Vol.13
No.5&6, May 1, 2013
Research Articles:
The robustness of magic state distillation against
errors in Clifford gates (pp03610378)
Tomas
JochymO'Connor, Yafei Yu, Bassam Helou, and Raymond Laflamme
Quantum error correction and faulttolerance have provided the
possibility for large scale quantum computations without a detrimental
loss of quantum information. A very natural class of gates for
faulttolerant quantum computation is the Clifford gate set and as such
their usefulness for universal quantum computation is of great interest.
Clifford group gates augmented by magic state preparation give the
possibility of simulating universal quantum computation. However,
experimentally one cannot expect to perfectly prepare magic states.
Nonetheless, it has been shown that by repeatedly applying operations
from the Clifford group and measurements in the Pauli basis, the
fidelity of noisy prepared magic states can be increased arbitrarily
close to a pure magic state~\cite{Bravyi}. We investigate the robustness
of magic state distillation to perturbations of the initial states to
arbitrary locations in the Bloch sphere due to noise. Additionally, we
consider a depolarizing noise model on the quantum gates in the decoding
section of the distillation protocol and demonstrate its effect on the
convergence rate and threshold value. Finally, we establish that faulty
magic state distillation is more efficient than faulttoleranceassisted
magic state distillation at low error rates due to the large overhead in
the number of quantum gates and qubits required in a faulttolerance
architecture. The ability to perform magic state distillation with noisy
gates leads us to conclude that this could be a realistic scheme for
future smallscale quantum computing devices as faulttolerance need
only be used in the final steps of the protocol.
Entanglement and output entropy of the diagonal
map
(pp03790392)
Meik Hellmund
We review some properties of the convex roof extension, a
construction used, e.g., in the definition of the entanglement of
formation. Especially we consider the use of symmetries of channels and
states for the construction of the convex roof. As an application we
study the entanglement entropy of the diagonal map for permutation
symmetric real $N=3$ states $\omega(z)$ and solve the case $z<0$ where
$z$ is the nondiagonal entry in the density matrix. We also report a
surprising result about the behavior of the output entropy of the
diagonal map for arbitrary dimensions $N$; showing a bifurcation at
$N=6$.
Trivial low energy states for commuting
Hamiltonians, and the quantum PCP conjecture
(pp03930429)
Matthew
Hastings
We consider the entanglement properties of ground states of Hamiltonians
which are sums of commuting projectors (we call these commuting
projector Hamiltonians), in particular whether or not they have
``trivial" ground states, where a state is trivial if it is constructed
by a local quantum circuit of bounded depth and range acting on a
product state. It is known that Hamiltonians such as the toric code only
have nontrivial ground states in two dimensions. Conversely, commuting
projector Hamiltonians which are sums of twobody interactions have
trivial ground states\cite{bv}. Using a coarsegraining procedure, this
implies that any such Hamiltonian with bounded range interactions in one
dimension has a trivial ground state. In this paper, we further explore
the question of which Hamiltonians have trivial ground states. We define
an ``interaction complex" for a Hamiltonian, which generalizes the
notion of interaction graph and we show that if the interaction complex
can be continuously mapped to a $1$complex using a map with bounded
diameter of preimages then the Hamiltonian has a trivial ground state
assuming one technical condition on the Hamiltonians holds (this
condition holds for all stabilizer Hamiltonians, and we additionally
prove the result for all Hamiltonians under one assumption on the
$1$complex). While this includes the cases considered by Ref.~\onlinecite{bv},
we show that it also includes a larger class of Hamiltonians whose
interaction complexes cannot be coarsegrained into the case of Ref.~\onlinecite{bv}
but still can be mapped continuously to a $1$complex. One motivation
for this study is an approach to the quantum PCP conjecture. We note
that many commonly studied interaction complexes can be mapped to a
$1$complex after removing a small fraction of sites. For commuting
projector Hamiltonians on such complexes, in order to find low energy
trivial states for the original Hamiltonian, it would suffice to find
trivial ground states for the Hamiltonian with those sites removed. Such
trivial states can act as a classical witness to the existence of a low
energy state. While this result applies for commuting Hamiltonians and
does not necessarily apply to other Hamiltonians, it suggests that to
prove a quantum PCP conjecture for commuting Hamiltonians, it is worth
investigating interaction complexes which cannot be mapped to
$1$complexes after removing a small fraction of points. We define this
more precisely below; in some sense this generalizes the notion of an
expander graph. Surprisingly, such complexes do exist as will be shown
elsewhere\cite{fh}, and have useful properties in quantum coding theory.
Realization of the probability laws in the quantum
central limit theorems by a quantum walk
(pp04300438)
Takuya Machida
Since a limit distribution of a discretetime quantum walk on the
line was derived in 2002, a lot of limit theorems for quantum walks with
a localized initial state have been reported. On the other hand, in
quantum probability theory, there are four notions of independence
(free, monotone, commuting, and boolean independence) and quantum
central limit theorems associated to each independence have been
investigated. The relation between quantum walks and quantum probability
theory is still unknown. As random walks are fundamental models in the
Kolmogorov probability theory, can the quantum walks play an important
role in quantum probability theory? To discuss this problem, we focus on
a discretetime 2state quantum walk with a nonlocalized initial state
and present a limit theorem. By using our limit theorem, we generate
probability laws in the quantum central limit theorems from the quantum
walk.
Faulttolerant quantum error correction code
conversion
(pp04390451)
Charles D.
Hill, Austin G. Fowler, David S. Wang, and Lloyd C.L. Hollenberg
In this paper we demonstrate how data encoded in a fivequbit quantum
error correction code can be converted, faulttolerantly, into a sevenqubit
Steane code. This is achieved by progressing through a series of codes,
each of which faulttolerantly corrects at least one error. Throughout
the conversion the encoded qubit remains protected. We found, through
computational search, that the method used to convert between codes
given in this paper is optimal.
