Quantum Search and Quantum Learning
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Presentation of the Bernstein and Vazirani’s Algorithm, Impatient Learning and Amplitude amplification by Brassard & Høyer
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Quantum Search and Quantum Learning
- 1. Quantum Learning Jean- Christophe Lavocat Definitions Hadamard Impatient Learning Amplitude Amplification Majority Problem Quantum Learning Quantum perceptron Jean-Christophe Lavocat David Meyer, Mathematics Department UCSD La Jolla July 6, 2009
- 2. Quantum Learning Jean- Christophe Lavocat Definitions Hadamard Impatient Learning Amplitude Amplification Majority Problem Table of Content 1 Definitions 2 Hadamard 3 Impatient Learning 4 Amplitude Amplification 5 Majority Problem
- 3. Quantum Learning Jean- Christophe Lavocat Definitions Hadamard Impatient Learning Amplitude Amplification Majority Problem Definitions Boolean Concepts Definition Concept : a concept is a map c : X → Z2 define on a discret set X. Extension (of the concept) : support of c−1 (1) ⊂ X We call Bn (where n ∈ N) the set of boolean functions : {0,1}n → {0,1} Concept : a function of Bn Concept class : a subset of Bn. We note C a concept class
- 4. Quantum Learning Jean- Christophe Lavocat Definitions Hadamard Impatient Learning Amplitude Amplification Majority Problem Definitions Oracles In quantum computations we have access to oracles : unitary black boxes that return some function output. Definition Oracle Membership Oracle : responds to x ∈ X with ¯c(x) where c ∈ C is the target concept : |x Mc |¯c(x) Equivalence Oracle : responds to c ∈ C with δc¯c : |c Ec |δc¯c
- 5. Quantum Learning Jean- Christophe Lavocat Definitions Hadamard Impatient Learning Amplitude Amplification Majority Problem Definitions Exact Learning Define the teacher as the membership oracle, the protocol is the agreement about admissible queries and possible answers. Definition : Exact Learning A concept class C is learnable according to the protocol P if there is a learning algorithm L such that, ∀f ∈ C and for any teacher Tf , L outputs a circuit h with probability 2/3 such that : ∀x ∈ Zn 2 h(x) = f(x)
- 6. Quantum Learning Jean- Christophe Lavocat Definitions Hadamard Impatient Learning Amplitude Amplification Majority Problem Formalization of Quantum Learning Algorithms Quantum Information Properties Properties ∀U ∈ Un, U |x,b = |x,b ⊕f(x) we have : U |x |− = (−1)f(x) |x |− We know that : H = 1 √ 2 1 1 1 −1 = H H|0 = |0 +|1 √ 2 H⊗n |0...0 = 1 √ 2n ∑ x∈Zn 2 |x
- 7. Quantum Learning Jean- Christophe Lavocat Definitions Hadamard Impatient Learning Amplitude Amplification Majority Problem Table of Content 1 Definitions 2 Hadamard 3 Impatient Learning 4 Amplitude Amplification 5 Majority Problem
- 8. Quantum Learning Jean- Christophe Lavocat Definitions Hadamard Impatient Learning Amplitude Amplification Majority Problem Bernstein and Vazirani’s Algorithm Description of the Problem Problem We take a ∈ Zn 2. We have an oracle that returns a.x mod 2. We study the following family of concept classes BVn : BVn = {pa : Zn 2 → Z2|pa(x) = a.x mod 2} We try to find a given an access to the oracle matrix. Classically the complexity is : Ω(n) Theorem The search can be performed with only one query to the oracle : H⊗n UaH⊗n |0 = |a
- 9. Quantum Learning Jean- Christophe Lavocat Definitions Hadamard Impatient Learning Amplitude Amplification Majority Problem Bernstein and Vazirani’s Algorithm Proof of the result Proof. We will consider concept classes such as : |X| = |C|. We then have the following membership query matrix AC whose column c is : (AC )_c = Uc 1 |X| ∑ x∈X |x . Therefore : UaH⊗n |0 = (UBV )c 1 √ 2n ∑ x∈X |x xa = (ABV )xa = (−1)x.a √ 2n We recognize H⊗n . So finally: H⊗n (ABV )a = (H⊗n ABV )a = (H⊗n H⊗n )a = Ia = |a
- 10. Quantum Learning Jean- Christophe Lavocat Definitions Hadamard Impatient Learning Amplitude Amplification Majority Problem Bernstein and Vazirani’s Algorithm Restriction of the result This result is true only for the Hadamard concept classes ( where the membership query matrix is an Orthogonal matrix with entries ±1 ). For the Grover’s Search the membership oracle is : Gn = {δa : Zn 2 → Z2|δa(x) = δax } Classically the complexity is : Ω(n) As : (ABV )xa = (−1)δxa √ 2n = 1 √ N NF† |0 0|F −2I xa Not unitary, and then, not part of Hadamard Class
- 11. Quantum Learning Jean- Christophe Lavocat Definitions Hadamard Impatient Learning Amplitude Amplification Majority Problem Table of Content 1 Definitions 2 Hadamard 3 Impatient Learning 4 Amplitude Amplification 5 Majority Problem
- 12. Quantum Learning Jean- Christophe Lavocat Definitions Hadamard Impatient Learning Amplitude Amplification Majority Problem Impatient Learning Context Context 1-query complexity is not always reached. Question : How well is it possible to do with just a simple query? Problem : If we can make any unitary transformation after a single query, how do we maximize the probability that a measurement returns |a ?
