ASSIGNMENT 1 CO 481/CS 467/PHYS 467 (Winter 2012) - UMD
ASSIGNMENT 1
Due in class on Thursday, January 12.
CO 481/CS 467/PHYS 467 (Winter 2012)
1. Universality of reversible logic gates.
(a) [3 points] The cccnot (triple-controlled not) gate is a four-bit reversible gate that flips its fourth bit if and only if the first three bits are all in the state 1. Show how to implement a cccnot gate using Toffoli gates. You may use additional workspace as needed. You may assume that bits in the workspace start with a particular value, either 0 or 1, provided you return them to that value. For a bonus point, give a circuit that works regardless of the values of any bits of workspace.
(b) [4 points] Show that a Toffoli gate cannot be implemented using any number of cnot gates, with any amount of workspace. Hence the cnot gate alone is not universal. (Hint: It may be helpful to think of the gates as performing arithmetic operations on integers mod 2.)
2. Computing reversibly. The function eq : {0, 1}3 {0, 1} determines whether its three input bits are equal, namely
1 if x = y = z eq(x, y, z) =
0 otherwise.
(a) [2 points] Show how to compute the function eq using and, or, not, and fanout gates. (b) [4 points] Show how to compute the function eq reversibly using Toffoli gates. You may
use ancilla bits initialized to either 0 or 1 provided you return them to that value. You may use gates other than Toffoli gates provided you explain how to implement any such gates using Toffoli gates.
3. Mach-Zehnder interferometer with a phase shift.
0
ei
1
Analyze the experiment depicted above using the mathematical model described in class. (Note
that the model from class differs slightly from the model described in the textbook; in particular,
you should use the matrix 1
2
11 1 -1
to model the beamsplitters.)
(a) [4 points] Compute the quantum state of the system just before reaching the detectors. Express your answer using Dirac notation.
(b) [2 points] Compute the probability that the "0" detector clicks as a function of , and plot your result for [0, 2].
1
4. Pauli operators.
(a) [2 points] Express each of the three Pauli operators,
X=
0 1
1 0
,
Y=
0 i
-i 0
,
Z=
1 0
0 -1
,
using Dirac notation in the computational basis.
(b) [3 points] Find the eigenvalues and the corresponding eigenvectors of each Pauli operator. Express the eigenvectors using Dirac notation.
(c) [2 points] Write the operator X Z as a matrix and using Dirac notation (in both cases using the computational basis).
(d) [2 points] What are the eigenspaces of the operator X Z? Express them using Dirac notation.
5. Product and entangled states. Determine which of the following states are entangled. If the state is not entangled, show how to write it as a tensor product; if it is entangled, prove this.
(a)
[2 points]
2 3
|00
+
1 3
|01
-
2 3
|11
(b)
[2 points]
1 2
(|00
- i|01
+ i|10
+ |11 )
(c)
[2 points]
1 2
(|00
- |01
+ |10
+ |11 )
6. Unitary operations and measurements. Consider the state
2
1
2
| = |00 + |01 - |11 .
3
3
3
(a) [2 points] Let | = (I H)| , where H denotes the Hadamard gate. Write | in the computational basis.
(b) [3 points] Suppose the first qubit of | is measured in the computational basis. What is the probability of obtaining 0, and in the event that this outcome occurs, what is the resulting state of the second qubit?
(c) [3 points] Suppose the second qubit of | is measured in the computational basis. What is the probability of obtaining 0, and in the event that this outcome occurs, what is the resulting state of the first qubit?
(d) [2 points] Suppose | is measured in the computational basis. What are the probabilities of the four possible outcomes? Show that they are consistent with the marginal probabilities you computed in the previous two parts.
2
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