Quantum algorithms based on span programs

Transcription

1 Quantum algorithms based on span programs Ben Reichardt IQC, U Waterloo [arxiv: ]

2 Model: Complexity measure: Quantum algorithms Black-box query complexity Span programs Witness size [KW 93] [RŠ 08] Optimal span program witness size is characterized by an SDP (Adv±) quantum query complexity characterized by the same SDP Span programs compose easily quantum algorithms compose easily optimal quantum algorithms for many formula-evaluation problems

3 Farhi, Goldstone, Gutmann 07 algorithm Theorem ([FGG 07]): A balanced binary AND-OR formula can be evaluated in time N ½+o(1). x 1 x 2 x 3 x 4 x 5 x 6 x 7 x 8 AND AND AND AND OR OR AND ϕ(x)

4 Farhi, Goldstone, Gutmann 07 algorithm Theorem ([FGG 07]): A balanced binary AND-OR formula can be evaluated in time N ½+o(1). Convert formula to a tree, and attach a line to the root Add edges above leaf nodes evaluating to one =0 =1

5 Farhi, Goldstone, Gutmann 07 algorithm Theorem ([FGG 07]): A balanced binary AND-OR formula can be evaluated in time N ½+o(1). Convert formula to a tree, and attach a line to the root Add edges above leaf nodes evaluating to one x 1 x 2 x 3 x 4 x 5 x 6 x 7 x 8 AND AND AND AND OR OR AND ϕ(x)

6 Continuous-time quantum walk [FGG 07] =0 =1

7 Quantum walk ψ t = e ia Gt ψ 0

8 Quantum walk ψ t = e ia Gt ψ 0 Wave transmits! output ϕ(x) = 1

9 Two generalizations: Theorem ([FGG 07]): A balanced binary AND- OR formula can be evaluated in time N ½+o(1). Unbalanced AND-OR Balanced, More gates Theorem ([ACRŠZ 07]): An approximately balanced AND-OR formula can be evaluated with O( N) queries (optimal!). Theorem ([RŠ 08]): A balanced formula φ over a gate set including all three-bit gates can be evaluated in O(Adv(φ)) queries (optimal!). A general AND-OR formula can be evaluated with N ½+o(1) queries. (Running time is poly-logarithmically slower in each case, after preprocessing.)

10 Span programs [Karchmer, Wigderson 93] Theorem ([R, Špalek 08]): Let Define f : {0, 1} n {0, 1} f k : {0, 1} nk {0, 1} by f f f f k levels Let P be a span program computing f. Then Q(f k )=O ( wsize(p ) k) f f f. quantum query complexity span program complexity measure Many optimal algorithms: for f any 3-bit function (e.g., AND, OR, PARITY, MAJ 3 ), and ~70 of ~200 different 4-bit functions Open problems: How can we find more good span programs? Are span programs useful for developing other qu. algorithms?

11 Time complexity = number of gates Query complexity = number of input bits that must be looked at e.g., Search: Quantum query complexity Q(f) classical query complexity of OR n is Θ(n) for both deterministic & randomized algorithms quantum query complexity is Q(OR n )=Θ( n), by Grover search Most quantum algorithms are based on good qu. query algorithms Provable lower bounds

12 Two methods to lower-bound Q(f) Polynomial method: Q(f) =Ω ( deg(f) ) for total functions Q(f) D(f) =O ( deg(f) 6 ) Adversary method: How much can be learned from a single query? Adv(f) = max adversary matrices Γ: Γ 0 Γ max j [n] Γ j Incomparable lower bounds: deg Adv Element Distinctness: n 2/3 n 1/3 Ambainis formula: 2 d 2.5 d (n=4 d )

13 Polynomial method Q(f) =Ω( deg(f)) Adversary bound Adv(f) = Q(f) =Ω(Adv(f)) max adversary matrices Γ: Γ 0 Γ max j [n] Γ j General adversary bound Adv ± (f) = [Høyer, Lee, Špalek 07] max adversary matrices Γ Γ max j [n] Γ j Q(f) =Ω(Adv ± (f)) deg Adv Adv ± Q Element Distinctness: n 2/3 n 1/3?? n 2/3 Ambainis formula: 2 d 2.5 d d d (n=4 d )

