arXiv subjects

Aryeh Lev Zabokritskiy

Publications and source records attributed to Aryeh Lev Zabokritskiy.

At least 19 recordsLinked to original sources

arXiv2026

From Round-Trip State Echo to Error Recovery: Snapshot-Resolved Quantum-Hardware Diagnostics

We examine which conclusions from quantum-hardware benchmarks persist when the same programs are run again. On eight fixed routes of IQM Emerald, round-trip state echo (RTSE), a diagnostic of state return after a round trip, gave mean scores near $0.84$ that differed by only $0.00125$ between two executions. Individual route estimates nevertheless changed by up to $0.06$ in both directions. The mean difference between RTSE and measurement-free teleportation along the same routes shrank from $0.046$ to $0.014$ and was no longer statistically resolved on repetition; none of the four prespecified communication targets met its replication rule. Recovery with an existing four-qubit single-deletion code also changed on repetition: after discarding one encoded qubit in the circuit model, the IQM mean fell from $0.738$ to $0.624$ and failed the replication test against the two-thirds operational reference. In a separate IonQ experiment, recovery means of $0.91$ and $0.92$ passed tests against that reference and against a deliberately incorrect decoder in both executions. These results concern the tested workloads and executions. They show why a nearly unchanged aggregate benchmark score needs to be interpreted alongside individual routes, direct task comparisons, and repeated executions.

quant-ph↗

arXiv2026

Stabilizer Learning under Global Depolarizing Noise with Polynomial Copies and $2^t$ Time

We study learning the stabilizer structure of an unknown pure $n$-qubit state observed through global depolarizing noise of unknown strength, with a fixed promised upper bound below one, using destructive two-copy Bell measurements. Let $0\le t\le n$ bound the stabilizer nullity, let $\varepsilon>0$ be the accuracy, and let $0<δ<1$ be the failure tolerance. With probability at least $1-δ$, a learner must return a commuting unsigned Pauli group containing every stabilizer of the pure state; averaging conjugation by that group must decrease the purity of the observed noisy state by at most $\varepsilon$. Tang and Steckmann gave such a learner with polynomially many copies and $2^t$ times polynomial time when the noise strength is below a threshold of at most about $0.16$, which tends to zero as $t/n\to1$. They asked whether other algorithms with these resources can exceed this threshold. We answer this question affirmatively for every fixed bound on global depolarizing noise below one and every $0\le t\le n$, including states with only one promised independent stabilizer. Our learner uses polynomially many copies and $2^t$ times polynomial classical running time. All resource bounds hold on every execution. Their polynomial factors depend on $n,1/\varepsilon,\log(1/δ)$, with degrees and constants depending on the fixed noise promise.

quant-ph↗

arXiv2026

Concatenation Locality under Successive Adjacent Transpositions

We study codes correcting up to $t\geq1$ successive adjacent transpositions, using block concatenation to support blockwise encoding and decoding. A fixed set $P$ of words of length $b$ over a $q$-symbol alphabet generates, for every $m$, the length-$mb$ code of all concatenations of $m$ words from $P$. For our explicit constructions, the prepared encoder and decoder depend only on the block parameters and are reused at every concatenation length. We first prove that correction at every concatenation length is certified by a single finite test at $2t+1$ blocks, and this test is tight over a $q$-symbol alphabet exactly when $\binom q2\geq2t$. For binary words and $b\geq2t+1$, we give an explicit construction with total redundancy at most $mt\log_2 b+O_t(m)$, matching the optimal per-block redundancy $t\log_2 b+O_t(1)$ among block sets correcting at every concatenation length. Among all binary $t$-correcting codes of length $mb$, the optimal total redundancy is $t\log_2(mb)+O_t(1)$. Using all concatenations of one block set makes our total redundancy grow linearly in $m$ and buys independent block encoding, with one encoder and decoder at every length. The construction encodes and decodes in time linear in $mb$ for fixed $b,t$, and recovers the $j$-th source block from the first $jb$ received symbols. We also prove that, whenever the length-$mb$ code corrects $t$ transpositions, the $j$-th source block is determined within $2tb$ received symbols beyond position $jb$, and that explicit binary block sets with rates tending to one as $b$ grows with $t$ fixed attain this bound in the worst case. Finally, we prove that every ordinary correcting code contains a constant fraction of its words whose arbitrary concatenations retain correction, for fixed $q,t$.

