Complexity Equals Action

Adam R. Browna, Daniel A. Robertsb, Leonard Susskinda,

Brian Swinglea, and Ying Zhaoa

aStanford Institute for Theoretical Physics and

Department of Physics, Stanford University,

Stanford, California 94305, USA

bCenter for Theoretical Physics and

Department of Physics, Massachusetts Institute of Technology,

Cambridge, Massachusetts 02139, USA

Abstract

We conjecture that the quantum complexity of a holographic state is dual to

the action of a certain spacetime region that we call a Wheeler-DeWitt patch.

We illustrate and test the conjecture in the context of neutral, charged, and

rotating black holes in AdS, as well as black holes perturbed with static shells

and with shock waves. This conjecture evolved from a previous conjecture that

complexity is dual to spatial volume, but appears to be a major improvement

over the original. In light of our results, we discuss the hypothesis that black

holes are the fastest computers in nature.

Complexity, Action, and Black Holes Adam Brown, Daniel Roberts, Leonard Susskind, Brian Swingle, Ying Zhao

Complexity Equals Action Adam Brown, Daniel Roberts, Leonard Susskind, Brian Swingle, Ying Zhao

arXiv:1509.07876

arXiv:1512.04993

Complexity Equals Action

Adam R. Browna, Daniel A. Robertsb, Leonard Susskinda,

Brian Swinglea, and Ying Zhaoa

aStanford Institute for Theoretical Physics and

Department of Physics, Stanford University,

Stanford, California 94305, USA

bCenter for Theoretical Physics and

Department of Physics, Massachusetts Institute of Technology,

Cambridge, Massachusetts 02139, USA

Abstract

We conjecture that the quantum complexity of a holographic state is dual to

the action of a certain spacetime region that we call a Wheeler-DeWitt patch.

We illustrate and test the conjecture in the context of neutral, charged, and

rotating black holes in AdS, as well as black holes perturbed with static shells

and with shock waves. This conjecture evolved from a previous conjecture that

complexity is dual to spatial volume, but appears to be a major improvement

over the original. In light of our results, we discuss the hypothesis that black

holes are the fastest computers in nature.

holographic duality

Complexity Equals Action

Adam R. Browna, Daniel A. Robertsb, Leonard Susskinda,

Brian Swinglea, and Ying Zhaoa

aStanford Institute for Theoretical Physics and

Department of Physics, Stanford University,

Stanford, California 94305, USA

bCenter for Theoretical Physics and

Department of Physics, Massachusetts Institute of Technology,

Cambridge, Massachusetts 02139, USA

Abstract

We conjecture that the quantum complexity of a holographic state is dual to

the action of a certain spacetime region that we call a Wheeler-DeWitt patch.

We illustrate and test the conjecture in the context of neutral, charged, and

rotating black holes in AdS, as well as black holes perturbed with static shells

and with shock waves. This conjecture evolved from a previous conjecture that

complexity is dual to spatial volume, but appears to be a major improvement

over the original. In light of our results, we discuss the hypothesis that black

holes are the fastest computers in nature.

holographic duality bulk (AdS)

Complexity Equals Action

Adam R. Browna, Daniel A. Robertsb, Leonard Susskinda,

Brian Swinglea, and Ying Zhaoa

aStanford Institute for Theoretical Physics and

Department of Physics, Stanford University,

Stanford, California 94305, USA

bCenter for Theoretical Physics and

Department of Physics, Massachusetts Institute of Technology,

Cambridge, Massachusetts 02139, USA

Abstract

We conjecture that the quantum complexity of a holographic state is dual to

the action of a certain spacetime region that we call a Wheeler-DeWitt patch.

