54 research outputs found
Propensity score models are better when post-calibrated
Theoretical guarantees for causal inference using propensity scores are
partly based on the scores behaving like conditional probabilities. However,
scores between zero and one, especially when outputted by flexible statistical
estimators, do not necessarily behave like probabilities. We perform a
simulation study to assess the error in estimating the average treatment effect
before and after applying a simple and well-established post-processing method
to calibrate the propensity scores. We find that post-calibration reduces the
error in effect estimation for expressive uncalibrated statistical estimators,
and that this improvement is not mediated by better balancing. The larger the
initial lack of calibration, the larger the improvement in effect estimation,
with the effect on already-calibrated estimators being very small. Given the
improvement in effect estimation and that post-calibration is computationally
cheap, we recommend it will be adopted when modelling propensity scores with
expressive models.Comment: 15 pages, 6 figure
The Groverian Measure of Entanglement for Mixed States
The Groverian entanglement measure introduced earlier for pure quantum states
[O. Biham, M.A. Nielsen and T. Osborne, Phys. Rev. A 65, 062312 (2002)] is
generalized to the case of mixed states, in a way that maintains its
operational interpretation. The Groverian measure of a mixed state of n qubits
is obtained by a purification procedure into a pure state of 2n qubits,
followed by an optimization process based on Uhlmann's theorem, before the
resulting state is fed into Grover's search algorithm. The Groverian measure,
expressed in terms of the maximal success probability of the algorithm,
provides an operational measure of entanglement of both pure and mixed quantum
states of multiple qubits. These results may provide further insight into the
role of entanglement in making quantum algorithms powerful.Comment: 6 pages, 2 figure
Entangled Quantum States Generated by Shor's Factoring Algorithm
The intermediate quantum states of multiple qubits, generated during the
operation of Shor's factoring algorithm are analyzed. Their entanglement is
evaluated using the Groverian measure. It is found that the entanglement is
generated during the pre-processing stage of the algorithm and remains nearly
constant during the quantum Fourier transform stage. The entanglement is found
to be correlated with the speedup achieved by the quantum algorithm compared to
classical algorithms.Comment: 7 pages, 4 figures submitted to Phys. Rev.
Algebraic analysis of quantum search with pure and mixed states
An algebraic analysis of Grover's quantum search algorithm is presented for
the case in which the initial state is an arbitrary pure quantum state of n
qubits. This approach reveals the geometrical structure of the quantum search
process, which turns out to be confined to a four-dimensional subspace of the
Hilbert space. This work unifies and generalizes earlier results on the time
evolution of the amplitudes during the quantum search, the optimal number of
iterations and the success probability. Furthermore, it enables a direct
generalization to the case in which the initial state is a mixed state,
providing an exact formula for the success probability.Comment: 13 page
Characterization of pure quantum states of multiple qubits using the Groverian entanglement measure
The Groverian entanglement measure, G(psi), is applied to characterize a
variety of pure quantum states |psi> of multiple qubits. The Groverian measure
is calculated analytically for certain states of high symmetry, while for
arbitrary states it is evaluated using a numerical procedure. In particular, it
is calculated for the class of Greenberger-Horne-Zeilinger states, the W states
as well as for random pure states of n qubits. The entanglement generated by
Grover's algorithm is evaluated by calculating G(psi) for the intermediate
states that are obtained after t Grover iterations, for various initial states
and for different sets of the marked states.Comment: 28 pages, 5 figure
Formation of Multipartite Entanglement Using Random Quantum Gates
The formation of multipartite quantum entanglement by repeated operation of
one and two qubit gates is examined. The resulting entanglement is evaluated
using two measures: the average bipartite entanglement and the Groverian
measure. A comparison is made between two geometries of the quantum register: a
one dimensional chain in which two-qubit gates apply only locally between
nearest neighbors and a non-local geometry in which such gates may apply
between any pair of qubits. More specifically, we use a combination of random
single qubit rotations and a fixed two-qubit gate such as the controlled-phase
gate. It is found that in the non-local geometry the entanglement is generated
at a higher rate. In both geometries, the Groverian measure converges to its
asymptotic value more slowly than the average bipartite entanglement. These
results are expected to have implications on different proposed geometries of
future quantum computers with local and non-local interactions between the
qubits.Comment: 7 pages, 5 figure
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