cs.IT↗

arXiv2026

Optimal GHZ extraction from MABK violations

We determine the least Greenberger-Horne-Zeilinger (GHZ) extractability compatible with a Mermin-Ardehali-Belinskii-Klyshko (MABK) Bell score for every $n\geq3$. Extractability is the largest squared overlap with a GHZ state obtainable by separate local quantum channels. Between the biseparable bound and the quantum maximum, its exact minimum is the affine interpolation from one half to one. This proves the bound conjectured for six or more parties. The result holds for arbitrary binary measurements and density operators on tensor products of finite- or infinite-dimensional local spaces. We also determine the exact minimum for qubits, which is strictly higher at every interior score. One qutrit and qubits elsewhere attain the unrestricted minimum with measurements fixed as the score varies and, at interior scores, realize the smallest product of ranks of the local reduced states among states attaining that minimum. For $n\geq4$, the attaining state with this rank product is unique at each interior score, up to local isometries and party permutations. On one qutrit and qubits elsewhere, for $n\geq4$ and scores a fixed fraction away from both endpoints, a sufficiently small excess of extractability above the minimum controls trace-norm distance to the attaining state of the same score, up to local unitaries. The dependence is of square-root order, with optimal exponent and explicit constants depending only on that fraction, uniformly in $n$. Equality and stability concern actual extractability; identical Bell statistics can accompany different extractabilities.

quant-ph↗

arXiv2026

Exact value and rigidity of the $2\times3$ magic rectangle

The $2\times3$ magic rectangle is a nonlocal game in which two players match entries subject to incompatible parity constraints. Its quantum value was suggested by numerical bounds and known quantum random access codes. We prove that this value is $(1+\sqrt{2/3})/2$ and classify all optimal strategies in arbitrary finite local dimensions, allowing general measurements: up to local isometries and ancillary systems, each contains a maximally entangled pair of four-dimensional systems with fixed measurements on the local state supports. The same value holds for commuting-operator strategies and for the almost-quantum relaxation, confirming the Adamson--Wallden conjecture for this rectangle. For finite-dimensional magic-square strategies, the largest score compatible with perfect prediction of Alice's complete first row by an adversary with quantum side information is one third plus two thirds of the rectangle value. We classify the strategies attaining this threshold with perfect prediction and give linear-order upper and lower bounds on one minus the best compatible guessing probability above it. The rectangle value is also the optimal average success probability of finite-dimensional three-bit random access codes that hide the full input parity, and we classify all optimal codes.

quant-ph↗

arXiv2026

Bosonic Loss Codes with Bounded Occupation

We study Fock-state codes on $q$ bosonic modes, with total excitation $N$ and at most $B$ excitations per mode, that exactly correct every pattern of at most $δN$ photon losses, for $q$ proportional to $N$. For every fixed cap $B$ and every ratio $q/N>1/B$, we prove explicit achievable rates, positive for all sufficiently small $δ>0$, in logical qubits per excitation. At $B=2$ and $q=N$, the rate is positive for every $δ<0.1355$ and is about $0.281$ at $δ=0.1$, compared with the previously established exact-code qutrit rate of about $0.275$. We also give an explicit upper bound on the dimension of every such exact code, including degenerate codes. For the pure-loss channel truncated to at most $δN$ losses and renormalized, we obtain approximate codes with exponentially small worst-case error. When the loss probability is positive and at most $1/(B+1)$, they improve our positive exact-code rate bounds; at $B=2$, $q=N$, and loss probability $0.1$, they remain positive up to $δ\approx0.158$.

