Vol.20
No.1&12,
February 1, 2020
Research Articles:
Some considerations on
quantum computing at sub-atomic scales and its impact in the future of
Moore's law
(pp1-13)
Nicolas
F. Lori, Jos Neves, Alex H. Blin, and Victor Alves
The contemporary development of
Quantum Computers has opened new possibilities for computation
improvements, but the limits of Moore's law validity are starting to
show. We analyze here the possibility that miniaturization will continue
to be the source of Moore's law validity in the near future, and our
conclusion is that miniaturization is no longer a reliable answer for
the future development of computer science, but instead we suggest that
lateralization is the correct approach. By lateralization, we mean the
use of biology as the correct format for the implementation of
ubiquitous computerized systems, a format that might in many
circumstances eschew miniaturization as an overly expensive useless
advantage whereas in other cases miniaturization might play a key role.
Thus, the future of computer science is not towards a miniaturization
that goes from the atom-scale (its present application scale) towards
the nucleus-scale, but rather in developing more integrated circuits at
the micrometer to nanometer scale, so as to better mimic and interact
with biological systems. We analyze some "almost sci-fi" approaches to
the development of better computer systems near the Bekenstein bound
limit, and unsurprisingly they fail to have any realistic feasibility.
Then, we use the difference between the classical vs. quantum version of
the Hammerstein-Clifford theorem to explain why biological systems
eschewed quantum computation to represent the world but have chosen
classical computation instead. Finally, we analyze examples of recent
work which indicate future possibilities of integration between
computers and biological systems. As a corollary of that choice by the
biological systems, we propose that the predicted lateralization-driven
evolution in computer science will not be based in quantum computers,
but rather in classical computers.
Quantum algorithm for
matrix functions by Cauchy's integral formula
(pp14-36)
Souichi
Takahira, Asuka Ohashi, Tomohiro Sogabe, and Tsuyoshi S. Usuda
For matrix
$A$,
vector $\vec{b}$
and function $f$,
the computation of vector $f(A)\vec{b}$
arises in many scientific computing applications. We consider the
problem of obtaining quantum state
$\ket{f}$ corresponding to vector
$f(A)\vec{b}$.
There is a quantum algorithm to compute state
$\ket{f}$
using eigenvalue estimation that uses phase estimation and Hamiltonian
simulation $\e^{\im A t}$.
However, the algorithm based on eigenvalue estimation needs
$\poly(1/\epsilon)$
runtime,
where $\epsilon$
is the desired accuracy of the output state. Moreover, if matrix
$A$ is
not Hermitian,
$\e^{\im A t}$
is not unitary and we cannot run eigenvalue estimation. In this paper,
we propose a quantum algorithm that uses
Cauchy's integral formula and the
trapezoidal rule as an approach that avoids eigenvalue estimation. We
show that the runtime of the
algorithm is
$\poly(\log(1/\epsilon))$ and the
algorithm outputs state $\ket{f}$
even if $A$
is not Hermitian.
Quantum entanglement,
supersymmetry, and the generalized Yang-Baxter equation
(pp37-64)
Pramod
Padmanabhan, Fumihiko Sugino, and Diego Trancanelli
Entangled states, such as the
Bell and GHZ states, are generated from separable states using matrices
known to satisfy the Yang-Baxter equation and its generalization. This
remarkable fact hints at the possibility of using braiding operators as
quantum
entanglers, and is part of a larger
speculated connection between topological and quantum entanglement. We
push the analysis of this connection forward, by showing that
supersymmetry
algebras can be used to construct large families of solutions of the
spectral parameter-dependent generalized Yang-Baxter equation. We
present a number of explicit examples and outline a general algorithm
for arbitrary numbers of
qubits.
The operators we obtain produce, in turn, all the entangled states in a
multi-qubit
system classified by the Stochastic Local Operations and Classical
Communication protocol introduced in quantum information theory.
Quantum period finding
based on the Bernstein-Vazirani algorithm
(pp65-84)
Xuexuan
Hao, Fengrong Zhang, Yongzhuang Wei, and Yong Zhou
Quantum period finding
algorithms have been used to analyze symmetric cryptography. For
instance, the 3-round
Feistel
construction and the Even-Mansour
construction could be broken in polynomial time by using quantum period
finding algorithms. In this paper, we firstly provide a new algorithm
for finding the nonzero period of a
vectorial
function with $O(n)$
quantum queries, which uses the Bernstein-Vazirani
algorithm as one step of the subroutine. Afterwards, we compare our
algorithm with
Simon's
algorithm. In some scenarios, such as the Even-Mansour
construction and the function satisfying
Simon's
promise, etc, our algorithm is more efficient than
Simon's
algorithm with respect to the
tradeoff
between quantum memory and time. On the other hand, we combine our
algorithm with Grover's algorithm for the key-recovery attack on the
FX
construction. Compared with the Grover-Meets-Simon algorithm proposed by
Leander and May at
Asiacrypt
2017, the new algorithm could save the quantum memory.
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