Floquet Codes

Figure

Description

Floquet codes are a family of dynamical quantum error-correcting codes where the code space is not defined by a fixed set of stabilizers but instead emerges from a periodic sequence of two-body measurements. The codespace changes at each time step but returns to itself after a complete measurement cycle, defining logical qubits that persist stroboscopically.

Introduced by Hastings and Haah (2021), the honeycomb Floquet code is the prototypical example. The physical qubits sit on vertices of a honeycomb lattice, and at each time step, two-body Pauli measurements (XX, YY, or ZZ) are performed on pairs of qubits sharing an edge. The three edge colors are measured in a repeating 3-step cycle.

Key advantages:

• Two-body measurements only: No multi-qubit stabilizer measurements needed (vs. weight-4 checks in surface code), simplifying hardware.
• Inherent fault tolerance: The dynamical structure naturally detects errors from measurement failures.
• Competitive thresholds: Comparable to surface code despite simpler measurements.
• Potential for qLDPC integration: Floquet techniques may help implement non-local codes on local hardware.

The main subtlety is that the instantaneous stabilizer group changes at each step — logical information is only well-defined when viewed over complete cycles.

Measurement Cycle

On a honeycomb lattice with edge coloring (r, g, b), the 3-step cycle measures:

$\text{Step 1: }XX\text{ on red edges}$ $\text{Step 2: }YY\text{ on green edges}$ $\text{Step 3: }ZZ\text{ on blue edges}$

The effective code after one complete cycle is equivalent to a topological code with distance set by the lattice size. The check operators from consecutive rounds combine to form the stabilizers of a toric/surface code.

Motivation

• Hardware simplification: Two-body measurements are natively available on most platforms (superconducting, neutral atom, trapped ion) — no ancilla overhead for syndrome extraction.
• Dynamic error detection: The time-varying stabilizer structure provides redundant error information across measurement rounds.
• Competitive performance: Achieves thresholds comparable to surface code with dramatically simpler measurements.
• Beyond surface code: Active area of research for next-generation error correction, with Google and others exploring Floquet codes as alternatives to the surface code.

Key Metrics

Metric	Value	Notes	Fidelity reference
Measurement weight	2	Only two-body measurements (vs. weight-4 for surface code)	Hastings and Haah 2021
Threshold (circuit-level)	~0.2–0.5%	Depends on noise model and decoder	Hastings and Haah 2021
Code distance	$\Theta (L)$	$L$ = linear lattice dimension	—
Encoding rate	$O(1/L^2)$	Same scaling as surface code for 2D	—

Scaling Considerations

• Hardware simplicity: Two-body measurements are natively available on most platforms (especially superconducting and neutral atom).
• Decoding: Requires decoders that handle the time-varying stabilizer structure; matching-based decoders adapted from surface code work well.
• Google interest: Active exploration for next-generation error correction beyond the surface code.

References

Original proposal

• M. B. Hastings and J. Haah, “Dynamically Generated Logical Qubits,” Quantum 5, 564 (2021) — arXiv:2107.02194

Related theory

• M. B. Hastings and J. Haah, “Measurement sequences for magic state distillation,” arXiv:2302.12292
• M. S. Kesselring et al., “Anyon condensation and the color code,” arXiv:2212.00042

Linked Papers

• hastings-2021-floquet

Related Entries

• surface-code-logical-qubit — Static topological code; Floquet codes can emulate its effective stabilizers
• color-code-logical-qubit — Alternative topological code with transversal gates
• qldpc-codes — Floquet techniques may help implement qLDPC codes on local hardware