Quantum chaos, quantum information, quantum complexity, quantum ...
Quantum chaos, quantum information, quantum complexity, quantum gravity Dan Roberts IAS
December 8, 2016
Quantum {chaos, information, complexity, gravity} Dan Roberts IAS
December 8, 2016
Quantum: many worlds
>>> import random >>> quantum = [’chaos’, ’information’, ’complexity’, ’gravity’ ] >>> random.choice(quantum)
Quantum chaos: butterfly effect For all simple Hermitian operators W , V , having O(1) energy and localized at x and y , this commutator should grow: −h[V , W (t)]2 iβ =hV W (t) W (t) V iβ + hW (t) V V W (t)iβ − hV W (t) V W (t)iβ − hW (t)V W (t) V iβ
At t |x − y |: I
norm of a perturbed thermal state
I
approaches hW W iβ hV V iβ = “1”
I
inner product of two different states
I
decays initially like ∼ 1 −
C0 λL (t−|x−y |/vB ) e N2
+ O(N −4 )
Quantum information: k-designs
Given an ensemble of unitary operators E = {pj , Uj } acting on H with probability distribution pj , the ensemble E is a unitary k-design if and only if X pj (Uj ⊗ · · · ⊗ Uj )ρ(Uj† ⊗ · · · ⊗ Uj† ) = | {z } | {z } j k k Z dU(U ⊗ · · · ⊗ U )ρ(U † ⊗ · · · ⊗ U † ) | {z } | {z } Haar k
for all quantum states ρ in H⊗k .
k
Quantum information: scrambling hA(0)B(t)C (0)D(t)iβ =
X
1 −βH A e iHt Be −iHt Z (β) tr {e
C e iHt De −iHt }
hA P † BP C P † DPiβ=0 = hAC ihBDi
P∈Pauli
Z Haar
dU hA U † BU C U † DUiβ=0 = −
d2
1 hAC ihBDi −1
Quantum complexity: counting How many circuits can we make with g gates acting on n qubits with C steps? There are g n2 choices to make at each step, therefore we can make at most # circuits = (gn2 )C . If we have a collection of different circuits we want to make E, then if we want to ensure we can make all of them, we can determine the minimal number of steps C≥
log |E| . log(gn2 ) [w/ Yoshida]
Quantum complexity: lower bounds The frame potential can bound the number of elements in an ensemble 2k 1 X (k) † ≥ 1 22nk . FE ≡ tr {U V } |E|2 |E| U,V ∈E
This means we can bound the complexity of the ensemble by measuring the frame potential (k)
C(E) ≥
C(E) ≥ (2k−1)2n−log
n
2kn log(2) − log FE . log(choices) X
A1 ,··· ,B1 ,···
Z
2 o dU A1 U † B1 U · · · Ak U † Bk˜U . β [w/ Yoshida]
Quantum gravity: black holes 1 Action(W) = 16πGN
Z W
√
1 −g (R − 2Λ) + 8πGN
Z
p |h|K ,
∂W
Action(W) π~ (Neutral) black holes are fastest computers in nature. Complexity =
Quantum gravity: tensor networks
W (t) = e −Ht W e Ht
x
t
Tensor network for a single localized operator.
Quantum gravity: tensor networks
W (t) = e −Ht W e Ht
x
t
Tensor network for a single localized operator.
This slide is intentionally left blank.
