Ancilla-error-transparent swap tests
in circuit quantum electrodynamics
Ondřej Černotík
Department of Optics, Palacký University Olomouc
Department of Condensed Matter Physics, Charles University, 27 May 2021

Ondrej Cernotík (UP Olomouc): Ancilla-error-transparent swap tests in circuit quantum electrodynamics
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Superconducting circuits are the workhorse of quantum
computing.
2
IBM Research
Google

Ondrej Cernotík (UP Olomouc): Ancilla-error-transparent swap tests in circuit quantum electrodynamics
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Superconducting quantum bits are enabled by the
nonlinearity of the Josephson junction.
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A. Blais et al., RMP 93, 025005 (2021)

Ondrej Cernotík (UP Olomouc): Ancilla-error-transparent swap tests in circuit quantum electrodynamics
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Losses and noise can be combatted using quantum error
correction.
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C. K. Andersen et al., Nat. Phys. 16, 875 (2020)
Z. Chen et al., arXiv:2102.06132 (2021)

Ondrej Cernotík (UP Olomouc): Ancilla-error-transparent swap tests in circuit quantum electrodynamics
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Alternatively, qubits can be encoded into harmonic
oscillators.
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A. Joshi et al., QST 6, 033001 (2021)
|0L⟩ =
∑
n
cn |n⟩
|1L⟩ =
∑
n
dn |n⟩
⟨0L |1L⟩ = ⟨0L |a†
a|1L⟩ = 0

Swap gates
Swap tests
Kerr-cat qubits

Kerr-cat qubits

Ondrej Cernotík (UP Olomouc): Ancilla-error-transparent swap tests in circuit quantum electrodynamics
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Coherent states can be used to encode quantum
information.
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a|α⟩ = α|α⟩
a| − α⟩ = − α| − α⟩
|0L⟩ = |α⟩
|1L⟩ = | − α⟩
Protected against bit
fl
ips
Amplitude damping

Ondrej Cernotík (UP Olomouc): Ancilla-error-transparent swap tests in circuit quantum electrodynamics
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Kerr nonlinearity and two-photon driving stabilise a cat
qubit.
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H = − Ka†2
a2
+ ϵ(a2
+ a†2
)
= − K(a†2
− α2
)(a2
− α2
) + Kα4
α = ϵ/K
A. Grimm et al., Nature 584, 205 (2020)
S. Puri et al., PRX 9, 041009 (2019)

Ondrej Cernotík (UP Olomouc): Ancilla-error-transparent swap tests in circuit quantum electrodynamics
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The noise of a Kerr-cat qubit is biased towards phase
fl
ips.
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Photon loss → phase
fl
ips κα2
S. Puri et al., PRX 9, 041009 (2019)
S. Puri et al., Sci. Adv. 6, eaay5901 (2020)
Photon gain → phase
fl
ips
Photon gain → leakage
Photon gain → bit
fl
ips ∼ exp(−α2
)

Swap gates

Ondrej Cernotík (UP Olomouc): Ancilla-error-transparent swap tests in circuit quantum electrodynamics
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Swap gates are important non-Clifford gates for bosonic
qubits.
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H.-K. Lau and A. A. Clerk, PRL 117, 100501 (2016)
Controlled-swap gate |0⟩⟨0| ⊗ I + |1⟩⟨1| ⊗ S
Exponential-swap gate exp(iθS) = cos θ I + i sin θ S
S|ϕ⟩|ψ⟩ = |ψ⟩|ϕ⟩

Ondrej Cernotík (UP Olomouc): Ancilla-error-transparent swap tests in circuit quantum electrodynamics
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Bosonic swap gates have been implemented but with
limited quality.
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Y. Y. Gao et al., Nature 566, 509 (2019)

