Known as decoherence, the unavoidable interaction of a quantum system with its surrounding environment is usually considered to be detrimental for quantum information processing. In this thesis the coherent, open loop control of such open systems is studied. Concepts from quantum control theory and the theory of open quantum system are adopted in order to fight decoherence and implement quantum gates in a noiseless manner. In particular, Lie algebraic methods and numerical optimization tools are used to investigate the control properties of a single spin interacting with a spin environment. We show that, independent of the size of the environment, every unitary transformation can be implemented on the system spin through a single control field. We proceed by investigating dynamical decoupling, a method to suppress the interactions with the environment, for finite- and for in- finite dimensional systems. We prove that every finite dimensional system can be protected from decoherence, even if the environment is infinite dimensional, whereas for noise described by a Lindblad master equation dynamical decoupling will never succeed. This will lead to a new method to distinguish decoherence from intrinsic noise terms. We further prove that not every infinite dimensional system can be protected from decoherence through dynamical decoupling. Afterwards we investigate dynamical decoupling of systems that are described by quadratic Hamiltonians, showing that such interactions can always be suppressed with two simple operations. In the last part we investigate the coherent control of a Lindblad master equation. We show that a strong noise process exhibiting a decoherence free subspace can substantially increase the number of unitary operations that can be implemented, allowing us to fully control parts of the system. Afterwards we develop a scheme to make Hamiltonians and Lindbladians commutative by adding an auxiliary system. The old, possibly non-commutative dynamics, is recovered through a non-selective measurement.
Known as decoherence, the unavoidable interaction of a quantum system with its surrounding environment is usually considered to be detrimental for quantum information processing. In this thesis the coherent, open loop control of such open systems is studied. Concepts from quantum control theory and the theory of open quantum system are adopted in order to fight decoherence and implement quantum gates in a noiseless manner. In particular, Lie algebraic methods and numerical optimization tools are used to investigate the control properties of a single spin interacting with a spin environment. We show that, independent of the size of the environment, every unitary transformation can be implemented on the system spin through a single control field. We proceed by investigating dynamical decoupling, a method to suppress the interactions with the environment, for finite- and for in- finite dimensional systems. We prove that every finite dimensional system can be protected from decoherence, even if the environment is infinite dimensional, whereas for noise described by a Lindblad master equation dynamical decoupling will never succeed. This will lead to a new method to distinguish decoherence from intrinsic noise terms. We further prove that not every infinite dimensional system can be protected from decoherence through dynamical decoupling. Afterwards we investigate dynamical decoupling of systems that are described by quadratic Hamiltonians, showing that such interactions can always be suppressed with two simple operations. In the last part we investigate the coherent control of a Lindblad master equation. We show that a strong noise process exhibiting a decoherence free subspace can substantially increase the number of unitary operations that can be implemented, allowing us to fully control parts of the system. Afterwards we develop a scheme to make Hamiltonians and Lindbladians commutative by adding an auxiliary system. The old, possibly non-commutative dynamics, is recovered through a non-selective measurement.
All physical quantum systems are in contact with the external world which is inevitably a source of noise. It is therefore necessary to take into account this dissipation when designing controls that accomplish useful tasks in quantum information processing. This thesis is on that overlap of open quantum systems and control theory; it looks at what dynamics can happen with the use of external driving, and how this driving can be chosen to accomplish a desired goal. The first result looks at how a given quantum dissipator can be manipulated, using coherent controls, into replicating the action of a different type of noise. This results in no-go theorems for how noise can be transformed based on its isotropy, with applications in simulating open systems. Another way of doing simulations is to use only unitary dynamics over the system and a finite dimensional ancilla, and it is proved that there always exists a dilation Hamiltonian that replicates the noisy dynamics continuously in time. This also highlights the fact that adding controls on noise can result in a different evolution than if the controls were done on the under- lying system-environment level. A conjecture is introduced and studied which states that both approaches are equivalent in the special case of the controls commuting with the Lindbladian. A way around this difficulty is to use a quantum system to compute in situ which controls are best for achieving a desired task on itself. This problem is studied in the context of reaching entangling gates on a quantum simulator in order to upgrade it into a quantum computer. The experimental cost of doing so is found to be polynomial in the number of qubits in simulations. The same underlying principle is also used to find error correcting codes tailored to the dissipation in a system.
For finite-dimensional quantum systems, such as qubits, a well established strategy to protect such systems from decoherence is dynamical decoupling. However many promising quantum devices, such as oscillators, are infinite dimensional, for which the question if dynamical decoupling could be applied remained open. Here we first show that not every infinite-dimensional system can be protected from decoherence through dynamical decoupling. Then we develop dynamical decoupling for continuous variable systems which are described by quadratic Hamiltonians. We identify a condition and a set of operations that allow us to map a set of interacting harmonic oscillators onto a set of non-interacting oscillators rotating with an averaged frequency, a procedure we call homogenization. Furthermore we show that every quadratic system-environment interaction can be suppressed with two simple operations acting only on the system. Using a random dynamical decoupling or homogenization scheme, we develop bounds that characterize how fast we have to work in order to achieve the desired uncoupled dynamics. This allows us to identify how well homogenization can be achieved and decoherence can be suppressed in continuous variable systems.
