This paperintroduces them-valued Swap-based quantum computations (QC)ofECA.Thefundamental m-valued quantum representations of:(1)m-valued orthonormal computational basis states quantum decision trees (QDTs) and(2)m-valued orthonormal composite basis states QDTsarealsointroduced asquantum modeling fortherepresentations andoperations oftheQECAdynamics. Thenewm-valued quantum representations andtheir associated operations canplayan important role intheautomated design offuture QECAcircuits that consume minimalpower inthequantum domain.
This paper introduces the m-valued Swap-based quantum computations (QC) of ECA. The fundamental m-valued quantum representations of: (1) m-valued orthonormal computational basis states quantum decision trees (QDTs) and (2) m-valued orthonormal composite basis states QDTs are also introduced as quantum modeling for the representations and operations of the QECA dynamics. The new m-valued quantum representations and their associated operations can play an important role in the automated design of future QECA circuits that consume minimal power in the quantum domain.
Measurement-based quantum computation (MBQC) is a paradigm for quantum computation where computation is driven by local measurements on a suitably entangled resource state. In this work we show that MBQC is related to a model of quantum computation based on Clifford quantum cellular automata (CQCAs). Specifically, we show that certain MBQCs can be directly constructed from CQCAs which yields a simple and intuitive circuit model representation of MBQC in terms of quantum computation based on CQCA. We apply this description to construct various MBQC-based Ans\"atze for parametrized quantum circuits, demonstrating that the different Ans\"atze may lead to significantly different performances on different learning tasks. In this way, MBQC yields a family of hardware-efficient Ans\"atze that may be adapted to specific problem settings and are particularly well suited for architectures with translationally invariant gates such as neutral atoms.
Quantum mechanical phenomena such as phase shifts, superposition, and entanglement show promise in use for computation. Suitable technologies for the modeling and design of quantum computers and other information processing techniques that exploit quantum mechanical principles are in the range of vision. Quantum algorithms that significantly speed up the process of solving several important computation problems have been proposed in the past. The most common representation of quantum mechanical phenomena are transformation matrices. However, the transformation matrices grow exponentially with the size of a quantum system and, thus, pose significant challenges for efficient representation and manipulation of quantum functionality. In order to address this problem, first approaches for the representation of quantum systems in terms of decision diagrams have been proposed. One very promising approach is given by Quantum Multiple-Valued Decision Diagrams (QMDDs) which are able to efficiently represent transformation matrices and also inherently support multiple-valued basis states offered by many physical quantum systems. However, the initial proposal of QMDDs was lacking in a formal basis and did not allow, e.g., the change of the variable order-an established core functionality in decision diagrams which is crucial for determining more compact representations. Because of this, the full potential of QMDDs or decision diagrams for quantum functionality in general has not been fully exploited yet. In this paper, we present a refined definition of QMDDs for the general quantum case. Furthermore, we provide significantly improved computational methods for their use and manipulation and show that the resulting representation satisfies important criteria for a decision diagram, i.e., compactness and canonicity. An experimental evaluation confirms the efficiency of QMDDs.
This thesis presents a model of Quantum Cellular Automata (QCA). The presented formalism is a natural quantization of the classical Cellular Automata (CA). It is based on a lattice of qudits, and an update rule consisting of local unitary operators that commute with their own lattice translations. One purpose of this model is to act as a theoretical model of quantum computation, similar to the quantum circuit model. The main advantage that QCA have over quantum circuits is that QCA make considerably fewer demands on the underlying hardware. In particular, as opposed to direct implementations of quantum circuits, the global evolution of the lattice in the QCA model does not assume independent control over individual \emph{qudits}. Rather, all qudits are to be addressed collectively in parallel. The QCA model is also shown to be an appropriate abstraction for space-homogeneous quantum phenomena, such as quantum lattice gases, spin chains and others. Some results that show the benefits of basing the model on local unitary operators are shown: computational universality, strong connections to the circuit model, simple implementation on quantum hardware, and a series of applications. A detailed discussion will be given on one particular application of QCA that lies outside either computation or simulation: single-spin measurement. This algorithm uses the techniques developed in this thesis to achieve a result normally considered hard in physics. It serves well as an example of why QCA are interesting in their own right.
In this paper we present a systematic view of Quantum Cellular Automata (QCA), a mathematical formalism of quantum computation. First we give a general mathematical framework with which to study QCA models. Then we present four different QCA models, and compare them. One model we discuss is the traditional QCA, similar to those introduced by Shumacher and Werner, Watrous, and Van Dam. We discuss also Margolus QCA, also discussed by Schumacher and Werner. We introduce two new models, Coloured QCA, and Continuous-Time QCA. We also compare our models with the established models. We give proofs of computational equivalence for several of these models. We show the strengths of each model, and provide examples of how our models can be useful to come up with algorithms, and implement them in real-world physical devices.
In Part I of this thesis, we present a new model of quantum cellular automata \n(QCA) based on local unitary operations. We will describe a set of desirable \nproperties for any QCA model, and show that all of these properties are \nsatisfied by the new model, while previous models of QCA do not. We will also \nshow that the computation model based on Local Unitary QCA is equivalent to \nthe Quantum Circuit model of computation, and give a number of applications of \nthis new model of QCA. We also present a physical model of classical CA, on \nwhich the Local Unitary QCA model is based, and Coloured QCA, which is an \nalternative to the Local Unitary QCA model that can be used as the basis for \nimplementing QCA in actual physical systems. \n \nIn Part II, we explore the quantum separability problem, where we are given a \ndensity matrix for a state over two quantum systems, and we are to determine \nwhether the state is separable with respect to these systems. We also look at \nthe converse problem of finding an entanglement witness, which is an \nobservable operator which can give a verification that a particular quantum \nstate is indeed entangled. Although the combined problem is known to be \nNP-hard in general, it reduces to a convex optimization problem, and by \nexploiting specific properties of the set of separable states, we introduce a \nclassical algorithm for solving this problem based on an Interior Point \nAlgorithm introduced by Atkinson and Vaidya in 1995. \n \nIn Part III, we explore the use of a low-depth AQFT (approximate quantum \nFourier transform) in quantum phase estimation. It has been shown previously \nthat the logarithmic-depth AQFT is as effective as the full QFT for the \npurposes of phase estimation. However, with sub-logarithmic depth, the phase \nestimation algorithm no longer works directly. In this case, results of the \nphase estimation algorithm need classical post-processing in order to retrieve \nthe desired phase information. A generic technique such as the method of \nmaximum likelihood can be used in order to recover the original phase. \n \nUnfortunately, working with the likelihood function analytically is \nintractable for the phase estimation algorithm. We develop some computational \ntechniques to handle likelihood functions that occur in phase estimation \nalgorithms. These computational techniques may potentially aid in the analysis \nof certain likelihood functions.