We illustrate and test the conjecture in the context of neutral, charged, and

rotating black holes in AdS, as well as black holes perturbed with static shells

and with shock waves. This conjecture evolved from a previous conjecture that

complexity is dual to spatial volume, but appears to be a major improvement

over the original. In light of our results, we discuss the hypothesis that black

holes are the fastest computers in nature.

holographic duality bulk (AdS)boundary (CFT)

Complexity Equals Action

Adam R. Browna, Daniel A. Robertsb, Leonard Susskinda,

Brian Swinglea, and Ying Zhaoa

aStanford Institute for Theoretical Physics and

Department of Physics, Stanford University,

Stanford, California 94305, USA

bCenter for Theoretical Physics and

Department of Physics, Massachusetts Institute of Technology,

Cambridge, Massachusetts 02139, USA

Abstract

We conjecture that the quantum complexity of a holographic state is dual to

the action of a certain spacetime region that we call a Wheeler-DeWitt patch.

We illustrate and test the conjecture in the context of neutral, charged, and

rotating black holes in AdS, as well as black holes perturbed with static shells

and with shock waves. This conjecture evolved from a previous conjecture that

complexity is dual to spatial volume, but appears to be a major improvement

over the original. In light of our results, we discuss the hypothesis that black

holes are the fastest computers in nature.

holographic duality bulk (AdS)boundary (CFT)

classical property

Complexity Equals Action

Adam R. Browna, Daniel A. Robertsb, Leonard Susskinda,

Brian Swinglea, and Ying Zhaoa

aStanford Institute for Theoretical Physics and

Department of Physics, Stanford University,

Stanford, California 94305, USA

bCenter for Theoretical Physics and

Department of Physics, Massachusetts Institute of Technology,

Cambridge, Massachusetts 02139, USA

Abstract

We conjecture that the quantum complexity of a holographic state is dual to

the action of a certain spacetime region that we call a Wheeler-DeWitt patch.

We illustrate and test the conjecture in the context of neutral, charged, and

rotating black holes in AdS, as well as black holes perturbed with static shells

and with shock waves. This conjecture evolved from a previous conjecture that

complexity is dual to spatial volume, but appears to be a major improvement

over the original. In light of our results, we discuss the hypothesis that black

holes are the fastest computers in nature.

holographic duality bulk (AdS)boundary (CFT)

classical propertygravitational theory

Complexity Equals Action

Adam R. Browna, Daniel A. Robertsb, Leonard Susskinda,

Brian Swinglea, and Ying Zhaoa

aStanford Institute for Theoretical Physics and

Department of Physics, Stanford University,

Stanford, California 94305, USA

bCenter for Theoretical Physics and

Department of Physics, Massachusetts Institute of Technology,

Cambridge, Massachusetts 02139, USA

Abstract

We conjecture that the quantum complexity of a holographic state is dual to

the action of a certain spacetime region that we call a Wheeler-DeWitt patch.

We illustrate and test the conjecture in the context of neutral, charged, and

rotating black holes in AdS, as well as black holes perturbed with static shells

and with shock waves. This conjecture evolved from a previous conjecture that

complexity is dual to spatial volume, but appears to be a major improvement

over the original. In light of our results, we discuss the hypothesis that black

holes are the fastest computers in nature.

holographic duality bulk (AdS)boundary (CFT)

classical propertygravitational theory

quantum property

Complexity Equals Action

Adam R. Browna, Daniel A. Robertsb, Leonard Susskinda,

Brian Swinglea, and Ying Zhaoa

aStanford Institute for Theoretical Physics and

Department of Physics, Stanford University,

Stanford, California 94305, USA

bCenter for Theoretical Physics and

Department of Physics, Massachusetts Institute of Technology,

Cambridge, Massachusetts 02139, USA

Abstract

We conjecture that the quantum complexity of a holographic state is dual to

the action of a certain spacetime region that we call a Wheeler-DeWitt patch.

