Quantum Information Theory as a Proof Technique

Quantum Information Theory as a Proof Technique

Fernando G.S.L. BrandãoETH Zürich

With B. Barak (MSR), M. Christandl (ETH), A. Harrow (MIT), J. Kelner (MIT), D. Steurer (Cornell), J. Yard (Station Q), Y. Zhou (CMU)

Caltech, February 2013

Simulating quantum is hard

More than 25% of DoE supercomputer power is devoted

to simulating quantum physics

Can we get a better handle on this simulation problem?

Simulating quantum is hard, secrets are hard to conceal

More than 25% of DoE supercomputer power is devoted

to simulating quantum physics

Can we get a better handle on this simulation problem?

All current cryptography is based on unproven hardness

assumptions

Can we have better security guarantees for our secrets?

Quantum Information Science…

… gives a path for solving both problems. But it’s a long

journey

QIS is at the crossover of computer science,

mathematics and physics

Physics CS

Math

The Two Holy Grails of QIS

Quantum Computation:

Use of well-controlled quantum systems for performing

computation

Exponential speed-ups over classical computing

E.g. Factoring (RSA)Simulating quantum systems

Quantum Cryptography:

Use of well-controlled quantum systems for secret key

distribution

Unconditional security based solely on the correctness of

quantum mechanics



Use of well controlled quantum systems for performing

computations.



Use of well-controlled quantum systems for secret key

distribution


quantum mechanics

State-of-the-art: 5 qubits computer

Can prove with high probability that 15 = 3 x 5



Use of well controlled quantum systems for performing

computations.



Use of well controlled quantum systems for secret key

distribution


quantum mechanics

State-of-the-art: 5 qubits computer

Can prove with high probability that 15 = 3 x 5

State-of-the-art: 100 km

Still many technological challenges

What If……one can never build a quantum computer?

Answer 1: Then we ought to know why: new physical principle that makesquantum computers impossible?

Answer 2: QIS is interesting and useful independent of building a quantum computer

What If……one can never build a quantum computer?

Answer 1: Then we ought to know why: new physical principle that makes quantum computers impossible?

Answer 2: QIS is interesting and useful independent of building a quantum computer

This talk

Outline

• Sum-Of-Squares Hierarchy and Entanglement • Mean-Field and the Quantum PCP Conjecture

• Conclusions

Outline


• Conclusions

Problem 1: For M in H(Cd) (d x d matrix) compute

Very Easy!

Problem 2: For M in H(Cd Cl), compute

Quadratic vs Biquadratic Optimization

Next:Best known algorithm (and best hardness result) using

ideas from Quantum Information Theory

Problem 1: For M in H(Cd) (d x d matrix) compute

Very Easy!

Problem 2: For M in H(Cd Cl), compute

Quadratic vs Biquadratic Optimization

Next:Best known algorithm (and best hardness result) using

ideas from Quantum Information Theory

Quantum MechanicsPure State: norm-one vector in Cd:

Mixed State: positive semidefinite matrix of unit trace:

Quantum Measurement: To any experiment with d outcomes we associate d PSD matrices {Mk} such that ΣkMk=I Born’s rule:

Dirac notation reminder:

Quantum EntanglementPure States:

If , it’s separable

otherwise, it’s entangled.

Mixed States:

If , it’s separable

otherwise, it’s entangled.

A Physical Definition of Entanglement

LOCC: Local quantum Operations and Classical Communication

Separable states can be created by LOCC:

Entangled states cannot be created by LOCC: non-classical correlations

The Separability Problem • Given

is it entangled?

• (Weak Membership: WSEP(ε, ||*||)) Given ρAB determine if it is separable, or ε-away from SEP

SEP Dε


is it entangled?


• Dual Problem: Optimization over separable states


is it entangled?


• Dual Problem: Optimization over separable states

• Relevance: Entanglement is a resource in quantum cryptography, quantum communication, etc…

Norms on Quantum States EDIT

How to quantify the distance in Weak-Membership?

