3,156 research outputs found
New Construction of Mutually Unbiased Bases in Square Dimensions
We show that k=w+2 mutually unbiased bases can be constructed in any square
dimension d=s^2 provided that there are w mutually orthogonal Latin squares of
order s. The construction combines the design-theoretic objects (k,s)-nets
(which can be constructed from w mutually orthogonal Latin squares of order s
and vice versa) and generalized Hadamard matrices of size s. Using known lower
bounds on the asymptotic growth of the number of mutually orthogonal Latin
squares (based on number theoretic sieving techniques), we obtain that the
number of mutually unbiased bases in dimensions d=s^2 is greater than s^1/14.8
for all s but finitely many exceptions. Furthermore, our construction gives
more mutually orthogonal bases in many non-prime-power dimensions than the
construction that reduces the problem to prime power dimensions.Comment: 10 page
Remark on multi-particle observables and entangled states with constant complexity
We show that every density matrix of an n-particle system prepared by a
quantum network of constant depth is asymptotically commuting with the
mean-field observables. We introduce certain pairs of hypersurfaces in the
space of density matrices and give lower bounds for the depth of a network
which prepares states lying outside those pairs. The measurement of an
observable which is not asymptotically commuting with the mean-field
observables requires a network of depth in the order of log n, if one demands
the measurement to project the state into the eigenspace of the measured
observable.Comment: 7 pages, Revte
Quantum algorithm for finding periodicities in the spectrum of a black-box Hamiltonian or unitary transformation
Estimating the eigenvalues of a unitary transformation U by standard phase
estimation requires the implementation of controlled-U-gates which are not
available if U is only given as a black box.
We show that a simple trick allows to measure eigenvalues of U\otimes
U^\dagger even in this case. Running the algorithm several times allows
therefore to estimate the autocorrelation function of the density of
eigenstates of U. This can be applied to find periodicities in the energy
spectrum of a quantum system with unknown Hamiltonian if it can be coupled to a
quantum computer.Comment: 3 pages, revte
The 2-local Hamiltonian problem encompasses NP
We show that the NP complete problems MAX CUT and INDEPENDENT SET can be
formulated as the 2-local Hamiltonian problem as defined by Kitaev. He
introduced the quantum complexity class BQNP as the quantum analog of NP, and
showed that the 5-local Hamiltonian problem is BQNP-complete. It is not known
whether the s-local Hamiltonian problem is BQNP-complete for s smaller than 5.
Therefore it is interesting to determine what problems can be reduced to the
s-local Hamiltonian problem. Kitaev showed that 3-SAT can be formulated as a
3-local Hamiltonian problem. We extend his result by showing that 2-locality is
sufficient in order to encompass NP.Comment: 7 page
Polynomial-Time Solution to the Hidden Subgroup Problem for a Class of non-abelian Groups
We present a family of non-abelian groups for which the hidden subgroup
problem can be solved efficiently on a quantum computer.Comment: 16 pages, LaTeX2e, 3 figure
Quasi-order of clocks and synchronism and quantum bounds for copying timing information
The statistical state of any (classical or quantum) system with non-trivial
time evolution can be interpreted as the pointer of a clock. The quality of
such a clock is given by the statistical distinguishability of its states at
different times. If a clock is used as a resource for producing another one the
latter can at most have the quality of the resource. We show that this
principle, formalized by a quasi-order, implies constraints on many physical
processes. Similarly, the degree to which two (quantum or classical) clocks are
synchronized can be formalized by a quasi-order of synchronism.
Copying timing information is restricted by quantum no-cloning and
no-broadcasting theorems since classical clocks can only exist in the limit of
infinite energy. We show this quantitatively by comparing the Fisher timing
information of two output systems to the input's timing information. For
classical signal processing in the quantum regime our results imply that a
signal looses its localization in time if it is amplified and distributed to
many devices.Comment: 13 pages, revtex, 1 figur
A Note on Non-Additive Quantum Codes
A method to combine two quantum error-correcting codes is presented. Even
when starting with additive codes, the resulting code might be non-additive.
Furthermore, the notion of the erasure space is introduced which gives a full
characterisation of the erasure-correcting capabilities of the codes. For the
special case that the two codes are unitary images of each other, the erasure
space and the pure erasure space of the resulting code can be calculated.Comment: 4 pages, RevTeX, no figures, preliminary repor
Quantum noise influencing human behaviour could fake effectiveness of drugs in clinical trials
To test the effectiveness of a drug one can advice two randomly selected
groups of patients to take or not to take it, respectively. It is well-known
that the causal effect cannot be identified if not all patients comply. This
holds even when the non-compliers can be identified afterwards since latent
factors like patient's personality can influence both his decision and his
physical response. However, one can still give bounds on the effectiveness of
the drug depending on the rate of compliance. Remarkably, the proofs of these
bounds given in the literature rely on models that represent all relevant
latent factors (including noise) by hidden classical variables. In strong
analogy to the violation of Bell's inequality, some of these bounds fail if
patient's behavior is influenced by latent quantum processes (e.g. in his
nervous system). Quantum effects could fake an increase of the recovery rate by
about 13% although the drug would hurt as many patients as it would help if
everyone took it. The other bounds are true even in the quantum case.
We do not present any realistic model showing this effect, we only point out
that the physics of decision making could be relevant for the causal
interpretation of every-day life statistical data.Comment: 17 pages, 3 figure
Quantum circuits for single-qubit measurements corresponding to platonic solids
Each platonic solid defines a single-qubit positive operator valued measure
(POVM) by interpreting its vertices as points on the Bloch sphere. We construct
simple circuits for implementing this kind of measurements and other simple
types of symmetric POVMs on one qubit. Each implementation consists of a
discrete Fourier transform and some elementary quantum operations followed by
an orthogonal measurement in the computational basis.Comment: 24 pages, Latex, 15 figure
Quantum BCH Codes
After a brief introduction to both quantum computation and quantum error
correction, we show how to construct quantum error-correcting codes based on
classical BCH codes. With these codes, decoding can exploit additional
information about the position of errors. This error model - the quantum
erasure channel - is discussed. Finally, parameters of quantum BCH codes are
provided
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