Download Lecture 8

Lecture 8
QUANTUM CIRCUITS
QUANTUM GATES
Classical single bit gate: NOT gate
This is the only non-trivial single bit classical gate.
Quantum single qubit gates
Quantum NOT gate
Question to the class: which 2x2 matrix represents this gate?
X
Questions for the class:
What other single qubit gates can you name?
How do they operate on the qubits?
What are their matrix representations?
Z gate: leaves |0> unchanged and flips the sign of |1>
Lecture 8 Page 1
L8.P2
Hadamard gate
More single qubit gates
Note: matrix U describing singe qubit gate must be unitary.
Phase gate:
Question to the class: what operation does this gate perform?
Lecture 8 Page 2
L8.P3
Question to the class:
Does this gate preserve normalization of a qubit?
Note: all gates do since
Why the T gate is called
while
appears in the definition?
The reason is historical. This gate is equivalent (up to unimportant global factor) to the
following gate:
Geometrical representation of a gubit
We can re-write
where
as
are real numbers.
Lecture 8 Page 3
L8.P4
global factor can be omitted since it has no observable effects
The numbers
and
define a point on the unit three-dimensional sphere:
This sphere is called the Bloch sphere; it provides means of visualizing the state of a
single qubit. The single qubit gates may be represented as rotations of a qubit on
the Bloch sphere. Unfortunately, there is no simple generalization of the Bloch
sphere for multiple qubits.
Class exercise: show
and
Lecture 8 Page 4
on the Bloch sphere.
L8.P5
Rotation operators
The Pauli matrices X, Y, and Z give rise to the rotation operators about the x, y,
and z axis of the Bloch sphere:
TWOTWO-QUBIT GATES
Controlled operations: "If A is true, then do B"
Controlled-NOT (CNOT) gate
Gate operations: if control qubit is
, then flip the target qubit.
Question for the class: what is the matrix representation for this gate?
Lecture 8 Page 5
L8.P6
In the same way
More on controlled operations
Suppose U is an arbitrary single qubit unitary operation. A controlled-U operation is a
two-qubit operation with a control qubit and a target qubit. If control qubit is set,
then U is applied to the target qubit.
Example: controlled-NOT gate is controlled-X gate.
Lecture 8 Page 6
L8.P7
Note that now we can write its matrix right away:
Question for the class: what does this circuit do?
Question for the class: what does this circuit do?
Lecture 8 Page 7
L8.P8
Classical computation on a quantum computer
Toffoli gate
control
target
Questions for the class:
1) How would you use Toffoli gate to implement NAND gate?
2) How would you use Toffoli gate to make a "copy"?
Lecture 8 Page 8
L8.P9
Universal quantum gates
A set of gates is said to be universal for quantum computation if any unitary
operation may be approximated to arbitrary accuracy by a quantum circuit involving
only those gates.
A unitary matrix U which acts on d-dimensional Hilbert space may be decomposed
into a product of two-level matrices; i.e. unitary matrices which act non-trivially only
on two-or-fewer vector components.
Single qubit + CNOT gates
Single qubit and CNOT gates together can be used to implement an arbitrary twolevel unitary operation on the state space of n qubits.
Suppose U is a two-level unitary matrix which acts non-trivially on the space
spanned by the computational basis states |s> and |t>, where s = s1 … sn and
t = t1 … tn. Let U be non-trivial 2 x 2 unitary submatrix of U.
Goal: to construct a circuit implementing U from single qubit and CNOT gates.
Use Gray codes: A Gray code connecting binary numbers s and t is a sequence of binary
numbers, starting with s and concluding with t, such that adjacent members of the list
differ in one bit.
Lecture 8 Page 9
L8.P10
Example: s=101001, t=110011.
Lecture 8 Page 10