Vol.16
No.1&12,
January 1, 2016
Research Articles:
Complexity of the XY antiferromagnet at fixed
magnetization
(pp00010018)
Andrew
M. Childs, David Gosset, and Zak Webb
We prove that approximating the ground energy of the
antiferromagnetic XY model on a simple graph at fixed magnetization
(given as part of the instance specification) is QMAcomplete. To show
this, we strengthen a previous result by establishing QMAcompleteness
for approximating the ground energy of the BoseHubbard model on simple
graphs. Using a connection between the XY and BoseHubbard models that
we exploited in previous work, this establishes QMAcompleteness of the
XY model.
Singlequery quantum algorithms for symmetric
problems
(pp00190038)
Orest
Bucicovschi, Daniel Copeland, David Meyer, and James Pommersheim
Given a unitary representation of a finite group on a
finitedimensional Hilbert space, we show how to find a state whose
translates under the group are distinguishable with the highest
probability. We apply this to several quantum oracle problems, including
the \GM\ problem, in which the product of an ordered $n$tuple of
group elements is to be determined by querying elements of the tuple.
For any finite group $G$, we give an algorithm to find the product of
two elements of $G$ with a single quantum query with probability
$2/G$. This generalizes Deutsch's Algorithm from $\Zs_2$ to an
arbitrary finite group. We further prove that this algorithm is optimal.
We also introduce the \HCEP, in which the oracle acts by conjugating by
an unknown element of the group. We show that for many groups, including
dihedral and symmetric groups, the unknown element can be determined
with probability $1$ using a single quantum query.
Quantum codes over a class of finite chain ring
(pp00390049)
Mustafa
Sari and Irfan Siap
In this study, we introduce a new Gray map which preserves the
orthogonality from the chain ring ${{F}_2}\left[ u \right]/\left(
{{u^s}} \right)$ to ${F}_2^s$ where $F_2$ is the finite field with two
elements. We also give a condition of the existence for cyclic codes of
odd length containing its dual over the ring $ {{F}_2}\left[ u
\right]/\left( {{u^s}} \right).$ By taking advantage of this Gray map
and the structure of the ring, we obtain two classes of binary quantum
error correcting (QEC) codes and we finally illustrate our results by
presenting some examples with good parameters.
Security analysis of QuantumReadout PUFs in the
case of challengeestimation attacks
(pp00500060)
Boris
Skoric
Quantum Readout (QR) of Physical Unclonable Functions (PUFs) is a
new technique for remotely authenticating objects, which has recently
been demonstrated experimentally. The security is based on basic quantum
information theoretic principles and holds under the assumption that the
adversary cannot clone or physically emulate PUFs. We analyse the
security of QR under a class of attacks called `digital emulation', in
which the adversary first performs state estimation on the challenge and
then bases his response on this estimate. We make use of a result by Bru\ss{}
and Macchiavello to derive an upper bound on the adversary's success
probability as a function of the Hilbert space dimension $K$ and the
photon number~$n$. We prove that QR is unconditionally secure against
digital emulation attacks when the challenges are Fock states. For nonFockstates
we provide a security proof under the condition that the attacker's
measurements commute with the particle number operator.
Quantumenhanced secure delegated classical
computing
(pp00610086)
Vedran
Dunjko, Theodoros Kapourniotis, and Elham Kashefi
We present a family of quantumlyenhanced protocols to achieve
unconditionally secure delegated classical computation where the client
and the server have both their classical and quantum computing capacity
limited. We prove the same task cannot be achieved using only classical
protocols. This extends the work of Anders and Browne on the
computational power of correlations to a security setting. In doing so
we are able to highlight the power of online quantum communication as we
prove the same task could not be achieved using preshared (offline)
quantum correlations.
Efficient approximation of diagonal unitaries over
the Clifford+T basis
(pp00870104)
Jonathan
Welch, Alex Bocharov, and Krysta M. Svore
We present an algorithm for the \emph{approximate} decomposition of
diagonal operators, focusing specifically on decompositions over the
Clifford+$T$ basis, that minimizes the number of phaserotation gates in
the synthesized approximation circuit. The equivalent $T$count of the
synthesized circuit is bounded by $k C_0 \log_2(1/\varepsilon) + E(n,k)$,
where $k$ is the number of distinct phases in the diagonal $n$qubit
unitary, $\varepsilon$ is the desired precision, $C_0$ is a quality
factor of the implementation method ($1<C_0<4$), and $E(n,k)$ is the
total entanglement cost (in $T$ gates). We determine an optimal decision
boundary in $(n,k,\varepsilon)$space where our decomposition algorithm
achieves lower entanglement cost than previous stateoftheart
techniques. Our method outperforms stateoftheart techniques for a
practical range of $\varepsilon$ values and diagonal operators and can
reduce the number of $T$ gates exponentially in $n$ when $k \ll 2^n$.
Gaussian (N, z)generalized YangBaxter operators
(pp01050114)
Eric
Rowell
We find unitary matrix solutions $\tilde{R}(a)$ to the
(multiplicative parameterdependent) $(N,z)$generalized YangBaxter
equation that carry the standard measurement basis to $m$level
$N$partite entangled states that generalize the $2$level bipartite
entangled Bell states. This is achieved by a careful study of solutions
to the YangBaxter equation discovered by Fateev and Zamolodchikov in
1982.
Natural information measures for contextual
probabilistic models
(pp01150133)
Federico
Holik, Angel Plastino, and Manuel Saenz
In this article we provide, from a novel perspective, arguments that
support the idea that, in the wake of Cox' approach to probability
theory, von Neumann's entropy should be the natural one in Quantum
Mechanics. We also generalize the pertinent reasoning to more general
orthomodular lattices, which reveals the structure of a general
nonBoolean information theory.
Quantum arithmetic and numerical analysis using
RepeatUntilSuccess circuits
(pp01340178)
Nathan
Wiebe and Martin Roetteler
We develop a method for approximate synthesis of singlequbit
rotations of the form $e^{i f(\phi_1,\ldots,\phi_k)X}$ that is based on
the RepeatUntilSuccess (RUS) framework for quantum circuit synthesis.
We demonstrate how smooth computable functions $f$ can be synthesized
from two basic primitives. This synthesis approach constitutes a
manifestly quantum form of arithmetic that differs greatly from the
approaches commonly used in quantum algorithms. The key advantage of our
approach is that it requires far fewer qubits than existing approaches:
as a case in point, we show that using as few as $3$ ancilla qubits, one
can obtain RUS circuits for approximate multiplication and reciprocals.
We also analyze the costs of performing multiplication and inversion on
a quantum computer using conventional approaches and find that they can
require too many qubits to execute on a small quantum computer, unlike
our approach.
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