or in the optical regime can be used to control Rydberg atom qubits [82,83]. Additionally, control schemes based on microwaves [84], and a combination of microwaves and gradient fields [85,86] have been proposed.
Unfortunately, compared with superconducting qubits and trapped ions, quantum gates with Rydberg atoms lack accuracy: Recent implementations of single-qubit gates feature errors slighlty below 1% [87, 88]. However, gate errors for multi-qubit gates are significantly worse. In advanced experimental setups, two-qubit gate errors after accounting for certain known error sources barely undercut values of 20% [89–91]. A detailed review of progress and challenges of quantum computation with neutral atom qubits is provided by Saffman [92]. Besides intrinsic limits on the fidelities of multi-qubit gates there are multiple experimental issues. Examples for most relevant issues in experiments with neutral atoms are Doppler dephasing, laser noise and atom loss during the operations.
There exist different approaches to generate entanglement between trapped Rydberg atoms [81,92]. Examples thereof are (i) Rydberg dressing [90, 93], where entanglement is generated via local spin exchange, (ii) dissipative entan- glement [94,95], where a combination of coherent and dissipative dynamics (see chapter3for details on dynamics of quantum systems) is used to prepare entangled states, and (iii) the Rydberg blockade [96–98], which is an effect based on the strong long-range interaction between Rydberg atoms. Entan- glement generation via the Rydberg blockade mechanism is an essential part of the thesis and will be discussed in detail in chapter7.
2.4
Other Candidates
Besides superconducting qubits, trapped ions and neutral atoms there are other hardware platforms that are potentially suitable for building future quantum computers. This section gives a superficial overview of these candi- dates and points to relevant literature for more information. In general, the remaining candidates can be categorized into semiconductor-based, photonic and molecular platforms.
1. Semiconductors:
Since semiconductors are of enormous industrial importance, fabrication and miniaturization has been perfected over the years and hence sub- stantiates the great potential of these platforms for large-scale quantum computing. The semiconductor platform can be divided into three main types: (i) Quantum dots [19,99], where trapped single electrons act as artificial atoms and their spin is used as a qubit. Operation of quantum dots shares some similarities with superconducting qubit circuits. (ii)
CHAPTER 2. PHYSICAL IMPLEMENTATIONS OF QUBITS 22
Color centers, referring to isolated defects in artificial diamond crystals [100, 101]. The diamond crystal serves as a trap for defects with nu- clear and electronic degrees of freedom that can be operated similar to trapped ions. (iii) Donor qubits in Silicon [102,103], where strongly bound electrons around single isolated dopants or defects in the material can be used to encode qubits.
2. Photonic platforms:
Aside from being used for control and communication channels in quan- tum systems, photons can also be used to host quantum information [104]. A fundamental challenge, however, is that there is no interac- tion between photons and hence multi-qubit gates cannot be imple- mented directly. Strategies to circumvent this limitation by simulating the required interactions involve special media, measurements or post- processing [105, 106]
3. Molecules:
Nuclear magnetic resonance (NMR), where the qubit is encoded in a nuclear spin 1/2, was one of the first candidates for quantum computing [107, 108]. Yet, scalability is severely limited and it is hence no longer pursued as a serious candidate for quantum information [109].
Chapter 3
Dynamics of Quantum Systems
In order to model and understand the dynamics of quantum systems it is necessary to formulate equations of motions that accurately incorporate all relevant effects. A fundamental criterion that differentiates between two ap- proaches is the existence of couplings between the quantum system and its environment. If such couplings do not exist – or at least are negligible for the time window of interest – the quantum system may be considered a closed system, which we will briefly review in the subsequent section3.1. If couplings to an environment are relevant and can no longer be neglected, the system is referred to as an open quantum system. Section 3.2 gives an overview of two methodologies that provide equations of motions for such open systems.
3.1
Closed Quantum Systems
The dynamics of a closed quantum system, often referred to as coherent or unitary, are solely governed by its Hamiltonian ˆH. The equation of motion for a specific quantum state |ψi is given by the Schr¨odinger equation i∂t|ψi =
ˆ
H|ψi (the convention ~ ≡ 1 is used in this thesis) with some initial condition |ψ(t0)i = |ψ0i. Equivalently, but more common for applications that we will study in part IIof the thesis, the Schr¨odinger equation is rephrased in terms of the unitary time evolution operator (the propagator ) ˆU (t),
∂tU (t) =ˆ −i ˆH(t) ˆU (t), U (0) = ˆˆ 1. (3.1) An equivalent way of representing the equation of motion is the von Neumann equation: It describes the evolution in terms of the system’s density operator that we introduced in equation (1.3). The von Neumann equation reads
˙ˆρ(t) = −i[ ˆH(t), ˆρ(t)], ρ(tˆ 0) = ˆρ0, (3.2) 23
CHAPTER 3. DYNAMICS OF QUANTUM SYSTEMS 24
where [ ˆA, ˆB] = ˆA ˆB− ˆB ˆA is the commutator of ˆA and ˆB, and ˆρ0 the initial state of the system at time t0. Often, the dynamics dictated by equation (3.1) can be greatly simplified by moving to a different basis, for instance to remove fast intrinsic oscillations from the Hamiltonian. We denote such a unitary, generally time-dependent, transformation by ˆV(t). It transforms Hamiltonian
ˆ
H according to
ˆ
Hv = ˆV†H ˆˆV + iV˙ˆ†V .ˆ (3.3) We will further refer to the original frame as the lab frame and label operators in the new frame with a subscript v. The form of the Schr¨odinger equation is in- variant under transformation (3.3), and the propagator ˆU (t) = ˆV (t) ˆUv(t) ˆV†(0) in the lab frame can directly be recovered from that in the new basis.