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

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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

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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

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Signal state (one qubit) Corresponsd to

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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

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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

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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

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

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Quantum Gate One-Input gate: NOT NOT By matrix NOT Gate Input Output

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

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

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

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