Introduction to Quantum logic (2)

1. Introduction to Quantum logic (2) 2002 .10 .10 Yong-woo Choi

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

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

4. Signal state (one qubit) Corresponsd to Corresponsd to Corresponsd to

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

6. Quantum Operations • Any linear operation that takes states satisfying and maps them to states satisfying 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. 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 Input Wire Output

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

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

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

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