distributed check prev notes... --- 1 Outline --- 2 Simulation = software implementation of a mathematical model Here: software implementation; other: use one quantum process to simulate another software implementation: don't need hardware, faster, cheaper; don't need measurement Grover with depolarizing noise "node" -> "compute node" or "machine"? Here: pure states, possibly entangled so not separable, represent using vector vs. mixed states, statistical ensemble of pure states, represent using density matrix vs. unentagled, separable - efficient simulation, solve problem classically, don't need quantum pure states, unitary matrix ops -> implemented directly without using matrices high level QC simulation: resulting quantum state is the same regardless of whether it is simulated at a low-level or high-level. --- 3 Classical Computing, Reversible Computing NOT is reversible quantum gates not physically present, represent an operation e.g. apply light or magnetic pulse quantum reversible ob unitary matrix evolution; measure value is not reversible Tolloli - AND could be any function, ob XOR is reversible --- 4 One Classical Bit, Two Classical Bits --- 5 Simulating One Classical Bit, Simulating Two (or more) Classical Bits --- 6 Simulating One Quantum Bit --- 7 Single Qubit General Unitary Operation show OpenQASM3 U gate, with gphase, ctrl ? + std lib ? --- 8 Simulating Two Quantum Bits in general, do everything twice, once with other bit 0, once with other bit 1 3 qubits, do everything 4 times, other bits 00, 01, 10, 11 X on one qubit: physically one op, simulation modify entire state array --- 9 Simulating Two Quantum Bits - Hadamard --- 10 Simulating n Quantum Bits - Hadamard --- 11 Simulating Measurements, One Qubit Simulating Measurements - generally not useful, just look at the probabilities --- 12 Simulating Measurements, Two Qubits - measure both --- 13 Simulating Measurements, Two Qubits - measure one --- 14 Measuring Entanglement - Theory --- 15 Measuring Entanglement - Code --- 16 Entanglement Example - Super Dense Coding secure against Eve in the middle, P=0.5 --- 17 Super Dense Coding with Hadamard Phase Shift Error --- 18 Addition - low-level implementation add - can change existing entanglement, create new entanglement classical operation, don't need quantum --- 19 Addition - high-level simulation --- 20 Grover's Algorithm - theory --- 21 Grover's Algorithm - low-level implementation --- 22 Grover's Algorithm - high-level simulation Grover - simulation: flip sign of m, we know the answer m, easy, no auxiliary qubit needed, real application: implement AES-128 or SHA256 in QASM, hard, 2**11 gate depth --- 23 Grover's Algorithm - Entanglement --- 24 Quantum Fourier Transform - low-level implementation Shor - factoring to break RSA, find hidden subgroup to break DH and ECC --- 25 Quantum Fourier Transform - high-level simulation --- 26 Future Work --- low-level corresponds more closely to physical Noise: noise, faults, decoherence, depolarization.