arXiv subjects

Anton Pakhunov

Publications and source records attributed to Anton Pakhunov.

5 recordsLinked to original sources

arXiv2026

No finite level of the NPA hierarchy is exact for the doubly-tilted CHSH functional near the critical tilt

Gigena, Panwar, Scala, Araujo, Farkas and Chaturvedi [npj Quantum Inf. 11, 82 (2025)] determined the quantum maximum of the doubly-tilted CHSH functionals, observed that the Navascues-Pironio-Acin level needed for exactness grows without evident bound toward the critical tilt, and asked whether any finite level suffices. We answer this in the negative. For the symmetric critical family $B_s=(1-s/2)(\langle A_0\rangle+\langle B_0\rangle)+\mathrm{CHSH}$ we prove: for every NPA level $k\ge 2$ there are an explicit rational $g_k>0$ and an $s^*_k>0$ with $c_k(s)\ge 4-s+g_k s^2$ on $(0,s^*_k]$; since the quantum value leaves the local bound only cubically, every finite level strictly overshoots on an interval: no finite level is exact on any neighbourhood of the critical point. Unconditionally $a_2>1/39$, $a_3>1/188$, $a_4>1/641$. The proof is a primal construction: an exactly feasible moment curve at each level, built from level-uniform structural laws and one level-independent signed witness -- a closed-form class function $y^*$ with $N_k^TΓ(y^*)N_k=u_k u_k^T$ at every level. The mechanism forces the sign: for $k\ge 3$ no quantum state and no smooth curve of quantum models can realize the gain direction, so the overshoot lives strictly in the non-quantum part of the NPA tangent cone. The proof is computer-assisted in the strict sense: finite exact-integer verifications with proven degree bounds are constituent parts of the argument; the chain has been re-verified against independent implementations, including a symbolic per-regime proof of the witness identity and a clean-room implementation written from the paper text alone. The one external input is the published quantum value of Gigena et al., cross-checked to twelve digits.

quant-ph↗

arXiv2026

A phase transition in the exactness of the NPA hierarchy at the critical doubly-tilted CHSH functional

Gigena et al. [npj Quantum Inf. 11, 82 (2025)] proved the exact quantum maximum of the doubly-tilted CHSH functional $B_{αβ}=α\langle A_0\rangle+β\langle B_0\rangle+\mathrm{CHSH}$ and observed that the NPA level required to reach it grows without evident bound toward the critical line $α+β=2$. We quantify the mechanism on the symmetric slice $s=2-α-β$: (i) the quantum value leaves the local bound cubically, $c_Q=4-s+s^3/6-s^4/36+O(s^5)$; (ii) each NPA level overshoots quadratically, $c_k(s)=4-s+a_k s^2+O(s^3)$, with the almost-quantum coefficient computed exactly, $a_{1+AB}=3/64$; (iii) the divergence of the required exact level is equivalent to positivity of the single sequence $(a_k)$ - proven for every $k$ in the companion paper. We prove the supercritical side completely: for all $α,β\ge 1$ and every level the hierarchy is exact, via three explicit rational certificates realizing an affine identity. The hierarchy's exactness thus undergoes a phase transition at the critical line. On the subcritical side we certify the first four levels in exact arithmetic (rational pseudo-moments beating $c_Q$, confirmed by Sturm's theorem). We identify the exact mechanism: rescaled to the critical corner, the limiting obstruction is the Motzkin polynomial, the classical nonnegative-but-not-sum-of-squares form, so the finite-level failure sits in the restricted-certificate regime. The phase boundary has a precise geometric reading via Nie's finite-convergence theorem and Marshall's boundary Hessian condition: a self-tested optimum is finitely NPA-certifiable whenever its boundary Hessian is nondegenerate (contact order two), which holds for the single tilt and fails exactly at the doubly-tilted cubic touch. Three verified errata in the published polynomial system of Gigena et al. are documented.

quant-ph↗

arXiv2026

Mycelium-Index: A Streaming Approximate Nearest Neighbor Index with Myelial Edge Decay, Traffic-Driven Reinforcement, and Adaptive Living Hierarchy

