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arXiv:1908.09253v1 [quant-ph] 25 Aug 2019

Holographic Code Rate

Noah Bray-Ali∗

Department of Physics, California State University, Dominguez Hil ls, California 90747 USA

David Chester

Department of Physics and Astronomy, University of California, Los Angeles, California 90095 USA and

Quantum Gravity Research, Los Angeles, California 90290 USA

Dugan Hammock, Marcelo M. Amaral, and Klee Irwin

Quantum Gravity Research, Los Angeles, California 90290 USA

Michael F. Rios

Dyonica, ICMQG, Los Angeles, California 90032 USA

(Dated: August 27, 2019)

Holographic codes grown with perfect tensors on regular hyperbolic tessellations using an inﬂation

rule protect quantum information stored in the bulk from errors on the boundary provided the code

rate is less than one. Hyperbolic geometry bounds the holographic code rate and guarantees quantum

error correction for codes grown with any inﬂation rule on all regular hyperbolic tessellations in a

class whose size grows exponentially with the rank of the perfect tensors for rank ﬁve and higher.

For the tile completion inﬂation rule, holographic triangle codes have code rate more than one but

all others perform quantum error correction.

I. INTRODUCTION

Holographic quantum error-correcting codes[1, 2]

merge quasi-crystals[3] and hyperbolic geometry[4, 5]

with quantum information[6–14] and holography[15–23].

One places rank-(p+ 1) perfect tensors Ta1a2...apap+1 on

the p-sided tiles of a tessellation of the hyperbolic plane

and contracts the tensors along the edges where tiles

meet: this leaves a single “bulk” index uncontracted for

each tile[1]. Starting from some simply connected set of

seed tiles, we grow the holographic code, layer by layer,

using an inﬂation rule[3].

The physical degrees of freedom of the code live on

the boundary of the growing tile set on the quasi-crystal

formed by the dangling edges of the tiles of the last

layer[1]. The logical degrees of freedom of the code live in

the bulk of the tile set on the tiles themselves. The per-

fect tensors map the physical Hilbert space isometrically

to the logical Hilbert space[24].

The quantum error-correcting property of a holo-

graphic code follows from a remarkable fact about hy-

perbolic geometry[25]. For a given growth rule τ(p, q )

on the {p, q}-hyperbolic tessellation with regular p-sided

tiles meeting qaround a vertex, there exists a ﬁnite code

rate

χτ(p,q)= lim

n→∞ Nbulk

Nboundary ,(1)

where, Nbulk is the number of logical degrees of freedom,

Nboundary is the number of physical degrees of freedom,

∗nbrayali1@csudh.edu

p q χτ C (p,q)χp,q χτ C(p,q )/χp,q

3 7 2.236 2.430 0.920

4 5 0.789 0.998 0.790

5 4 0.519 0.676 0.768

7 3 0.447 0.541 0.826

TABLE I. Code rate χτ(p,q )of holographic codes grown with

tile completion on the {p, q}-tiling of the hyperbolic plane by

regular p-gons meeting qaround a vertex together with χp,q

code rate bound from hyperbolic geometry. The code rate is

greater than one for triangle codes, is less than than one for

p-gon codes with pgreater than three, and obeys the bound

for all p.

and nis the number of layers of the code. In particular,

there exist growth rules and tilings such that the code

rate χτ(p,q)is less than one (Table I,Fig. 1, 2). For such

holographic codes, quantum erasures of a non-zero frac-

tion of physical degrees of freedom on the boundary do

not harm the quantum information stored in the logical

degrees of freedom of the bulk[1, 14].

We have found a simple geometric upper bound χp,q on

the code rate for all pand qsuch that 1

p+1

q<1

2, which is

the condition that the tiling lives in the hyperbolic plane.

