New Variants of the Quantum de Fine4 Theorem with Applica:ons

Transcription

1 New Variants of the Quantum de Fine4 Theorem with Applica:ons Fernando G.S.L. Brandão ETH Zürich Based on joint work with Aram Harrow Arxiv: Shanghai, 27/08/2012

2 Symmetric States ρ AB1...B n D(A B 1... B n ) is permutamon symmetric in the B subsystems if permutamon π. ρ AB1...B n = ρ ABπ (1)...B π (n ) for every ρ AB1...B n A B 1 B 2 B n- 1 B 4 B 3 B n = A B 1 B 2 B 3 B 4 ρ AB1...B n B n- 1 B n

3 Quantum de Fine4 Theorem (Christandl, Koenig, Mitchson, Renner 05) Let symmetric in the B subsystems. Then ρ AB1...B n be a state min ρ AB1...B µ k µ(dσ )ρ σ σ k 1 d 2 k n Final installment in a long sequence of works: (Hudson, Moody 76), (Stormer 69), (Raggio, Werner 89), (Caves, Fuchs, Schack 01), (Koenig, Renner 05),

4 Quantum de Fine4 Theorem (Christandl, Koenig, Mitchson, Renner 05) Let symmetric in the B subsystems. Then ρ AB1...B n be a state min ρ AB1...B µ k µ(dσ )ρ σ σ k 1 d 2 k n Final installment in a long sequence of works: (Hudson, Moody 76), (Stormer 69), (Raggio, Werner 89), (Caves, Fuchs, Schack 01), (Koenig, Renner 05), ρ AB Def. We say is k- extendible if there is a symmetric extension on B subsystems of it. ρ AB1...B n

5 Quantum de Fine4 Theorem as Monogamy of Entanglement all quantum states 2-extendible 3-extendible 137-extendible separable states= extendible

6 Quantum de Fine4 Theorem (Christandl, Koenig, Mitchson, Renner 05) Let symmetric in the B subsystems. Then ρ AB1...B n be a state min ρ AB1...B µ k µ(dσ )ρ σ σ k 1 d 2 k n In many applicamons parameters are not good enough. But unfortunately they are essenmally "ght

7 Quantum de Fine4 Theorem (Christandl, Koenig, Mitchson, Renner 05) Let symmetric in the B subsystems. Then ρ AB1...B n be a state min ρ AB1...B µ k µ(dσ )ρ σ σ k 1 d 2 k n In many applicamons parameters are not good enough. But unfortunately they are essenmally "ght Way Forward: Prove new versions of de Finef with beger error term, but for a coarser nomon of approximamon

8 Relaxed and Improved Quantum de Fine4 Theorems So far we know two examples of this approach: (Renner 07) ExponenMal de Finef Theorem: error term exp(- Ω(n- k)). Target state convex combinamon of almost i.i.d. states. (B., Christandl, Yard 10): de Finef theorem for k = 1 with error term O(log(dim)). Error measured in 1- LOCC norm.

9 Relaxed and Improved Quantum de Fine4 Theorems So far we know two examples of this approach: (Renner 07) ExponenMal de Finef Theorem: error term exp(- Ω(n- k)). Target state convex combinamon of almost i.i.d. states. (B., Christandl, Yard 10): de Finef theorem for k = 1 with error term O(log(dim)). Error measured in 1- LOCC norm. Can we push this approach further? Is it worth doing so?

10 Outline Quantum de Fine4 Theorem for Local Measurements Op:mality O(n 1/2 ) unentangled proofs for SAT Subexponen:al Algorithm for Small Set Expansion Efficient PreNy Good Tomography Proof Quantum de Fine4 Theorem without Symmetry Calcula:ng groundenergy dense local hamiltonians Evidence Against Quantum PCP

11 Quantum de Fine4 for Local Measurements Thm Part 1 For any state ρ AB1...B n D(A B 1... B n ) symmetric in the B subsystems and ν a distribumon over quantum operamons {Λ A,ν } ν : min max E Λ A,ν Λ ( B ρ AB σ ) AB σ SEP Λ B ν 1 2ln(2)log K n with Λ A,ν : D(A) D(K) channels with output dim K

