Bifluxon Qubit

Figure

Description

The bifluxon qubit is a fluxon-parity-protected superconducting qubit that achieves simultaneous protection against both charge noise (energy relaxation) and flux noise (dephasing) through a topological mechanism based on Aharonov-Casher interference. The circuit consists of a Cooper-pair box (CPB) shunted by a superinductor, forming a superconducting loop where the logical qubit states are encoded in the odd and even parity of fluxons (magnetic flux quanta) threading the loop.

Protection arises when the offset charge on the CPB island is tuned to an odd number of electrons, $(2n+1)e$. At this operating point, Aharonov-Casher interference causes the effective Josephson potential to become $\cos (\phi /2)$ rather than $\cos (\phi )$, where $\phi$ is the superconducting phase difference across the junction. This $\cos (\phi /2)$ potential has a double-well structure with degenerate minima at $\phi =0$ and $\phi =2\pi$ (corresponding to zero and one fluxon). The qubit states $|0⟩$ and $|1⟩$, localized in these separate wells, occupy disjoint regions of phase space, providing exponential suppression of matrix elements connecting them. This means that both charge fluctuations (which couple via $\hat{n}$) and flux fluctuations (which couple via $\hat{\phi }$) have exponentially small matrix elements between the logical states.

The bifluxon is distinct from other protected qubit designs: unlike the 0-π qubit (which requires precise symmetry between two circuit branches) or the $\cos (2\phi )$ qubit (which engineers a doubled Josephson potential), the bifluxon uses a single Josephson junction with a superinductive shunt, biased to the charge-parity sweet spot. This simpler circuit makes it more experimentally accessible.

Hamiltonian

The bifluxon Hamiltonian is:

$H=4E_C(\hat{n}-n_g{)}^2+\frac{1}{2}{E}_L{\hat{\phi }}^2-E_J\cos (\hat{\phi }-{\phi }_{\text{ext}})$

where $E_C$ is the charging energy of the CPB island, $E_L$ is the inductive energy of the superinductor, $E_J$ is the Josephson energy, $n_g$ is the offset charge, and ${\phi }_{\text{ext}}=2\pi {\Phi }_{\text{ext}}/{\Phi }_0$ is the external flux bias. At $n_g=(2n+1)/2$ (odd charge parity), Aharonov-Casher interference transforms the effective potential:

$V_{\text{eff}}(\phi )\approx \frac{1}{2}{E}_L{\phi }^2-E_J^{\text{eff}}\cos (\phi /2)$

The $\cos (\phi /2)$ term creates a double-well potential with minima at $\phi =0$ and $\phi =2\pi$, separated by a barrier of height $\sim E_J^{\text{eff}}$.

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
Qubit frequency	~5 GHz	Typical operating frequency	—
Operating temperature	~20 mK	Dilution refrigerator	—

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: theoretical proposals suggest replacing the single junction with arrays of $\cos (\phi /2)$ elements to exponentially enhance protection, but this adds significant fabrication complexity.
• Comparison to alternatives: the 0-π qubit and $\cos (2\phi )$ qubit offer similar protection goals with different circuit topologies and trade-offs. The bifluxon’s advantage is its simpler single-junction design; its disadvantage is sensitivity to quasiparticle poisoning at the charge sweet spot.

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

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