Vol.14
No.7&8, May
1, 2014
Research Articles:
Generalizing and
derandomizing Gurvits's approximation algorithm for the permanent (pp05410559)
Scott Aaronson and
Travis Hance
Around 2002, Leonid Gurvits gave a striking randomized algorithm to
approximate the permanent of an $n\times n$ matrix A. The
algorithm runs in $O( n^{2}/\varepsilon^{2})$ time, and approximates $\operatorname*{Per}(
A)$ to within $\pm\varepsilon\left\Vert A\right\Vert ^{n}$ additive
error. A major advantage of Gurvits's algorithm is that it works for
arbitrary matrices, not just for nonnegative matrices. \ This makes
it highly relevant to quantum optics, where the permanents of
boundednorm complex matrices play a central role. (In particular,
$n\times n$ permanents arise as the transition amplitudes for $n$
identical photons.) Indeed, the existence of Gurvits's algorithm
is why, in their recent work on the \textit{hardness} of quantum optics,
Aaronson and Arkhipov (AA) had to talk about sampling problems rather
than ordinary decision problems. In this paper, we improve Gurvits's
algorithm in two ways. First, using an idea from quantum optics, we
generalize the algorithm so that it yields a better approximation when
the matrix A has either repeated rows or repeated columns.
Translating back to quantum optics, this lets us classically estimate
the probability of any outcome of an AAtype experiment  even
an outcome involving multiple photons "bunched" in the same mode  at
least as well as that probability can be estimated by the experiment
itself. It also yields a general upper bound on the probabilities of
"bunched" outcomes, which resolves a conjecture of Gurvits and might be
of independent physical interest. Second, we use $\varepsilon$biased
sets to derandomize Gurvits's algorithm, in the special case where the
matrix A is nonnegative. More interestingly, we generalize the
notion of $\varepsilon$biased sets to the complex numbers, construct
"complex $\varepsilon$biased sets", then use those sets to derandomize
even our generalization of Gurvits's algorithm to the case (again for
nonnegative A) where some rows or columns are identical.
Whether Gurvits's algorithm can be derandomized for general A
remains an outstanding problem.
Distillation
protocols for Fourier states in quantum computing
(pp05600576)
Cody Jones
Fourier states are multiqubit registers that facilitate phase
rotations in faulttolerant quantum computing. We propose distillation
protocols for constructing the fundamental, $n$qubit Fourier state with
error $O(2^{n})$ at a cost of $O(n \log n)$ Toffoli gates and Clifford
gates, or any arbitrary Fourier state using $O(n^2)$ gates. We analyze
these protocols with methods from digital signal processing. These
results suggest that phase kickback, which uses Fourier states, could be
the current lowestoverhead method for generating arbitrary phase
rotations.
Quantum
computation of prime number functions
(pp05770588)
Jose I. Latorre
and German Sierra
We propose a quantum circuit that creates a pure state corresponding
to the quantum superposition of all prime numbers less than $2^n$, where
$n$ is the number of qubits of the register. This Prime state can
be built using Grover's algorithm, whose oracle is a quantum
implementation of the classical MillerRabin primality test. The
Prime state is highly entangled, and its entanglement measures
encode number theoretical functions such as the distribution of twin
primes or the Chebyshev bias. This algorithm can be further combined
with the quantum Fourier transform to yield an estimate of the prime
counting function, more efficiently than any classical algorithm and
with an error below the bound that allows for the verification of the
Riemann hypothesis. Arithmetic properties of prime numbers are then, in
principle, amenable to experimental verifications on quantum systems.
Robust
variations of secret sharing through noisy quantum channel
(pp05890607)
XiuBo
Chen, Gang Xu, Yuan Su, and YiXian Yang
In this paper, the perfect secret sharing in quantum cryptography is
investigated. On one hand, the security of a recent protocol [Adhikari
et al. Quantum Inform. \& Comput. 12 (2012) 02530261] is reexamined.
We find that it violates the requirement of information theoretic
security in the secret sharing and suffers from the information leakage.
The cryptanalysis including several specific attack strategies are
given, which shows that a dishonest participant can steal half or all of
the secrets without being detected. On the other hand, we design a new
quantum secret sharing protocol. The security of protocol is rigorously
proved. It meets the fundamental requirement of information theoretic
security. Furthermore, the security analysis including both the outside
attacks and participant attacks is given in details. It is shown that
our proposed protocol can achieve perfect secret sharing.
Uselessness for
an Oracle model with internal randomness
(pp06080624)
Aram W.
