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Information on mathematical topics related to quantum computation, specifically the probability of finding a qubit in a certain state after interference and the use of the hadamard transform. It includes questions on single qubit interference probabilities, the effect of decoherence on these probabilities, and the implementation and usefulness of the hadamard transform in quantum algorithms.
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Tuesday 3 June 2003 9 to 11
Attempt THREE questions.
There are four questions in total. The questions are of equal weight.
1 A quantum network which describes single qubit interference can be represented as follows:
H H
ฯ
(1) What is the probability P 0 (ฯ) that a qubit initially in state | 0 ใ will be found in state | 0 ใ at the output if it is measured in the {| 0 ใ, | 1 ใ} basis?
(2) Now suppose, that after the phase gate and before the second Hadamard gate, the qubit undergoes decoherence by interacting with an environment in state | e ใ so that: | 0 ใ| e ใ 7 โ | 0 ใ| e 0 ใ, (1)
| 1 ใ| e ใ 7 โ | 1 ใ| e 1 ใ, (2)
where | e 0 ใ and | e 1 ใ are the new states of the environment which are normalized but not necessarily orthogonal. The decoherence modifies P 0 (ฯ) which becomes a function of ฯ and of the scalar product ใ e 0 | e 1 ใ. Writing ใ e 0 | e 1 ใ = veiฮฑ^ express P 0 as a function of ฯ, v, and ฮฑ.
(3) Suppose the decoherence takes place between the first Hadamard gate and the phase gate, how different is the expression for P 0 (ฯ, v, ฮฑ)? (4) Deutschโs algorithm with an oracle f : { 0 , 1 } 7 โ { 0 , 1 }, is implemented by the following network, where the central two-qubit gate is the oracle implementing the operation, | x ใ| y ใ 7 โ | x ใ| x โ f (y) ใ and | ยฏ 0 ใ = โ^12 (| 0 ใ โ | 1 ใ) :
f
0
0 0
Measurement
Assume that only the first (top) qubit is affected by decoherence as described by Eqs.(1) and (2). How reliably can you tell whether f is constant or balanced?
Paper 47
3 Let A and B be two 2 ร 2 matrices. The inner product of A and B is defined
as 12 Tr
. Show that the identity ฯ 0 =
, the bit flip ฯ 1 =
, the
phase flip ฯ 3 =
, and the bit and phase flip ฯ 2 = iฯ 1 ฯ 3 =
0 โi i 0
form an
orthonormal basis in a space of 2 ร 2 matrices, i.e. any 2 ร 2 matrix E can be written as
k=
Tr (ฯkE) ฯk. (3)
Suppose error E entangles a qubit with its environment according to the rules
| 0 ใ| e ใ 7 โ | 0 ใ| e 00 ใ + | 1 ใ| e 01 ใ | 1 ใ| e ใ 7 โ | 0 ใ| e 10 ใ + | 1 ใ| e 11 ใ,
where | e ใ, | enm ใ, n, m = 0, 1 are the states of the environment which are not necessarily orthogonal or normalized. The r.h.s. of the two equations above can be conveniently written in the matrix form as ( | e 00 ใ | e 01 ใ | e 10 ใ | e 11 ใ
Using Eq.(4) together with Eq.(3), or otherwise, show that for any pure state of the qubit | ฮจ ใ, the action of the error E can be represented as
| ฮจ ใ| e ใ 7 โ
k=
(ฯk| ฮจ ใ) | ek ใ,
for some states of the environment | ek ใ which are not necessarily orthonormal. Express | ek ใ in terms of | emn ใ.
Suppose we are given a noisy single qubit channel in which the probability of a phase flip error is q and no other errors occur. Consider an arrangement, shown in the diagram below, in which the qubit in some unknown state of the form | ฯ ใ = ฮฑ| 0 ใ + ฮฒ| 1 ใ is encoded into a three qubit state
ฮฑ| ยฏ 0 ใ| ยฏ 0 ใ| ยฏ 0 ใ + ฮฒ| ยฏ 1 ใ| ยฏ 1 ใ| ยฏ 1 ใ,
where | ยฏ 0 ใ = โ^12 (| 0 ใ + | 1 ใ) and | ยฏ 1 ใ = โ^12 (| 0 ใ โ | 1 ใ). The three qubits are transmitted
through the channel and subsequently decoded.
Encoding Noisy Channel Decoding
H
H
H
H
H
H
0
0
ฯ ฯ
What is the probability of successful recovery of the input state ฮฑ| 0 ใ + ฮฒ| 1 ใ?
Paper 47
4 The quantum Fourier transform on the group ZN acts on a Hilbert space of dimension N = 2n. It is defined by linearity and its action on an orthonormal basis, {| 0 ใ, | 1 ใ, | 2 ใ,... , | N โ 1 ใ}:
QFn : | x ใ 7 โ โ^1 N
y=
e
2 ฯixy N (^) | y ใ.
The single qubit unitary transformation Rk is defined as
Rk =
0 e
2 ฯi 2 k
(1) Show that QFn can be implemented by a quantum network of size O(n^2 ) built from Hadamard gates and controlled Rk gates for k = 1, 2 ,... , n.
(2) If U 1 , U 2 , ...Um and V 1 , V 2 , ...Vm are unitary operators with ||Uk โ Vk|| < for k = 1, 2 ,... , m, show that
||U 1 U 2 ...Um โ V 1 V 2 ...Vm|| < m,
where the operator norm is defined as, ||A||^2 = (^) ||ฯsup||=1 ใ ฯ |A+A| ฯ ใ.
Suppose that an approximate QFn network is built with approximate Hadamard and approximate controlled Rk gates which implement unitary operations Gโฒ^ that approximate the specified gate operators G in the sense that
||Gโฒ^ โ G|| โค (^) n^14.
Show that the resulting network operation Un satisfies
||Un โ QFn|| = 0
n^2
Comment briefly on the practical implications.
Paper 47
