Dissipatively Stabilized 0-n Fock Qubits for Noise-Biased Quantum Computing

Technical Summary: Dissipatively Stabilized 0-n Fock Qubits for Noise-Biased Quantum Computing

Problem Statement
Noise-biased qubits, which exhibit exponentially suppressed bit-flip errors relative to phase-flip errors, offer a promising pathway to fault-tolerant quantum computing with reduced overhead. However, a critical limitation arises during syndrome extraction: entangling gates between noise-biased data qubits (e.g., cat qubits) and ancilla qubits can propagate errors. If the ancilla is not itself noise-biased (e.g., a standard transmon), back-action errors can introduce bit flips on the data qubit, compromising the error correction scheme. Existing solutions, such as using cat codes for ancillas, require complex Hamiltonian engineering, while hybrid approaches using standard transmons introduce additional decay and dephasing that limit the achievable noise bias.

Methodology
The authors propose a new architecture: the 0-n Fock qubit. This qubit encodes logical states $|0_L\rangle$ and $|1_L\rangle$ into the ground state $|0\rangle$ and the $n$-th excited state $|n\rangle$ of an anharmonic oscillator (e.g., a transmon or Kerr nonlinear oscillator).

1. Noise Bias Mechanism: The encoding protects against single-photon loss. A loss event transitions $|n\rangle$ to $|n-1\rangle$, moving the system out of the logical subspace rather than causing a direct logical bit flip. A logical bit flip requires $\lceil n/2 \rceil$ sequential intrinsic decay events. The authors propose stabilizing the system within the $\{|0\rangle, |n\rangle\}$ manifold using engineered dissipation. This involves creating a frequency-selective bath that induces decay for lower intermediate levels ($0 \le m \le \lfloor n/2 \rfloor - 1$) and gain for upper intermediate levels ($\lceil n/2 \rceil \le m \le n-1$), while leaving $|0\rangle$ and $|n\rangle$ invariant.
2. Experimental Implementation: To realize this frequency-selective dissipation, the authors propose coupling the qubit to a linear chain of $M$ LC resonators terminated by a lossy mode (a multimode filter). By engineering the couplings between resonators, the effective spectral density of the bath is shaped into a bandpass filter.
   • Decay: Achieved via a beam-splitter interaction between the qubit and the first filter mode.
   • Gain: Achieved via a two-mode squeezing interaction.
   • Selectivity: Due to the anharmonicity of the oscillator, transitions between adjacent Fock levels occur at distinct frequencies. By tuning the drive frequencies of the interactions, transitions in the lower half of the ladder can be placed within the filter's passband (inducing decay), while upper transitions lie outside (or vice versa for gain).
3. Gate Operations: The paper analyzes the use of the 0-n Fock qubit as an ancilla for controlled-X (CX) gates on stabilized cat qubits. These gates utilize simple, native dispersive coupling between the Fock and cat qubits, avoiding complex Hamiltonian engineering.

Key Contributions and Results

• Analytical Demonstration of Noise Bias: The authors derive that the effective bit-flip rate $\lambda_{bf}$ scales exponentially with the number of levels $n$ (specifically as $(r)^{(n+1)/2}$, where $r$ is the ratio of intrinsic to engineered dissipation rates). Conversely, phase-flip rates scale linearly with $n$. This establishes a hardware-level noise bias analogous to cat qubits but realized in a discrete Fock basis.
• Feasibility of Dissipation Engineering: Numerical simulations of the multimode filter demonstrate that a small number of modes ($M \sim 7$) is sufficient to achieve strong spectral selectivity and suppress unwanted engineered dissipation on non-target transitions, using realistic circuit parameters.
• Gate Performance:
  • Without stabilization, bit-flip probabilities in CX gates grow linearly with $n$ due to pulse errors populating intermediate levels.
  • With engineered dissipation, the system relaxes to a metastable state concentrated in $|0\rangle$ and $|n\rangle$. For $N \ge 10$ levels ($n \ge 9$), the authors numerically demonstrate that logical bit-flip probabilities for CX gates can be suppressed to below $10^{-8}$ under realistic experimental parameters (including intrinsic decay, heating, and pulse errors).
• Logical Error Rates: The authors simulate a repetition code using the 0-n Fock qubit as an ancilla and cat qubits as data. With a code distance of $d=9$ and $N \ge 10$, they achieve logical error rates in the megaquop regime ($p^*_L = 10^{-6}$). This performance is achieved in an experimentally realistic parameter regime where the cat qubit storage-mode lifetime is $\sim 2$ ms.

Significance
The paper claims that the 0-n Fock qubit provides a viable and compact route to realizing bias-preserving architectures. Its primary significance lies in bridging the gap between the high noise bias of cat qubits and the experimental simplicity of transmon-based systems. By using a Fock qubit ancilla, the architecture:

1. Preserves the noise bias of cat data qubits during syndrome extraction, preventing ancilla-induced bit flips from dominating the error budget.
2. Avoids the complex Hamiltonian engineering required for fully bosonic cat-cat gates.
3. Utilizes simple dispersive couplings for gate operations, which are native to superconducting circuits.
4. Demonstrates that frequency-selective stabilization can be achieved with a small number of filter modes, making it compatible with current superconducting circuit technology.

The authors conclude that this hybrid cat-Fock architecture offers a practical path toward low logical error rates without the stringent requirements of fully bosonic interactions or the bias limitations of standard transmon ancillas.