Quantum Computing and Quantum Programming Language Choon Oh Lee ISILab, KAIST

Motivation Moore’s Law Number of transistors per square inch doubles every two years. Expected be broken down in 2020 Atomic scales are reached Incoherent by any particle Solutions Fatalists: Noise-based Computing Optimists: Quantum Computing

various candidates (Ion trap, NMR, etc.) Bit vs. Qubit Value Realization Computation Universal component or NAND and NOR gate 1 various candidates (Ion trap, NMR, etc.) Controlled Not gate 5 V 0 V 1 1 S Q. Circuit ADDER S Measuring 1 1

How’s qubit possible? The legendary experiment By David Wineland and Christopher Monroe Steps Pin Barium ion in vacuum room Optical freezing on ion to deactivate it Expose ion on laser pulse for certain time Barium ion had superposition Slightly push ion using laser beam One ion existed on two different spots Remarks Superposition is collapsed in 25~50 microsecond NIST scientists succeed to make first controlled not gate based on this experiment

Conceptual Models Quantum Circuit Matrix Mechanics Computational Model Algorithmic Model Quantum Circuit Matrix Mechanics A B D M C M 4 by 4 2 by 2 2 by 2 2 by 2 2 by 2 2 by 1 2 by 1

Quantum Gates Pauli Gates Identity (I) Not Gate (X) Y Gate Z Gate

Quantum Gates Hadamard Gate (H) Make a qubit superpositioned.

Quantum Gates Controlled Not Gate (cNot) Apply NOT gate on second bit when first bit is 1.

Quantum Gates Measurement (M) Technically, it’s not a gate It measures a qubit to determine its value Output of measurement would be 0 or 1 Probability to be 0: Probability to be 1: After it measures qubit, qubit becomes just bit

Algorithms Quantum Teleportation Qubits cannot be copied or moved How to teleport a qubit Prepare two entangled qubits, Alice and Bob take each qubit initially Alice wants to send an another qubit to Bob

Algorithms Quantum Teleportation H M M Z Originalqubit Alice’s qubit Bob’s qubit Z Same qubit

Algorithms Quantum Teleportation Initial state H M M Z Originalqubit Alice’s qubit M Bob’s qubit Z Same qubit

Algorithms Quantum Teleportation Previous state Current state H M M Z Originalqubit H M Alice’s qubit M Bob’s qubit Z Same qubit

Algorithms Quantum Teleportation Previous state Current state H M M Z Originalqubit H M Alice’s qubit M Bob’s qubit Z Same qubit

Algorithms Quantum Teleportation Rearrange current state H M M Z Originalqubit H M Alice’s qubit M Bob’s qubit Z Same qubit

Algorithms Quantum Teleportation Measure first two qubits H M M Z Originalqubit H M Alice’s qubit M Bob’s qubit Z Same qubit

Algorithms Grover’s Search Algorithm Searching desired items from database Instead of checking every items in database, algorithm increases probabilities that desired items can be found decreasing others’ Idea is simple, but point is quantum parallelism!

Algorithms Grover’s Search Algorithm 1 2 3 4 5 6 7 Probability 1 2 3 4 5 6 7 Probability (amplitude)2 1. Apply Hadamard gates to make superposition 1 2 3 4 5 6 7 Probability (amplitude)2

Algorithms Grover’s Search Algorithm 1 2 3 4 5 6 7 Probability 1 2 3 4 5 6 7 Probability (amplitude)2 2. Apply function f 1 2 3 4 5 6 7 Probability (amplitude)2

Algorithms Grover’s Search Algorithm 1 2 3 4 5 6 7 Probability 1 2 3 4 5 6 7 Probability (amplitude)2 Average line 3. Flip probability based on average point 1 2 3 4 5 6 7 Probability (amplitude)2 Average line

Algorithms Shor’s Factoring Algorithm Great algorithm that attracted the world’s attention to quantum computing Proposed shortcut to break RSA system The source of RSA’s power is hardness of factorization of a big number which is product of two prime numbers Best known way to factor a number, N Find a and b where N divides (a2 – b2) If N doesn’t divide (a + b) and (a – b), Then, gcd(N, (a + b)) is one of factors of N Another factor is then N / gcd(N, (a + b))

Algorithms Shor’s Factoring Algorithm Algorithm input: an odd integer n Choose g in [2…n-1] randomly Calculate d = gcd(g, n) If d ≠ 1, return d Calculate r where gr = 1 (mod n) If r is even, and n doesn’t divide gr/2+1 or gr/2-1, then return gcd(n, gr/2+1) Re-find r or g Shor used superpositioned qubits to represent g and QFT to calculate r