Stairway Codes: Floquetifying Bivariate Bicycle Codes and Beyond

TL;DR

Stairway codes are introduced, a family of high-rate Floquet protocols obtained by Floquetifying Abelian two-block group algebra codes, a class that includes the bivariate bicycle codes and reduces the design of new codes within this family to the selection of favorable periodic boundary conditions.

Abstract

Floquet codes define fault-tolerant protocols through periodic measurement sequences that drive a dynamically evolving stabilizer group. They provide a natural framework for hardware supporting two-qubit parity measurements but no unitary entangling gates. However, few known constructions achieve both high encoding rates and high thresholds. We close this gap by introducing Stairway codes, a family of high-rate Floquet protocols obtained by Floquetifying Abelian two-block group algebra codes, a class that includes the bivariate bicycle codes. By representing the static code as a foliated ZX-calculus network within a $(w{-}1)$-dimensional space-time lattice and rotating the time axis, we decompose its weight-$w$ stabilizers into a periodic sequence of pairwise measurements. This reduces the design of new codes within this family to the selection of favorable periodic boundary conditions. We identify instances with competitive parameters, analyze their distance under circuit-level noise, and demonstrate logical error rates surpassing those of other Floquet codes at comparable encoding rates. Remarkably, our construction requires fewer than 300 physical qubits to match the distance and encoding rate of semi-hyperbolic Floquet codes that use over 1300 qubits.

Figures (17)

• Figure 1: a) Outline of our results. We create a new construction from Abelian two-block group algebra (2BGA) codes using the same technique used in Ref. Bombin_2024 to connect the toric code and the CSS honeycomb code. Abelian 2BGA codes lin2023twoblockgroupalgebra are a generalization of the toric code that includes the bicycle kovalev2013bicycle_codes and bivariate bicycle Bravyi_2024BB_codes code families. b) Outline of this work, with corresponding sections for each stage.
• Figure 2: a) Z- and X-spiders: The basic building blocks for the ZX-calculus framework. b) Representing basic quantum operations using the ZX-calculus. c+d) We define an abbreviated symbol for the pairwise parity measurement of $XX$ or $ZZ$ observables. By dropping the measurement result, we can write two-qubit measurements as two same-color spiders. e) Basic split and merge relations on ZX-calculus nodes. f) A representation of a multiqubit parity measurement. Dropping the measurement outcome port in these circuits corresponds to reinterpreting a Pauli measurement as a projection. We refer to Ref. Backens_2014ZXBombin_2024 for a rigorous introduction to ZX-calculus.
• Figure 3: The interpretation of a $2n$-legged spider as an $n\rightarrow n$ operator is that of a partial projection onto a 2-dimensional subspace. The two-qubit pairwise parity measurements implement the target partial projection for one outcome configuration and the orthogonal projections for the others. Here we present different ways to factor a four-, six- and eight-spider into pairwise measurements. The decomposition of the 8-spider intentionally includes one more measurement than necessary, introducing a local detector, a redundancy among the four measurement outcomes. Here, the symbol $\simeq$ denotes that equality up to a Pauli-frame, once a combination of measurement outcomes is fixed.
• Figure 4: a) Example of converting the "Gross code" Bravyi_2024BB_codes into a $D$-dimensional lattice. In this case, $D=w-2=4$, thus there are four basis vectors $j_1, \dots, j_4$. The figure displays the code's Tanner graph; for visual clarity, only four long-range connections are shown. One cell is highlighted in purple and its eight neighboring cells are highlighted in red. Four unit vectors are indicated by red arrows. b) The periodicity vectors $p_1,\dots, p_4$ of the generating sublattice $\Lambda$. These vectors are associated to combinations of unit vectors which are identified as summing to zero. c) The basic lattice cell represented in the ZX-calculus formalism. Vertical lines represent timelike connections while non-vertical lines represent spacelike connections. Each stabilizer spider have ${w}$ connections, and represents a $w$-wise parity measurement as presented in Fig. \ref{['fig:ZX-calculus']}f. Each data spider have $\frac{w}{2}{+}2$ connections. While the example in this figure is a weight-6 code, in this manuscript we focus on weight-8 codes.
• Figure 5: Geometry of the spacetime lattice. The green trajectory, resembling the path of a stairway, indicates the worldline of a physical qubit in a Stairway code. Every step has a positive projection onto the time vector. Thus, the qubit consistently advances in time. The $D$-dimensional spatial layers containing original stabilizer checks are shown as grey planes. Notably, the periodicity vector is chosen to be orthogonal to the time vector, preventing closed timelike curves and keeping the time coordinate well-defined.