Download Holographic quantum error-correcting code

Holographic quantum error-correcting code
- exactly solvable toy models for the AdS/
CFT correspondence
arXiv:1503.06237
Beni Yoshida (Caltech)
joint work with
Daniel Harlow
Fernando Pastawski
John Preskill
Elementary introduction to
our paper !!
Introduction
I.
Classical
Limit
AdS/CFT correspondence (Maldacena 98)
A. Flattening a hyperbolic spac
(d+1)-dim string theory
I. Classical Lim
B. Rotational symmetry
A. Flattening
bulk
O(x)
(x, r)
boundary
B. Rotational
O(x)
(x, r)
• Powerful framework to study stronglyinteracting systems
• Advanced our understanding of quantum
gravity
d-dim CFT
Holographic entanglement entropy
Ryu-Takayanagi formula (06)
Holographic
entanglement
entropy
the boundary field theory, find
bulk
bulk
To compute entropy
of region A in
minimal
Bulk/Boundary
duality
to
Geometry/Entanglement
duality
•
surface
minimal
bulk
surface
minimal area of the bulk surface
with the same boundary:
To compute entropy of region A in
1
field theory,
find
S (the
A)boundary
=
min
∂m =∂A area(m ) +!
minimal4G
area
N of the bulk surface
with the same boundary:
bulk
Ryu and Takayanagi 2006
1
S ( A) =Recover,min
) +!
∂m =∂A area(m
for
example,
in
1+1
Entanglement
4
G
N
dimensional
conformal field
boundary
theory:
Ryu
and Takayanagi 2006
c
SRecover,
( A( L)) for
= example,
log( L / ain) 1+1
+!
3
boundary
Geometry conformal
(Space-time)
dimensional
field
theory:
c
S ( A( L)) = log( L / a) +!
3
MERA (Vidal 07)
• Powerful numerical method to study strongly-correlated systems.
Entanglement Renormalization and Holography
In AdS/CFT, the emergent
dimension
of space can be
remove short-range
regarded as a
entanglement
renomalization scale.
Entanglement
renormalization, run
backwards, prepares a
region of length L in circuit
depth O(log L).
View the bulk space as a
prescription for building up
the boundary state
MERA = Multiscale entanglement renormalization
(Swingle, 2009). ansatz
AdS/CFT as a tensor network (Swingle 09)
AdS/CFT correspondence can be explained by a tensor network ?
MERA$
AdS$metric$
S(⇢A )
|`A |
Class$of$Quantum$Many6Body$States$That$Can$
Be$Efficiently$Simulated$Guifre$Vidal$(2008)$
Part I : A simple toy model
The bulk-locality paradox
A note on discrete, Abelian and non-Abelian lattice gauge theory, b
Rindler-wedge reconstruction
A bulk operator
can be represented by some integral of local
d reconstruction in the causal wedge. On the left is a spacetime diagram,
to
perturbatively
construct
operators
in
the
CFT
which
obe
boundary
operators
A if and
is contained inside
cetime extent
of the causal
wedge C[A] supported
associated withon
a boundary
sub-only if
thin a boundary time slice
⌃. The point
x lies
within
C[A] and thus
any
motion,
with
the
boundary
conditions
set by the dictionary (
the
causal
wedge
of
A.
reconstructed on A. On the right is a bulk time slice containing x and ⌃,
that
allx can
bulk
interactions are suppressed by inver
y similar to that of ourwill
tensorassume
networks. The
point
simultaneously
wedges, so (x) has multiple
representations
in theset
CFT.
N , which
will also
the AdS radius in Planck units. At lea
p
L results in a straightforward prescription for the CFT
procedure
ure of AdS/CFT, that a bulk local observable can be represented
p
field
we simply
have
operators in multiple
ways, (x);
is illustrated
in figure
11. The idea
L
me CFT subregion A defines a subregion in the bulk, the causal Z
geodesic
y point x 2 C[A], bulk
quantumline
field theory ensures that any bulk
(x) =
dY K(x; Y )O(Y ),
can be represented in the CFT as some nonlocal operator on A. Sd 1 ⇥R
is called the AdS-Rindler reconstruction of the operator [34, 35].
i +distinct
|the integral
i + ·wedges
· ·is over
ulk point x can| lie within
causal
associated
with
where
the conformal
boundary and K(x, Y )
regions, the bulk operator (x) can have distinct representations in
function”. The smearing function obeys the bulk wave equation
rent spatial support.
causal wedge
tooperator
(1.1) corresponding
as we taketoxthe
tobulk
theoperator
boundary. It can be chosen to on
uniqueness of A
the CFT
ed as indicating thatand
(x) Yis aare
logical
operator separated,
preserving a code
spacelike
which we illustrate for AdS3 in t
lbert space of the CFT. This code subspace is protected against
1; the
point the
x isboundary
represented
by
a boundary integral over the g
arts of the boundary are|0̃i
“erased.” |If
operator
corre1̃i
|
i+|
i + ···
Bulk locality paradox
All the bulk operators must correspond to identity operators on the
boundary.
