Span-program-based G ( k ) G ( 1 ) k 1 quantum algorithm for . - PowerPoint PPT Presentation

G ( ρ ) ρ NOT Span-program-based G ( ρ k ) G ( ρ 1 ) ρ k ρ 1 quantum algorithm for . . . . . . OR formula evaluation G ( ρ 1 ) G ( ρ 2 ) G ( ρ 3 ) ρ 1 ρ 2 ρ 3 MAJ Ben Reichardt Robert Š palek G ( ρ k ) G ( ¬ ρ k ) G ( ρ 1 ) G ( ¬ ρ 1 ) ρ 1 ρ k Caltech Google . . . . . . . . . EQUAL | t � e u r t A f o f o 0 n a s n p s m u l o c ) ) ) ) ρ ρ

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

Def: Read-once formula φ on gate set S = Tree of nested gates from S , with x 7 x 8 each input appearing once x 1 x 2 x 3 x 4 x 6 x 9 x 10 x 11 x 12 OR x 1 x 5 AND AND AND OR Ex: S = {AND, OR}: OR Gates cannot have fan-out! AND (unlike in a circuit ) ϕ ( x ) Problem: Evaluate φ (x).

Classical complexity of formula evaluation x N x 1 x 2 · · · Θ (N) OR Balanced AND-OR x 3 x 4 x 5 x 6 x 7 x 8 x 1 x 2 AND AND AND AND Θ (N 0.753… ) (fan-in two) OR OR [Snir ‘85, Saks & Wigderson ‘86, Santha ‘95] AND . . . x 1 x 1 x 1 Balanced MAJ 3 Ω ((7/3) depth ) = R 2 (f) = O((2.6537…) depth ) MAJ MAJ MAJ MAJ MAJ MAJ MAJ MAJ MAJ MAJ MAJ MAJ [Jayram, Kumar, Sivakumar ’03] MAJ ϕ ( x )

Classical Quantum x N x 1 x 2 · · · Θ (N) Θ ( √ N) [Grover ‘96] OR Balanced AND-OR x 3 x 4 x 5 x 6 x 7 x 8 x 1 x 2 AND AND AND AND Θ (N 0.753… ) Θ ( √ N) (fan-in two) OR OR [Farhi & Goldstone & Gutmann ’07, ACR Š Z ‘07] [S‘85, SW‘86, S‘95] AND . . . x 1 x 1 x 1 Ω ((7/3) d ), Θ ( 2 d =N log 3 2 ) Balanced MAJ 3 MAJ MAJ MAJ MAJ MAJ MAJ MAJ MAJ MAJ O((2.6537…) d ) MAJ MAJ MAJ [JKS ’03] [R Š ‘07] . . . MAJ ϕ ( x )

x 1 x 2 x 3 x 4 x 5 x 6 x 7 x 8 Two generalizations of [FGG ‘07] AND-OR NAND NAND NAND NAND algorithm: • NAND Theorem ([FGG ‘07, CCJY ‘07]) : NAND A balanced binary AND-OR formula can be NAND evaluated in time N ½ +o(1) . Balanced, more gates [R Š , STOC‘08] Unbalanced AND-OR • Theorem: A balanced (“adversary- [ACR Š Z, FOCS‘07] bound-balanced”) formula φ over a • Theorem : gate set including all three-bit gates • An “approximately balanced” AND-OR (and more…) can be evaluated in O(ADV( φ )) queries (optimal!) . formula can be evaluated with O( √ N) queries (optimal for read-once!) . • A general AND-OR formula can be evaluated with N ½ +o(1) queries.

Recursive 3-bit majority tree 3 d . . . • Best quantum lower bound is x 1 x 1 x 1 [LLS‘05] ADV( ϕ ) = 2 d � � Ω MAJ MAJ MAJ MAJ MAJ MAJ MAJ MAJ MAJ • Expand majority into {AND, OR} gates: d MAJ MAJ MAJ MAJ 3 ( x 1 , x 2 , x 3 ) = ( x 1 ∧ x 2 ) ∨ ( x 3 ∧ ( x 1 ∨ x 2 )) MAJ ϕ ( x ) ∴ {AND, OR} formula size is ≤ 5 d ∴ O( √ 5 d ) = O(2.24 d )-query algorithm [FGG, ACR Š Z ‘07] [R Š ‘07] algorithm • Theorem: A balanced (“adversary- • New: O(2 d )-query quantum algorithm bound-balanced”) formula φ over a gate set including all three-bit gates (and more…) can be evaluated in O(ADV( φ )) queries (optimal!) .

