Quantum Computation Theory Research Articles

Forecasting Fossil Fuel Energy Consumption for Power Generation Using QHSA-Based LSSVM Model

Accurate forecasting of fossil fuel energy consumption for power generation is important and fundamental for rational power energy planning in the electricity industry. The least squares support vector machine (LSSVM) is a powerful methodology for solving nonlinear forecasting issues with small samples. The key point is how to determine the appropriate parameters which have great effect on the performance of LSSVM model. In this paper, a novel hybrid quantum harmony search algorithm-based LSSVM (QHSA-LSSVM) energy forecasting model is proposed. The QHSA which combines the quantum computation theory and harmony search algorithm is applied to searching the optimal values of and C in LSSVM model to enhance the learning and generalization ability. The case study on annual fossil fuel energy consumption for power generation in China shows that the proposed model outperforms other four comparative models, namely regression, grey model (1, 1) (GM (1, 1)), back propagation (BP) and LSSVM, in terms of prediction accuracy and forecasting risk.

Algebraic Effects, Linearity, and Quantum Programming Languages

We develop a new framework of algebraic theories with linear parameters, and use it to analyze the equational reasoning principles of quantum computing and quantum programming languages. We use the framework as follows: we present a new elementary algebraic theory of quantum computation, built from unitary gates and measurement; we provide a completeness theorem or the elementary algebraic theory by relating it with a model from operator algebra; we extract an equational theory for a quantum programming language from the algebraic theory; we compare quantum computation with other local notions of computation by investigating variations on the algebraic theory.

A New Generalized Schur-Weyl Duality

Schur-Weyl duality theory has many applications in quantum information and quantum computation theory. In this paper, we first give an equivalent relationship about the subgroup of unitary group U(d) and give several examples of subgroup of U(d) which satisfy the equivalent relationship. Next, we establish a new generalized Schur-Weyl which is an improvement of the classical Schur-Weyl duality and a generalized Schur-Weyl duality proved recently.

Can Hidden Variables Theories Meet Quantum Computation?

We study the relation between hidden variables theories and quantum computation. We discuss an inconsistency between a hidden variables theory and controllability of quantum computation. To derive the inconsistency, we use the maximum value of the square of an expected value. We propose a solution of the problem by using new hidden variables theory. Also we discuss an inconsistency between hidden variables theories and the double-slit experiment as the most basic experiment in quantum mechanics. This experiment can be an easy detector to Pauli observable. We cannot accept hidden variables theories to simulate the double-slit experiment in a specific case. Hidden variables theories may not depicture quantum detector. This is a quantum measurement theoretical profound problem.

Quantum Algorithms and Quantum-Inspired Algorithms

With the decreasing development of the classical computation,quantum computation becomes the increasing hot topic recently.In this paper,the fundamental theory of quantum computation is briefly introduced firstly.Then,we give the state of the art of quantum computation and research principles in this field from two different research directions(quantum algorithms and quantum-inspired algorithms).At last,some trends of quantum computation in the future research are given.Through the review and prospect of the whole field,this paper could give a significant guide for the future development of quantum computation.

The resource theory of stabilizer quantum computation

Recent results on the non-universality of fault-tolerant gate sets underline the critical role of resource states, such as magic states, to power scalable, universal quantum computation. Here we develop a resource theory, analogous to the theory of entanglement, that is relevant for fault-tolerant stabilizer computation. We introduce two quantitative measures—monotones—for the amount of non-stabilizer resource. As an application we give absolute bounds on the efficiency of magic state distillation. One of these monotones is the sum of the negative entries of the discrete Wigner representation of a quantum state, thereby resolving a long-standing open question of whether the degree of negativity in a quasi-probability representation is an operationally meaningful indicator of quantum behavior.

