408 research outputs found
Design of Quantum Circuits for Galois Field Squaring and Exponentiation
This work presents an algorithm to generate depth, quantum gate and qubit
optimized circuits for squaring in the polynomial basis. Further, to
the best of our knowledge the proposed quantum squaring circuit algorithm is
the only work that considers depth as a metric to be optimized. We compared
circuits generated by our proposed algorithm against the state of the art and
determine that they require fewer qubits and offer gates savings that
range from to . Further, existing quantum exponentiation are
based on either modular or integer arithmetic. However, Galois arithmetic is a
useful tool to design resource efficient quantum exponentiation circuit
applicable in quantum cryptanalysis. Therefore, we present the quantum circuit
implementation of Galois field exponentiation based on the proposed quantum
Galois field squaring circuit. We calculated a qubit savings ranging between
to and quantum gate savings ranging between to
compared to identical quantum exponentiation circuit based on existing squaring
circuits.Comment: To appear in conference proceedings of the 2017 IEEE Computer Society
Annual Symposium on VLSI (ISVLSI 2017
Fast Quantum Modular Exponentiation
We present a detailed analysis of the impact on modular exponentiation of
architectural features and possible concurrent gate execution. Various
arithmetic algorithms are evaluated for execution time, potential concurrency,
and space tradeoffs. We find that, to exponentiate an n-bit number, for storage
space 100n (twenty times the minimum 5n), we can execute modular exponentiation
two hundred to seven hundred times faster than optimized versions of the basic
algorithms, depending on architecture, for n=128. Addition on a neighbor-only
architecture is limited to O(n) time when non-neighbor architectures can reach
O(log n), demonstrating that physical characteristics of a computing device
have an important impact on both real-world running time and asymptotic
behavior. Our results will help guide experimental implementations of quantum
algorithms and devices.Comment: to appear in PRA 71(5); RevTeX, 12 pages, 12 figures; v2 revision is
substantial, with new algorithmic variants, much shorter and clearer text,
and revised equation formattin
Remarks on Quantum Modular Exponentiation and Some Experimental Demonstrations of Shor's Algorithm
An efficient quantum modular exponentiation method is indispensible for
Shor's factoring algorithm. But we find that all descriptions presented by
Shor, Nielsen and Chuang, Markov and Saeedi, et al., are flawed. We also remark
that some experimental demonstrations of Shor's algorithm are misleading,
because they violate the necessary condition that the selected number ,
where is the number of qubits used in the first register, must satisfy , where is the large number to be factored.Comment: 12 pages,5 figures. The original version has 6 pages. It did not
point out the reason that some researchers took for granted that quantum
modlar exponentiation is in polynomial time. In the new version, we indicate
the reason and analyze some experimental demonstrations of Shor's algorithm.
Besides, the author Zhenfu Cao is added to the version for his contribution.
arXiv admin note: text overlap with arXiv:1409.735
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