Academic literature on the topic 'Quantum Logic'

The application of Moore's Law would not be feasible by using the computing systems fabrication principles that are prevalent today. Fundamental changes in the field of computing are needed to keep Moore's Law operational. Different quantum technologies are available to take the advancement of computing into the future. Logic in quantum technology uses gates that are very different from those used in contemporary technology. Limiting itself to reversible operations, this thesis presents different methods to realize these logic gates. Two methods using Generalized Ternary Gates and Muthukrishnan Stroud Gates are presented for synthesis of ternary logic gates. Realizations of well-known quantum gates like the Feynman gate, Toffoli Gate, 2-qudit and 3-qudit SW AP gates are shown. In addition a new gate, the Inverse SW AP gate, is proposed and its realization is also presented.

Since Quantum Computer is almost realizable on large scale and Quantum Technology is one of the main solutions to the Moore Limit, Quantum Logic Synthesis (QLS) has become a required theory and tool for designing Quantum Logic Circuits. However, despite its growth, there is no any unified aproach to QLS as Quantum Computing is still being discovered and novel applications are being identified. The intent of this study is to experimentally explore principles of Quantum Logic Synthesis and its applications to Inductive Machine Learning. Based on algorithmic approach, I first design a Genetic Algorithm for Quantum Logic Synthesis that is used to prove and verify the methods proposed in this work. Based on results obtained from the evolutionary experimentation, I propose a fast, structure and cost based exhaustive search that is used for the design of a novel, least expensive universal family of quantum gates. The results form both the evolutionary and heuristic search are used to formulate an Inductive Learning Approach based on Quantum Logic Synthesis with the intended application being the humanoid behavioral robotics. The presented approach illustrates a successful algorithmic approach, where the search algorithm was able to invent/discover novel quantum circuits as well as novel principles in Quantum Logic Synthesis.

This thesis describes research on the design of quantum logic circuits suitable for the experimental demonstration of a three-qubit quantum computation prototype. The design is based on a proposal for optically controlled, solid-state quantum logic gates. In this proposal, typically referred to as SFG model, the qubits are stored in the electron spin of donors in a solid-state substrate while the interactions between them are mediated through the optical excitation of control particles placed in their proximity. After a brief introduction to the area of quantum information processing, the basics of quantum information theory required for the understanding of the thesis work are introduced. Then, the literature on existing quantum computation proposals and experimental implementations of quantum computational systems is analysed to identify the main challenges of experimental quantum computation and typical system parameters of quantum computation prototypes. The details of the SFG model are subsequently described and the entangling characteristics of SFG two-qubit quantum gates are analysed by means of a geometrical approach, in order to understand what entangling gates would be available when designing circuits based on this proposal. Two numerical tools have been developed in the course of the research. These are a quantum logic simulator and an automated quantum circuit design algorithm based on a genetic programming approach. Both of these are used to design quantum logic circuits compatible with the SFG model for a three-qubit Deutsch-Jozsa algorithm. One of the design aims is to realise the shortest possible circuits in order to reduce the possibility of errors accumulating during computation, and different design procedures which have been tested are presented. The tolerance to perturbations of one of the designed circuits is then analysed by evaluating its performance under increasing fluctuations on some of the parameters relevant in the dynamics of SFG gates. Because interactions in SFG two-qubit quantum gates are mediated by the optical excitation of the control particles, the solutions for the generation of the optical control signal required for the proposed quantum circuits are discussed. Finally, the conclusions of this work are presented and areas for further research are identified.