We illustrate and test the conjecture in the context of neutral, charged, and

rotating black holes in AdS, as well as black holes perturbed with static shells

and with shock waves. This conjecture evolved from a previous conjecture that

complexity is dual to spatial volume, but appears to be a major improvement

over the original. In light of our results, we discuss the hypothesis that black

holes are the fastest computers in nature.

holographic duality bulk (AdS)boundary (CFT)

classical propertygravitational theory

quantum propertynon-gravitational theory

Complexity Equals Action

Adam R. Browna, Daniel A. Robertsb, Leonard Susskinda,

Brian Swinglea, and Ying Zhaoa

aStanford Institute for Theoretical Physics and

Department of Physics, Stanford University,

Stanford, California 94305, USA

bCenter for Theoretical Physics and

Department of Physics, Massachusetts Institute of Technology,

Cambridge, Massachusetts 02139, USA

Abstract

We conjecture that the quantum complexity of a holographic state is dual to

the action of a certain spacetime region that we call a Wheeler-DeWitt patch.

We illustrate and test the conjecture in the context of neutral, charged, and

rotating black holes in AdS, as well as black holes perturbed with static shells

and with shock waves. This conjecture evolved from a previous conjecture that

complexity is dual to spatial volume, but appears to be a major improvement

over the original. In light of our results, we discuss the hypothesis that black

holes are the fastest computers in nature.

holographic duality bulk (AdS)boundary (CFT)

classical propertygravitational theory

quantum propertynon-gravitational theory

?

Complexity Equals Action

Adam R. Browna, Daniel A. Robertsb, Leonard Susskinda,

Brian Swinglea, and Ying Zhaoa

aStanford Institute for Theoretical Physics and

Department of Physics, Stanford University,

Stanford, California 94305, USA

bCenter for Theoretical Physics and

Department of Physics, Massachusetts Institute of Technology,

Cambridge, Massachusetts 02139, USA

Abstract

We conjecture that the quantum complexity of a holographic state is dual to

the action of a certain spacetime region that we call a Wheeler-DeWitt patch.

We illustrate and test the conjecture in the context of neutral, charged, and

rotating black holes in AdS, as well as black holes perturbed with static shells

and with shock waves. This conjecture evolved from a previous conjecture that

complexity is dual to spatial volume, but appears to be a major improvement

over the original. In light of our results, we discuss the hypothesis that black

holes are the fastest computers in nature.

classical propertygravitational theory

Complexity Equals Action

Adam R. Browna, Daniel A. Robertsb, Leonard Susskinda,

Brian Swinglea, and Ying Zhaoa

aStanford Institute for Theoretical Physics and

Department of Physics, Stanford University,

Stanford, California 94305, USA

bCenter for Theoretical Physics and

Department of Physics, Massachusetts Institute of Technology,

Cambridge, Massachusetts 02139, USA

Abstract

We conjecture that the quantum complexity of a holographic state is dual to

the action of a certain spacetime region that we call a Wheeler-DeWitt patch.

We illustrate and test the conjecture in the context of neutral, charged, and

rotating black holes in AdS, as well as black holes perturbed with static shells

and with shock waves. This conjecture evolved from a previous conjecture that

complexity is dual to spatial volume, but appears to be a major improvement

over the original. In light of our results, we discuss the hypothesis that black

holes are the fastest computers in nature.

