Departement of Physics, University of Basel
Dr. Rakesh P. Tiwari ([email protected])
Nanophysics — Fall 2014
Exercise 1
(1) Entangled Bell states
In the lecture we defined the Bell states as
∣β00 ⟩ = √12 (∣00⟩ + ∣11⟩),
∣β01 ⟩ = √12 (∣01⟩ + ∣10⟩),
∣β10 ⟩ = √12 (∣00⟩ − ∣11⟩),
∣β11 ⟩ = √12 (∣01⟩ − ∣10⟩).
Imagine that one of the qubits of a Bell state is sent to Alice and the other one from the same
Bell state is sent to Bob. If both Alice and Bob apply a Hadamard gate H to their qubit, show
that two of the Bell states are interchanged, and two of the states are left unchanged by this
1 1
]
transformation. Remember: H = √12 [
1 −1
(2) Deutsch’s Algorithm
Consider a binary classical function f (x) ∶ {0, 1} → {0, 1}. A quantum circuit that implements Deutsch’s Algorithm is shown below
|0>
x
H
x
H
Uf
|1>
y y+f(x)
H
ψ0
ψ1
ψ2
ψ3
where Uf ∶ ∣x, y⟩ → ∣x, y + f (x)⟩. The input state ∣ψ0 ⟩ = ∣01⟩.
(a) Write down the state ∣ψ1 ⟩ obtained when the input state is sent through two Hadamard
gates (H).
(b) Show that
∣0⟩−∣1⟩
∣0⟩+∣1⟩
⎧
⎪
⎪± ( √2 ) ( √2 ) if f (0) = f (1)
∣ψ2 ⟩ = ⎨ ∣0⟩−∣1⟩ ∣0⟩−∣1⟩
⎪
√
√
⎪
⎩± ( 2 ) ( 2 ) if f (0) ≠ f (1)
,
(1)
√ ), as mentioned in the lecture.
(c) Show that ∣ψ3 ⟩ = ±∣f (0) ⊕ f (1)⟩ ( ∣0⟩−∣1⟩
2
(3) Two qubit gate
The SWAP-gate is defined through the relation
USWAP ∣ψ1 ψ2 ⟩ = ∣ψ2 ψ1 ⟩,
where two qubits swap their states. This two-qubit gate is not by itself sufficient for universal
quantum computation,
but if the gate is pulsed for half-a-period, the resulting ‘square-root√
1/2
of-SWAP’ (or SWAP ≡ USWAP ) becomes useful since one can obtain an XOR-gate (up to
some single-qubit rotations).
1/2
(a) Show that the matrix form of the USWAP takes the following form in the computational
basis (∣00⟩, ∣01⟩, ∣10⟩, ∣11⟩):
⎛1
⎜0
1/2
USWAP = ⎜
⎜0
⎝0
0
0
1
2 (1 + i)
1
2 (1 − i)
1
2 (1 − i)
1
2 (1 + i)
0⎞
0⎟
⎟
0⎟
1⎠
0
0
√
(b) Using the state ∣ψ⟩ = ∣10⟩ as an input of SWAP gate, what is the output state? Are the
input and/or output states entangled?
(c) Repeat (b) using states ∣ψ⟩ = ∣00⟩, ∣ψ⟩ = ∣01⟩ and ∣ψ⟩ = ∣11⟩ as input states.