cos(2φ) Qubit

Figure

Description

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.

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.

Motivation

• Dual protection: First superconducting qubit proposal with simultaneous exponential suppression of both charge and flux noise — breaks the transmon/fluxonium tradeoff.
• Disjoint support: Qubit states have exponentially small overlap in both charge and flux basis, making all local noise channels exponentially suppressed.
• Potential QEC bypass: If realized, could eliminate the need for quantum error correction at the physical level for memory operations.
• Scalable protection: Protection improves exponentially with barrier height, which is a tunable circuit parameter.

Key Metrics

Metric	Value	Notes	Fidelity reference
Predicted $T_1$ enhancement	$10^2$–$10^4$ ×	Over transmon, from exponential suppression	Smith et al. 2020
Protection type	Charge + flux	Simultaneous protection (unique among SC qubits)	Smith et al. 2020
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.

References

Original proposal

• W. C. Smith et al., “Superconducting circuit protected by two-Cooper-pair tunneling,” npj Quantum Inf. 6, 8 (2020) — arXiv:1905.01206

Related theory

• A. Kitaev, “Protected qubit based on a superconducting current mirror,” arXiv:cond-mat/0609441 (2006)
• P. Brooks, A. Kitaev, and J. Preskill, “Protected gates for superconducting qubits,” Phys. Rev. A 87, 052306 (2013) — arXiv:1302.4122

Linked Papers

• smith-2020-superconducting-circuit-protected

Related Entries

• 0-pi-qubit — Related protected qubit with $\cos (\varphi )$ potential at $\pi$-sweet spot
• fluxonium — Uses superinductance but has $2\pi$-periodic potential
• heavy-fluxonium-qubit — Heavy variant approaching protected regime
• blochnium — Single-junction superinductance qubit
• transmon — conventional transmon that cos(2φ) qubit improves upon via disjoint support
• ferbo-qubit — alternative dual-protected design using fermion-boson hybridization instead of multiple bosonic modes
• bifluxon-qubit — alternative protected qubit using fluxon parity encoding