The result takes the following form:

χp,q =ℓp,q

ap,q

,(2)

where, ℓp,q = 2 cosh−1(cos(π/p)/sin(π/q)) is the length

of the side of the regular p-gon tile and ap,q = 2πp(1/2−

1/p −1/q) is its area[4]. Notice that the condition

1

p+1

q<1

2is necessary and suﬃcient to make the area

of the tiles ap,q positive and the length of their sides ℓp,q

2

real. Remarkably, for all p > 3, there exists an exponen-

tially growing range (qmin (p), qmax(p)) such that, for q

in this range, the simple geometric bound χp,q guaran-

tees quantum error correction for all holographic codes

grown with any growth rule from any simply connected

seed tiles in the hyperbolic plane.

FIG. 1. Quantum error correction threshold (QEC) together

with code rates of holographic codes grown with tile comple-

tion on the {p, q}-tiling of the hyperbolic plane by regular

p-gons meeting qaround a vertex (1/p + 1/q < 1/2) plotted

as a function of the ratio of the growth rate bound set by

hyperbolic geometry to the growth rate for p= 3,4,5,6,7

(Square, Pentagon, Hexagon, Heptagon).

FIG. 2. Quantum error correction threshold (QEC) together

with code rates of holographic codes grown with tile comple-

tion on the {p, q}-tiling of the hyperbolic plane by regular

p-gons meeting qaround a vertex (1/p + 1/q < 1/2) plotted

as a function of the ratio of the growth rate bound set by hy-

perbolic geometry to the growth rate for q= 3,4,5,6,7 (Dual

Triangle, Dual Square, Dual Pentagon, Dual Hexagon, Dual

Heptagon).

Sec. II ﬁnds the code rate bound using hyperbolic ge-

ometry. In Sec. III, we ﬁnd the best code rate bound and

the range of codes for which the bound guarantees quan-

tum error correction. Sec. IV shows the holographic code

rate for the tile completion growth rule obeys the bound

from hyperbolic geometry (Fig. 3,4). Using this growth

rule, we ﬁnd that all codes with p > 3 perform quantum

error correction but the code rate is greater than one for

holographic triangle codes (Fig. 1,2). In Sec. V we pro-

vide details of the quasi-crystalline interpretation which

allows the holographic code rate to be computed for all

regular tilings using the tile completion growth rule and

we make contact with other growth rules and interpreta-

tions introduced in the literature[1, 3].

II. CODE RATE BOUND

The code rate of holographic codes has an upper bound

from hyperbolic geometry. To show this, we use the fol-

lowing result from plane geometry, known as the isoperi-

metric inequality[26]:

A(4π+kA)≤L2,(3)

where, Lis the length of the curve bounding a region of

the plane with area Aand k= 1,0,−1 for the hyper-

bolic, Euclidean, and elliptical plane, respectively. The

name of the inequality refers to the isoperimetric prob-

lem of ﬁnding the curve with given length which bounds

the largest area in the plane. The classical solution is

simply the circle, consisting of all the points with ﬁxed

geodesic distance from a given point in the plane. In-

deed, the circle saturates Eq. (3), as one can show using

plane geometry.[27]

Let us apply this to the holographic code growing on

the regular {p, q}-tiling of the hyperbolic plane. At a

given layer, the code spans Nbulk tiles in the bulk with

Nboundary edges along its boundary. The length of the

boundary L=ℓp,q Nboundary and the area of the bulk A=

ap,qNbulk follow from the fact that the tiles are regular

with each edge having the same length ℓp,q and each tile

having the same area ap,q .

Using the isoperimetric inequality, we ﬁnd the code

rate bound for holographic codes. First, plug the length

L=ℓp,qNboundar y and area A=ap,q Nbulk into the

isoperimetric inequality for the hyperbolic plane. Next,

take the square root of both sides. At last, divide both

sides by Nboundaryap,q to ﬁnd:

Nbulk

Nboundary s1 + 4π

Nbulkap,q

≤ℓp,q

ap,q

.(4)

Finally, to establish the code rate bound (2), we pass

to the limit of inﬁnite layer. The number of tiles Nbulk

goes to inﬁnity in this limit and the left hand side of the

inequality (4) becomes the code rate, establishing the

bound on code rate from hyperbolic geometry.

