arXiv subjects

Aryeh Lev Zabokritskiy

Publications and source records attributed to Aryeh Lev Zabokritskiy.

At least 19 recordsLinked to original sources

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. We prove that its quantum value is $(1+\sqrt{2/3})/2$ and classify all optimal strategies in arbitrary finite local dimensions, allowing general measurements. Every optimal strategy contains a maximally entangled pair of four-dimensional systems, with fixed measurements on the local state supports up to local isometries and ancillary systems. The attaining construction was previously known from quantum random access codes. An exact sum-of-squares identity gives the upper bound, and its equality relations determine the measurement algebra needed for the classification. The rectangle determines the largest magic-square score compatible with perfect prediction of Alice's first row by an adversary with quantum side information. We classify all finite-dimensional strategies attaining this score with perfect prediction and show that one minus the best compatible guessing probability has linear order in the score excess above it. The same rigidity theorem classifies all optimal three-bit random access codes that hide the full input parity.

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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 number of parties greater than two. Extractability is the largest squared overlap with a GHZ state obtainable by separate local quantum channels. Between the biseparable bound and the quantum maximum, the exact minimum is the affine interpolation from one half to one. The bound holds for normal states on tensor products of arbitrary local dimension and arbitrary binary measurements. We settle the previously unresolved range of six or more parties with an analytic proof that is uniform from four parties onward. We also determine the exact minimum when every local system is a qubit, and show that it lies strictly above the unrestricted bound at every interior score. One qutrit and qubits at all remaining parties attain every point of the bound with measurements fixed as the score varies. For at least four parties and strict interior scores, we classify all states attaining the bound with the minimum product of local support dimensions. For at least four parties on one qutrit and qubits elsewhere, we also prove that, when the extractability excess above the affine minimum is small, the trace-norm distance from the attaining mixture, up to local unitaries, is bounded by a constant times the square root of that excess. The exponent $1/2$ is optimal, and the constants are independent of the number of parties on every fixed interior interval of normalized scores.

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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.

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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.

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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.

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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.

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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.

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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.

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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.

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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.

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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.

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arXiv2026

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

A literal-systematic $q$-ary code leaves its $n$ information symbols unchanged and appends deletion-prone checks. For $n\ge2$, a known lower bound requires three such checks to correct two arbitrary deletions. We give a uniform construction meeting this minimum over a power-of-two alphabet $q<2^{32}n^{10}$, with polynomial-time encoding and decoding. It combines boundary authentication, symmetric field checks, and a pointwise computable coloring of a sparse conflict graph. For one adjacent burst of length at most two, we give a closed-form construction with three check symbols over $q_{\mathrm{adj}}<(n+2)^4$. Alphabet size and fixed-alphabet bit redundancy remain separate objectives.

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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.

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arXiv2026

Perfect codes as exact minimizers of quadratic discrepancy in q-ary Hamming spaces

Total quadratic ball discrepancy measures the deviation of codeword counts from uniformity over all Hamming balls. We prove that, whenever a Hamming space and cardinality admit a perfect code, the discrepancy minimizers among all subsets of that cardinality are precisely the perfect codes. This extends Barg's binary minimizing result and characterizes equality over arbitrary finite alphabets. At the necessary parameter sets for correcting one or two errors, we give explicit lower bounds whose attainment is equivalent to perfect tiling, even when existence is unresolved. For alphabets of size at least four, the numerical values follow from known universal energy bounds; we establish their discrepancy normalization and the attainment criterion. The ternary discrepancy potential falls outside the completely monotonic regime of those bounds, and separate exact certificates settle the ternary Hamming and Golay cases.

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arXiv2026

Quantitative tiling stability from quadratic discrepancy in Hamming spaces

Quadratic ball discrepancy measures how uniformly a code meets Hamming balls over all centers and radii. Perfect codes minimize this quantity among codes of the same cardinality. We prove a quantitative refinement: at the parameters of every nontrivial perfect code, excess discrepancy is at least a positive multiple of the discrepancy at the correction radius. The latter equals the squared deviation of the covering multiplicity from one, divided by the square of the code cardinality. The comparison applies to all codes of that cardinality. For one-error parameters, the coefficient is at least one, which is sharp uniformly over the parameters; explicit coefficients depending on the parameters can be much larger. For two-error parameters of length at least five, we obtain a positive coefficient against an algebraic benchmark under the sphere-packing and Lloyd integrality conditions, even when existence of a perfect code is unknown. The multiplicity defect also measures the failure of uniform ball noise to smooth a code distribution. Consequently, the same discrepancy excess gives explicit bounds on holes, overlaps, and divergence from the uniform distribution.

