Beginners Guide to Quantum Computing Graduate Seminar Presentation

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

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

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

Qubit State l “Other” states l 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 l (1/2)^0. 5|0> + (1/2)^0. 5|1> l (1/2)^0. 5 (|0> +|1>) l [(1/2)^0. 5, (1/2)^0. 5] l [(1/2)^0. 5+i/2, i/2] l [sin(x), cos(x)] l [sin(x)+cos(x)i, 0]

Qubit Systems l Two qubits (q 0, q 1) => Four probabilities ¡ |00>, |01>, |10>, |11> l Tensor ¡ [a, b] l. N product * [c, d] = [ac, ad, bc, bd] qubits => 2^N probabilities l Exponential growth!

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

Entangled States l Cannot be represented as tensor product of two qubits l [(1/2)^0. 5, 0, 0, (1/2)^0. 5] (Bell state) l Measure 1 qubit, “fixes” other qubit!

Unitary Operators l 1 -qubit ops ¡ effect both complex probabilities ¡ 2 x 2 matrix of complex numbers ¡ UUT = I (reversible) l Examples ¡ THISXYZ ¡ T=[1 0][0 (1+i)/(2^0. 5)]

(Walsh)-Hadamard Gate l. H = [(1/2)^0. 5, (1/2)^0. 5] [(1/2)^0. 5, -(1/2)^0. 5] l Applying to N qubits generates superposition – 2^N possibilities equally likely l True random number generator

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

Shor’s Algorithm l Finding prime factors (RSA) l Input N (integer) in binary (e. g. , 128 -bit) l Randomly choose x, 1<x<N l Find smallest r such that x^r % N = 1 l If r is even and x^(r/2) % N != N-1 l Factors are at least one of gcd(x^(r/2)-1, N) & gcd(x^(r/2)+1, N)

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

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

Shor’s Algorithm - Analysis l Benefits ¡ Uses O(log N)^3 time ¡ Uses O(log N) space ¡ Implemented on 7 qubit machine l Cons ¡ Probabilistic – est. 25% failure rate ¡ More qubits required than bits

Grover’s Algorithm l Unordered search – O(N) l Quantum results – O(N^0. 5) l Search matrix ¡ e. g. , identity with -1 at desired location l Rotation ¡ Applied l Optimal matrix N^0. 5 times yields minimal failure rate for quantum

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

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

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

Questions?