III. BEST CODE RATE BOUND

The results for the best code rate bound χp,qopt (p)and

optimal number of tiles qopt (p) for 3 ≤p≤10 are shown

in Table II. For 8 ≤p≤11, the optimum bound comes

3

p345678910

qopt(p) 14 9 7 6 6 5 5 5

χp,qopt(p)1.614 0.776 0.500 0.365 0.285 0.233 0.196 0.169

TABLE II. Best code rate bounds χp,qopt(p)and number of

tiles qopt(p) meeting around a vertex for holographic codes

grown with regular p-sided tiles on the hyperbolic plane.

p4 5 6 7

qmin(p) 5 4 4 3

qmax(p) 36 199 952 4468

q1(p) 45 216 971 4491

TABLE III. Boundaries of the range (qmin (p), qmax (p)) within

which the code rate bound χp,q guarantees quantum error cor-

rection for all holographic codes, and the asymptotically exact

estimate q1(p) = πcosh(π(p−2)/2)/cosh(π/p) for qmax (p).

with ﬁve tiles around a vertex. For tiles with more sides

12 ≤p≤30 the optimum number of tiles drops to four

while the best code rate bound falls from χ12,4= 0.132

to χ30,4= 0.043. For tiles with a huge number of sides

p > 30 the optimum bound occurs for three tiles meeting

around a vertex, the least possible, and the optimum

bound χp,3falls to zero O(1/p) inversely with pas pgoes

to inﬁnity.

We note that the best code rate bound is less than one

for all pgreater than three. This shows that any perfect

tensor of rank ﬁve and higher has at least one hyper-

bolic tessellation on which every holographic code grown

with the tensor has code rate less than one. Thus we are

guaranteed by hyperbolic geometry the existence of holo-

graphic codes that perform quantum error-correction,

provided we can construct perfect tensors with rank ﬁve

and higher.

Further, there exists the range (qmin (p), qmax (p))

within which χp,q <1 for all p > 3. In particular, we

report this range together with the code rate bounds χp,q

and an analytic estimate q1(p) for qmax (p) for 4 ≤p≤7

in Table III. Notice that χp,3decreases with pand,

since the bound χ7,3= 0.541 is less than one, it fol-

lows that the minimum qmin(p) must be three for all

p≥7. The maximum qmax(p) clearly grows quickly al-

ready for the small pin Table III. Meanwhile the mini-

mum qmin(p) = 1+⌊2+4/(p−2)⌋is simply the minimum

number of tiles around a vertex allowed by hyperbolic

geometry: the range includes all geometrically allowed

tilings at the lower end.

For large p, we ﬁnd how qmax (p) grows by setting

χp,q = 1 and expanding sin(π/q) = π/q +... and

ap,q =π(p−2)/2 + ..., where, the omitted terms are

sub-leading in 1/q. The result is the asymptoticaly ex-

act estimate q1(p) = πcosh(π(p−2)/2)/cos(π/p) which

grows O(exp(πp/2)) exponentially for large p. We see in

Table 2 that already for small pthe estimate gives a good

FIG. 3. Ratio of the code rate to the code rate bound from

hyperbolic geometry for holographic codes grown with tile

completion on the p, q-tiling of the hyperbolic plane by regu-

lar p-gons meeting qaround a vertex (1/p+1/q < 1/2) plotted

as a function of the ratio of the growth rate bound from hype-

bolic geometric to the growth rate for p= 3,4,5,6,7 (Square,

Pentagon, Hexagon, Heptagon).

approximation to qmax (p).

