Product-State Approximations to Quantum - PowerPoint Presentation

Slide1

Product-State Approximations to Quantum Groundstates

Fernando

G.S.L.

Brand

ão

Imperial -> UCL

Based on joint work with

A. Harrow

Paris, April 2013Slide2

Quantum Many-Body Systems

Quantum Hamiltonian

Interested in computing properties such as

minimum energy

,

correlations functions

at

zero

and

finite temperature

,

dynamical properties

, …Slide3

Quantum Hamiltonian Complexity…analyzes quantum many-body physics through the computational lens

Relevant for

condensed matter physics

,

quantum chemistry, statistical mechanics, quantum information2. Natural generalization of the study of

constraint satisfaction problems in theoretical computer scienceSlide4

Constraint Satisfaction Problems vs Local Hamiltonians

k

-

a

rity CSP:

Variables {x1, …, xn}, alphabet Σ

Constraints:Assignment: Unsat :=Slide5

Constraint Satisfaction Problems vs Local Hamiltonians

k

-

a

rity CSP:

Variables {x1, …, xn}, alphabet Σ

Constraints:Assignment: Unsat :=

k

-local Hamiltonian H:

n

qu

dits in Constraints:qUnsat :=

E0 : min eigenvalue

H

1

qu

d

itSlide6

C. vs Q. Optimal Assignments

Finding

o

ptimal

assignment of CSPs can be hardSlide7

C. vs Q. Optimal Assignments

Finding

o

ptimal

assignment of CSPs can be hard

Finding optimal assignment of quantum CSPs can be even harder(BCS Hamiltonian groundstate, Laughlin states for FQHE

,…)Slide8

C.

vs

Q. Optimal Assignments

Finding

o

ptimal assignment of CSPs can be hardFinding optimal assignment of quantum CSPs can be even harder

(BCS Hamiltonian groundstate, Laughlin states for FQHE,…)

Main difference:

Optimal Assignment can be a highly entangled state (unit vector in )Slide9

Optimal Assignments:Entangled States

Non-entangled state:

e.g.

Entangled states:

e.g.

To describe a general entangled state of

n

spins requires

exp

(O(n))

bitsSlide10

How Entangled?

Given bipartite entangled state

t

he reduced state on A is mixed:

The

more mixed

ρ

A, the

more entangled ψAB:Quantitatively: E(ψAB) := S(ρ

A) = -tr(ρ

A log ρA)

Is there a relation between the amount of entanglement in the ground-state and the computational complexity of the model? Slide11

NP ≠ Non-Polynomial

NP

is the class of problems for which one can check the correctness of a potential solution efficiently (in polynomial time)

E.g.

Graph Coloring: Given a graph and 3 colors, color the graph such that no two neighboring vertices have the same color

3-coloringSlide12

NP ≠ Non-Polynomial

NP

is the class of problems for which one can check the correctness of a potential solution efficiently (in polynomial time)

E.g.

Graph Coloring: Given a graph and 3 colors, color the graph such that no two neighboring vertices have the same color

3-coloring

The million dollars question:

Is P = NP?Slide13

NP-hardnessA problem is

NP-hard

if any other problem in NP can be reduced to it in polynomial time.

E.g. 3-SAT:

CSP with binary variables x1, …, xn

and constraints {Ci}, Cook-Levin Theorem: 3-SAT is NP-hard

Slide14

NP-hardnessA problem is

NP-hard

if any other problem in NP can be reduced to it in polynomial time.

E.g. 3-SAT:

CSP with binary variables x1, …, xn

and constraints {Ci}, Cook-Levin Theorem: 3-SAT is NP-hard

E.g. There is an efficient mapping between graphs and 3-SAT formulas such that given a graph G

and the associated 3-SAT formula S G is 3-colarable

<-> S is satisfiable Slide15

NP-hardnessA problem is

NP-hard

if any other problem in NP can be reduced to it in polynomial time.

E.g. 3-SAT:

CSP with binary variables x1, …, xn

and constraints {Ci}, Cook-Levin Theorem: 3-SAT is NP-hard

E.g. There is an efficient mapping between graphs and 3-SAT formulas such that given a graph G

and the associated 3-SAT formula S G is 3-colarable

<-> S is satisfiableNP-complete: NP-hard + inside NP

Slide16

Complexity of qCSP

Since computing the ground-energy of local Hamiltonians is a generalization of solving CSPs,

the problem is at least NP-hard.