Trapped atomic ions are one of the most promising systems for building a quantum computer -- all of the fundamental operations needed to build a quantum computer have been demonstrated in such systems. The challenge now is to understand and reduce the operation errors to below the 'fault-tolerant threshold' (the level below which quantum error correction works), and to scale up the current few-qubit experiments to many qubits. This thesis describes experimental work concentrated primarily on the first of these challenges. We demonstrate high-fidelity single-qubit and two-qubit (entangling) gates with errors at or below the fault-tolerant threshold. We also implement an entangling gate between two different species of ions, a tool which may be useful for certain scalable architectures. We study the speed/fidelity trade-off for a two-qubit phase gate implemented in ⁴³Ca^+ hyperfine trapped-ion qubits. We develop an error model which describes the fundamental and technical imperfections / limitations that contribute to the measured gate error. We characterize and minimise various error sources contributing to the measured fidelity, allowing us to account for errors due to the single-qubit operations and state readout (each at the 0.1% level), and to identify the leading sources of error in the two-qubit entangling operation. We achieve gate fidelities ranging between 97.1(2)% (for a gate time t_g = 3.8 μs) and 99.9(1)% (for t_g = 100 μs), representing respectively the fastest and lowest-error two-qubit gates reported between trapped-ion qubits by nearly an order of magnitude in each case. We also characterise single-qubit gates with average errors below 10^-4 per operation, over an order of magnitude better than previously achieved with laser-driven operations. Additionally, we present work on a mixed-species entangling gate. We entangle of a single ⁴⁰Ca^+ ion and a single ⁴³Ca^+ ion with a fidelity of 99.8(5)%, and perform full tomography of the resulting entangled state. We describe how this mixed-species gate mechanism could be used to entangle ⁴³Ca^+ and ⁸⁸Sr^+, a promising combination of ions for future experiments.

Quantum mechanics is basically a mathematical recipe on how to construct physical models. Historically its origin and main domain of application has been in the microscopic regime, although it strictly seen constitutes a general mathematical framework not limited to this regime. Since it is a statistical theory, the meaning and role of probabilities in it need to be defined and understood in order to gain an understanding of the predictions and validity of quantum mechanics. The interpretational problems of quantum mechanics are also connected with the interpretation of the concept of probability. In this thesis the use of probability theory as extended logic, in particular in the way it was presented by E. T. Jaynes, will be central. With this interpretation of probabilities they become a subjective notion, always dependent on one's state of knowledge or the context in which they are assigned, which has consequences on how things are to be viewed, understood and tackled in quantum mechanics. For instance, the statistical operator or density operator, is usually defined in terms of probabilities and therefore also needs to be updated when the probabilities are updated by acquisition of additional data. Furthermore, it is a context dependent notion, meaning, e.g., that two observers will in general assign different statistical operators to the same phenomenon, which is demonstrated in the papers of the thesis. It is also presented an alternative and conceptually clear approach to the problematic notion of "probabilities of probabilities", which is related to such things as probability distributions on statistical operators. In connection to this, we consider concrete numerical applications of Bayesian quantum state assignment methods to a three-level quantum system, where prior knowledge and various kinds of measurement data are encoded into a statistical operator, which can then be used for deriving probabilities of other measurements. The thesis also offers examples of an alternative quantum state assignment technique, using maximum entropy methods, which in some cases are compared with the Bayesian quantum state assignment methods. Finally, the interesting and important problem whether the statistical operator, or more generally quantum mechanics, gives a complete description of "objective physical reality" is considered. A related concern is here the possibility of finding a "local hidden-variable theory" underlying the quantum mechanical description. There have been attempts to prove that such a theory cannot be constructed, where the most well-known impossibility proof claiming to show this was given by J. S. Bell. In connection to this, the thesis presents an idea for an interpretation or alternative approach to quantum mechanics based on the concept of space-time.<br>QC 20100810

