The recent progress in quantum technologies shines a bright light on the future of quantum computation. However, resource estimation for quantum computations remains a key challenge. The resource I study in this thesis is known as Magic or Non-stabilizerness and it represents the key requirement for quantum computational advantage in computation. Recent studies in quantum information suggested wide classes of quantifiers for non-stabilizerness.

In this thesis, I develop novel techniques for quantifying non-stabilizerness with the tools fromquantuminformation theory. I am specifically interested inmaking quantum resource estimation computationally as efficient as possible so that quantum resource estimation can become a routine step in both numerical and experimental exploration of quantumcomputing.

I show that by measuring the spreading of the local information in the quantumsystem, we can quantify non-stabilizerness. The measures of such information-spreading can be classified into two categories, entropic-based measures, such as mutual information, and correlator-based measures such as Out-of-Time Ordered Correlators. I investigated both classes of measures and I related them both numerically and analytically to the estimation of non-stabilizerness.

Finally, I relate non-stabilizerness quantification to classical variational methods. Classical methods designed to approximate quantum states are by construction not restricted by non-stabilizerness. The question therefore remained how well these techniques can capture quantumresources. I provide a systematic benchmark for both classical and quantum approximate methods of expressing quantum states and their tradeoff between non-stabilizerness expressivity and ground-state energy accuracy. We observed that having better energy accuracy is necessary but insufficient to have better accuracy in non-stabilizerness.

This Thesis forms a bridge between quantum information resource theory and condensed matter physics and offers a stepping stone towards further exchange between these two fields.
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The spectrometers are widely used in various fields ranging from astronomy to biology. Recently, lots of efforts have been made to miniaturize the size of the spectrometer to meet the needs of fields such as real-time health monitoring. Therefore, achieving a high-resolution, broadband on-chip spectrometer is an intriguing but challenging topic. Photonic crystals are constructed by material with periodic refractive indices. They allow us to modify the interaction between electromagnetic fields and dielectric media. In this work, using a special kind of photonic crystal called “zipper cavity”, we design a wavelength tunable filter, which can be used as the key component for a filter-based spectrometer. By FEM simulation, we design a zipper cavity with a resolution of sub-nanometer and a wavelength tunable range of 21 nm. The prototypes of the devices are fabricated and measured. The loss rate is evaluated and the tunability of the wavelength is examined.

Recently, quantum networks have emerged as a focal point of research and discussion due to their promise in overcoming the limitations of classical networks, offering unparalleled capabilities in secure communication, quantum computation, and distributed quantum information processing. Simply speaking, a quantum network is a network in which the nodes are capable of storing and processing quantum information and communicate via quantum channels. Unlike classical computers and networks, in which components at nodes and the interconnections between them are mostly made of electronics circuits, there is not a homogeneous system for all the applications in a quantum network. This in particular, increases the need for implementation of heterogeneous quantum systems. An important milestone in the development of hybrid quantum networks are the interconnections between different components at the nodes, which serve as quantum channels. A quantumchannel is an interconnection between two quantum systems that can route carriers of quantum information, while preserving their coherence over the routing process. Finding such a channel, with carriers having the ability to couple to different quantum systems is not trivial. Although quanta of mechanical vibrations - known as "phonons" - have shown great potential for this task, as they can couple to many different types of quantumsystems.
The aim of this thesis is to design an integrated platform using highly confined GHz phonons, with scalability as a primary consideration. This platform serves as an on-chip phononic quantum channel, enabling the ability to perform on-chip operations directly on single phonons on a chip. Moreover, the designed structures in this thesis are advantageous for advanced quantum acoustics experiments and pave the way towards having full coherent control on phonons on a chip.@en

