189 research outputs found
Entanglement Patterns in Mutually Unbiased Basis Sets for N Prime-state Particles
A few simply-stated rules govern the entanglement patterns that can occur in
mutually unbiased basis sets (MUBs), and constrain the combinations of such
patterns that can coexist (ie, the stoichiometry) in full complements of p^N+1
MUBs. We consider Hilbert spaces of prime power dimension (as realized by
systems of N prime-state particles, or qupits), where full complements are
known to exist, and we assume only that MUBs are eigenbases of generalized
Pauli operators, without using a particular construction. The general rules
include the following: 1) In any MUB, a particular qupit appears either in a
pure state, or totally entangled, and 2) in any full MUB complement, each qupit
is pure in p+1 bases (not necessarily the same ones), and totally entangled in
the remaining p^N-p. It follows that the maximum number of product bases is
p+1, and when this number is realized, all remaining p^N-p bases in the
complement are characterized by the total entanglement of every qupit. This
"standard distribution" is inescapable for two qupits (of any p), where only
product and generalized Bell bases are admissible MUB types. This and the
following results generalize previous results for qubits and qutrits. With
three qupits there are three MUB types, and a number of combinations (p+2) are
possible in full complements. With N=4, there are 6 MUB types for p=2, but new
MUB types become possible with larger p, and these are essential to the
realization of full complements. With this example, we argue that new MUB
types, showing new entanglement characteristics, should enter with every step
in N, and when N is a prime plus 1, also at critical p values, p=N-1. Such MUBs
should play critical roles in filling complements.Comment: 27 pages, one figure, to be submitted to Physical Revie
Unified derivations of measurement-based schemes for quantum computation
We present unified, systematic derivations of schemes in the two known
measurement-based models of quantum computation. The first model (introduced by
Raussendorf and Briegel [Phys. Rev. Lett., 86, 5188 (2001)]) uses a fixed
entangled state, adaptive measurements on single qubits, and feedforward of the
measurement results. The second model (proposed by Nielsen [Phys. Lett. A, 308,
96 (2003)] and further simplified by Leung [Int. J. Quant. Inf., 2, 33 (2004)])
uses adaptive two-qubit measurements that can be applied to arbitrary pairs of
qubits, and feedforward of the measurement results. The underlying principle of
our derivations is a variant of teleportation introduced by Zhou, Leung, and
Chuang [Phys. Rev. A, 62, 052316 (2000)]. Our derivations unify these two
measurement-based models of quantum computation and provide significantly
simpler schemes.Comment: 14 page
On the Impossibility to Extend Triples of Mutually Unbiased Product Bases in Dimension Six
An analytic proof is given which shows that it is impossible to extend any
triple of mutually unbiased (MU) product bases in dimension six by a single MU
vector. Furthermore, the 16 states obtained by removing two orthogonal states
from any MU product triple cannot figure in a (hypothetical) complete set of
seven MU bases. These results follow from exploiting the structure of MU
product bases in a novel fashion, and they are among the strongest ones
obtained for MU bases in dimension six without recourse to computer algebra.Comment: 12 pages, identical to published versio
Universal quantum computation on a semiconductor quantum wire network
Universal quantum computation (UQC) using Majorana fermions on a 2D
topological superconducting (TS) medium remains an outstanding open problem.
This is because the quantum gate set that can be generated by braiding of the
Majorana fermions does not include \emph{any} two-qubit gate and also the
single-qubit phase gate. In principle, it is possible to create these
crucial extra gates using quantum interference of Majorana fermion currents.