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arXiv2026

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

Quantum-hardware scores vary across workloads and execution times. We study these variations through repeated communication and deletion-recovery experiments on superconducting and trapped-ion hardware. Our opening test is round-trip state echo (RTSE): prepare one of four tetrahedral qubit states, move it along a route and back by swaps, undo the preparation at the root, and measure the return probability. Repeating identical native communication programs on IQM Emerald changed the differences between tasks on the same routes, while the aggregate RTSE estimates differed by only $0.00125$. A separate two-window experiment found a drop in root return probability from route length 2 to 10 for both RTSE and its remote-inverse do-nothing predecessor; the prespecified directional test did not support better retention by RTSE. Across 64 fixed combinations of placement, input state, and deleted position, the IQM recovery mean fell from $0.738$ to $0.624$. On IonQ's five submitted virtual wires, with advertised all-to-all connectivity, recovery means of $0.911$ and $0.923$ exceeded the prespecified two-thirds reference in both windows. The corresponding differences from a control using the decoder's adjoint were $0.446$ and $0.443$. These measurements characterize complete compiled workloads at dated executions. Recovery is tested after subsystem discard in the circuit model, and each hardware interface is analyzed separately.

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arXiv2026

Prescribed-Difference Matchings with Four and Eight Holes: Fourier Filters for Compatible Boundaries

Let $s\geq 2$ and let $v_1,\ldots,v_{2^{s-1}}\in F_2^s\setminus\{0\}$ have sum zero. The prescribed-difference matching problem asks whether $F_2^s$ can be partitioned into pairs whose differences, counted with multiplicity, are exactly these vectors. Fix a hyperplane $H\leq F_2^s$. An $h$-hole instance is one in which exactly $h$ prescribed differences lie in $H$, counted with multiplicity. We prove that every four-hole instance has a solution for $s\geq 3$ and every eight-hole instance has a solution for $s\geq 4$. Thus no restriction on the multiplicities or on the sum of the holes is needed beyond the global zero-sum condition. After the internal pairs are placed, their endpoints are deleted from the two affine halves determined by $H$, and the remaining vectors must be paired with the prescribed crossing differences. A locally valid placement of the internal pairs need not admit such a completion. We encode the possible completions by coefficients of signed determinants and use the Walsh transform to sum over all disjoint legal placements while keeping the crossing profile fixed. Nonvanishing of this sum guarantees that at least one placement extends to a full matching. The four-hole and zero-sum eight-hole theorems are proved theoretically. The general eight-hole proof uses three finite exact verifications, independent of the crossing profile: a local filter calculation, a classification of the remaining eight-hole configurations, and integer certificates for those configurations. The verification software, exact inputs, and recorded outputs are archived in a versioned Zenodo record.

math.CO↗

arXiv2026

Independent Sets in Multiset Profile Graphs via Weighted Local Covers

Let $G_q(d)$ be the unit-transfer graph on the nonnegative integer vectors whose $q$ coordinates sum to $d$, equivalently on the multiplicity profiles of size-$d$ multisets over $q$ symbols. The prime-checksum conjecture predicts that, for prime $q$ and all sufficiently large $d$, a largest independent set is a fiber of the natural cyclic checksum. We develop a finite-state weighted local-cover method for $G_q(d)$: translated induced subgraphs give local independence inequalities, while capped anchor profiles reduce the covering conditions for infinitely many degrees to a finite rational linear system. This method gives new proofs of the known cases $q=3$ and $q=4$ and determines $α(G_q(d))$ exactly for $q=5$ and $q=7$ in every degree, thereby proving the next two odd-prime cases of the conjecture. In the complementary regime where $d$ is fixed and $q$ grows, the same method gives an explicit upper bound for $α(G_q(5))$ for every $q\ge7$, determines $α(G_q(d))$ exactly when $q$ is a power of two and $d\in\{6,8,10\}$, and yields an asymptotically sharp upper bound through three terms for every fixed $d\ge7$. The finite systems arising in these arguments are verified in exact arithmetic and supported by independently checkable certificates.

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