Beginner’s Guide to Quantum Computing Graduate Seminar Presentation Oct. 5, 2007

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 Quantum Mechanics Demo

Qubits Representation of basic physical property  Spin of atom  Orientation of photon Computational Basis  Ket notation  |0> |1>

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(x+yi) = (x^2 + y^2)^0.5

Example Qubits (1/2)^0.5|0> + (1/2)^0.5|1> (1/2)^0.5 (|0> +|1>) [(1/2)^0.5,(1/2)^0.5] [(1/2)^0.5+i/2,i/2] [sin(x),cos(x)] [sin(x)+cos(x)i,0]

Qubit Systems Two qubits (q0, q1) => Four probabilities  |00>, |01>, |10>, |11> Tensor product  [a,b] * [c,d] = [ac, ad, bc, bd] N qubits => 2^N probabilities Exponential growth!

Measurement Reduces qubit to classical bit [1,0] (|0>) or [0, 1] (|1>) Can measure 1 qubit and leave rest alone

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!

Unitary Operators 1-qubit ops  effect both complex probabilities  2x2 matrix of complex numbers  UU T = I (reversible) Examples  THISXYZ  T=[1 0][0 (1+i)/(2^0.5)]

(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 – 2^N possibilities equally likely True random number generator

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

Shor’s Algorithm Finding prime factors (RSA) Input N (integer) in binary (e.g., 128-bit) Randomly choose x, 1<x<N Find smallest r such that x^r % 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)

Factoring 15 Randomly pick 8 8^4 % 15 = 1 gcd(8^2-1,15) = 3 gcd(8^2+1,15) = 5

Shor’s Algorithm – Quantum Part Finding r Superposition N qubits Apply x^r % N on all qubits  Effectively calculates r for all values from 0 to N-1 Find minimum value (1)

Shor’s 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

Grover’s Algorithm Unordered search – O(N) Quantum results – O(N^0.5) Search matrix  e.g., identity with -1 at desired location Rotation matrix  Applied N^0.5 times yields minimal failure rate Optimal for quantum

Phase Estimation Unitary op has eigenvalue e^(2*PI*i*X) Estimate X Basis for Grover’s algorithm and Shor’s algorithm Shor’s algorithm achieves exponential speed-up Grover’s algorithm quadratic

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

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

Questions?