{"id":"2657d298-ce96-4b49-afee-5f658f90c45c","arxiv_id":"2412.04181","paper_version":1,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":5.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":0,"one_line_summary":"Pruning bivariate bicycle codes produces open-boundary codes with the same locality and distance and half the logical qubits, proven for cyclic hypergraph products and shown on explicit examples.","lead":"The authors introduce a pruning procedure that turns periodic-boundary bivariate bicycle quantum codes into local codes on open lattices, halving the logical qubit count while preserving distance. The construction is proven for hypergraph-product cyclic codes and demonstrated on honeycomb color code examples, with fold-transversal gates carried over.","discovery_kind":"new_method","skeptic_critique":{"model":"deepseek-v4-flash","headline":"The algebraic pruning in Lemma 4 is likely correct, but the paper gives no proof that the remaining stabilizers are local on an open-boundary lattice; for l=6, A=1+x+x^2, kept rows wrap across the cut, connecting endpoints at distance 5.","rationale":"The reader's weakest assumption was that Lemma 4's one-sentence proof of the deletion correspondence is under-specified. Read charitably, that correspondence is actually checkable: the pruned matrices are obtained by deleting specific rows and columns of the parent check matrices, and the commutation H_X H_Z^T = 0 follows from the identity Ared ⊗ Bred + Ared ⊗ Bred = 0. So the deletion itself is not the core problem. The load-bearing gap is the locality half of Theorem 1: no explicit open-boundary embedding is given, and the geometry of cyclic shifts can place the support of a kept stabilizer on both sides of the cut. This is a correctness risk for the practical claim, not merely a stylistic issue. It is also fixable, for example by deleting a different set of consecutive rows and columns, which is why a full rejection is not warranted. The bivariate examples and fold-transversal claims have additional reproducibility gaps, but the locality of the main construction is the most central unresolved point. The verdict stays CONDITIONAL, consistent with the reader's assessment, with the concrete test above determining whether the concern actually lands.","tokens_in":12190,"tokens_out":20089,"duration_ms":215181,"concrete_test":"Implement Lemma 4 for l=m=6, A(x)=B(x)=1+x+x^2 (both divide x^6-1). Fix S_6 as multiplication by x, i.e., (S f)_i = f_{i-1} mod 6. Build Ared by deleting rows 0 and 1 of A(S_6), Bred by deleting columns 4 and 5 of B(S_6), and form the pruned H_X and H_Z exactly as in Lemma 4. Place the 36 horizontal and 16 vertical qubits on a 6x6 rectangle with open boundaries and compute, for every remaining check, the maximum L1 distance between any two qubits in its support. If any check has support spanning both sides of a cut (e.g., positions 5 and 0), the locality claim fails for this instance. If all supports have diameter at most deg A + deg B, rerun the same test with the opposite cyclic-shift convention to determine whether the result depends on the convention, and report the maximum diameter and whether any edge crosses the boundary.","verdict_should_be":"UNCHANGED","load_bearing_attack":"Theorem 1 has two parts: the parameter formula (Proposition 1) and the claim that the pruned code is 'as local as Q on a lattice with open boundary conditions.' Proposition 1 follows from the hypergraph-product distance formula once one checks that C^T_Ared and C_Bred are trivial, which is routine. The locality claim, however, is not proved. Lemma 4 only says 'Effectively, we just removed columns and rows,' but no embedding of the remaining Tanner graph into an open-boundary lattice is given, and no argument shows that every remaining stabilizer support is contained in a bounded neighborhood. This matters: if S_l is the standard multiplication-by-x shift, then for A(x)=1+x+x^2 and l=6 (allowed since 3 divides l), deleting the first rA=2 rows leaves the rows indexed 4 and 5, whose supports are {4,5,0} and {5,0,1} modulo 6. After cutting the periodic boundary, those stabilizers connect positions 0 and 5, whose distance in the open 1D layout is 5, not O(deg A). The parameter formula may still hold algebraically, but