Quantum Cellular Automaton for Simulating Static Magnetic Fields

This paper proposes a new quantum cellular automaton (quantum CA) for simulating macroscopic electromagnetic fields. The final target is to simulate the macroscopic electromagnetic fields on a quantum computer. In the proposal approach, Maxwell's historical model, which was explained the nature of electromagnetic fields in 1861, has been modified for a quantum CA model to simulate the electromagnetic fields. Then, a state transition rule for CA is determined by a strategy based on the quantum mechanics and the quantum computation theories. One of originalities of the proposed approach is that a system using quantum gates is superstructed to simulate magneto-static fields from currents. First, a modified Maxwell model for applying to quantum CA was shown. Second, by using the quantum computation theory, a quantum gate system to simulate the magnetic fields from currents was described. Finally, the proposed approach was applied to an example of magnetostatic field simulation.

Geometry of discrete quantum computing

Conventional quantum computing entails a geometry based on the description of an n-qubit state using 2n infinite precision complex numbers denoting a vector in a Hilbert space. Such numbers are in general uncomputable using any real-world resources, and, if we have the idea of physical law as some kind of computational algorithm of the universe, we would be compelled to alter our descriptions of physics to be consistent with computable numbers. Our purpose here is to examine the geometric implications of using finite fields Fp and finite complexified fields (based on primes p congruent to 3 (mod4)) as the basis for computations in a theory of discrete quantum computing, which would therefore become a computable theory. Because the states of a discrete n-qubit system are in principle enumerable, we are able to determine the proportions of entangled and unentangled states. In particular, we extend the Hopf fibration that defines the irreducible state space of conventional continuous n-qubit theories (which is the complex projective space ) to an analogous discrete geometry in which the Hopf circle for any n is found to be a discrete set of p + 1 points. The tally of unit-length n-qubit states is given, and reduced via the generalized Hopf fibration to , the discrete analogue of the complex projective space, which has irreducible states. Using a measure of entanglement, the purity, we explore the entanglement features of discrete quantum states and find that the n-qubit states based on the complexified field have pn(p − 1)n unentangled states (the product of the tally for a single qubit) with purity 1, and they have pn + 1(p − 1)(p + 1)n − 1 maximally entangled states with purity zero.

Quantum Computing Without Wavefunctions: Time-Dependent Density Functional Theory for Universal Quantum Computation

We prove that the theorems of TDDFT can be extended to a class of qubit Hamiltonians that are universal for quantum computation. The theorems of TDDFT applied to universal Hamiltonians imply that single-qubit expectation values can be used as the basic variables in quantum computation and information theory, rather than wavefunctions. From a practical standpoint this opens the possibility of approximating observables of interest in quantum computations directly in terms of single-qubit quantities (i.e. as density functionals). Additionally, we also demonstrate that TDDFT provides an exact prescription for simulating universal Hamiltonians with other universal Hamiltonians that have different, and possibly easier-to-realize two-qubit interactions. This establishes the foundations of TDDFT for quantum computation and opens the possibility of developing density functionals for use in quantum algorithms.

Time complexity and gate complexity

We formulate and investigate the simplest version of time-optimal quantum computation theory (t-QCT), where the computation time is defined by the physical one and the Hamiltonian contains only one- and two-qubit interactions. This version of t-QCT is also considered as optimality by sub-Riemannian geodesic length. The work has two aims: one is to develop a t-QCT itself based on physically natural concept of time, and the other is to pursue the possibility of using t-QCT as a tool to estimate the complexity in conventional gate-optimal quantum computation theory (g-QCT). In particular, we investigate to what extent is true the statement: time complexity is polynomial in the number of qubits if and only if so is gate complexity. In the analysis, we relate t-QCT and optimal control theory (OCT) through fidelity-optimal computation theory (f-QCT); f-QCT is equivalent to t-QCT in the limit of unit optimal fidelity, while it is formally similar to OCT. We then develop an efficient numerical scheme for f-QCT by modifying Krotov's method in OCT, which has monotonic convergence property. We implemented the scheme and obtained solutions of f-QCT and of t-QCT for the quantum Fourier transform and a unitary operator that does not have an apparent symmetry. The former has a polynomial gate complexity and the latter is expected to have exponential one because a series of generic unitary operators has a exponential gate complexity. The time complexity for the former is found to be linear in the number of qubits, which is understood naturally by the existence of an upper bound. The time complexity for the latter is exponential. Thus the both targets are examples satisfyng the statement above. The typical characteristics of the optimal Hamiltonians are symmetry under time-reversal and constancy of one-qubit operation, which are mathematically shown to hold in fairly general situations.

