Adversary-Based Parity Lower Bounds with Small Probability Bias

Transcription

1 Adversary-Based Parity Lower Bounds with Small Probability Bias Badih Ghazi Adversary-Based Parity Lower Bounds with Small Probability Bias 1 / 10

2 Setup Computing the parity function in the query model with error probability 1/2 q(n) where q(n) = o(1). Adversary-Based Parity Lower Bounds with Small Probability Bias 2 / 10

3 Setup Computing the parity function in the query model with error probability 1/2 q(n) where q(n) = o(1). Using the polynomial method: Ω(n) for any q(n) 0. Adversary-Based Parity Lower Bounds with Small Probability Bias 2 / 10

4 Setup Computing the parity function in the query model with error probability 1/2 q(n) where q(n) = o(1). Using the polynomial method: Ω(n) for any q(n) 0. Additive adversary arguments: Adversary-Based Parity Lower Bounds with Small Probability Bias 2 / 10

5 Setup Computing the parity function in the query model with error probability 1/2 q(n) where q(n) = o(1). Using the polynomial method: Ω(n) for any q(n) 0. Additive adversary arguments: Reichardt s characterization of Q(f ) in terms of Adv ± (f ) holds only in the bounded error case. Q 1/2 q(n) (Parity n ) Adv ± 1/2 q(n) (Parity n) = Ω((q(n)) 2 n) Adversary-Based Parity Lower Bounds with Small Probability Bias 2 / 10

6 Setup Computing the parity function in the query model with error probability 1/2 q(n) where q(n) = o(1). Using the polynomial method: Ω(n) for any q(n) 0. Additive adversary arguments: Reichardt s characterization of Q(f ) in terms of Adv ± (f ) holds only in the bounded error case. Q 1/2 q(n) (Parity n ) Adv ± 1/2 q(n) (Parity n) = Ω((q(n)) 2 n) Decays rapidly with q(n). Adversary-Based Parity Lower Bounds with Small Probability Bias 2 / 10

7 Setup Computing the parity function in the query model with error probability 1/2 q(n) where q(n) = o(1). Using the polynomial method: Ω(n) for any q(n) 0. Additive adversary arguments: Reichardt s characterization of Q(f ) in terms of Adv ± (f ) holds only in the bounded error case. Q 1/2 q(n) (Parity n ) Adv ± 1/2 q(n) (Parity n) = Ω((q(n)) 2 n) Decays rapidly with q(n). Gives a trivial constant bound even for q(n) = 1/ n. Adversary-Based Parity Lower Bounds with Small Probability Bias 2 / 10

8 Setup Computing the parity function in the query model with error probability 1/2 q(n) where q(n) = o(1). Using the polynomial method: Ω(n) for any q(n) 0. Additive adversary arguments: Reichardt s characterization of Q(f ) in terms of Adv ± (f ) holds only in the bounded error case. Q 1/2 q(n) (Parity n ) Adv ± 1/2 q(n) (Parity n) = Ω((q(n)) 2 n) Decays rapidly with q(n). Gives a trivial constant bound even for q(n) = 1/ n. Question: Can we get a better lower bound using adversary-based arguments? Adversary-Based Parity Lower Bounds with Small Probability Bias 2 / 10

9 Outline Adversary-Based Parity Lower Bounds with Small Probability Bias 3 / 10

10 Outline Show a lower bound of Ω(n) even for exponentially small q(n). Adversary-Based Parity Lower Bounds with Small Probability Bias 3 / 10

11 Outline Show a lower bound of Ω(n) even for exponentially small q(n). Proof is based on a quantum reduction to the t-fold search problem, with t = θ(n). Adversary-Based Parity Lower Bounds with Small Probability Bias 3 / 10

12 Outline Show a lower bound of Ω(n) even for exponentially small q(n). Proof is based on a quantum reduction to the t-fold search problem, with t = θ(n). Adaptation of the proof of Cleve et. al to our setup. Adversary-Based Parity Lower Bounds with Small Probability Bias 3 / 10

13 Outline Show a lower bound of Ω(n) even for exponentially small q(n). Proof is based on a quantum reduction to the t-fold search problem, with t = θ(n). Adaptation of the proof of Cleve et. al to our setup. Holds even for weak algorithms for parity. Adversary-Based Parity Lower Bounds with Small Probability Bias 3 / 10

14 The quantum reduction Notation For any S [n], χ S : Characteristic vector of S x S : Restriction of x to the subset S. Adversary-Based Parity Lower Bounds with Small Probability Bias 4 / 10