The first part of this thesis provides a mathematical description for bipartite quantum correlations, aiming to analyze the geometry of several sets of correlations. We explain why quantum entanglement can be used to simulate shared randomness: C_loc(Γ) ⊆ C_q^d(Γ) for a sufficiently large d. The known bound for this dimension d in the literature is d ≥ dim(C_loc(Γ))+1, but we improve this by showing that the inclusion is always true for d ≥ dim(C_loc(Γ)). For the proof of this bound, we show that the set C_private(Γ) of correlations using private randomness is connected, which allows the use of an improved version of Carathéodory’s Theorem. In the second part of this thesis, we define and analyze a see-saw method to determine the state and measurement operators that reconstruct both the correlation itself as its entanglement dimension, by solving consecutive semidefinite programs. One of the strengths of the algorithm is its generality: it applies to different dimensions, question sets, and answer sets. Some numerical experiments demonstrated that the method can indeed reconstruct quantum correlations, although some highly entangled correlations failed to be reconstructed due to the computationallimitations. The numerical experiments motivated several new theorems, for example the fact that every correlation with |A|= 1 or |B|= 1 has entanglement dimension 1, which means that it can be written as a private randomness correlation. The proof of this result is based on the earlier described improvement for the dimension d.

This thesis investigates two types of classical capacities of both classical and quantum channels, giving rise to four different settings. The first type of classical capacity investigated is the ordinary capacity of a channel to transmit classical information with a probability of error which becomes arbitrarily small as the channel is used arbitrarily many times. The second type of classical capacity investigated is the capacity of a channel to transmit information with zero probability of error, called the zero-error classical capacity. The first setting which is studied is the ordinary capacity of a classical channel. The noisy channel coding theorem is proven in two different ways: one using the Markov inequality and the Law of Large Numbers and one using typical sets. The additivity of this capacity is also discussed. The second setting is the zero-error capacity of classical channel. Lower and upper bounds on this capacity are proven, and its superadditivity is discussed. The third setting is the ordinary classical capacity of a quantum channel. The Holevo-Schumacher-Westmoreland theorem is proven using typical subspaces and the packing lemma, and the superadditivity of the Holevo information is discussed in terms of entanglement at the encoder. The fourth and last setting investigated is that of the zero-error classical capacity of a quantum channel. It is shown that this capacity can be achieved using only pure input states and that this capacity never exceeds the ordinary classical capacity. Moreover, a detailed investigation of superactivation of the zero-error classical capacity is presented. A topic for further research would be an exposition of the analogous concepts in the case of the quantum capacity of quantum channels. Another topic for further research would be an explicit construction of two quantum channels whose zero-error classical capacity is superactivated.

The defining challenge of connecting NISQ quantum computers over large distances is efficient microwave to optical transduction. In this work, we demonstrate a new platform for piezo-optomechanical conversion by combining a crystalline silicon nanobeam photonic crystal cavity and a suspended thin-film lithium niobate acoustic resonator. The goal is to combine the excellent optomechanical properties of the silicon nanobeam cavity with coherent excitations in a piezoelectric resonator in order to achieve large effective coupling between the microwave excitation and the optical cavity. In order to fabricate this hybrid silicon-on-lithium niobate device, we explore a technique termed ‘slapping’ where a loosely connected suspended nanostructure is patterned such that a tapered optical fiber can be used to rip it away and place it in an arbitrary location on another chip. We report a single photon optomechanical coupling rate g_0 = 5.1 kHz and a single photon microwave to optical efficiency of η_{μ→o} = 6.1 × 10−6 at 25μW of input optical power in a device limited by a suboptimal optical interface.

