Vol.15
No.1&12,
January 1, 2015
Research Articles:
Improving quantum algorithms
for quantum chemistry
(pp00010021)
Matthew
B. Hastings, Dave Wecker, Bela Bauer, and Matthias Troyer
We present several improvements to the standard TrotterSuzuki based
algorithms used in the simulation of quantum chemistry on a quantum
computer. First, we modify how JordanWigner transformations are
implemented to reduce their cost from linear or logarithmic in the
number of orbitals to a constant. Our modification does not require
additional ancilla qubits. Then, we demonstrate how many operations can
be parallelized, leading to a further linear decrease in the parallel
depth of the circuit, at the cost of a small constant factor increase in
number of qubits required. Thirdly, we modify the term order in the
TrotterSuzuki decomposition, significantly reducing the error at given
TrotterSuzuki timestep. A final improvement modifies the Hamiltonian to
reduce errors introduced by the nonzero TrotterSuzuki timestep. All of
these techniques are validated using numerical simulation and detailed
gate counts are given for realistic molecules.
A model of quantumvon Neumann
hybrid cellular automata: principles and simulation of quantum coherent
superposition and dehoerence in cytoskeletal microtubules
(pp00220036)
Manuel
Alfonseca, Alfonso Ortega, Marina de la Cruz, Stuart R. Hameroff, and
Rafael LahozBeltra
Although experimental evidence suggests the influence of quantum
effects in living organisms, one of the most critical problems in
quantum biology is the explanation of how those effects that take place
in a microscopic level can manifest in the macroscopic world of living
beings. At present, quantum decoherence associated with the wave
function collapse is one of the most accepted mechanisms explaining how
the classical world of living beings emerges from the quantum world.
Whatever the cause of wave function collapse, there exist biological
systems where a biological function arises as a result of this collapse
(e.g. birds navigation, plants photosynthesis, sense of smell, etc.), as
well as the opposite examples (e.g. release of energy from ATP molecules
at actomyosin muscle) where a biological function takes place in a
quantum coherent environment. In this paper we report the modelling and
simulation of quantum coherent superposition in cytoskeletal
microtubules including decoherence, thus the effect of the collapse of
the microtubule coherent state wave function. Our model is based on a
new class of hybrid cellular automata (QvN), capable of performing as
either a quantum cellular automata (QCA) or as a classical von Neumann
automata (CA). These automata are able to simulate the transition or
reduction from a quantum microscopic level with superposition of several
quantum states, to a macroscopic level with a single stable state. Our
results illustrate the significance of quantum biology explaining the
emergence of some biological functions. We believe that in the future
quantum biology will have a deep effect on the design of new devices,
e.g. quantum hardware, in electrical engineering.
Detection loophole attacks on
semideviceindependent quantum and classical protocols
(pp00370049)
Michele
Dall'Arno, Elsa Passaro, Rodrigo Gallego, Marcin Pawlowski, and Antonio
Acin
Semideviceindependent quantum protocols realize information tasks
 e.g. secure key distribution, random access coding, and randomness
generation  in a scenario where no assumption on the internal working
of the devices used in the protocol is made, except their dimension.
These protocols offer two main advantages: first, their implementation
is often less demanding than fullydeviceindependent protocols. Second,
they are more secure than their devicedependent counterparts. Their
classical analogous is represented by random access codes, which provide
a general framework for describing onesided classical communication
tasks. We discuss conditions under which detection inefficiencies can be
exploited by a malicious provider to fake the performance of
semideviceindependent quantum and classical protocols  and how to
prevent it.
A limit theorem for a 3period
timedependent quantum walk
(pp00500060)
F.
Alberto Grunbaum and Takuya Machida
We consider a discretetime 2state quantum walk on the line. The
state of the quantum walker evolves according to a rule which is
determined by a coinflip operator and a positionshift operator.
In this paper we take a 3periodic time evolution as the rule. For such
a quantum walk, we get a limit distribution which expresses the
asymptotic behavior of the walker after a long time. The limit
distribution is different from that of a timeindependent quantum walk
or a 2period timedependent quantum walk. We give some analytical
results and then consider a number of variants of our model and indicate
the result of simulations for these ones.
Group theoretic, Lie algebraic
and Jordan algebraic formulations of the SIC existence problem
(pp00610094)
D. M.
Appleby, Christopher A. Fuchs, and Huangjun Zhu
Although symmetric informationally complete positive operator valued
measures (SIC POVMs, or SICs for short) have been constructed in every
dimension up to 67, a general existence proof remains elusive. The
purpose of this paper is to show that the SIC existence problem is
equivalent to three other, on the face of it quite different problems.