15 The quantum reduction Notation For any S [n], χ S : Characteristic vector of S x S : Restriction of x to the subset S. Observation [Scott Aaronson]: for all x {0, 1} n, we have 1 2 n ( 1) Parn(x S) H n χ S = x S [n] Adversary-Based Parity Lower Bounds with Small Probability Bias 4 / 10

16 The quantum reduction Notation For any S [n], χ S : Characteristic vector of S x S : Restriction of x to the subset S. Observation [Scott Aaronson]: for all x {0, 1} n, we have 1 2 n ( 1) Parn(x S) H n χ S = x S [n] Given an algorithm that produces such a superposition with r(n) queries, get an algorithm that recovers x with r(n) queries. Adversary-Based Parity Lower Bounds with Small Probability Bias 4 / 10

17 The quantum reduction Notation For any S [n], χ S : Characteristic vector of S x S : Restriction of x to the subset S. Observation [Scott Aaronson]: for all x {0, 1} n, we have 1 2 n ( 1) Parn(x S) H n χ S = x S [n] Given an algorithm that produces such a superposition with r(n) queries, get an algorithm that recovers x with r(n) queries. Need to deal with garbage. Adversary-Based Parity Lower Bounds with Small Probability Bias 4 / 10

18 The quantum reduction for coherent algorithms A parity algorithm is coherent if on inputs x and S, it takes the state x χ S z to x χ S z + Par n (x S ). Adversary-Based Parity Lower Bounds with Small Probability Bias 5 / 10

19 The quantum reduction for coherent algorithms A parity algorithm is coherent if on inputs x and S, it takes the state x χ S z to x χ S z + Par n (x S ). The reduction Start with the state x 0 n 1. Adversary-Based Parity Lower Bounds with Small Probability Bias 5 / 10

20 The quantum reduction for coherent algorithms A parity algorithm is coherent if on inputs x and S, it takes the state x χ S z to x χ S z + Par n (x S ). The reduction Start with the state x 0 n 1. Hadamard the last n + 1 qubits: 1 2 n+1 (z,χ S ) {0,1} n+1 ( 1) z x χ S z Adversary-Based Parity Lower Bounds with Small Probability Bias 5 / 10

21 The quantum reduction for coherent algorithms A parity algorithm is coherent if on inputs x and S, it takes the state x χ S z to x χ S z + Par n (x S ). The reduction Start with the state x 0 n 1. Hadamard the last n + 1 qubits: 1 2 n+1 (z,χ S ) {0,1} n+1 ( 1) z x χ S z Apply the coherent parity algorithm and Hadamard the last (n + 1) qubits: 1 2 n+1 (z,χ S ) {0,1} n+1 ( 1) z+parn(x S) x H n χ S H 1 z = x x 1 Adversary-Based Parity Lower Bounds with Small Probability Bias 5 / 10

22 The quantum reduction for coherent algorithms A parity algorithm is coherent if on inputs x and S, it takes the state x χ S z to x χ S z + Par n (x S ). The reduction Start with the state x 0 n 1. Hadamard the last n + 1 qubits: 1 2 n+1 (z,χ S ) {0,1} n+1 ( 1) z x χ S z Apply the coherent parity algorithm and Hadamard the last (n + 1) qubits: 1 2 n+1 (z,χ S ) {0,1} n+1 ( 1) z+parn(x S) x H n χ S H 1 z = x x 1 Measure the middle n qubits in the standard basis: get x with probability 1! Adversary-Based Parity Lower Bounds with Small Probability Bias 5 / 10

23 The quantum reduction for coherent algorithms Claim Let ΣO be any finite set. If there exists a coherent algorithm A that computes Parityn using r(n) queries, then for any function f : {0, 1} n Σ O, there is an algorithm B f that computes f exactly using r(n) queries. Adversary-Based Parity Lower Bounds with Small Probability Bias 6 / 10

24 The quantum reduction for general algorithms Claim Let t n 4e, t = θ(n). Let q(n) = Ω(e t/16 ) If A computes Parity n with error probability p x (n) for every x {0, 1} n and for all x with x = t, 1 2 n p x S (n) 1/2 q(n) S {0,1} n Then, A makes Ω(n) queries. Adversary-Based Parity Lower Bounds with Small Probability Bias 7 / 10

25 The quantum reduction for general algorithms Claim Let t n 4e, t = θ(n). Let q(n) = Ω(e t/16 ) If A computes Parity n with error probability p x (n) for every x {0, 1} n and for all x with x = t, Then, A makes Ω(n) queries. Corollary 1 2 n p x S (n) 1/2 q(n) S {0,1} n Q 1 2 e c n(parity n) = Ω(n) for any constant c Adversary-Based Parity Lower Bounds with Small Probability Bias 7 / 10

