PreprintPDF Available
Preprints and early-stage research may not have been peer reviewed yet.

Abstract and Figures

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.
Content may be subject to copyright.
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 )p,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 =p,q
ap,q
,(2)
where, p,q = 2 cosh1(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 p,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=p,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 p,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=p,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
p,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 p10 are shown
in Table II. For 8 p11, 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(p2)/2)/cosh(π/p) for qmax (p).
with five tiles around a vertex. For tiles with more sides
12 p30 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 p7
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
p7. The maximum qmax(p) clearly grows quickly al-
ready for the small pin Table III. Meanwhile the mini-
mum qmin(p) = 1+2+4/(p2)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(p2)/2 + ..., where, the omitted terms are
sub-leading in 1/q. The result is the asymptoticaly ex-
act estimate q1(p) = πcosh(π(p2)/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 p30 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)p,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 p30,
requires us to search the finite number of codes with
qqopt(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 p30
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) 3,7= 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 p3
and p4 dangling edges while for codes grown with more
than three tiles meeting around a vertex the two types of
tiles have p3 and p2 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, ~uis
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) = p3 (p3)(q3) 1
p2 (p2)(q3) 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)=
(p3, 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
1q4!(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) = 1p6
1p5!(7)
Here, the basis is such that the tile vector is ~
t= (1,1)T
and the edge vector is ~eτ C (p,3) = (p4, p3)T.We notice
that the determinant of these matrices is one while the
trace is 2γ(p, q) = (p2)(q2) 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)21,(8)
where, γ(p, q) = 1
2trMτ C(p,q)= ((p2)(q2) 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 p7, we find the
p-gon code with three tiles around a vertex grows slowest
giving growth rate bound λτ C (p,3). Finally, for q7 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) = (p6, 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 = ((p3)(q3) 1, λτ C(p,q)(p3)).
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 )) = ((p3) + (p2))/((p
3)2+ (p2)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 (p2)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)
p,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 p,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 p3. 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/(p2)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”).
[1] F. Pastawski, B. Yoshida, D. Harlow, and J. Preskill,
Holographic quantum error-correcting codes: toy
models for the bulk/boundary correspondence,
Journal of High Energy Physics 2015, 149 (2015),
hep-th/1503.06237.
[2] A. Jahn, M. Gluza, F. Pastawski, and J. Eisert, Hologra-
phy and criticality in matchgate tensor networks, Science
Advances 5(2019).
[3] L. Boyle, B. Dickens, and F. Flicker, Conformal qua-
sicrystals and holography (2018), hep-th/1805.02665.
[4] W. Thurston, Three-dimensional geometry and topology
(Princeton University Press, Princeton, NJ, 1997) pp. 55–
85.
[5] A. J. Koll´ar, M. Fitzpatrick, and A. A. Houck, Hy-
perbolic lattices in circuit quantum electrodynamics,
Nature 571, 45 (2019), quant-ph/1802.09549.
[6] P. W. Shor, Fault-tolerant quantum computation, in
Proc. 37th Conf. on Foundations of Comp. Sci. (1996)
pp. 56–65, quant-ph/960511.
[7] A. R. Calderbank and P. W. Shor, Good quantum error-
correcting codes exist, Phys. Rev. A 54, 1098 (1996).
[8] A. Steane, Multiple-particle interfer-
7
ence and quantum error correction,
Proc. Roy. Soc. (London) A 452, 2551 (1996),
quant-ph/9601029.
[9] D. Aharonov and M. Ben-Or, Fault-tolerant
quantum computation with constant error, in
Proc. 29th ACM Symp. on Theory of Computing,
STOC ’97 (ACM, New York, NY, USA, 1997) pp.
176–188.
[10] R. Cleve, D. Gottesman, and H.-K. Lo, How to share a
quantum secret, Phys. Rev. Lett. 83, 648 (1999).
[11] M. D. Reed, L. DiCarlo, S. E. Nigg, L. Sun, L. Frun-
zio, S. M. Girvin, and R. J. Schoelkopf, Realization of
three-qubit quantum error correction with superconduct-
ing circuits, Nature 482, 382 (2012).
[12] W. Helwig, W. Cui, J. I. Latorre, A. Riera, and H.-K.
Lo, Absolute maximal entanglement and quantum secret
sharing, Phys. Rev. A 86, 052335 (2012).
[13] W. Helwig, Absolutely maximally entangled qudit graph
states (2013), quant-ph/13062879.
[14] R. J. Harris, N. A. McMahon, G. K. Brennen, and T. M.
Stace, Calderbank-shor-steane holographic quantum
error-correcting codes, Phys. Rev. A 98, 052301 (2018).
