Yoked surface codes

TL;DR

This work introduces yoked surface codes that concatenate surface codes with high-rate multi-dimensional quantum parity-check outer codes to dramatically increase logical-qubit density without requiring long-range connectivity. By measuring row- and column-parity checks via lattice surgery and exploiting a ZX-calculus perspective, the authors show how 1D and 2D outer codes can double or quadruple the inner code distance, respectively, while using a conservative protection strategy for correlated errors. They develop and validate a gap-based decoding framework using complementary gap distributions to efficiently simulate large, many-round outer-code cycles, and provide scaling laws and footprint estimates that project substantial qubit savings in the teraquop regime at a physical error rate of $p=10^{-3}$. The proposed hierarchical-memory architectures—with hot and cold storage variants—offer significant density gains (roughly 2×–3× more logical qubits per physical qubit) and are attractive because they avoid introducing new connectivity requirements, though they rely on advanced tooling and full-scale simulations to confirm overhead reductions. Overall, yoked surface codes present a promising, practical path to drastically reducing surface-code overhead while remaining compatible with near-term superconducting-qubit architectures.

Abstract

We nearly triple the number of logical qubits per physical qubit of surface codes in the teraquop regime by concatenating them into high-density parity check codes. These "yoked surface codes" are arrayed in a rectangular grid, with parity checks (yokes) measured along each row, and optionally along each column, using lattice surgery. Our construction assumes no additional connectivity beyond a nearest neighbor square qubit grid operating at a physical error rate of $10^{-3}$.

Figures (16)

• Figure 1: From left to right: unyoked, 1D, and 2D yoked surface code patches. In each row of 1D yoked surface codes, we measure multi-body logical $X$- and $Z$-type stabilizers. In 2D yoked surface codes, we additionally measure multi-body logical $X$- and $Z$-type stabilizers in each column. The $Z$-type stabilizers are applied to a permutation of the 2D code to commute with the $X$-type stabilizers. Grey patches represent overhead introduced by the stabilizers (i.e. "yokes"). Dark patches represent the workspace required to measure the row/column stabilizers. There is also overhead due to interstitial space between patches for lattice surgery. Concatenated code parameters, along with approximate overall qubit footprints (including the various overheads) labeled below. Note that the $[[192, 176, 2]]$ outer code is a collection of eight $[[24, 22, 2]]$ 1D parity check code blocks. All logical qubits can be reliably stored for about a trillion operations assuming a physical error rate of $10^{-3}$. The relative savings of yoked surface codes over unyoked surface codes grows as the target error rate decreases.
• Figure 2: Topological diagrams of multi-body logical measurements, with connections between blocks stretched out to show the topology. Left: a multi-body $Z$-measurement. The correlation surface shows the equivalence of a short spacelike "hook" error to two data errors. Right: the same multi-body $Z$-measurement with protection against correlated hook errors. We can increase protection against the hook error by extending the distance between the boundaries that it connects. Naively, this would increase the overall qubit footprint of the circuit. However, we can orient this extension in time, trading a smaller qubit footprint for a longer syndrome extraction cycle.
• Figure 3: The patch rotation construction from litinski2019gameofsurfacecodes, with the last step omitted, leaving the patch shifted as part of the rotation.
• Figure 4: Checking the $X^{\otimes n}$ and $Z^{\otimes n}$ stabilizers of 1D yoked surface codes using lattice surgery. The process occupies $2n$ surface code patches for $8d$ rounds, where $n$ is the block length of the outer code. Time flows left to right. The corresponding ZX diagram is shown to the left.
• Figure 5: Checking two row and birow stabilizers of 2D yoked surface codes using lattice surgery. This process occupies $3w$ surface code patches for $25d$ rounds, where $w = \sqrt{n}$ is the the width of the outer array. Time flows left to right. The corresponding ZX diagram is shown to the left. In that diagram, top pipes correspond to every other wire beginning from the top, while bottom pipes correspond to every other wire beginning second from the top.