Complexity Equals Action
Adam R. Browna , Daniel A. Robertsb , Leonard Susskinda ,Brian Swinglea , and Ying Zhaoa

aComplexity Equals Action

arXiv:1509.07876
Adam Brown, Daniel Roberts, Leonard Susskind, Brian Swingle, Ying Zhao

Stanford Institute for Theoretical Physics andarXiv:1512.04993
Action, and Black Holes
Department of Physics,Complexity,
Stanford
University,
Adam Brown, Daniel Roberts, Leonard Susskind, Brian Swingle, Ying Zhao

holographic
dualityComplexity Equals Action
Adam R. Browna , Daniel A. Robertsb , Leonard Susskinda ,Brian Swinglea , and Ying Zhaoaa

Stanford Institute for Theoretical Physics andDepartment of Physics, Stanford University,holographic
dualitybulk (AdS)

Complexity Equals Action
Adam R. Browna , Daniel A. Robertsb , Leonard Susskinda ,Brian Swinglea , and Ying Zhaoa

aStanford Institute for Theoretical Physics andDepartment of Physics, Stanford University,

boundary (CFT)

holographic
dualitybulk (AdS)

Complexity Equals Action
Adam R. Browna , Daniel A. Robertsb , Leonard Susskinda ,Brian Swinglea , and Ying ZhaoaaStanford Institute for Theoretical Physics andDepartment of Physics, Stanford University,

boundary (CFT)

holographicduality

bulk (AdS)

Complexity Equals Action
Adam R. Browna , Daniel A. Robertsb , Leonard Susskinda ,aproperty
Brian Swinglea , and Ying ZhaoclassicalaStanford Institute for Theoretical Physics andDepartment of Physics, Stanford University,

boundary (CFT)

holographic
dualitybulk (AdS)

Complexity Equals Action
Adam R. Browna , Daniel A. Robertsb , Leonard Susskinda ,aproperty
Brian Swinglea , and Ying Zhaoclassicalgravitational theoryaStanford Institute for Theoretical Physics andDepartment of Physics, Stanford University,

boundary (CFT)

holographic
dualitybulk (AdS)

Complexity Equals Action
Adam R. Browna , Daniel A. Robertsb , Leonard Susskinda ,aquantum property
property
Brian Swinglea , and Ying Zhaoclassicalgravitational theory

aStanford Institute for Theoretical Physics andDepartment of Physics, Stanford University,

boundary (CFT)

holographic
dualitybulk (AdS)

Complexity Equals Action
Adam R. Browna , Daniel A. Robertsb , Leonard Susskinda ,aquantum property
property
Brian Swinglea , and Ying Zhaoclassicalnon-gravitational theoryagravitational theory

Stanford Institute for Theoretical Physics andDepartment of Physics, Stanford University,

boundary (CFT)

holographicdualitybulk (AdS)

Complexity Equals Action?

Adam R. Browna , Daniel A. Robertsb , Leonard Susskinda ,aquantum property
property
Brian Swinglea , and Ying Zhaoclassicalnon-gravitational theory
agravitational theory

Stanford Institute for Theoretical Physics andDepartment of Physics, Stanford University,Complexity Equals Action
Adam R. Browna , Daniel A. Robertsb , Leonard Susskinda ,aproperty
Brian Swinglea , and Ying Zhaoclassicalgravitational theory

aStanford Institute for Theoretical Physics andDepartment of Physics, Stanford University,
Stanford, California 94305, USAb

Center for Theoretical Physics andDepartment of Physics, Massachusetts Institute of Technology,Cambridge, Massachusetts 02139, USA
AbstractComplexity Equals Action
Adam R. Browna , Daniel A. Robertsb , Leonard Susskinda ,‘size’ofawormhole
aproperty
Brian Swingle , and Ying Zhaoclassicalfor black hole in AdS

agravitational theory

Stanford Institute for Theoretical Physics andDepartment of Physics, Stanford University,
Stanford, California 94305, USAb

Center for Theoretical Physics andDepartment of Physics, Massachusetts Institute of Technology,
Cambridge, Massachusetts 02139, USA
Abstract‘size’ of wormhole
for black hole in AdS