1. Euclidean Norm (Hilbert-Schmidt): ||X||2 = tr(XTX)1/2

2. Trace Norm: ||X||1 = tr((XTX)1/2)

||ρ – σ||1 = 2 max 0<M<I tr(M(ρ – σ))

3. 1-LOCC Norm: ||ρAB – σAB||1-LOCC = 2 max 0<M<I tr(M(ρ – σ)) : {M, I - M} in 1-LOCC





||ρ – σ||1 = 2 max 0<M<I tr(M(ρ – σ))

3. 1-LOCC Norm: ||ρAB – σAB||1-LOCC = 2 max 0<M<I tr(M(ρ – σ)) : {M, I - M} in 1-LOCC





||ρ – σ||1 = 2 max 0<M<I tr(M(ρ – σ))

3. 1-LOCC Norm: ||ρAB – σAB||1-LOCC = 2 max 0<M<I tr(M(ρ – σ)) : M in 1-LOCC

M in 1-LOCC

Previous Work

When is ρAB entangled? - Decide if ρAB is separable or ε-away from separable

Beautiful theory behind it (PPT, entanglement witnesses, etc)

Horribly expensive algorithms

State-of-the-art: 2O(|A|log|B|) time complexity for either ||*||2 or ||*||1

Same for estimating hSEP (no better than exaustive search!)

Hardness Results

When is ρAB entangled? - Decide if ρAB is separable or ε-away from separable

(Gurvits ’02, Gharibian ‘08) NP-hard with ε=1/poly(|A||B|) for ||*||1 or ||*||2

(Harrow, Montanaro ‘10) No exp(O(log1-ν|A|log1-μ|B|)) time algorithm with ε=Ω(1) and any ν+μ>0 for ||*||1, unless ETH fails

ETH (Exponential Time Hypothesis): SAT cannot be solved in 2o(n) time(Impagliazzo&Paruti ’99)

Quasipolynomial-time Algorithm

(B., Christandl, Yard ‘11) There is a exp(O(ε-2log|A|log|B|)) time algorithm for WSEP(||*||, ε) (in ||*||2 or ||*||1-LOCC)

2 norm natural from geometrical point of view

1-LOCC norm natural from operational point of view (distant lab paradigm)



Corollary 1: Solving WSEP(||*||2, ε) is not NP-hard for ε = 1/polylog(|A||B|), unless ETH fails

Contrast with:

(Gurvits ’02, Gharibian ‘08) Solving WSEP(||*||2, ε) NP-hard for ε = 1/poly(|A||B|)



Corollary 2: For M in 1-LOCC, can compute hSEP(M) within additive error ε in time exp(O(ε-2log|A|log|B|))

Contrast with:

(Harrow, Montanaro ’10) No exp(O(log1-ν|A|log1-μ|B|)) algorithm for hSEP(M) with ε=Ω(1), for separable M

Algorithm: SoS Hierarchy

Polynomial optimization over hypersphereSum-Of-Squares (Parrilo/Lasserre) hierarchy: gives sequence of SDPs that approximate hSEP(M)

- Round k takes time dim(M)O(k)

- Converge to hSEP(M) when k -> ∞

SoS is the strongest SDP hierarchy known for polynomial optimization (connections with SoS proof system, real algebraic geometric, Hilbert’s 17th problem, …)

We’ll derive SoS hierarchy by a quantum argument



- Round k SDP has size dim(M)O(k)




- Round k SDP has size dim(M)O(k)


SoS is the strongest SDP hierarchy known for polynomial optimization (connections with SoS proof system, real algebraic geometric, Hilbert’s 17th problem, …)

We’ll derive SoS hierarchy by a quantum argument

Classical Correlations are Shareable Given separable state

Consider the symmetric extension

Def. ρAB is k-extendible if there is ρAB1…Bk s.t for all j in [k], tr\ Bj (ρAB1…Bk) = ρAB

A

B1B2B3B4

Bk

…

Entanglement is Monogamous(Stormer ’69, Hudson & Moody ’76, Raggio & Werner ’89)ρAB separable iff ρAB is k-extendible for all k

SoS as optimization over k-extensions

(Doherty, Parrilo, Spedalieri ‘01) k-level SoS SDP for hSEP(M) is equivalent to optimization over k-extendible states(plus PPT (positive partial transpose) test):