If so, the AdS/CFT correspondence seems boring ...
AdS/CFT is a quantum code ?
Solution: The AdS/CFT correspondence can be viewed as a
quantum error-correcting code.
Lattice gauge theory
Lattice Lattice
gauge theory
gauge theory
Beni Yoshida
Beni Yoshida
Beni Yoshida
(Dated: April 12, 2015)
(Dated: April
12, 2015)
(Dated:
April 12, 2015)
note on discrete,
Abelian
andAbelian
non-Abelian
lattice
theory,
based
on Kogut’
A
note gauge
on
discrete,
non-Abelian
lattice
gauge
theory,
based o
n and A
non-Abelian
lattice
theory,
based
onand
Kogut’s
1979gauge
review.
OAB
OBC
OCA
OAB
OABOCA
OBC
OBC
OCA
They are different operators, but act in the same manner in a low
energy subspace.
cf. Quantum secret-sharing code
Entanglement wedge reconstruction
Operators in the entanglement wedge can be reconstructed (?)
* Entanglement wedge may extend over the horizon (vs firewall).
* Along the spirit of the Ryu-Takayanagi formula.
Holographic code
Minimal model
Let us construct the simplest toy model !
• 5 qubits on the boundary
• 1 qubit on the bulk
Minimal model
Let us construct the simplest toy model !
• 5 qubits on the boundary
• 1 qubit on the bulk
Causal wedge reconstruction implies
A bulk operator must have representations on ABC, BCD, ....
Minimal model
Let us construct the simplest toy model !
• 5 qubits on the boundary
• 1 qubit on the bulk
Entanglement wedge reconstruction implies
A bulk operator must have representations on ABD, BCE, ....
Desired properties
A bulk operator must have representations on any region
with three qubits.
Beni Yoshida
Five qubit code
(Dated: April 14, 2015)
a|0i + b|1i
Abelian and non-Abelian lattice gauge theory, based on Kogut’s 1979 review.
Encode one logical qubit into five physical qubits.
a|0̃i + b|1̃i
input
a|0i + b|1i
a|0̃i + b|1̃i
Pauli X, Z
OAB
OBC
output
OAB
OBC
OCA
logical Pauli X, Z
OCA
Logical Pauli X,Z have representations on any region with
three qubits.
p
L
“Distance 3” quantum
code
Six-qubit tensor
Isometry from the tensor
injecting a Pauli operator
into one tensor leg
pushing into three
tensor legs
X
Y
=
X
Y
The tensor gives rise to an isometry along any bipartition.
Holographic Code
Causal wedge reconstruction
D
Input
A
A
Entanglement wedge reconstruction
D
C
Input
A
B
Part II : Generic properties
Perfect tensors
: (C ) ! (C )
is anA isometry since
B
U (x0 , r, t = 0)U † = O(xoutput
ris0 Nrik+1
= ,...,i
ct. 2n . This map(1.3)
0 , t)
Perfect
state
to(a)orthogonal.
Such/ tensor
an isometry (b)
can be viewed as
the boundary CFT operator
O(x, t) can becode
written
as k -spin input states ar
error-correcting
where
VB
U
A
UB =
=
A
pure
state
with
maximal
entanglement
in
any
bipartition
• O(x, t) = U O0 (t
states
where
k
input
legs
can
be
viewed
as
logical
le
†
= 0)U
(1.4)
constructed from a perfect tensor, will be called a pe
This
contraction
the map
on B
as depicted
in Fig defines
1(b).
A
A
state (2n
spins)
Perfect
tensorBthe
(2n legs)
ator O0 (tPerfect
= 0) depends
only
on O(x, t) where
x is within
EPR pair
the bulk operator (x0 , r, t = 0) can be represented by boundary
(a)
(b)
ausal wedge.