- 13. Quantum Learning Jean- Christophe Lavocat Definitions Hadamard Impatient Learning Amplitude Amplification Majority Problem Impatient Learning Measurement maximization Measurement We have a quantum system whose state is in the set of unit vectors {|vi |i ∈ ZN}. The task is to perform a measurement that maximize the probability of correctly guessing which state the system was in before the measurement was made. If we use a basis {|ei |i ∈ ZN}, the probability to measure |ei given a state |v before the measurement is : | v|ei |2 If we assume the system has been prepared in the state {|xi }, the quantity we want to improve is : ∑ i=1..N | vi |ei |2
- 14. Quantum Learning Jean- Christophe Lavocat Definitions Hadamard Impatient Learning Amplitude Amplification Majority Problem Impatient Learning Algebras manipulations Matrices optimization In the third part of the article, they describe how to maximize the measurement of a state. Before making the measurement on the matrix A we can apply a unitary matrix. We consider the matrices B ∼ A (∃S ∈ U(n), B = SA) If we use the diagonal projection (∀M ∈ Mn,∃U,V orthogonal and d(M) diagonal such that : M = Ud(M)V): ||M||2 = max i d(M)ii Equivalently, we try to maximize the quantity ||d(B)||2 over {B|B ∼ A}.
- 15. Quantum Learning Jean- Christophe Lavocat Definitions Hadamard Impatient Learning Amplitude Amplification Majority Problem Impatient Learning Algebras manipulations Result The main result is : for any inversible matrix A with a decomposition A = S−1 B (S: unitary, B:positive hermitian), the closest point to In in the U(N)-orbit of A is B. Bd(B) = d(B)B† (1) In order to find such a B we can use the square root of the Gram matrix of A : G = A† A. One can prove that there is always a unitary matrix such that : √ G = SA
- 16. Quantum Learning Jean- Christophe Lavocat Definitions Hadamard Impatient Learning Amplitude Amplification Majority Problem Impatient Learning Algorithm Impatient Learning : Algorithm In order to have the correct answer with the most probable measurement : Prepare the query register in an equal superposition of state F† |0 Query the membership oracle (the cth column of AC is UC F† |0 Apply a unitary transformation SC such that BC = SC AC satisfies (1) Measure the resulting state SC F† |0 in the computational basis Success probability For initial equal superposition of states, it is optimal, and the probability of learning is |(BC )¯c¯c|2
- 17. Quantum Learning Jean- Christophe Lavocat Definitions Hadamard Impatient Learning Amplitude Amplification Majority Problem Table of Content 1 Definitions 2 Hadamard 3 Impatient Learning 4 Amplitude Amplification 5 Majority Problem
- 18. Quantum Learning Jean- Christophe Lavocat Definitions Hadamard Impatient Learning Amplitude Amplification Majority Problem Amplitude Amplification Context Context Single membership query does not provide enough information to learn with a probability close to 1. We will try to apply quantum amplification techniques invented by Brassard & Høyer to increase the probability.