IV. HOLOGRAPHIC CODE RATE

The code rate of the holographic code grown on any

hyperbolic tessellation with any growth rule obeys the

upper bound from hyperbolic geometry. We check this

fact for the tile completion growth rule for which the code

rate may be computed analytically for all regular tilings

of the hyperbolic plane (Table I and Fig. 3,4). Along the

way, we show that holographic triangle codes have code

rate greater than one and holographic p-gon codes with p

greater than three have code rate less than one (Fig. 1,2).

FIG. 4. Ratio of the code rate to the code rate bound from

hyperbolic geometry for holographic codes grown with tile

completion on the p, q-tiling of the hyperbolic plane by regu-

lar p-gons meeting qaround a vertex (1/p+1/q < 1/2) plotted

as a function of the ratio of the growth rate bound from hy-

perbolic geometric to the growth rate for q= 3,4,5,6,7 (Dual

Triangle, Dual Square, Dual Pentagon, Dual Hexagon, Dual

Heptagon).

4

Here is how to grow the holographic code using the tile

completion growth rule[3]. Start from a simply connected

set of seed tiles which form the zero-th layer. The ﬁrst

layer of the code is made of all the tiles that share a vertex

with a seed tile. Similarly, the second layer consists of all

the tiles which share a vertex with a tile in the ﬁrst layer

but which are not in the seed layer, and so on, layer by

layer.[28]

The basic fact about the code rate χτ C (p,q)for holo-

graphic codes grown with tile completion on the {p, q}-

tiling of the hyperbolic plane is that it decreases as a

function of pat ﬁxed qand as a function of qat ﬁxed

p(Section V). Here we will simply use this fact to show

that the code rate obeys the bound from hyperbolic ge-

ometry and that all codes with p > 3 perform quantum

error correction.

Given that the code rate χτ C (p,q)falls with qat ﬁxed p

and with pat ﬁxed q, it is enough to show the code rates

are less than one for just three codes (Table I): the hep-

tagon code grown on the {7,3}-tiling, the pentagon code

grown on the {5,4}-tiling, and the square code grown on

the {4,5}-tiling. These tilings have code rate less than

one and we turn now to showing that they dominate the

code rate of holographic p-gon codes with pgreater than

three.

To show that all holographic codes grown with tile

completion have code rate bounded by one of these three

codes we begin by analyzing codes with seven or more

edges around a tile. Codes with more than seven edges

around a tile have lower code rate than the heptagon code

with the same number of tiles meeting around a vertex,

since the code rate falls when we increase the number

of sides around a tile while keeping ﬁxed the number of

tiles around a vertex. Similarly, these heptagon codes

have lower code rate than the heptagon code grown with

three tiles around a vertex, since the code rate falls with

increasing number of tiles around a vertex while keeping

ﬁxed the number of edges around a tile. The code rate

χτ C(7,3) = 0.447 is less than one, and so we have shown

that the holographic p-gon codes with pgreater than or

equal to seven all have code rate less than one.

By the same reasoning, we analyze the code rate of

codes with six edges around a tile and show that they are

less than one. The hexagon code with four tiles around

a vertex has the largest code rate among hexagon codes.

Now, this hexagon code has code rate which is lower than

that of the pentagon code grown on the {5,4}-tiling since

that tiling has the same number of tiles around vertex

but a smaller number of edges around a tile. So, for

hexagon codes the code rate must be less than that of the

pentagon code grown on the {5,4}tiling of the hyperbolic

plane.

Finally, we consider the code rate of pentagon and

square codes to show that they too have code rate less

than one. The pentagon code grown on the {5, q}-tiling

with qgreater than four and the square code grown on

the {4, q}-tiling with q greater than ﬁve have lower code

rate than the pentagon code grown on the {5,4}-tiling

and the square code grown on the {4,5}-tiling, respec-

tively. Again, the reasoning is that the code rate falls

when we increase the number of tiles around a vertex for

a ﬁxed number of edges around a tile. The code rates

χτ C(4,5) = 0.789 and χτ C (5,4) = 0.519 are less than one,

and thus we have shown that holographic square, pen-

tagon, and hexagon codes have code rate less than one.