12 Quantum de Fine4 for Local Measurements Thm Part 1 For any state ρ AB1...B n D(A B 1... B n ) symmetric in the B subsystems and ν a distribumon over quantum operamons {Λ A,ν } ν : min max E Λ A,ν Λ ( B ρ AB σ ) AB σ SEP Λ B ν 1 2ln(2)log K n with Λ A,ν : D(A) D(K) channels with output dim K Obs: For {Λ A,ν } = {id}, we recover result of (B.,Christandl, Yard 10) as max I Λ B ρ AB σ AB Λ B ( ) 1 = ( ρ AB σ AB ) 1 LOCC

13 Quantum de Fine4 for Local Measurements Thm Part 1 For any state ρ AB1...B n D(A B 1... B n ) symmetric in the B subsystems and ν a distribumon over quantum operamons {Λ A,ν } ν : min max E Λ A,ν Λ ( B ρ AB σ ) AB σ SEP Λ B ν 1 2ln(2)log K n with Λ A,ν : D(A) D(K) channels with output dim K Obs2: Semi- classical quantum de Finef Thm

14 Quantum de Fine4 for Local Measurements (part 2) Thm Part 2 For any symmetric state there is a measure μ s.t. ρ B1...B n D(B 1... B n ) ( ) 1 max I B1 Λ B2... Λ Bk ρ B1...B Λ 2,...,Λ k µ(dσ )σ k k with 4k 2 log B n l L i=1 Λ Bj (X) = tr(m i,bj X) i i quantum- classical channels Compare with dimension independent classical de Finef (Diaconis, Freedman 80)

15 Short Quantum Proofs Given samsfiable 3- SAT instance on n variables, what s the size of the smallest proof for it? (Remainder 3- SAT: (x i or x j or x k ) and and (x p or x q or x s ))

16 Short Quantum Proofs Given samsfiable 3- SAT instance on n variables, what s the size of the smallest proof for it? (Remainder 3- SAT: (x i or x j or x k ) and and (x p or x q or x s )) Classically we need Ω(n) bits, unless there is a subexponenmal Mme algorithm for SAT Quantumly we need Ω(n) qubits, unless there is a quantum subexponenmal algorithm for SAT (Marriog and Watrous 05)

17 Short Quantum Proofs Given samsfiable 3- SAT instance on n variables, what s the size of the smallest proof for it? (Remainder 3- SAT: (x i or x j or x k ) and and (x p or x q or x s )) Classically we need Ω(n) bits, unless there is a subexponenmal Mme algorithm for SAT Quantumly we need Ω(n) qubits, unless there is a quantum subexponenmal algorithm for SAT (Marriog and Watrous 05) But what if we have a quantum state, but with the promise that parts of it are not entangled?

18 The Power of Unentanglement (Chen, Drucker 10, based on Aaronson et al 07): One can convince a quantum verifier that a n- variable SAT instance is samsfiable by sending m = O(n 1/2 polylog(n)) states, each of O(log(n)) qubits, assuming the promise that the states are not entangled with each other. The verifier only measures locally each state and post- process the classical outcomes. ψ 1 ψ 2 ψ 3 ψ 4 ψ m ( C 2 ) O(log(n)) M 1 M 2 M 3 M 4 M k Classical post- processing

19 The Power of Unentanglement (in complexity theory jargon.) BellQMA k (m, c, s): Analogue of QMA (or NP) where prover sends m unentangled proofs, each of k qubits, to the verifier, who measures each of the proofs and classical post- process the outcomes to decide whether to accept. - YES instance: there is a proof that makes him accept with prob. > c. - NO instance: no proof is accepted with prob. > s

20 The Power of Unentanglement (in complexity theory jargon.) BellQMA k (m, c, s): Analogue of QMA (or NP) where prover sends m unentangled proofs, each of k qubits, to the verifier, who measures each of the proofs and classical post- process the outcomes to decide whether to accept. - YES instance: there is a proof that makes him accept with prob. > c. - NO instance: no proof is accepted with prob. > s (Chen, Drucker 10) n- variable SAT is in BellQMA O(log(n)) (n 1/2 polylog(n), 2/3, 1/3) Implies BellQMA k (m) is not in QMA o(km 2- ε ) unless there is a subexponenmal algorithm For SAT. Square root advantage of unentanglement

