La théorie du calcul quantique se situe à la frontière de la physique quantique et de l’informatique. Par conséquent, les deux domaines contribuent à la rendre d’autant plus riche en apportant leurs propres méthodes et outils mathématiques. La présente thèse tente de mettre en évidence cette particularité en traitant des problématiques qui vont la physique expérimentale aux modèles de calcul. Le but est d’offrir de nouvelles possibilités pour démontrer un avantage quantique. Après une brève introduction aux notions de base de la mécanique quantique, certains aspects liés à l’informatique sont discutés. Le formalisme des classes de complexité quantiques ainsi que le concept du calcul quantique en variables continues sont décrits. Ensuite, le modèle connu comme instantaneous quantum computing est traduit en variables continues, le rendant attrayant d’un point de vue expérimental. Le chapitre conclut sur une discussion concernant un protocole hybride impliquant l’algorithme de Grover dans le cadre des communications quantiques. La dernière partie de la thèse s’intéresse à des problématiques issues de la physique expérimentale. Le lien entre l’effet Hong-Ou-Mandel et la fonction de Wigner d’un état à deux photons est mise en évidence, et un protocole expérimental est décrit en conséquence. La suite traite du domaine des circuits supraconducteurs et envisage de possibles expériences. Il est montré comment utiliser un qubit de flux pour manipuler un centre coloré du diamant. Il est également décrit comment sonder le modèle de Rabi dans le régime de couplage ultra fort en utilisant un qubit supplémentaire faiblement couplé. / The theory of quantum computing lies at the very boundary between quantum physics and computer science. As such, both fields bring their own methods and mathematical tools to make quantum computing even richer. The present thesis attempts to reflect this specificity by addressing questions ranging from experimental physics to computational models. The goal is to provide novel ways of demonstrating quantum advantage. After a short introduction to basic notions of quantum mechanics, some computer science aspects are discussed. We describe the powerful formalism of quantum complexity classes and the concept of quantum computations based on continuous variables. We then translate the model of instantaneous quantum computing to continuous variables, which is experimentally appealing. The chapter concludes with a discussion on a hybrid protocol involving Grover’s algorithm in a quantum communication framework. The last part of the thesis is devoted to experimentally driven issues. A fundamental connection between the Hong-Ou-Mandel experiment and the Wigner function of two-photon states is derived and a verification protocol is designed accordingly. We then move to the field of superconducting circuits to discuss proposals for future experiments. We show how to use a flux qubit to manipulate a NV color center. We also describe how to use to probe the Rabi model in the ultra strong coupling regime using an additional weakly coupled qubit.

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付記する学位プログラム名: 京都大学卓越大学院プログラム「先端光・電子デバイス創成学」 / 京都大学 / 新制・課程博士 / 博士(理学) / 甲第24123号 / 理博第4851号 / 新制||理||1694(附属図書館) / 京都大学大学院理学研究科物理学・宇宙物理学専攻 / (主査)教授 高橋 義朗, 教授 石田 憲二, 教授 田中 耕一郎 / 学位規則第4条第1項該当 / Doctor of Science / Kyoto University / DFAM

