REVIEW
Logical information localisation in stabiliser codes via single-qubit measurements
Quantum Physics · Submitted 30 Sep 2026 · 56 references
Pith's verdict: Proves an exact localisation threshold at p=1/2 for the planar surface code with a clean percolation-plus-disjointness argument, and ships two algorithms that genuinely beat the prior state of the art.
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Referee report
The paper
The paper claims
The planar surface code can still deliver its logical qubit when up to half its qubits are lost, and fails beyond that.
The reading
What carries the argument
The central object is the g-SPF pair, a pair of logical operators X and Z that avoid all lost qubits and anti-commute qubit-wise on at most g qubits; measuring them qubit-wise localises the logical information onto those g qubits. The proof chain runs through four steps: edge-disjoint open paths in the lattice survive as disjoint logical representatives after loss (percolation, Lemma 9 cited from Grimmett); Harris' inequality joins the left-to-right and top-to-bottom path events; Corollary 11 converts the number of edge-disjoint open paths into a lower bound on the disjointness of surviving X and Z representatives; and Lemma 2, a variant of the scrubbing lemma from Ref. [30], bounds the minimum support intersection by n divided by the product of the two disjointnesses. Since the number of qubit-wise anti-commutations satisfies α(X, Z) ≤ |supp(X) ∩ supp(Z)|, a constant bound on the intersection yields a constant-g pair.
Extended reading notes
Core claim
Theorem 1: for the family of planar surface codes under i.i.d. qubit loss, there is an exact localisation threshold at p = 1/2. For every p < 1/2 there exists a strictly positive constant g = g(p) such that the probability of finding a g-SPF pair converges to one as the code size grows, where a g-SPF pair is a pair of logical X and Z representatives with no support on lost qubits that anti-commute qubit-wise on at most g qubits. For every p > 1/2 the g-SPF success rate converges to zero for any constant g. Practically, this means that below 1/2 loss a sufficiently large surface code allows a readout of the logical qubit by one round of single-qubit measurements that concentrate the information onto O(1) physical qubits. The authors stress that this localisation threshold coincides with the known erasure threshold even though localisation and erasure decoding are not obviously the same task.
Load-bearing premise
The lower-bound half of Theorem 1 rests on the percolation concentration statement that for every p < 1/2 the number of edge-disjoint open left-to-right paths in the Λ × (Λ−1) lattice is at least β(Λ−1) with probability exponentially close to one; if that bound failed for some p < 1/2, the whole chain from percolation to a constant-g pair would collapse.
Editorial extensions
If this is right
- For any loss rate below 1/2, a sufficiently large planar surface code lets one read out the logical qubit onto O(1) physical qubits by single-qubit measurements, with probability converging to one.
- For any loss rate above 1/2, no constant-size localisation set survives, so p = 1/2 is a sharp cutoff rather than a soft crossover.
- The localisation threshold equals the standard erasure threshold even though localisation and erasure decoding are distinct tasks, suggesting the erasure threshold governs more than mere survival of the logical qubit.
- The exact D-LoFi and heuristic H-LoFi algorithms reproduce this threshold numerically and extend analyses to surface codes of distance 9 (81 qubits) and graph codes of 51 qubits, well beyond earlier limits.
- Because g = 1 already shows threshold behaviour numerically, the attainable localisation set may in fact be a single qubit for all p < 1/2, a stronger statement than the theorem proves.
- Relaxing a pre-designated target qubit to an arbitrary one measurably raises the success rate, so flexible target choice is strictly better than fixing a target in advance.
Reading between the lines
- The proof uses support intersection as a proxy for qubit-wise anti-commutation, which is a loose bound: two operators with large overlapping supports can still anti-commute on very few qubits. A commutation-aware notion of disjointness could sharpen the constant g, plausibly down to g = 1, which the numerics already suggest.
- Because every stabiliser code is local-Clifford equivalent to a graph code, the localisation threshold plausibly transfers to graph-code versions of the surface code, making the result relevant to photonic fusion and teleportation schemes that express codes as graphs.
- The exact coincidence of the localisation threshold with the erasure threshold is a hint that, in this setting, localisation via g-SPF may be equivalent to erasure decodability; if that equivalence held broadly, it would give a clean necessary-and-sufficient characterisation of post-loss localisation by LOCC.
- The upper-bound half of the theorem is inherited from the exactness of the erasure threshold, so the genuinely new content is the lower-bound half; a failure of either the percolation concentration bound or the disjointness argument would collapse the threshold claim at p = 1/2.