(Stormer ’69, Hudson & Moody ’76, Raggio & Werner ’89) give alternative proof that hierarchy converges

(Barak, B., Harrow, Kelner, Steurer, Zhou ‘12)gives a proof of equivalence

SoS as optimization over k-extensions

(Doherty, Parrilo, Spedalieri ‘01) k-level SoS SDP for hSEP(M) is equivalent to optimization over k-extendible states(plus PPT (positive partial transpose) test):

(Stormer ’69, Hudson & Moody ’76, Raggio & Werner ’89) give alternative proof that hierarchy converges(before the hierarchy was even defined :-)

(Barak, B., Harrow, Kelner, Steurer, Zhou ‘12)gives a proof of equivalence

How close to separable are k-extendible states?

(Christandl, Koenig, Mitschison, Renner ‘05) De Finetti Bound

If ρAB is k-extendible

But there are k-extendible states ρAB s.t.

Improved de Finetti Bound

If ρAB is k-extendible:

for 1-LOCC or 2 normk=2ln(2)ε-2log|A| rounds of SoS solves the problemwith a SDP of size |A||B|k = exp(O(ε-2log|A| log|B|))


(Christandl, Koenig, Mitschison, Renner ‘05) De Finetti Bound

If ρAB is k-extendible

But there are k-extendible states ρAB s.t.

Improved de Finetti Bound (B, Christandl, Yard ‘11)


for 1-LOCC or 2 norm


Improved de Finetti Bound (B, Christandl, Yard ’11)


for 1-LOCC or 2 norm

k=2ln(2)ε-2log|A| rounds of SoS solves WSEP(ε) with a SDP of size

|A||B|k = exp(O(ε-2log|A| log|B|))SEP Dε

Proving…Improved de Finetti Bound (B, Christandl, Yard ’11)

If ρAB is k-extendible:for 1-LOCC or 2 norm

Proof is information-theoreticMutual Information: I(A:B)ρ := H(A) + H(B) – H(AB) H(A)ρ := -tr(ρ log(ρ))

Proving…


Let ρAB1…Bk be k-extension of ρAB

2log|A| > I(A:B1…Bk) = I(A:B1)+I(A:B2|B1)+…+I(A:Bk|B1…Bk-1)

For some l<k: I(A:Bl|B1…Bl-1) < 2 log|A|/k

(chain rule)



Proving…


Let ρAB1…Bk be k-extension of ρAB

2log|A| > I(A:B1…Bk) = I(A:B1)+I(A:B2|B1)+…+I(A:Bk|B1…Bk-1)

For some l<k: I(A:Bl|B1…Bl-1) < 2 log|A|/k

(chain rule)

What does it imply?



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.

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

information.• I(A:B|C)ρ ≈ 0 doesn’t imply ρ is

approximately separablein 1-norm (Ibinson, Linden, Winter ’08)

Good news• I(A:B|C) still defined• Chain rule, etc. still hold• I(A:B|C)ρ=0 implies ρ is

separable (Hayden, Jozsa, Petz, Winter‘03)

information theory

Chain rule:2log|A| > I(A:B1…Bk) = I(A:B1)+I(A:B2|B1)+…+I(A:Bk|B1…Bk-1)

Then

Proving the Bound

Thm (B, Christandl, Yard ‘11) For ||*||1-LOCC or ||*||2

Conditional Mutual Information Bound

• Coding Theory Strong subadditivity of von Neumann entropy as state redistribution rate (Devetak, Yard ‘06)

• Large Deviation Theory Hypothesis testing for entanglement (B., Plenio ‘08)

( )

hSEP equivalent to

1. Injective norm of 3-index tensors

2. Minimum output entropy quantum channel

3. Optimal acceptance probability in QMA(2)

4. 2->4 norm of projectors:

Unique Games Conjecture(Unique Games Conjecture, Khot ‘02) For every ε>0 it’s NP-hard to tell for a system of equations xi + xj = c mod k

YES) More than 1-ε fraction constraints satisfiableNO) Less than ε fraction satisfiable

Unique Games Conjecture(Unique Games Conjecture, Khot ‘02) For every ε>0 it’s NP-hard to tell for a system of equations xi + xj = c mod k

