457 research outputs found
Gate-Level Simulation of Quantum Circuits
While thousands of experimental physicists and chemists are currently trying
to build scalable quantum computers, it appears that simulation of quantum
computation will be at least as critical as circuit simulation in classical
VLSI design. However, since the work of Richard Feynman in the early 1980s
little progress was made in practical quantum simulation. Most researchers
focused on polynomial-time simulation of restricted types of quantum circuits
that fall short of the full power of quantum computation. Simulating quantum
computing devices and useful quantum algorithms on classical hardware now
requires excessive computational resources, making many important simulation
tasks infeasible. In this work we propose a new technique for gate-level
simulation of quantum circuits which greatly reduces the difficulty and cost of
such simulations. The proposed technique is implemented in a simulation tool
called the Quantum Information Decision Diagram (QuIDD) and evaluated by
simulating Grover's quantum search algorithm. The back-end of our package,
QuIDD Pro, is based on Binary Decision Diagrams, well-known for their ability
to efficiently represent many seemingly intractable combinatorial structures.
This reliance on a well-established area of research allows us to take
advantage of existing software for BDD manipulation and achieve unparalleled
empirical results for quantum simulation
Synthesis and Optimization of Reversible Circuits - A Survey
Reversible logic circuits have been historically motivated by theoretical
research in low-power electronics as well as practical improvement of
bit-manipulation transforms in cryptography and computer graphics. Recently,
reversible circuits have attracted interest as components of quantum
algorithms, as well as in photonic and nano-computing technologies where some
switching devices offer no signal gain. Research in generating reversible logic
distinguishes between circuit synthesis, post-synthesis optimization, and
technology mapping. In this survey, we review algorithmic paradigms ---
search-based, cycle-based, transformation-based, and BDD-based --- as well as
specific algorithms for reversible synthesis, both exact and heuristic. We
conclude the survey by outlining key open challenges in synthesis of reversible
and quantum logic, as well as most common misconceptions.Comment: 34 pages, 15 figures, 2 table
DDMF: An Efficient Decision Diagram Structure for Design Verification of Quantum Circuits under a Practical Restriction
Recently much attention has been paid to quantum circuit design to prepare
for the future "quantum computation era." Like the conventional logic
synthesis, it should be important to verify and analyze the functionalities of
generated quantum circuits. For that purpose, we propose an efficient
verification method for quantum circuits under a practical restriction. Thanks
to the restriction, we can introduce an efficient verification scheme based on
decision diagrams called
Decision Diagrams for Matrix Functions (DDMFs). Then, we show analytically
the advantages of our approach based on DDMFs over the previous verification
techniques. In order to introduce DDMFs, we also introduce new concepts,
quantum functions and matrix functions, which may also be interesting and
useful on their own for designing quantum circuits.Comment: 15 pages, 14 figures, to appear IEICE Trans. Fundamentals, Vol.
E91-A, No.1
Symbolic Quantum Simulation with Quasimodo
The simulation of quantum circuits on classical computers is an important
problem in quantum computing. Such simulation requires representations of
distributions over very large sets of basis vectors, and recent work has used
symbolic data-structures such as Binary Decision Diagrams (BDDs) for this
purpose. In this tool paper, we present Quasimodo, an extensible, open-source
Python library for symbolic simulation of quantum circuits. Quasimodo is
specifically designed for easy extensibility to other backends. Quasimodo
allows simulations of quantum circuits, checking properties of the outputs of
quantum circuits, and debugging quantum circuits. It also allows the user to
choose from among several symbolic data-structures -- both unweighted and
weighted BDDs, and a recent structure called Context-Free-Language Ordered
Binary Decision Diagrams (CFLOBDDs) -- and can be easily extended to support
other symbolic data-structures.Comment: 15 pages; 35th International Conference on Computer Aided
Verification (CAV 2023
Hard Mathematical Problems in Cryptography and Coding Theory
In this thesis, we are concerned with certain interesting computationally hard problems and the complexities of their associated algorithms. All of these problems share a common feature in that they all arise from, or have applications to, cryptography, or the theory of error correcting codes. Each chapter in the thesis is based on a stand-alone paper which attacks a particular hard problem. The problems and the techniques employed in attacking them are described in detail. The first problem concerns integer factorization: given a positive integer . the problem is to find the unique prime factors of . This problem, which was historically of only academic interest to number theorists, has in recent decades assumed a central importance in public-key cryptography. We propose a method for factorizing a given integer using a graph-theoretic algorithm employing Binary Decision Diagrams (BDD). The second problem that we consider is related to the classification of certain naturally arising classes of error correcting codes, called self-dual additive codes over the finite field of four elements, . We address the problem of classifying self-dual additive codes, determining their weight enumerators, and computing their minimum distance. There is a natural relation between self-dual additive codes over and graphs via isotropic systems. Utilizing the properties of the corresponding graphs, and again employing Binary Decision Diagrams (BDD) to compute the weight enumerators, we can obtain a theoretical speed up of the previously developed algorithm for the classification of these codes. The third problem that we investigate deals with one of the central issues in cryptography, which has historical origins in the theory of geometry of numbers, namely the shortest vector problem in lattices. One method which is used both in theory and practice to solve the shortest vector problem is by enumeration algorithms. Lattice enumeration is an exhaustive search whose goal is to find the shortest vector given a lattice basis as input. In our work, we focus on speeding up the lattice enumeration algorithm, and we propose two new ideas to this end. The shortest vector in a lattice can be written as . where are integer coefficients and are the lattice basis vectors. We propose an enumeration algorithm, called hybrid enumeration, which is a greedy approach for computing a short interval of possible integer values for the coefficients of a shortest lattice vector. Second, we provide an algorithm for estimating the signs or of the coefficients of a shortest vector . Both of these algorithms results in a reduction in the number of nodes in the search tree. Finally, the fourth problem that we deal with arises in the arithmetic of the class groups of imaginary quadratic fields. We follow the results of Soleng and Gillibert pertaining to the class numbers of some sequence of imaginary quadratic fields arising in the arithmetic of elliptic and hyperelliptic curves and compute a bound on the effective estimates for the orders of class groups of a family of imaginary quadratic number fields. That is, suppose is a sequence of positive numbers tending to infinity. Given any positive real number . an effective estimate is to find the smallest positive integer depending on such that for all . In other words, given a constant . we find a value such that the order of the ideal class in the ring (provided by the homomorphism in Soleng's paper) is greater than for any . In summary, in this thesis we attack some hard problems in computer science arising from arithmetic, geometry of numbers, and coding theory, which have applications in the mathematical foundations of cryptography and error correcting codes
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