimeter of a circle of radius . In hyperbolic space, such a circle has perimeter . The growth is exponentially faster than its Euclidean counterpart . In spherical space on the other hand, the perimeter of a circle of radius is only (for ). Informally speaking, there is more room in the angular direction in hyperbolic space than in Euclidean space just like there is less room in the angular direction in spherical space than in Euclidean space. One can make this statement more precise by considering regular tessellations.

3.2 Regular tessellations

Results of this section come from Ref. [Cox54]

. We will classify regular tessellations in three groups: spherical, Euclidean and hyperbolic.

Definition 1.

A regular tessellation is called spherical (respectively Euclidean, respectively hyperbolic) if it can be embedded with regular faces in spherical (respectively Euclidean, respectively hyperbolic) space.

Informally speaking, if a tessellation is too small to fit in Euclidean space, it curves inwards and yields a spherical tessellation. If it is too big, it yields a hyperbolic tessellation.
In the Euclidean case, the faces of the tessellation can be scaled by multiplying all lengths by a given positive real . In the spherical and hyperbolic cases, however, the volumes of faces are imposed by the combinatorics of the tessellation: the furthest the tessellation is from being Euclidean, the greatest volumes of faces are.

3.2.1 Schläfli symbol

A convenient way to describe regular tessellations is via their Schläfli symbols, which are defined recursively for positive integers:

  • {} refers to a regular -sided polygon.

  • {,} refers to a regular tessellation by regular -sided polygons such that each vertex is incident to regular -sided polygons.
    One obtains a tessellation of the Euclidean plane if , or of the hyperbolic plane if . Finally if , then can represent either a tessellation of the two-dimensional sphere or a 3-dimensional polyhedron.
    There are five regular 3-dimensional polyhedrons called the Platonic solids: the regular icosahedron ; the regular octahedron ; the regular tetrahedron ; the cube and the regular dodecahedron, .

  • If {,} and {,} are 3-dimensional polyhedrons111The condition that is also a 3-dimensional polyhedron is necessary, for instance, to ensure that the dual tessellation is well-defined., then {,,} refers to a regular tessellation by {,}-polyhedrons such that each edge of the tessellation is incident to {,}-polyhedrons. Note that the terminology honeycomb is sometimes used instead of tessellation to insist on the 3-dimensionality. The terminology mosaic can be encountered as well. We will use tessellation in the sequel regardless of the dimension.
    Similarly as before, the nature of the tessellation depends on the relation between the integers . If , one obtains a tessellation of the Euclidean 3-dimensional space. If , one gets a tessellation of the hyperbolic 3-dimensional space. Finally, if , it can represent either a tessellation of the spherical 3-dimensional space or a 4-dimensional polytope.
    There are six regular 4-dimensional polytopes: {3,3,5} is the 600-cell, {3,3,4} is the 4-orthoplex, {3,4,3} is the 24-cell, {3,3,3} is the regular 4-simplex, {4,3,3} is the 4-dimensional hypercube, and {5,3,3} is the 120-cell.

  • If {,,} and {,,} are 4-dimensional polytopes, {,,,} refers to a regular tessellation by {,,}-polytopes such that each 2-face of the tessellation is incident to {,,}-polytopes.
    If , it is a tessellation of the 4-dimensional Euclidean space. If , it is a tessellation of hyperbolic 4-space.
    There are five regular tessellations of hyperbolic 4-space: {3,3,3,5}, {4,3,3,5}, {5,3,3,5}, {5,3,3,4} and {5,3,3,3}.

Given a tessellation or a polytope described by Schläfli symbol {,…,}, the tessellation or polytope described by {,…,} is called the dual tessellation or polytope. It is the tessellation obtained by mapping every -face to an -face. Note that duality doesn’t change the hyperbolic, Euclidean or spherical type of a tessellation.