UA
or
B
B
T
A
T
U=B =
T
VB
B
UB
duality
A
B
A
B
A
Figure
1. (a)
DualityConsider
of unitarya operators
a perfect state. (b) Dist
troduce the notion of
perfect
tensors.
system ofin 2n
number of states per spin. A wavefunction | i is said to be a
We then introduce the notion of perfect tensors. A perfect stat
duced density matrix ⇢A for all A such that |A|  n is maximally
by a tensor as follows:
⇢A / I A
for all |A|  n
v 1 X
v 1 2. X
v(a)
1 Perfect state. (b)
Figure
X
| i=
i1 =0 i2 =0
· · · (2.1)Ti1 i2 ...in |i1 i2 . . . in i
in =0
For
readers
from
quantum
information
science,
it
y matrix on A. So, for any bipartition, the entanglement entropy
Duality
unitary
a dual
unitaryofoperator
I ⌦operators
VB that acts only on B such t
• Given UA, there always exists VB such that UA ⌦ I| i = I ⌦ VB | i.
This duality is often referred to as a gate teleportation wh
state teleportation with EPR pairs.
This
duality between
“gate
teleportation”
VB
UA
=
Choi-Jamiolkowski isomorphism where a state is interpret
An important consequence of the duality is that, in an
B number
A
A a maximal
of EPRB pairs. Here an EPR pair refe
Pv 1
EPR pair
1
two-spin state: |EP Ri = pv j=0 |ji ⌦ |ji. Consider a pe
in a bipartition into A and B such that |A|  n. Then t
will lead to bulk/
UA
UB =
T
transformation
T UB which
VB acts exclusively on B such t
= I⌦
boundary duality...
duality
of |A| decoupled
copies of EPR pairs supported over spin
B
A decoupledB spinsA in B. Namely,
| i = |EP Ri
⌦|A|
⌦ |0i
⌦2n 2|A|
Distillation of EPR pairs
UA
VB
=
• By applying a unitary only on B, EPR pairs can be distilled
B
A
A
B
EPR pair
only on B !
UB =
B
A
T
B
A
UB
distillation
will lead to the RT
formula...
Almost perfect tensors
• Perfect tensors are very rare !
• For qubits, there are only 2-leg and 6-leg perfect tensors.
• In general, to increase the number of legs, we need to
increase the spin (number of internal states).
• But a random tensor does a similar job !
• Take a random quantum state and construct a tensor
• It is almost maximally entangled along any cut (canonical
typicality, i.e. Page’s theorem)
Holographic state
Holographic quantum state / code
physical legs
(a)
(b)
logical legs
holographic
state White dots representholographic
Figure 3. (a) The
holographic state.
physical legs. code
(b) The holographic
code. Red dots connected by dotted lines represent logical legs.
u-Takayanagi formula exactly.
Ryu-Takayanagi
formula,
it’s exact
sult 1. Consider
a wavefunction
| i characterized
by !hyperboli
sors. Then its entanglement entropy SA on a connected region A
re
[Claim] Entanglement entropy for A (connected region) is equal to the
geodesic length.
SA = log2 v ·
spin dim
A
A
length
is theAdS$metric$
length of the geodesic line of A.
In fact, the above result holds
S(⇢A )for any tiling of a space with nonp
ong as the distance functions have no local minima.
To obtain the result,
|`A | we use the correspondence between local m
ns of EPR pairs. Let us first consider a wavefunction which is obtain
perfect tensors as depicted in Fig. 6(a). By applying local unitar
lusively acting on A, one can decouple spins which are not entan
ill EPR pairs bridging across A and B. In Fig. 6(a), we first dis
lying some appropriate unitary transformation on spins in A wh
Geodesic line from local moves
Imagine an “algorithm” to find a geodesic line by local updates.
local moves
B
geodesic
local move
A
(Even-leg hyperbolic space does not have a local minima).
Local move = distillation of EPR pairs
Fig. 6(b). In general, local moves correspond to some distillation processes sin
moves always bring a line over the tensor so that it reduces the number of cuts
seen,
in a distill
perfectEPR
statepairs
in a and
bipartition
into“junks”.
A and B such that |A|  n,
Local
moves
decouple
•have
distill |A| copies of EPR pairs by applying local unitary transformations only
(a)
distillation of
EPR pairs
B
A
(b)
B
A
B
A
B
local move
A
Figure 6. (a) Distillation of two EPR pairs. (b) The corresponding local mov
entangled with A as depicted in Fig. 7. This leaves physical spins which are
along the geodesic. We then apply similar procedure on A to obtain multiple
map
entanglement
irs bridging AGeometric
and B. This leaves
pairs of
of physical
spins which are sitting along
desic. The number of EPR pairs is exactly the number of cuts made by the
, i.e. •the
length line
of the
geodesic.
is possible
lines of A and B
Geodesic
between
twoItregions
A andthat
B =geodesic
EPR pairs
er.
