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Quantum Computing. From Bits to Qubits Wayne Viers and Josh Lamkins. Quantum Computing Algorithms. Shor’s Algorithm - integer factorization Grovers Algorithm - searching an unsorted database. How Hard is Factoring?. RSA-129 - 129 Decimal Digits factored in 1994 using 1600 machines.

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quantum computing

Quantum Computing

From Bits to Qubits

Wayne Viers and Josh Lamkins

quantum computing algorithms
Quantum Computing Algorithms
  • Shor’s Algorithm - integer factorization
  • Grovers Algorithm - searching an unsorted database
how hard is factoring
How Hard is Factoring?
  • RSA-129 - 129 Decimal Digits factored in 1994 using 1600 machines.
  • RSA-768 - 768 bits (232 decimal digits) factored in 2009 using ~2000 years of computational power.
best known algorithms
Best Known Algorithms
  • b = log2n
  • Brute Force Method

for i in range of (2, sqrt(n))

if is_factor(i,n)

return i

RSA modulus

Running Time

# bits

Linear in the sqrt(n)

O(2b/2)

Assume Constant

best known algorithms1
Best Known Algorithms

*

  • Best Known
    • General Number Field Sieve
    • Exponential Running Time
      • O(b1/3 log b2/3)
  • Best Known for Quantum Computer
    • Shor’s Algorithm

*

Classical Computer

shor s algorithm
Shor’s Algorithm
  • Named after mathematician Peter Shor.
  • Quantum Algorithm for integer factorization formulated in 1994.
  • Simply put, it solves the following problem:
    • “Given an integer N, find its prime factors.”
shor s algorithm efficiency
Shor’s Algorithm Efficiency
  • On a quantum computer, to factor an integer N, takes O((log N)3).
  • This demonstrates that integer factorization problems can be efficiently solved on a quantum computer.
complexity class
Complexity Class
  • BQP - (Bounded-error Quantum Polynomial Time) is the class of decision problems solvable by a quantum computer in polynomial time, with an error probability of at most 1/3 for all instances.
shor s algorithm 2 parts
Shor’s Algorithm - 2 Parts
  • A reduction, which can be done on a classicalcomputer, of the factoring problem to the problem of order-finding.
  • A quantum algorithm to solve the order-finding problem.
shor s algorithm another version
Shor’s Algorithm - Another Version
  • The classical part of the algorithm turns the factoring problem into the problem of finding the period of a function - this part may be implemented in a classic computer.
  • The second part finds the period using a Quantum Fourier Transform and is responsible for the quantum speedup.
shor s algorithm classical part
Shor’s Algorithm - Classical Part
  • Pick a random number a<N
  • Compute gcd(a, N) (this may be done using the Euclidian Algorithm).
  • If gcd(a, N) 1, then there is a nontrivial factor of N and we are done.
  • Otherwise, use the period-finding subroutine to find r, the period of the following function:
    • f(x) = ax mod n
  • If r is odd, go back to step 1.
  • If ar/2= -1(mod N), go back to step 1.
  • gcd(ar/2+/- 1,N) is a nontrivial factor of N, we are done.

Quantum Part of the Algorithm

shor s algorithm quantum part
Shor’s Algorithm - Quantum Part
  • The Fourier Transform relates a functions time domain with its frequency domain. The component frequencies spread across the frequency spectrum, are represented as peaks in the frequency domain.
shor s algorithm quantum part1
Shor’s Algorithm - Quantum Part
  • Fourier analysis converts time (or space) to frequency and vice versa; a FFT rapidly computes such transformations by factorizing the DFT matrix into a product of sparse (mostly zero) factors.
  • The Quantum Fourier transform is the classical discrete Fourier transform applied to the vector of amplitudes of a quantum state.
grover s
Grover’s
  • Can be used to perform a search on an unsorted database
  • O(N1/2) run time with only O(logN) storage space
  • Has exponential speedup over the fastest possible classical algorithm
grover s1
Grover’s
  • The algorithm more specifically inverts functions → Finds y given F(x) and vice versa
  • In the case of searching databases a function can be determined whose solution is an element in a database.
other grover s applications
Other Grover’s Applications
  • Can be used to estimate the mean or median for a large set of numbers
  • Can be used to solve the collision problem (finding two inputs for a function that map to the same value)
grover s2
Grover’s
  • A probabilistic algorithm
  • Probability of failure is decreased by running the algorithm many times
grover s algorithm steps
Grover’s Algorithm Steps
  • Setup
    • Unsorted database of N entries
    • logN qubits
    • Have a function that maps each database entry to 0 or 1, with only one index mapping to 1
    • The qubits are initialized to all the possible function inputs
grover s algorithm steps1
Grover’s Algorithm Steps
  • Takes a function and a set of its possible inputs
  • Searches through those

inputs

  • Finds the single one that

causes the algorithm to

return true.