3.2.2 The {4,3,3,5} regular tessellation of hyperbolic 4-space

In this work we will focus on the {4,3,3,5} regular tessellation of hyperbolic 4-space. The 4-faces {4,3,3} of this tessellation are 4-dimensional hypercubes, which are particularly nice. In particular, one can exploit the fact that the coordinate axes are symmetry axes of the hypercube to find a nice description of a discrete subgroup of corresponding to the {4,3,3,5} regular tessellation. The other regular tessellations of hyperbolic 4-space could lead to similar constructions but it would require more work to make the computations explicit. We will not consider them in this work.

3.3 Isometry group of the tessellation

We consider a hypercube centered at the origin of the hyperboloid model and such that all four coordinate axes are symmetry axes of the hypercube. We denote this hypercube by in the sequel (a 4-dimensional hypercube is also called a tesseract). Since 3-faces of the hypercube are orthogonal to coordinate axes, there exist direct isometries of hyperbolic 4-space sending any one of them onto the opposite one. These direct isometries are elements of acting nontrivially on only two coordinates. For example direct isometries sending the 3-faces orthogonal to the first coordinate axis onto each other are given by the following matrices:


The angle between two 3-faces of the hypercube depends on the parameter : the larger is, the smaller the angle. We will compute the value of such that this angle equals . Indeed in the {4,3,3,5} regular tessellation of hyperbolic 4-space, five hypercubes meet along each 2-face, which means that the dihedral angle between two 3-faces of the same hypercube must be . Note that the dihedral angle between 3-faces is sometimes called dichoral angle to insist on higher dimension. We will use the terminology dihedral angle in the sequel regardless of dimension.

Definition 2 (Ratcliffe [Rat06]).

The Lorentzian inner product denoted is the bilinear map defined on by:

Two vectors

, are Lorentz orthogonal if .

Let , respectively , be the 3-face of orthogonal in hyperbolic 4-space to the first, respectively second, axis and such that its second coordinate in the hyperboloid model is non-negative. Points of have coordinates of the form for some . Similarly points of have coordinates of the form with . We deduce that is Lorentz orthogonal to and is Lorentz orthogonal to .

Definition 3 (Ratcliffe [Rat06]).

The Lorentzian norm of a vector is the complex number denoted satisfying and such that is either positive imaginary, or positive.
Note that if is positive imaginary, denotes its modulus.

Definition 4 (Ratcliffe [Rat06]).

The space-like angle between two space-like vectors and is defined by: and .

Lemma 5.

Let and

be two hyperplanes of hyperbolic space. Let

, respectively , be a vector Lorentz orthogonal to , respectively . Let be the angle between and , and the space-like angle between and . Then,

Proof.

We use Lorentz transformations to reduce to the case where the origin of the hyperboloid belongs to . Indeed both and are invariant under Lorentz transformations. We can then work in the hyperbolic plane with two lines and intersecting at the origin of the hyperboloid.
Using another Lorentz transformation and possibly renaming and , we can assume that is the first coordinate axis and is
with . Then is Lorentz orthogonal to and is Lorentz orthogonal to , and therefore . Other choices of and such as e.g. can lead to . ∎

For our choice of and , we have

Since we want to build a {4,3,3,5} tessellation, five hypercubes have to be incident to each 2-face of the hypercube. This imposes and therefore . This leads to and we finally obtain

the golden ratio and its square root.

We denote by the discrete subgroup of generated by the four direct isometries sending a 3-face of the hypercube onto the opposite 3-face. Note that there are eight such direct isometries but they are pairwise inverse of each other.

By definition, for , is an -face of the tessellation of hyperbolic 4-space if there exists and an -face of the centred hypercube T such that .
This tessellation of hyperbolic 4-space has an infinite number of -faces for every . To build a code with a finite number of qubits, we need a tessellation having a finite number of 2-faces. We use in the sequel number theoretical tools to construct quotients of the -tessellated hyperbolic 4-space.

4 Compact -tessellated manifolds

We want to define a quantum code by identifying physical qubits with 2-faces of a tessellation. To obtain a finite code, we need to consider tessellations of compact manifolds. We will therefore consider the tessellation of compact manifolds obtained as quotients of hyperbolic 4-space. These manifolds are called arithmetic because they are quotients of by arithmetic subgroups of . We first review the definitions of a number field and its ring of integers. We then use these tools to associate an arithmetic subgroup to every ideal of the ring of integers .