21 The Limited Power of Unentanglement (Chen, Drucker 10) n- variable SAT is in BellQMA O(log(n)) (n 1/2 polylog(n), 2/3, 1/3) Implies BellQMA k (m) is not in QMA o(km 2- ε ) unless there is a subexponenmal algorithm For SAT. Square root advantage of unentanglement

22 The Limited Power of Unentanglement (Chen, Drucker 10) n- variable SAT is in BellQMA O(log(n)) (n 1/2 polylog(n), 2/3, 1/3) Implies BellQMA k (m) is not in QMA o(km 2- ε ) unless there is a subexponenmal algorithm For SAT. Square root advantage of unentanglement Cor. BellQMA k (m) is in QMA O(km 2 ) The square root advantage is all there is! Proof Idea: Instead of sending ψ m, prover sends ρ A1...A O(km 2 ). Verifier symmetrizes ρ, traces out all except m subsystems and runs original verificamon protocol. By Thm part 2 this works.

23 The Limited Power of Unentanglement (Chen, Drucker 10) n- variable SAT is in BellQMA O(log(n)) (n 1/2 polylog(n), 2/3, 1/3) Implies BellQMA k (m) is not in QMA o(km 2 ) unless there is a subexponenmal algorithm For SAT. Square root advantage of unentanglement Cor. BellQMA k (m) is in QMA O(km 2 ) The square root advantage is all there is! Proof Idea: Instead of sending ψ m, prover sends ρ A1...A O(km 2 ). Verifier symmetrizes ρ, traces out all except m subsystems and runs original verificamon protocol. By Thm part 2 this works.

24 Op:miza:on over Separable For a m- parmte matrix M define States h SEP(m) (M ) := max ψ 1,..., ψ m ψ 1,...,ψ m M ψ 1,...,ψ m Acceptance probability of BellQMA(m) is equivalent to esmmamng h SEP (M) for a Bell operator M. - - Chen and Drucker protocol translates into hardness result for esmmamng h SEP (M) De Finef bound translates into algorithms results for h SEP (M)

25 Subexponen:al Algorithm for Small Set Expansion (in passing) The result has implicamons to polynomial opmmizamon: O(log(n)) rounds of Lasserre hierarchy are enough for opmmizing polynomials over the hypersphere. Generalizes similar result by (Powers, Reznick 00) for the hypercube.

26 Subexponen:al Algorithm for Small Set Expansion (in passing) The result has implicamons to polynomial opmmizamon: O(log(n)) rounds of Lasserre hierarchy are enough for opmmizing polynomials over the hypersphere. Generalizes similar result by (Powers, Reznick 00) for the hypercube. (Barak, B., Harrow, Kelner, Steurer, Zhou 12) Algorithm (B, Christandl, Yard 10) and hardness result (Aaronson et al 07 + Harrow, Montanaro 10) for the quantum problem have implicamons to Small Set Expansion and Unique Games problems: Route to prove quasi- polynomial hardness of unique games and for giving a quasi- polynomial Mme alg. for it

27 Subexponen:al Algorithm for Small Set Expansion (in passing) The result has implicamons to polynomial opmmizamon: O(log(n)) rounds of Lasserre hierarchy are enough for opmmizing polynomials over the hypersphere. Generalizes similar result by (Powers, Reznick 00) for the hypercube. (Barak, B., Harrow, Kelner, Steurer, Zhou 12) Algorithm (B, Christandl, Yard 10) and hardness result (Aaronson et al 07 + Harrow, Montanaro 10) for the quantum problem have implicamons to Small Set Expansion and Unique Games problems: Route to prove quasi- polynomial hardness of unique games and for giving a quasi- polynomial Mme alg. for it - e.g. de Finef bound can be used to show Lasserre Hierarchy achieves the subexponenmal Mme algorithm of (Arora, Barak, Steurer 10) for SSE.

