Related papers: An algorithm for dividing quaternions
The computation of generalized inverses of quaternion matrices is a fundamental problem in quaternion linear algebra, with wide-ranging applications in signal processing, image restoration, and multidimensional data analysis. This paper…
This paper presents arithmetic operations like addition, subtraction and multiplications in Modulo-4 arithmetic, and also addition, multiplication in Galois field, using multi-valued logic (MVL). Quaternary to binary and binary to…
The Quantum Fourier Transformation (QFT) is a well-known subroutine for algorithms on qubit-based universal quantum computers. In this work, the known QFT circuit is used to derive an efficient circuit for the multidimensional QFT. The…
Two algorithms for computing $P(n,m)$, the number of integer partitions of $n$ into exactly $m$ parts, are described, and using a combination of these two algorithms, the resulting algorithm is $O(n^{3/2})$. The second algorithm uses a list…
This paper shows that it is possible to improve the computational cost, the memory requirements and the accuracy of Quick Fourier Transform (QFT) algorithm for power-of-two FFT (Fast Fourier Transform) just introducing a slight modification…
We study the bit complexity of two methods, related to the Euclidean algorithm, for computing cubic and quartic analogs of the Jacobi symbol. The main bottleneck in such procedures is computation of a quotient for long division. We give…
We investigate factorizability of a quadratic split quaternion polynomial. In addition to inequality conditions for existence of such factorization, we provide lucid geometric interpretations in the projective space over the split…
Quasiprobabilistic cutting techniques allow us to partition large quantum circuits into smaller subcircuits by replacing non-local gates with probabilistic mixtures of local gates. The cost of this method is a sampling overhead that scales…
A new method for computing sums on a quantum computer is introduced. This technique uses the quantum Fourier transform and reduces the number of qubits necessary for addition by removing the need for temporary carry bits. This approach also…
The two currently fastest general-purpose integer factorization algorithms are the Quadratic Sieve and the Number Field Sieve. Both techniques are used to find so-called smooth values of certain polynomials, i.e., values that factor…
Quantum computing not only holds the potential to solve long-standing problems in quantum physics, but also to offer speed-ups across a broad spectrum of other fields. However, due to the noise and the limited scale of current quantum…
To obtain accurate results in numerical computation, high-precision arithmetic is a straightforward approach. However, most processors lack hardware support for floating-point formats beyond double precision (FP64). Double-word arithmetic…
Quantum bits have technological imperfections. Additionally, the capacity of a component that can be implemented feasibly is limited. Therefore, distributed quantum computation is required to scale up quantum computers. This dissertation…
We present a novel right-to-left long division algorithm based on the Montgomery modular multiply, consisting of separate highly efficient loops with simply carry structure for computing first the remainder (x mod q) and then the quotient…
The problem of efficient multiplication of large numbers has been a long-standing challenge in classical computation and has been extensively studied for centuries. It appears that the existing classical algorithms are close to their…
The technique for hardware multiplication based upon Fourier transformation has been introduced. The technique has the highest efficiency on multiplication units with up to 8 bit range. Each multiplication unit is realized on base of the…
The roots of -1 in the set of biquaternions (quaternions with complex components, or complex numbers with quaternion real and imaginary parts) are studied and it is shown that there is an infinite number of non-trivial complexified…
A common starting point of traditional quantum algorithm design is the notion of a universal quantum computer with a scalable number of qubits. This convenient abstraction mirrors classical computations manipulating finite sets of symbols,…
Tensor networks have proven to be a valuable tool, for instance, in the classical simulation of (strongly correlated) quantum systems. As the size of the systems increases, contracting larger tensor networks becomes computationally…
A digital computer is generally believed to be an efficient universal computing device; that is, it is believed able to simulate any physical computing device with an increase in computation time of at most a polynomial factor. This may not…