However, it is not clear if the motion of the various order parameter defects
(vortices, domain walls, \emph{etc.}), to which the Majorana fermions are bound
in a TS medium, can be quantum coherent. We show that these obstacles can be
overcome using a semiconductor quantum wire network in the vicinity of an
-wave superconductor, by constructing topologically protected two-qubit
gates and any arbitrary single-qubit phase gate in a topologically unprotected
manner, which can be error corrected using magic state distillation. Thus our
strategy, using a judicious combination of topologically protected and
unprotected gate operations, realizes UQC on a quantum wire network with a
remarkably high error threshold of as compared to to
in ordinary unprotected quantum computation.Comment: 7 pages, 2 figure
Computation by measurements: a unifying picture
The ability to perform a universal set of quantum operations based solely on
static resources and measurements presents us with a strikingly novel viewpoint
for thinking about quantum computation and its powers. We consider the two
major models for doing quantum computation by measurements that have hitherto
appeared in the literature and show that they are conceptually closely related
by demonstrating a systematic local mapping between them. This way we
effectively unify the two models, showing that they make use of interchangeable
primitives. With the tools developed for this mapping, we then construct more
resource-effective methods for performing computation within both models and
propose schemes for the construction of arbitrary graph states employing
two-qubit measurements alone.Comment: 13 pages, 18 figures, REVTeX
Information-Disturbance Theorem for Mutually Unbiased Observables
We derive a novel version of information-disturbance theorems for mutually
unbiased observables. We show that the information gain by Eve inevitably makes
the outcomes by Bob in the conjugate basis not only erroneous but random
Threshold Error Penalty for Fault Tolerant Computation with Nearest Neighbour Communication
The error threshold for fault tolerant quantum computation with concatenated
encoding of qubits is penalized by internal communication overhead. Many
quantum computation proposals rely on nearest-neighbour communication, which
requires excess gate operations. For a qubit stripe with a width of L+1
physical qubits implementing L levels of concatenation, we find that the error
threshold of 2.1x10^-5 without any communication burden is reduced to 1.2x10^-7
when gate errors are the dominant source of error. This ~175X penalty in error
threshold translates to an ~13X penalty in the amplitude and timing of gate
operation control pulses.Comment: minor correctio
On fault-tolerance with noisy and slow measurements
It is not so well-known that measurement-free quantum error correction
protocols can be designed to achieve fault-tolerant quantum computing. Despite
the potential advantages of using such protocols in terms of the relaxation of
accuracy, speed and addressing requirements on the measurement process, they
have usually been overlooked because they are expected to yield a very bad
threshold as compared to error correction protocols which use measurements.
Here we show that this is not the case. We design fault-tolerant circuits for
the 9 qubit Bacon-Shor code and find a threshold for gates and preparation of
(30% of the best known result for the
same code using measurement based error correction) while admitting up to 1/3
error rates for measurements and allocating no constraints on measurement
speed. We further show that demanding gate error rates sufficiently below the
threshold one can improve the preparation threshold to .
We also show how these techniques can be adapted to other
Calderbank-Shor-Steane codes.Comment: 11 pages, 7 figures. v3 has an extended exposition and several
simplifications that provide for an improved threshold value and resource
overhea
Constructing Mutually Unbiased Bases in Dimension Six
The density matrix of a qudit may be reconstructed with optimal efficiency if
the expectation values of a specific set of observables are known. In dimension
six, the required observables only exist if it is possible to identify six
mutually unbiased complex 6x6 Hadamard matrices. Prescribing a first Hadamard
matrix, we construct all others mutually unbiased to it, using algebraic
computations performed by a computer program. We repeat this calculation many
times, sampling all known complex Hadamard matrices, and we never find more
than two that are mutually unbiased. This result adds considerable support to
the conjecture that no seven mutually unbiased bases exist in dimension six.Comment: As published version. Added discussion of the impact of numerical
approximations and corrected the number of triples existing for non-affine
families (cf Table 3
A method of enciphering quantum states
In this paper, we propose a method of enciphering quantum states of two-state
systems (qubits) for sending them in secrecy without entangled qubits shared by
two legitimate users (Alice and Bob). This method has the following two
properties. First, even if an eavesdropper (Eve) steals qubits, she can extract
information from them with certain probability at most. Second, Alice and Bob
can confirm that the qubits are transmitted between them correctly by measuring
a signature. If Eve measures m qubits one by one from n enciphered qubits and
sends alternative ones (the Intercept/Resend attack), a probability that Alice
and Bob do not notice Eve's action is equal to (3/4)^m or less. Passwords for
decryption and the signature are given by classical binary strings and they are
disclosed through a public channel. Enciphering classical information by this
method is equivalent to the one-time pad method with distributing a classical
key (random binary string) by the BB84 protocol. If Eve takes away qubits,
Alice and Bob lose the original quantum information. If we apply our method to
a state in iteration, Eve's success probability decreases exponentially. We
cannot examine security against the case that Eve makes an attack with using
entanglement. This remains to be solved in the future.Comment: 21 pages, Latex2e, 10 epsf figures. v2: 22 pages, added two
references, several clarifying sentences are added in Sec. 5, typos
corrected, a new proof is provided in Appendix A and it is shorter than the
old one. v3: 23 pages, one section is adde
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