- 19. Quantum Learning Jean- Christophe Lavocat Definitions Hadamard Impatient Learning Amplitude Amplification Majority Problem Amplitude Amplification Theorem Theorem (Brassard, Høyer, Mosca, Tapp - 2002) ξ ∈ X a concept. H1 is the subspace spanned by the extension of ξ, (ξ−1 (1)). Π is the projection on H1. For any unitary transformation W, let p(W) = |ΠW |0 |2 (propability that the state W |0 be in the subspace of the extension). If p(W) > 0 we set θ2 = p(W) (0 < θ π/2). Applying the unitary transformation WUδ0 W† Uξ amplifies the probability of measuring the state in H1 : p((WUδ0 W† Uξ)m W) max{1 −p(W),p(W)} where m = π 4θ − 1 2 (nearest integer)
- 20. Quantum Learning Jean- Christophe Lavocat Definitions Hadamard Impatient Learning Amplitude Amplification Majority Problem Amplitude Amplification Application If we let ξ = δc, H1 = span{|c } and Wc = ScUcF† Amplified Impatient Learning : Algorithm In order to have the correct answer with the most probable measurement : Prepare the query register in an equal superposition of state F† |0 Query the membership oracle (the cth column of AC is UC F† |0 ) Apply a unitary transformation SC (we get : SC UcF† |0 = WC |0 ) Apply m times WC Uδ0 W † C Uδc (with sin θ = |(BC )¯c¯c|) Measure the resulting state SC F† |0 in the computational basis
- 21. Quantum Learning Jean- Christophe Lavocat Definitions Hadamard Impatient Learning Amplitude Amplification Majority Problem Amplitude Amplification Complexity Theorem For problems with BC having constant diagonal s, amplified learning suceeds with probability max 1 −s2 ,s2 . The sample complexity is then 2m +1 = O(1). However we use also the equivalence oracle (m queries).
- 22. Quantum Learning Jean- Christophe Lavocat Definitions Hadamard Impatient Learning Amplitude Amplification Majority Problem Table of Content 1 Definitions 2 Hadamard 3 Impatient Learning 4 Amplitude Amplification 5 Majority Problem
- 23. Quantum Learning Jean- Christophe Lavocat Definitions Hadamard Impatient Learning Amplitude Amplification Majority Problem Majority Problem Context Context The majority problem is equivalent to the perceptron learning. For each a ∈ Zn define a function ma : ZN 2 → Z2 : ma(x) = 1 if wt(a −x) n/2 0 otherwise We write M AJn the majority concept class (size 2n ) Theorem Amplified Impatient Learning solves Majority with O( √ n) (given acces to membership and equivalence oracle)
- 24. Quantum Learning Jean- Christophe Lavocat Definitions Hadamard Impatient Learning Amplitude Amplification Majority Problem Majority Problem Case : N odd Case N odd If N is odd only one additional query. Indeed if a and b are complementary bit strings then ma(x) = 1 −mb(x). If we write M = M AJn−1 and M the associated complementary matrix, we have M M M M We could note that the matrice is not inversible
- 25. Quantum Learning Jean- Christophe Lavocat Definitions Hadamard Impatient Learning Amplitude Amplification Majority Problem Majority Problem Zn 2 group algebra Decomposition in X matrices By induction on n we can see that the matrices representation of Zn 2 could be decomposed as : M = ∑ x∈Zn 2 vx Xx where Xx = Xx1...xn = Xx1 ⊗...⊗Xxn and where |v = ∑ x∈Zn 2 vx |x M is symmetric and has a constant diagonal. Diagonalization with H If M ∈ Zn 2 , M is diagonalized by H (Z=HXH).
- 26. Quantum Learning Jean- Christophe Lavocat Definitions Hadamard Impatient Learning Amplitude Amplification Majority Problem Majority Problem Norm maximization Norm maximization We can show that the query matrix of the Majority problem is part of the Zn 2 group algebra. If c is a given concept, we call ϕ(c −x) = ϕ(b) = Θ(n/2 −wt(b)) (Heaviside function). We have AC = 1 √ 2n ∑ b∈Zn 2 (−1)ϕ(b) Xb and thus, the eigenvalues are : λc = 1 √ 2n ∑ b∈Zn 2 (−1)ϕ(b) (−1)b.c
- 27. Quantum Learning Jean- Christophe Lavocat Definitions Hadamard Impatient Learning Amplitude Amplification Majority Problem Majority Problem Norm maximization Norm maximization Calling λn,k = λc the eigenvalue of any c ∈ Zn 2 of weight k. Using standard combinatorial techniques we have : λn,k = −(−1)k/2 √ 2n 1.3...(k−1) (n−1).(n−3)...(n−k+1))) if k is even λn,k−1 if k is odd Thus, if n is even, the smallest eigenvalue is : |λn,n/2| = −(−1)√ 2n n/2 n/4 if n ≡ 0 mod 4 − 2√ 2n (n−2)/2 (n−2)/4 if n ≡ 2 mod 4
- 28. Quantum Learning Jean- Christophe Lavocat Definitions Hadamard Impatient Learning Amplitude Amplification Majority Problem Majority Problem Norm maximization Norm maximization Using Stirling’s formule ( n! ∼ √ 2πn n e n ) we have : |λn,n/2| lim −−−→n→∞ 1/ √ n The eigenvalues’ average, s, is then Ω(1/ √ n). Result So the claimed result is proved : The quantum query complexity of Majority is O(1/s) = O( √ n)