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Quantum Information Theory as a Proof Technique

Fernando G.S.L. Brand ão ETH Zürich W ith 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. Quantum Information Theory as a Proof Technique. Simulating quantum is hard.

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Quantum Information Theory as a Proof Technique

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  1. Fernando G.S.L. Brandão ETH 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 Quantum Information Theory as a Proof Technique

  2. 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?

  3. 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?

  4. 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

  5. The Two Holy Grails of QIS Quantum Computation: Use of well-controlled quantum systemsfor 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

  6. The Two Holy Grails of QIS Quantum Computation: Use of well controlled quantum systemsfor performing computations. Exponential speed-ups over classical computing Quantum Cryptography: Use of well-controlled quantum systems for secret key distribution Unconditional security based solely on the correctness of quantum mechanics State-of-the-art: 5qubitscomputer Can prove with high probability that 15 = 3 x 5

  7. The Two Holy Grails of QIS Quantum Computation: Use of well controlled quantum systemsfor performing computations. Exponential speed-ups over classical computing Quantum Cryptography: Use of well controlled quantum systems for secret key distribution Unconditional security based solely on the correctness of quantum mechanics State-of-the-art: 5qubitscomputer Can prove with high probability that 15 = 3 x 5 State-of-the-art: 100 km Still many technological challenges

  8. 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 usefulindependent of building a quantum computer

  9. 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 usefulindependent of building a quantum computer This talk

  10. Outline • Sum-Of-Squares Hierarchy and Entanglement • Mean-Field and the Quantum PCP Conjecture • Conclusions

  11. Outline • Sum-Of-Squares Hierarchy and Entanglement • Mean-Field and the Quantum PCP Conjecture • Conclusions

  12. Quadratic vs Biquadratic Optimization • Problem 1: For M in H(Cd) (d x d matrix) compute • Very Easy! • Problem 2: For M in H(CdCl), compute Next: Best known algorithm (and best hardness result) using ideas from Quantum Information Theory

  13. Quadratic vs Biquadratic Optimization • Problem 1: For M in H(Cd) (d x d matrix) compute • Very Easy! • Problem 2: For M in H(CdCl), compute Next: Best known algorithm (and best hardness result) using ideas from Quantum Information Theory

  14. Quantum Mechanics • Pure State: norm-one vector in Cd: • Mixed State: positive semidefinite matrix of unit trace: • Quantum Measurement: To any experiment with d outcomes we associate dPSD matrices {Mk} such that ΣkMk=I • Born’s rule: Dirac notation reminder:

  15. Quantum Entanglement • Pure States: • If , it’s separable • otherwise, it’sentangled. • Mixed States: • If , it’s separable • otherwise, it’s entangled.

  16. 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-classicalcorrelations

  17. The Separability Problem • Given • is it entangled? • (Weak Membership: WSEP(ε, ||*||)) Given ρAB • determine if it is separable, or ε-away from SEP D ε SEP

  18. The Separability Problem • Given • is it entangled? • (Weak Membership: WSEP(ε, ||*||)) Given ρAB • determine if it is separable, or ε-away from SEP • Dual Problem: Optimization over separable states

  19. The Separability Problem • Given • is it entangled? • (Weak Membership: WSEP(ε, ||*||)) Given ρAB • determine if it is separable, or ε-away from SEP • Dual Problem: Optimization over separable states • Relevance: Entanglement is a resource in quantum • cryptography, quantum communication, etc…

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

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

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

  23. 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!)

  24. 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) Noexp(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)

  25. Quasipolynomial-time Algorithm (B., Christandl, Yard ‘11) There is aexp(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)

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

  27. Quasipolynomial-time Algorithm (B., Christandl, Yard ‘11) There is aexp(O(ε-2log|A|log|B|)) time algorithm for WSEP(||*||, ε) (in ||*||2 or ||*||1-LOCC) Corollary 2: For M in 1-LOCC, can computehSEP(M) within additive error ε in timeexp(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

  28. Algorithm: SoS Hierarchy • Polynomial optimization over hypersphere • Sum-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