Operator growth: light cones C (x, t) = −h[V , W (t)]2 iβ
[w/ Stanford/Susskind]
Operator growth: commutator C (x, t) = −h[Z1 (t), Z8 ]2 iβ 3
2
1
0
0
2
4
6
8
t
H=−
P
i
Zi Zi+1 + gXi + hZi [w/ Stanford/Susskind]
Operator growth: chaotic eigenvalues Eigenvalues of [Z1 (t), Z8 ] 2 1.5 1 0.5 0 -0.5 -1 -1.5 -2
0
2
4
6
8
t
H=−
P
i
Zi Zi+1 − 1.05Xi + 0.5Zi [w/ Gur-Ari]
Operator growth: integrable eigenvalues Eigenvalues of [Z1 (t), Z8 ] 2 1.5 1 0.5 0 -0.5 -1 -1.5 -2
0
2
4
6
8
t
H=−
P
i
Zi Zi+1 + Xi [w/ Gur-Ari]
December 8, 2016
Quantum {chaos, information, complexity, gravity} Dan Roberts IAS
December 8, 2016
Quantum: many worlds
>>> import random >>> quantum = [’chaos’, ’information’, ’complexity’, ’gravity’ ] >>> random.choice(quantum)
Quantum chaos: butterfly effect For all simple Hermitian operators W , V , having O(1) energy and localized at x and y , this commutator should grow: −h[V , W (t)]2 iβ =hV W (t) W (t) V iβ + hW (t) V V W (t)iβ − hV W (t) V W (t)iβ − hW (t)V W (t) V iβ
At t |x − y |: I
norm of a perturbed thermal state
I
approaches hW W iβ hV V iβ = “1”
I
inner product of two different states
I
decays initially like ∼ 1 −
C0 λL (t−|x−y |/vB ) e N2
+ O(N −4 )
Quantum information: k-designs
Given an ensemble of unitary operators E = {pj , Uj } acting on H with probability distribution pj , the ensemble E is a unitary k-design if and only if X pj (Uj ⊗ · · · ⊗ Uj )ρ(Uj† ⊗ · · · ⊗ Uj† ) = | {z } | {z } j k k Z dU(U ⊗ · · · ⊗ U )ρ(U † ⊗ · · · ⊗ U † ) | {z } | {z } Haar k
for all quantum states ρ in H⊗k .
k
Quantum information: scrambling hA(0)B(t)C (0)D(t)iβ =
X
1 −βH A e iHt Be −iHt Z (β) tr {e
C e iHt De −iHt }
hA P † BP C P † DPiβ=0 = hAC ihBDi
P∈Pauli
Z Haar
dU hA U † BU C U † DUiβ=0 = −
d2
1 hAC ihBDi −1
Quantum complexity: counting How many circuits can we make with g gates acting on n qubits with C steps? There are g n2 choices to make at each step, therefore we can make at most # circuits = (gn2 )C . If we have a collection of different circuits we want to make E, then if we want to ensure we can make all of them, we can determine the minimal number of steps C≥
log |E| . log(gn2 ) [w/ Yoshida]
Quantum complexity: lower bounds The frame potential can bound the number of elements in an ensemble 2k 1 X (k) † ≥ 1 22nk . FE ≡ tr {U V } |E|2 |E| U,V ∈E
This means we can bound the complexity of the ensemble by measuring the frame potential (k)
C(E) ≥
C(E) ≥ (2k−1)2n−log
n
2kn log(2) − log FE . log(choices) X
A1 ,··· ,B1 ,···
Z
2 o dU A1 U † B1 U · · · Ak U † Bk˜U . β [w/ Yoshida]
Quantum gravity: black holes 1 Action(W) = 16πGN
Z W
√
1 −g (R − 2Λ) + 8πGN
Z
p |h|K ,
∂W
Action(W) π~ (Neutral) black holes are fastest computers in nature. Complexity =
Quantum gravity: tensor networks
W (t) = e −Ht W e Ht
x
t
Tensor network for a single localized operator.
Quantum gravity: tensor networks
W (t) = e −Ht W e Ht
x
t
Tensor network for a single localized operator.
This slide is intentionally left blank.
Operator growth: light cones C (x, t) = −h[V , W (t)]2 iβ
[w/ Stanford/Susskind]
Operator growth: commutator C (x, t) = −h[Z1 (t), Z8 ]2 iβ 3
2
1
0
0
2
4
6
8
t
H=−
P
i
Zi Zi+1 + gXi + hZi [w/ Stanford/Susskind]
Operator growth: chaotic eigenvalues Eigenvalues of [Z1 (t), Z8 ] 2 1.5 1 0.5 0 -0.5 -1 -1.5 -2
0
2
4
6
8
t
H=−
P
i
Zi Zi+1 − 1.05Xi + 0.5Zi [w/ Gur-Ari]
Operator growth: integrable eigenvalues Eigenvalues of [Z1 (t), Z8 ] 2 1.5 1 0.5 0 -0.5 -1 -1.5 -2
0
2
4
6
8
t
H=−
P
i
Zi Zi+1 + Xi [w/ Gur-Ari]
