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

2. Classical Logic Circuits
Circuit behavior is governed implicitly
by classical physics
Signal states are simple bit vectors,
e.g. X =
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 algebra over Hilbert Space and 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
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 y Two-Input Gate: Controlled NOT (Feynman gate) By matrix
CNOT
concatenate y
By matrix =

11. Quantum Gate 3-Input gate: Controlled CNOT (Toffoli gate) By matrix=

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