classical propertygravitational theory

‘size’ of wormholefor black hole in AdS

‘size’ of wormholefor black hole in AdS

space

time

‘size’ of wormholefor black hole in AdS

space

time

outside

‘size’ of wormholefor black hole in AdS

space

time

inside

outside

‘size’ of wormholefor black hole in AdS

space

time

inside

outside

‘size’ of wormholefor black hole in AdS

r = 1

r = 0

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 = 1

r = 0‘size’ of wormhole

for black hole in AdS

r = 1

r = 0‘size’ of wormhole

for black hole in AdS

‘other’ r = 1

r = 1

r = 0‘size’ of wormhole

for black hole in AdS

‘other’ r = 1

r = 1

r = 0‘size’ of wormhole

for black hole in AdS

‘other’ r = 1

rh

r = 0‘size’ of wormhole

for black hole in AdS

timeL timeR

rh

r = 0‘size’ of wormhole

for black hole in AdS

timeL timeR

rh

r = 0‘size’ of wormhole

for black hole in AdS

timeL timeR

rh

r = 0‘size’ of wormhole

for black hole in AdS

timeL timeR

rh

r = 0‘size’ of wormhole

for black hole in AdS

timeL timeR

rh

r = 0‘size’ of wormhole

for black hole in AdS

timeL timeR

wormhole length ⇠ tL + tR

rh

r = 0‘size’ of wormhole

for black hole in AdS

timeL timeR

wormhole length ⇠ tL + tR

rh

r = 0‘size’ of wormhole

for black hole in AdS

timeL timeR

wormhole length ⇠ tL + tR

rh

r = 0

wormhole length ⇠ tL + tR

rh

CFTRCFTL

r = 0

wormhole length ⇠ tL + tR

CFTRCFTL

|TFDi =X

i

e��Ei/2|EiiL|EiiR

r = 0

wormhole length ⇠ tL + tR

CFTRCFTL

|TFDi =X

i

e��Ei/2|EiiL|EiiR

| (tL, tR)i =X

i

e��Ei/2+iEi(tL+tR)|EiiL|EiiR

r = 0

wormhole length ⇠ tL + tR

CFTRCFTL

|TFDi =X

i

e��Ei/2|EiiL|EiiR

What is CFT dual to linear growth of wormhole?

| (tL, tR)i =X

i

e��Ei/2+iEi(tL+tR)|EiiL|EiiR

r = 0

wormhole length ⇠ tL + tR

CFTRCFTL

|TFDi =X

i

e��Ei/2|EiiL|EiiR

What is CFT dual to linear growth of wormhole?

COMPLEXITY?

| (tL, tR)i =X

i

e��Ei/2+iEi(tL+tR)|EiiL|EiiR

COMPLEXITY?

computational gate complexity of a quantum state

COMPLEXITY?

computational gate complexity of a quantum statee.g. | (tL, tR)i =

X

i

e��Ei/2+iEi(tL+tR)|EiiL|EiiR

COMPLEXITY?

computational gate complexity of a quantum state

DEFINITION?

e.g. | (tL, tR)i =X

i

e��Ei/2+iEi(tL+tR)|EiiL|EiiR

COMPLEXITY?

computational gate complexity of a quantum state

starting in a reference stateDEFINITION?e.g. |TFDi =

X

i

e��Ei/2|EiiL|EiiR

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 ✏

e.g. | (tL, tR)i =X

i

e��Ei/2+iEi(tL+tR)|EiiL|EiiR

COMPLEXITY?

computational gate complexity of a quantum state

starting in a reference stateDEFINITION?e.g. |TFDi =

X

i

e��Ei/2|EiiL|EiiR

how many fundamental gates

are needed to make the statee.g. to within an accuracy ✏

e.g.

• can be exponentially large (Hilbert Space is huge) • expected to grow linearly (at early times)

| (tL, tR)i =X

i

e��Ei/2+iEi(tL+tR)|EiiL|EiiR

e.g. unitaries each of which act only on two-qubits

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 match including delicate cancellations!

• single/multiple/localized perturbations• entomb black hole in inert shell

tRtL

tRtL

r = 0

r = 0

r = 1r = 1

tL tR

(old) maximal slice

(new/improved) WdW patch

how to characterize size of wormhole?

volume

action

1

G`AdS

1

~

tRtL

tRtL

r = 0

r = 0

r = 1r = 1

tL tR

(old) maximal slice

(new/improved) WdW patch

how to characterize size of wormhole?

volume

action

1

G`AdS

1

~

tRtL

tRtL

r = 0

r = 0

r = 1r = 1

tL tR

(old) maximal slice

(new/improved) WdW patch

how to characterize size of wormhole?

volume

action

1

G`AdS

1

~

volume action1

G`AdS

1

~vs.

tRtL

tRtL

r = 0

r = 0

r = 1r = 1

tL tR

(old) maximal slice

(new/improved) WdW patch

how to characterize size of wormhole?

volume

action

1

G`AdS

1

~tRtL

tRtL

r = 0

r = 0

r = 1r = 1

tL tR

(old) maximal slice

(new/improved) WdW patch

how to characterize size of wormhole?

volume

action

1

G`AdS

1

~

volume action1

G`AdS

1

~vs.

