Tutorial on the adversary method for quantum and classical lower bounds. Sophie Laplante LRI, Université Paris-Sud Orsay

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1 Tutorial on the adversary method for quantum and classical lower bounds Sophie Laplante LRI, Université Paris-Sud Orsay

2 Adversary method lower bounds Exact complexity Strong lower bound Semidefinite Programming [BSS03] Kolmogorov method [LM04] Quantum query complexity Polynomial method [BBCMW01] Spectral method [BSS03] Weighted adversary [A03] Weak lower bound Adversary method [A02] 2

3 Summary of this talk Deterministic method - General statement - Combinatorial bound DT (f) R max i R i max{ m l, m l } Quantum method - General statement - Combinatorial bound [A02] Q ε (f) Weighted method [A03, LM04] Q ε (f) c ε 2 min x,y,i Spectral method [BSS03] Q ε (f) c ε 2 c ε R i P rogress t(i) c ε mm 2 ll p(x)p(y)p x,i (y)p y,i (x) q(x, y) Γ max i Γ i 3

4 Classical decision tree model To compute a boolean function f : {0,1} n {0,1}, x 1 = 0 x 1 =? x 1 = 1 Model : decision tree Cost : Number of queries to input Query complexity of f : depth of shallowest decision tree for f x 2 =? x 3 =? x 2 = 0 x 2 = 1 x 3 = 0 x 3 = 1 x 3 =? x 3 = 0 x 5 =? x 3 = 1 x 5 =? x 5 = 0 x 5 = 0 x 5 = 1 x 5 = 1 ACC REJ REJ ACC 4

5 Quantum query model ancilla Computation U 0 Query O x Computation U 1 Query: Unitary transformation maps i, b to i, b x i Computation : ψ T = U T O x O x U 0 0 O x that Query O x Query O x Computation U T Output: Measure 1st qubit of Error probability at most 1/3 ψ T Same model with stochastic matrices for randomized query complexity output 5

6 Estimating number of pieces by their size 6

7 Adversary method, deterministic setting Goal: separate {x : f(x)=1} from {y : f(y)=0} with minimum queries to input bits Subrelation Ri : pairs for which query i is useful Queries number of Ri needed to cover R. R {x : f(x)=1} {y : f(y)=0} y x xi yi DT (f) R max i R i Ri 7

8 Deterministic lower bound left degree of R m right degree m left degree of all Ri l right degree l R m X, m Y Ri l X, l Y DT (f) R max i R i max{ m l, m l } R {x : f(x)=1} {y : f(y)=0} y x Ri xi yi 8

9 Progress on x, y, i after a query At time t, progress towards distinguishing (x,y) Ri by making one query : Deterministic case DProgresst x,y (i) = Randomized case RProgresst x,y (i) = 2 min{p x t (i), p y t (i)} Quantum case? (next slide) Progresst x,y (i) { 1 if query(x,t) = i 0 otherwise 9

10 Quantum progress State of the system after query t, on input x is written ψt x Fix (x,y) R. At beginning, At each time step, ψ x t ψ y t ψ x t+1 ψ y t+1 2 ψ x 0 ψ y 0 = 1 Amplitude 2 of i in query t, input x p x t (i)p y t (i) ψ x 0 ψ y 0 At the end, i:x i y i ψ x T ψ y T 2 ε(1 ε) T 2 p x t (i)p y t (i) 1 2 ε(1 ε) i Progress t x,y(i) c ε 10

11 Quantum progress on x, y, i after a query At time t, progress towards distinguishing (x,y) Ri by making one query : Deterministic case DProgresst x,y (i) = Randomized case RProgresst x,y (i) = Quantum case QProgresst x,y (i) = 2 p x t (i)p y t (i) Progresst x,y (i) { 1 if query(x, t) = i 0 otherwise 2 min{p x t (i), p y t (i)} 11

12 Ambainis unweighted method Claim: Proof: P rogress t (i) 2 l X l Y i P rogress t (i) i = x,y,i = 2 x,y,i P rogress x,y t (i) p x t (i)p y t (i) 2 x i y p x t (i) y p y t (i) i x Cauchy Schwarz X terms p x t (i) = 1 i l terms 2 l X l Y 13