Theory of quantum computation and philosophy of mathematics. Part I

The aim of this paper is to present some basic notions of the theory of quantum computing and to compare them with the basic notions of the classical theory of computation. I am convinced, that the results of quantum computation theory (QCT) are not only interesting in themselves, but also should be taken into account in discussions concerning the nature of mathematical knowledge. The philosophical discussion will however be postponed to another paper. QCT seems not to be well-known among philosophers (at least not to the degree it deserves), so the aim of this paper is to provide the necessary technical preliminaries presented in a way accessible to the general philosophical audience.

Quantum algorithms for algebraic problems

Quantum computers can execute algorithms that dramatically outperform classical computation. As the best-known example, Shor discovered an efficient quantum algorithm for factoring integers, whereas factoring appears to be difficult for classical computers. Understanding what other computational problems can be solved significantly faster using quantum algorithms is one of the major challenges in the theory of quantum computation, and such algorithms motivate the formidable task of building a large-scale quantum computer. This article reviews the current state of quantum algorithms, focusing on algorithms with superpolynomial speedup over classical computation and, in particular, on problems with an algebraic flavor.

Efficient algebraic representation of quantum circuits

A novel and powerful method for representing the unitary matrix of a quantum circuit as an algebraic equation is presented. The method forms a product of sum of Kronecker products (POSOK’s) equation whereby simple mathematical properties can be used to manipulate the algebraic equation to reduce the quantum circuit and prove known circuit equivalences. The method is simple enough to be done by hand or simple automation. The benefits of such a method apply to quantum simulation where the quantum circuit can be easily parsed and reduction techniques can reduce simulation time. Also, having an algebraic expression that can be manipulated through a set of mathematical rules has applications in quantum logic synthesis and quantum computational theory.

Twisted Graph States for Ancilla-driven Universal Quantum Computation

We introduce a new paradigm for quantum computing called Ancilla-Driven Quantum Computation (ADQC) which combines aspects both of the quantum circuit [D. Deutsch. Quantum computational networks. Proc. Roy. Soc. Lond A, 425, 1989] and the one-way model [R. Raussendorf and H. J. Briegel. A one-way quantum computer. Physical Review Letters, 86, 2001] to overcome challenging issues in building large-scale quantum computers. Instead of directly manipulating each qubit to perform universal quantum logic gates or measurements, ADQC uses a fixed two-qubit interaction to couple the memory register of a quantum computer to an ancilla qubit. By measuring the ancilla, the measurement-induced back-action on the system performs the desired logical operations. The underlying mathematical model is based on a new entanglement resource called twisted graph states generated from non-commuting operators, leading to a surprisingly powerful structure for parallel computation compared to graph states obtained from commuting generators. [M. Hein, J. Eisert, and H.J. Briegel. Multi-party entanglement in graph states. Physical Review A, 69, 2004. quant-ph/0307130]. The ADQC model is formalised in an algebraic framework similar to the Measurement Calculus [V. Danos, E. Kashefi, and P. Panangaden. The measurement calculus. Journal of ACM, 2007]. Furthermore, we present the notion of causal flow for twisted graph states, based on the stabiliser formalism, to characterise the determinism. Finally we demonstrate compositional embedding between ADQC and both the one-way and circuit models which will allow us to transfer recently developed theory and toolkits of measurement-based quantum computing and quantum circuit models directly into ADQC.