4.1 Number fields and rings of integers

Definition 6.

A number field is a finite degree field extension of the field of rational numbers .

Theorem 7 (e.g. Marcus, [Mar77]).

Every number field has the form for some algebraic number . If is a root of an irreducible polynomial over having degree n, then

Since is a root of , which is irreducible over , we have

Definition 8.

A complex number is an algebraic integer if it is a root of a monic (leading coefficient equal to 1) polynomial with coefficients in .

Definition 9.

The ring of integers of a number field is the subset of its algebraic integers. It is denoted .

Propositon 10 (e.g. Marcus, [Mar77] p.15).

Let satisfy and let be the quadratic number field . Then,

Applying this characterization to the case yields:

where is the golden ratio .

4.2 Arithmetic subgroups

Since and its square root are algebraic numbers, is a number field. Its ring of integers is , and therefore every matrix of is with coefficients in the ring .

Definition 11.

A number field is totally real if all its embeddings in are embeddings in .

In order to obtain the same asymptotic behaviour of the code parameters , and as in Refs [GL14] and [Mur16], we need to work with a totally real number field. This is not the case of , however. We therefore conjugate matrices of in such a way that all their entries now belong to a totally real number field. It is sufficient to ensure that the four matrices generating have their entries in a totally real number field.

We remark that .

Therefore, defining , the group defined as has all its matrices with coefficients in the number field , and in fact in its ring of integers .

Definition 12.

The norm of an ideal of a ring is the cardinal of the quotient .

The ring has a family of ideals whose norms are unbounded. Indeed the norm of the ideal of generated by is . On top of this, there are other ideals in . For example, the ideal generated by has norm 5.

Definition 13.

Let be an ideal of a ring . Let be a matrix group with coefficients in . The principal congruence subgroup of level of is the kernel of the reduction modulo morphism. It is denoted .

Hence to each ideal of corresponds a normal subgroup of .

We denote by the quotient of by . By definition is the set of orbits of under the action of . Note that we use the notation and not because acts on on the left. naturally inherits the hyperbolic structure of . It also naturally inherits the tessellation of by projection.

Definition 14.

Let be a subgroup of a group . The index of in , denoted , is the cardinal of the quotient .

Lemma 15.

The number of 2-faces of the tessellation of is proportional to .

Proof.

It is sufficient to show that the number of 4-faces of the tessellation of is proportional to .
The action of on the set of 4-faces of -tessellated naturally induces an action of on . In particular, is naturally isomorphic to . Since is the group of direct symmetries of the hypercube, it is finite (with cardinality 192). Therefore there are 4-faces in the tessellation of . ∎

It is shown in Ref. [Mur16] that . This provides an upper bound on the size of the quantum code associated with an ideal .

4.3 From a tessellated 4-manifold to a quantum code

We will paraphrase in this section the correspondence exposed in Ref. [GL14] between a family of coverings and a family of quantum codes. From each 4-dimensional -tessellated manifold , a code is constructed: qubits are identified with 2-faces of , -type stabilizers are identified with 1-faces (edges) of and -type stabilizers are identified with 3-faces of . Each -type, respectively -type, stabilizer acts by an Pauli matrix, respectively a Pauli matrix, on every qubit it is incident to. The codespace is the common (+1)-eigenspace of the set of stabilizers. The length of the code, i.e. its number of physical qubits, is the number of 2-faces of the tessellation. It is proportional to the volume of . The dimension of the code, i.e. its number of logical qubits, is the second Betti number of , i.e. the rank of its second homology group. The minimum distance of the code is the minimal number of 2-faces forming a homologically nontrivial 2-cycle in . It is proportional to the least area of a homologically nontrivial surface of . These proportionality coefficients do not depend on the ideal . With this correspondence, the asymptotic behaviour of , and is understood in terms of the family of manifolds independently of the tessellation.