Superconductor electronics based on the Single Flux Quantum (SFQ) technology are considered a strong contender for the ‘beyond CMOS’ future of digital circuits because of the high speed and low power dissipation associated with them. In fact, digital operations beyond tens of GHz have been routinely demonstrated in the SFQ technology. These circuits have widespread applications such as high-speed analog-to-digital conversion, digital signal processing, high speed computing and in emerging topics such as control circuitry for superconducting quantum computing. Rapid Single Flux Quantum (RSFQ) circuits have emerged as a promising candidate within the SFQ technology, with information encoded in picosecond wide, milli-volt voltage pulses. As is the case with any integrated circuit technology, scalability of RSFQ circuits is essential to realizing their applications. These circuits, based on the Josephson junction, require a DC bias current for the correct operation. The DC bias current requirement increases with circuit complexity, and this has multiple implications on circuit operation. Large currents produce magnetic fields that can interfere with logic operation. Furthermore, the heat load delivered to the superconducting chip also increases with current which could result in the circuit becoming ‘normal’ and not superconducting. These problems make reduction of the bias current necessary. Serial Biasing (SB) is a bias current reduction technique, that has been proposed in the past. In this technique, a digital circuit is partitioned into multiple identical islands and bias current is provided to each island in a serial manner. While this scheme is promising, there are multiple challenges such as design of the driver-receiver pair circuit resulting in robust and wide operating bias margins, current management on the floating islands, etc. This thesis investigates SB in a systematic manner, focusing on the design and measurement of the fundamental components of this technique with an emphasis on reliability and scalability. It presents works on circuit techniques achieving high speed serially biased RSFQ circuits with robust operating margins and the experimental evidence to support the ideas. It develops a framework for serial biasing that could be used by electronic design tools to automate design and synthesis of complex RSFQ circuits. It also investigates Passive Transmission Lines (PTLs) for use as passive interconnects between library cells in a complex design, reducing the DC bias current required by the active circuitry.

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Majorana fermions promise potential applications in quantum computing, superconductivity, and related fields. In this thesis, an analysis of A. Y. Kitaev's “Kitaev Chain”, a quasi-one-dimensional quantum wire in contact with a p-wave superconductor, designed as a model exhibiting unpaired Majoranas, is performed. Described by tunneling of spinless fermions between quantum dots, and formation of Cooper pairs on neighboring dots, Kitaev's chain Hamiltonian serves as a basis for emergent Majorana Zero Modes (zero energy Majorana fermions localized at either end of the chain) and artificial gauges (phases) to appear. By exact diagonalization, energy spectra and wavefunctions of a chain of spinless fermions on discrete quantum dots described by Kitaev's Hamiltonian are generated. By transforming the system into a basis of Majorana fermions and "bond fermions", where Majoranas on neighboring dots are paired, emergent Majorana Zero Modes (MZMs) are found at the ends of the chain. These emergent MZMs are paired in a non-local, zero energy bond fermion, which is found to allow degenerate energy states of the system to occur. Joining the ends of the chain by allowing tunneling and pairing of fermions on end sites, a ring topology is considered, where an "artificial gauge" emerges. This artificial gauge, or phase, causes a phase change on tunneling and Cooper pairing Hamiltonian matrix elements as a result of operator ordering within the Hamiltonian's ring terms. These required operator orderings are derived by comparison of energy spectra of the Kitaev ring in the fermion and bond fermion bases. Matching of calculated energy spectra in the Majorana and fermionic bases is used to confirm the presence of the artificial gauge, where this phase is found to be necessary in order to maintain a consistent energy spectra across the transformation between bases. This analysis is performed in order to understand the concept of Majorana Zero Modes and the emergence of Majorana fermions in 1D chains. By doing so, it is determined what Majorana fermions are, where they come from, and why Majorana Zero Modes are considered to be zero energy. These results contribute to the understanding of Kitaev chains and rings, as well as serve as a starting point for discussions regarding physical implications of the artificial gauge's effect, fermion statistics, and the emergence of Majorana Zero Modes in quasi-one-dimensional systems.