‘size’ of wormhole
for black hole in AdS

time
space

‘size’ of wormhole
for black hole in AdS

outside

time
space

‘size’ of wormhole
for black hole in AdS

insideoutside

time
space

‘size’ of wormhole
for black hole in AdS
r=0

insider=1

outside

time
space

‘size’ of wormhole
for black hole in AdS
r=1

r=0

‘size’ of wormhole
for black hole in AdS
r=1

r=0

‘size’ of wormhole
for black hole in AdS
r=1

r=0

‘size’ of wormhole
for black hole in AdS
r=1r=0‘size’ of wormhole
for black hole in AdS
r=1

r=0

‘other’ r = 1

‘size’ of wormhole
for black hole in AdS
r=1

r=0

‘other’ r = 1

‘size’ of wormhole
for black hole in AdS
r=1

r=0

‘other’ r = 1rh‘size’ of wormhole
for black hole in AdS
time L

r=0

timeR

rh

‘size’ of wormhole
for black hole in AdS
time L

r=0

timeR

rh

‘size’ of wormhole
for black hole in AdS
time L

r=0

timeRrh‘size’ of wormhole
for black hole in AdS
time L

r=0

timeRrh‘size’ of wormhole
for black hole in AdS
time L

r=0

timeR

rh

r=0

‘size’ of wormhole
for black hole in AdS
time L

timeRrhwormhole length ⇠ tL + tR

r=0‘size’ of wormhole
for black hole in AdS
time L

timeRrh

wormhole length ⇠ tL + tR

r=0‘size’ of wormhole
for black hole in AdS
time L

timeR

rh

wormhole length ⇠ tL + tR

r=0

CFTL

CFTRrhwormhole length ⇠ tL + tR

r=0

CFTL

CFTR|TFDi =XieEi /2|Ei iL |Ei iR

wormhole length ⇠ tL + tR

r=0

CFTL

CFTR|TFDi =| (tL , tR )i =X

e

Ei /2eEi /2+iEi (tL +tR )i

Xi|Ei iL |Ei iR|Ei iL |Ei iR

wormhole length ⇠ tL + tR

r=0

CFTL

CFTR|TFDi =| (tL , tR )i =X

e

Ei /2eEi /2+iEi (tL +tR )i

Xi|Ei iL |Ei iR|Ei iL |Ei iR

What is CFT dual to linear growth of wormhole?
wormhole length ⇠ tL + tR

r=0

CFTL

CFTR|TFDi =| (tL , tR )i =X

e

Ei /2eEi /2+iEi (tL +tR )i

X|Ei iL |Ei iRi|Ei iL |Ei iR

What is CFT dual to linear growth of wormhole?
COMPLEXITY?
wormhole length ⇠ tL + tR

COMPLEXITY?
computational gate complexity of a quantum state

COMPLEXITY?
computational gate complexity of a quantum stateXe.g. | (tL , tR )i = e E /2+iE (t +t ) |Ei iL |Ei iR
iiiLRCOMPLEXITY?
computational gate complexity of a quantum stateXe.g. | (tL , tR )i = e E /2+iE (t +t ) |Ei iL |Ei iRii

DEFINITION?

iL

R

COMPLEXITY?
computational gate complexity of a quantum stateXe.g. | (tL , tR )i = e E /2+iE (t +t ) |Ei iL |Ei iRii

LRi

DEFINITION? starting in a reference state
Xe.g. |TFDi = e E /2 |Ei iL |Ei iRii

how many fundamental gatese.g. unitaries each of which
act only on two-qubits
are needed to make the statee.g. to within an accuracy ✏COMPLEXITY?
computational gate complexity of a quantum stateXe.g. | (tL , tR )i = e E /2+iE (t +t ) |Ei iL |Ei iRiiLRi

DEFINITION? starting in a reference state
Xe.g. |TFDi = e E /2 |Ei iL |Ei iRii

how many fundamental gatese.g. unitaries each of which
act only on two-qubits
are needed to make the statee.g. to within an accuracy ✏• can be exponentially large (Hilbert Space is huge)• expected to grow linearly (at early times)

complexity ~ size of wormhole?