  29. Algorithm: SoS Hierarchy • Polynomial optimization over hypersphere • Sum-Of-Squares (Parrilo/Lasserre) hierarchy: • gives sequence of SDPs that approximate hSEP(M) • - Round k SDP has size dim(M)O(k) • Converge to hSEP(M) when k -> ∞

  30. Algorithm: SoS Hierarchy • Polynomial optimization over hypersphere • Sum-Of-Squares (Parrilo/Lasserre) hierarchy: • gives sequence of SDPs that approximate hSEP(M) • - Round k SDP has size 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

  31. Classical Correlations are Shareable Given separable state Consider the symmetric extension B1 B2 B3 B4 A … Bk Def. ρAB is k-extendible if there is ρAB1…Bks.t for all j in [k], tr\ Bj(ρAB1…Bk) = ρAB

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

  33. 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

  34. 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

  35. How close to separable are k-extendible states? (Christandl, Koenig, Mitschison, Renner ‘05) De Finetti Bound If ρABis k-extendible But there are k-extendible states ρABs.t. Improved de Finetti Bound IfρABis k-extendible: for 1-LOCC or 2 norm k=2ln(2)ε-2log|A| rounds of SoS solves the problem with a SDP of size |A||B|k= exp(O(ε-2log|A| log|B|))

  36. How close to separable are k-extendible states? (Christandl, Koenig, Mitschison, Renner ‘05) De Finetti Bound If ρABis k-extendible But there are k-extendible states ρABs.t. Improved de Finetti Bound (B, Christandl, Yard ‘11) IfρABis k-extendible: for 1-LOCC or 2 norm

  37. How close to separable are k-extendible states? Improved de Finetti Bound (B, Christandl, Yard ’11) IfρABis k-extendible: 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|)) ε SEP D

  38. Proving… Improved de Finetti Bound (B, Christandl, Yard ’11) IfρABis k-extendible: for 1-LOCC or 2 norm Proof is information-theoretic Mutual Information: I(A:B)ρ := H(A) + H(B) – H(AB) H(A)ρ:= -tr(ρ log(ρ))

  39. Proving… Improved de Finetti Bound (B, Christandl, Yard ’11) IfρABis k-extendible: for 1-LOCC or 2 norm Proof is information-theoretic Mutual Information: I(A:B)ρ := H(A) + H(B) – H(AB) H(A)ρ:= -tr(ρ log(ρ)) Let ρAB1…Bkbe 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)

  40. Proving… Improved de Finetti Bound (B, Christandl, Yard ’11) IfρABis k-extendible: for 1-LOCC or 2 norm Proof is information-theoretic Mutual Information: I(A:B)ρ := H(A) + H(B) – H(AB) H(A)ρ:= -tr(ρ log(ρ)) Let ρAB1…Bkbe 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?

  41. 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.

  42. 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. information theory • 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 impliesρ is separable • (Hayden, Jozsa, Petz, Winter‘03)

  43. Proving the Bound Chain rule: 2log|A| > I(A:B1…Bk) = I(A:B1)+I(A:B2|B1)+…+I(A:Bk|B1…Bk-1) Then Thm(B, Christandl, Yard ‘11)For ||*||1-LOCC or ||*||2

  44. 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) • ()

  45. hSEPequivalent to Injective norm of 3-index tensors Minimum output entropy quantum channel Optimal acceptance probability in QMA(2) 4. 2->4 norm of projectors:

  46. 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 satisfiable NO) Less than εfraction satisfiable

  47. 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 satisfiable NO) Less than εfraction satisfiable (Raghavedra ‘08) UGC implies 2-level SoSgives the best approximation algorithmfor allConstraints Satisfaction Problems (max-cut, vertex cover, …) Majorbarrier in current knowledge of algorithms for combinatorial problems (Arora, Barak, Steurer ‘10) exp(nO(ε)) time algorithm for UG

  48. 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: V M

  49. 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: V (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 M

  50. Quantum Bound on SoS Implies (Barak, B., Harrow, Kelner, Steurer, Zhou ‘12) 1. For n x n matrix A can compute with O(log(n)ε-3) rounds of SoS a number xs.t. 2. nO(ε)-level SoS solves Small Set Expansion Alternative 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

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