quantum-computing.pdf

# In practice this is not a problem rerun until prob of

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

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10 Is this leading to something real?
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

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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.
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?

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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
15 Quantum Measurement Demo

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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 ]
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 , ...
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• Fall '13
• Quantum computer, Integer factorization, Shor's algorithm, Qubit, quantum computers, Quantum information science

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