complexity ~ size of wormhole?
EVIDENCE:

complexity ~ size of wormhole?
EVIDENCE:• both expected to grow linearly (at early times)• can be exponentially large (max out at same time)complexity ~ size of wormhole?
EVIDENCE:• both expected to grow linearly (at early times)• can be exponentially large (max out at same time)• perturb black hole with boundary operator

complexity ~ size of wormhole?
EVIDENCE:• both expected to grow linearly (at early times)• can be exponentially large (max out at same time)• perturb black hole with boundary operator
in CFT, leads to increase in complexity (chaos)in wormhole, leads to shockwave, increases size

complexity ~ size of wormhole?
EVIDENCE:• both expected to grow linearly (at early times)• can be exponentially large (max out at same time)• perturb black hole with boundary operator
in CFT, leads to increase in complexity (chaos)in wormhole, leads to shockwave, increases size
the two increases match
including delicate cancellations!

complexity ~ size of wormhole?
EVIDENCE:• both expected to grow linearly (at early times)• can be exponentially large (max out at same time)• perturb black hole with boundary operator
in CFT, leads to increase in complexity (chaos)in wormhole, leads to shockwave, increases size
the two increases match
including delicate cancellations!
• single/multiple/localized perturbations

complexity ~ size of wormhole?
EVIDENCE:• both expected to grow linearly (at early times)• can be exponentially large (max out at same time)• perturb black hole with boundary operator
in CFT, leads to increase in complexity (chaos)in wormhole, leads to shockwave, increases size
the two increases matchincluding delicate cancellations!
• single/multiple/localized perturbations• entomb black hole in inert shelltLL

how to characterize
size of wormhole?

tRR

rr ==00

rr ==1
1=11
rr =r=0

11G`
G`AdS
AdS

volume1~actiontLtRtLtR(old)maximal slice(new/improved)
WdW patch

tL 1

actionvolume~1
G`AdS (old)tRmaximal slicetL(new/improved)
WdW patch

tR

vs.

tL

1

action(new/improved)~ patchWdW

tL 1

actionvolume~1G`AdS (old)maximal slice

tRvs.

tL1

action(new/improved)~ patchWdW

prefer action because:tLtR• naturally “dimensionless” (no arbitrary length scale)(new/improved)
WdW patch

tL 1

actionvolume~1G`AdS (old)maximal slice

tRvs.

tL

1

action(new/improved)~ patchWdW

prefer action because:tLtR• naturally “dimensionless” (no arbitrary length scale)(new/improved)• d(action)/dt = 2M (independent of size and dimension)
WdW patch

tL 1

actionvolume~1G`AdS (old)maximal slice

tRvs.

tL1action(new/improved)~ patchWdW

prefer action because:tLtR• naturally “dimensionless” (no arbitrary length scale)(new/improved)• d(action)/dt = 2M (independent of size and dimension)
WdW patchconnect totL 1

actionvolume~1
G`AdS (old)maximal slice

tR

vs.

tL

1

action(new/improved)~ patchWdW

prefer action because:tLtR• naturally “dimensionless” (no arbitrary length scale)(new/improved)• d(action)/dt = 2M (independent of size and dimension)
WdW patchconnect toblack holes saturate limit implies

Complexity Equals Action?

Adam R. Browna , Daniel A. Robertsb , Leonard Susskinda ,FURTHER WORK:
Brian Swinglea , and Ying Zhaoa• precise definition of complexity?
• precise definition of action?
aStanford
Institute forinTheoretical
Physics and• relate imprecision
two definitions?
• reference state?
(“complexityformation”)
Departmentof Physics,
StanfordofUniversity,• classicalproofCalifornia
that black94305,
holes maximize
action?
Stanford,USA• more general black holes?
b• higher-derivative
theories andsingularities?
Center for TheoreticalPhysics
and• principleof least
computation?
Departmentof Physics,Massachusetts
Institute of Technology,• complexity
andhorizon
transparency?
Cambridge, Massachusetts 02139, USA• lots of puzzles!
Abstract