Is it in NP? Or is it harder?The fact that the optimal assignment is a highly entangled state might make things harder…

Slide17

The Local Hamiltonian Problem

Problem

Given a local

H

amiltonian H, decide if E0(H)=0 or

E0(H)>ΔE0(H) : minimum eigenvalue of H

Thm (Kitaev ‘99) The local Hamiltonian problem is QMA-complete for

Δ = 1/poly(n)Slide18

The Local Hamiltonian Problem

Problem

Given a local

H

amiltonian H, decide if E0(H)=0 or

E0(H)>ΔE0(H) : minimum eigenvalue of H

Thm (Kitaev ‘99) The local Hamiltonian problem is QMA-complete for

Δ = 1/poly(n)(analogue Cook-Levin thm)

QMA is the quantum analogue of NP, where the proof and the computation are quantum.

Input

Witness

U

1

….

U

5

U

4

U

3

U

2Slide19

The meaning of it

It’s widely believed

QMA ≠ NP

Thus, there is generally no

efficient classical description of groundstates of local Hamiltonians Even very simple models are QMA-completeE.g.

(Aharonov, Irani, Gottesman, Kempe ‘07) 1D models

“1D systems as hard as the general case” Slide20

The meaning of it

It’s widely believed

QMA ≠ NP

Thus, there is generally no

efficient

classical description of groundstates of local Hamiltonians Even very simple models are QMA-completeE.g. (Aharonov, Irani,

Gottesman, Kempe ‘07) 1D models

“1D systems as hard as the general case” What’s the role of the acurracy Δ on the hardness?

… But first what happens classically?Slide21

PCP TheoremPCP Theorem

(

Arora

et al

’98, Dinur ‘07): There is a ε > 0

s.t.it’s NP-complete to determine whether for a CSP with m constraints, Unsat = 0 or Unsat > εm

NP-hard even for Δ=Ω

(m)Equivalent to the existence of Probabilistically Checkable

Proofs for NP. Central tool in the theory of hardness of approximation (optimal threshold for 3-SAT (7/8-factor), max-clique (

n1-ε-factor))

(obs: Unique Game Conjecture is about the existence of strong form of PCP)Slide22

PCP TheoremPCP Theorem

(

Arora

et al

’98, Dinur ‘07): There is a ε > 0

s.t.it’s NP-complete to determine whether for a CSP with m constraints, Unsat = 0 or Unsat > εm

NP-hard even for Δ=Ω

(m)Equivalent to the existence of Probabilistically Checkable

Proofs for NP. Central tool in the theory of hardness of approximation (optimal threshold for 3-SAT (7/8-factor), max-clique (

n1-ε-factor))

(obs: Unique Game Conjecture is about the existence of strong form of PCP)Slide23

PCP TheoremPCP Theorem

(

Arora

et al

’98, Dinur ‘07): There is a ε > 0

s.t.it’s NP-complete to determine whether for a CSP with m constraints, Unsat = 0 or Unsat > εm

NP-hard even for Δ=Ω

(m)Equivalent to the existence of Probabilistically Checkable

Proofs for NP. Central tool in the theory of hardness of approximation (optimal threshold for 3-SAT (7/8-factor), max-clique (

n1-ε-factor))

(obs: Unique Game Conjecture is about the existence of strong form of PCP)Slide24

PCP TheoremPCP Theorem

(

Arora

et al

’98, Dinur ‘07): There is a ε > 0

s.t.it’s NP-complete to determine whether for a CSP with m constraints, Unsat = 0 or Unsat > εm

NP-hard even for Δ=Ω

(m)Equivalent to the existence of Probabilistically Checkable

Proofs for NP. Central tool in the theory of hardness of approximation (optimal threshold for 3-SAT (7/8-factor), max-clique (

n1-ε-factor))

Slide25

Quantum PCP?The

qPCP

conjecture

: There is ε

> 0 s.t. the following problem is QMA-complete: Given 2-local Hamiltonian H

with m local terms determine whether (i) E0(H)=0 or (ii) E0

(H) > εm.

(Bravyi, DiVincenzo, Loss, Terhal

‘08) Equivalent to conjecture for O(1)-local Hamiltonians over qdits.Equivalent to estimating mean groundenergy

to constant accuracy

(eo(H) := E0(H)/m)

- And related to estimating energy at constant temperature - At least NP-hard (

by PCP Thm) and in QMASlide26

Quantum PCP?The

qPCP

conjecture

: There is ε

> 0 s.t. the following problem is QMA-complete: Given 2-local Hamiltonian H

with m local terms determine whether (i) E0(H)=0 or (ii) E0

(H) > εm.