the code would not have the advertised locality. The paper does not specify a shift convention or a placement rule that rules this out; a different choice of which consecutive rows to delete may repair the issue, but that choice is not made or analyzed.","agreement_with_reader":"partial"},"referee_report":{"model":"deepseek-v4-flash","summary":"This paper introduces a notion of \"pruning\" quantum stabilizer codes—deleting selected qubits and stabilizer generators while preserving commutativity—as a way to convert bivariate bicycle codes from periodic to open boundary conditions. For hypergraph products of cyclic codes, i.e., bivariate bicycle codes specified by univariate polynomials A(x) and B(y) with A(x)|(x^l-1) and B(y)|(y^m-1), the paper states Theorem 1 and Proposition 1: there exists a pruned code with parameters [[lm+(l-r_A)(m-r_B), r_A r_B, min(d_A,d_B)]] that is \"as local as\" the parent code on a lattice with open boundary conditions, halving the number of logical qubits while preserving distance and roughly preserving the physical qubit count. The paper also gives explicit examples of pruning non-hypergraph bivariate bicycle codes, including a honeycomb color code family, and discusses how fold-transversal gates can be carried over to the pruned codes.","tokens_in":12462,"tokens_out":16669,"duration_ms":163708,"significance":"Conditional on the proof being completed, the construction is appealing: it offers a concrete route from periodic bivariate bicycle codes to open-boundary codes with preserved distance and locality, which is directly relevant to hardware implementations. The algebraic part is clean and uses standard tools: Lemma 1 and Proposition 1 follow from cyclic-code theory and the Tillich-Zemor hypergraph product formula, and the parameter computation is transparent. The paper is also honest about its exploratory scope, presenting the non-hypergraph cases as examples rather than as a general theorem. Its main weakness is that the geometric claim underlying Theorem 1—the open-boundary locality of the pruned Tanner graph—is not actually proved, and the gate-compatibility proposition is asserted rather than demonstrated. These gaps are local in the sense that they may be fixable, but they are load-bearing for the paper's central claims.","major_comments":[{"comment":"The proof that C_Ared ⊠ C^T_Bred is \"as local as\" Q on a lattice with open boundary conditions is missing. The one-sentence proof of Lemma 4 only says that rows and columns of the check matrices are removed; it does not specify which deleted rows and columns correspond to which removed qubits and stabilizers, nor does it give an explicit embedding of the remaining Tanner graph into an open-boundary lattice. The following claim, \"by construction as local ... but does not utilize the periodic boundary conditions,\" therefore does not follow from the stated argument. Concretely, for l=6 and A(x)=1+x+x^2, r_A=2, deleting the first two rows of A(S_6) leaves row 4 with support {4,5,0} modulo 6; if the open boundary is placed between positions 5 and 0, that stabilizer has diameter 5 in the open layout, not r_A+1. A different cut or a different choice of which r_A consecutive rows to delete may repair this, but the paper neither specifies nor analyzes such a choice. Since open-boundary locality is the central novelty of Theorem 1, an explicit placement rule and a proof that every remaining stabilizer has bounded support in the open layout are required.","section":"Section 3, Lemma 4 and following paragraph"},{"comment":"The compatibility of the ZX-duality pi with pruning is asserted without proof. The statement that the restriction to the remaining physical qubits \"still yields a unitary on the logical level\" is not formalized: when pruning removes one member of a two-qubit orbit, one must specify how the fold-transversal physical gate is modified, and then prove that the modified operator preserves the stabilizer group of the pruned code and induces the claimed logical action on the surviving logical qubits. The appeal to Figures 4b and 5c is not a substitute for this argument. A concise proof for the general hypergraph-product case of Lemma 4 should be supplied.","section":"Section 4, Proposition 2"}],"minor_comments":[{"comment":"The