Reflections for quantum query algorithms Reflections Ben Reichardt - PowerPoint PPT Presentation

Reflections for quantum query algorithms Reflections Ben Reichardt University of Waterloo

Reflections for quantum query algorithms Reflections Ben Reichardt University of Waterloo

Reflections for quantum query algorithms Reflections Theorem: An optimal quantum query algorithm for evaluating any boolean function can be built out of two fixed reflections R 1 R 2 R 1 R 2 R 1 R 2

Goal: Evaluate f: {0,1} n → {0,1} using | x ∈ { 0 , 1 } n | j x j

j 1 x j 1 j 2 x j 2 y y r r e e u u x x q q … U 0 U 1 U T f ( x )

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

Quantum query complexity | x ∈ { 0 , 1 } n | ( − 1) x j | j � | j � | 1 � + | 2 � �→ ( − 1) x 1 | 1 � + ( − 1) x 2 | 2 � y y r r e e u u x x q q … U 0 U 1 U T f ( x ) w/ prob. ≥ 2/3

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

U 0 O x U 1 O x U 2 O x Clearly, w.l.o.g., • may assume U t is independent of t T � | t + 1 � � t | ⊗ U t + c.c. U = t =0 • or, may assume U t is a reflection ∀ t � 1 | ⊗ U † R t = | 1 � � 0 | ⊗ U t + | 0 � t

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

⇒ 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) Input For f=OR: Minimal certificate 00110 {3} 00000 {1,2,3,4,5}

� C ( f ) = max p x [ j ] min { � 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 � p x [ j ] 2 Adv( f ) = max min { � p x ∈ R n } x 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)

• Adversary method 1 − 2 √ ǫ (1 − ǫ ) Q ǫ ( f ) ≥ Adv( f ) 2 • Bennett, Bernstein, Brassard, Vazirani 9701001 • Ambainis ’00 • Høyer, Neerbek, Shi ’02 • Ambainis 0305028 • Barnum, Saks & Szegedy ’03 • Laplante & Magniez 0311189 • Zhang 0311060 • Barnum, Saks ’04 • Š palek & Szegedy 0409116 [BBCMW 9802049]

General adversary bound � Adv ± ( f ) = p x [ j ] 2 max min { � p x ∈ R n } x j � p x [ j ] p y [ j ] = 1 if f ( x ) � = f ( y ) s.t. j : x j � = y j [Høyer, Lee, Š palek 0611054]

General adversary bound u xj � 2 � Adv ± ( f ) = � � min max x { � u xj ∈ R m } j � � u xj , u yj � = 1 if f ( x ) � = f ( y ) s.t. j : x j � = y j [Høyer, Lee, Š palek 0611054]

Theorem: The general adversary lower bound on quantum query complexity is also an upper bound u xj � 2 � Adv ± ( f ) = � � min max x { � u xj ∈ R m } j � � u xj , u yj � = 1 if f ( x ) � = f ( y ) s.t. j : x j � = y j 1. Simple understanding of quantum query complexity: • No unitaries, measurements, or time dependence • Equivalent to span programs [Karchmer, Wigderson ’93] Span programs Quantum algorithms ≈ query complexity witness size (up to a constant factor, for boolean functions)

Query complexity under composition g • Deterministic = D ( f ) D ( g ) g • Certificate ≤ C ( f ) C ( g ) f . . • Randomized ≤ R ( f ) R ( g ) O (log n ) . g Theorem: Adv ± ( f ◦ � g ) = Adv ± ( f )Adv ± ( g ) [HL Š ’06, R’09] � � ⇒ 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 ◦ · · · ◦ � Adv ± ( f 1 ) · · · Adv ± ( f d ) � � f d ) = Θ

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

R = 2 s S = 2 t n i o p Π R( Π ) points R( Δ ) θ Δ points R( Π ) R( Δ ) is a rotation by angle 2 θ , eigenvalues e ±2i θ

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

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

Two subspaces will not generally lie at a fixed angle Π Δ Jordan’s Lemma (1875)   0 0  · · ·      R ( Π ) R ( ∆ ) = cos 2 θ − sin 2 θ  0 0   cos 2 θ sin 2 θ    · · ·   0 0

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

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

Effective Spectral Gap Lemma: • Let P Θ be the projection onto eigenvectors of R( Π )R( Δ ) with phase less than 2 Θ in magnitude • Then for any v with Δ v = 0, � � � P Θ Π � v � ≤ Θ � � v � Proof: Jordan’s Lemma ⇒ Up to a change in basis, � � β | ⊗ ( 1 0 ∆ = β | β � 0 0 ) cos 2 θ β � � sin θ β cos θ β � β | β � � β | ⊗ Π = sin 2 θ β sin θ β cos θ β � β d β | β � ⊗ ( 0 ∆ | v � = 0 ⇒ | v � = 1 ) � � � cos θ β ⇒ Π | v � = d β | β � ⊗ sin θ β P Θ � sin θ β : | θ β | ≤ Θ β

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

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 � 1 | 0 � + j | j � | u yj � | y j � √ A ± 10 with f(y)=1 3. Starting at , alternate R( Δ ) with the input oracle | 0 �

� � 1 � | 0 � + j | j, u yj , y j � � � u xj , u yj � = 1 if f ( x ) � = f ( y ) √ A ± 10 ∆ = Proj : f ( y ) = 1 j : x j � = y j Lemma: v ∈ ∆ ⊥ ⇒ � P Θ Π � Π x = | 0 � � 0 | + P j | j � � j | ⊗ I ⊗ | x j � � x j | v � ≤ Θ � � v � � The analysis: Case f(x)=1: | 0 �        close to � 1 | 0 � + j | j � | u xj � | x j � √ A ± 10 ⇒ doesn’t move!

� � 1 � | 0 � + j | j, u yj , y j � � � u xj , u yj � = 1 if f ( x ) � = f ( y ) √ A ± 10 ∆ = Proj : f ( y ) = 1 j : x j � = y j Lemma: v ∈ ∆ ⊥ ⇒ � P Θ Π � Π x = | 0 � � 0 | + P j | j � � j | ⊗ I ⊗ | x j � � x j | v � ≤ Θ � � v � � The analysis: Case f(x)=1: Case f(x)=0: | 0 � | 0 �               close to = √ A ± � � � � 1 | 0 � − 10 | j, u xj , ¯ x j � | 0 � + j | j � | u xj � | x j � Π x √ A ± 10 j ⇒ doesn’t move! = v ∈ ∆ ⊥ � ⇒ Ω (1/Adv ± ) effective spectral gap

Theorem: Optimal quantum Theorem: The general query algorithms can be built out adversary bound on quantum of two alternating reflections query complexity is tight Summary R O x R O x R O x Corollary: Characterization of Corollary: Quantum query quantum query complexity for algorithms are equivalent to read-once boolean formulas. span programs. Strong direct-product theorems? Open problems ✓ 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?