4.4 Gates
4.4.1 Multiple qubit gates
We can generalise these gates to multiple qubits. The classical two bit gates that we are used to such as AND, OR and their NOT counterparts, NAND and NOR,
take two inputs and give a single output. Two qubit gates have two inputs and two outputs. The type of gates that are relevant to us are called controlled gate and to explain them, let us define our terms. There are two types of input qubit, a control qubit and a target qubit. These correspond to the qubit that makes a decision and the one that is affected. In this way, if we begin with two inputs, then we are left with two outputs, but the target qubit has undergone some operation. As an example, let us consider the CNOT (controlled NOT) gate, which as we shall see is equivalent to a kind of classical XOR (exclusive OR) gate. The truth table for the CNOT gate with two qubits labelled C (control) and T (target) and the subscripts b (before) and a (after) is as follows
Cb Tb Ca Ta 0 0 0 0 0 1 0 1 1 0 1 1 1 1 1 0 (4.29)
We see that the target qubit is flipped i.e. a NOT operation is performed on it, if and only if the control qubit is 1. Due to this fact, the only way to have an output of 1 in the target qubit is if the inputs are opposites. Thus the CNOT gate is the same as the classical XOR gate if the output is stored in the target bit. As mentioned earlier, we can represent this operation as a unitary matrix
UCN OT ≡ 0 1 0 0 1 0 0 0 0 0 1 0 0 0 0 1 , (4.30)
or in the much more compact notation |A, Bi → |A, B⊕Ai, where ⊕ is a modulo two addition. We have mentioned that quantum gates are unitary. This means that they are reversible and since the gates shown so far have all been comprised of real elements, they are reversible by repetition of the gate. Classical gates like the normal XOR gate, where the result is not stored in the second bit, is not reversible since it is not possible to determine the states of the inputs from the output. Another important gate, which we will use later is the CPHASE (controlled phase) gate. This gate, as the name suggests, affects the phase of the target qubit. As a matrix,
in Chapter 7.
This is the end of the background chapters. We have discuss open and closed quantum systems, from which we learnt much about the master equation formalism that can be used is the unconditional way to obtain averages and in the quantum jump formalism that allows us to model single experimental runs. This stands us in good stead since we shall use them throughout the research chapters. In addition we explored physical qubits with a focus of quantum dots. We learnt about their electronic structure, but also that while they may be complicated to model exactly, they can in fact be modelled as atoms and when considering low lying energy states, the complexity reduces to that of a two level system. Let us begin the research chapters.
T
fluctuating charges and correlations in photon statistics resulting from their presence. This is done with the aid of two quantum dots that are grown in such an environment. We shall assume that the dots are identical. The aim of this chapter is to show that the common, correlated, noise that is generated by charges fluctuating in the vicinity of two QDs can be detected by optically driving the QDs and then analysing the emitted photons. To this end, we shall determine the cross correlation function g(2)(τ) of the emitted light and show that knowledge about the nature of the charge environment can be revealed; including how common it is to both qubits. It will be demonstrated that in some regimes it is possible to determine the number of charges and the rate of their fluctuation. For ease of reading and in order to pro- vide context, let us recap some of the main points that we saw in Chapter 4, which will be of most use to us now. Quantum dots are semiconductors heterostructures, which by their nature, have a Fermi energy inside the band gap. When the tem- perature is relatively low, the gap can be small enough in doped samples that the conduction band may be thermally occupied. The crystal structure of any material is rarely perfect and vacancies or impurities in the structure of a semiconductor lead to local alterations to the band structure. As a result charges can become trapped in lower lying states [72]. This shall form the basis of our common environmental disturbance. Depending on the temperature, the charges in these traps will ran- domly hop into and out of them [73]. This leaves us with a system of traps that are either filled or not. As such they may then be modelled as two state fluctuators. In the vicinity of QD excitons, such fluctuators lead to random telegraph noise in theexciton energy and thus, via the DC Stark shift, so too in the frequency of emitted photons; [5] see Figs. 5.1 and 5.2.
Figure 5.1: The qubit energy shift for an unoccupied (left) and charged (right) trap. The exciton creation energy is denoted byω and the charge-qubit interaction strength is δ.
Figure 5.2: Demonstrative graph of telegraph noise. The fluctuating charge switches stochastically between states 1 (charged) and 0 (uncharged).