Quantum discord and quantum phase transition in
spin1/2 frustrated Heisenberg chain
(pp04520468)
ChuHui Fan,
HengNa Xiong, Yixiao Huang, and Zhe Sun
By using the concept of the quantum discord (QD), we study the
spin1/2 antiferromagnetic Heisenberg chain with nextnearestneighbor
interaction. Due to the $SU(2)$ symmetry and $Z_{2}$ symmetry in this
system, we obtain the analytical result of the QD and its geometric
measure (GMQD), which is determined by the twosite correlators. For the
4site and 6site cases, the connection between GMQD (QD) and the
eigenenergies was revealed. From the analytical and numerical results,
we find GMQD (QD) is an effective tool in detecting the both the
firstorder and the infiniteorder quantumphasetransition points for
the finitesize systems. Moreover, by using the entanglement excitation
energy and a universal frustration measure we consider the frustration
properties of the system and find a nonlinear dependence of the GMQD on
the frustration.
Nonmonogamy of quantum discord and upper bounds
for quantum correlation
(pp04690478)
XiJun Ren and
Heng Fan
We consider a monogamy inequality of quantum discord in a pure
tripartite state and show that it is equivalent to an inequality between
quantum mutual information and entanglement of formation of two parties.
Since this inequality does not hold for arbitrary bipartite states,
quantum discord can generally be both monogamous and polygamous. We also
carry out numerical calculations for some special states. The upper
bounds of quantum discord and classical correlation are also discussed
and we give physical analysis on the invalidness of a previous
conjectured upper bound of quantum correlation. Our results provide new
insights for further understanding of distributions of quantum
correlations.
Measurementinduced nonlocality for an arbitrary
bipartite state
(pp04790489)
Sayyed Y.
Mirafzali, Iman Sargolzahi, Ali Ahanj, Kurosh Javidan, and Mohsen
Sarbishaei
Measurementinduced nonlocality is a measure of nonlocalty introduced by
Luo and Fu [Phys. Rev. Lett \textbf{106}, 120401 (2011)]. In this paper,
we study the problem of evaluation of Measurementinduced nonlocality
(MIN) for an arbitrary $m\times n$ dimensional bipartite density matrix
$\rho$ for the case where one of its reduced density matrix, $\rho^{a}$,
is degenerate (the nondegenerate case was explained in the preceding
reference). Suppose that, in general, $\rho^{a}$ has $d$ degenerate
subspaces with dimension $m_{i} (m_{i} \leq m , i=1, 2, ..., d)$. We
show that according to the degeneracy of $\rho^{a}$, if we expand $\rho$
in a suitable basis, the evaluation of MIN for an $m\times n$
dimensional state $\rho$, is degraded to finding the MIN in the $m_{i}\times
n$ dimensional subspaces of state $\rho$. This method can reduce the
calculations in the evaluation of MIN. Moreover, for an arbitrary
$m\times n$ state $\rho$ for which $m_{i}\leq 2$, our method leads to
the exact value of the MIN. Also, we obtain an upper bound for MIN which
can improve the ones introduced in the above mentioned reference.
Finally, we explain the evaluation of MIN for $3\times n$ dimensional
states in details.
Optimal BaconShor codes
(pp04900510)
John Napp and
John Preskill
We study the performance of BaconShor codes, quantum subsystem codes
which are well suited for applications to faulttolerant quantum memory
because the error syndrome can be extracted by performing twoqubit
measurements. Assuming independent noise, we find the optimal block size
in terms of the bitflip error probability $p_X$ and the phase error
probability $p_Z$, and determine how the probability of a logical error
depends on $p_X$ and $p_Z$. We show that a single BaconShor code block,
used by itself without concatenation, can provide very effective
protection against logical errors if the noise is highly biased ($p_Z/p_X\gg
1)$ and the physical error rate $p_Z$ is a few percent or below. We also
derive an upper bound on the logical error rate for the case where the
syndrome data is noisy.
Perfect state transfer on signed
graphs
(pp05110530)
John Brown,
Chris Godsil, Devlin Mallory, Abigail Raz, and Christino Tamon
We study perfect state transfer of quantum walks on signed graphs.
Our aim is to show that negative edges are useful for perfect state
transfer. First, we show that the signed join of a negative $2$clique
with any positive $(n,3)$regular graph has perfect state transfer even
if the unsigned join does not. Curiously, the perfect state transfer
time improves as $n$ increases. Next, we prove that a signed complete
graph has perfect state transfer if its positive subgraph is a regular
graph with perfect state transfer and its negative subgraph is periodic.
This shows that signing is useful for creating perfect state transfer
since no complete graph (except for the $2$clique) has perfect state
transfer. Also, we show that the doublecover of a signed graph has
perfect state transfer if the positive subgraph has perfect state
transfer and the negative subgraph is periodic.Here, signing is useful
for constructing unsigned graphs with perfect state transfer. Finally,
we study perfect state transfer on a family of signed graphs called the
exterior powers which is derived from a manyfermion quantum walk on
graphs.
Entanglement transfer between
atomic qubits and thermal fields
(pp05310540)
ShiBiao Zheng
and RongXin Chen
We investigate entanglement reciprocation between atomic qubits and
cavity fields initially in a thermal state. We show that the
entanglement between the atomic qubits can be fully transferred to the
mixed fields through displacement operation and resonant atomcavity
interaction. This is a rare
example, in which quantum systems in mixed states can be used as the
memory for entanglement. The entanglement can be retrieved by another
atomic pair. Apart from fundamental interest, the results are useful for
implementation of quantum networking with atomfield interface in the
microwave regime.
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