28 Tomography µ(dσ )σ k Suppose we have for unknown μ. We can perform tomography by measuring l copies. CondiMoned on obtained outcomes X we get w.h.p. a post- selected state µ X (dσ )σ k l where up to error exp(- lε 2 ), μ X only has support on states that are poly(d)ε- close to a state compamble with stamsmcs.

29 Tomography µ(dσ )σ k Suppose we have for unknown μ. We can perform tomography by measuring l copies. CondiMoned on obtained outcomes X we get w.h.p. a post- selected state µ X (dσ )σ k l where up to error exp(- lε 2 ), μ X only has support on states that are poly(d)ε- close to a state compamble with stamsmcs. Standard de Finef allows us to apply same reasoning to general ω n (by symmetrizing it, tracing out n- k copies and measuring l of the remaining k copies). Same conclusion as before, but now μ X has support on good states up to error exp(- lε 2 ) + d 2 k/n.

30 Tomography But we need to measure l = O(poly(d)) copies. ExponenMal in the number of qubits! Makes sense, since we need Suppose we have µ(dσ )σ k an exponenmal for unknown μ. We can number of parameters (in log(d)) to describe perform tomography by measuring l copies. CondiMoned on the state. obtained outcomes X we get w.h.p. a post- selected state Can we improve on this? µ X (dσ )σ k n where up to error exp(- lε 2 ), μ X only has support on states that are poly(d)ε- close to a state compamble with stamsmcs. Standard de Finef allows us to apply same reasoning to general ω n (by symmetrizing it, tracing out n- k copies and measuring l of the remaining k copies). Same conclusion as before, but now μ X has support on good states up to error exp(- lε 2 ) + d 2 k/n.

31 PreNy Good Tomography (Aaronson 06 Pregy- good- tomography thm) µ(dσ )σ k Given and a distribumon over measurements ν, suppose we measure l = O(poly(1/ε)log(d)) copies and get outcomes X. Let ρ X be any state compamble with X. Then w.h.p. the post- selected state can be wrigen as µ X (dσ )σ k l s.t., up to error ε, μ X only has support on states σ s.t. Pr M ~ν ( M(σ ) M(ρ X ) 1 > ε α ) < ε β

32 PreNy Good Tomography (Aaronson 06 Pregy- good- tomography thm) µ(dσ )σ k Given and a distribumon over measurements ν, suppose we measure l = O(poly(1/ε)log(d)) copies and get outcomes X. Let ρ X be any state compamble with X. Then w.h.p. the post- selected state can be wrigen as µ X (dσ )σ k l s.t., up to error ε, μ X only has support on states σ s.t. Pr M ~ν ( M(σ ) M(ρ X ) 1 > ε α ) < ε β Cor. The same works for general ω n with extra error O(k 2 log(d)/n)

33 PreNy Good Tomography (Aaronson 06 Pregy- good- tomography thm) µ(dσ )σ k Given and a distribumon over measurements ν, suppose we measure l = O(poly(1/ε)n) copies and get outcomes X. Let ρ X be any state compamble with X. Then w.h.p. the post- selected state can be wrigen as µ X (dσ )σ k l s.t., up to error ε, μ X only has support on states σ s.t. Pr M ~ν ( M(σ ) M(ρ X ) 1 > ε α ) < ε β Cor. The same works for general ω n with extra error O(k 2 log(d)/n)

34 Proof (of part 1) Let π AB1...B n K := E µ ( Λ Λ A,µ B 1... Λ )( Bn ρ ) AB1...B n µ µ K

35 Proof (of part 1) Let π AB1...B n K := E µ ( Λ Λ A,µ B 1... Λ )( Bn ρ ) AB1...B n µ µ K On one hand: max I(A : B 1...B n K) π = max E I(A : B 1...B n ) πµ Λ 1,...,Λ n Λ 1,...,Λ n µ log K

36 Proof (of part 1) Let π AB1...B n K := E µ On one hand: ( Λ Λ A,µ B 1... Λ )( Bn ρ ) AB1...B n µ µ K max I(A : B 1...B n K) π = max E I(A : B 1...B n ) πµ Λ 1,...,Λ n Λ 1,...,Λ n µ log K On the other hand we will show: max I(A : B 1...B n K) π Λ 1,...,Λ n n 2 ln2 min max E σ SEP Λ µ 2 Λ A,µ Λ( ρ σ ) 1