Having shown that holographic codes grown with tile

completion perform quantum error correction on all reg-

ular hyperbolic tessellations with tiles having more than

three sides, we turn to checking the bound on the holo-

graphic code rate from hyperbolic geometry for all such

codes including those grown on triangular tilings. Recall

that there exists a best code rate bound which occurs

when we tune to the optimum number of tiles qopt(p)

meeting around a vertex while keeping the number of

sides paround a tile ﬁxed. Now, the tile completion code

rate falls with qat ﬁxed p, so the ratio of the code rate

to the code rate bound has its maximum when the num-

ber of tiles qmax (p)≤qopt(p) is less than or equal to the

number of tiles that gives the best code rate bound. This

is the key to the analysis as it presents us with a ﬁnite

number of codes to analyze for each p.

We break the analysis of the ratio of the code rate

to the code rate bound further into two cases: codes

with p > 30 for which the best code rate bound oc-

curs for the dual triangle code {p, 3}with three tiles

around each vertex and codes with p≤30 for which

the best code rate bound comes from using a larger, but

still ﬁnite number of tiles around each vertex. For the

p > 30 case, we look at the ratio of the dual trian-

gle code rate to the code rate bound from hyperbolic

geometry. It rises with pand tends to the ﬁnite limit

limp→∞(χτ C (p,3)/χp,3) = π/(3 ln 3) ≈0.953 which is

less than one. Thus the holographic p-gon codes with

pgreater than thirty obey the code rate bound from hy-

perbolic geometry, as they must.

The analysis of the ratio of the code rate to the code

rate bound for codes with number of sides per tile p≤30,

requires us to search the ﬁnite number of codes with

q≤qopt(p) to establish that the code rate bound is

obeyed. In fact, the ratio of code rate to code rate

bound reaches its maximum among codes with p≤30

for the holographic triangle code grown with tile com-

pletion on the {3,7}-tiling. This code has the small-

est growth rate among triangle codes and gives the ratio

χτ C(3,7) /χ3,7= 2.236/2.430 ≈0.920, still less than one

(Fig. 3,4 and Table I). Thus, in all cases the code rate of

holographic codes grown with tile completion on regular

tilings of the hyperbolic plane obeys the code rate bound

from hyperbolic geometry.

V. TILE COMPLETION DETAILS

We give here the details of the code rate of the holo-

graphic code grown with tile completion growth rule on

the hyperbolic plane. After a ﬁnite number of layers, the

5

holographic code takes a quasi-crystal form with two unit

cell types[3]. The cells each have one tile but diﬀer in the

number of dangling edges. For codes grown with three

tiles meeting around a vertex, the two types have p−3

and p−4 dangling edges while for codes grown with more

than three tiles meeting around a vertex the two types of

tiles have p−3 and p−2 dangling edges. In either case,

we ﬁnd a growth rule relating the number of cells of each

type in the current layer with the number of cells of each

type in the next layer.

The growth rule for holographic codes grown with tile

completion on the {p, q}-tiling of the hyperbolic plane

may be put in matrix form Mτ C(p,q )[1, 3]. We compile

the number of cells of each type into the growth vector

~u and ﬁnd that the tile completion growth rule leads

to the linear relationship ~u′=Mτ C(p,q)~u, where, ~u′is

the vector containing the number of cells of each type

after applying the growth rule. Further, after a ﬁnite

number of layers, the matrix becomes square with rank

two, determinant det Mτ C (p,q)equal to one, and integer

matrix elements. In sum, the growth matrix becomes

part of the group SL(2, Z ) of rank two square matrices

with unit determinant and integer coeﬃcients[3].