[15] J. D. Brown and M. Hennaux, Central charges in the
canonical realization of asymptotic symmetries: An ex-
ample from three-dimensional gravity, Commun. Math.
Phys. 104, 207 (1986).
[16] E. Witten, 2+1-dimensional gravity as an exactly soluble
system, Nucl. Phys. B 311, 46 (1988).
[17] A. Strominger, Black hole entropy from near-horizon mi-
crostates, JHEP 9802, 009, hep-th/9712251.
[18] E. Witten, Anti de sitter space and holography,
Adv.Theor.Math.Phys. 2, 253 (1998).
[19] G. Vidal, Class of quantum many-body
states that can be efficiently simulated,
Phys. Rev. Lett. 101, 110501 (2008).
[20] X.-L. Qi, Exact holographic mapping and emergent
space-time geometry (2013), hep-th/1309.6282.
[21] B. Yoshida, Information storage capacity of discrete spin
systems, Ann. Phys. 338, 134 (2013).
[22] J. I. Latorre and G. Sierra, Holographic codes (2015),
quant-ph/150206618.
[23] F. Pastawski and J. Preskill, Code properties from holo-
graphic geometries, Phys. Rev. X 7, 021022 (2017).
[24] To see the isometry, split the perfect tensor index
list into two pieces A={ai1, ai2,...aik}and B=
{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
the perfect tensor as a linear map TAB from the Hilbert
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 )TAB=
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.
ResearchGate has not been able to resolve any citations for this publication.
Article
Full-text available
After close to two decades of research and development, superconducting circuits have emerged as a rich platform for both quantum computation and quantum simulation. Lattices of superconducting coplanar waveguide (CPW) resonators have been shown to produce artificial materials for microwave photons, where weak interactions can be introduced either via non-linear resonator materials or strong interactions via qubit-resonator coupling. Here, we introduce a technique using networks of CPW resonators to create a new class of materials which constitute regular lattices in an effective hyperbolic space with constant negative curvature. We show numerical simulations of a class of hyperbolic analogs of the kagome lattice which show unusual densities of states with a spectrally-isolated degenerate flat band. We also present an experimental realization of one of these lattices, exhibiting the aforementioned band structure. This paper represents the first step towards on-chip quantum simulation of materials science and interacting particles in curved space.
Article
Full-text available
The AdS/CFT correspondence conjectures a holographic duality between gravity in a bulk space and a critical quantum field theory on its boundary. Tensor networks have come to provide toy models to understand such bulk-boundary correspondences, shedding light on connections between geometry and entanglement. We introduce a versatile and efficient framework for studying tensor networks, extending previous tools for Gaussian matchgate tensors in 1+1 dimensions. Using regular bulk tilings, we show that the critical Ising theory can be realized on the boundary of both flat and hyperbolic bulk lattices. Within our framework, we also produce translation-invariant critical states by an efficiently contractible network dual to the multi-scale entanglement renormalization ansatz. Furthermore, we explore the correlation structure of states emerging in holographic quantum error correction. We hope that our work will stimulate a comprehensive study of tensor-network models capturing bulk-boundary correspondences.
Article
Full-text available
Almheiri, Dong, and Harlow [hep-th/1411.7041] proposed a highly illuminating connection between the AdS/CFT holographic correspondence and operator algebra quantum error correction (OAQEC). Here we explore this connection further. We derive some general results about OAQEC, as well as results that apply specifically to quantum codes which admit a holographic interpretation. We introduce a new quantity called `price', which characterizes the support of a protected logical system, and find constraints on the price and the distance for logical subalgebras of quantum codes. We show that holographic codes defined on bulk manifolds with asymptotically negative curvature exhibit `uberholography', meaning that a bulk logical algebra can be supported on a boundary region with a fractal structure. We argue that, for holographic codes defined on bulk manifolds with asymptotically flat or positive curvature, the boundary physics must be highly nonlocal, an observation with potential implications for black holes and for quantum gravity in AdS space at distance scales small compared to the AdS curvature radius.
Article
Full-text available
We study the existence of absolutely maximally entangled (AME) states in quantum mechanics and its applications to quantum information. AME states are characterized by being maximally entangled for all bipartitions of the system and exhibit genuine multipartite entanglement. With such states, we present a novel parallel teleportation protocol which teleports multiple quantum states between groups of senders and receivers. The notable features of this protocol are that (i) the partition into senders and receivers can be chosen after the state has been distributed, and (ii) one group has to perform joint quantum operations while the parties of the other group only have to act locally on their system. We also prove the equivalence between pure state quantum secret sharing schemes and AME states with an even number of parties. This equivalence implies the existence of AME states for an arbitrary number of parties based on known results about the existence of quantum secret sharing schemes.
