efficient measure of scalability
DESCRIPTION
Efficient measure of scalability. ( through fidelity decay ). Cecilia L ó pez, Benjamin L é vi, Joseph Emerson, David Cory Department of Nuclear Science & Engineering, Massachusetts Institute of Technology. Other proposals: less information but at a lower cost. Fidelity decay. . . - PowerPoint PPT PresentationTRANSCRIPT
Efficient measure of scalability
Cecilia López, Benjamin Lévi, Joseph Emerson, David Cory
Department of Nuclear Science & Engineering, Massachusetts Institute of Technology
( through fidelity decay )
Definitions
Identifying errors through fidelity decay
Target
Control of the systemWe must fight against errors.We need to identify errors.
Quantum process tomography
Inefficient!
Other proposals: less information but at a lower cost
Fidelity decay
tttt
tt
EUtUEt
UtUt
ttTrtf
:evolution Perturbed
:evolution Perfect
)()1(
)()1(
)]()([)(
)cos()exp()sin()exp(
)sin()exp()cos()exp()()()()(
)()()()()(
jt
jt
jt
jt
jt
jt
jt
jtj
tii
iR
is a random rotation
that spans U(2):
Definitions
Using randomness to explore the Hilbert space
with , , drawn randomly.
)(ktR
We use a random operator as the evolution operator U:
)()1(
)()1(
)()1(
nttt
tttt
tt
RRRwith
ERtREt
RtRt
)cos()exp()sin()exp(
)sin()exp()cos()exp()()()()(
)()()()()(
jt
jt
jt
jt
jt
jt
jt
jtj
tii
iR
is a random rotation
that spans U(2):
Definitions
Using randomness to explore the Hilbert space
with , , drawn randomly.
n
jkj
kz
jzkj
n
j
jzjH
1
)()(,
1
)(
E is the error arising from an imperfect implementation of the Identity operator:
)exp( iHE
with j, j,k small.
)(ktR
We use a random operator as the evolution operator U:
)()1(
)()1(
)()1(
nttt
tttt
tt
RRRwith
ERtREt
RtRt
R
rr tf
Rtftf
1
)(1
)()(
(an ensemble of realizations)
Type of errors
Type of errors: how constant is E?
Coherent: The parameters j, j,k remain constant.
Incoherent: The parameters j, j,k change after certain time – the correlation time.
Long correlation time order of the experiment length
Short correlation time order of the implementation of a gate length
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.
Type of errors: how are the non-null coefficients in H ?
n
jkj
kz
jzkj
n
j
jzjH
1
)()(,
1
)( )exp( iHE
General results
The decay is essentially exponential:
Numerically:
General results
strength error weak, for 1)( 0 ttftf
nNtN
tf 2,1
)( for
We can fit
NNfttf
11)exp()( 0
At long times, the state is completely randomized:
General results
The decay is essentially exponential:
Numerically:
General results
General results
The decay is essentially exponential:
Numerically:
General results
strength error weak, for 1)( 0 ttftf
nNtN
tf 2,1
)( for
We can fit
NNfttf
11)exp()( 0
At long times, the state is completely randomized:
Analytically:
Confirmed by expressions for H with one-qubit terms only.
General resultsThe initial decay rate
22,1
2
2,
2
2,1
2
2
0
3
4
3
1
00002/
isolate
n
jkkj
n
kk
n
jj
I
22,1
2
2,1
21
0
1
3
4
2/2/00
NN
IIn
kk isolate
strength error weak, for 1)( 0 ttftf
n
jkkj
n
jj
1
2,
1
2
0
9
8
3
2
0000
Promising!
Inefficient!
Hard to engineer!
The initial decay rate Locality of errors
tftf :e.g. qubits,few aonly offidelity the measure We )2,1()2,1(0
)2,1( )(
n
kkIP
n
kkPI
n
kkkPP
n
kk
I
I
3
22,1
2,2
22
)2,1(
)2(0
)1(0
3
22,1
2,1
21
)2,1(
)2(0
)1(0
3
22,1
2,2
2,1
22
21
)2,1(
)2(0
)1(0
2
2,1
21
)1(
)1(0
3
1
002/
3
1
2/00
3
4
3
2
0000
3
2
00
][][ ,,3)2,1()2,1()2,1(
2,1)2,1( nTrTrf and with
For instance:
22,1
)2()1()2,1(
9
4 PP
Advantages: Initial state preparation is less critical Less measurements
General results
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.
Conclusions
(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.
References
Questions?
J. Emerson et al., PRL 89, 284102 (2002)D. Poulin et al., PRA 68, 022302 (2003)
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)