Explicitly, the growth matrix takes the following form

for pand qboth greater than three:

Mτ C(p>3,q>3) = p−3 (p−3)(q−3) −1

p−2 (p−2)(q−3) −1!(5)

where, we express the growth matrix in the basis in which

the tile vector is~

t= (1,1)Tand the edge vector ~eτ C(p,q)=

(p−3, p −2)T. These vectors list in a column the number

of tiles and dangling edges, respectively, for each type

of cell. Similarly, for triangle codes we ﬁnd the growth

matrix for codes where the number of triangles meeting

around a vertex qis greater than six:

Mτ C(3,q>6) = 0 1

−1q−4!(6)

Here, the basis is such that the tile vector is ~

t= (1,1)T

and the edge vector is ~eτ C (3,q)= (0,1)T.Finally for dual

triangle codes in which three tiles meet around each ver-

tex, we ﬁnd the growth matrix for tiles with the number

of edges around each tile pgreater than six:

Mτ C(p>6,3) = 1p−6

1p−5!(7)

Here, the basis is such that the tile vector is ~

t= (1,1)T

and the edge vector is ~eτ C (p,3) = (p−4, p−3)T.We notice

that the determinant of these matrices is one while the

trace is 2γ(p, q) = (p−2)(q−2) −2 for all pand q.

The growth rate of the holographic code grown with

tile completion on the {p, q}-tiling of the hyperbolic plane

is simply the largest eigenvalue of the growth matrix

Mτ C(p,q)[3]:

λτ C(p,q)=γ(p, q ) + pγ(p, q)2−1,(8)

where, γ(p, q) = 1

2trMτ C(p,q)= ((p−2)(q−2) −2)/2

is half the trace of the growth matrix. Note that the

growth rate is greater than one and is not rational for

1/p + 1/q < 1/2. Further, the growth rate grows with p

at ﬁxed qand with qat ﬁxed pand, in fact, is symmetric

under exchange of pand q:λτ C (p,q)=λτ C(q ,p).

Hyperbolic geometry provides a natural lower bound

on the growth rate of holographic codes grown with tile

completion. The growth rate bound for triangle codes

and dual triangle codes is λτ C(3,7) , while for square codes

and dual square codes it is λτ C (4,5). These bounds follow

from the fact that the growth rate λτ C(p,q )is symmetric

with respect to interchange of pand q, grows with pat

ﬁxed qand with qat ﬁxed p, and that we must have 1/p+

1/q < 1/2 for the tiling to live in the hyperbolic plane.

Similarly pentagon and dual pentagon codes have growth

rate bound λτ C (5,4) and hexagon and dual hexagon codes

have growth rate bound λτ C(6,4) . For p≥7, we ﬁnd the

p-gon code with three tiles around a vertex grows slowest

giving growth rate bound λτ C (p,3). Finally, for q≥7 the

triangle code with qtiles around a vertex gives the growth

rate bound λτ C (3,q).

As the number of layers goes to inﬁnity, the growth vec-

tor then tends to ~uτ C(p,q )the eigenvector of the growth

matrix Mτ C(p,q )with eigenvalue equal to the growth

rate[1]. The number of cells at layer nof each type

forms a geometric sequence with ratio between consec-

utive terms equal to the growth rate and the initial value

tending to the corresponding component of the growth

vector ~uτ C(p,q). Explicitly, we ﬁnd the growth vector is

~uτ C(p,3) = (p−6, λτ C (p,3) −1)for codes where three tiles

meet around each vertex. For triangle codes, the growth

vector becomes ~uτ C (3,q)= (1, λτ C (3,q)). For codes with

more than three tiles meeting around a vertex and more

than three sides per tile, the growth vector takes the form

~up,q = ((p−3)(q−3) −1, λτ C(p,q)−(p−3)).