Long-range
B
B
A
local unitary on B
Short-range
A
local unitary on A
7. The geometric map of entanglement. White dots represent physical spins after
tary transformations on A and B.
depicted in Fig. 4(a). For all the open legs on the bulk, w
egs
ofatwo
holographic
states
along
somestates,
circlesbetween
atresulting
the sames
sents
pure
state
i which
has
entanglement
states.
Since
we| have
injected
mixed
the
Aeach
hole
and
wormhole
penWe
legs
from
copy
of holographic
together.
Thir
ing
out
the
second
system
call
such
ablack
state
simply
a black
holestates
since its
properties
eometry
with The
two cutting
AdS spaces
areviewed
connected
a worm
black hole.
circlewhich
may be
as anby
event
hor
T
r
(|
ih
|)
=
⇢
(3.1)
As
a
mixed
state
As
a
purified
state
esulting
tensor
network
represents
a
pure
state
|
i
which has
B
any information inside it.
wo systems
A and
B.view
By tracing
outhole
theas
second
system of tw
One
may
also
the
black
a
contraction
(a)
(b)
on A which is identical to the one obtained by injecting
sider two identical copies of holographic states denoted by A
n legs
of
a
single
copy
of
holographic
state.
T
r
(|
ih
|)
=
⇢
B
legs of two holographic states
along some circles at the sa
A
horizon
open legs from each copy of holographic states together.
ne obtains the mixed
state ⇢ on A which is identical to the on
black hole
wormhole
geometry with two AdS spaces
which are connected
by a w
B
maximally mixed states to open legs of a single copy of hologra
resulting tensor network represents a pure state | i which
two systems A and B. By tracing out the second system
–8–
T rB (| ih |) = ⇢
Figure
4. Wormhole
geometry for a black hole.
inject maximally
mixed
state
one obtains the mixed state ⇢ on A which is identical to the
Entanglement in a black hole
• EPR pairs along the wormhole (ER=EPR ?)
• RT formula with a black hole
A
B1
B2
re 8. The geometric map of entanglement for the wormhole geometry (color onl
ws represent local moves and distillation of EPR pairs.
Holographic state
(multi-partition)
rmula
servescircuits,
as a generic
upper
bound on does
the bipartite
entanglement
of athe
wavefunc
om
MERA
so the
wavefunction
not necessarily
saturates
RT
ising itfrom
MERA
circuits,
so the wavefunction
does
notapproximately
necessarily saturates
Yet,
is not
difficult
to construct
a wavefunction
that
satisfiesthe
rmula.
Yet, it
is not
difficult
to construct
a wavefunction
that approximately
Takayanagi
formula
for bipartite
entanglement
along specific
cuts by somesati
Multi-partite
entanglement
he
Ryu-Takayanagi
entanglement
specific cuts
by s
ensor
network. For formula
instance,for
onebipartite
may achieve
this goal along
by distributing
EPR
ERA•tensor
network.
For
instance,
one may
achieve
by ,where
distributing
di↵erent
scales into
acreate
scale
invariant
way so
that Sthis
log(L)
It length
is not
difficult
a wavefunction
with
but... L is E
A /goal
airs
length scales
in a scale
way so that
SA /tensors
log(L)with
where
h of at
A.di↵erent
Such distributions
of EPR
pairsinvariant
may be possible
by using
he
length
A. Such
of EPR pairs
may be
possible
by using
tensors
ctures
or of
placing
thedistributions
so-called disentanglers
at each
layer
of a MERA
circuit.
eg)do
distribute
in a circ
ee structures
or placing
the so-called
disentanglers
at
layer
of
MERA
ever
such simple
constructions
with EPR
pairs
noteach
giveEPR
rise pairs
to awavefuncHowevermulti-partite
such simple entanglement
constructions which
with EPR
pairs
give rise to
wavef
treedo not
h genuine
atensor
holographic
wavefunction
must
ons
genuine
multi-partite
entanglement
a holographic
wavefunction
To with
see this,
let us
recall a result
by Patrickwhich
Hayden
and his friends
on the m
ossess.
see this,
let us recallunder
a result
Patrick Hayden
and his friends
y of the To
tripartite
entanglement
theby
gauge/gravity
correspondence.