The logic Lq, introduced within this thesis, is a logic for quantum information. The purpose was, in fact, to describe logically the qubit structure (that is, the intrinsic quantum superposition of a two-level quantum state) and the maximal quantum entanglement of two qubits. The logic Lq is obtained via Sambin’s reflection principle of Basic logic, by which the metalinguistic links among assertions reflect (by solving a definitional equation) into logical connectives among propositions. However, while in Basic logic the metalanguage is classical, in our case it is quantum. In the quantum metalanguage each atomic assertion carries along an assertion degree, a complex number, which is interpreted as a probability amplitude. It is just the presence of assertion degrees that allows the introduction of the connective “quantum superposition” in Lq. This connective is a generalization of the logical conjunction “and”. It is labelled by complex numbers indicating the weight by which each proposition contributes to the compound proposition. The truth-values (or truth-degrees) are the squared modules of the assertion degrees, and their range is the real interval [0,1]. Then, the logic Lq is many-valued. Differently from fuzzy logics, however, the truth-degrees are interpreted here as quantum-mechanical probabilities. The logic Lq keeps the three main properties of Basic logic, namely symmetry, reflection and visibility. This choice has been dictated by the following considerations: 1) The no-cloning and no-erase theorems of quantum information do not allow the corresponding logic to have the structural rules of weakening and contraction, with which they disagree. This fact rules out, in the search of a logic for quantum information, every kind of structural logic. 2) Choosing Basic logic instead of Linear logic (the other main sub-structural logic) was due to the fact that without visibility the connective “quantum entanglement” cannot be introduced. Furthermore, we looked for a logic of quantum information which was endowed with a deductive calculus, (in particular a sequent calculus). The logic Lq appears, in so far, as the only one which can take into account all the above desiderata. The interpretation of the assertions of the quantum metalanguage is given in terms of quantum states (the quantum metalanguage “is” the Hilbert space). The interpretation of the propositions of Lq is given in terms of (non-hermitian) operators which are weak measurements. Then, the interpretation of Lq is based on a generalization of the concepts already proposed by Birkhoff and von Neumann in “orthodox” quantum logic. The difference stands in the fact that the interpretation of Lq is not given in terms of projectors, but in terms of weak measurements, which do not give rise to an abrupt collapse of quantum wave functions. This allows a logical description of quantum superposition, because the latter is not destroyed. The possibility of interpreting propositions as weak measurements is due to the fact that we introduced a quantum metalanguage. In fact, in the interpretation of propositions, the complex factors multiplying the projection operators are nothing else than the assertion degrees. Some results of this thesis are: a) The adoption of a new kind of metalanguage, the quantum metalanguage, where the metalinguistic links are quantum correlations, and assertions have a complex assertion-degree. b) The introduction, through the reflection principle, of new (quantum) connectives, like “quantum superposition”, and “quantum entanglement”. c) The introduction of a new dual operation (which is a generalization of Sambin-Girard logical duality) to take into account the dual Hilbert space occurring in the interpretation. d) A quantum cut rule, which is interpreted as a quantum projective measurement. As the cut is a meta-rule, it follows that a quantum machine cannot perform a self-measurement and destroy itself. e) A new meta-rule, not equivalent to the cut, named “EPR rule” (to remind the Einstein-Podolsky-Rosen paradox). This rule allows to prove simultaneously two entangled theorems. f) The formulation of the “qubit theorem”, which is the logical description of the preparation of the optical qubit state. g) The lattice of propositions of Lq is, in the case of two qubits, orthomodular and non-distributive. Then, Lq is a quantum logic. It should be noticed that Lq is the first logic which is sub-structural, many-valued and quantum at the same time.<br>La logica introdotta in questa tesi, detta Lq, è una logica dell’ informazione quantistica. Lo scopo, infatti, era quello di descrivere logicamente la struttura del qubit (cioè, la sovrapposizione quantistica intrinseca di uno stato quantico a due livelli) e l’intreccio (entanglement) quantistico massimale di due qubits. La logica Lq è ottenuta tramite il principio di riflessione di Sambin della logica di Base, secondo il quale i legami metalinguistici tra asserzioni si riflettono (risolvendo un’ equazione definitoria) in connettivi logici tra proposizioni. Comunque, mentre nella logica di Base il metalinguaggio è classico, nel nostro caso è quantistico. Nel metalinguaggio quantistico, ciascuna asserzione atomica è dotata di un grado di asserzione, un numero complesso che viene interpretato come un’ ampiezza di probabilità. E’ proprio la presenza dei gradi di asserzione che permette l’introduzione del connettivo logico di “sovrapposizione quantistica” in Lq. Quest’ultimo è una generalizzazione del connettivo di congiunzione “and” dotato di indici complessi indicanti con quale “peso” ciascuna proposizione contribuisce alla formazione della proposizione composta. I valori (o gradi) di verità sono i moduli quadrati dei gradi di asserzione, con un range che è l’intervallo reale [0,1]. Pertanto, la logica Lq è polivalente. I gradi di verità, differentemente dalle logiche fuzzy, sono qui interpretati come probabilità quantistiche. Nella logica Lq si mantengono le tre importanti proprietà della logica di Base, cioè simmetria, riflessione e visibilità. Questa scelta è stata dettata dalle seguenti considerazioni: 1) I teoremi di no-cloning e no-erase dell’informazione quantistica non permettono di avere, nella logica corrispondente, le regole strutturali di indebolimento e contrazione, che sono in antitesi con i suddetti teoremi. Pertanto, nella ricerca di una logica dell’ informazione quantistica, ogni logica strutturale deve essere esclusa a priori. 2) La scelta tra le due più importanti logiche sub-strutturali, cioè la logica di Base e la logica Lineare, in favore della prima, è dovuta al fatto che, in assenza di visibilità, il connettivo logico “quantum entanglement” non può essere introdotto. Inoltre, si è cercata una logica dell’ informazione quantistica che avesse un calcolo deduttivo (in particolare il calcolo dei sequenti). La logica Lq sembra essere, finora, l’ unica logica dell’ informazione quantistica che possa soddisfare questi desiderata. L’ interpretazione delle asserzioni del metalinguaggio quantistico è data in termini di stati quantistici (il metalinguaggio quantistico “è” lo spazio di Hilbert). L’ interpretazione delle proposizioni di Lq è data in termini di operatori non-hermitiani, che sono misure deboli. L’ interpretazione di Lq si basa su una generalizzazione dei concetti già proposti da Birkhoff e von Neumann nella logica quantistica “ortodossa”, dove le proposizioni sono interpretate come operatori di proiezione. La differenza consiste nel fatto che in Lq le proposizioni sono interpretate invece come misure deboli, che, diversamente dalle misure proiettive, non danno luogo ad un brusco collasso della funzione d’onda. Questo permette una descrizione logica della sovrapposizione quantistica, perché essa non viene distrutta. La possibilità di interpretare le proposizioni come misure deboli, è dovuta al fatto che abbiamo introdotto un metalinguaggio quantistico. Infatti, il grado di asserzione si riflette, nell’ interpretazione delle proposizioni, con la presenza un fattore moltiplicativo complesso sui proiettori. Alcuni risultati di questa tesi sono: a) L’ adozione di un nuovo tipo di metalinguaggio, il metalinguaggio quantistico, dove i legami metalinguistici sono correlazioni quantistiche, e le asserzioni hanno un grado di asserzione complesso. b) L’ introduzione, tramite il principio di riflessione, di nuovi connettivi logici “quantistici”, quali la “sovrapposizione quantistica” e l’ “entanglement”. c) L’ introduzione di una nuova operazione duale, che è una generalizzazione della dualità logica di Sambin-Girard, che tiene conto, nell’ interpretazione, dello spazio duale di Hilbert. d) Una regola del taglio quantistica, che viene interpretata come misura quantistica proiettiva. Poiché il taglio è una meta-regola, ne consegue che una macchina quantistica non può effettuare una auto-misura e quindi auto-distruggersi. e) Una nuova meta-regola, non equivalente al taglio, detta regola EPR (rifacentesi al paradosso di Einstein-Podolsky-Rosen). Questa regola permette di dimostrare simultaneamente due teoremi entanglati. f) La formulazione del “teorema del qubit”, che è la descrizione logica della preparazione dello stato quantistico del qubit ottico. g) Il fatto che il reticolo delle proposizioni di Lq nel caso di due qubits è orto-modulare non-distributivo. Quindi Lq è una logica quantistica. E’ da notare il fatto che Lq è la prima logica ad essere contemporaneamente sub-strutturale, a molti valori di verità, e quantistica.

