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Introduction to Quantum logic (2) 2002 .10 .10 Yong-woo Choi 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 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 algebra over Hilbert Space and can be represented by unitary matrices with complex elements Signal state (one qubit) 0 1 0 0 1 1 Corresponsd to 1 0 Corresponsd to 0 1 Corresponsd to 1 0 0 0 1 0 1 1 2 2 0 1 1 More than one qubit If we concatenate two qubits 0 0 1 1 0 0 1 1 0 1 1 2 2 0 1 1 2 2 We have a 2-qubit system with 4 basis states 0 0 00 1 0 10 0 1 01 1 1 11 And we can also describe the state as 0 0 00 0 1 01 1 0 10 11 11 Or by the vector 0 0 0 1 1 0 1 1 1 2 2 2 2 2 2 2 2 0 0 0 0 0 0 1 0 1 1 0 1 1 0 1 1 1 1 Tensor product Quantum Operations Any linear operation that takes states 0 0 1 1 satisfying 0 1 1 2 2 and maps them to states 0 0 1 1 satisfying must be UNITARY t UU I t U U 1 0 1 1 2 2 Find the quantum gate(operation) t UU I t U U 1 From upper statement We now know the necessities 1. A matrix must has inverse , that is reversible. 2. Inverse matrix is the same as U t. So, reversible matrix is good candidate for quantum gate Quantum Gate One-Input gate: Wire c0 0 c1 1 c0 0 c1 1 WIRE Input By matrix 1 0 c0 c0 0 1 c1 c1 Wire Output Quantum Gate One-Input gate: NOT c0 0 c1 1 c1 0 c0 1 NOT Input By matrix 0 1 c0 c1 1 0 c1 c0 NOT Gate Output Quantum Gate Two-Input Gate: Controlled NOT (Feynman gate) x y CNOT x x x x y y x y concatenate By matrix 1 0 0 0 1 0 0 0 0 0 0 1 y 0 0 1 0 00 00 01 01 10 = 11 11 10 x 0 x x x 1 x x Quantum Gate 3-Input gate: Controlled CNOT (Toffoli gate) x y C 2 NOT z By matrix 1 0 0 0 0 0 0 0 x x x y y y xy z z xy z 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 1 0 0 0 0 0 1 0 0 0 0 0 1 1 1 1 0 0 1 1 0 0 1 1 0 1 0 1 0 1 0 1 = 0 0 0 0 1 1 1 1 0 0 1 1 0 0 1 1 0 1 0 1 0 1 1 0 Quantum circuit G1 G2 Tensor Product Matrix multiplication G1 G2 G3 G4 G5 G6 G7 G8 G3 G4 G5 G6 G7 G8 (G 6 G 7 G8) (G3 G 4 G5) (G1 G 2)