37 Remember π AB1...B n K := E µ We have: max Λ 1,...,Λ n I(A : B 1...B n K) π = max Λ 1,...,Λ n = max Λ 1,...,Λ n 1 Proof (of part 1) ( Λ Λ A,µ B 1... Λ )( Bn ρ ) AB1...B n µ µ K ( I(A : B 1 K) π I(A : B n KB 1...B n 1 ) π ) ( I(A : B 1 K) π max I(A : B n KB 1...B n 1 ) ) π Λ n

38 Remember π AB1...B n K := E µ We have: max Λ 1,...,Λ n I(A : B 1...B n K) π = max Λ 1,...,Λ n = max Λ 1,...,Λ n 1 Thus it suffices to prove Proof (of part 1) ( Λ Λ A,µ B 1... Λ )( Bn ρ ) AB1...B n µ µ K ( I(A : B 1 K) π I(A : B n KB 1...B n 1 ) π ) ( I(A : B 1 K) π max I(A : B n KB 1...B n 1 ) ) π max I(A : B n KB 1...B n 1 ) π 1 Λ n 2ln2 min max 2 E Λ A,µ Λ ( Bn ρ ABn σ ) ABn σ SEP Λ n µ 1 Λ n

39 Proof (of part 1) Note that I(A : B n KB 1...B n 1 ) π = E µ I(A : B k B 1...B k 1 ) πµ with π µ := ( Λ A,µ Λ B1... Λ )( Bn ρ ) AB1...B n Moreover I(A : B n B 1...B n 1 ) πµ = q i I(A : B n ) πi,µ ( ) ρ i for π i,µ := Λ A,µ Λ Bk with ({q i, ρ i } depend on Λ 1,, Λ n- 1, but not on Λ n ). By Pinsker s ineq. and convexity of x 2 : i ( ) ρ AB = q i ρ i i Thus: I(A : B n B 1,..., B n 1 ) πµ 1 2ln2 Λ & ) A, j Λ Bk ( ρ AB q i ρ i,a ρ i,bn + ' * I(A : B n B 1,..., B n 1 ) πµ 1 2ln2 max Λ B k i E Λ Λ & ( ρ q ρ ρ A, j µ Bk ' AB i i,a i,bn i 2 1 ) + * 2 1

40 Proof (part 2) Similar tricks, but applied to mulmparmcle mutual informamon: I(A 1 :... : A k ) := S( ρ A1...A k ρ A1... ρ ) Ak and using the useful inequality (Yang, Horodecki 3, Oppenheim, Song 07) I(A 1 :... : A k ) = I(A 1 : A 2 )+ I(A 3 : A 1 A 2 ) I(A k : A 1...A k 1 )

41 Part 2: de Fine4 with no symmetry

42 Quantum de Fine4 with no symmetry p 1,...,n j1 =a 1,..., j t =a t Thm Let p 1,,n be a prob. distribumon over Σ n. Let be the probability condimoned on observing (j 1,, j t ) = (a 1,, a t ). Then: E j 1,..., j t E a1,...,a t E i1,...,i k p i1...i k j 1 =a 1,..., j t =a t p i1 j 1 =a 1,..., j t =a t... p ik j 1 =a 1,..., j t =a t 1 2ln(2)k(k 1)log Σ t 1 Based on bound by (Raghavendra, Tan 11) (proposed in the context of bounding convergence of Lasserre hierarchy for certain CSP)

43 Quantum de Fine4 with no symmetry p 1,...,n j1 =a 1,..., j t =a t Thm Let p 1,,n be a prob. distribumon over Σ n. Let be the probability condimoned on observing (j 1,, j t ) = (a 1,, a t ). Then: E j 1,..., j t E a1,...,a t E i1,...,i k p i1...i k j 1 =a 1,..., j t =a t p i1 j 1 =a 1,..., j t =a t... p ik j 1 =a 1,..., j t =a t 1 2ln(2)k(k 1)log Σ t 1 Quantum Version: p 1,...,n = Λ n (ρ) for informamonally complete POVM Λ. Same theorem (for trace norm) with error term poly(d k k(k 1)logd ) t 1