Converting formula into a tree G ( ρ ) ρ NOT G ( ρ k ) G ( ρ 1 ) ρ k ρ 1 . . . . . . NAND OR G ( ρ 1 ) G ( ρ 2 ) G ( ρ 3 ) ρ 1 ρ 2 ρ 3 MAJ 3 G ( ρ 1 ) G ( ¬ ρ 2 ) G ( ¬ ρ 1 ) G ( ρ 2 ) G ( ρ k ) G ( ¬ ρ k ) G ( ρ 1 ) G ( ¬ ρ 1 ) ρ 2 ρ 1 ρ 1 ρ k . . . . . . . . . PARITY EQUAL (with appropriate edge weights)

⇒ ⇒ • Main Theorem: • φ (x)=1 A G has λ =0 eigenvector with Ω (1) support on the root. G ( ρ 1 ) G ( ρ 2 ) G ( ρ 3 ) ρ 1 ρ 2 ρ 3 • φ (x)=0 A G has no eigenvectors overlapping the root with | λ |<1/O(ADV( φ )). MAJ 3 G ( ρ 1 ) G ( ¬ ρ 2 ) G ( ¬ ρ 1 ) G ( ρ 2 ) ρ 2 ρ 1 PARITY x 1 x 2 x 3 x 1 x 2 x 3 x 1 x 2 x 3 MAJ 3 x 4 x 4 x 4

⇒ ⇒ • Main Theorem: • φ (x)=1 A G has λ =0 eigenvector with Ω (1) support on the root. • φ (x)=0 A G has no eigenvectors overlapping the root with | λ |<1/O(ADV( φ )). quantitative bounds needed to analyze the running time

⇒ ⇒ • Main Theorem: • φ (x)=1 A G has λ =0 eigenvector with Ω (1) support on the root. • φ (x)=0 A G has no eigenvectors overlapping the root with | λ |<1/O(ADV( φ )). Fast Quantum Algorithm: • Start at the root • Apply phase estimation to the quantum walk with precision 1/O(ADV( φ )) • If measured phase is 0, output “ φ (x)=1.” Otherwise, output “ φ (x)=0.” Running time is O(ADV( φ )) Precision- δ phase estimation on a unitary U, starting at an corr. e-state, returns the e-value | eigenvector � | λ � λ ± δ eigenvalue ± δ to precision δ , except w/ prob. 1/4. It uses O(1/ δ ) calls to c-U.

Computation of Eigenvalue-zero ⇔ formula eigenvector of tree x= 1 0 0 0 1 1 0 0 0 0 1 11 1 0 1 Input dependence • Substitutions define G(0 N ); to 1 1 0 1 1 0 0 1 define graph G(x), delete edges to all leaves evaluating to x i =1. 0 1 1 1 NAND 00 1 1 0 Q: What is an eigenvalue-0 eigenvector of a graph? 01 1 1 A: Assignment of coefficients 10 1 to each vertex, such that sum 11 0 of neighboring coefficients adds up to 0. (each edge labeled by the evaluation of the NAND sub-formula above it)

Computation of Eigenvalue-zero ⇔ formula eigenvector of tree Induction Claim: Each edge x= 1 0 0 0 1 1 0 0 0 0 1 11 1 0 1 gives a “dual-rail” encoding for the evaluation of the sub- formula above that edge… sub-formula φ v The λ =0 eigenvector v “output edge” of G( φ v ,x) is: Supported here ⇔ φ v (x)=false Supported here r ⇔ φ v (x)=true

3-Majority gate gadget G ( ϕ 1 ) G ( ϕ 2 ) G ( ϕ 3 ) ϕ 1 ϕ 2 ϕ 3 1 1 1 MAJ 3 ω 2 ω 1 1 1 1 MAJ 3 1 ω = e 2 π i/ 3