Restrictions on Transversal Encoded Quantum Gate Sets

Transversal gates play an important role in the theory of fault-tolerant quantum computation due to their simplicity and robustness to noise. By definition, transversal operators do not couple physical subsystems within the same code block. Consequently, such operators do not spread errors within code blocks and are, therefore, fault tolerant. Nonetheless, other methods of ensuring fault tolerance are required, as it is invariably the case that some encoded gates cannot be implemented transversally. This observation has led to a long-standing conjecture that transversal encoded gate sets cannot be universal. Here we show that the ability of a quantum code to detect an arbitrary error on any single physical subsystem is incompatible with the existence of a universal, transversal encoded gate set for the code.

The quantum revolution: a historical perspective

Quantum mechanics is one of the great success stories of modern physics, making sense of the very small just as Einstein's theory of relativity made sense of the very large. But, for most students, the ideas that make quantum mechanics powerful can be confusing and counterintuitive. This volume in the Greenwood Guides to Great Ideas in Science series provides a history of quantum mechanics from the early breakthroughs of Planck and Einstein, at the beginning of the 20th century, to the present frontiers of quantum computing and quantum gravity. The approach is entirely non-technical, and is aimed at the general reader who may not have much mathematical background but who has a strong curiosity about some of the most important developments in modern science. Quantum Mechanics: A Historical Perspective traces the history of this powerful theory, including:; The early discoveries by Max Planck and Albert Einstein regarding the quantization of radiation; The early quantum theory, including Neils Bohr's theory of the atom; The birth of modern quantum mechanics through the work of Heisenberg, Schrodinger, Born, Dirac and others; Applications of quantum mechanics in chemistry, nuclear physics, electronics, and many other areas; Recent work in quantum computation and quantum information theory The book emphasizes the fact that despite the great success of quantum mechanics, many exciting intellectual frontiers remain open for further researchers to explore. It includes a glossary, a timeline, and a bibliography of accessible resources for further research.

Perturbative gadgets at arbitrary orders

Adiabatic quantum algorithms are often most easily formulated using many-body interactions. However, experimentally available interactions are generally two-body. In 2004, Kempe, Kitaev, and Regev introduced perturbative gadgets, by which arbitrary three-body effective interactions can be obtained using Hamiltonians consisting only of two-body interactions. These three-body effective interactions arise from the third order in perturbation theory. Since their introduction, perturbative gadgets have become a standard tool in the theory of quantum computation. Here we construct generalized gadgets so that one can directly obtain arbitrary $\mathit{k}$-body effective interactions from two-body Hamiltonians. These effective interactions arise from the $\mathit{k}$th order in perturbation theory.

Simulation of n-qubit quantum systems. IV. Parametrizations of quantum states, matrices and probability distributions