44 Quantum PCP The PCP conjecture: There is a ε > 0 s.t. it s QMA- complete to determine whether the mean groundenergy of a 2- local Hamiltonian on n qubits is 0 or more than ε. (Bravyi, divincenzo Loss, Terhal XX) Equivalent to conjecture for O(1)- local Hamiltonians over qdits. (Reminder) 2- Local Hamiltonian: Mean energy: e 0 := E 0 / E E 0 : groundenergy E : # of edges in G i~ G j H ij By PCP theorem, the problem is at least NP- hard H ij i j

45 Quantum PCP: Non- trivial approxima:on in NP Cor: There is a constant c > 0 such that the following problem is NP- complete: Given a 2- Local Hamiltonian on n qudits with interacmon graph of average degree deg, decide whether e 0 = 0 or e 0 > cd^3/deg. Proof Outline: NP- hardness follows from PCP theorem + Raz parallel repemmon theorem InteresMng part: There is always a product state assignment with energy no bigger than e 0 + cd 3 /deg (proof by condimoning de Finef bound )

46 Quantum PCP: Non- trivial approxima:on in NP Cor: There is a constant c > 0 such that the following problem is NP- complete: Given a 2- Local Hamiltonian on n qudits with interacmon graph of average degree deg, decide whether e 0 = 0 or e 0 > cd^3/deg. This gives evidence against the PCP conjecture. At least it can only hold for ε < cd 3 /deg. Note that classical PCP holds for graphs for which Ω(1) = ε >> poly(σ)/deg = o(1).

47 Summary Playing with I(A:B K) leads to new (semi- classical) quantum de Finef Theorems with (i) beger error or (ii) valid for general quantum states.

48 Summary Playing with I(A:B K) leads to new (semi- classical) quantum de Finef Theorems with (i) beger error or (ii) valid for general quantum states. The improved error de Finef Thm is useful in (a) showing hardness of a constant error approximamon of quantum value of games and (b) give a matching simulamon to the BellQMA protocol for SAT of Chen and Drucker.

49 Summary Playing with I(A:B K) leads to new (semi- classical) quantum de Finef Theorems with (i) beger error or (ii) valid for general quantum states. The improved error de Finef Thm is useful in (a) showing hardness of a constant error approximamon of quantum value of games and (b) give a matching simulamon to the BellQMA protocol for SAT of Chen and Drucker. The de Finef Thm with no symmetry is useful in (a) giving a PTAS for esmmamng the energy of dense Hamiltonians and (b) give evidence against the quantum PCP conjecture.

50 Open Quesitons 1. Get more de Finef Theorems (perhaps combine semi- classical ones with exponenmal de Finef?)

51 Open Quesitons 1. Get more de Finef Theorems (perhaps combine semi- classical ones with exponenmal de Finef?) 2. Prove NP- hardness of quantum value for 2- players game

52 Open Quesitons 1. Get more de Finef Theorems (perhaps combine semi- classical ones with exponenmal de Finef?) 2. Prove NP- hardness of quantum value for 2- players game 3. (Dis)prove that QMA k (2) is in QMA O(k) 2.

53 Open Quesitons 1. Get more de Finef Theorems (perhaps combine semi- classical ones with exponenmal de Finef?) 2. Prove NP- hardness of quantum value for 2- players game 3. (Dis)prove that QMA k (2) is in QMA O(k) (Dis)prove Quantum PCP conjecture

54 Open Quesitons 1. Get more de Finef Theorems (perhaps combine semi- classical ones with exponenmal de Finef?) 2. Prove NP- hardness of quantum value of games 3. (Dis)prove that QMA k (2) is in QMA O(k) (Dis)prove Quantum PCP conjecture 5. Further develop the connecmon of BellQMA (opmmizamon of Bell observables over separable states) to the hardness of Small Set Expansion and Unique Games problems: - Can we prove quasi- polynomial hardness of UG by this approach? - Can we prove convergence of O(log(n)) rounds of Lasserre hierarchy for Small Set Expansion by this approach?

55 Thanks!