We present mycelium-index, a streaming approximate nearest neighbor (ANN) index for high-dimensional vector spaces, inspired by the adaptive growth patterns of biological mycelium. The system continuously adapts its topology through myelial edge decay and reinforcement, a traffic-driven living hierarchy, and hybrid deletion combining O(1) bypass for cold nodes with O(k) beam-search repair for hub nodes. Experimental evaluation on SIFT-1M demonstrates that mycelium achieves 0.927 +/- 0.028 recall@5 under FreshDiskANN's 100%-turnover benchmark protocol -- within the measurement confidence interval of FreshDiskANN's ~0.95 -- while using 5.7x less RAM (88 MB vs. >500 MB) and achieving 4.7x higher QPS (2,795 vs. ~600). On the static index, at ef=192, mycelium matches HNSW M=16 recall (0.962 vs. 0.965) at 5.2x less RAM (163 MB vs. 854 MB). Performance optimizations including NEON SIMD distance computation, Vec-backed node storage, and bitset visited tracking yield a cumulative 2.7x QPS improvement. A systematic study of ten streaming repair mechanisms finds that geometric heuristics universally fail in high dimensions, while topological mechanisms succeed -- a principle we term the topological repair invariance of high-dimensional ANN graphs.

cs.LG↗

arXiv2026

Analytical Theory of Greedy Peeling for Bivariate Bicycle Codes and Two-Shot Streaming Decoding

We present an analytical theory of greedy peeling decoding for bivariate bicycle (BB) codes under circuit-level noise. The deferred greedy decoder achieves 330x latency reduction over belief propagation (BP) at p = 10^{-3} while maintaining identical logical error rate. Our main theoretical contribution is a closed-form collision resolution factor A_0 = |true collisions| / |birthday collisions|, derived from XOR syndrome analysis with no free parameters, that quantifies the fraction of detector-sharing fault pairs genuinely blocking iterative peeling. For the [[144,12,12]] Gross code, A_0 = 0.8685 (within 0.5% of the empirical value), with shared-2 pairs (4-cycles) always resolving under peeling. We show A_0 depends on the mean fault-graph degree d-bar rather than code size: A_0 = 0.87 for d-bar = 52 (Gross family) versus A_0 = 0.76 for d-bar = 17 ([[32,8,6]]). We establish a syndrome code stopping distance d_S = n/4.5 for the Gross family and demonstrate that [[32,8,6]] (d_S = 4) enables two-shot streaming decoding: T = 2 rounds achieve 89% peeling success with 1.29 +/- 0.03 LER ratio versus T = 12, at estimated latency ~50 ns. The full formula P_peel = exp(-A_0 * gamma_analytic * exp(-BTp) * n * p^2) is validated across five BB codes, four noise levels, and four values of T with R^2 = 0.86. Cross-platform reproduction of the Kunlun [[18,4,4]] experiment matches their hardware LER within 0.73 percentage points.

quant-ph↗

arXiv2026

Belief Propagation Convergence Prediction for Bivariate Bicycle Quantum Error Correction Codes

Decoding Bivariate Bicycle (BB) quantum error correction codes typically requires Belief Propagation (BP) followed by Ordered Statistics Decoding (OSD) post-processing when BP fails to converge. Whether BP will converge on a given syndrome is currently determined only after running BP to completion. We show that convergence can be predicted in advance by a single modulo operation: if the syndrome defect count is divisible by the code's column weight w, BP converges with high probability (100% at p <= 0.001, degrading to 87% at p = 0.01); otherwise, BP fails with probability >= 90%. The mechanism is structural: each physical data error activates exactly w stabilizers, so a defect count not divisible by w implies the presence of measurement errors outside BP's model space. Validated on five BB codes with column weights w = 2, 3, and 4, mod-w achieves AUC = 0.995 as a convergence classifier at p = 0.001 under phenomenological noise, dominating all other syndrome features (next best: AUC = 0.52). The false positive rate scales empirically as O(p^2.05) (R^2 = 0.98), confirming the analytical bound from Proposition 2. Among BP failures on mod-w = 0 syndromes, 82% contain weight-2 data error clusters, directly confirming the dominant failure mechanism. The prediction is invariant under BP scheduling strategy and decoder variant, including Relay-BP - the strongest known BP enhancement for quantum LDPC codes. These results apply directly to IBM's Gross code [[144, 12, 12]] and Two-Gross code [[288, 12, 18]], targeted for deployment in 2026-2028.

quant-ph↗