Reflections Reflections

Transcription

1 Reflections Reflections for quantum query algorithms Ben Reichardt University of Waterloo

2 Reflections Reflections for quantum query algorithms Ben Reichardt University of Waterloo

3 Reflections for quantum query algorithms Theorem: An optimal quantum query algorithm for Reflections evaluating any boolean function can be built out of two fixed reflections R1 R2 R1 R2 R1 R2

4 Goal: Evaluate f: {0,1} n {0,1} using j x {0, 1} n x j

5 U0 j1 query x xj1 U1 j2 query x xj2 UT f(x)

6 Query complexity models: Deterministic Randomized - bounded-, zero- or one-sided error Nondeterministic (Certificate complexity) Quantum

7 Quantum query complexity j x {0, 1} n ( 1) x j j ( 1) x 1 1 +( 1) x 2 2 U0 query x U1 query x UT f(x) w/ prob. 2/3

8 Theorem: An optimal quantum query algorithm for evaluating any boolean function can be built out of two fixed reflections R Ox R Ox R Ox

9 U0 Ox U1 Ox U2 Ox Clearly, w.l.o.g., may assume Ut is independent of t U = T t +1 t U t + c.c. t=0 or, may assume Ut is a reflection t R t = 1 0 U t U t

10 Theorem: An optimal quantum query algorithm for evaluating any boolean function can be built out of two fixed reflections R Ox R Ox R Ox Theorem: The general adversary lower bound on quantum query complexity is also an upper bound

11 A certificate for input x is a set of positions whose values fix f. (Given a certificate for the input, it suffices to read those bits) For f=or: Input Minimal certificate {3} {1,2,3,4,5}

12 C(f) = min max p x [j] { p x {0,1} n } x j p x [j]p y [j] 1 if f(x) f(y) s.t.j:x j y j Adv(f) = min max p x [j] 2 { p x R x n } j p x [j]p y [j] 1 if f(x) f(y) s.t.j:x j y j Adv(f) is a semi-definite program (SDP)

13 Adversary method Q ɛ (f) 1 2 ɛ(1 ɛ) 2 Adv(f) Bennett, Bernstein, Brassard, Vazirani Ambainis 00 Høyer, Neerbek, Shi 02 Ambainis Barnum, Saks & Szegedy 03 Laplante & Magniez Zhang Barnum, Saks 04 Špalek & Szegedy [BBCMW ]

14 Adv ± (f) = s.t. General adversary bound min { p x R n } max x j:x j y j p x [j]p y [j] = 1 j p x [j] 2 if f(x) f(y) [Høyer, Lee, Špalek ]

15 Adv ± (f) = s.t. General adversary bound min { u xj R m } max x j:x j y j u xj,u yj = 1 j u xj 2 if f(x) f(y) [Høyer, Lee, Špalek ]

16 Adv ± (f) = Theorem: The general adversary lower bound on quantum query complexity is also an upper bound s.t. min { u xj R m } max x j:x j y j u xj,u yj = 1 j u xj 2 if f(x) f(y) 1. Simple understanding of quantum query complexity: No unitaries, measurements, or time dependence Equivalent to span programs [Karchmer, Wigderson 93] Quantum algorithms query complexity Span programs witness size (up to a constant factor, for boolean functions)

17 Query complexity under composition g Deterministic = D(f)D(g) g f Certificate C(f)C(g). Randomized R(f)R(g) O(log n) g Theorem: [HLŠ 06, R 09] Adv ± (f g) = Adv ± (f)adv ± (g) Q(f g) = Θ ( Q(f)Q(g) ) Composition of optimal algorithms for f and for g via tensor product of SDP vector solutions Characterizes query complexity for read-once formulas Q(f 1 f d )=Θ ( Adv ± (f 1 ) Adv ± (f d ) )

18 A. Query model B. Adversary lower bounds C. Spectra of reflections D. Adversary upper bound Q(f) =Θ(Adv ± (f))

19 points Π R S R(Π) points θ R(Δ) Δ points R(Π)R(Δ) is a rotation by angle 2θ, eigenvalues e ±2iθ

20 Two subspaces will not generally lie at a fixed angle Π Δ

21 Two subspaces will not generally lie at a fixed angle Π Δ Jordan s Lemma (1875) Any two projections can be simultaneously block-diagonalized with blocks of dimension at most two

22 Two subspaces will not generally lie at a fixed angle Π Δ Jordan s Lemma (1875) R(Π)R( ) = 0 cos 2θ sin 2θ sin 2θ cos 2θ 0 0

23 Effective Spectral Gap Lemma: Let PΘ be the projection onto eigenvectors of R(Π)R(Δ) with phase less than 2Θ in magnitude Then for any v with Δv = 0, P Θ Π v Θ v v Π θ Π v

24 Effective Spectral Gap Lemma: Let PΘ be the projection onto eigenvectors of R(Π)R(Δ) with phase less than 2Θ in magnitude Then for any v with Δv = 0, P Θ Π v Θ v v Π v Π Π v Π v θ > Θ P Θ Π v =0 θ Θ P Θ Π v = Π v

25 Effective Spectral Gap Lemma: Let PΘ be the projection onto eigenvectors of R(Π)R(Δ) with phase less than 2Θ in magnitude Then for any v with Δv = 0, P Θ Π v Θ v Proof: Jordan s Lemma Up to a change in basis, = Π = β β ( 10 β 00 ( ) ) β β cos 2 θ β sin θ β cos θ β β sin θ β cos θ β sin 2 θ β v =0 v = β d β β ( 0 1 ) P Θ Π v = β d β β sin θ β ( cos θβ : θ β Θ sin θ β )

26 A. Query model B. Adversary lower bounds C. Spectra of reflections D. Adversary upper bound Q(f) =Θ(Adv ± (f))

27 The algorithm: 1. Begin with an SDP solution: u xj, u yj =1 if f(x) f(y) j:x j y j 2. Let Δ = projection to the span of the vectors j u A yj y j ± j with f(y)=1 3. Starting at 0, alternate R(Δ) with the input oracle

28 = Proj { A ± Π x = P j j j I x j x j j, u } yj,y j j : f(y) = 1 u xj,u yj = 1 if f(x) f(y) j:x j y j Lemma: v P Θ Π v Θ v The analysis: Case f(x)=1: 0 close to A ± j j u xj x j doesn t move!

29 = Proj { A ± Π x = P j j j I x j x j j, u } yj,y j j : f(y) = 1 u xj,u yj = 1 if f(x) f(y) j:x j y j Lemma: v P Θ Π v Θ v The analysis: Case f(x)=1: 0 close to A ± j j u xj x j doesn t move! Case f(x)=0: 0 = = Π x ( 0 10 A ± j v j, u xj, x j ) Ω(1/Adv ± ) effective spectral gap

30 Summary Theorem: Optimal quantum query algorithms can be built out of two alternating reflections R Ox R Ox R Ox Corollary: Characterization of quantum query complexity for read-once boolean formulas. Theorem: The general adversary bound on quantum query complexity is tight Corollary: Quantum query algorithms are equivalent to span programs. Open problems Strong direct-product theorems? Query complexity for non-boolean functions and state generation? Composition for non-boolean functions? Upper and lower bounds for zero-error quantum query complexity? Tight characterizations for communication complexity?