quant-ph↗

arXiv2026

The strengthened quantum Hamming bound of Yu, Lai, and Oh

We prove a conjecture of Yu, Lai, and Oh: their strengthened quantum Hamming bounds, a family that interpolates between Hamming-type bounds and the Singleton bound, hold for all quantum codes of dimension $K>1$, including impure and nonadditive codes. The result covers every integer local dimension $p\ge2$, not necessarily a prime power, every integer $t\ge1$, both distances $2t+1$ and $2t+2$, and every interpolation parameter $e\in\{0,\ldots,t\}$. The case $e=0$ implies the ordinary quantum Hamming bound for general codes, recently proved by Zhang and Chen. Yu, Lai, and Oh proved their bounds for pure codes and for general codes of distance three or four. More generally, for every integer $p\ge2$ and every quantum code of length $n$, dimension $K>1$ and distance $d\ge3$, we prove $Kf(0)\le p^n\mathbb{E} f(X)$, where $X\sim\operatorname{Bin}(n,1-p^{-2})$, for every real polynomial $f$ of degree less than $d$ that is nonnegative on $\{0,\ldots,n\}$ and positive at zero. For pure codes this inequality is implicit in the argument of Yu, Lai, and Oh. The proof uses the linear-programming method with quantum weight enumerators.

quant-ph↗

arXiv2026

Systematic Codes Correcting Two Deletions with Three Redundant Symbols over Large Alphabets

We construct systematic $q$-ary codes correcting two deletions with three redundant symbols over polynomial-size alphabets. For every information length $n\ge2$, our first construction has length $n+3$ over an alphabet of size $q=2^s<2^{32}n^7$ for some integer $s$, with polynomial-time encoding and decoding. A cardinality bound of Liu and Xing implies that three redundant symbols are necessary, so the construction is optimal in this measure. We also establish the existence of systematic codes with the same redundancy over alphabets of size $O(n^4)$, leaving efficient encoding and decoding at that alphabet scale open. For a single burst of at most two adjacent deletions, we give a polynomial-time systematic construction with three redundant symbols over a power-of-two alphabet of size $q<(n+2)^4$. A converse bound for adjacent deletions shows that three redundant symbols are again optimal for $n\ge2$.

cs.IT↗

arXiv2026

How Far Can You Do Nothing On a Quantum Computer?

We present a route-resolved comparative assessment of Rigetti's Cepheus-1-108Q and IBM Heron-r2 processors using the established do-nothing state-transfer protocol. Rather than proposing a new protocol, we use this deterministic, low-complexity task as a high-resolution spatial probe. For each evaluated initial qubit, we report two complementary quantities: the largest tested radius within which every evaluated shortest route satisfies the operational success rule, and the longest successful route identified within the evaluated route family. To address the question "How far can you do-nothing on a quantum computer?", we prepare a quantum state, route it outward using SWAP gates, apply the inverse preparation at the destination, and route it back for measurement. We compare the measured probability of returning $0$ with the empirical cutoff $2/3$. For this circuit ordering, that comparison does not certify above-classical unknown-state transfer. While this trivial state-transfer protocol serves as the most intuitive baseline, actively preserving a quantum state across a physical lattice proves to be a non-trivial task that exposes the information to cumulative relaxation, dephasing, and environmental cross-talk. In the highlighted IBM QPU case, we identify an isotropic radius of 10 and a successful path of swap distance 27, whereas the highlighted Rigetti Cepheus case exhibits an isotropic radius of 1 but selected above-threshold routes reaching swap distance 8. These results reveal a sharp distinction between uniform spatial reliability and best-route performance. The presented quantities are empirical and conditional on the evaluated route families, finite-shot decision rule, calibration state, and execution time; they are not architecture-wide constants.

quant-ph↗

arXiv2026

Efficient Non-Uniform Quantum Hermite Transform through Adaptive Sampling

On the span of the first $N$ oscillator modes, Gauss--Hermite quadrature gives an exact change of basis between mode coefficients and $N$ weighted position space samples. We implement this transform with $O(N\operatorname{polylog}(N,1/\varepsilon))$ logical gates and polylogarithmic quantum width. The operator-error bound $\varepsilon$ holds on arbitrary superpositions and includes all auxiliary registers. The construction uses signed averages on adaptive windows to convert uniform-grid samples into weighted Hermite-root samples. Their varying widths control the amplification cost, giving the near-linear bound.

