Bifluxon Qubit

Figure

Description

The bifluxon qubit is a fluxon-parity-protected superconducting qubit based on Aharonov-Casher interference. Its loop contains a split Cooper-pair box—two Josephson junctions separated by a small island—in series with a superinductor. The logical states occupy the even- and odd-fluxon sectors of that loop.

When the island offset charge is one electron modulo $2e$ ($n_g=1/2$ in Cooper-pair units), the two paths for a single fluxon to cross the split junction interfere destructively. Single-fluxon tunnelling is then forbidden in the symmetric-junction limit, while double-fluxon tunnelling remains allowed. The resulting low-energy dynamics are equivalent to a $4\pi$-periodic, $\cos (\varphi /2)$ Josephson element. Even- and odd-parity logical wavefunctions occupy disjoint sets of wells near $\varphi =2\pi m$, suppressing relaxation matrix elements. Delocalization within each parity sector can additionally reduce flux-noise dephasing.

This is symmetry protection in a two-dimensional circuit Hilbert space, not a literal single-junction $\cos (\varphi /2)$ device. Junction asymmetry restores a residual single-fluxon amplitude, charge offsets must remain near the interference point, and the demonstrated first-generation device did not yet realize full dephasing protection.

Hamiltonian

With the high-frequency common mode eliminated, Kalashnikov et al. write the two-dimensional circuit Hamiltonian in the island-charge basis as

$$H=\sum _n\left[4E_C(n-n_g{)}^2|n⟩⟨n|-E_J\cos \left(\frac{\varphi }{2}\right)({\sigma }_n^{+}+{\sigma }_n^{-})\right]-4E_{CL}{\partial }_{\varphi }^2+\frac{E_L}{2}(\varphi -{\varphi }_{\mathrm{ext}}{)}^2,$$

where $n$ counts Cooper pairs on the island, ${\sigma }_n^{+}=|n+1⟩⟨n|$, $E_C$ is the island charging energy, $E_{CL}$ is the charging energy of the superinductor mode, and $E_L=({\Phi }_0/2\pi {)}^2/L$. The loop phase is $\varphi$ and ${\varphi }_{\mathrm{ext}}=2\pi {\Phi }_{\mathrm{ext}}/{\Phi }_0$.

At $n_g=1/2$, the charge-sector symmetry makes the single-phase-slip matrix element vanish between adjacent fluxon wells. That selection rule is the origin of the effective $4\pi$ periodicity; replacing the full Hamiltonian by an ordinary one-dimensional fluxonium potential would miss the interference mechanism. Junction imbalance adds the leading symmetry-breaking term $\delta H=\Delta E_J\sin (\varphi /2)\sin \phi$ (after the common mode is frozen).

Motivation

The central challenge in superconducting quantum computing is decoherence from environmental noise, primarily $1/f$ charge noise and $1/f$ flux noise. Conventional qubits (transmon, fluxonium) are protected against one noise type by operating at sweet spots, but remain vulnerable to the other. The bifluxon offers a path to simultaneous protection against both noise channels through a topological mechanism (fluxon-parity encoding) rather than materials engineering. If the protection can be scaled up — for instance, by replacing the single junction with arrays of $\cos (\phi /2)$ elements — the bifluxon could achieve coherence times orders of magnitude beyond current transmon levels, potentially exceeding the threshold for quantum error correction with much lower overhead.

Experimental Status

First demonstration — Kalashnikov et al. (2020):

• Fabricated a bifluxon circuit with a CPB shunted by a granular aluminum superinductor.
• Demonstrated tenfold increase in energy relaxation time ($T_1$ up to 100 μs) when the offset charge was tuned from the unprotected point to the charge-parity sweet spot.
• Measured charge-noise dephasing time $T_2^{\ast }>1\mu \text{s}$.
• Confirmed the Aharonov-Casher protection mechanism by observing the charge-dependent oscillation of $T_1$.
• Published in PRX Quantum 1, 010307 (2020).

Key Metrics

Metric	Value	Notes	Fidelity reference
$T_1$ (protected)	~100 μs	At charge-parity sweet spot	Kalashnikov et al. 2020
$T_1$ (unprotected)	~10 μs	Away from sweet spot	Kalashnikov et al. 2020
$T_2^{\ast }$	>1 μs	Charge-noise limited	Kalashnikov et al. 2020
$E_J/E_C$	~1	CPB regime (charge-sensitive)	Kalashnikov et al. 2020
$E_L/E_J$	≪1	Large superinductance required	Kalashnikov et al. 2020

Scaling Considerations

• Superinductor quality: the protection level scales with the inductance of the superinductive shunt. Current granular aluminum superinductors achieve $L\sim 1\mu \text{H}$, but higher values with lower loss are needed for stronger protection.
• Charge stability: the protection mechanism requires stable tuning to the odd-charge parity point. Charge jumps (quasiparticle poisoning) can move the system away from the sweet spot, temporarily disabling protection.
• Gate implementation: universal gates on a protected qubit are inherently difficult because the same protection that suppresses noise also suppresses intentional control signals. Proposals include using the charge degree of freedom for $X$ gates and flux for $Z$ gates, but experimental gate demonstrations are still pending.
• $\cos (\phi /2)$ arrays: the demonstration proposes combining several split-CPB $\cos (\phi /2)$ elements to enhance dephasing protection, but this adds fabrication and control complexity.
• Comparison to alternatives: the 0-π qubit and $\cos (2\phi )$ qubit pursue related protection goals with different circuit symmetries. The bifluxon uses a split CPB plus one superinductive branch, but remains sensitive to junction asymmetry and quasiparticle-induced charge-parity changes.

References

Proposal and demonstration

• K. Kalashnikov et al., “Bifluxon: Fluxon-Parity-Protected Superconducting Qubit,” PRX Quantum 1, 010307 (2020) | arXiv:1910.03769

Linked Papers

• kalashnikov-2020-bifluxon

Evergreen context

• josephson-junction-as-nonlinear-element — the split CPB supplies the two phase-slip paths whose interference protects fluxon parity.
• charge-noise-sweet-spot — explains why the odd-charge operating point is useful and why charge-parity jumps remain dangerous.
• noise-bias-and-asymmetric-error-channels — frames the distinction between suppressing relaxation and achieving full dephasing protection.

Related Entries

• fluxonium — related circuit topology with superinductive shunt
• 0-pi-qubit — alternative protected qubit using circuit symmetry
• cos2phi-qubit — alternative protected qubit with doubled Josephson potential
• ferbo-qubit — another protected superconducting qubit design
• heavy-fluxonium-qubit — fluxonium variant with enhanced protection