Microwave entanglement created using swap tests with biased noise
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The document discusses the creation and verification of microwave entanglement using swap tests, which conditionally project systems onto symmetric or antisymmetric subspaces with the aid of an ancilla qubit. It highlights the role of errors in ancilla qubits and their impact on entanglement measurement, presenting a method to mitigate these errors through biased noise. Additionally, it outlines the experimental realization using a Kerr-cat qubit and includes numerical simulations that address the effects of cross-Kerr interactions and decoherence.
![Microwave entanglement created using swap tests with biased noise
Ondřej Černotík,1,
* Iivari Pietikäinen,1
Shruti Puri,2
S. M. Girvin,2
and Radim Filip1
1
Department of Optics, Palacký University Olomouc, Olomouc, Czechia
2
Yale Quantum Institute, Yale University, New Haven, CT, USA
* ondrej.cernotik@upol.cz
References
[1] R. Filip, PRA 65, 062320 (2002).
[2] A. Grimm et al., Nature 584, 205 (2020).
Acknowledgments
O.Č., I.P., and R.F. are supported by the projects LTAUSA19099
and 8C20002 of MEYS ČR and 20-16577S of GA ČR. S.P. and
S.M.G. are supported by the grant No. ARO W911NF-18-1-0212.
Entanglement manipulation with swap tests
Swap tests allow generation and verification of entanglement by conditionally projecting two
systems onto their symmetric or antisymmetric subspace. These projections are heralded by a
measurement on the ancilla qubit.
Generation of entanglement
Entanglement verification
For the input state
the probability for detecting the state is
The circuit on the left uses the ancilla measurement
to project the two modes onto their symmetric [for
] or anti-symmetric [for
] subspace. It thus conditionally
prepares one of the Bell states
with probability .
We can now define the quantity . Whenever , the projection
of the state onto the anti-symmetric subspace is larger than the projection on the
symmetric subspace. The quantity can therefore act as a witness of singlet-like
entanglement [1].
Swap tests with general pure states
Coherent states
We consider two coherent states, . The witness is shown below for the
choice (a) and (b). The different interference pattern is due to different overlap
between the original and phase-shifted states.
0 /2 3 /2 2
Phase
1.0
0.5
0.0
0.5
1.0
Witness (a)
= 0.5
= 1
= 2
= 5
0 /2 3 /2 2
Phase
(b)
Ancilla errors
Errors of the ancilla qubit reduce the quality of
the observed interference. Phase flips (with
probability ) reduce the contrast as shown
here for . For each qubit, the contrast is
reduced by a factor of . Bit flips also
introduce phase errors; using ancillas with
noise biased towards phase flips thus
eliminates one type of error.
0 /8
Phase
1.0
0.5
0.0
0.5
1.0
Witness
p = 0
p = 0.05
p = 0.1
0 /8
Phase
The circuit above combines the preparation and verification of entanglement in an
interferometric scheme. For two general pure states with overlap , the first ancilla
measurement conditionally creates the singlet-like state
with probability .
After the phase shift , we can evaluate the statistics of the second ancilla measurement. We
obtain the witness
where For the Fock states (which are
orthogonal, ), we obtain the witness . For more general states, however, we
get more complex interference patterns that depend on the overlap between the input states
via the functions
Experimental realization in 3D circuit QED
Noise bias can be achieved with a Kerr-cat qubit [2]. A SNAIL device, serving as the Kerr cat
mediates beam-splitter coupling between two microwave cavities as described by the
Hamiltonian
In the mean-field approximation, , this Hamiltonian describes a beam splitter
between the two cavity fields with a phase controlled by the state of the Kerr cat,
(see poster by Iivari Pietikäinen for details). The scheme can then be implemented with two 50:50
controlled-phase beam splitters (CPBS) and modified coherent states, ,
, where is the beam-splitter phase.
