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Introduction to Quantum Information Processing QIC 710 / CS 667 / PH 767 / CO 681 / AM 871 Lectures 17-19 (2011) Richard Cleve DC 2117 cleve@cs.uwaterloo.ca 1 Complexity class NP 2 Complexity classes Recall: • P (polynomial time): problems solved by O(nc)-size classical circuits (decision problems and uniform circuit families) • BPP (bounded error probabilistic polynomial time): problems solved by O(nc)-size probabilistic circuits that err with probability £ ¼ • BQP (bounded error quantum polynomial time): problems solved by O(nc)-size quantum circuits that err with probability £ ¼ • PSPACE (polynomial space): problems solved by algorithms that use O(nc) memory. 3 Summary of previous containments P Í BPP Í BQP Í PSPACE Í EXP EXP We now consider further structure between P and PSPACE PSPACE Technically, we will restrict BQP our attention to languages (i.e. {0,1}-valued problems) BPP Many problems of interest can P be cast in terms of languages For example, we could define FACTORING = {(x,y) : $ 2 £ z £ y, such that z divides x} 4 NP Define NP (non-deterministic polynomial time) as the class of languages whose positive instances have “witnesses” that can be verified in polynomial time Example: Let 3-CNF-SAT be the language consisting of all 3-CNF formulas that are satisfiable 3-CNF formula: is satisfiable iff there exists such that No sub-exponential-time algorithm is known for 3-CNF-SAT But poly-time verifiable witnesses exist (namely, b1, ..., bn) 5 Other “logic” problems in NP • k-DNF-SAT: * But, unlike with k-CNF-SAT, this one is known to be in P • CIRCUIT-SAT: 1 Λ Ø Λ Ø Ø output bit 0 Ø Λ Ø Λ Λ Ø Ø * All known 1 Λ Λ Λ Λ algorithms 1 Ø Ø Λ Λ exponential- 0 Λ Λ Ø Λ time 6 “Graph theory” problems in NP • k-COLOR: does G have a k-coloring ? • k-CLIQUE: does G have a clique of size k ? • HAM-PATH: does G have a Hamiltonian path? • EUL-PATH: does G have an Eulerian path? 7 “Arithmetic” problems in NP • FACTORING = {(x, y) : $ 2 £ z £ y, such that z divides x} • SUBSET-SUM: given integers x1, x2 , ..., xn, y, do there exist i1, i2 , ..., ik Î{1, 2,... , n} such that xi1+ xi2 + ... + xik = y? • INTEGER-LINEAR-PROGRAMMING: linear programming where one seeks an integer-valued solution (its existence) 8 P vs. NP All of the aforementioned problems have the property that they reduce to 3-CNF-SAT, in the sense that a polynomial- time algorithm for 3-CNF-SAT can be converted into a poly- time algorithm for the problem Example: algorithm for 3-COLOR algorithm for 3 -CNF-SAT If a polynomial-time algorithm is discovered for 3-CNF-SAT then a polynomial-time algorithm for 3-COLOR follows And this holds for any problem X Î NP 9 P vs. NP For any problem X Î NP algorithm for X algorithm for 3 -CNF-SAT A problem that has this property is said to be NP-hard Polynomial-time algorithm any NP-Hard problem implies P=NP NP-hard problems: 3-CNF-SAT, CIRCUIT-SAT, 3-COLOR , k-CLIQUE, HAM-PATH, SUBSET-SUM, INT-LIN-PROG, … Some problems in P: k-DNF-SAT, 2-COLOR, 3-CLIQUE, EUL-PATH, ... 10 FACTORING vs. NP Is FACTORING NP-hard too? PSPACE If so, then every problem in NP is solvable by a poly-time quantum algorithm! 3-CNF-SAT NP co-NP But FACTORING has not been shown to be NP-hard FACTORING P Moreover, there is “evidence” that it is not NP-hard: FACTORING Î NPÇco-NP If FACTORING is NP-hard then NP = co-NP 11 FACTORING vs. co-NP FACTORING = {(x, y) : $ 2 £ z £ y, s.t. z divides x} co-NP: languages whose negative PSPACE instances have “witnesses” that can be verified in poly-time NP co-NP Question: what is a good witness for the negative instances? FACTORING P Answer: the prime factorization p1, p2 , ..., pm of x will work Can verify primality and compare p1, p2 , ..., pm with y, all in poly-time 12 Grover’s quantum search algorithm 13 Quantum search problem Given: a black box computing f : {0,1}n à {0,1} Goal: determine if f is satisfiable (if $x Î {0,1}n s.t. f(x) = 1) In positive instances, it makes sense to also find such a satisfying assignment x n Classically, using