cos(2φ) Qubit

The cos(2φ) qubit is a proposed superconducting qubit with intrinsic protection against both charge noise and flux noise simultaneously, achieved through a potential energy that is $\pi$-periodic in the superconducting phase $\varphi$ rather than $2\pi$-periodic. The qubit states are encoded in two degenerate minima of this $\cos (2\varphi )$ potential, and transitions between them require tunneling through a large barrier — exponentially suppressing both bit-flip and phase-flip errors.

Figure

Description

Most superconducting qubits are protected against one type of noise at the cost of sensitivity to another. The transmon suppresses charge noise by operating at large $E_J/E_C$, but gains sensitivity to flux noise. The flux qubit can be biased to a sweet spot for flux noise but remains charge-sensitive. The cos(2φ) qubit breaks this tradeoff.

The key insight: if the Josephson potential is $\cos (2\varphi )$ rather than $\cos (\varphi )$, the two qubit states $|0⟩$ and $|1⟩$ have wavefunctions localized in different wells separated by $\pi$ in phase space. Crucially, the charge and flux matrix elements connecting $|0⟩$ and $|1⟩$ are both exponentially small in the barrier height — the operators $\hat{\varphi }$ and $\hat{n}$ have disjoint support on the two qubit states.

Proposed implementations use circuits with pairs of Josephson junctions arranged to cancel the $\cos (\varphi )$ term and retain only $\cos (2\varphi )$, typically involving a superinductance and carefully tuned junction asymmetry.

Hamiltonian

Effective Hamiltonian:

$H=4E_C{\hat{n}}^2+\frac{1}{2}{E}_L{\hat{\varphi }}^2-E_J\cos (2\hat{\varphi })$

where $E_C$ is the charging energy, $E_L$ is the inductive energy from the superinductance, and $E_J$ is the effective Josephson energy of the $\cos (2\varphi )$ element.

The qubit states are the symmetric and antisymmetric combinations of wavefunctions localized near $\varphi =0$ and $\varphi =\pi$:

$|0⟩\approx \frac{1}{\sqrt{2}}(|\varphi \approx 0⟩+|\varphi \approx \pi ⟩)$ $|1⟩\approx \frac{1}{\sqrt{2}}(|\varphi \approx 0⟩-|\varphi \approx \pi ⟩)$

Protection mechanism: $⟨0|\hat{n}|1⟩\sim e^{-S}$ and $⟨0|\hat{\varphi }|1⟩\sim e^{-S}$ where $S$ is the WKB tunneling action through the barrier.

Performance Metrics

Metric	Value	Notes	Fidelity reference
Predicted $T_1$ enhancement	$10^2$–$10^4$ ×	Over transmon, from exponential suppression	kalashnikov-2019-cos2phi
Protection type	Charge + flux	Simultaneous protection (unique among SC qubits)	kalashnikov-2019-cos2phi
Required inductance	>100 nH	Superinductance (granular aluminum or JJ array)	—
Experimental status	Not yet realized	Circuit complexity is the barrier	—

Scaling Considerations

• Circuit complexity: Requires precise cancellation of $\cos (\varphi )$ terms, demanding high junction symmetry and stable superinductance.
• Not yet demonstrated: Remains theoretical. The required parameter regime is at the edge of current fabrication capabilities.
• Potential payoff: If realized, would be the first superconducting qubit with built-in protection against all local noise channels, potentially eliminating the need for QEC at the physical level for memory operations.

Related Entries

• 0-pi-qubit
• fluxonium
• heavy-fluxonium-qubit
• blochnium