14 The general adversary bound is nearly tight Theorem 1: For any f : {0, 1} n {0, 1} and Q(f) =Ω(Adv ± (f)) [HLŠ 07] ( Q(f) =O Adv ± log Adv ± ) (f) (f) log log Adv ± (f) Nearly tight characterization of quantum query complexity; the general adversary bound is always (almost) optimal Adv deg Adv ± Q Element Distinctness: n 1/3 n 2/3 n 2/3 /log n n 2/3 Ambainis formula: 2.5 d 2 d d d (n=4 d ) Simpler, greedier semi-definite program than [Barnum, Saks, Szegedy 03]

15 Two steps to proving Theorem 1 Theorem 1: For any f : {0, 1} n {0, 1} ( Q(f) =O Adv ± log Adv ± ) (f) (f) log log Adv ± (f) Theorem 2: For any boolean function f, inf wsize(p ) = P computing f Adv± (f) Theorem 3: For any span program P computing f, ( ) log wsize(p ) Q(f) =O wsize(p ) log log wsize(p )

16 Theorem 2: inf wsize(p ) = P computing f Adv± (f) = O(Q(f)) ( ) log wsize(p ) Theorem 3: If P computes f, Q(f) =O wsize(p ) log log wsize(p ) Span programs are equivalent to quantum computers! (up to a log factor) Model: Complexity measure: Quantum algorithms query complexity Span programs witness size

17 Theorem 2: inf P computing f wsize(p ) = Adv± (f) = O(Q(f)) Theorem 3: If P computes f, ( ) log wsize(p ) Q(f) =O wsize(p ) log log wsize(p ) Thm. [RŠ 08]: Q(fP k )=O ( wsize(p ) k) Thm. [HLŠ 07, R 09]: Adv ± (f k )=O ( Adv ± (f) k) Using Theorem 2, implies optimal qu. algorithm for evaluating balanced formulas over any finite gate set

18 Def.: Read-once formula ' on gate set S = Tree of nested gates from S, with each input appearing once Problem: Evaluate '(x). x 7 x 8 x 1 x 2 x 3 x 4 x 6 x 10 x OR x 9 1 x 11 x 12 Example: S = {AND, OR}: x 5 AND AND AND OR OR AND ϕ(x)

19 x1 Classical Quantum x 1 x 2 x N OR Θ(n) Θ( n) [Grover 96] Balanced AND-OR x 1 x 2 x 3 x 4 x 5 x 6 x 7 x 8 AND AND AND AND Θ(n ) (fan-in two) Θ( n) OR AND OR (fan-in two case) [S 85, SW 86, S 95] [Snir 85] [Saks, Wigderson 86] [Santha 95] [Farhi, Goldstone, Gutmann 07] [Ambainis, Childs, R, Špalek, Zhang 07] General read-once AND-OR x1... x1 Ω(n 0.51 ) [HW 91] Conj.: [Heiman, Ω(D(f) Wigderson ]) [SW 86] Ω( n), n 2O( log n) [Barnum, Saks 04] [ACRŠZ 07] Balanced MAJ 3 MAJ MAJ MAJ MAJ MAJ MAJ MAJ MAJ MAJ MAJ MAJ MAJ MAJ ϕ(x) Ω(2.333 d ), O(2.654 d ) [JKS 03] [Jayram, Kumar, Sivakumar 03] Θ(2 d ). [RŠ 08]

20 OR n (Search) Balanced AND-OR Classical Θ(n) Θ(n ) Quantum Θ( n) Θ( n) General read-once AND-OR Ω(n 0.51 ) Conj.: Ω(D(f) ) [SW 86] Ω( n), n 2O( (log n)) Balanced MAJ 3 Ω(2.333 d ), O(2.654 d ) Θ(2 d ) Approximately balanced formula over an arbitrary finite gate set Unbalanced formulas??? Θ(Adv±(f)) [R 09] Query complexity now understood, but not time-complexity

21 Two steps to proving Theorem 1 Theorem 1: For any f : {0, 1} n {0, 1} ( Q(f) =O Adv ± log Adv ± ) (f) (f) log log Adv ± (f) Theorem 2: For any boolean function f, inf wsize(p ) = P computing f Adv± (f) Theorem 3: For any span program P computing f, ( ) log wsize(p ) Q(f) =O wsize(p ) log log wsize(p )

22 t Definition: Span program P on n bits V 1,0 vector space V 0 target vector t V n,1 subspaces V 1,0 V 1,1 V 2,0 V 2,1 V n,0 V n,1 P computes f P : {0, 1} n {0, 1} f P (x) =1 t Span j V j,xj Example: V=C 2 t V 1,0 = V 2,0 = 0 V 1,1 V 2,1 = f P = AND 2