YES) More than 1-ε fraction constraints satisfiableNO) Less than ε fraction satisfiable

(Raghavedra ‘08) UGC implies 2-level SoS gives the best approximation algorithm for all Constraints Satisfaction Problems (max-cut, vertex cover, …)

Major barrier in current knowledge of algorithms for combinatorial problems

(Arora, Barak, Steurer ‘10) exp(nO(ε)) time algorithm for UG

Small Set Expansion Conjecture(Small Set Expansion Conjecture, Raghavendra, Steurer ‘10) For every ε, δ>0 it’s NP-hard to tell for a graph G = (V, E) whether

YES) Φ(M) < ε for a region M of size ≈ δ|V|,

NO) Φ(M) > 1-ε for all regions M of size ≈ δ|V|,

Expansion:

M

V

Small Set Expansion Conjecture

(Raghavendra, Steurer ‘10) Small Set Expansion ≈ Unique Games(Barak, B, Harrow, Kelner, Steurer, Zhou ‘12) Rough estimate 2->4 norm of projector onto top eigenspace of graphs ≈ Small Set Expansion

Expansion:

M

V

(Small Set Expansion Conjecture, Raghavendra, Steurer ‘10) For every ε, δ>0 it’s NP-hard to tell for a graph G = (V, E) whether

YES) Φ(M) < ε for a region M of size ≈ δ|V|,

NO) Φ(M) > 1-ε for all regions M of size ≈ δ|V|,

Quantum Bound on SoS Implies

1. For n x n matrix A can compute with O(log(n)ε-3) rounds of SoS a number x s.t.

2. nO(ε) -level SoS solves Small Set ExpansionAlternative algorithm to (Arora, Barak, Steurer ’10)

3. Improvement in bound (additive -> multiplicative error) would solve SSE in exp(O(log2(n))) time

4. Other quantum arguments show cannot compute rough approximation of ||P||2->4 in less than exp(O(log2(n))) time under ETH

5. Improvement (hardness specific P) would imply exp(O(log2(n))) time lower bound to Unique Games under ETH

(Barak, B., Harrow, Kelner, Steurer, Zhou ‘12)



















4. Other quantum arguments show one cannot compute rough approximation of ||P||2->4 in less than exp(O(log2(n))) time (under ETH)

5. Improvement (hardness for P from graphs) would imply exp(O(log2(n))) time lower bound to Unique Games (under ETH)







5. Improvement (hardness for P from graphs) would imply exp(O(log2(n))) time lower bound on Unique Games (under ETH)







5. Improvement (hardness for P from graphs) would imply exp(O(log2(n))) time lower bound on Unique Games (under ETH)

(Barak, B., Harrow, Kelner, Steurer, Zhou ‘12)Both improvements can be casted as open problems in quantum information theory

Maybe to solve UGC we should learn more about QIT?

Outline


• Conclusions

Quantum Mechanics, again

Dynamics in quantum mechanics is given by a

Hamiltonian H:

Equilibrium properties are also determined by H

Thermal state:

Groundstate:

Constraint Satisfaction Problems vs Local Hamiltonians

k-arity CSP:

Variables {x1, …, xn}, alphabet Σ

Constraints:

Assignment:

Unsat :=

Constraint Satisfaction Problems vs Local Hamiltonians

k-arity CSP:

Variables {x1, …, xn}, alphabet Σ

Constraints:

Assignment:

Unsat :=

k-local Hamiltonian H:

n qudits in

Constraints:

qUnsat := E0 : min eigenvalue

H1

qudit

C. vs Q. Optimal AssignmentsFinding optimal assignment of CSPs can be hard

C. vs Q. Optimal AssignmentsFinding optimal assignment of CSPs can be hard

Finding 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 )

Mean-Field……consists in approximating groundstate by a product state

is a CSP

Successful heuristic in Quantum Chemistry Condensed matter

Folklore: Mean-Field good when Many-particle interactions Low entanglement in state

Mean-Field……consists in approximating groundstate by a product state

is a CSP

Successful heuristic in Quantum Chemistry Condensed matter

Folklore: Mean-Field good when Many-particle interactions Low entanglement in state

Can we make folklore more rigorous?

