How Quantum Computing This Lower Bound Is . . . Can Help With - PowerPoint PPT Presentation

Known Advantages of . . . Need to Consider . . . We Consider Only . . . Non-Quantum Lower . . . How Quantum Computing This Lower Bound Is . . . Can Help With (Continuous) How Quantum . . . Auxiliary Quantum . . . Optimization Main Algorithm: First . . . Main Algorithm: . . . Christian Ayub, Martine Ceberio, and Vladik Kreinovich Home Page Title Page Department of Computer Science University of Texas at El Paso, El Paso, Texas 79968, USA, ◭◭ ◮◮ cayub@miners.utep.edu, mceberio@utep.edu, vladik@utep.edu ◭ ◮ Page 1 of 24 Go Back Full Screen Close Quit

Known Advantages of . . . Need to Consider . . . 1. Known Advantages of Quantum Computing We Consider Only . . . • It is known that quantum computing enables us to Non-Quantum Lower . . . drastically speed up many computations. This Lower Bound Is . . . How Quantum . . . • One example of such a problem is the problem of look- Auxiliary Quantum . . . ing for a given element in an unsorted n -element array. Main Algorithm: First . . . • With non-quantum computations: Main Algorithm: . . . Home Page – to be sure that we have found this element, – we need to spend at least n computational steps. Title Page ◭◭ ◮◮ • Indeed, if we use fewer than n steps: ◭ ◮ – this would mean that we only look at less than n elements of the array, and thus, Page 2 of 24 – we may miss the element that we are looking for. Go Back Full Screen Close Quit

Known Advantages of . . . Need to Consider . . . 2. Advantages of Quantum Computing (cont-d) We Consider Only . . . • Grover’s quantum-computing algorithm allows us to Non-Quantum Lower . . . reduce the time to c · √ n . This Lower Bound Is . . . How Quantum . . . • So, we reduce the non-quantum computation time T to Auxiliary Quantum . . . √ T q ∼ T. Main Algorithm: First . . . Main Algorithm: . . . Home Page Title Page ◭◭ ◮◮ ◭ ◮ Page 3 of 24 Go Back Full Screen Close Quit

Known Advantages of . . . Need to Consider . . . 3. Need to Consider Optimization Problems We Consider Only . . . • In many applications, we also need to solve continuous Non-Quantum Lower . . . optimization problems. This Lower Bound Is . . . How Quantum . . . • We want to find an object or a strategy for which the Auxiliary Quantum . . . given objective function attains its maximum. Main Algorithm: First . . . • An object is usually characterized by its parameters Main Algorithm: . . . Home Page x 1 , . . . , x n . Title Page • For each x i , we usually know the bounds: x i ≤ x i ≤ x i . ◭◭ ◮◮ • Let f ( x 1 , . . . , x n ) denote the value of the objective func- ◭ ◮ tion corresponding to the parameters x 1 , . . . , x n . Page 4 of 24 • In most practical situations, the objective function is Go Back several (at least two) times continuously differentiable. Full Screen Close Quit

Known Advantages of . . . Need to Consider . . . 4. General Optimization Problem We Consider Only . . . • Ideal case: find x opt 1 , . . . , x opt for which f ( x 1 , . . . , x n ) Non-Quantum Lower . . . n attains its maximum on the box This Lower Bound Is . . . How Quantum . . . def B = [ x 1 , x 1 ] × . . . × [ x n , x n ] . Auxiliary Quantum . . . • In practice, we can only attain values approximately. Main Algorithm: First . . . Main Algorithm: . . . • So, in practice, we are looking for the values x d 1 , . . . , x d n Home Page which are maximal with given accuracy ε > 0: Title Page � � f ( x d 1 , . . . , x d n ) ≥ ( x 1 ,...,x n ) ∈ B f ( x 1 , . . . , x n ) max − ε. ◭◭ ◮◮ ◭ ◮ • We show that for this problem, quantum computing √ Page 5 of 24 reduced computation time T to T q ∼ T · ln( T ). Go Back Full Screen Close Quit

Known Advantages of . . . Need to Consider . . . 5. We Consider Only Guaranteed Global Opti- We Consider Only . . . mization Algorithms Non-Quantum Lower . . . • Of course, there are many semi-heuristic ways to solve This Lower Bound Is . . . the optimization problem. How Quantum . . . Auxiliary Quantum . . . • For example, we can start at some point x = ( x 1 , . . . , x n ) Main Algorithm: First . . . and use gradient techniques to reach a local maximum. Main Algorithm: . . . • However, these methods only lead to a local maximum. Home Page • If we want to make sure that we reached the actual Title Page ( global ) maximum: ◭◭ ◮◮ – we cannot skip some subdomains of the box B , ◭ ◮ – we have to analyze all of them. Page 6 of 24 • How can we do it? Go Back Full Screen Close Quit