To each ideal I of the ring of integers corresponds a -tessellated manifold and a quantum error correcting code . Like in Ref. [GL14], the rate of this family of codes is non-vanishing and its minimum distance d asymptotically satisfies for an . Similarly to Ref. [Mur16], we can consider the spin group , which is a double covering of . Defining principal congruence subgroups at the level of the spin group , Murillo shows that the minimum distance of the corresponding codes satisfies [Mur16]. We note that the arithmetic manifolds defined at the level of the spin group are not strictly speaking the same as the ones defined at the level of the indefinite orthogonal group. Indeed the arithmetic subgroups of by which hyperbolic 4-space is quotiented are different. To derive the lower bound on the minimum distance, the whole construction has to be done at the level of the spin group. Doing so does not alter the rate of the family of codes nor its local structure. Therefore it does not modify the local decoders designed in Section 5. We can now state the main theorem:

Theorem 16.

There exists a family of homological quantum error correcting codes defined from -tessellated hyperbolic 4-manifolds. This family has non-vanishing rate . The minimum distance of its codes grows at least like .

4.4 Estimates for the number of physical qubits

The family of codes used to state Theorem 16 has the drawback of being sparse. We show now that the smallest value of corresponding to a proper ideal of is of the order of . However there are normal subgroups of which are not constructed from an ideal of . Finding such normal subgroups with small index in would lead to quantum codes with a reasonable, i.e. small enough to be practical, number of physical qubits. Even though the control over minimum distance is lost when considering non arithmetic normal subgroups, the rate of the family of codes and the local decoders are valid for any normal subgroup. Moreover it could be interesting to use the technique of Ref. [BVC17]

to interpolate between arithmetic hyperbolic 4-dimensional codes and

e.g. Euclidean 4-dimensional codes. This can be done by refining the hyperbolic tessellation by a Euclidean tessellation of the hypercubes.
Since has dimension 10, the number of hypercubes in the -tessellated manifold is proportional to . Therefore the number of qubits of the quantum error correcting code is also proportional to . Indeed in the tessellation, each hypercube is incident to 24 squares and each square is incident to 5 hypercubes. Thus the number of qubits is times the number of hypercubes. In , the smallest proper ideals we have found have norm 4, 5, 9, 11. If we ignore the ideal of norm 4 because of possible difficulties when has characteristic 2, the smallest norm is 5. It corresponds to a number of qubits of the order of .

5 Local decoders

In this section, we design efficient decoding algorithms for the family of codes constructed in the previous section. These decoders are tailored for the whole -tessellated hyperbolic 4-space. Of course we want to apply these decoders to codes with a finite number of physical qubits, i.e. to quotients of the -tessellated hyperbolic 4-space.
In this work we don’t address the finite size effects in every detail but make the two following remarks instead. First, the injectivity radii of the arithmetic hyperbolic manifolds associated with the golden code family scale logarithmically with their volumes. In terms of decoding, this implies that decoding a number of errors logarithmic in the number of physical qubits is strictly equivalent in the arithmetic hyperbolic manifolds and in the whole -tessellated hyperbolic 4-space. In other words our decoders provably succeed for any error pattern of weight logarithmic in the number of physical qubits. Second, the same decoder will succeed with high probability random error patterns of weight linear in the number of physical qubits, for instance if the qubits are affected independently by depolarizing noise.
The advantage of our decoders over the generic hyperbolic 4-dimensional decoder by Hastings [Has16] is their high locality. Indeed Hastings’ decoder is local at the level of a ball of radius where is constant but unknown. Since in hyperbolic 4-space the number of 2-faces in a ball of radius grows like , even small values of can lead to an unpractical degree of locality. For instance the authors of [BDMT16] use the value to implement a version of Hastings’ decoder in a 4-dimensional toric code setting. With such a small value of the analysis of the performance of Hastings’ decoder probably does not apply. The analysis of our decoders, on the other hand, is valid at a level of locality that is computationally practical.

