Efficient measure of scalability

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Efficient measure of scalability
( through fidelity decay )
Cecilia López, Benjamin Lévi, Joseph Emerson,
David Cory Department of Nuclear Science
Engineering, Massachusetts Institute of
Technology
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Definitions
Identifying errors through fidelity decay
Target
We must fight against errors. We need to identify
errors.
?
Control of the system
? Quantum process tomography
Inefficient!
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Definitions
Using randomness to explore the Hilbert space
We use a random operator as the evolution
operator U
is a random rotation that spans U(2)
with ??, ?, ? drawn randomly.
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Definitions
Using randomness to explore the Hilbert space
We use a random operator as the evolution
operator U
is a random rotation that spans U(2)
with ??, ?, ? drawn randomly.
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Type of errors
Type of errors how constant is E?
Type of errors how are the non-null coefficients
in H? ?
? Uniform All the qubits perceive the same
error ?j ?, ?j,k ?
? Gaussian The qubits react independently the
?j, ?j,k are drawn from a Gaussian distribution
with center ?, ? and dispersion ??, ??
respectively.
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General results
General results
?? The decay is essentially exponential
Numerically
?
?At long times, the state is completely
randomized
?
We can fit
?
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General results
General results
?? The decay is essentially exponential
Numerically
8
General results
General results
?? The decay is essentially exponential
Numerically
?
?At long times, the state is completely
randomized
?
We can fit
?
Analytically
Confirmed by expressions for H? with one-qubit
terms only.
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General results
The initial decay rate ??
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The initial decay rate ?? ? ? Locality of
errors
Promising!
Inefficient!
Hard to engineer!
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For instance
Advantages ? Initial state preparation is less
critical ? Less measurements
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General results
Conclusions
? The decay is essentially exponential ? The
fidelity decay rate is related to type and
strength of the noise ? The initial decay rate ?
is independent of the type of errors ? ? can
be used to address the question of the locality
of errors ? The locality of errors is key to
determine whether we need non-local gates to
correct them the need of non-local gates would
imply the lack of scalability of that particular
system.
(analytically for one-qubit terms, numerically
including two-qubit terms)
? We are working on the experimental
implementation of this scheme in liquid NMR, with
a 4-qubit molecule.
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References
Questions?
On the fidelity as a useful tool J. Emerson et
al., quant-ph/0503243 (2005) C. A. Ryan et al.,
quant-ph/0506085 (2005) On the mathematical
background for our calculations P. W. Brouwer
and C. W. J. Beenakker, J. Math. Phys. 37, 4904
(1996) P. A. Mello, J. Phys. A 23, 4061 (1990) S.
Samuel, J. Math. Phys. 21, 2695 (1980)
J. Emerson et al., PRL 89, 284102 (2002) D.
Poulin et al., PRA 68, 022302 (2003)