• naturally “dimensionless” (no arbitrary length scale)

tRtL

tRtL

r = 0

r = 0

r = 1r = 1

tL tR

(old) maximal slice

(new/improved) WdW patch

how to characterize size of wormhole?

volume

action

1

G`AdS

1

~tRtL

tRtL

r = 0

r = 0

r = 1r = 1

tL tR

(old) maximal slice

(new/improved) WdW patch

how to characterize size of wormhole?

volume

action

1

G`AdS

1

~

prefer action because:

volume action1

G`AdS

1

~vs.

• naturally “dimensionless” (no arbitrary length scale) • d(action)/dt = 2M (independent of size and dimension)

tRtL

tRtL

r = 0

r = 0

r = 1r = 1

tL tR

(old) maximal slice

(new/improved) WdW patch

how to characterize size of wormhole?

volume

action

1

G`AdS

1

~tRtL

tRtL

r = 0

r = 0

r = 1r = 1

tL tR

(old) maximal slice

(new/improved) WdW patch

how to characterize size of wormhole?

volume

action

1

G`AdS

1

~

prefer action because:

volume action1

G`AdS

1

~vs.

• naturally “dimensionless” (no arbitrary length scale) • d(action)/dt = 2M (independent of size and dimension)

connect to

tRtL

tRtL

r = 0

r = 0

r = 1r = 1

tL tR

(old) maximal slice

(new/improved) WdW patch

how to characterize size of wormhole?

volume

action

1

G`AdS

1

~tRtL

tRtL

r = 0

r = 0

r = 1r = 1

tL tR

(old) maximal slice

(new/improved) WdW patch

how to characterize size of wormhole?

volume

action

1

G`AdS

1

~

prefer action because:

volume action1

G`AdS

1

~vs.

• naturally “dimensionless” (no arbitrary length scale) • d(action)/dt = 2M (independent of size and dimension)

connect to

black holes saturate limit implies

tRtL

tRtL

r = 0

r = 0

r = 1r = 1

tL tR

(old) maximal slice

(new/improved) WdW patch

how to characterize size of wormhole?

volume

action

1

G`AdS

1

~tRtL

tRtL

r = 0

r = 0

r = 1r = 1

tL tR

(old) maximal slice

(new/improved) WdW patch

how to characterize size of wormhole?

volume

action

1

G`AdS

1

~

prefer action because:

FURTHER WORK:• precise definition of complexity? • precise definition of action? • relate imprecision in two definitions? • reference state? (“complexity of formation”) • classical proof that black holes maximize action? • more general black holes? • higher-derivative theories and singularities? • principle of least computation? • complexity and horizon transparency? • lots of puzzles!

Complexity Equals Action

Adam R. Browna, Daniel A. Robertsb, Leonard Susskinda,

Brian Swinglea, and Ying Zhaoa

aStanford Institute for Theoretical Physics and

Department of Physics, Stanford University,

Stanford, California 94305, USA

bCenter for Theoretical Physics and

Department of Physics, Massachusetts Institute of Technology,

Cambridge, Massachusetts 02139, USA

Abstract

We conjecture that the quantum complexity of a holographic state is dual to

the action of a certain spacetime region that we call a Wheeler-DeWitt patch.

We illustrate and test the conjecture in the context of neutral, charged, and

rotating black holes in AdS, as well as black holes perturbed with static shells

and with shock waves. This conjecture evolved from a previous conjecture that

complexity is dual to spatial volume, but appears to be a major improvement

over the original. In light of our results, we discuss the hypothesis that black

holes are the fastest computers in nature.

?