Quantum computing aims to outperform classical computers for some specific calculations. It relies on the additional resources given by the qubit superposition states, which can be used to design more efficiently. Reversibility of quantum computation may help in solving NP problems, which are easy in one direction but hard in the opposite sense. Quantum computers can enhance data processing and analysis tasks. The potential impact of quantum algorithms on optimization problems, simulation of quantum systems, and machine learning tasks such as clustering and classification. It also includes efforts to standardize quantum hardware and software interfaces, enabling interoperability and accelerating the adoption of quantum computing. The barriers we face while adopting Quantum computing are the accuracy of the information, lack of technical experts, and organizational interest in adopting the new process. We find one more critical barrier is the lack of standards of secure communication techniques while adopting quantum computing. While numerous challenges lie ahead, the promise of quantum computing in solving complex problems and driving innovation is too significant to ignore. We will examine the existing quantum algorithms and applications that have demonstrated quantum advantage or potential utility in various domains. The quantum computing field is expected to transition from physical qubits to error-corrected logical qubits. AI is likely to be unleashed by quantum computing for formidable progress advances in speed, cost, and algorithm learning function. Quantum computing solves one or two problems, with little practical use, and costs a fortune to run. Additionally, we will assess the challenges associated with scaling up quantum systems, addressing the issues of quantum coherence, error correction, and noise reduction.
Over the last decade, a large number of quantum algorithms have been discovered that outperform their classical counterparts. However, depending on the main techniques used, most of them fall into only three categories. The first category uses quantum Fourier transform to achieve super-polynomial speedup in group-theoretical problems. The second category uses amplitude amplification to achieve polynomial speedup in search problems. The third category simulates quantum many-body systems. Finding a new class of quantum algorithms has proven a challenging task. This dissertation explores several new approaches to developing quantum algorithms. These approaches include span programs, electrical flows and nonsparse Hamiltonian simulation. We demonstrate their power by successfully applying them to some useful problems, including tree detection, effective resistance estimation and curve fitting. All of these algorithms are time-efficient, and some of them are proven to be (nearly) optimal.Span program is a linear-algebraic computation model originally proposed to prove circuit lower bounds. Recently, it is found to be closely related to quantum query complexity. We develop a span-program-based quantum algorithm for the following variant of the tree containment problem. Let
T be a fixed tree. Given the
ntimesn adjacency matrix of a graph
G, we need to decide whether
G contains
T as a subgraph, or
G does not contain
T as a minor, under the promise that one of these cases holds. We call this problem is the subgraph/not-a-minor problem for
T. We show that this problem can be solved by a quantum algorithm with
O(n) query complexity and
tildeO(n) time complexity. This query complexity is optimal, and this time complexity is tight up to poly-logarithmic factors. Electrical network theory has many applications to the design and analysis of classical algorithms. Its connection to quantum computation, however, remains mostly unclear. We give two quantum algorithms for approximating the effective resistance between two given vertices in an electrical network. Both of them have time complexity polynomial in
logn,
d,
c,
1/phi and
1/epsilon, where
n is the number of vertices,
d is the maximum degree of the vertices,
c is the ratio of the largest to the smallest edge resistance,
phi is the conductance of the network, and
epsilon is the relative error. Our algorithms have exponentially better dependence on
n than classical algorithms. Furthermore, we prove that the polynomial dependence on the inverse conductance is necessary. As a consequence, our algorithms cannot be greatly improved. Finally, as a by-product, our second algorithm also produces a quantum state encoding the electrical flow between two given vertices in a network, which might be of independent interest. Our algorithms are developed by using quantum tools to analyze the algebraic properties of graph-related matrices. While the first one relies on inverting the Laplacian matrix, the second one relies on projecting onto the kernel of the weighted incidence matrix. It is hopeful that more quantum algorithms could be designed in similar way. Curve fitting is the process of constructing a (simple continuous) curve that has the best fit to a series of data points. It is a common practice in many fields of science and engineering, because it can help us to understand the relationships among two or more variables, and to infer the values of a function where no data are available. We propose efficient quantum algorithms for estimating the best-fit parameters and the quality of least-square curve fitting. The running times of our algorithms are polynomial in
logn,
d,
kappa,
nu,
chi,
1/Phi and
1/epsilon, where
n is the number of data points to be fitted,
d is the dimension of the feature vectors,
kappa is the condition number of the design matrix,
nu and
chi are some parameters reflecting the variances of the design matrix and response vector,
Phi is the fit quality, and
epsilon is the tolerable error. Our algorithms have exponentially better dependence on
n than classical algorithms. They are designed by combining the techniques of phase estimation and density matrix exponentiation for nonsparse Hamiltonian simulation.
Quantum computers promise to efficiently solve not only problems believed to be intractable for classical computers, but also problems for which verifying the solution is also considered intractable. This raises the question of how one can check whether quantum computers are indeed producing correct results. This task, known as quantum verification , has been highlighted as a significant challenge on the road to scalable quantum computing technology. We review the most significant approaches to quantum verification and compare them in terms of structure, complexity and required resources. We also comment on the use of cryptographic techniques which, for many of the presented protocols, has proven extremely useful in performing verification. Finally, we discuss issues related to fault tolerance, experimental implementations and the outlook for future protocols.
We study quantum algorithms that learn properties of a matrix using queries that return its action on an input vector. We show that for various problems, including computing the trace, determinant, or rank of a matrix or solving a linear system that it specifies, quantum computers do not provide an asymptotic speedup over classical computation. On the other hand, we show that for some problems, such as computing the parities of rows or columns or deciding if there are two identical rows or columns, quantum computers provide exponential speedup. We demonstrate this by showing equivalence between models that provide matrix-vector products, vector-matrix products, and vector-matrix-vector products, whereas the power of these models can vary significantly for classical computation.
Quantum computation is a subject born out of the combination between physics and computer science. It studies how the laws of quantum mechanics can be exploited to perform computations much more efficiently than current computers (termed classical computers as oppose to quantum computers). The thesis starts by introducing ideas from quantum physics and theoretical computer science and based on these ideas, introducing the basic concepts in quantum computing. These introductory discussions are intended for non-specialists to obtain the essential knowledge needed for understanding the new results presented in the subsequent chapters. After introducing the basics of quantum computing, we focus on the recently proposed quantum algorithm for linear systems. The new results include i) special instances of quantum circuits that can be implemented using current experimental resources; ii) detailed quantum algorithms that are suitable for a broader class of linear systems. We show that for some particular problems the quantum algorithm is able to achieve exponential speedup over their classical counterparts.
Quantum computing is a modern new paradigm of computation similarly exploiting the standards of quantum mechanics like superposition, entanglement and quantum interference to finish computations a good way to be unfathomable for classical computers. This paper follows the trails to the theoretical foundations of quantum computing and offers with a few essential algorithms for quantum computing like Shor's set of rules for factorization, Grover's set of rules for search, Quantum Fourier Transform and Quantum Approximate Optimization Algorithm (QAOA). This paper specializes in how complicated the computation is, mathematical ideas underlying such algorithms and their feasible makes use of in cryptography, optimization and simulation in quantum. By sifting thru the interaction among quantum mechanics and algorithmic design, this paintings offers a few perception into the mechanisms that permit for quantum speedup, the cutting-edge obstacles due to hardware obstacles and mistakess charges and what may be carried out withinside the destiny each theoretically and empirically.