Eigenvalue-zero Computation of ⇔ eigenvector of MAJ 3 gate graph • Induction hypothesis: λ =0 eigenvectors w 3 w 1 w 2 on sub-formula graphs G( φ i ) compute the sub-formulas v 1 v 2 v 3 • Constraints     α s α r   1 1 1 0 0 α c � 1 �   1 1 1 ω 2 α v 1     ω ω 2  = 0 = 1 0 1 0 α w 1       ω 2 0 1 ω α v 2    1 0 0 1 1 ω α w 2   A G α v 3 s α w 3 c • When can α r be nonzero (i.e., gadget evaluates to true)? 1. Only depends on first constraint eq.’s ( ) α v 1 , α v 2 , α v 3 2. Need , but α v 1 + α v 2 + α v 3 � = 0 r α v 1 + ωα v 2 + ω 2 α v 3 = 0 3. Can only have if input i evaluates to true α v i � = 0 • At least two inputs φ i must be true to satisfy both ✓ MAJ 3 constraints nontrivially.

General graph gadgets Input edges Arbitrary weighted bipartite graph Induction Claim: Each edge (p,v) gives a “dual-rail” encoding… v The λ =0 eigenvector Supported on v Output edge of G( φ v ,x) is: ⇔ φ v (x)=false p Supported on p ⇔ φ v (x)=true

Span program definition • Substitution rules defining G come from span programs . [Karchmer, Wigderson ’93] • Def: A span program P is: • A target vector t in vector space V over C , • Input vectors v j each associated with a literal from { x 1 , x 1 , . . . , x n , x n } Span program P computes f P : {0,1} n → {0,1}, f P (x) = 1 ⇔ t lies in the span of { true v j } • Ex. 1: P: x 1 x 2 x 3 t = ( 1 ( 1 ( 1 ( 1 0 ) a ) b ) c ) with a,b,c distinct and nonzero. ➡ f P = MAJ 3 •

Span program ⇔ Bipartite graph gadget with t=(1,0,…,0) b 1 b 2 b 3 E.g., MAJ 3 : a 3 a 2 a 1 x 1 x 2 x 3 � � 1 1 1 1 b 1 1 c t = a b c 0 1 a b O b C a O input edges In general:   1 A 0   t =   . .   . . . .   constraints 0 output edge

⇔ Composing span programs • Given span programs for g, h 1 , …, h k , immediately get s.p. for f = g ◦ ( h 1 , . . . , h k ) SP composition Graph gadget composition inputs • Ex.: MAJ 3 (x 1 , x 2 , MAJ 3 (x 4 ,x 5 ,x 6 )): b c t x 1 x 2 x 3 x 4 x 5 1 a � � 1 1 1 1 0 0 0 MAJ 3 a b c 0 0 0 0 0 0 0 1 1 1 1 b c a b c 0 0 0 0 a MAJ 3 output

⇔ Composing span programs • Given span programs for g, h 1 , …, h k , immediately get s.p. for f = g ◦ ( h 1 , . . . , h k ) SP composition Graph gadget composition • Ex.: MAJ 3 (x 1 , x 2 , MAJ 3 (x 4 ,x 5 ,x 6 )): t x 1 x 2 x 3 x 4 x 5 1 � � 1 1 1 1 0 0 0 a b c 0 0 0 0 0 0 0 1 1 1 1 b c b c a b c 0 0 0 0 a a

⇔ ∴ ⇔ Eigenvalue-zero lemmas • Define: G P (x) by deleting edges to true input literals • Lemma: f P (x)=1 ⇔ ∃ λ =0 eigenvector of A G P (x) supported on a O . b 2 b 1 b 3 b m … TRUE TRUE a 3 a m a 2     a 1 1 A 0     t =   .   .   .     0 b O . . . c 1 c C a O �� �� . . . . . . . . �

Eigenvalue-zero lemmas • Define: G P (x) by deleting edges to true input literals • Lemma: f P (x)=1 ⇔ ∃ λ =0 eigenvector of A G P (x) supported on a O . • Lemma: Delete output edge (a O , b O ). Then f P (x)=0 ⇔ ∃ λ =0 eigenvector supported on b O . Proof: f P (x) is false ⇔ |t 〉 not in span of true columns of A | t � span of true columns of A 0 . . . b O �� �� . . . . . . . . �