26 General parity algorithms A takes the state x χ S z 0 0 w to a x,s x χ S z Par n (x S ) J x,s + b x,s x χ S z Par n(x S ) K x,s where b x,s 2 = p x S (n), a x,s 2 + b x,s 2 = 1 and J x,s and K x,s are unit vectors. Adversary-Based Parity Lower Bounds with Small Probability Bias 8 / 10

27 General parity algorithms A takes the state x χ S z 0 0 w to a x,s x χ S z Par n (x S ) J x,s + b x,s x χ S z Par n(x S ) K x,s where b x,s 2 = p x S (n), a x,s 2 + b x,s 2 = 1 and J x,s and K x,s are unit vectors. Apply a CNOT gate and uncompute A: x χ S z + Par n (x S ) 0 (w+1) + 2b x,s M x,s,z where M x,s,z satisfies the properties M x,s,0 = M x,s,1 and { M x,s,0 } x,s is orthonormal. Adversary-Based Parity Lower Bounds with Small Probability Bias 8 / 10

28 General parity algorithms A takes the state x χ S z 0 0 w to a x,s x χ S z Par n (x S ) J x,s + b x,s x χ S z Par n(x S ) K x,s where b x,s 2 = p x S (n), a x,s 2 + b x,s 2 = 1 and J x,s and K x,s are unit vectors. Apply a CNOT gate and uncompute A: x χ S z + Par n (x S ) 0 (w+1) + 2b x,s M x,s,z where M x,s,z satisfies the properties M x,s,0 = M x,s,1 and { M x,s,0 } x,s is orthonormal. x χ S z + Par n (x S ) 0 (w+1) : Output of a coherent parity algorithm Not necessarily orthogonal to Mx,S,z Adversary-Based Parity Lower Bounds with Small Probability Bias 8 / 10

29 The quantum reduction for general algorithms Apply Hadamard gate, the above algorithm and another Hadamard gate: x x 1 0 (w+1) + ψ ψ 2 2 = 1 ( 1) z b 2 n x,s M x,s,z 2 2 (z,χ S ) {0,1} n+1 = 4 2 n p x S (n) χ S {0,1} n Adversary-Based Parity Lower Bounds with Small Probability Bias 9 / 10

30 The quantum reduction for general algorithms Apply Hadamard gate, the above algorithm and another Hadamard gate: x x 1 0 (w+1) + ψ ψ 2 2 = 1 ( 1) z b 2 n x,s M x,s,z 2 2 (z,χ S ) {0,1} n+1 = 4 2 n p x S (n) χ S {0,1} n Pr[Obtaining x] 4q 2 (n) whenever 1 2 n p x S (n) ( 1 2 q(n)) S {0,1} n Adversary-Based Parity Lower Bounds with Small Probability Bias 9 / 10

31 The quantum reduction for general algorithms Holevo s theorem? Adversary-Based Parity Lower Bounds with Small Probability Bias 10 / 10

32 The quantum reduction for general algorithms Holevo s theorem? n queries to the oracle might give n log n classical bits of information. Adversary-Based Parity Lower Bounds with Small Probability Bias 10 / 10

33 The quantum reduction for general algorithms Holevo s theorem? n queries to the oracle might give n log n classical bits of information. The t-fold search problem Given x {0, 1} n and promised that x = t, find the subset J [n] of 1 s of x. Adversary-Based Parity Lower Bounds with Small Probability Bias 10 / 10

34 The quantum reduction for general algorithms Holevo s theorem? n queries to the oracle might give n log n classical bits of information. The t-fold search problem Given x {0, 1} n and promised that x = t, find the subset J [n] of 1 s of x. For every t n 4e and every ɛ = 1 Ω(e t/8 ), Q ɛ (t-fold search) = Ω( tn). Proof using the earliest version of the multiplicative adversary method(ambianis 2005, Spalek 2007) Adversary-Based Parity Lower Bounds with Small Probability Bias 10 / 10

35 The quantum reduction for general algorithms Holevo s theorem? n queries to the oracle might give n log n classical bits of information. The t-fold search problem Given x {0, 1} n and promised that x = t, find the subset J [n] of 1 s of x. For every t n 4e and every ɛ = 1 Ω(e t/8 ), Q ɛ (t-fold search) = Ω( tn). Proof using the earliest version of the multiplicative adversary method(ambianis 2005, Spalek 2007) Conclude: The claim holds for all q(n) = Ω(e t/16 ). Adversary-Based Parity Lower Bounds with Small Probability Bias 10 / 10