Thesis: Ph. D., Massachusetts Institute of Technology, Department of Electrical Engineering and Computer Science, 2015.<br>This electronic version was submitted by the student author. The certified thesis is available in the Institute Archives and Special Collections.<br>Cataloged from student-submitted PDF version of thesis.<br>Includes bibliographical references (pages 151-161).<br>InGaAs is a promising candidate as an n-type channel material for future CMOS due to its superior electron transport properties. Great progress has taken place recently in demonstrating InGaAs MOSFETs for this goal. Among possible InGaAs MOSFET architectures, the recessed-gate design is an attractive option due to its scalability and simplicity. In this thesis, a novel self-aligned recessed-gate fabrication process for scaled InGaAs Quantum-Well MOSFETs (QW-MOSFETs) is developed. The device architectural design emphasizes scalability, performance and manufacturability by making extensive use of dry etching and Si-compatible materials. The fabrication sequence yields precise control of all critical transistor dimensions. This work achieved InGaAs MOSFETs with the shortest gate length (Lg=20 nm), and MOSFET arrays with the smallest contact size (Lc=40 nm) and smallest pitch size (Lp=150 nm), at the time when they were made. Using a wafer bonding technique, InGaAs MOSFETs were also integrated onto a silicon substrate. The fabricated transistors show the potential of InGaAs to yield devices with well-balanced electron transport, electrostatic integrity and parasitic resistance. A device design optimized for transport exhibits a transconductance of 3.1 mS/[mu]m, a value that matches the best III-V high-electron-mobility transistors (HEMTs). The precise fabrication technology developed in this work enables a detailed study of the impact of channel thickness scaling on device performance. The scaled III-V device architecture achieved in this work has also enabled new device physics studies relevant for the application of InGaAs transistors for future logic. A particularly important one is OFF-state leakage. For the first time, this work has unambiguously identified band-to-band tunneling (BTBT) amplified by a parasitic bipolar effect as the cause of excess OFF-state leakage current in these transistors. This finding has important implications for future device design<br>by Jianqiang Lin.<br>Ph. D.

