Vol.12
No.11&12,
November 1, 2012
Research Articles:
Hamiltonian simulation using
linear combinations of unitary operations
(pp09010924)
Andrew M.
Childs and Nathan Wiebe
We present a new approach to simulating Hamiltonian dynamics based on
implementing linear combinations of unitary operations rather than
products of unitary operations. The resulting algorithm has superior
performance to existing simulation algorithms based on product formulas
and, most notably, scales better with the simulation error than any
known Hamiltonian simulation technique. Our main tool is a general
method to nearly deterministically implement linear combinations of
nearby unitary operations, which we show is optimal among a large class
of methods.
Classical
simulation of dissipative fermionic linear optics
(pp09250943)
Sergey Bravyi
and Robert Konig
Fermionic linear optics is a limited form of quantum computation which
is known to be efficiently simulable on a classical computer. We revisit
and extend this result by enlarging the set of available computational
gates: in addition to unitaries and measurements, we allow dissipative
evolution governed by a Markovian master equation with linear Lindblad
operators. We show that this more general form of fermionic computation
is also simulable efficiently by classical means. Given a system of $N$~fermionic
modes, our algorithm simulates any such gate in time $O(N^3)$ while a
singlemode measurement is simulated in time $O(N^2)$. The steady state
of the Lindblad equation can be computed in time $O(N^3)$.
Limitation for
linear maps in a class for detection and quantification of bipartite
nonclassical correlation
(pp09440952)
Akira
SaiToh, Roabeh Rahimi, and Mikio Nakahara
Eigenvaluepreservingbutnotcompletelyeigenvaluepreserving (EnCE)
maps were previously introduced for the purpose of detection and
quantification of nonclassical correlation, employing the
paradigm where nonvanishing quantum discord implies the existence of
nonclassical correlation. It is known that only the matrix transposition
is nontrivial among Hermiticitypreserving (HP) linear EnCE maps when we
use the changes in the eigenvalues of a density matrix due to a partial
map for the purpose. In this paper, we prove that this is true even
among notnecessarily HP (nnHP) linear EnCE maps. The proof utilizes a
conventional theorem on linear preservers. This result imposes a strong
limitation on the linear maps and promotes the importance of nonlinear
maps.
Largescale
multipartite entanglement in the quantum optical frequency comb
(pp09530969)
Reihaneh
Shahrokhshahi and Olivier Pfister
We show theoretically that multipartite entanglement is generated on a
massive scale in the spectrum, or optical frequency comb, of a single
optical parametric oscillator (OPO) emitting well above threshold. In
this system, the quantum dynamics of the strongly depleted pump field
are responsible for the onset of the entanglement by correlating the
twomode squeezed, bipartiteentangled pairs of OPO signal fields. (Such
pairs are independent of one another in the undepleted, classical pump
approximation.) We verify the multipartite nature of the entanglement by
evaluating the van LoockFurusawa criterion for a particular set of
entanglement witnesses deduced from physical considerations.
Lowoverhead
surface code logical Hadamard (pp09700982)
Austin G.
Fowler
We present an improved lowoverhead implementation of surface code
logical Hadamard ($H$). We describe in full detail logical $H$ applied
to a single distance7 doubledefect logical qubit in an otherwise idle
scalable array of such qubits. Our goal is to provide a clear
description of logical $H$ and to emphasize that the surface code
possesses lowoverhead implementations of the entire Clifford group.
Improved bounds
on negativity of superpositions
(pp09830988)
ZhiHao Ma,
ZhiHua Chen, Shuai Han, ShaoMing Fei, and Simone Severini
We consider an alternative formula for the negativity based on a simple
generalization of the concurrence. We use the formula to bound the
amount of entanglement in a superposition of two bipartite pure states
of arbitrary dimension. Various examples indicate that our bounds are
tighter than the previously known results.
