In practice this is not a problem rerun until prob of

This preview shows page 9 - 17 out of 43 pages.

In practice this is not a problem: rerun until prob of error is less than prob of earth exploding, etc.
Image of page 9

Subscribe to view the full document.

10 Is this leading to something real?
Image of page 10
11 Database Search Problem Database Search Problem: To Solve: guess random answers and check them There are n possible answers Each guess takes the same time to check Example Uses: Searching for an entry in an unsorted array Guessing your friend's password Attacking symmetric ciphers (AES, 3DES) Classical Running Time: O(n) linear search Average Case: (n+1)/2 guesses to find answer
Image of page 11

Subscribe to view the full document.

12 Enter Lov Grover in 1996 Grover's Algorithm Quantum algorithm for solving the database search problem (i.e., for doing linear search) Takes O( N) time (yes, sqrt(N) time!) Uses O(log 2 N) qubits of storage Probability of measuring wrong answer: O(1/N) Algorithm is optimal I will not explain Grover's Algorithm. Instead, we will cover Shor's Algorithm.
Image of page 12
13 Let's Break RSA Break RSA == find factors of large integer N No known polynomial classical algorithms General Number Field Sieve: ~ O(2 b ) b is number of bits in N (e.g., b =512) Could use Grover Linear search for factors in O( 2 b ) = O( 2 b/2 ) time Still too slow! New Proposed Quantum Approach Try all factors in parallel, pick the right one! Does this work?
Image of page 13

Subscribe to view the full document.

14 Itty Bitty Living Space Despite popular rumors, quantum computers cannot easily “try everything in parallel” Sure, you can try everything in parallel But when you measure the outcome, you get something random ! In this case, you'd get a random (non-)divisor Which is not what we want! Using the power of quantum computing to look for a needle in a haystack is not efficient! So we must exploit the structure of the problem
Image of page 14
15 Quantum Measurement Demo
Image of page 15

Subscribe to view the full document.

16 Needles In Haystacks Quantum computing can give you a random answer back So the trick is to make sure that even a random answer will help you By making sure that all answers have some important property that contributes to what you really want to know Look: if you think about quantum computing in terms of “parallel universes” (and whether you do or don’t is up to you), there’s no feasible way to detect a single universe that’s different from all the rest. Such a lone voice in the wilderness would be drowned out by the vast number of suburb-dwelling, Dockers-wearing conformist universes. What one can hope to detect, however, is a joint property of all the parallel universes together — a property that can only be revealed by a computation to which all the universes contribute. [ Scott Aaronson ]
Image of page 16
17 Digression: Periodic Sequences Powers of Two: 2, 4, 8, 16, 32, 64, 128, 256, ... Not a periodic sequence. Powers of Two Mod 15: 2, 4, 8, 1 , 2, 4, 8, 1 , ...
Image of page 17
You've reached the end of this preview.
  • Fall '13
  • Quantum computer, Integer factorization, Shor's algorithm, Qubit, quantum computers, Quantum information science

{[ snackBarMessage ]}

What students are saying

  • Left Quote Icon

    As a current student on this bumpy collegiate pathway, I stumbled upon Course Hero, where I can find study resources for nearly all my courses, get online help from tutors 24/7, and even share my old projects, papers, and lecture notes with other students.

    Student Picture

    Kiran Temple University Fox School of Business ‘17, Course Hero Intern

  • Left Quote Icon

    I cannot even describe how much Course Hero helped me this summer. It’s truly become something I can always rely on and help me. In the end, I was not only able to survive summer classes, but I was able to thrive thanks to Course Hero.

    Student Picture

    Dana University of Pennsylvania ‘17, Course Hero Intern

  • Left Quote Icon

    The ability to access any university’s resources through Course Hero proved invaluable in my case. I was behind on Tulane coursework and actually used UCLA’s materials to help me move forward and get everything together on time.

    Student Picture

    Jill Tulane University ‘16, Course Hero Intern