Finally, the code rate of holographic codes grown with

the tile rule on the {p, q}-tiling of the hyperbolic plane

by regular p-gons meeting qaround a vertex, where, p

and qare numbers such that 1/p + 1/q < 1/2 takes the

form[1]:

χτ C(p,q)=λτ C (p,q)

λτ c(p,q)−1

~uτ C(p,q )·~

t

~uτ C(p,q )·~eτ C (p,q)

.(9)

Here, we have summed the geometric series Nbulk =

Pn

k=1 λk

τ C(p,q)(~uτ C (p,q)·~

t) = λn+1

τ C(p,q)/(λτ C (p,q)−

1)(~uτ C(p,q)·~

t) + ... to obtain the number of tiles in the

bulk at layer n, and omitted terms which are sublead-

ing in powers of the growth rate. Similarly, the number

of dangling edges on the boundary tends to Nboundar y =

λn

τ C(p,q)(~uτ C (p,q)·~eτ C(p,q)), as the number of layers ngoes

to inﬁnity. Taking the ratio Nbulk/Nboundary and passing

to the limit of inﬁnite layer number gives the code rate

in Eq. (9). We note that this agrees with the form of

the code rate found using a diﬀerent growth rule in [1].

In fact, the form applies for any growth rule which gen-

erates a quasi-crystal with a ﬁnite number of cell types

6

after all but a ﬁnite number of layers[3].

Using the explicit form for the code rate, we show that

triangle codes have code rate greater than one. We com-

pute the triangle code rate in terms of the growth rate:

χτ C(3,q)=λτ C(3,q)+ 1

λτ C(3,q)−1.(10)

The fact that the growth rate is greater than one then

gives the result that the code rate for holographic triangle

codes grown with the tile rule must also be greater than

one. Along the way, we notice that the code rate χτ C (3,q)

of triangle codes drops as qincreases, since increasing q

increases the growth rate λτ C(3,q)which drives the code

rate down to one, according to Eq. (10).

Moving on to p-gon codes with pgreater than three,

we show that the code rate χτ C (p,q)drops as pincreases

at ﬁxed qand with increasing qat ﬁxed p. We begin by

noting that the code rate drops to zero as 1/p as pgoes

to inﬁnity at ﬁxed q. To show this, we note the growth

rate goes to inﬁnity in this limit and takes the factor

λτ C(p,q)/(λτ C (p,q)−1) to one. Similarly, the edge vector

becomes parallel to the tile vector ~eτC (p,q)=p~

t, up to

corrections of order 1/p. Thus, the growth vector can-

cels from the ratio (~uτ C (p,q)·~

tτ C(p,q))/(~uτ C (p,q)·~eτ C(p,q))

giving the result that the code rate fall to zero as 1/p as

pgoes to inﬁnity at ﬁxed q.

Continuting the analysis of the code rate for p-gon

codes with pgreater than three, we note that the

code rate falls to the non-zero, q-independent function

(~eτ C(p,q)·~

t)/(~eτ C(p,q)·~eτ C(p,q )) = ((p−3) + (p−2))/((p−

3)2+ (p−2)2) as qgoes to inﬁnity at ﬁxed p. To see

this, we look at how the growth rate λτ C (p,q)grows

in this limit. We ﬁnd it grows to inﬁnity as (p−2)q

so that again the factor λτ C (p,q)/(λτ C (p,q)−1) in the

code rate goes to one. Similarly, the growth vector be-

comes parallel to the edge vector ~uτ C (p,q)=q~eτ C (p,q),

up to corrections of order 1/q. Thus the number of

tiles meeting around a vertex qcancels from the ratio

(~uτ C(p,q)·~

t)/(~uτ C(p,q)·~eτ C (p,q)) giving the result that the

code rate falls to the non-zero, qindependent function

(~eτ C(p,q)·~

t)/(~eτ C(p,q)·~eτ C (p,q)) in the limit that qgoes to

inﬁnity at ﬁxed p. Thus, we have shown that the holo-

graphic code grown with tile completion on the {p, q}-

tiling of the hyperbolic plane by regular p-gons meeting

qaround a vertex has code rate χτ C (p,q)which falls as p

increases at ﬁxed qand as qincreases at ﬁxed p, for all

pand qsuch that 1/p + 1/q < 1/2.