(Bravyi, DiVincenzo, Loss, Terhal

‘08) Equivalent to conjecture for O(1)-local Hamiltonians over qdits.Equivalent to estimating mean groundenergy

to constant accuracy

(eo(H) := E0(H)/m)

- And related to estimating energy at constant temperature - At least NP-hard (

by PCP Thm) and in QMASlide27

Quantum PCP?The

qPCP

conjecture

: There is ε

> 0 s.t. the following problem is QMA-complete: Given 2-local Hamiltonian H

with m local terms determine whether (i) E0(H)=0 or (ii) E0

(H) > εm.

(Bravyi, DiVincenzo, Loss, Terhal

‘08) Equivalent to conjecture for O(1)-local Hamiltonians over qdits.Equivalent to estimating mean groundenergy

to constant accuracy

(eo(H) := E0(H)/m)

- And related to estimating energy at constant temperature - At least NP-hard (

by PCP Thm) and in QMASlide28

Quantum PCP?The

qPCP

conjecture

: There is ε

> 0 s.t. the following problem is QMA-complete: Given 2-local Hamiltonian H

with m local terms determine whether (i) E0(H)=0 or (ii) E0

(H) > εm.

(Bravyi, DiVincenzo, Loss, Terhal

‘08) Equivalent to conjecture for O(1)-local Hamiltonians over qdits.Equivalent to estimating mean groundenergy

to constant accuracy

(eo(H) := E0(H)/m)

- Related to estimating energy at constant temperature - At least

NP-hard (by PCP Thm) and in QMASlide29

Quantum PCP?The

qPCP

conjecture

: There is ε

> 0 s.t. the following problem is QMA-complete: Given 2-local Hamiltonian H

with m local terms determine whether (i) E0(H)=0 or (ii) E0

(H) > εm.

(Bravyi, DiVincenzo, Loss, Terhal

‘08) Equivalent to conjecture for O(1)-local Hamiltonians over qdits.Equivalent to estimating mean groundenergy

to constant accuracy

(eo(H) := E0(H)/m)

- Related to estimating energy at constant temperature - At least

NP-hard (by PCP Thm) and in QMASlide30

Quantum PCP?

NP

QMA

qPCP

?

?Slide31

Previous Work and Obstructions

(

Aharonov

, Arad, Landau,

Vazirani ‘08) Quantum version of 1 of 3 parts of Dinur’s proof of the PCP thm

(gap amplification)But: The other two parts (alphabet and degree reductions) involve massive copying of information; not clear how to do it with a highly entangled assignmentSlide32

Previous Work and Obstructions

(

Aharonov

, Arad, Landau,

Vazirani ‘08) Quantum version of 1 of 3 parts of Dinur’s proof of the PCP thm

(gap amplification)But: The other two parts (alphabet and degree reductions) involve massive copying of information; not clear how to do it with a highly entangled assignment(Bravyi

, Vyalyi ’03; Arad ’

10; Hastings ’12; Freedman, Hastings ’13; Aharonov

, Eldar ’13, …) No-go for large class of commuting Hamiltonians and almost commuting Hamiltonians But: Commuting case might always be in NPSlide33

Going Forward

Can we understand why got stuck in quantizing the classical proof?

Can we prove partial no-go beyond commuting case?

Yes, by considering the simplest possible reduction from quantum Hamiltonians to CSPs. Slide34

Mean-Field……consists in approximating

groundstate

by a product state

is a CSP

Successful heuristic in

Quantum

Chemistry (Hartree-Fock)

Condensed matter (e.g. BCS theory)Folklore: Mean-Field good when Many-particle interactions Low entanglement in stateIt’s a mapping from quantum Hamiltonians to CSPs Slide35

Approximation in NP

(B., Harrow ‘12)

Let

H

be a 2-local Hamiltonian on qudits with interaction graph

G(V, E) and |E| local terms.Slide36

Approximation in NP

(B., Harrow ‘12)

Let

H

be a 2-local Hamiltonian on qudits with interaction graph

G(V, E) and |E| local terms.Let {Xi} be a partition of the sites with each Xi having

m sites.