23 Example t t P: P : V 1,1 V 1,0 = V 2,0 = 0 V 1,1 V 2,1 V 1,0 = V 2,0 = 0 V 2,1 f P = f P = AND 2 but P seems better wsize(p, 11) > wsize(p, 11)

24 Span programs in coordinates Span program P: target t V 1,0 V 1,1 V n,0 V n,1 A = v 1,0,1 v 1,0,m v 1,1,1 v 1,1,m v n,0,1 v n,0,m v n,1,1 v n,1,m Π(x) = projection onto available coordinates Then f P (x) =1 t Range(AΠ(x)) Def.: If f(x) =1, let wsize(p, x) = min w : AΠ(x) w = t w 2 (intuition: want a short witness)

25 f P (x) =1 = t Range(AΠ(x)) wsize(p, x) = min w : AΠ(x) w = t w 2 (intuition: want a short witness) f P (x) =0 = t / Range(AΠ(x)) w orthogonal to Range(AΠ(x)) with t w 0 w 0 t Range(AΠ(x)) = span of available columns of A wsize(p, x) = min w : t w =1 w Π(x)A=0 A w 2 (intuition: if t is close to the span of available columns of A, then wsize should be large) Definition: wsize(p ) = max x wsize(p, x)

26 Define a span program P as follows: Vector space V = C Example: Search (OR) Target vector t = n 1/4 ( V 1,0 V 1,1 A = 0 1 V 2,0 V 2,1 0 1 V n,0 V n,1 0 1 ) f P = OR n wsize(p, 0 n )= n w =1/n 1/4 wsize(p, ) = n w = (0,n 1/4, 0,..., 0)... wsize(p )= n

27 f P (x) =1 = wsize(p, x) = min w : AΠ(x) w = t w 2 (intuition: want a short witness) f P (x) =0 = wsize(p, x) = min w : t w =1 w Π(x)A=0 A w 2 (intuition: if t is close to the span of available columns of A, then wsize should be large) Definition: wsize(p ) = max x Why is this the right definition? wsize(p, x) 1. Negating a span program leaves wsize invariant 2. Composing span programs: wsize is multiplicative 3. Leads to quantum algorithms Q(fP k )=O ( wsize(p ) k) [RŠ 08] Q(f P )=Õ( wsize(p ) ) (Theorem 3)

28 Proof of Theorem 2 Theorem 2: For any boolean function f, inf wsize(p ) = P : f P =f Adv± (f)

29 Theorem 2: For any boolean function f, Proof: inf wsize(p ) = P : f P =f Adv± (f) We look for span programs where the rows of A correspond to {x : f(x) = 0} target is all 1s vector 1 1 t =.. 1 V 1,0 V 1,1 V n,0 V n,1 v 1,0,1 v 1,0,m v 1,1,1 v 1,1,m v n,0,1 v n,0,m v n,1,1 v n,1,m 0 0 f 1 (0) x = and in the row corresponding to x, the columns available for input x are all zero (Such span programs are said to be in canonical form [KW 93].) This form guarantees that f(x) = 0 f P (x) =0 ( w = x itself is the witness)

30 x = Example: AND t = f P = AND n V 1,1 V 2,1 V n,1...

31 t = V 1,0 V 1,1 V n,0 V n,1 v 1,0,1 v 1,0,m v 1,1,1 v 1,1,m v n,0,1 v n,0,m v n,1,1 v n,1,m v x1 0 0 v xn f 1 (0) x = in the xth row, the columns available for input x are all 0; hence Now consider a f(x) = 0 f P (x) =0 y f 1 (1) We want to find vectors v y1,..., v yn such that x f 1 (0), 1= v xj v yj j:x j y j The witness size is max x v xj 2 j

32 t = V 1,0 V 1,1 V n,0 V n,1 v 1,0,1 v 1,0,m v 1,1,1 v 1,1,m v n,0,1 v n,0,m v n,1,1 v n,1,m v x1 0 0 v xn f 1 (0) x = = inf P : f P =f wsize(p ) inf max { v xj } : if f(x) f(y), P x j:x j y v xj v yj =1 j v xj 2 j = min max x, j X x, j X 0: (x,y), P x j j:x j y x,j X y,j =1 j (Cholesky decomposition) = Adv ± (f) (SDP duality)