The variational quantum eigensolver (VQE) approach is currently one of the most promising strategies for simulating chemical systems on quantum hardware. In this work, I will describe a new quantum algorithm and a new set of classical algorithms based on VQE. The quantum algorithm, ADAPT-VQE, shows promise in mitigating many of the known limitations of VQEs: Ansatz ambiguity, local minima, and barren plateaus are all addressed to varying degrees by ADAPT-VQE. The classical algorithm family, O2DX-UCCSD, draws inspiration from VQEs, but is classically solvable in polynomial time. This group of algorithms yields equations similar to those of the linearized coupled cluster theory (LCCSD) but is more systematically improvable and, for X = 3 or X = ∞, can break single bonds, which LCCSD cannot do. The overall aim of this work is to showcase the richness of the VQE algorithm and the breadth of its derivative applications. / Doctor of Philosophy / A core goal of quantum chemistry is to compute accurate ground-state energies for molecules. Quantum computers promise to simulate quantum systems in ways that classical computers cannot. It is believed that quantum computers may be able to characterize molecules that are too large for classical computers to treat accurately. One approach to this is the variational quantum eigensolver, or VQE. The idea of a VQE is to use a quantum computer to measure the molecular energy associated with a quantum state which is parametrized by some classical set of parameters. A classical computer will use a classical optimization scheme to update those parameters before the quantum computer measures the energy again. This loop is expected to minimize the quantum resources needed for a quantum computer to be useful, since much of the work is outsourced to classical computers. In this work, I describe two novel algorithms based on the VQE which solve some of its problems.

Propositional satisfiability (SAT) has long been a cornerstone of computer science, especially in areas like hardware verification and AI. Whereas solving SAT problems has received significant attention, the task of enumerating all satisfying solutions—known as the All-Solution Satisfiability Problem (AllSAT)—remains comparatively underexplored. This thesis addresses the gap in research on AllSAT and its extension to All-Satisfiability Modulo Theories (AllSMT), where both Boolean and first-order logic atoms come into play. We started by introducing a novel algorithm for enumerating disjoint partial models without introducing blocking clauses, by integrating Conflict-Driven Clause Learning, Chronological Backtracking, and implicant shrinking techniques. We further extend this algorithm to handle projected enumeration and AllSMT, incorporating theory reasoning to address the complexities of richer logical frameworks. Dealing with the enumeration of non-Conjunctive Normal Form (non-CNF) formulas poses significant challenges. Traditional transformations such as Tseitin and Plaisted-Greenbaum hinder the production of small partial satisfying assignments, affecting the effectiveness of enumeration. In this thesis, we discuss a novel approach that combines Plaisted-Greenbaum transformation with Negative Normal Form (NNF) preprocessing, dramatically reducing the overall size of partial assignments and thus improving both efficiency and effectiveness of enumeration. Beyond the novel procedures mentioned above, we demonstrate the practical applications of AllSAT and AllSMT in several novel domains. These include (i) Weighted Model Integration (WMI), where AllSMT is used to efficiently reduce the number of integrals required by WMI algorithms based on predicate abstraction; (ii) knowledge compilation (KC), where theory lemmas generated from AllSMT prune inconsistent paths to create canonical decision diagrams; and (iii) quantum computing, where SMT and OMT are used to encode prime factorization problems of biprime numbers into D-Wave quantum annealers.

This PhD work took place in the framework of theoretical research aimed at implementation of quantum computing schemes and algorithms in solid state devices. The electron and nuclear spins of dopant atoms implanted in silicon crystals, that already lie at the core of commercial diodes and the photovoltaic industry, are able to store quantum information longer than anything else in the solid state. Controlled manipulations of silicon qubits depend on the ability to tune the nanoscopic donor electron state: we provide a complete theoretical picture that includes, within the insightful and analytic framework of effective mass theory, the effects of the non-trivial silicon conduction band and the different lattice distortions caused by the implantation of the donor species. Calibration of the multi-valley bulk theory to account for binding energies and electron-nuclear hyperfine couplings allows improved estimates of the exchange splittings between two neighbouring donors, that provide the simplest handle for tuning two-qubit operations. Further refinements to our approach lead to exceptional agreement with experimental measurements of Stark effects, where an external electric field is used to enable local single qubit manipulations within global driving fields: we set reliable thresholds on such gating speeds across all group V donors. Finally, we propose a scalable scheme for silicon quantum computing that relies on the coherent transfer of information from Si:Bi donors, that are established as excellent memory qubits, to surface quantum dots that are easier to manipulate, within a topological surface code which enables outstanding tolerance to errors. Analysis of the optimal working regimes and inclusion of the leading sources of decoherence allow us to set out a robust design of the basic building block of future realizations.