X

1

X

3

X

2

m < O(log(n))Slide37

Approximation in NP

(B., Harrow ‘12)

Let

H

be a 2-local Hamiltonian on qudits with interaction graph

G(V, E) and |E| local terms.Let {Xi} be a partition of the sites with each Xi having

m sites.

X

1

X

3

X

2

m < O(log(n))

E

i

: expectation over X

i

deg

(G)

: degree of G

Φ

(X

i

)

: expansion of X

i

S(X

i

)

: entropy of

groundstate

in X

iSlide38

Approximation in NP

(B., Harrow ‘12)

Let

H

be a 2-local Hamiltonian on qudits with interaction graph

G(V, E) and |E| local terms.Let {Xi} be a partition of the sites with each Xi having

m sites. Then there are products states ψi in X

i s.t.

E

i

: expectation over Xideg(G) : degree of GΦ

(Xi) : expansion of XiS(X

i) : entropy of groundstate in Xi

X

1

X

3

X

2

m < O(log(n))Slide39

Approximation in NP

(B., Harrow ‘12)

Let

H

be a 2-local Hamiltonian on qudits with interaction graph

G(V, E) and |E| local terms.Let {Xi} be a partition of the sites with each Xi having

m sites. Then there are products states ψi in X

i s.t.

E

i

: expectation over Xideg(G) : degree of GΦ

(Xi) : expansion of XiS(X

i) : entropy of groundstate in Xi

X

1

X

3

X

2

Approximation in terms of

3

parameters

:

Average expansion

Degree interaction graph

Average entanglement

groundstateSlide40

Approximation in terms of average expansion

Average Expansion:

Well known fact:

‘s divide and conquer

Potential hard instances must be based on expanding graphs

X

1

X

3

X

2

m < O(log(n))Slide41

Approximation in terms of degree

N

o classical analogue:

(PCP + parallel repetition)

For all

α, β,

γ

> 0 it’s NP-complete to determine whether a CSP C is

s.t. Unsat = 0 or Unsat > α Σ

β/deg(G

)γ

Parallel repetition: C -> C’ i

. deg(G’) = deg(G)k ii. Σ

’ = Σk

ii. Unsat(G’) > Unsat

(G)

(

Raz

‘00)

even showed

Unsat

(G’

) approaches 1 exponentially fastSlide42

Approximation in terms of degree

N

o classical analogue:

(PCP + parallel repetition)

For all

α, β,

γ

> 0 it’s NP-complete to determine whether a CSP C is

s.t. Unsat = 0 or Unsat > α Σ

β/deg(G

)γ

Q. Parallel repetition: H -> H’

i. deg(H’) = deg(H)k

????? ii. d’ =

dk iii. e

0(H

’) > e

0

(

H

)Slide43

Approximation in terms of degree

N

o classical analogue:

(PCP + parallel repetition)

For all

α, β,

γ

> 0 it’s NP-complete to determine whether a CSP C

is s.t. Unsat = 0 or Unsat > α Σ

β/deg(G)

γ Contrast: It’s in NP determine whether a Hamiltonian H is

s.t e0(H)=0 or

e0(H) > αd3/4/deg(G)1/8Quantum generalizations of PCP

and parallel repetition cannot both be true (assuming QMA not in NP)Slide44

Approximation in terms of degree

Bound:

Φ

G

< ½ -

Ω

(1/

deg) impliesHighly expanding graphs (ΦG

-> 1/2) are not hard instancesObs: (Aharonov, Eldar ‘13)

k-local, commuting modelsSlide45

Approximation in terms of degree

1-D

2-D

3-D

∞-D

…shows mean field becomes exact in high dim

Rigorous justification to folklore in condensed matter physicsSlide46

Approximation in terms of average entanglement

Mean field works well if entanglement of

groundstate

satisfies a

subvolume

law

:

Connection of

amount of entanglement in groundstate and computational complexity of the model

X

1

X

3

X

2

m < O(log(n))Slide47

Approximation in terms of average entanglement

S

ystems with

low entanglement

are expected to be

easySo far only precise in 1D:

Area law for entanglement -> MPS description

Here:Good: arbitrary lattice, only subvolume law

Bad: Only mean energy approximated wellSlide48

New Classical Algorithms for Quantum Hamiltonians

Following same approach we also obtain

polynomial time algorithms

for approximating the

groundstate energy ofPlanar Hamiltonians, improving on

(Bansal, Bravyi, Terhal ‘07)Dense Hamiltonians, improving on (Gharibian

, Kempe ‘10)Hamiltonians on graphs

with low threshold rank, building on (Barak, Raghavendra, Steurer ‘10)