VI. CONCLUSION

Holographic codes living on the tiles of the {p, q}tes-

sellations of the hyperbolic plane with p-sided regular

tiles meeting qaround a vertex have code rate χτ(p,q)≤

ℓp,q/ap,q for a code grown layer by layer using inﬂation

rule τ(p, q). Here the length of the sides of the tiles ℓp,q

and their area ap,q combine to give an upper bound on

the code rate from hyperbolic geometry.

We ﬁnd the tiling with the best code rate bound for

holographic codes on hyperbolic tessellations with reg-

ular p-sided tiles for all p≥3. The best bound falls

quickly with pas does the optimal number qopt (p) of tiles

meeting around each vertex. We show that there exists

the range (qmin(p), qmax(p)) within which all holographic

codes grown on regular {p, q }-tilings of the hyperbolic

plane perform quantum error correction. In particular

we ﬁnd qmin(p) = 1 + ⌊2 + 4/(p−2)⌋includes all tilings

allowed by hyperbolic geometry for all p > 3 and we show

that qmax(p) grows O(exp(πp/2)) exponentially with p.

Finally, the code rate of the holographic codes grown

with tile completion on the {p, q}-tessellation of the hy-

perbolic plane was computed. The triangle codes have

code rate greater than one and cannot perform quantum

error correction while those with tiles having more than

three sides have code rate less than one and can perform

quantum error correction. The computed code rates obey

the upper bound from hyperbolic geometry.

ACKNOWLEDGMENTS

The authors gratefully acknowledge helpful discussions

with Latham Boyle, Alan Fuchs, Ray Aschheim, and

Fang Fang. Funding was provided by Quantum Gravity

Research (Research Proposal: “Quasi-Crystalline Tensor

Networks”).

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{aik+1 , aik+2 ,...aip+ 1 }, with k=|A|less than or equal

to half of p+ 1 the rank of the perfect tensor. Then view

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space HAto the Hilbert space HB. For any such split,

the perfect tensor preserves the overlap between states

up to a proportionality constant: PB(TAB )∗TA′B=

1/dim(HA)δAA′, where, dim(HA) is the dimension of HA.

For example, when the dimension of HAand HBmatch,

then the linear map TAB is proportional to a unitary

operator. For the purposes of holography and quantum

error-correction, the tensors need only preserve the over-

lap between states for index splits which have the “block”

form ij=ij+1 + 1 modulo (p+ 1) for j= 1,2,...,p+ 1.

In fact the holographic code rate for a given hyperbolic

tessellation is the same for codes grown with perfect and

block perfect tensors[14].

[25] The quantum error correction property we have in mind

is simply the existence of a ﬁnite, non-zero threshold for

the fraction of quantum erasures of physical degrees of

freedom below which the logical degrees of freedom can

be recovered, in the limit of large code size. A necessary

condition for this quantum error-correcting property is

that the code rate be less than one. Conversely, numerical

simulations suggest the existence of the non-zero erasure

threshold for holographic codes with code rate less than

one[1, 14].

[26] R. Courant and D. Hilbert, Methods of mathematical

physics (Wiley, New York, 1953) pp. 166–167.

[27] The circle of radius sin the hyperbolic plane has circum-

ference L(s) = 2πsinh(s) = 4πsinh(s/2) cosh(s/2) and

area A(s) = 4πsinh2(s/2).Using the relation between

hyperbolic sine and cosine 1 + sinh2(s/2) = cosh2(s/2),

we ﬁnd the circumference and area of the circle in the hy-

perbolic plane obey the equality L(s)2=A(s)(4π+A(s)),

saturating the bound in Eq. (3).

[28] Tile completion on the {p, q}-tiling gives the same code

as vertex completion on the {q, p}-tiling by duality. The

duality transformation joins the centers of the regular p-

gon tiles with segments. These segments form faces with

qsegments around a face, and pfaces meeting at each

point where the segments end.