Ondrej Cernotík (UP Olomouc): Ancilla-error-transparent swap tests in circuit quantum electrodynamics
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A controlled-phase beam splitter can be implemented
with a Kerr-cat ancilla.
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Hint = − ζ(a†
bc†
+ ab†
c)
≃ ∓ ζα(a†
b + ab†
)
|0L⟩⟨0L | ⊗ U†
BS
+ |1L⟩⟨1L | ⊗ UBS
ωa ωb
ωc
−ωa + ωb − ωc

Ondrej Cernotík (UP Olomouc): Ancilla-error-transparent swap tests in circuit quantum electrodynamics
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A controlled beam splitter can be turned into a controlled-
swap gate.
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ωa ωb
−ωa + ωb
+
+|1L⟩⟨1L | ⊗ U2
BS
|0L⟩⟨0L | ⊗ I
ωa ωb
ωc
−ωa + ωb − ωc

Ondrej Cernotík (UP Olomouc): Ancilla-error-transparent swap tests in circuit quantum electrodynamics
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The full system contains additional interactions and
imperfections.
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H = − Kc†2
c2
+ ϵ(c†2
+ c2
) − χ(a†
a + b†
b − N)(c†
c − α2
)
+ζ1(t)(a†
b + b†
a) − ζ2(t)(a†
bc†
+ ab†
c)
+ losses and thermal noise of the Kerr-cat qubit
CPBS BS Free evolution

Ondrej Cernotík (UP Olomouc): Ancilla-error-transparent swap tests in circuit quantum electrodynamics
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The full system contains additional interactions and
imperfections.
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CPBS BS Free evolution

Ondrej Cernotík (UP Olomouc): Ancilla-error-transparent swap tests in circuit quantum electrodynamics
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The controlled-swap gate inherits the Kerr cat's noise bias.
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Z error
non-Z error

Swap tests
| + ⟩ = |0⟩ + |1⟩
X

Ondrej Cernotík (UP Olomouc): Ancilla-error-transparent swap tests in circuit quantum electrodynamics
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A swap test is an important protocol in quantum
information science.
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| + ⟩ = |0⟩ + |1⟩
X
R. Filip, PRA 65, 062320 (2002)
overlap measurement
state puri
fi
cation
purity estimation
entanglement witness
P± =
1
2
(I ± S)
Projection onto (anti-)symmetric space

Ondrej Cernotík (UP Olomouc): Ancilla-error-transparent swap tests in circuit quantum electrodynamics
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Swap test performance suffers from ancilla errors.
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Phase
fl
ips: incorrect result
Solution: repeated swap test
| + ⟩ | − ⟩ | + ⟩
Z
(I + S)ρ(I + S)
ρ
Bit
fl
ips: imperfect swap
X
≠ (I + S)ρ(I + S)
| + ⟩

Ondrej Cernotík (UP Olomouc): Ancilla-error-transparent swap tests in circuit quantum electrodynamics
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Swap tests can be used for precise phase estimation.
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|n⟩
|0⟩
|n⟩|0⟩ − |0⟩|n⟩
e−inφ
|n⟩|0⟩ − |0⟩|n⟩
Δ = p+ − p− = − cos(nφ)

Ondrej Cernotík (UP Olomouc): Ancilla-error-transparent swap tests in circuit quantum electrodynamics
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High sensitivity is possible also with classical input states.
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Input states: |α⟩, | − α⟩ |α⟩, |0⟩

Ondrej Cernotík (UP Olomouc): Ancilla-error-transparent swap tests in circuit quantum electrodynamics
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Kerr-cat qubits open new paths in quantum optics and
quantum information.
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Noise bias Ef
fi
cient quantum gates
| + ⟩ = |0⟩ + |1⟩
X
Low-noise swap tests

Ondrej Cernotík (UP Olomouc): Ancilla-error-transparent swap tests in circuit quantum electrodynamics
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UP Olomouc
Iivari Pietikäinen Radim Filip
Yale Quantum Institute
Shruti Puri Steven M. Girvin
Funding