Since the codes we consider are CSS, it is possible to decode -type and -type errors independently, and this is what our algorithm does. Because correcting these two types of errors on a qubit is sufficient to correct an arbitrary single-qubit error, we can state our decoding theorem as follows.

Theorem 17.

There exists a constant such that for any error corrupting less than physical qubits, the decoding algorithm returns a set of qubits such that and differ by a sum of stabilizers.

Since stabilizers act trivially on the codespace, Theorem 17 implies that any codestate corrupted on at most physical qubits is perfectly recovered by the active error correction procedure.
Moreover, standard results in percolation theory show that for a random error model where each qubit is affected independently and identically with a depolarizing node, then below some constant noise threshold, the error will affect qubits that belong to small connected components of the tessellation of size . This is because the tessellation has constant degree. In that case, using the same ideas as in [FGL17], the decoding algorithm will correct the error with high probability.

Theorem 18.

There exists a constant such that if each qubit is independent and identically affected by an or a error with probability , then the decoding algorithm corrects the error with high probability.

5.1 Decoding -errors

As mentioned, the algorithm successively decodes -errors then -errors. It succeeds if it recovers the right error patterns, up to some element of the stabilizer group. We first consider -errors. A -decoder takes as input a syndrome on -type stabilizers and outputs a set of -errors consistent with this syndrome. For golden codes, -type stabilizers are defined by edges in the tessellation. The error pattern is by definition the set of 2-faces corresponding to qubits having a -error. The syndrome is the boundary of the error pattern. Since every boundary is a cycle, the syndrome consists of several loops of edges.

Definition 19.

A path of edges from vertex to vertex is minimal if no other path of edges from vertex to vertex is shorter.

The -decoder follows from following lemma:

Lemma 20.

In the tessellation, every loop of edges has at least one subpath of length at most 8 which is not minimal.

Lemma 20 is proven in the appendix.

With Lemma 20 at hand, it is now easy to design a local decoder:

  • From every edge of the syndrome, explore every path of edges in the syndrome of length at most 8.

  • If such a path is not minimal, flip qubits to decrease its length.

  • Iterate, until no non-minimal path of length at most 8 can be found.

While the complexity of the -decoder appears at first sight to be quadratic in the size of the syndrome, it can be made linear if one only explores in the step paths that were not already explored during the -th round of the algorithm. Moreover, as long as the error weight is below the injectivity radius of the manifold, or if the error consists of many such small connected components, then the syndrome weight is proportional to the error weight. In other words, the decoding algorithm has a complexity linear in the error weight.

5.2 Decoding -errors

We now turn our attention to decoding -errors. An -decoder takes as input a syndrome on -type stabilizers and outputs a set of -errors consistent with this syndrome. For golden codes, -type stabilizers are defined by polyhedrons (3-faces) in the tessellation. It is more convenient for us to work with edges than with polyhedrons. We therefore consider the dual tessellation. With this point of view, -type stabilizers are defined by edges in the dual tessellation.

The -decoder follows from the following lemma:

Lemma 21.

In the tessellation, every loop of edges admits at least one subpath incident to a single 4-face and which is not minimal.

Lemma 21 is proven in the appendix.

With Lemma 21 at hand, it is now easy to design a local decoder:

  • From every edge of the syndrome, explore every path of edges in the syndrome incident to a single 4-face.

  • If such a path is not minimal, flip qubits to decrease its length.

  • Iterate, until no non-minimal path incident to a single 4-face can be found.

The complexity of this -decoder is linear in the size of the error for the same reason as the -decoder.

6 Conclusion and perspectives

In this work, we have presented a variant of the quantum LDPC code family due to Guth and Lubotzky. Like theirs, our family is also obtained by considering tessellations of hyperbolic 4-space, but the crucial new feature of our construction is that the tessellation is regular. We then exploit this regularity to design an efficient and explicit decoding algorithm that provably corrects arbitrary errors of weight and decodes with high probability random independent and identically distributed errors provided the error rate is below some constant threshold.