Entanglement is known today as a key resource in many protocols from quantum computation and quantum information theory. However, despite the successful demonstration of several protocols, such as teleportation or quantum key distribution, there are still many open questions of how entanglement affects the efficiency of quantum algorithms or how it can be protected against noisy environments. The investigation of these and related questions often requires a search or optimization over the set of quantum states and, hence, a parametrization of them and various other objects. To facilitate this kind of studies in quantum information theory, here we present an extension of the Feynman program that was developed during recent years as a toolbox for the simulation and analysis of quantum registers. In particular, we implement parameterizations of hermitian and unitary matrices (of arbitrary order), pure and mixed quantum states as well as separable states. In addition to being a prerequisite for the study of many optimization problems, these parameterizations also provide the necessary basis for heuristic studies which make use of random states, unitary matrices and other objects. Program summary Program title: FEYNMAN Catalogue identifier: ADWE_v4_0 Program summary URL: http://cpc.cs.qub.ac.uk/summaries/ADWE_v4_0.html Program obtainable from: CPC Program Library, Queen's University, Belfast, N. Ireland Licensing provisions: Standard CPC licence, http://cpc.cs.qub.ac.uk/licence/licence.html No. of lines in distributed program, including test data, etc.: 24 231 No. of bytes in distributed program, including test data, etc.: 1 416 085 Distribution format: tar.gz Programming language: Maple 11 Computer: Any computer with Maple software installed Operating system: Any system that supports Maple; program has been tested under Microsoft Windows XP, Linux Classification: 4.15 Does the new version supersede the previous version?: Yes Nature of problem: During the last decades, quantum information science has contributed to our understanding of quantum mechanics and has provided also new and efficient protocols, based on the use of entangled quantum states. To determine the behavior and entanglement of n-qubit quantum registers, symbolic and numerical simulations need to be applied in order to analyze how these quantum information protocols work and which role the entanglement plays hereby. Solution method: Using the computer algebra system Maple, we have developed a set of procedures that support the definition, manipulation and analysis of n-qubit quantum registers. These procedures also help to deal with (unitary) logic gates and (nonunitary) quantum operations that act upon the quantum registers. With the parameterization of various frequently-applied objects, that are implemented in the present version, the program now facilitates a wider range of symbolic and numerical studies. All commands can be used interactively in order to simulate and analyze the evolution of n-qubit quantum systems, both in ideal and noisy quantum circuits. Reasons for new version: In the first version of the FEYNMAN program [1], we implemented the data structures and tools that are necessary to create, manipulate and to analyze the state of quantum registers. Later [2,3], support was added to deal with quantum operations (noisy channels) as an ingredient which is essential for studying the effects of decoherence. With the present extension, we add a number of parametrizations of objects frequently utilized in decoherence and entanglement studies, such that as hermitian and unitary matrices, probability distributions, or various kinds of quantum states. This extension therefore provides the basis, for example, for the optimization of a given function over the set of pure states or the simple generation of random objects. Running time: Most commands that act upon quantum registers with five or less qubits take ⩽10 seconds of processor time on a Pentium 4 processor with ⩾ 2 GHz or newer, and about 5–20 MB of working memory (in addition to the memory for the Maple environment). Especially when working with symbolic expressions, however, the requirements on CPU time and memory critically depend on the size of the quantum registers, owing to the exponential growth of the dimension of the associated Hilbert space. For example, complex (symbolic) noise models, i.e. with several symbolic Kraus operators, result for multi-qubit systems often in very large expressions that dramatically slow down the evaluation of e.g. distance measures or the final-state entropy, etc. In these cases, Maple's assume facility sometimes helps to reduce the complexity of the symbolic expressions, but more often only a numerical evaluation is possible eventually. Since the complexity of the various commands of the FEYNMAN program and the possible usage scenarios can be very different, no general scaling law for CPU time or the memory requirements can be given.

Multiple System-Decomposition Method for Avoiding Quantum Decoherence

Decomposition of a composite system C into different subsystems, A + B or D + E, may help in avoiding decoherence. For example, the environment-induced decoherence for an A+B system need not destroy entanglement present in the D + E system (A + B = C = D + E). This new approach opens some questions also in the foundations of the quantum computation theory that might eventually lead to a new model of quantum computation.

Spin‐Based Optical Quantum Information Processing

AbstractWe shall present a review of semiconductor spin‐based implementation schemes for the realization of quantum information/computation solid‐state devices. After briefly describing the fundamentals of quantum computation theory, we shall introduce and discuss potential implementation schemes based on the spin degrees of freedom in semiconductor nanostructures. More specifically, we shall describe an implementation scheme for quantum information processing in which the spin degrees of freedom of electrons confined to a quantum dot are the computational degrees of freedom, and spins are manipulated/controlled through interaction between trionic states created by interband optical transitions by ultrafast sequences of multicolor laser pulses. We will also review briefly an adiabatic method for operating the two‐qubit gate that avoids the main imperfections present in real quantum dots: exciton decay, hole mixing, and phonon decoherence.