sentence \"Plugging this into Lemma 2\" should refer to Theorem 2, since the hypergraph product parameter formula is Theorem 2, not Lemma 2.","section":"Section 3, proof of Proposition 1"},{"comment":"The symbol l is used both for the number of stabilizer generators in Definition 1 and for the horizontal lattice size in Section 2.4; this overloaded notation should be disambiguated.","section":"Section 2.4 and Definition 1"},{"comment":"The statement that the pruned examples were constructed by computer search over certain convex regions is not accompanied by any details of the search or by a verification method; please add enough information for the reader to reproduce these claims.","section":"Example 2"}],"recommendation":"major_revision","confidential_remarks":"This is a well-written exploratory paper whose parameter algebra is sound, but the main theorem currently rests on a proof sketch for the open-boundary locality claim. I believe the missing locality proof and the missing proof of Proposition 2 are both fixable within the manuscript's scope, so I recommend major revision rather than rejection. The bivariate examples are interesting but not fully general, and the paper is appropriately transparent about this."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"Short version: this is a useful, honest methods paper. It gives a simple pruning construction that converts hypergraph-product bivariate bicycle codes into codes with open boundary conditions, halving the logical dimension and keeping distance, and it demonstrates the idea on two non-hypergraph-product families—the honeycomb color code and the A=1+x+xy+y^-1 family. The univariate theorem is the real result; the bivariate examples are exploratory.\n\nWhat did well: Proposition 1 is clean and the parameter computation follows from standard cyclic-code facts and the Tillich–Zémor formula. I verified the logic: the reduced codes C_Ared and C^T_Bred have the claimed dimensions, and the hypergraph product gives [[lm+(l-rA)(m-rB), rA rB, min(dA,dB)]]. The remark that pruning a trivial code can yield a nontrivial code is a nice sanity check. The paper also honestly says the general bivariate question is open and cites the related work it overlaps with.\n\nThe soft spots: Lemma 4 is a one-sentence sketch. The paper never specifies the placement rule that makes the pruned code local on an open-boundary lattice. The stress-test example with A=1+x+x^2 and l=6 is a fair illustration: if you delete the first two rows and cut the cycle at the usual place, surviving checks wrap across the cut. A cyclic shift of the layout probably fixes this, but the authors don't say so, and the claim 'by construction' needs a real argument. The bivariate examples are figure- and search-based, with no code or explicit check matrices; that's a reproducibility gap, but the examples look plausible and the color-code connection is a useful pointer. The fold-transversal section is informal but the compatibility statement is reasonable.\n\nWho it's for: people building qLDPC hardware or studying boundaries of bicycle codes. It's not a breakthrough, but it's a concrete step and the parameter trade-off is worth knowing. I'd send it to peer review; the referee should ask for a rigorous proof of the locality claim, a precise statement of the deletion/placement rule, and either code or explicit stabilizer lists for the bivariate examples. With those fixes it would be a solid contribution.","headline":"Useful methods paper with a sound univariate pruning result, exploratory bivariate examples, and a locality proof that needs spelling out.","tokens_in":13000,"tokens_out":12780,"would_cite":true,"duration_ms":124482,"reading_group":"maybe","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["81P70"],"pacs":["03.67.Pp"],"model":"deepseek-v4-flash","headline":"By deleting selected qubits and stabilizers from bivariate bicycle codes, one obtains a code on an open-boundary lattice with the same distance and locality but half the logical qubits.","keywords":["quantum error correction","bivariate bicycle codes","qLDPC codes","open boundary conditions","pruning","hypergraph product codes","fold-transversal gates","cyclic codes"],"falsifier":"Take a small example, say $\\ell = m = 6$ with $A(x) = 1 + x + x^2$ and $B(y) = 1 + y + y^2$, construct $A_{\\mathrm{red}}$ and $B_{\\mathrm{red}}$ as in Lemma 4, and explicitly form the hypergraph product; check whether its X- and Z-check matrices satisfy $H_X H_Z^T = 0$ and whether its distance equals $\\min(d_A,d_B)$. Any failure—non-commuting checks, or a lower distance—would disprove the theorem.","tokens_in":11992,"feed_emoji":"⚛️","tokens_out":10553,"duration_ms":85478,"temperature":0.7,"pith_summary":"Bivariate bicycle codes are quantum low-density parity-check codes whose local checks live on a 2D lattice, but the lattice has periodic boundary conditions, which complicates hardware implementation. This paper introduces a pruning operation—deleting selected qubits and stabilizers while keeping the remaining checks commuting—and proves that every bivariate bicycle code that is a hypergraph product of two classical cyclic codes can be pruned onto a lattice with open boundary conditions. For $A(x)$ dividing $x^\\ell - 1$ and $B(y)$ dividing $y^m - 1$, the pruned code has the same distance, roughly the same physical qubit count, and exactly half the logical qubits, with parameters $[[\\ell m + (\\ell - r_A)(m - r_B), r_A r_B, \\min(d_A,d_B)]]$. The paper also gives examples of genuinely bivariate (non-hypergraph-product) codes that can be pruned, including the honeycomb color code, and shows that some fold-transversal gates survive the pruning. The importance is that this moves low-overhead bivariate bicycle codes one step closer to planar hardware.","feed_headline":"Pruning cuts logical qubits in half, keeps distance and locality","feed_subtitle":"New construction removes periodic boundary conditions from bivariate bicycle codes without distance loss.","key_machinery":"The key object is the pruning operation on stabilizer codes: choose subsets of qubits and of stabilizer generators, restrict the surviving generators to the surviving qubits, and require the restricted X- and Z-checks to commute. For cyclic codes, the construction uses reduced parity-check matrices: $A_{\\mathrm{red}}$ deletes the first $r_A$ rows of $A(S_\\ell)$, and $B_{\\mathrm{red}}$ deletes the last $r_B$ columns of $B(S_m)$. The pruned code is then the hypergraph product $C_{A_{\\mathrm{red}}} \\boxtimes C^T_{B_{\\mathrm{red}}}$, which is shown to be a literal row-and-column deletion of the original bivariate bicycle code's check matrices. This explicit deletion description is what guarantees the open-boundary layout: the deleted rows and columns are exactly the qubits and stabilizers whose checks would wrap around the periodic boundary, and since hypergraph products of cyclic codes automatically satisfy the CSS commutativity condition, the remaining code is valid and local.","core_discovery":"The central discovery is Theorem 1: if $Q$ is a bivariate bicycle code built from univariate polynomials $A(x)$ and $B(y)$ with $A(x) \\mid (x^\\ell - 1)$ and $B(y) \\mid (y^m - 1)$, then there exists a pruned bivariate bicycle code that is as local as $Q$ on a lattice with open boundary conditions. Concretely, the pruned code is the hypergraph product $C_{A_{\\mathrm{red}}} \\boxtimes C^T_{B_{\\mathrm{red}}}$, where $A_{\\mathrm{red}}$ and $B_{\\mathrm{red}}$ are the parity-check matrices of the original cyclic codes with redundant rows (respectively columns) deleted. Proposition 1 gives its parameters as $[[\\ell m + (\\ell - r_A)(m - r_B), r_A r_B, \\min(d_A,d_B)]]$, so the distance is unchanged, the physical overhead is nearly unchanged, and the logical dimension drops from $2 r_A r_B$ to $r_A r_B$. The same construction, applied when the divisibility condition fails by first replacing $A$ and $B$ with their greatest common divisors with $x^\\ell - 1$ and $y^m - 1$, yields pruned codes with identical parameters. For the non-hypergraph-product regime, explicit prunings of the honeycomb color code and of a code with parameters $[[30,2,4]]$ and $[[66,2,6]]$ are exhibited.","pith_inferences":["If the row-and-column deletion argument can be extended to all bivariate bicycle codes, the high-rate codes from the main motivating family could run on planar hardware; the examples here suggest the obstruction is practical rather than structural.","The halving of the logical dimension may reflect a general