For on
egativity of thesubsets
tripartite
entanglement
the gauge/gravity
correspondence.
n-overlapping
of spins
A, B, C, under
the tripartite
entanglement
is defined
hree
non-overlapping subsets of spins A, B, C, the tripartite entanglement is defi
s
follows
• Negativity of tripartite information (any “holographic state”)
I(A, B, C) = SA + SB + SC SAB SAC SBC + SABC .
(1.1)
I(A, B, C) = SA + SB + SC SAB SAC SBC + SABC .
(
neric quantum state, this quantity can be negative, positive or zero. Let us
-- EPR pair,
then
I(A,B,C)=0.
or
a
generic
quantum
state,
this
can be
negative,
positive
zero. Le
ider a wavefunction | i which is quantity
dual to some
classical
gravity
with or
negative
owand
consider
a wavefunction
i which
isA,
dual
to some
classical
gravity
with nega
--three
GHZconnected
state, |then
I(A,B,C)>0.
e,
consider
regions
B, C.
We assume
that
a wavefuncurvature,
andformula,
considermeaning
three connected
A, B, C.entropy
We assume
that a wavef
fies the RT
that theregions
entanglement
is proportional
Multi-partition and residual regions
A
|B|
B
large
C
A
B
C
A
B
small
C
Multi-partition and residual regions
A
B
C
A
B
C
A
B
D
RC
RA
A
|B|
C
B
large
small
C
Multi-partition and residual regions
A
B
C
A
B
C
A
D
A
|B|
RD
C
B
large
C
D
RC
RA
B
RB
A
B
small
C
Multi-partition and residual regions
A
B
C
A
D
RC
RA
A
|B|
C
B
large
B
C
A
B
D
D
R
RD
A
C
B
C
RB
A
B
small
C
Multi-partition and residual regions
• Identified segment of geodesic lines = EPR pairs
• Residual regions = Multipartite entanglement ?
A
B
C
A
D
RC
RA
A
|B|
C
B
large
B
C
A
B
D
D
R
RD
A
C
B
C
RB
A
B
small
C
Residual regions in holographic state
(a)
(b)
RC
C
C
R
RA
B
B
A
A
D
C
C
RC
D
R
RA
B
B
A
A
Residual regions in holographic state
(a)
(a)
(b)
(b)
RC
RC
C
C
C
C
R
R
RA
RA
B
B
B
B
A
A
D
D
RA
RA
A
A
A
A
C
C
C
C
RC
RC
D
D
R
R
BB
B
B
A
A
s upper bounded by the minimal number of cuts one needs to make to connect t
dpoints of A, i.e. the length of the geodesic line of A. Here, the Ryu-Takayan
mula serves as a generic upper bound on the bipartite entanglement of a wavefunct
sing from MERA circuits, so the wavefunction does not necessarily saturates the
mula. Yet, it is not difficult to construct a wavefunction that approximately satis
Perfect tensor (state) is the key for negative tripartite
•
Ryu-Takayanagi formula for bipartite entanglement along specific cuts by so
entanglement !
ERA tensor network. For instance, one may achieve this goal by distributing E
Split invariant
2n-perfectway
state
rs at di↵erent length scales in a-- scale
so into
that four
SA /subsets
log(L) A,
where L
D. pairs may be possible by using tensors w
length of A. Such distributionsB,ofC,EPR
D
e structures or placing the so-called disentanglers at each layer of a MERA circu
However such simple constructions
with 0<
EPR
do not
-- Assume
|A|,pairs
|B|, |C|,
|D| <give
n+1rise to wavefu
ns with genuine multi-partite entanglement which a holographic wavefunction m
sess. To see this, let us recall a result by Patrick Hayden and his friends on
Then the tripartite entanglement is always
ativity of the tripartite entanglement under the gauge/gravity correspondence. F
negative !
ee non-overlapping
subsets
of spins A, B, C, the tripartite entanglement is defin
A
C
B
follows
Negativity of tripartite entanglement
I(A, B, C) = SA + SB + SC
SAB
SAC
SBC + SABC .
(1
a generic quantum state, this quantity can be negative, positive or zero. Let
Summary of the talk
• A toy model of the AdS/CFT correspondence
Causal wedge and entanglement wedge reconstruction
• The RT formula is exact for a single connected region
Negativity of tripartite entanglement
• Toward the AdS/CFT limit ?
• Dynamics, fast scrambling ?
Von Neumann vs Peter Shor
Entanglement entropy (1964)
vs
Quantum'Error'Correc.ng'Codes'(1995)'
Quantum code (1995)