We note that both the dimension 4 and hyperbolicity present advantages for decoding. Placing the qubits on dimension 2 yields syndromes which are cycles of dimension or codimension 1 and a decoder should simply try to shorten such cycles, which can be done efficiently by means of a local algorithm as we demonstrated. This algorithm is also more efficient in hyperbolic space since the syndrome weight increases linearly with the error weight (for small errors). This is arguably simpler than pairing vertices as required in surface codes. Another advantage of 1-dimensional syndromes is that they contain redundant information, which should be helpful when considering more realistic scenarios where syndrome measurements are not assumed to be ideal.

Future work should focus on simulating the performance of hyperbolic 4-dimensional codes with respect to different error models. Although the code family based on quotienting by arithmetic subgroups is arguably out of reach for simulations, it will be interesting to consider quotienting by different normal subgroups, yielding codes of more reasonable size. While the bounds on the minimum distance would not apply anymore in that case, we expect the behaviour of the decoding algorithm to be essentially identical for the usual error models.

Acknowledgements

We thank Gilles Zémor for introducing the authors to the construction of [GL14] and for fruitful discussions on tessellations and homological codes. We thank Nicolas Bergeron for mentioning the arithmeticity of the discrete isometry groups corresponding to regular tessellations of hyperbolic 4-space. We also thank Benjamin Audoux, Alain Couvreur, Antoine Grospellier, Anirudh Krishna and Jean-Pierre Tillich for useful discussions on quantum codes.

7 Appendix

7.1 Proof of the -decoder Lemma

Before proving Lemma 20, we first establish a 2-dimensional version of it. Even though this 2-dimensional version is irrelevant to decoding homological quantum codes, it allows us to illustrate the main ideas with figures and may help the reader understand the key role of hyperbolicity in Lemma 20.

Lemma 22.

In the tessellation of hyperbolic plane, every loop of edges has at least one subpath of length at most 4 which is not minimal.

Equivalently, in the tessellation of hyperbolic plane every loop of edges admits at least one of the two subpaths depicted on Figure 1.

(a) Subpath 1
(b) Subpath 2
(c) By flipping one qubit, we replace the red edges of the syndrome by the green one and thus decrease the syndrome weight.
(d) Flipping two qubits, we replace the red edges of the syndrome by the green ones and thus decrease the syndrome weight.
Figure 1: In the tessellation of hyperbolic plane, every loop of edges contains one of the two subpaths in red. These two subpaths are not minimal: they can be replaced by the shorter green ones by flipping one or two qubits. (source for image: [tes])
Proof.

It is sufficient to prove it on a single loop of edges of the tessellation of hyperbolic plane. We choose an arbitrary orientation on this loop. An edge is written if it is oriented from to . To each edge we assign a cone defined as the set of points of hyperbolic plane closer to than to any other edge incident to . The cone divides the hyperbolic plane in two regions: the outside of the cone and the inside of the cone.

We suppose by contradiction that there exists a loop of edges in the tessellation of hyperbolic plane such that every subpath of of length at most 4 is minimal. Figure 2 shows by an exhaustive search that for any edge , there exists in such that contains . By immediate induction, it is then possible to construct a sequence of edges in such that implies that contains . This contradicts the fact that is a loop. ∎

(a) Subpath 3
(b) Subpath 4
(c) Subpath 5
(d) Subpath 6
(e) Subpath 7
(f) Subpath 8
Figure 2: The dark blue cone assigned to the last edge of the path contains the light blue cone assigned to the first edge of the path. Every minimal path of length 4 contains one of these six subpaths (or a subpath symmetric to it). Therefore if every length 4 subpath is minimal, it is impossible to form a loop. (source for image: [tes])
Proof of Lemma 20.