Original abstract
Stabiliser path finding (SPF) has previously been introduced as a method to localise logical information in a stabiliser code undergoing loss onto a single pre-specified target qubit, using only one round of single-qubit measurements. When working with limited resources and flying qubits, this fast read-out of logical information is a helpful tool for fault-tolerant communication. In this work, we provide a broad analytical and computational study of localisation via SPF. We introduce $g$-SPF, where the task is to localise the logical information onto a set of at most $g$ unspecified target qubits. Through analytical arguments based on percolation theory and the disjointness of stabiliser codes, we prove that for i.i.d. qubit loss with probability $p<1/2$ and sufficiently large planar surface codes, localisation via $g$-SPF for constant $g$ succeeds with a probability converging to one, which establishes a localisation threshold. Furthermore, we propose and implement two algorithms to solve SPF. The first is exact and formulates SPF as an integer linear program, whereas the second is heuristic and formulates SPF as a decoding problem. We validate both algorithms by numerically reproducing the localisation threshold for the surface code and demonstrate that the heuristic algorithm is considerably faster. Our work significantly reduces the time required to solve SPF compared to current state-of-the-art algorithms, allowing us to study localisation in substantially larger stabiliser codes than previously considered in the literature. Together, these theoretical and computational results open the door to various applications, such as fault-tolerant teleportation and efficient logical fusion.
The compiled record
The paper's own sentences, arranged, kept beside the review and separate from it. Each label is drawn from a closed vocabulary; the number beside it is the probability the classifier gave that label. Numbers in the tables are the authors', checked by code. No sentence in this record was written by a model.
The abstract, sentence by sentence
Stabiliser path finding (SPF) has previously been introduced as a method to localise logical information in a stabiliser code undergoing loss onto a single pre-specified target qubit, using only one round of single-qubit measurements. When working with limited resources and flying qubits, this fast read-out of logical information is a helpful tool for fault-tolerant communication. In this work, we provide a broad analytical and computational study of localisation via SPF. We introduce $g$-SPF, where the task is to localise the logical information onto a set of at most $g$ unspecified target qubits. Through analytical arguments based on percolation theory and the disjointness of stabiliser codes, we prove that for i.i.d. qubit loss with probability $p<1/2$ and sufficiently large planar surface codes, localisation via $g$-SPF for constant $g$ succeeds with a probability converging to one, which establishes a localisation threshold. Furthermore, we propose and implement two algorithms to solve SPF. The first is exact and formulates SPF as an integer linear program, whereas the second is heuristic and formulates SPF as a decoding problem. We validate both algorithms by numerically reproducing the localisation threshold for the surface code and demonstrate that the heuristic algorithm is considerably faster. Our work significantly reduces the time required to solve SPF compared to current state-of-the-art algorithms, allowing us to study localisation in substantially larger stabiliser codes than previously considered in the literature. Together, these theoretical and computational results open the door to various applications, such as fault-tolerant teleportation and efficient logical fusion.
background contribution method result interpretation main claim
| # | sentence | role | the authors' own work | states the main conclusion | strength |
|---|---|---|---|---|---|
| s0 | Stabiliser path finding (SPF) has previously been introduced as a method to localise logical information in a stabiliser code undergoing loss onto a single pre-specified target qubit, using only one round of single-qubit measurements. | background 1.00 | 0.05 | 0.04 | asserted 1.00 |
| s1 | When working with limited resources and flying qubits, this fast read-out of logical information is a helpful tool for fault-tolerant communication. | background 0.86 | 0.06 | 0.03 | asserted 0.87 |
| s2 | In this work, we provide a broad analytical and computational study of localisation via SPF. | contribution 1.00 | 0.91 | 0.04 | asserted 0.88 |
| s3 | We introduce $g$-SPF, where the task is to localise the logical information onto a set of at most $g$ unspecified target qubits. | contribution 0.97 | 0.94 | 0.08 | asserted 0.97 |
| s4 | Through analytical arguments based on percolation theory and the disjointness of stabiliser codes, we prove that for i.i.d. qubit loss with probability $p<1/2$ and sufficiently large planar surface codes, localisation via $g$-SPF for constant $g$ succeeds with a probability converging to one, which establishes a localisation threshold. | result 0.99 | 0.94 | 0.70 | asserted 1.00 |
| s5 | Furthermore, we propose and implement two algorithms to solve SPF. | contribution 0.51 | 0.95 | 0.04 | asserted 1.00 |
| s6 | The first is exact and formulates SPF as an integer linear program, whereas the second is heuristic and formulates SPF as a decoding problem. | method 0.93 | 0.94 | 0.03 | asserted 1.00 |
| s7 | We validate both algorithms by numerically reproducing the localisation threshold for the surface code and demonstrate that the heuristic algorithm is considerably faster. | result 0.97 | 0.97 | 0.11 | asserted 1.00 |
| s8 | Our work significantly reduces the time required to solve SPF compared to current state-of-the-art algorithms, allowing us to study localisation in substantially larger stabiliser codes than previously considered in the literature. | result 0.53 | 0.94 | 0.13 | asserted 1.00 |
| s9 | Together, these theoretical and computational results open the door to various applications, such as fault-tolerant teleportation and efficient logical fusion. | interpretation 0.85 | 0.43 | 0.22 | asserted 0.58 |
What was examined, and what was not
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Figures
Figures reproduced from arXiv:2609.39980 by the authors.
References
Discussion
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