arXiv:1908.09253v1 [quant-ph] 25 Aug 2019Holographic Code Rate
Noah Bray-Ali∗
Department of Physics, California State University, Dominguez Hills, 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 q around 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
χτC(p,q)
χp,q
χτC(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 q around 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 p greater than three, and obeys the bound
for all p.
and n is 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 p and q such that 1
p
+ 1
q
< 12 , 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
< 12 is 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 q around 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 q around 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, L is the length of the curve bounding a region of
the plane with area A and 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,qNboundary 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,qNboundary and area A = ap,qNbulk 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
√
1 +
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
p
3
4
5
6
7
8
9
10
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.
p
4
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) = π cosh(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,3 falls to zero O(1/p) inversely with p as p goes
to infinity.
We note that the best code rate bound is less than one
for all p greater 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,3 decreases with p and,
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 p in 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) = π cosh(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 p the 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 q around 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 q around 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 χτC(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 p at fixed q and as a function of q at 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 χτC(p,q) falls with q at fixed p
and with p at 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 p greater 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
χτC(7,3) = 0.447 is less than one, and so we have shown
that the holographic p-gon codes with p greater 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 q greater 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
χτC(4,5) = 0.789 and χτC(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 p around a tile fixed. Now, the tile completion code
rate falls with q at 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 p and tends to the finite limit
limp→∞(χτC(p,3)/χp,3) = π/(3 ln 3) ≈ 0.953 which is
less than one. Thus the holographic p-gon codes with
p greater 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
χτC(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 MτC(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′ = MτC(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 detMτC(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 p and q both greater than three:
MτC(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)T and the edge vector ~eτC(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 q is greater than six:
MτC(3,q>6) =
(
0
1
−1 q − 4
)
(6)
Here, the basis is such that the tile vector is ~t = (1, 1)T
and the edge vector is ~eτC(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 p greater than six:
MτC(p>6,3) =
(
1 p− 6
1 p− 5
)
(7)
Here, the basis is such that the tile vector is ~t = (1, 1)T
and the edge vector is ~eτC(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 p and 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
MτC(p,q)[3]:
λτC(p,q) = γ(p, q) +
√
γ(p, q)2 − 1,
(8)
where, γ(p, q) = 12 trMτC(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 q and with q at fixed p and, in fact, is symmetric
under exchange of p and q: λτC(p,q) = λτC(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 λτC(3,7), while for square codes
and dual square codes it is λτC(4,5). These bounds follow
from the fact that the growth rate λτC(p,q) is symmetric
with respect to interchange of p and q, grows with p at
fixed q and with q at 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 λτC(5,4) and hexagon and dual hexagon codes
have growth rate bound λτC(6,4). For p ≥ 7, we find the
p-gon code with three tiles around a vertex grows slowest
giving growth rate bound λτC(p,3). Finally, for q ≥ 7 the
triangle code with q tiles around a vertex gives the growth
rate bound λτC(3,q).
As the number of layers goes to infinity, the growth vec-
tor then tends to ~uτC(p,q) the eigenvector of the growth
matrix MτC(p,q) with eigenvalue equal to the growth
rate[1]. The number of cells at layer n of 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 ~uτC(p,q). Explicitly, we find the growth vector is
~uτC(p,3) = (p− 6, λτC(p,3) − 1)for codes where three tiles
meet around each vertex. For triangle codes, the growth
vector becomes ~uτC(3,q) = (1, λτC(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, λτC(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 q around a vertex, where, p
and q are numbers such that 1/p+ 1/q < 1/2 takes the
form[1]:
χτC(p,q) =
λτC(p,q)
λτc(p,q) − 1
~uτC(p,q) · ~t
~uτC(p,q) · ~eτC(p,q)
.
(9)
Here, we have summed the geometric series Nbulk =
∑n
k=1 λ
k
τC(p,q)(~uτC(p,q)
· ~t) = λn+1
τC(p,q)/(λτC(p,q) −
1)(~uτC(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 Nboundary =
λn
τC(p,q)(~uτC(p,q) ·~eτC(p,q)), as the number of layers n goes
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:
χτC(3,q) =
λτC(3,q) + 1
λτC(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 χτC(3,q)
of triangle codes drops as q increases, since increasing q
increases the growth rate λτC(3,q) which drives the code
rate down to one, according to Eq. (10).
Moving on to p-gon codes with p greater than three,
we show that the code rate χτC(p,q)drops as p increases
at fixed q and with increasing q at fixed p. We begin by
noting that the code rate drops to zero as 1/p as p goes
to infinity at fixed q. To show this, we note the growth
rate goes to infinity in this limit and takes the factor
λτC(p,q)/(λτC(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 (~uτC(p,q) ·~tτC(p,q))/(~uτC(p,q) ·~eτC(p,q))
giving the result that the code rate fall to zero as 1/p as
p goes to infinity at fixed q.
Continuting the analysis of the code rate for p-gon
codes with p greater than three, we note that the
code rate falls to the non-zero, q-independent function
(~eτC(p,q) ·~t)/(~eτC(p,q) ·~eτC(p,q)) = ((p−3)+(p−2))/((p−
3)2 + (p − 2)2) as q goes to infinity at fixed p. To see
this, we look at how the growth rate λτC(p,q) grows
in this limit. We find it grows to infinity as (p − 2)q
so that again the factor λτC(p,q)/(λτC(p,q) − 1) in the
code rate goes to one. Similarly, the growth vector be-
comes parallel to the edge vector ~uτC(p,q) = q~eτC(p,q),
up to corrections of order 1/q. Thus the number of
tiles meeting around a vertex q cancels from the ratio
(~uτC(p,q) ·~t)/(~uτC(p,q) ·~eτC(p,q)) giving the result that the
code rate falls to the non-zero, q independent function
(~eτC(p,q) ·~t)/(~eτC(p,q) ·~eτC(p,q)) in the limit that q goes 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
q around a vertex has code rate χτC(p,q) which falls as p
increases at fixed q and as q increases at fixed p, for all
p and q such 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 q around 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 p as 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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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
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area A(s) = 4π sinh2(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
q segments around a face, and p faces meeting at each
point where the segments end.