quant-ph↗

arXiv2026

Quantum Leakage from Shamir Sharing with Correlated Resources

We study leakage from Shamir secret sharing when each device uses its share to produce one qubit from an initial quantum resource independent of the secret and sharing randomness. We prove exponentially small trace distance between the leakage states of any two secrets, when reconstruction requires a sufficiently large fraction of the shares, for fixed stationary chains satisfying an upper operator mixing condition and for resources generated by fixed finite quantum memories with primitive transfer channels. These guarantees permit arbitrary extensions of the device marginal as adversary side information and hold uniformly over prime fields larger than the number of shares. For independent device--reference pairs, which provide the quantitative input to the argument, we determine the optimal exponential base when reconstruction requires all shares: $2\sqrt2/π$ for a qubit and $2/π$ for a classical bit, in both cases allowing arbitrary finite references. A comparison of selected marginals with product states converts the bounds for independent device--reference pairs into security guarantees for the correlated resources.

quant-ph↗

arXiv2026

Correcting Connectivity in Arc-Based QUBO Models for Fixed-Fleet Vehicle Routing

We revisit a degree-only arc Hamiltonian for fixed-fleet, homogeneous, uncapacitated vehicle routing. Its local penalties define only a cycle cover, so ground states may contain customer cycles disconnected from the depot. We construct a polynomial-size quadratic unconstrained binary optimization (QUBO) repair using capped single-commodity flow and prove connected, cost-optimal ground-state routings under explicit penalty assumptions. For $N-1$ customers and $K$ nonempty routes, the unreduced encoding uses exactly $|E|(1+\lceil\log_2(N-K+1)\rceil)$ logical problem qubits. A reversible compute-phase-uncompute implementation evaluates the flow penalties in $O(N^2\log N+N\log^2N)$ logical gates on a complete graph, using $O(\log N)$ reusable workspace and no product register. On complete loopless graphs, a position encoding that arranges all routes in one sequence separated by depot occurrences uses fewer problem qubits and written terms when the number of bits per flow register grows. Under the same reversible accounting model, the flow formulation admits a smaller logical-gate upper bound. Exact Hamiltonian and circuit audits and a benchmark of 1,200 cost matrices verify the formulation and quantify the connectivity gap. A 32,000-shot Amazon Braket task on IQM Emerald characterizes depth-one termwise Ising circuits for a diagnostic $N=4$, $K=1$ counterexample. In the degree-only circuit, 78.05% of selected $p=1$ shots realize the invalid disconnected ground state; the reduced 14-qubit flow-augmented circuit yields no fully feasible sample. These device results characterize mapped Hamiltonians and compilation, not an asymptotic advantage in solving routing problems.

quant-ph↗

arXiv2026

Asymptotically unit-rate storage codes from binary BCH codes

A storage code on a graph assigns a symbol to each vertex so that the symbol can be recovered from its neighbors. We prove that the binary full-parity storage codes on triangle-free coset graphs of primitive BCH codes have rate tending to one for every fixed error-correction capability of at least two. The conclusion also holds for an increasing error budget under an explicit growth condition. The proof bounds the binary rank of convolution matrices associated with polynomial maps. Low coordinate degree forces cancellation in the polynomial representing the image, and a polynomial rank bound converts this cancellation into a quantitative estimate for the storage rate. For the double-error-correcting family over $\mathbb{F}_{2^m}$, we prove that the storage deficiency is $Θ(((1+\sqrt{5})/4)^m)$. This refinement follows from an exact Fibonacci formula for a polynomial coefficient rank and a matching exponential lower bound after finite-field evaluation.

cs.IT↗

arXiv2026

Binary Multiple-Node-Erasure-Correcting Codes over Complete Graphs: Constructions, q-Ary Metric Balls, and Duality

We study linear codes whose coordinates are the ordinary edges and self-loops of complete undirected graphs; a node erasure removes all coordinates incident with a failed vertex. The construction results are binary. For triple-node erasures, we extend the published cyclic construction by allowing a suitable cyclic check slope to depend on the prime graph length. An explicit determinant test proves that one of three fixed slope choices works at infinitely many prime lengths, unconditionally, and gives redundancy $3n-2$, one bit above the graph Singleton bound. We also give Singleton-optimal triple-node codes at $n=6,8,10,12$, together with a general ordinary-edge framework that isolates the remaining loop-completion problem. When $2$ is primitive modulo an odd prime $n$, a binary multi-slope construction corrects every $ρ$-node erasure for $2\leqρ<n$, with redundancy $ρn-(ρ-1)$ in the range $2\leqρ\leq(n+1)/2$. Returning to arbitrary prime powers, we derive exact generating transforms and inclusion--exclusion formulas for node-metric ball volumes, fixed-radius asymptotics, and packing, existence, and covering bounds. Finally, for the complementary clique-erasure metric, we obtain an exact weight enumerator and a Singleton-optimal node--clique duality.

