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 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
• 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
Quantum circuit
G1
G3
G6
G4
G7
G2
G5
G8