Download Problem 1: Gate Teleportation The set of Clifford gates consisting of

Problem Set 5
Problem 1: Gate Teleportation
The set of Clifford gates consisting of CNOT gates, single qubit Pauli X,Y,Z and single
qubit Hadamard H gates


1 1
1

H= √ 
2 1 −1
and the phase S gate

S=

1 0
.
0 i
is not universal for quantum computation. However supplemented by the T or π/8 gate


1 0

T=
0 eiπ/4
the set is universal. In this exercise we will show how to implement the T gate using special
T ancilla’s by a construction which is called gate teleportation.
a. Verify the action of the following two circuits which are called 1-bit Z-teleportation
and 1-bit X-teleportation in Figure below.
FIG. 1: So-called one-bit teleportation circuits. The measurement denoted by the meter is a
measurement in the Z-basis and determines whether to do a Pauli on the output qubit.
b. Imagine applying the desired T gate on the outgoing qubit of the X-teleportation
circuit. Show how to commute the T gate backwards through this circuit so that we obtain
a circuit which uses a single T ancilla, namely the state T H|0⟩ =
√1 (|0⟩ + eiπ/4 |1⟩),
2
a CNOT
gate and a classically controlled SX gate where S is the phase gate. The circuit we obtain
should take a qubit |ψ⟩ as input and outputs T |ψ⟩ hence it implements the T gate.
1
c. Can you produce a similar circuit for the S gate, i.e. a circuit which uses some
single-qubit S ancilla (what is it?) and Clifford gates such as H , CNOT and Pauli’s, but
not the S gate itself?
Comment: The motivation for these constructions comes from physical implementations which sometimes allow one only to implement (a subset of ) the Clifford gates (e.g.
topological quantum computation using Ising anyons, computation using the surface code).
These constructions show how to do universal computation with these gates and a supply
of special ancillas such as the T-ancilla. In addition, these constructions are important
when one constructs noise- or fault-tolerant quantum circuits.
Problem 2: Q and P phase space representation
a. Calculate the P-representation of a pure number state ρ = |n⟩⟨n|.
You will see that a number state does not have a well-defined P-representation! What is
the physical interpretation of this irregularity?
b. Evaluate the Q-representation of a squeezed coherent state |β, ζ⟩(pure state). Hint: A
squeezed coherent state |β, ζ⟩ can be obtained by first acting with the displacement operator
D(β) on the vacuum followed by the squeeze operator S(ζ), i.e.,
|β, ζ⟩ = S(ζ)D(β)|0⟩.
(1)
c. So far, we have discussed pure states. Now try to obtain the Q-function of a thermal
field (mixed state)
ρ=
∞
1 ∑ ( n̄ )m
|m⟩⟨m|.
n̄ + 1 m=0 n̄ + 1
(2)
where n̄ = ⟨n|ρ|n⟩ is the average number of photons.
Problem 3: Wigner Function
A Schrödinger cat state is defined as:
)
N (
|ψ⟩ = √ |αeiϕ ⟩ + |αe−iϕ ⟩ ,
2
which shows the quantum-mechanical superposition of two coherent state |αe±iϕ ⟩.
a. Find the normalization constant N .
2
(3)
b. Show that the Wigner phase space representation of the cat state has the following
form and give an expression for each of the terms:
W|ψ⟩ =
)
N2 (
W|αeiϕ ⟩ + W|αe−iϕ ⟩ + Wint .
2
(4)
Describe the physics behind each term in this equation (you may even plot the Wigner
function).
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