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MODULE CODE : PHAS0070
ASSESSMENT : PHAS0070A7PB
PATTERN
MODULE NAME : Quantum Computation and Communication
LEVEL: : Postgraduate
DATE : 01-June-2020
TIME : 10:00
TIME ALLOWED : 2 hrs 30 mins
Year
2019/20
Answer THREE questions. Note: only three answers will be
marked.
The numbers in square brackets show the provisional
allocation of maximum marks per question or part of
question.
The following may be assumed:
x = X =
0 1
1 0
!
, y = Y =
0 i
i 0
!
, z = Z =
1 0
0 1
!
1. (a) Write down the truth table for a three bit Modified-Toffoli gate [2]
acting on classical bits A,B and C such that the state of bit B
is flipped if and only if both the bits A and C have value 0.
The gate keeps the bits A and C always unchanged.
(b) Construct the 8⇥ 8 matrix operator T which corresponds to [4]
the action of the Modified-Toffoli gate (as described above in
part (a)) in the computational basis.
(c) Show that the matrix operator T obtained above is a unitary [4]
operator.
(d) Now suppose that A,B and C are three qubits, as opposed to
being three classical bits. Show that the action of T on the [5]
state 1p
(|0iA + |1iA) 1p2(|0iB + |1iB) 1p2(|0iC + |1iC) results in an
unentangled state. Show that the action of T can entangle
qubits B and C for some other initial state of A,B and C.
(e) Show , using a circuit diagram, how the two qubit unitary [4]
operation
U =
266664
1 0 0 0
0 1 0 0
0 0 1 0
0 0 0 ei
377775
can be implemented using local unitary operations and
Controlled-NOT operations.
(f) How many bits of information can be stored in a d-state [1]
system?
2. (a) Discuss the demonstration of the non-local character of [7]
quantum mechanics using the 3-qubit
Greenberger-Horne-Zeilinger (GHZ) state
|GHZ iABC = 1p
2
(|0iA|0iB|0iC + |1iA|1iB|1iC)
(b) Consider three Bell states as follows:
|+iAB = 1p
2
(|0iA |0iB + |1iA |1iB),
|+iCD = 1p
2
(|0iC |0iD + |1iC |1iD)
and
|+iEF = 1p
2
(|0iE |0iF + |1iE |1iF )
Let the joint state of 6 qubits A,B,C,D,E and F be
|+iAB ⌦ |+iCD ⌦ |+iEF . If a measurement is performed on
the qubits A,C and E , and the outcome
|W iACE =
1p
3
(|0iA |1iC |0iE+! |0iA |0iC |1iE+!2 |1iA |0iC |0iE ),
is obtained where ! = e i2⇡3 , write down the state to which the [2]
qubits B,D and F are projected.
(c) Consider the case when two distant parties Alice and Bob
share two copies (labelled ⇢12 and ⇢34) of the mixed
entangled state
⇢i j = P| +ih +|i j + (1 P)|✓ih✓|i j
in which | +ii j = 1p2(|0ii |1ij + |1ii |0ij) and |✓ii j = |0ii |0ij (P is
real and 0  P  1).
Alice holds qubits 1 and 3, while Bob holds qubits 2 and 4.
Each of Alice and Bob perform CNOT gates: Alice with qubit
1 as control and qubit 3 as target, and Bob with qubit 2 as
control and qubit 4 as target, and subsequently measure the
target qubits to obtain an outcome of |1i3|1i4. Show that they [7]
obtain the pure maximally entangled state | +i12 of qubits 1
and 2.
(d) State and prove the quantum no-cloning theorem. [4]
[Part marks]
3. (a) Alice and Bob share an entangled state of two three level
systems A and B with levels |0i , |1i and |2i. The state is
given by
|+i = 1p
3
(|0i |0i + |1i |1i + |2i |2i).
Two unitary operators written in the {|0i , |1i , |2i} basis are
R =
2664 1 0 00 ! 0
0 0 !2
3775 ,
where ! = ei2⇡/3 and
P =
2664 0 0 11 0 0
0 1 0
3775 .
Find R3 and P2. [2]
(b) Find the states generated from |+i through the action of [9]
RmPn ⌦ I, where m and n each can take values 0, 1 or 2.
(c) A controlled permutation or CPERM gate on a four qubit
system cyclically permutes the state of the second, third and
fourth qubits (i.e., 2! 3, 3! 4, 4! 2) if the state of the first
is |1i. Otherwise it leaves them unchanged. Consider a
CPERM gate acting on the initial state of four qubits to be
1p
2
(|0i + |1i)1|⇠i2|⇣i3|i4, where |⇠i, |⇣i and |i are arbitrary
states except for the fact that their overlaps are
h⇠|⇣i = ⌘, h⇣|i = ✓, h|⇠i = . Find the reduced density matrix [5]
of the first qubit after the CPERM gate (express the answer in
terms of the overlaps ⌘, ⇣ and ).