Known Advantages of . . . Need to Consider . . . 6. Non-Quantum Lower Bound We Consider Only . . . • Let us select some size δ > 0 (to be determined later). Non-Quantum Lower . . . = x i − x i This Lower Bound Is . . . def • Let us divide each interval [ x i , x i ] into N i δ How Quantum . . . subintervals of width δ . Auxiliary Quantum . . . • This divides the whole box B into into Main Algorithm: First . . . n Main Algorithm: . . . x i − x i = V � N = N 1 · . . . · N n = δ n subboxes . Home Page δ i =1 Title Page • Here, V is the volume of the original box B : ◭◭ ◮◮ def V = ( x 1 − x 1 ) · . . . · ( x n − x n ) . ◭ ◮ Page 7 of 24 • We can have functions which are 0 everywhere except for one subbox at which this function grows to 1 . 1 · ε . Go Back • On this subbox, the function is approximately quadratic. Full Screen • We have a bound S on the second derivative. Close Quit

Known Advantages of . . . Need to Consider . . . 7. Non-Quantum Lower Bound (cont-d) We Consider Only . . . • This function starts with 0 at a neighboring subbox. Non-Quantum Lower . . . • So, it cannot grow faster than S · x 2 on this subbox. This Lower Bound Is . . . How Quantum . . . • Thus, to reach a value larger than ε , we need to select Auxiliary Quantum . . . δ for which S · ( δ/ 2) 2 = 1 . 1 · ε , i.e., the value δ ∼ ε 1 / 2 . Main Algorithm: First . . . • For this value δ , we get V/δ n ∼ ε − ( n/ 2) subboxes: Main Algorithm: . . . Home Page – if we do not explore some values of the optimized function at each of the subboxes, Title Page – we may miss the subbox that contains the largest ◭◭ ◮◮ value. ◭ ◮ • Thus, we will not be able to localize the point at which Page 8 of 24 the function attains its maximum. Go Back • So, to locate the global maximum, we need at least as Full Screen many computation steps as there are subboxes. Close • So, we need at least time ∼ ε − ( n/ 2) . Quit

Known Advantages of . . . Need to Consider . . . 8. This Lower Bound Is Reachable We Consider Only . . . • Let us show that there exists an algorithm that always Non-Quantum Lower . . . locates the global maximum in time ∼ ε − ( n/ 2) . This Lower Bound Is . . . How Quantum . . . • Let us divide the box B into subboxes of linear size δ . Auxiliary Quantum . . . • For each subbox b , each of its sides has size ≤ δ . Main Algorithm: First . . . • Thus, each component x i differs from the midpoint’s Main Algorithm: . . . x m def = ( x m 1 , . . . , x m Home Page n ) by no more than δ/ 2: Title Page def = x i − x m | ∆ x i | ≤ δ/ 2 , where ∆ x i i . ◭◭ ◮◮ • Thus, by using known formulas from calculus, we can ◭ ◮ conclude that for each point x = ( x 1 , . . . , x n ) ∈ b : Page 9 of 24 f ( x 1 , . . . , x n ) = f ( x m 1 + ∆ x 1 , . . . , x m n + ∆ x n ) = Go Back n n n � � � f ( x m 1 , . . . , x m n ) + c i · ∆ x i + c ij · ∆ x i · ∆ x j . Full Screen i =1 i =1 j =1 Close Quit

Known Advantages of . . . Need to Consider . . . 9. This Lower Bound Is Reachable (cont-d) We Consider Only . . . ∂ 2 f = ∂f Non-Quantum Lower . . . def def ( x m 1 , . . . , x m • Here c i n ) , and c ij = ( ξ 1 , . . . , ξ n ) ∂x i ∂x i ∂x j This Lower Bound Is . . . for some ξ 1 , . . . , ξ n ) ∈ b. How Quantum . . . Auxiliary Quantum . . . • We assumed that the function f is twice continuously differentiable. Main Algorithm: First . . . Main Algorithm: . . . • So, all its second derivatives are continuous. Home Page • Thus, there exists a general bound S on all the values Title Page of all second derivatives: | c ij | ≤ S . ◭◭ ◮◮ • Because of these bounds, the quadratic terms in the ◭ ◮ above formula are bounded by n 2 · S · ( δ/ 2) 2 = O ( δ 2 ). Page 10 of 24 Go Back Full Screen Close Quit

Known Advantages of . . . Need to Consider . . . 10. These Estimations Lead to a Global Piece- We Consider Only . . . Wise Linear Approximate Function Non-Quantum Lower . . . • By considering only linear terms on each subbox, we This Lower Bound Is . . . get an approximate piece-wise linear function f ≈ ( x 1 , . . . , x n ). How Quantum . . . Auxiliary Quantum . . . • On each subbox b : Main Algorithm: First . . . n � f ≈ ( x 1 , . . . , x n ) = f ( x m 1 , . . . , x m n ) + c i · ∆ x i . Main Algorithm: . . . Home Page i =1 Title Page • For each x = ( x 1 , . . . , x n ), we have ◭◭ ◮◮ | f ( x 1 , . . . , x n ) − f ≈ ( x 1 , . . . , x n ) | ≤ n 2 · S · ( δ/ 2) 2 . ◭ ◮ Page 11 of 24 Go Back Full Screen Close Quit