Numerical simulations
In full numerical simulations, we also include the effect of cross-Kerr interaction between the
cavity fields and the SNAIL cat as well as decoherence of the SNAIL cat. The total Hamiltonian is
where the last term describes the cross-Kerr coupling. We subtract the mean-field contribution
from the cavity fields, , and from the cat,
Phase flips of the SNAIL cat are the dominant error and the above analysis holds. The second
largest imperfection stems from the cross-Kerr which introduces additional dephasing of the cat
and the cavity fields.](https://image.slidesharecdn.com/cernotik-210126115621/85/Microwave-entanglement-created-using-swap-tests-with-biased-noise-1-320.jpg)
Microwave entanglement created using swap tests with biased noise
- 1. Microwave entanglement created using swap tests with biased noise Ondřej Černotík,1, * Iivari Pietikäinen,1 Shruti Puri,2 S. M. Girvin,2 and Radim Filip1 1 Department of Optics, Palacký University Olomouc, Olomouc, Czechia 2 Yale Quantum Institute, Yale University, New Haven, CT, USA * ondrej.cernotik@upol.cz References [1] R. Filip, PRA 65, 062320 (2002). [2] A. Grimm et al., Nature 584, 205 (2020). Acknowledgments O.Č., I.P., and R.F. are supported by the projects LTAUSA19099 and 8C20002 of MEYS ČR and 20-16577S of GA ČR. S.P. and S.M.G. are supported by the grant No. ARO W911NF-18-1-0212. Entanglement manipulation with swap tests Swap tests allow generation and verification of entanglement by conditionally projecting two systems onto their symmetric or antisymmetric subspace. These projections are heralded by a measurement on the ancilla qubit. Generation of entanglement Entanglement verification For the input state the probability for detecting the state is The circuit on the left uses the ancilla measurement to project the two modes onto their symmetric [for ] or anti-symmetric [for ] subspace. It thus conditionally prepares one of the Bell states with probability . We can now define the quantity . Whenever , the projection of the state onto the anti-symmetric subspace is larger than the projection on the symmetric subspace. The quantity can therefore act as a witness of singlet-like entanglement [1]. Swap tests with general pure states Coherent states We consider two coherent states, . The witness is shown below for the choice (a) and (b). The different interference pattern is due to different overlap between the original and phase-shifted states. 0 /2 3 /2 2 Phase 1.0 0.5 0.0 0.5 1.0 Witness (a) = 0.5 = 1 = 2 = 5 0 /2 3 /2 2 Phase (b) Ancilla errors Errors of the ancilla qubit reduce the quality of the observed interference. Phase flips (with probability ) reduce the contrast as shown here for . For each qubit, the contrast is reduced by a factor of . Bit flips also introduce phase errors; using ancillas with noise biased towards phase flips thus eliminates one type of error. 0 /8 Phase 1.0 0.5 0.0 0.5 1.0 Witness p = 0 p = 0.05 p = 0.1 0 /8 Phase The circuit above combines the preparation and verification of entanglement in an interferometric scheme. For two general pure states with overlap , the first ancilla measurement conditionally creates the singlet-like state with probability . After the phase shift , we can evaluate the statistics of the second ancilla measurement. We obtain the witness where For the Fock states (which are orthogonal, ), we obtain the witness . For more general states, however, we get more complex interference patterns that depend on the overlap between the input states via the functions Experimental realization in 3D circuit QED Noise bias can be achieved with a Kerr-cat qubit [2]. A SNAIL device, serving as the Kerr cat mediates beam-splitter coupling between two microwave cavities as described by the Hamiltonian In the mean-field approximation, , this Hamiltonian describes a beam splitter between the two cavity fields with a phase controlled by the state of the Kerr cat, (see poster by Iivari Pietikäinen for details). The scheme can then be implemented with two 50:50 controlled-phase beam splitters (CPBS) and modified coherent states, , , where is the beam-splitter phase. Numerical simulations In full numerical simulations, we also include the effect of cross-Kerr interaction between the cavity fields and the SNAIL cat as well as decoherence of the SNAIL cat. The total Hamiltonian is where the last term describes the cross-Kerr coupling. We subtract the mean-field contribution from the cavity fields, , and from the cat, Phase flips of the SNAIL cat are the dominant error and the above analysis holds. The second largest imperfection stems from the cross-Kerr which introduces additional dephasing of the cat and the cavity fields.