Quantum Processing Simulation Dalibor Hrg Vienna, June 18, 2004.

Moore’s law, classical computers

Technology and computation NANOTECHNOLOGY

Development Impact on big mathematical questions (P=PSPACE, P=NP), theoretical research! We still don’t know if quantum computers are stronger than classical computers! von Neumann architecture? Quantum memory is needed! (in progress) “Quantum cryptography” is demonstrated! (problem with error corection codes and speed) “Quantum teleportation”, Quantum communication methods (demonstrated, in progress)

Classical and Quantum computer State of classical computer of quantum computer bits qubits in 2 bits: in 2 qubits: Transformation of states classical computer quantum computer Boolean circuits: Unitary operators: (Quantum circuits)

Classical and Quantum algorithms (C,C++,C#,… ) A problem Asembler Machine code Boolean circuits Pseudo code quantum Mathematical model Quantum circuits (?) Grover, Shor Deutsch-Jozsa, Simon ? EASY HARD

Quantum algorithms Grover’s algorithm (1997.) - searching unsorted database of N elements in steps - on classical computer, steps are needed - if sorted, there exist classical algorithm with steps Deutsch-Jozsa problem (1992.) - finding global property of some Boolean function with N variables (function is constant or balanced) - complexity of quantum algorithm - complexity of classical algorithm

Grover’s algorithm

Deutsch-Jozsa problem For state amplitude is Function constant if: Function balanced if:

QPS Application Quantum gates (unitary operators). Act on selected qubits of quantum register. All states of register are seen here! Quantum register. State of a qubit is colored: (blue, state is 0), (red, state is 1), (green, superposition of 0 and 1).

Characteristics of the QPS Windows application, C#, .NET Framework 1.1 Grover’s and Deutsch-Jozsa algorithm simulation (up to 8 qubits). Implementation of the most useful operators (H, Pauli X, Pauli Z, Oracle, WH, Grover). Easy to use interface (selecting qubits and operators) For education and further research on quantum algorithms (handy tool).

Memory for Walsh-Hadamard operation Simulation problems? Number of qubits: N Number of states in register: Needed memory for all states: Needed memory for Walsh-Hadamard (interference) operation: Number of qubits Number of states Memory for all states Memory for Walsh-Hadamard operation 4 16 0.18 KB 2 KB 8 256 4 KB 0.5 MB 12 1024 80 KB 128 MB 65536 1.5 MB 32 GB 32 4294967296 160 GB 13.7 10 GB 64 1.84 10 1.23 10 GB 31.2 10 GB

Conclusion Quantum algorithms can be simulated, but inefficiantly (memory used and time needed) on classical computers. Impossibility to implement quantum parallelism is a main reason for inefficient simulation ( > 10 qubits on classical PC, 256-512 MB RAM). QPS is useful in education and research (handy tool).