Can we put limitations on use of mean-field and related schemes?

The Local Hamiltonian Problem and Quantum Complexity Theory

ProblemGiven a local Hamiltonian H, decide if E0(H)=0 or E0(H)>Δ

E0(H) : minimum eigenvalue of H

The Local Hamiltonian Problem and Quantum Complexity Theory

ProblemGiven a local Hamiltonian H, decide if E0(H)=0 or E0(H)>Δ

E0(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

U1

…. U5U4 U3 U2

The meaning of itIt’s believed QMA ≠ NP

Thus there is generally no efficient classical description of groundstates of local Hamiltonians

What’s the role of the promise gap Δ on the hardness?

…. But first, what happens for CSP?

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)









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 to estimate the energy at constant temperature

- At least NP-hard (by PCP Thm) and in QMA
















Quantum PCP?

NP

QMAqPCP?

?

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

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 NP

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.

X1

X3X2

m < O(log(n))


Let {Xi} be a partition of the sites with each Xi having m sites. Then there are products states ψi in Xi s.t.

Ei : expectation over Xi

deg(G) : degree of GΦ(Xi) : expansion of Xi

S(Xi) : entropy of groundstate in Xi

X1

X3X2

m < O(log(n))


Let {Xi} be a partition of the sites with each Xi having m sites. Then there are products states ψi in Xi s.t.

Ei : expectation over Xi

deg(G) : degree of GΦ(Xi) : expansion of Xi

S(Xi) : entropy of groundstate in Xi

X1

X3X2

Approximation in terms of 3 parameters:

1. Average expansion2. Degree interaction graph3. Average entanglement groundstate

Approximation in terms of degree

No 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 Σβ/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 fast




Contrast: It’s in NP determine whether a Hamiltonian H is s.t E0(H)=0 or E0(H) > αd3/4/deg(G)1/8

Quantum generalizations of PCP and parallel repetition cannot both be true (assuming QMA not in NP)




Bound: ΦG < ½ - Ω(1/deg) implies

Highly expanding graphs (ΦG -> 1/2) are not hard instances


1-D

2-D

3-D

∞-D

…shows mean field becomes exact in high dim

Rigorous justification to folklore in condensed matter physics

Approximation in terms of average entanglement

The problem is in NP if entanglement of groundstate satisfies a subvolume law:

Connection of amount of entanglement in groundstate and computational complexity of the model

X1

X3X2

m < O(log(n))

New Classical Algorithms for Quantum Hamiltonians

Following same approach we also obtain polynomial time algorithms for approximating the groundstate energy of

1. Planar Hamiltonians, improving on (Bansal, Bravyi, Terhal ‘07)2. Dense Hamiltonians, improving on (Gharibian, Kempe ‘10)3. 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.

Proof Idea: Monogamy of Entanglement

Cannot be highly entangled with too many neighbors

Entropy quantifies how entangled can be

Proof uses information-theoretic techniques (chain rule of conditional mutual information, informationally complete POVMs, etc) to make this intuition precise

Inspired by classical information-theoretic ideas for bounding convergence of SoS hierarchy for CSPs (Tan, Raghavendra ‘10, Barak, Raghavendra, Steurer ‘10)

Quantum Information Method• Condensed Matter Physics: Tensor Network States (Cirac, Hastings, Verstraete, Vidal, …)

• Mathematical Physics: Area Law from Exponential Decay of Correlations (B., Horodecki ‘12)

• Computational Complexity: Lower bounds on LP extensions for Travel Salesman Problem (Fiorini et al ‘11)

• Compressed Sensing: Better low rank matrix recovery methods (Gross et al ‘10)

• Etc, see (Drucker, de Wolf ‘09) for more

ConclusionsQIT useful to bound SoS hierarchy

- Quasi-polynomial algorithm for deciding entanglement - Connections 2->4 norm, Small Set Expansion - New approach to resolve UGC (improve quantum SoS bound and/or quantum hardness)

QIT useful to bound efficiency of mean-field theory

- Cornering quantum PCP - Poly-time algorithms for planar and dense Hamiltonians

Thank you!

Quantum Information Theory as a Proof Technique

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