In cavity optomechanics, optical fields are coupled to the displacement of mechanical resonators. While it is interesting to study fundamental aspects of this interaction, it is the ability to link this mechanical displacement to various other degrees of freedom that inspires many applications in the field. In these applications, the mechanical resonator can be used as a handle to an external influence for the purpose of sensing, but also as a transducer between two otherwise detached degrees of freedom. The latter approach is the focus of this work. A particularly interesting regime for such a transduction process is between a few-gigahertz microwave tone and optical photons at telecom wavelengths around 1550 nm, connecting the operating regimes of long-range telecommunication with that of superconducting quantum nodes. Bridging the gap between these domains is an essential step towards any size of quantum network based on superconducting nodes, as the losses encountered in microwave transmission lines prohibits the connection of such nodes over length scales extending beyond a few meters. As such, a transducer between the few-gigahertz and optical telecom domains would enable the use of low-loss optical channels to connect remote superconducting nodes, given that the quantum information is preserved throughout the conversion process. One approach of realizing such a converter makes use of a gigahertz-frequency mechanical mode as the transducing element, which is coupled to the optical telecom domain using the optomechanical interaction and to the microwave domain using the electromechanical interaction. In this work, we aim to unify both of these interactions in a single device by designing and fabricating optomechanical devices from the III/V semiconductors, which, alongside their good optical properties, are also piezoelectric. In Chapter 2, we motivate our material choice and introduce the relevant properties of the materials, as well as their impact on the fabrication process. Following the material discussion, we then set out in Chapter 3 to realize a microwave-to-optics converter made of an optomechanical crystal in gallium arsenide, which we resonantly couple to a interdigital transducer using surface-acoustic-waves. With this device, we demonstrate the first microwave-to-optics conversion using a mechanical mode with an average number of thermal excitations below one. As well as verifying the coherence of the conversion process, our experiments also highlight the limitations arising from the material choice due to absorption-induced incoherent heating of the mechanical mode. The material choice itself then becomes the central topic of Chapter 4, where we opt for a gallium phosphide, a relative of gallium arsenide, which prominently features a larger bandgap. With this material we show non-classical correlations between photons an phonons. We enter a a regime that was previously inaccessible to devices made from piezoelectric materials, a promising step towards realizing the noise-requirements for microwave-to-optics converters. In Chapter 5, we use the insights from the two previous chapters to design a new type of electro-opto-mechanical resonator, specifically aimed at microwave-to-optics conversion. We miniaturize the electromechanical interface and use strongly coupled mechanical resonators for the transfer of excitations between the microwave and mechanical mode. We fabricate and characterise initial devices as well as demonstrate the validity of the functioning principle. Finally in Chapter 6, we reflect on the results of the previous chapters and highlight some potential advantages of other approaches. @en

In recent years, cavity optomechanics has attracted lots of attention. In this system, photons inside a cavity interact with a macroscopic harmonic oscillator through the radiation pressure force. Depending on the type of structure, opto-mechanical systems can be used in, for example, precision sensing, including force and acceleration measurements, testing fundamental physics such as quantum mechanics at a macroscopic scale and the link between quantum physics and gravity. Also, it potentially plays an important role in quantum information, where optical photons can be used to transmit quantum information in a long range and the mechanical oscillation is used to couple to other types of local qubits where quantum information is processed.

Previously, many of the aforementioned applications have been demonstrated and are shown it to be a promising system. The mechanical oscillator has been cooled down to close to its ground state. Furthermore, for the opto-mechanical system with high quality oscillator, the macroscopic harmonic oscillator is possible to be cooled down to its quantum ground state from room temperature, opening up a new regime where quantum experiments on massive objects can be done at room temperature. Moreover, non-classical states between photons and the harmonic oscillator have been achieved, making it close to real applications in quantum information and approaching the testing and realizing quantum entanglement of large objects. In an opto-mechanical system, there are different structures. Two examples, which are both of great interested, are the Fabry-Pérot cavity with photonic crystal and the nanobeam. In a Fabry-Pérot cavity, which is the prototype of most of the cavity opto-mechanics system, there are two reflectors. One is fixed, and another one acts as an harmonic oscillator. To enlarge the interaction, the oscillating reflector is made of thin photonic crystal slab. For the nanobeam cavity, the mechanical resonator is the nanobeam itself, and it also forms a cavity where the optical field is trapped inside. This two systems have their own strengths and drawbacks, which are suitable for different applications.

In this work, both types are explored. In the Fabry-Pérot cavity, the set-up is usually bulky. Is it possible to make the footprint of the whole set-up smaller such that it can be integrated into a chip? If the answer is yes, it would be a great boost to its sensing application where easy-to-use devices are usually needed. Moreover, for the photonic crystal, is there a limitation on its thickness? Previously, in our group, we have noticed that the reflectivity of a photonic crystal drops sharply if the thickness is reduced, though the simulation still yields a reflectivity close to unity. The results of these two problems are presented in chapter 3 and chapter 4, respectively. The results for the nanobeam cavity are shown in chapter 5. Designs are proposed, aiming at increasing the photon-phonon coupling while keeping the photon decay rate unchanged.