13 Ambainis unweighted method Claim: P rogress t (i) 2 l X l Y i Corollary: Q ε (f) c ε R i P rogress t(i) m X m Y c ε 2 lx l Y Q ε (f) c ε mm 2 ll R max{m X, m Y } m X m Y 13

14 Example: OR function m=1 l=1 m =n l = Deterministic queries max{m/l, m /l } = n Quantum queries = n mm ll 14

15 Unweighted lmax lower bound left degree of R m right degree m R {x : f(x)=1} {y : f(y)=0} left degree of Ri li right degree li R m X, m Y P rogress t (i) 2 max i {l i X l i Y } i Q ε (f) c ε 2 mm max i {l i l i } x Ri xi yi y 15

16 Example: Connectivity k k+1 X X k'+1 k' Deterministic queries max{m/l, m /l } = n 2 m n 2 (x) li n ( ) li 1 m n 2 li =1 li =n Quantum queries mm ll = n 3/2 16

17 Weighted method 1/8 1/8 1/8 3/16 1/8 1/16 1/8 1/8 17

18 Weight scheme for adversary method q(x,y) x q(x,y) y p(x) p' x,i (y) i i' i'' i''' p' y,i (x) p(y) x,y R q(x, y) = 1, x X p(x) = 1, y Y p x,i(y) = 1 18

19 Weighted adversary method Consider three distributions over pairs (x,y) R P (x, y) = i P (x, y) = i p(x)p x t (i)p x,i(y), p(y)p y t (i)p y,i(x), Claim: Q(x, y) = q(x, y) (x, y) R i P rogress x,y t (i) 2max i q(x, y) p(x)p x,i (y)p(y)p y,i (x) P (x, y)p (x, y) x,y P (x, y) P (x, y) x,y x,y 1 = Q(x, y) x,y Proof: x, y P (x, y)p (x, y) Q(x, y) x, y p x t (i)p y q(x, y) t (i) i min i p(x)p(y)p x,i (y)p y,i (x) 19

20 Weighted adversary method Consider three distributions over pairs (x,y) R P (x, y) = i P (x, y) = i p(x)p x t (i)p x,i(y), p(y)p y t (i)p y,i(x), Q(x, y) = q(x, y) Claim: (x, y) R i P rogress x,y t (i) 2max i q(x, y) p(x)p x,i (y)p(y)p y,i (x) Corollary: Recall that (x, y) R Q ε (f) Q ε (f) c ε 2 min x,y,i c ε i P rogressx,y t (i) p(x)p(y)p x,i (y)p y,i (x) q(x, y) 19

21 Example: Ambainis function f(x 1 x 2 x 3 x 4 ) = 1 if x 1 x 2 x 3 x 4 1 if x 1 x 2 x 3 x 4 0 otherwise q(x,y) !! 3/80 3/80 1/40 1/ p(x) = p(y) =1/8 Q ε (f) c ε 2 min x,y,i p(x)p(y)p x,i (y)p y,i (x) q(x, y) 20

22 Example: Ambainis function f(x 1 x 2 x 3 x 4 ) = 1 if x 1 x 2 x 3 x 4 1 if x 1 x 2 x 3 x 4 0 otherwise !! 0001 p' x,i (y) 3/4 1/4 p' y,i (x) 3/ !! !! q(x,y) 3/80 1/40 p x,i (y)p y,i (x) q(x, y) p(x)p(y) 1/8 1/ !! 1 1/ !! Q ε (f) c ε 2 min x,y,i 1/40 p(x)p(y)p x,i (y)p y,i (x) q(x, y) 20 1/8 = 20/8 = 5/2 20

23 Spectral method Nonnegative weights matrix Γ[x, y] x Γ[x, y] y Γ[x, y] = 0 if f(x) = f(y) { 0 if xi = y Γ i [x, y] = i Γ[x, y] otherwise Γ = max u Γv u,v u = v =1 Q ε (f) c ε 2 Idea: q(x,y) derived from p(x), p(y) derived from u, v maximizing u*tv Γ max i Γ i " p y,i(x), p x,i(y) derived from ui, vi maximizing ui*tivi 21