Introduction The Search Algorithm Grovers Algorithm References. Grovers Algorithm. Quantum Parallelism. Joseph Spring.

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1 Quantum Parallelism Applications

2 Outline 1 2 One or Two Points 3 4

3 Quantum Parallelism We have discussed the concept of quantum parallelism and now consider a range of applications. These will include: The Deutsch-Josza Algorithm One of the first of this type of algorithm suggesting potential benefits from quantum algorithms based on quantum parallelism and interference over classical algorithms Shor s Algorithm This has the potential to make any public key cryptography algorithm based on either the DLP or the IFP redundant. No more RSA, El-Gamal,... A search algorithm offering potential benefits over classical alternatives. Is this the end of symmetric key cryptography?

4 Background Grovers algorithm (1996) is a quantum algorithm designed to search an unsorted database containing N entries, for a specific unique item. It runs in a time of the order O( N) and uses O(logN) space. Like Shor s Algorithm it is a probabalistic with high probability of producing the correct answer. It is said to offer quadratic speedup over corresponding classical search algorithms. Note this is not exponential speedup It is still a considerable improvement on alternative algorithms gaining significant advantage as the size N of the database increases If there exists M copies of the item being searched for then the search time is of the order O( N M )

5 Background Classical Algorithms A search algorithm is a procedure for finding an element that satisfies given criteria, from a collection of geven elements. The elements may be part of a database (a colletion of records/items), a search space (defined by a mathematical formula or procedure) or a combination of the two (Hamiltonian path in graph theory - visiting each vertice in a graph once only) Classical Search Algorithm Linear/Sequential Search. Start at beginning and compare each entry to criteria. Average Time O(N) Binary Search (sorted database), start in middle, compare to criteria, work with upper half or lower half until item found. Average Time O(logN) Travelling Salesman: (Shortest Route), O(N), Backtracking, Alpha-Beta Minimax Search (Chess Games),...

6 Outline One or Two Points 1 2 One or Two Points 3 4

7 Search Algorithm Procedure One or Two Points Let S = {e i } N=2n 1 i=0 denote the search space. Let M N denote the number of elements that satisfy the query. The Problem: Identify one element satisfying the query. Procedure: Select an element e i of S and determine whether or not the element satisfies the query. Let f (x) denote a function with 0 x N = 2 n 1 such that f (x) = { 0ifx does NOT satisfy the query 1ifx does satisfy the query

8 Search Algorithm The Oracle O One or Two Points This is a Black Box accepting an n-qubit input (indexed by i for element e i ) The Oracle another input referred to as an oracle qubit. This is set to q and flipped to q if the oracle recognises a solution. q {0, 1} The black box performs the following transformation i q O q f (i)

9 Search Algorithm The Oracle O One or Two Points Suppose that the oracle qubit is set to. Then: i = 1 2 i ( 0 1 ) O { 1 2 i ( 0 1 ) if f (i) =0, so e i is not a solution 1 2 i ( 0 1 ) =( 1) f (i) i if f (i) =1, so e i is a solution Since the oracle qubit does not change overall, we describe the transformation O as: i O ( 1) f (i) i

10 Outline 1 2 One or Two Points 3 4

11 Algorithm involves the following steps: 1. Initialise the n qubits to Apply Hadamard H n 3. Apply The Grover iteration O( N) times Step a) Apply the Oracle Step b) Apply Hadamard H n to the output Step c) Perform Conditional Phase Shift i ( 1) f (j) j Step d) Apply Hadamard H n to the output 4. Perform Measurement

12 Steps 1 and 2 Step 1 ψ 0 = Step 2 ψ 1 = H n = H 0 H 0 H 0...H 0 = = 1 2 n/2 Σ z z For general ψ = x 1 x 2 x 3...x n ψ = H n x 1 x 2 x 3...x n = H x 1 H x 2 H x 3...H x n = 1 2 n/2 Σ z ( 1) x z z (Lemma 3, Deutsch-Josza slides)

13 Outline 1 2 One or Two Points 3 4

14 Circuit 1 - One Round of the G

15 The G We note that: The Phase Shift Gate leaves i alone if f (i) =0 and transforms i to i when f (i) > 0 Lemma It may be shown that G =(2 ψ ψ I)O = f ψ O Let φ = N 1 Σ α k k k=0 Then f ψ1 φ = N 1 Σ β r r with β r = 2α α r and α = 1 N 1 r=0 N Σ α k k=0

16 Step 3 - Circuit

17 Outline 1 2 One or Two Points 3 4

18 Geometric Interpretation The discussion on geometric interpretation of Grovers algorithm in (for example) [2], [4] (or wikipaedia) motivates the idea that repeated use of the Grovers Iteration G, brings the vector G ψ, ( ψ denoting the initial state), closer and closer to b, the state corresponding to a uniform superposition of all solutions to the query/search problem. This produces, with high probability, one of the outcomes superposed in b. N O( M ) iterations are required to rotate ψ close to b, explaining the efficiency of the algorithm.

19 Question Question: So what impact will this have on the demise, if any of symmetric key cryptography? Conjecture: The speed up obtained is not exponential and so symmetric key problems remain outside of P complexity. Comments?

20 [1] L. K. Grover, A fast quantum mechanical algorithm for database search, Proceedings, 28th Annual ACM Symposium on the Theory of Computing, (May 1996) p. 212, See also: [2] D. C. Marinescu & G. M. Marinescu, Approaching Quantum Computing, Pearson Prentice Hall, [3] N. D. Mermin, Quantum omputer Science, Cambridge University Press, 2007 [4] M. A. Nielsen & I. L. Chuang, Quantum Computation and Quantum Information, Cambridge University Press, 2000.