Harrow and David J. Rosenbaum
We consider a generalization of the standard oracle model in which
the oracle acts on the target with a permutation selected according to
internal random coins. We describe several problems that are impossible
to solve classically but can be solved by a quantum algorithm using a
single query; we show that such infinityvsone separations between
classical and quantum query complexities can be constructed from much
weaker separations. We also give conditions to determine when oracle
problems  either in the standard model, or in any of the
generalizations we consider 
cannot be solved
with success probability better than random guessing would achieve. In
the oracle model with internal randomness where the goal is to gain any
nonzero advantage over guessing, we prove (roughly speaking) that k
quantum queries are equivalent in power to 2k classical queries,
thus extending results of Meyer and Pommersheim.
Implementation
of a multiqubit phase gate with one qubit simultaneously controlling n
qubits in the iontrap system
(pp06250632)
Xiu Lin
In this paper, we present a scheme for implementing a multiqubit
phase gate with one control qubit simultaneously controlling n target
qubits in the iontrap system. In our scheme, there is no energy
exchange between the internal and external degrees of freedom in the
course of operation, and the vibrational mode is only virtually excited.
The system is insensitive to changes in the vibrational motion. The
proposed scheme is experimentally feasible based on the present iontrap
techniques.
Classical simulation complexity of extended Clifford circuits
(pp06330648)
Richard
Jozsa and Marrten Van den Nest
Clifford gates are a winsome class of quantum operations combining
mathematical elegance with physical significance. The GottesmanKnill
theorem asserts that Clifford computations can be classically
efficiently simulated but this is true only in a suitably restricted
setting. Here we consider Clifford computations with a variety of
additional ingredients: (a) strong vs. weak simulation, (b) inputs being
computational basis states vs. general product states, (c) adaptive vs.
nonadaptive choices of gates for circuits involving intermediate
measurements, (d) single line outputs vs. multiline outputs. We
consider the classical simulation complexity of all combinations of
these ingredients and show that many are not classically efficiently
simulatable (subject to common complexity assumptions such as P not
equal to NP). Our results reveal a surprising proximity of classical to
quantum computing power viz. a class of classically simulatable quantum
circuits which yields universal quantum computation if extended by a
purely classical additional ingredient that does not extend the class of
quantum processes occurring.
Fast quantum
modular exponentiation architecture for Shor's factoring alogrithm
(pp06490682)
Archimedes
Pavlidis and Dimitris Gizopoulos
We present a novel and efficient, in terms of circuit depth, design
for Shor's quantum factorization algorithm. The circuit effectively
utilizes a diverse set of adders based on the Quantum Fourier transform
(QFT) Draper's adders to build more complex arithmetic blocks: quantum
multiplier/accumulators by constants and quantum dividers by constants.
These arithmetic blocks are effectively architected into a quantum
modular multiplier which is the fundamental block for the modular
exponentiation circuit, the most computational intensive part of Shor's
algorithm. The proposed modular exponentiation circuit has a depth of
about $2000n^2$ and requires $9n+2$ qubits, where $n$ is the number of
bits of the classic number to be factored. The total quantum cost of the
proposed design is $1600n^3$. The circuit depth can be further decreased
by more than three times if the approximate QFT implementation of each
adder unit is exploited.
On the geometry
of stabilizer states
(pp06830720)
Hector J.
Garcia, Igor L. Markov, and Andrew W. Cross
Largescale quantum computation is likely to require massive quantum
error correction (QEC). QEC codes and circuits are described via the
stabilizer formalism, which represents stabilizer states by
keeping track of the operators that preserve them. Such states are
obtained by {\em stabilizer circuits} (consisting of CNOT, Hadamard and
Phase gates) and can be represented compactly on conventional computers
using $\Theta (n^2)$ bits, where $n$ is the number of qubits
\cite{Gottes98}. As an additional application, the work in~\cite{Aaron,
AaronGottes} suggests the use of superpositions of stabilizer states to
represent arbitrary quantum states. To aid in such applications and
improve our understanding of stabilizer states, we characterize and
count nearestneighbor stabilizer states, quantify the distribution of
angles between pairs of stabilizer states, study succinct stabilizer
superpositions and stabilizer bivectors, explore the approximation of
nonstabilizer states by single stabilizer states and short linear
combinations of stabilizer states, develop an improved innerproduct
computation for stabilizer states via synthesis of compact canonical
stabilizer circuits, propose an orthogonalization procedure for
stabilizer states, and evaluate several of these algorithms empirically.
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