cs.IT↗

arXiv2026

The Exact Second Generalized Covering Radius of Binary Primitive Triple-Error-Correcting BCH Codes

Let $C_m:=\mathrm{BCH}(3,m)$ be the binary primitive triple-error-correcting BCH code of length $2^m-1$. We determine its second generalized covering radius exactly: $R_2(C_m)=8$ for every $m\geq5$. Equivalently, every two-dimensional syndrome subspace is contained in the binary span of at most eight parity-check columns, and eight columns are necessary in the worst case. This matches the known lower bound.

cs.IT↗

arXiv2026

Asymptotically Tight Bounds for Generalized Covering Radii of Binary Primitive BCH Codes at All Higher Orders

We study the generalized covering radii of binary primitive BCH codes, which measure how many parity-check columns suffice to span several prescribed syndromes. For the four-error-correcting family of length $2^m-1$, the second radius is exactly $11$ for $m\geq55$, and it is either $11$ or $12$ for $m\geq16$. For every fixed error parameter at least two, we give an explicit stable bound with two adjacent possible values for the second radius and an arithmetic criterion for exactness. At every higher order, we obtain explicit lower and upper bounds whose additive gap is bounded independently of the order for each fixed error parameter, once explicit field-size conditions hold. The bounds are therefore asymptotically tight as the order grows. The higher-order upper bound retains the common-core support count established by Xiong, Yip, and Zullo; our refinement reduces its sufficient field-size threshold at large order. The refinement combines componentwise mixed degrees with the ordinary degree of a multicone. The upper bounds come from constructing syndrome representations with a common set of parity-check columns.

cs.IT↗

arXiv2026

Coding for Multiple Reverse-Complement and Palindromic Duplications

Reverse-complement (RC) and palindromic (PAL) duplications copy a length-$k$ block, reverse the copy, and insert it immediately after the original block; an RC duplication also complements the copied symbols. We study $q$-ary codes correcting $t$ such operations performed sequentially, so a later operation may copy symbols created by an earlier one. For fixed $q\geq2$ and $k,t\geq1$, every length-$n$ code $C$ for either channel satisfies $n-\log_q|C|\geq t\log_q n-O_{q,k,t}(1)$; for fixed $q,k$ and $1\leq t=t(n)=o(n)$ the lower bound is $t\log_q(n/t)-O_{q,k}(t)$. For a single RC error over an even alphabet with a fixed-point-free complement, the previously known RC-specific construction applies at odd $k$ and does not cover even $k$. For every even $k$ and any involutive complement, we give a coordinate-wise bijection that turns each RC duplication into a PAL duplication. Applying this bijection to every codeword therefore converts any $t$-error-correcting RC code into a PAL code of the same size, and conversely; encoders and decoders transfer by adding linear-time coordinate passes. For every even $k$, we also determine the maximum number of distinct descendants produced by exactly two errors from one source word. Words alternating between any two distinct alphabet symbols attain this maximum for PAL, and the bijection gives the RC maximizers. For both PAL and RC at fixed even $k$, we prove the existence of two-error-correcting codes with redundancy $4\log_q n+O_{q,k}(1)$. In the binary two-error problem, the converse gives $2\log_2 n-O_k(1)$, leaving a factor-two gap in the best existence bounds.

cs.IT↗

arXiv2026

Auxiliary Codes and the Generalized Packing-Covering Conjecture

The generalized packing--covering conjecture asks whether, at every order, the packing radius of a linear code is at most its covering radius. We prove the conjecture for every linear code of redundancy at most fourteen over every finite field, extending the previously established range of redundancies at most seven. We also prove the bound on generalized Hamming weights $d_t(C)\le2R_t(C)+1$ whenever the alphabet size $q$ satisfies $q\ge R_t(C)$. Both results use an auxiliary-code criterion that converts a covering property in the syndrome space into a weight bound. For binary primitive BCH codes, the packing radius is strictly smaller than the covering radius for every fixed error parameter and order, both at least two, once the extension degree is sufficiently large; this follows from existing covering bounds.

cs.IT↗