Although it is still not clear whether these reformulations of the
problem will make it more tractable, we believe that the fact that SICs
have these connections to other areas of mathematics is of some
intrinsic interest. Specifically, we reformulate the SIC problem in
terms of (1) Lie groups, (2) Lie algebras and (3) Jordan algebras (the
second result being a greatly strengthened version of one previously
obtained by Appleby, Flammia and Fuchs). The connection between these
three reformulations is nontrivial: It is not easy to demonstrate their
equivalence directly, without appealing to their common equivalence to
SIC existence. In the course of our analysis we obtain a number of other
results which may be of some independent interest.
Controlling the coherence in a
pure dephasing model for arbitrary prescibed time span
(pp00950104)
Lucio
Fassarella
We present an openloop unitary strategy to control the coherence in
a pure dephasing model (related to the phaseflip channel) that is able
to recover, for whatever prescribed time span, the initial coherence at
the end of the control process. The strategy's key idea is to steer the
quantum state to the subset of invariant states and keep it there the
necessary time, using a fine tuned control Hamiltonian.
Timeaveraged limit measure of
the Wojcik model
(pp01050133)
Takako
Endo and Norio Konno
We investigate ``the Wojcik model" introduced and studied by Wojcik
et al. \cite{wojcik}, which is a onedefect quantum walk (QW) having a
single phase at the origin. They reported that giving a phase at one
point causes an astonishing effect for localization. Three types of
measures have important roles in the study of QWs: timeaveraged limit
measure, weak limit measure, and stationary measure. The first two
measures imply a coexistence of localized behavior and the ballistic
spreading in the QW. As Konno et al. \cite{segawa} suggested, the
timeaveraged limit and stationary measures are closely related to each
other for some QWs with one defect. In this paper, we focus on a
relation between the two measures for the Wojcik model. The stationary
measure was already obtained by our previous work \cite{watanabe}. Here,
we get the timeaveraged limit measure by several methods. Our results
show that the stationary measure is a special case of the timeaveraged
limit measure.
Polyestimate: a library for
nearinstantaneous surface code analysis
(pp01340144)
Austin
G. Fowler
The surface code is highly practical, enabling arbitrarily reliable
quantum computation given a 2D nearestneighbor coupled array of qubits
with gate error rates below approximately 1\%. We describe an open
source library, Polyestimate, enabling a user with no knowledge of the
surface code to specify realistic physical quantum gate error models and
obtain logical error rate estimates. Functions allowing the user to
specify simple depolarizing error rates for each gate have also been
included. Every effort has been made to make this library userfriendly.
Polyestimate provides data essentially instantaneously that previously
required hundreds to thousands of hours of simulation, statements which
we discuss and make precise. This advance has been made possible through
careful analysis of the error structure of the surface code and
extensive presimulation.
Minimum weight perfect matching
of faulttolerant topological quantum error correction in average O(1)
parallel time
(pp01450158)
Austin
G. Fowler
Consider a 2D square array of qubits of extent $L\times L$. We
provide a proof that the minimum weight perfect matching problem
associated with running a particular class of topological quantum error
correction codes on this array can be exactly solved with a 2D square
array of classical computing devices, each of which is nominally
associated with a fixed number $N$ of qubits, in constant average time
per round of error detection independent of $L$ provided physical error
rates are below fixed nonzero values, and other physically reasonable
assumptions. This proof is applicable to the fully faulttolerant case
only, not the case of perfect stabilizer measurements.
Efficient Clifford+T
approximation of singlequbit operators
(pp01590180)
Peter
Selinger
We give an efficient randomized algorithm for
approximating an arbitrary element of $\SU(2)$ by a product of
Clifford+$T$ operators, up to any given error threshold $\epsilon>0$.
Under a mild hypothesis on the distribution of primes, the algorithm's
expected runtime is polynomial in $\log(1/\epsilon)$. If the operator to
be approximated is a $z$rotation, the resulting gate sequence has
$T$count $K+4\log_2(1/\epsilon)$, where $K$ is approximately equal to
$10$. We also prove a worstcase lower bound of $K+4\log_2(1/\epsilon)$,
where $K=9$, so that our algorithm is within an additive constant of
optimal for certain $z$rotations. For an arbitrary member of $\SU(2)$,
we achieve approximations with $T$count $K+12\log_2(1/\epsilon)$. By
contrast, the SolovayKitaev algorithm achieves $T$count
$O(\log^c(1/\epsilon))$, where $c$ is approximately $3.97$.
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