From Code Surgery to Lifted Surgery: How Abelian Group Algebra Codes Cut Syndrome Rounds

Why code surgery's time overhead became the bottleneck

Quantum low-density parity-check codes emerged as leading candidates for low-overhead fault-tolerant quantum computing because they encode many logical qubits per block while retaining good error correction [2]. Code surgery generalises lattice surgery to these codes, deforming the error-correcting code with an auxiliary system so that syndrome measurements on the deformed code return the value of a desired logical operator [5]. The procedure is space-efficient, but it carries a significant time overhead: noisy syndrome measurements must be repeated a number of times proportional to the code distance to ensure fault tolerance [1]. This O(d) scaling means that as codes grow to achieve lower logical error rates, logical operations become proportionally slower, creating a tension between the space savings that motivate QLDPC codes and the time cost of computing with them [5].

Fast surgery was proposed to remove this scaling by using auxiliary systems with additional properties that protect against syndrome measurement errors, enabling constant-time fault-tolerant logical measurements [1]. However, straightforward fast-surgery constructions incur large qubit overhead, so the overall space-time overhead matches standard code surgery schemes [1]. Practical constructions of auxiliary systems with the required properties remained unknown in general, apart from techniques specialised to hypergraph product codes [5]. This is the gap that lifted surgery addresses for Abelian group algebra codes, a widespread family underpinning several proposed QLDPC architectures [1].

How lifted surgery preserves symmetry and enables parallel measurement

Lifted surgery exploits algebraic symmetries already present in group algebra codes to construct practical fast surgery schemes [1]. The group algebra canonically decomposes into a product of local rings, and the codespace and every linear map over the group algebra decompose accordingly into independent blocks [1]. This block decomposition exposes the structure and symmetries of the logical subspace and provides the natural framework in which lifted surgery is defined [1]. By working within this decomposition, the authors characterise which logical operations are addressable and construct explicit surgery maps that preserve the underlying group symmetry [1].

The approach yields two addressability constraints: every measured set splits as a direct sum over blocks, and within a block each measured set is invariant under the group action [1]. These constraints mean that lifted surgery cannot address arbitrary logical subspaces, but they also make the search for large instances tractable and offer a natural route towards efficient hardware implementations [1]. The authors identify well-behaved subfamilies, including scalar surgery where a single group algebra element describes the chain maps, and show that for codes described by Koszul complexes, scalar surgery preserves the Koszul form [1]. This algebraic structure is what allows lifted surgery to achieve fast, parallel, and addressable logical measurements in a single round of syndrome extraction [1].

Quantum radial codes: tenfold fewer rounds with comparable logical performance

The paper presents quantum radial codes with parameters [[90,8,10]] and [[198,8,16]] for which surgery is fast, parallel, and addressable, allowing arbitrary sets of independent logical operators of the same Pauli type to be measured in a single round of syndrome extraction [1]. For the [[90,8,10]] code, the authors benchmark lifted surgery under circuit-level depolarising noise and find logical performance comparable to standard code surgery while requiring ten times fewer rounds of syndrome measurement [1]. This means that the time cost of logical measurement drops by an order of magnitude without sacrificing the error suppression that makes the code useful. The [[198,8,16]] code demonstrates that the construction scales to larger distances, with 45 orbits for rank-1 subspace measurements [1].

The merged codes produced by lifted surgery on the [[90,8,10]] quantum radial code have parameters [[135,7,≤10]] for rank-1 subspace measurements and [[135,6,≤10]] for rank-2, with maximum check weight and qubit degree of 11 and 12 respectively [1]. These overheads are determined by noting that for a code with n data qubits, m_X X-type checks, and m_Z Z-type checks, the merged code has n+m_X data qubits, m_Z+n Z-type checks, and m_X X-type checks when performing a Z-type logical measurement [1]. The check weights and qubit degrees remain bounded, preserving the LDPC property that makes these codes attractive for hardware implementation [1].

How lifted surgery compares with other fast-surgery and lattice-surgery approaches

Constant-time surgery on 2D hypergraph product codes achieves O(1) time overhead and near-constant space overhead by amortising d surgery operations over O(d) time, but this approach is specialised to hypergraph product codes and relies on different algebraic structures than lifted surgery [5]. The hypergraph product construction uses chain complexes and mapping cones, with meta-checks providing the redundancy needed for single-shot measurement [5]. Lifted surgery instead exploits the block decomposition of Abelian group algebra codes, which are a distinct family that includes bivariate bicycle codes and quantum radial codes [1]. Both approaches achieve constant-time logical measurement, but they apply to different code families and use different algebraic techniques.

On the experimental side, lattice surgery has been demonstrated on distance-three surface codes and repetition codes with superconducting qubits, showing that merge and split operations can be composed to realise logical state routing and Clifford gates [3][6]. These experiments operate on 2D planar architectures with nearest-neighbour connectivity, whereas lifted surgery targets QLDPC codes that require long-range connectivity [1][2]. The experimental demonstrations validate the building blocks of lattice surgery but do not address the time overhead of repeated syndrome extraction that lifted surgery aims to eliminate [3][6]. Time-dynamic circuits for shift automorphisms in QLDPC codes show that dynamically varying syndrome measurement circuits can reduce logical error rates compared to SWAP-based approaches, achieving a 38-fold reduction in logical bit-flip error for the [[144,12,12]] gross code at p=10^-3 [4]. This is a complementary approach to reducing time overhead in QLDPC codes, focusing on automorphism gates rather than logical measurement [4].

What remains uncertain and where the claims stop

The results are limited to Abelian group algebra codes and the specific [[90,8,10]] and [[198,8,16]] instances constructed in the paper, under a circuit-level depolarising noise model [1]. The addressability constraints mean that lifted surgery cannot measure arbitrary logical subspaces: every measured set must split as a direct sum over blocks and be invariant under the group action within each block [1]. This restricts the set of logical operations that can be performed in a single round, and the authors note that different group algebra representations of the same binary code can lead to different sets of lifted surgery operations [1]. The construction also requires CSS-type measurements, so non-CSS operations such as standard surgery, automorphism gates, or Y-state distillation will still be needed for universal computation [1].

Integrating lifted surgery into a fault-tolerant architecture will require compilation strategies that minimise the use of non-CSS measurements and make optimal use of the parallel measurements available for certain codes [1]. The authors also note that optimised hardware layouts for performing lifted surgery operations remain to be developed, and that lifted surgery is particularly suitable for platforms allowing dynamic reconfiguration such as neutral atoms, trapped ions, and spin qubits [1]. The decoding algorithm uses a most-likely error decoder implemented with integer linear programming, with a two-hour timeout for simulations, which may limit scalability to larger code instances [1]. Whether the tenfold reduction in syndrome rounds translates to comparable reductions in overall algorithm runtime depends on the relative cost of syndrome extraction versus other operations, which is not addressed in the current benchmarks [1].

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This research page is built on 6 studies (4 peer-reviewed, 2 preprints) — published from 2021 to 2026, 5 from 2024 or later, collectively cited 753 times — selected as the most relevant from 13 studies that passed quality screening, drawn from 89 papers retrieved from a database of over 500 million.

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