trade-off: changing a periodic-boundary local code into an open-boundary local code with the same check weights seems to force the loss of roughly half the logical qubits.","The computer search over convex lattice regions used for the examples could be systematized into an algorithm for finding open-boundary prunings of arbitrary bivariate bicycle codes.","Combining pruning with the boundary gauge operators studied in the works on extracting topological orders of Pauli stabilizer codes might yield open-boundary codes with better parameters than direct pruning alone."],"forward_implications":["Every hypergraph-product bivariate bicycle code satisfying the divisibility condition can be converted to an open-boundary code with the same distance and locality, at the cost of half its logical qubits.","The conversion preserves fold-transversal gates: the ZX-duality that implements the logical S/CZ-type gates restricts to the pruned code, so fault-tolerant Clifford operations remain available.","When $A(x)$ does not divide $x^\\ell - 1$, one can first replace $A$ by $\\gcd(A, x^\\ell - 1)$ and $B$ by $\\gcd(B, y^m - 1)$; the resulting code has the same parameters and then satisfies the divisibility condition, so pruning still applies.","The explicit examples—the honeycomb color code and the codes with parameters $[[30,2,4]]$ and $[[66,2,6]]$—show that pruning is not restricted to hypergraph products, suggesting a more general mechanism."],"supporting_citations":[{"why":"Introduces bivariate bicycle codes, the family of codes the paper prunes.","marker":"[KP13]"},{"why":"Provides the hypergraph product construction and the parameter formula used to derive Proposition 1.","marker":"[TZ14]"},{"why":"Supplies the practically interesting high-rate bivariate bicycle codes and the fold-transversal gate context motivating open-boundary variants.","marker":"[BCG+23]"},{"why":"Develops the logical-operator and fold-transversal-gate framework for bivariate bicycle codes that the paper extends to pruned codes.","marker":"[ES24]"},{"why":"Provides the logical operator basis and phase-type gate action used to explain why the fold-transversal gate remains logical after pruning.","marker":"[QWV23]"},{"why":"Identifies the honeycomb color code family used as the non-hypergraph-product pruning example.","marker":"[BMD06]"}],"fun_headline_variants":["Pruning halves logical qubits, preserves distance and locality","Open boundaries for bivariate bicycle codes without distance loss","Pruning qLDPC codes: open boundaries, same distance","Pruned bicycle codes: halved qubits, kept distance and locality","New pruning method opens boundaries, halves logical qubits"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"Everything rests on the claim in Lemma 4 that the reduced hypergraph product code is literally obtained from the original code by deleting specified rows and columns of the check matrices, so that commutativity and parameters follow without any additional adjustment.","fun_headline_variants_meta":{"raw":{"variants":["Pruning halves logical qubits, preserves distance and locality","Open boundaries for bivariate bicycle codes without distance loss","Pruning qLDPC codes: open boundaries, same distance","Pruned bicycle codes: halved qubits, kept distance and locality","New pruning method opens boundaries, halves logical qubits"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000673,"raw_usage":{"total_tokens":3098,"prompt_tokens":1015,"completion_tokens":2083,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":631,"completion_tokens_details":{"reasoning_tokens":1999}},"tokens_in":631,"tokens_out":2083,"duration_ms":15037,"temperature":1.0,"reasoning_tokens":1999,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-11T21:40:24.438221+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"Take a small example, say $\\ell = m = 6$ with $A(x) = 1 + x + x^2$ and $B(y) = 1 + y + y^2$, construct $A_{\\mathrm{red}}$ and $B_{\\mathrm{red}}$ as in Lemma 4, and explicitly form the hypergraph product; check whether its X- and Z-check matrices satisfy $H_X H_Z^T = 0$ and whether its distance equals $\\min(d_A,d_B)$. Any failure—non-commuting checks, or a lower distance—would disprove the theorem.","supporting_citations":[],"review_version":1}