It follows the exact same line. To each edge , we assign a cone defined as the set of points of hyperbolic 4-space closer to than to any other edge incident to . The exhaustive search is done numerically on the edge graph of the tessellation. We find that every minimal path of length 8 contains a subpath such that the cone assigned to its last edge contains the cone assigned to its first edge. Therefore in order to form a loop, at least one subpath of length at most 8 has to not be minimal. The decoder consists in flipping qubits in order to shorten this subpath. ∎

7.2 Proof of the -decoder Lemma

Before proving Lemma 21 we prove a 2-dimensional version of it. Even though the 2-dimensional version is irrelevant to decoding homological quantum codes, it allows us to illustrate the main ideas with figures and may help the reader understand the key role of hyperbolicity in Lemma 21.

Lemma 23.

In the tessellation of hyperbolic plane, every loop of edges admits at least one subpath incident to a single pentagon and which is not minimal.

Equivalently, in the tessellation of hyperbolic plane every loop of edges has at least one subpath consisting of three edges incident to the same pentagon. After flipping the qubit corresponding to this pentagon, this subpath of length 3 (or 4) is replaced by a subpath of length 2 (or 1) and thus the syndrome weight is reduced.

(a) In red: initial syndrome
(b) In red: decreased syndrome
Figure 3: Every loop of edges in the tessellation of hyperbolic plane contains a subpath of three edges incident to the same pentagon. Flipping the qubit corresponding to this pentagon reduces the syndrome weight. (source for image: [tes])
Proof.

We consider a loop of edges in the tessellation.
As shown in Figure 3(a), there exists a geodesic line in the tessellation which intersects the loop at two of its vertices and . and define a partition of into two subpaths. We denote these two subpaths by and . Without loss of generality, assume that the geodesic line is extremal with respect to in the sense that every edge in is incident to a pentagon incident to an edge of . This is illustrated in Figure 3(b). Without loss of generality, assume that the edge of incident to does not belong to the extremal geodesic line .
If there exists a pentagon such that every edge in is incident to , then is not minimal because the path in the geodesic line going from to consists of a single edge. It is thus shorter than and Lemma 23 is proven in this case.
If such a pentagon doesn’t exist, we denote by the last vertex of such that every vertex between and in is incident to a single pentagon (see Figure 3(c)). We consider the subpath of going from to . is incident to a single pentagon. It has length 3. We denote by the vertex of at edge-distance 1 from . The path consisting of the edge and the edge has length 2 (see Figure 3(d)). It is shorter than . ∎

(a) The geodesic line intersects the loop at vertices and .
(b) The geodesic line is extremal: every edge of is incident to a pentagon incident to .
(c) Vertx is the last vertex of incident to the green pentagon. We define as the subpath of from to and as the vertex of at distance 1 from .
(d) The path going from to through is shorter than . Path is incident to a single pentagon.
Figure 4: Every loop in the tessellation has a subpath consisting of three edges incident to the same pentagon. is not minimal since it can be replaced by a path of length 2. A similar property holds for loops in the tessellation. (source for image: [tes])

We are now ready to prove Lemma 21.

Proof of Lemma 21.

The proof is very similar to the proof of Lemma 23. We consider a loop of edges in the tessellation.
As shown in Figure 3(a), there exists a geodesic hyperplane in the tessellation which intersects the loop at two of its vertices and . Vertices and define a partition of into two subpaths. We denote these two subpaths by and . Without loss of generality, assume that the geodesic hyperplane is extremal with respect to in the sense that every edge in is incident to a 4-face incident to an edge of . This is illustrated in Figure 3(b). Without loss of generality, assume that the edge of incident to does not belong to the extremal geodesic hyperplane .
If there exists a 4-face such that every edge in is incident to , then is not minimal. Indeed the path in the geodesic hyperplane going from to is shorter than and Lemma 21 is proven in this case.
If such a 4-face doesn’t exist, we denote by the last vertex of such that every vertex between and in is incident to a single 4-face (see Figure 3(c)). We consider the subpath of going from to . is incident to a single 4-face. We denote by the vertex of at edge-distance 1 from . We define as one of the shortest path in going from to concatenated with the single edge path going from to (see Figure 3(d)). An exhaustive search on the 1-skeleton of a 120-cell shows that is always shorter than . ∎

References