33 Proof of Theorem 3 Theorem 3: For any span program P, Q(f P )=O ( ) log wsize(p ) wsize(p ) log log wsize(p ) 1. Correspondence between P and bipartite graphs G P (x) 2. Eigenvalue-zero eigenvectors imply an effective spectral gap around zero 3. Quantum algorithm for detecting eigenvectors of structured graphs

34 ( ) A Def.: Let G 0 (x) be the bipartite graph with biadjacency matrix* 1 Π(x) ( ) t A Let G 1 (x) 0 1 Π(x) Bipartite graph with edge weights given by A Input vertices & edges Output vertex & edges, weights given by t each weight-one input edge is present iff corresponding input bit x j = 0 G 1 (x) G 0 (x) *Adjacency matrix is ( 0 B B 0 )

35 ( ) A Def.: Let G 0 (x) be the bipartite graph with biadjacency matrix* 1 Π(x) ( ) t A Let G 1 (x) 0 1 Π(x) Lemma: G 1 (x) has an eigenvalue-zero eigenvector overlapping the output vertex f P (x) =1 G 0 (x) has an eigenvalue-zero eigenvector overlapping ( )( ) t A 1 Proof: For G 1 (x): 0= 0 1 Π(x) v t f P (x) =0 Π(x) v = v and t = A v f P (x) =1 Note: After normalizing the eigenvector, squared overlap on the output vertex is 1/(1+wsize(P,x)). Small wsize large overlap

36 2 Small-eigenvalue analysis When f P (x) =0, G 0 (x) having an eigenvalue-zero eigenvector with large (δ) squared overlap on t implies that G 1 (x) has a large ( δ) effective spectral gap around zero. G 0 (x) ( A ) 1 Π(x) G 1 (x) ( t A ) 0 1 Π(x) Idea: Think of G 1 (x) as a perturbation of G 0 (x).

37 2 Small-eigenvalue analysis Theorem. Let G be a weighted bipartite graph on V = T U. Assume that for some δ > 0 and t C T, the adjacency matrix has an eigenvalue-zero eigenvector ψ with t ψ T 2 ψ 2 δ Let G be the same as G except with a new vertex v added to the U side, and for i T the new edge (τ i,v) weighted by i t. Take { λ } a complete set of orthonormal eigenvectors of A G. Then for all Λ 0, λ: λ Λ v λ 2 8Λ2 δ Good eigenvalue-zero eigenvector (Think δ =1/wsize(P ) 2, Λ =1 /wsize(p ) ) Large effective spectral gap

38 3 Quantum algorithm Scale the target vector down by 1/ wsize(p). When f P (x) = 1, there is an eigenvector with large ( ½) squared overlap on the output vertex When f P (x) = 0, there is a spectral gap of 1/wsize(P) around zero Algorithm: Start at the output vertex and measure the adjacency matrix (as a Hamiltonian). Output 1 iff the measurement returns 0. Key technical step: Since we have no control over the norm of the matrix, need [Cleve, Gottesman, Mosca, Somma, Yonge-Mallo 09] to simulate the measurement with a log / log log overhead factor.

39 Summary Main corollaries Theorem 2: For any boolean function f, inf wsize(p ) = P computing f Adv± (f) Theorem 3: For any span program P, ( ) log wsize(p ) Q(f) =O wsize(p ) log log wsize(p ) The general adversary bound is (almost) optimal for every total or partial function f : {0, 1} n poly(log n) {0, 1} Optimal quantum algorithm for evaluating balanced formulas over any finite gate set Span programs are (almost) equivalent to quantum computers

40 Recipe for finding optimal quantum query algorithms Find a solution to: Adv ± (f) = min max x X j x {X j 0}: (x,y), P x j j:x j y x X j y =1 j ( ) Take the Cholesky decomposition: { v xj } : v xj v yj = x X j y Use the entries of the vectors to weight the edges of a graph, and run phase estimation on the quantum walk 1 n B G = x x j v xj. x:f(x)=0 j=1 1 But how can we find good solutions to ( )?

41 Open problems Can the log overhead factor be removed? Is Adv± tight in the continuous-query model? Functions with non-binary domains? f : {1, 2,..., k} n {0, 1} Is there a good classical algorithm for evaluating span programs? Any algorithm faster than O(wsize(P) 6 ) would give a better relationship between classical and quantum query complexities. D(f) =O(Q(f) 6 ) Our results apply to both total and partial functions, though (e.g., Simon s prob.) Robustness? More explicit and time-efficient algorithms

42 Blank slide Thank you!