The scaling of effort to achieve control objectives with system size is an important consideration in the development of many quantum devices, especially in those intended for quantum information processing. This thesis investigates the scaling with system size of effort required to control manybody quantum systems, and asks whether this is sufficiently favourable to enable supremacy in quantum computation. This is tested using quantum control theory and numerical control optimisation, which are well established tools for investigating the dynamics of driven quantum systems, in a variety of theoretical models related to state-of-the-art quantum devices. Gates acting on a chain of coupled oscillators are found to achievable in times that scale approximately linear with chain length, where the control is over only a single oscillator. The scalability of a scheme for implementing quantum gates in a many-body quantum simulator is shown to be favourable enough to allow for them to be configured to perform quantum computations. A state measurement protocol is proposed as part of the scheme. The scalability is validated through numerical simulation of simulator models composed of up to 9 qubits. The reachable set of operations for systems with quadratic Hamiltonians and infinite-dimensional Hilbert spaces is investigated. The passive operations in quantum optics are proven to be unreachable for single-mode systems with ‘unstable’ Hamiltonians. Further characterisation of the reachable set is made through numerical simulation of control optimisation. Some progress is made in extending the reachability result to n-modes. A non-Markovian noise model is used in simulating control optimisation of a dynamical decoupling scheme, which would be impossible to simulate with Markovian dynamics. The popular QuTiP Python library has been extended to allow simulation of control optimisation with a range of dynamical models. A description of these software modules and their method of use is given. Many of the software tools developed for the study are made available through open-source repositories. Some outlook is given for the use of in-situ open-loop control in optimising controls in quantum system experiments.
Abstract For finite-dimensional quantum systems, such as qubits, a well established strategy to protect such systems from decoherence is dynamical decoupling. However many promising quantum devices, such as oscillators, are infinite dimensional, for which the question if dynamical decoupling could be applied remained open. Here we first show that not every infinite-dimensional system can be protected from decoherence through dynamical decoupling. Then we develop dynamical decoupling for continuous variable systems which are described by quadratic Hamiltonians. We identify a condition and a set of operations that allow us to map a set of interacting harmonic oscillators onto a set of non-interacting oscillators rotating with an averaged frequency, a procedure we call homogenization. Furthermore we show that every quadratic system-environment interaction can be suppressed with two simple operations acting only on the system. Using a random dynamical decoupling or homogenization scheme, we develop bounds that characterize how fast we have to work in order to achieve the desired uncoupled dynamics. This allows us to identify how well homogenization can be achieved and decoherence can be suppressed in continuous variable systems.
Complex quantum systems, such as a quantum computer, will always be coupled in some way to the environment. This can cause what's called decoherence, a destructive process by which information is lost from the system into the environment. In this bachelor thesis paper, we discuss decoherence-free subspaces within networks of coupled quantum harmonic oscillators (or QHOs). We investigate where such noiseless subspaces (or NSs) occur most frequently in an ensemble of Erdos-Renyi networks, for which we do not yet consider the influence of the bath. We then proceed by adding the bath into the equation, using some of the theory of open quantum systems. Specifically, we derive the Lindblad master equation and show its form for the case of our networks. Consequently, we simulate the behavior of the moments of the position operators for the graphs with 3 nodes, both by means of the full Lindblad equation, and by first tracing out those moments to obtain their differential equations. We compare those two results to each other, and also look back to the situation before adding the bath to see if the NSs are still present. From the results of the simulations, we can conclude several things. Firstly, we see that for ensembles with probability of connection p very close to either 0 or 1, both to number of noiseless modes and the probability of finding at least one is largest. This is credited to their relatively high degrees of symmetry. Secondly, in the results of the density matrix and moment simulations, we see that, indeed, the noiseless modes are preserved when considering the influence of the bath. Furthermore, we can conclude that simulation of the density matrix for the case of coupled QHOs in a network is in many cases not stable; the cutoff at a finite level s needed to simulate an otherwise infinite- dimensional operator leads to non-positivity of the density matrix. Therefore, it is best to simulate the moments from their respective differential equations, as opposed to the full Lindblad master equation. Finally, the differential equations for the moments and their solutions show that there are indeed noiseless clusters for eigenmodes perpendicular to the center of mass, as predicted.
Quantum communication provides essentially secure transmission of information and quantum sensing provides measurement sensitivities better than classical. These powers come because of unique quantum processes, especially superposition and entanglement. Although there is a significant theoretical and experimental development, the phenomenon of decoherence is still one of the main limiting factors in performance and scalability of quantum devices. The importance of this study is that it considers an active approach to decoherence as one of the key impediments to scalable quantum technologies. Although the research on the noise sources and error mechanisms has been done extensively, the characterization is not enough to be applied to the real world. This study makes use of a theoretical and computation effort to explore scalable quantum technologies integrated decoherence control strategies. The research paper is based on the open quantum systems framework, in which quantum systems are represented as finite-dimensional systems that interact with the external environment. The findings of the study highlight that active decoherence control plays an important role in enhancing performance of open quantum systems. Without any control, the simulated quantum system is characterized by a fast decoherence time, coherence times are short, about 20 u ms, and the physical error rate is about 2.5%. The hybrid strategy is a feasible way to achieve fault-tolerant quantum operation and the transition point between theory-based control architectures and experimental hardware limitations, thus speeding up the development of reliable large-scale quantum technologies. The findings affirm that each of the isolated methods like dynamical decoupling, quantum error correction, or reservoir engineering can only be a partial solution when used alone. Adaptive and real-time control schemes responding dynamically to changing noise environments also require further research to develop them.