probabilistic procedures, order 2 queries are necessary to succeed—even with probability ¾ (say) n Grover’s quantum algorithm that makes only O(Ö2 ) queries Query: |x1ñ |x1ñ |xnñ Uf |xnñ [Grover ’96] |yñ |y Å f(x1,...,xn)ñ 14 Applications of quantum search The function f could be realized as a 3-CNF formula: PSPACE Alternatively, the search could be for a certificate for any problem in NP NP co-NP 3-CNF-SAT The resulting quantum algorithms appear to be quadratically more efficient FACTORING than the best classical P algorithms known* Subtlety in that the search space might have to be redefined to achieve this 15 Prelude to Grover’s algorithm: two reflections = a rotation Consider two lines with intersection angle q: reflection 2 q2 q2 q1 reflection 1 q q1 Net effect: rotation by angle 2q, regardless of starting vector 16 Grover’s algorithm: description I Basic operations used: |x1ñ |x1ñ Uf |xñ|-ñ = (-1) f(x) |xñ|-ñ |xnñ Uf |xnñ |yñ |y Å f(x1,...,xn)ñ Implementation? X X |x1ñ |x1ñ X X X X |xnñ U0 |xnñ |yñ |y Å [x = 0...0]ñ U0 |xñ|-ñ = (-1) [x = 0...0]|xñ|-ñ H H H Hadamard H 17 Grover’s algorithm: description II iteration 1 iteration 2 ... |0ñ |0ñ H Uf H U0 H Uf H U0 H |-ñ • construct state H |0...0ñ|-ñ • repeat k times: apply -H U0 H Uf to state n 3. measure state, to get xÎ{0,1} , and check if f (x) =1 (The setting of k will be determined later) 18 Grover’s algorithm: analysis I Let A = {x Î{0,1}n : f (x) = 1} and B = {x Î{0,1}n : f (x) = 0} and N = 2 and a = |A| and b = |B| n Let and Consider the space spanned by |Añ and |Bñ |Añ ß goal is to get close to this state H|0...0ñ |Bñ Interesting case: a << N 19 Grover’s algorithm: analysis II |Añ Algorithm: (-H U0 H Uf )k H |0...0ñ H|0...0ñ |Bñ Observation: Uf is a reflection about |Bñ: Uf |Añ = -|Añ and Uf |Bñ = |Bñ Question: what is -H U0 H ? U0 is a reflection about H|0...0ñ Partial proof: -H U0 H H|0...0ñ = -H U0 |0...0ñ = -H (- |0...0ñ) = H |0...0ñ 20 Grover’s algorithm: analysis III |Añ Algorithm: (-H U0 H Uf )k H |0...0ñ 2q 2q 2q 2q H|0...0ñ q |Bñ Since -H U0 H Uf is a composition of two reflections, it is a rotation by 2q, where sin(q)=Öa/N » Öa/N When a = 1, we want (2k+1)(1/ÖN) » p/2 , so k » (p/4)ÖN More generally, it suffices to set k » (p/4)ÖN/a Question: what if a is not known in advance? 21 Unknown number of solutions 1 solution 2 solutions 3 solutions 1 probability success 0 number of iterations p√N/2 4 solutions 6 solutions 100 solutions success probability very close to zero! Choose a random k in the range to get success probability > 0.43 22 Optimality of Grover’s algorithm 23 Optimality of Grover’s algorithm I Theorem: any quantum search algorithm for f : {0,1}n à {0,1} must make ( n) queries to f W Ö2 (if f is used as a black-box) Proof (of a slightly simplified version): f(x) Assume queries are of the form |x ñ f (-1) |x ñ and that a k-query algorithm is of the form |0...0ñ U0 f U1 f U2 f U3 f Uk where U0, U1, U2, ..., Uk, are arbitrary unitary operations 24 Optimality of Grover’s algorithm II n Define fr : {0,1} à {0,1} as fr (x) = 1 iff x = r Consider |0ñ U0 fr U1 fr U2 fr U3 fr Uk |ψr,kñ versus |0ñ U0 I U1 I U2 I U3 I Uk |ψr,0ñ We’ll show that, averaging over all r Î {0,1} , || |ψr,kñ - |ψr,0ñ || £ 2k / n Ö2n 25 Optimality of Grover’s algorithm III Consider |0ñ U0 I U1 I U2 fr U3 fr Uk |ψr,iñ k-i i Note that |ψr,kñ - |ψr,0ñ = (|ψr,kñ - |ψr,k-1ñ) + (|ψr,k-1ñ - |ψr,k-2ñ) + ... + (|ψr,1ñ - |ψr,0ñ) which implies || |ψr,kñ - |ψr,0ñ || £ || |ψr,kñ - |ψr,k-1ñ || + ... + || |ψr,1ñ - |ψr,0ñ || 26 Optimality of Grover’s algorithm IV query i query i+1 |0ñ U0 I U1 I U2 fr U3 fr Uk |ψr,iñ query i query i+1 |0ñ U0 I U1 I U2 I U3 fr Uk |ψr,i-1ñ || |ψr,iñ - |ψr,i-1ñ || = |2ai,r|, since query only negates |rñ Therefore, || |ψr,kñ - |ψr,0ñ || £ 27 Optimality of Grover’s algorithm V n Now, averaging over all r Î {0,1} , (By Cauchy-Schwarz) n ( n) Therefore, for some r Î {0,1} , the number of queries k must be W Ö2 , in order to distinguish fr from the all-zero function This completes the proof 28

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