Article
Full-text available
Quantum computers could be used to solve certain problems exponentially faster than classical computers, but are challenging to build because of their increased susceptibility to errors. However, it is possible to detect and correct errors without destroying coherence, by using quantum error correcting codes. The simplest of these are three-quantum-bit (three-qubit) codes, which map a one-qubit state to an entangled three-qubit state; they can correct any single phase-flip or bit-flip error on one of the three qubits, depending on the code used. Here we demonstrate such phase- and bit-flip error correcting codes in a superconducting circuit. We encode a quantum state, induce errors on the qubits and decode the error syndrome--a quantum state indicating which error has occurred--by reversing the encoding process. This syndrome is then used as the input to a three-qubit gate that corrects the primary qubit if it was flipped. As the code can recover from a single error on any qubit, the fidelity of this process should decrease only quadratically with error probability. We implement the correcting three-qubit gate (known as a conditional-conditional NOT, or Toffoli, gate) in 63 nanoseconds, using an interaction with the third excited state of a single qubit. We find 85 ± 1 per cent fidelity to the expected classical action of this gate, and 78 ± 1 per cent fidelity to the ideal quantum process matrix. Using this gate, we perform a single pass of both quantum bit- and phase-flip error correction and demonstrate the predicted first-order insensitivity to errors. Concatenation of these two codes in a nine-qubit device would correct arbitrary single-qubit errors. In combination with recent advances in superconducting qubit coherence times, this could lead to scalable quantum technology.
Article
We expand the class of holographic quantum error-correcting codes by developing the notion of block perfect tensors, a wider class that includes previously defined perfect tensors. The relaxation of this constraint opens up a range of other holographic codes. We demonstrate this by introducing the self-dual Calderbank-Shor-Steane (CSS) heptagon holographic code, based on the 7-qubit Steane code. Finally, we show promising thresholds for the erasure channel by applying a straightforward, optimal erasure decoder to the heptagon code and benchmark it against existing holographic codes.
Article
We propose a family of exactly solvable toy models for the AdS/CFT correspondence based on a novel construction of quantum error-correcting codes with a tensor network structure. Our building block is a special type of tensor with maximal entanglement along any bipartition, which gives rise to an exact isometry from bulk operators to boundary operators. The entire tensor network is a quantum error-correcting code, where the bulk and boundary degrees of freedom may be identified as logical and physical degrees of freedom respectively. These models capture key features of entanglement in the AdS/CFT correspondence; in particular, the Ryu-Takayanagi formula and the negativity of tripartite information are obeyed exactly in many cases. That bulk logical operators can be represented on multiple boundary regions mimics the Rindler-wedge reconstruction of boundary operators from bulk operators, realizing explicitly the quantum error-correcting features of AdS/CFT recently proposed by Almheiri et. al in arXiv:1411.7041.
Article
In this paper, we propose an {\it exact holographic mapping} which is a unitary mapping from the Hilbert space of a lattice system in flat space (boundary) to that of another lattice system in one higher dimension (bulk). By defining the distance in the bulk system from two-point correlation functions, we obtain an emergent bulk space-time geometry that is determined by the boundary state and the mapping. As a specific example, we study the exact holographic mapping for (1+1)-dimensional lattice Dirac fermions and explore the emergent bulk geometry corresponding to different boundary states including massless and massive states at zero temperature, and the massless system at finite temperature. We also study two entangled one-dimensional chains and show that the corresponding bulk geometry consists of two asymptotic regions connected by a worm-hole. The quantum quench of the coupled chains is mapped to dynamics of the worm-hole. In the end we discuss the general procedure of applying this approach to interacting systems, and other open questions.
Article
Absolutely maximally entangled (AME) states are multipartite entangled states that are maximally entangled for any possible bipartition. In this paper, we study the description of AME states within the graph state formalism. The graphical representation provides an intuitive framework to visualize the entanglement in graph states, which makes them a natural candidate to describe AME states. We show two different methods of determining bipartite entanglement in graph states and use them to define various AME graph states. We further show that AME graph states exist for all number of parties, and that any AME graph states shared between an even number of parties can be used to describe quantum secret sharing schemes with a threshold or ramp access structure directly within the graph states formalism.
Article
By disentangling the hamiltonian constraint equations, 2 + 1 dimensional gravity (with or without a cosmological constant) is shown to be exactly soluble at the classical and quantum levels. Indeed, it is closely related to Yang-Mills theory with purely the Chern-Simons action, which recently has turned out to define a soluble quantum field theory. 2 + 1 dimensional gravity has a straightforward renormalized perturbation expansion, with vanishing beta function. 2 + 1 dimensional quantum gravity may provide a testing ground for understanding the role of classical singularities in quantum mechanics, may be related to the discrete series of Virasoro representations in 1 + 1 dimensions, and may be a useful tool in studying three-dimensional geometry.