Introduced by C.C. Chang in the 1950s, MV algebras are to many-valued (Łukasiewicz) logics what boolean algebras are to two-valued logic. More recently, effect algebras were introduced by physicists to describe quantum logic. In this thesis, we begin by investigating how these two structures, introduced decades apart for wildly different reasons, are intimately related in a mathematically precise way. We survey some connections between MV/effect algebras and more traditional algebraic structures. Then, we look at the categorical structure of effect algebras in depth, and in particular see how the partiality of their operations cause things to be vastly more complicated than their totally defined classical analogues. In the final chapter, we discuss coordinatization of MV algebras and prove some new theorems and construct some new concrete examples, connecting these structures up (requiring a detour through effect algebras!) to boolean inverse semigroups.

While classical accounts of identity (‘classical’ here pertaining to both logics and physics) are generally well understood, the advent of quantum theory, specifically quantum statistics, has cast shadow over these conceptions. Dealing with the consequently surfacing problems is a philosophically rich and interesting enterprise. I begin this thesis by providing an exegesis of the roles played by, and features of, identity in logics, classical physics, and quantum physics. Therein I consider how under a quantal description of reality, classical notions of identity and individuality break down. In the second chapter, I address how this problem has launched an arc of thought in analytic metaphysics and formal philosophy motivating the development of non-standard formal frameworks with which philosophical sense can be made of quantal objects. Among these, I explore and critically evaluate quaset theory, quasi-set theory, and non-reflexive Schr¨odinger logics, identifying some significant problems with quaset theory that arise in defining cardinality and later, pointing out a problem with Schr¨odinger logics in their modelling of the continuity between quantal and classical treatments of the world. The queer character of identity in the quantal regime motivates a turn to vagueness which I introduce in the third chapter, providing a brief outline of vagueness and the sorites paradox. Further, I reflect on the fundamental nature of vagueness, outlining and evaluating the semantic and ontic conceptions thereof. In the final chapter, I proceed to explicate and assess notions that identity and quantal objects can be vague. I shall discuss accounts according to which the vagueness of identity and quantum objects is posited as a feature of nature emerging in quantum systems — the ontic vagueness of identity — finding that these ideas are flawed and/or rely on misinterpretations of vagueness. Finally, I present an argument which suggests how the vagueness of identity can arise as an artifact of the differing treatments of identity in the quantal and classical regimes in which the vagueness involved can be semantic rather than ontic.