Find the entanglement of qubit 1 with the joint state of qubits [2]
2, 3 and 4 as quantified by the von Neumann entropy.
(d) State one advantage and one disadvantage of using a photon [2]
as a qubit.
[Part marks]
4. (a) Alice and Bob intend to share a secret random key following a
modified version of the BB84 protocol where 6 states instead
of 4 states are sent. The bit string that Alice intends to
encode and send is b = 11001111011010000 while the basis
that she randomly uses is a = xzyyxzzxyxzxyzzxy (here, x
stands for the x basis whose base kets are |+i = 1p
2
(|0i + |1i)
and |i = 1p
2
(|0i |1i), y stands for the y basis whose base
kets are |i = 1p
2
(|0i + i |1i) and |⌦i = 1p
2
(|0i i |1i), while z
stands for z basis whose base kets are |0i and |1i
respectively. Note that here, x , y and z are mere symbols for
the bases and not operators. Write down a sequence of [3]
quantum states that Alice should transmit to Bob. In this
protocol, if Eve measures all the states transmitted, what is
the fraction of the check bits in which Alice and Bob will see [1]
errors? Explain the argument leading to the fraction. [2]
(b) Consider the following operators acting on a collection of
seven qubits labelled A,B,C,D,E ,F and G:
M1 = IA ⌦ IB ⌦ IC ⌦ ZD ⌦ ZE ⌦ ZF ⌦ ZG,
M2 = ZA ⌦ ZB ⌦ IC ⌦ ZD ⌦ ZE ⌦ IF ⌦ IG,
M3 = ZA ⌦ IB ⌦ ZC ⌦ ID ⌦ ZE ⌦ IF ⌦ ZG,
M4 = IA ⌦ IB ⌦ IC ⌦ XD ⌦ XE ⌦ XF ⌦ XG,
M5 = XA ⌦ XB ⌦ IC ⌦ XD ⌦ XE ⌦ IF ⌦ IG,
M6 = XA ⌦ IB ⌦ XC ⌦ ID ⌦ XE ⌦ IF ⌦ XG.
In the expressions M1 M6, the operators XA,ZA and IA are
the Pauli X ,Z and I operators acting on the qubit A, and the
other operators are similarly defined (e.g., ID means identity
acting on qubit D). Expressions for the Pauli matrices are
given in the rubric at the beginning of the paper.
Show that [M1,M2] = 0, [M4,M6] = 0. [2]
Show that [M1,M4] = 0, [M2,M5] = 0. [4]
Based on the mutual commutation of the operators Mj and
the fact that M2j = I, where I = IA ⌦ IB ⌦ IC ⌦ ID ⌦ IE ⌦ IF ⌦ IG,
justify why the state [4]
|0Li = (I+M6)(I+M5)(I+M4)(I+M3)(I+M2)(I+M1)|0000000iABCDEFG,
is a simultaneous eigenstate of all Mj with eigenvalue +1. In
the above,
|0000000iABCDEFG = |0iA⌦|0iB⌦|0iC⌦|0iD⌦|0iE⌦|0iF⌦|0iG.
The state |0Li is used as a logical state of a qubit in the
Steane code for quantum error correction, while the operators
Mj are used as syndrome measurement operators.
Which Mj operator will have an eigenvalue 1 for an error [2]
IA ⌦ IB ⌦ IC ⌦ ID ⌦ IE ⌦ ZF ⌦ IG|0Li (the result of a phase flip of
the qubit F )?
Justify why (you do not need to perform explicit calculations) [2]
X¯ = XA ⌦ XB ⌦ XC ⌦ XD ⌦ XE ⌦ XF ⌦ XG will commute with all
the operators Mj (for j = 1, .., 6).
[Part marks]
5. (a) Considering Grover’s algorithm for a 3 qubit register (i.e., 8
elements to search from) and making your own selection of
the marked element, find the amplitude of the marked state [5]
after one iteration of the algorithm.
(b) Using a 2 qubit register and the unknown bit string a = 11,
write down explicitly (i.e, by writing a superposition of the 4 [5]
computational basis states |00i, |01i, |10i and |11i, and
subjecting that superposition to a series of transformations)
an implementation of the Bernstein-Vazirani algorithm.
(c) Consider the state
| ()i = 1p
8
7X
j=0
eij |ji
of a 3-qubit register (i.e., each j is realized in terms of its
binary representation j1j2j3 where jm 2 {0, 1}, using three
qubits). Show that the above is a factorizable state of the 3 [4]
qubits.
(d) Two superconducting qubits interact through a mediating
microwave cavity via the effective Hamiltonian
H = X ⌦ X + Y ⌦ Y
Starting from a state |0i|1i of two qubits show that the above [6]
can be used to realise an entangling quantum gate.