Dirac
fourpotential tuningsbased quantum transistor utilizing the Lorentz
force (pp09891010)
Agung
Trisetyarso
We propose a mathematical model of \textit{quantum} transistor in which
bandgap engineering corresponds to the tuning of Dirac potential in the
complex fourvector form. The transistor consists of $n$relativistic
spin qubits moving in \textit{classical} external electromagnetic
fields. It is shown that the tuning of the direction of the external
electromagnetic fields generates perturbation on the potential
temporally and spatially, determining the type of quantum logic gates.
The theory underlying of this scheme is on the proposal of the
intertwining operator for Darboux transfomations on onedimensional
Dirac equation amalgamating the \textit{vectorquantum gates duality} of
Pauli matrices. Simultaneous transformation of qubit and energy can be
accomplished by setting the $\{\textit{control, cyclic}\}$operators
attached on the coupling between onequbit quantum gate: the chose of \textit{cyclic}operator
swaps the qubit and energy simultaneously, while \textit{control}operator
ensures the energy conservation.
Finite geometry behind the
HarveyChryssanthacopoulos fourqubit magic rectangle
(pp10111016)
Metod Saniga
and Michel Planat
A ``magic rectangle" of eleven observables of four qubits, employed by
Harvey and Chryssanthacopoulos (2008) to prove the BellKochenSpecker
theorem in a 16dimensional Hilbert space, is given a neat
finitegeometrical reinterpretation in terms of the structure of the
symplectic polar space $W(7,2)$ of the real fourqubit Pauli group. Each
of the four sets of observables of cardinality five represents an
elliptic quadric in the threedimensional projective space of order two
(PG$(3,2)$) it spans, whereas the remaining set of cardinality four
corresponds to an affine plane of order two. The four ambient PG$(3,
2)$s of the quadrics intersect pairwise in a line, the resulting six
lines meeting in a point. Projecting the whole configuration from this
distinguished point (observable) one gets another, complementary ``magic
rectangle" of the same qualitative structure.
Multicopy
programmable discriminators between two unknown qudit states with
grouptheoretic approach (pp10171033)
Tao Zhou, Jing
Xin Cui, Xiaohua Wu, and Gui Lu Long
The discrimination between two unknown states can be performed by a
universal programmable discriminator, where the copies of the two
possible states are stored in two program systems respectively and the
copies of data, which we want to confirm, are provided in the data
system. In the present paper, we propose a grouptheretic approach to
the multicopy programmable state discrimination problem. By equivalence
of unknown pure states to known mixed states and with the representation
theory of $U(n)$ group, we construct the Jordan basis to derive the
analytical results for both the optimal unambiguous discrimination and
minimumerror discrimination. The POVM operators for unambiguous
discrimination and orthogonal measurement operators for minimumerror
discrimination are obtained. We find that the optimal failure
probability and minimumerror probability for the discrimination between
the mean input mixd states are dependent on the dimension of the unknown
qudit states. We applied the approach to generalize the results of He
and Bergou (2007) from qubit to qudit case, and we further solve the
problem of programmable dicriminators with arbitrary copies of unknown
states in both program and data systems.
Faulttolerant
ancilla preparation and noise threshold lower bounds for the 23qubit
Golay code (pp10341080)
Adam Paetznick
and Ben W. Reichardt
In faulttolerant quantum computing schemes, the overhead is often
dominated by the cost of preparing codewords reliably. This cost
generally increases quadratically with the block size of the underlying
quantum errorcorrecting code. In consequence, large codes that are
otherwise very efficient have found limited faulttolerance
applications. Faulttolerant preparation circuits therefore are an
important target for optimization. We study the
Golay code, a $23$qubit quantum errorcorrecting code that protects the
logical qubit to a distance of seven. In simulations, even using a na{\"i}ve
ancilla preparation procedure, the Golay code is competitive with other
codes both in terms of overhead and the tolerable noise threshold. We
provide two simplified circuits for faulttolerant preparation of Golay
codeencoded ancillas. The new circuits minimize error propagation,
reducing the overhead by roughly a factor of four compared to standard
encoding circuits. By adapting the malignant set counting technique to
depolarizing noise, we further prove a threshold above $\threshOverlap$
noise per gate.
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