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Ancilla-error-transparent swap tests in circuit quantum electrodynamics

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Poster presented at the 737th WE Heraeus Seminar Advances in Scalable Hardware Platforms for Quantum Computing

Postdoctoral researcher in theoretical physics

- 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.

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