Beginner

1
Beginners Guide to Quantum Computing

• Graduate Seminar Presentation
• Oct. 5, 2007

2
Introduction

• Quantum Computation and Quantum Information by
  Nielsen and Chuang
• Answer Guide Tech Report
• An Introduction to Quantum Computing for
  Non-Physicists ACM Computing Surveys, Sept.
  2000
• Quantum Mechanics Demo

3
Qubits

• Representation of basic physical property
• Spin of atom
• Orientation of photon
• Computational Basis
• Ket notation
• 0gt 1gt

4
Qubit State

• Other states
• Complex probabilities
• x yi i is square root of -1
• Sum of square of absolute values of probabilities
  1
• Absolute value of complex number is distance from
  origin in complex plane
• abs(xyi) (x2 y2)0.5

5
Example Qubits

• (1/2)0.50gt (1/2)0.51gt
• (1/2)0.5 (0gt 1gt)
• (1/2)0.5,(1/2)0.5
• (1/2)0.5i/2,i/2
• sin(x),cos(x)
• sin(x)cos(x)i,0

6
Qubit Systems

• Two qubits (q0, q1) gt Four probabilities
• 00gt, 01gt, 10gt, 11gt
• Tensor product
• a,b c,d ac, ad, bc, bd
• N qubits gt 2N probabilities
• Exponential growth!

7
Measurement

• Reduces qubit to classical bit
• 1,0 (0gt) or 0, 1 (1gt)
• Can measure 1 qubit and leave rest alone

8
Entangled States

• Cannot be represented as tensor product of two
  qubits
• (1/2)0.5, 0, 0, (1/2)0.5 (Bell state)
• Measure 1 qubit, fixes other qubit!

9
Unitary Operators

• 1-qubit ops
• effect both complex probabilities
• 2x2 matrix of complex numbers
• UUT I (reversible)
• Examples
• THISXYZ
• T1 00 (1i)/(20.5)

10
(Walsh)-Hadamard Gate

• H (1/2)0.5, (1/2)0.5
  (1/2)0.5, -(1/2)0.5
• Applying to N qubits generates superposition
  2N possibilities equally likely
• True random number generator

11
Review

• Benefits
• Massive parallelism
• Exponential state space growth
• Problems
• Measurement collapses state
• Reversible computation
• No copying

12
Shors Algorithm

• Finding prime factors (RSA)
• Input N (integer) in binary (e.g., 128-bit)
• Randomly choose x, 1ltxltN
• Find smallest r such that xr N 1
• If r is even and x(r/2) N ! N-1
• Factors are at least one of gcd(x(r/2)-1,N)
  gcd(x(r/2)1,N)

13
Factoring 15

• Randomly pick 8
• 84 15 1
• gcd(82-1,15) 3
• gcd(821,15) 5

14
Shors Algorithm Quantum Part

• Finding r
• Superposition N qubits
• Apply xr N on all qubits
• Effectively calculates r for all values from 0 to
  N-1
• Find minimum value (1)

15
Shors Algorithm - Analysis

• Benefits
• Uses O(log N)3 time
• Uses O(log N) space
• Implemented on 7 qubit machine
• Cons
• Probabilistic est. 25 failure rate
• More qubits required than bits

16
Grovers Algorithm

• Unordered search O(N)
• Quantum results O(N0.5)
• Search matrix
• e.g., identity with -1 at desired location
• Rotation matrix
• Applied N0.5 times yields minimal failure rate
• Optimal for quantum

17
Phase Estimation

• Unitary op has eigenvalue e(2PIiX)
• Estimate X
• Basis for Grovers algorithm and Shors algorithm
• Shors algorithm achieves exponential speed-up
• Grovers algorithm quadratic

18
Simulation

• Java using dense matrices
• 12-qubit op requires 5-25 seconds
• 12-qubit Inverse QFT requires 2 minutes
• 10-qubit ripple carry adder requires 2 min
• Allows combination quantum and classical

19
Future

• Quantum algorithm other than phase estimation?
• Quantum computer larger than 16 qubits?
• Quantum data structures?
• Quantum subprocessor?

20
Questions?