In all cases we prove that a product state does a good job and use efficient algorithms for CSPs. Slide49

Proof Idea: Monogamy of Entanglement

Cannot be highly entangled with too many neighbors

Entropy quantifies how entangled it can be

Proof uses

information-theoretic techniques

to make this intuition precise

I

nspired by classical information-theoretic ideas for bounding convergence of

SoS

hierarchy for CSPs

(Tan,

Raghavendra

‘10, Barak,

Raghavendra

, Steurer ‘10)Slide50

Tool: Information Theory

Mutual Information

Pinsker’s

inequality

Conditional Mutual Information

Chain Rule for some t<kSlide51

Conditioning DecouplesIdea that almost works

(c.f.

Raghavendra-Tan ‘11)1. Choose

i, j1, …, jk

at random from {1, …, n}.Then there exists t<k such that

i

j

1

j

2

j

kSlide52

Conditioning Decouples2. Conditioning on subsystems

j

1

, …,

jt causes, on average, error <k/n and leaves a distribution q

for which

j

1

j

t

j

2Slide53

Conditioning Decouples2. Conditioning on subsystems

j

1

, …,

jt causes, on average, error <k/n and leaves a distribution q

for which which implies

j

1

j

t

j

2Slide54

Conditioning Decouples2. Conditioning on subsystems

j

1

, …,

jt causes, on average, error <k/n and leaves a distribution q

for which which implies By Pinsker’s:

j

1

j

t

j

2Slide55

Conditioning Decouples2. Conditioning on subsystems

j

1

, …,

jt causes, on average, error <k/n and leaves a distribution q

for which which implies By Pinsker’s:

j

1

j

t

j

2

Choosing

k =

εnSlide56

Quantum Information?

Nature isn't classical, dammit, and if you want to make a simulation of Nature, you'd better make it quantum mechanical, and by golly it's a wonderful problem, because it doesn't look so easy.Slide57

Quantum Information?

Nature isn't classical, dammit, and if you want to make a simulation of Nature, you'd better make it quantum mechanical, and by golly it's a wonderful problem, because it doesn't look so easy.

Bad news

Only definition

I(A:B|C)=H(AC)+H(BC)-H(ABC)-H(C)

Can’t condition on quantum informationI(A:B|C)ρ

≈ 0 doesn’t imply ρAB

is approximately product by meas. C

(Ibinson, Linden, Winter ’08)

Good newsI(A:B|C) still defined

Chain rule, etc. still

holdI(A:B|C)ρ

=0 implies ρAB is product by measuring

C

(Hayden, Jozsa, Petz

, Winter‘03)

i

nformation theorySlide58

Really Good News: Informationally Complete Measurements

There exists an

i

nformationally

-complete measurement M(ρ) = Σ

k tr(Mkρ) |k><k| s.t. for

ρ, σ in D(C

d)

and for all k and ρ1…k, σ1…k in D

((Cd)

k) Slide59

Proof Overview

Measure

εn

qudits with M and condition on outcomes.Incur error

ε.Most pairs of other qudits would have mutual information ≤

log(d) / ε deg(G)

if measured.Thus their state is within distance

d3(log(d) / ε deg(G))1/2

of product.

Witness is a global product state. Total error isε +

d6(log(d) / ε

deg(G))1/2.Choose ε to balance these terms

.General case follows by coarse graining sites

(can replace log(d) by E

i

H(X

i

)

)Slide60

Proof Overview

Let

…

p

revious argument

q

:

probability distribution obtained conditioning on z

j1

, …,

z

jtSlide61

Proof Overview

σ

:

probability distribution obtained by measuring M on j

1

, …, jt and conditioning on outcome

info complete measurementSlide62

Conclusions

Can approximate mean energy in terms of

degree

and

amount of entanglement: Monogamy of entanglement in groundstates

Mean field exact in the limit of large dimensionsNo-go against qPCP + “quantum parallel repetition”

Tools from information theory are usefulSlide63

Open Questions

Go

beyond mean

field

Is there a meaningful notion of parallel repetition for qCSP?

Does every groundstate have subvolume entanglement after constant-depth-circuit renormalization?Find

more classes of Hamiltonians with efficient algorithms

(dis)prove qPCP conjecture!

Mean field exact in the limit of large dimensionsNo-go

against qPCP + “quantum parallel repetition”

Tools from information theory are useful!