119 research outputs found
Factoring Safe Semiprimes with a Single Quantum Query
Shor's factoring algorithm (SFA), by its ability to efficiently factor large
numbers, has the potential to undermine contemporary encryption. At its heart
is a process called order finding, which quantum mechanics lets us perform
efficiently. SFA thus consists of a \emph{quantum order finding algorithm}
(QOFA), bookended by classical routines which, given the order, return the
factors. But, with probability up to , these classical routines fail, and
QOFA must be rerun. We modify these routines using elementary results in number
theory, improving the likelihood that they return the factors.
The resulting quantum factoring algorithm is better than SFA at factoring
safe semiprimes, an important class of numbers used in cryptography. With just
one call to QOFA, our algorithm almost always factors safe semiprimes. As well
as a speed-up, improving efficiency gives our algorithm other, practical
advantages: unlike SFA, it does not need a randomly picked input, making it
simpler to construct in the lab; and in the (unlikely) case of failure, the
same circuit can be rerun, without modification.
We consider generalizing this result to other cases, although we do not find
a simple extension, and conclude that SFA is still the best algorithm for
general numbers (non safe semiprimes, in other words). Even so, we present some
simple number theoretic tricks for improving SFA in this case.Comment: v2 : Typo correction and rewriting for improved clarity v3 : Slight
expansion, for improved clarit
Analysis of circuit imperfections in BosonSampling
BosonSampling is a problem where a quantum computer offers a provable speedup
over classical computers. Its main feature is that it can be solved with
current linear optics technology, without the need for a full quantum computer.
In this work, we investigate whether an experimentally realistic BosonSampler
can really solve BosonSampling without any fault-tolerance mechanism. More
precisely, we study how the unavoidable errors linked to an imperfect
calibration of the optical elements affect the final result of the computation.
We show that the fidelity of each optical element must be at least , where refers to the number of single photons in the scheme. Such
a requirement seems to be achievable with state-of-the-art equipment.Comment: 20 pages, 7 figures, v2: new title, to appear in QI
Quantum Computing and Quantum Algorithms
The field of quantum computing and quantum algorithms is studied from the ground up. Qubits and their quantum-mechanical properties are discussed, followed by how they are transformed by quantum gates. From there, quantum algorithms are explored as well as the use of high-level quantum programming languages to implement them. One quantum algorithm is selected to be implemented in the Qiskit quantum programming language. The validity and success of the resulting computation is proven with matrix multiplication of the qubits and quantum gates involved
4-bit Factorization Circuit Composed of Multiplier Units with Superconducting Flux Qubits toward Quantum Annealing
Prime factorization (P = M*N) is considered to be a promising application in
quantum computations. We perform 4-bit factorization in experiments using a
superconducting flux qubit toward quantum annealing. Our proposed method uses a
superconducting quantum circuit implementing a multiplier Hamiltonian, which
provides combinations of M and N as a factorization solution after quantum
annealing when the integer P is initially set. The circuit comprises multiple
multiplier units combined with connection qubits. The key points are a native
implementation of the multiplier Hamiltonian to the superconducting quantum
circuit and its fabrication using a Nb multilayer process with a Josephson
junction dedicated to the qubit. The 4-bit factorization circuit comprises 32
superconducting flux qubits. Our method has superior scalability because the
Hamiltonian is implemented with fewer qubits than in conventional methods using
a chimera graph architecture. We perform experiments at 10 mK to clarify the
validity of interconnections of a multiplier unit using qubits. We demonstrate
experiments at 4.2 K and simulations for the factorization of integers 4, 6,
and 9.Comment: Main text (9 pages, 5 figures) and Appendix (8 pages, 7 figures).
Submitted in IEEE Transactions on Applied Superconductivity (under review
Duality symmetry and the form fields of M-theory
In previous work we derived the topological terms in the M-theory action in
terms of certain characters that we defined. In this paper, we propose the
extention of these characters to include the dual fields. The unified treatment
of the M-theory four-form field strength and its dual leads to several
observations. In particular we elaborate on the possibility of a twisted
cohomology theory with a twist given by degrees greater than three.Comment: 12 pages, modified material on the differentia
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