1. Introduction to Quantum logic (2) 2002.10.10 Yong-woo Choi

2. Classical Logic Circuits v Circuit behavior is governed implicitly by classical physics v Signal states are simple bit vectors, e.g. X = 01010111 v Operations are defined by Boolean Algebra

3. Quantum Logic Circuits v Circuit behavior is governed explicitly by quantum mechanics v Signal states are vectors interpreted as a superposition of binary “qubit” vectors with complex-number coefficients v Operations are defined by linear algebra over Hilbert Space and can be represented by unitary matrices with complex elements

4. Signal state (one qubit) Corresponsd to

5. More than one qubit v If we concatenate two qubits We have a 2-qubit system with 4 basis states We have a 2-qubit system with 4 basis states And we can also describe the state as And we can also describe the state as Or by the vector Tensor product

6. Quantum Operations v Any linear operation that takes states satisfying satisfying and maps them to states and maps them to states satisfying satisfying must be UNITARY must be UNITARY

7. Find the quantum gate(operation) From upper statement We now know the necessities 1. A matrix must has inverse, that is reversible. 1. A matrix must has inverse, that is reversible. 2. Inverse matrix is the same as. 2. Inverse matrix is the same as. So, reversible matrix is good candidate for quantum gate

8. Quantum Gate  One-Input gate: Wire WIRE   By matrix Wire Input Output

9. Quantum Gate  One-Input gate: NOT NOT   By matrix NOT Gate Input Output

10. Quantum Gate  Two-Input Gate: Controlled NOT (Feynman gate) CNOT   By matrix = concatenate y

11. Quantum Gate  3-Input gate: Controlled CNOT (Toffoli gate) NOT NOT   By matrix =

12. Quantum circuit G1 G2 G3 G4 G5 G7 G6 G8 TensorProduct Matrix multiplication multiplication