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arXiv2026

Vanilla Exact Synthesis of CNOT Circuits is NP-hard

Exact CNOT synthesis asks for a minimum-size CNOT circuit implementing an invertible linear transformation. Although several related synthesis models have been shown to be computationally hard, their hardness proofs rely on additional structure such as restricted qubit connectivity, encoded inputs, or unrestricted intermediate variables. The complexity of the most basic setting---identity input, a fixed number of labelled qubits, no ancillas, and all-to-all CNOT connectivity---had remained unresolved. In this work, we prove that the decision version of this vanilla exact CNOT synthesis problem is NP-complete, and consequently that its optimization version is NP-hard. Our proof gives a polynomial-time reduction from the Hamiltonian-path problem on grid graphs in two steps. First, we isometrically embed the grid graph into a hypercube via a unary encoding map. We then encode this hypercube Hamiltonian path problem into vanilla exact CNOT synthesis. The main challenge is that CNOT synthesis specifies only the final parity matrix and cannot directly enforce the intermediate vertex visits required by a Hamiltonian path. To overcome this difficulty, we introduce extra recorder qubits that encode the required intermediate vertex visits into the final transformation, forcing any CNOT circuit implementation to realize the intended path structure. Beyond CNOT synthesis, our result directly implies hardness for several related problems, including the shortest word problem over $\mathrm{GL}(n,2)$, distance computation on Cayley graphs over $\mathrm{GL}(n,2)$, minimization of sequential XOR programs, and exact synthesis of phase polynomial circuits.

quant-ph↗
arXiv2026

Quantum Codes from $r$-Nearly Self-Orthogonal Linear Codes via Jordan Canonical Form over $\mathbb{F}_{q^2}$

We introduce a Jordan-canonical-form framework for constructing $q$-ary quantum stabilizer codes from arbitrary classical linear codes over $\F_{q^2}$. The framework does not require the classical linear code $\mathcal{C}$ to satisfy the dual-containing condition (i.e., self-orthogonality). Given a classical code $\mathcal{C}=[n,k,d]_{q^2}$ with parity-check matrix $H$, we measure the obstruction to Hermitian self-orthogonality by the rank $r=(n-k)-\dim_{\F_{q^2}}(\mathcal{C}^{\perp_h}\cap \mathcal{C})$. The ingredient code $\mathcal{C}$ is $r$-nearly dual containing, or, equivalently, $\mathcal{C}^{\perp_h}$ is $r$-nearly self-orthogonal, by which we mean that $r=\Rank(HH^{\dagger})=\dim_{\F_{q^2}}(\mathcal{C}^{\perp_h})-\dim_{\F_{q^2}}(\mathcal{C}^{\perp_h}\cap \mathcal{C})$. By systematically reducing the rank of the Hermitian inner-product matrix $A=HH^{\dagger}$ through rank-one perturbations along the Jordan basis $W=P^{-1}$ of the decomposition $A=PJ_AP^{-1}$, we construct an explicit Hermitian self-orthogonal code $\mathcal{C}_{\mathrm{so}}=[n+r,n-k]_{q^2}$. A sufficient distance-preservation criterion guarantees that the resulting $q$-ary quantum code has parameters $[[n+r,2k-n+r,\geq d]]_q$. Applying this construction to classical codes produces several record quantum codes that improve or supplement the best-known parameters in Grassl's tables.

cs.IT↗
arXiv2026

Separating QMA from QCMA with a classical oracle

We construct a classical oracle proving that, in a relativized setting, the set of languages decidable by an efficient quantum verifier with a quantum witness (QMA) is strictly bigger than those decidable with access only to a classical witness (QCMA). The separating classical oracle we construct is for a decision problem we coin spectral Forrelation -- the oracle describes two subsets of the boolean hypercube, and the computational task is to decide if there exists a quantum state whose standard basis measurement distribution is well supported on one subset while its Fourier basis measurement distribution is well supported on the other subset. This is equivalent to estimating the spectral norm of a "Forrelation" matrix between two sets that are accessible through membership queries. Our lower bound derives from a simple observation that a query algorithm with a classical witness can be run multiple times to generate many samples from a distribution, while a quantum witness is a "use once" object. This observation allows us to reduce proving a QCMA lower bound to proving a sampling hardness result which does not simultaneously prove a QMA lower bound. To prove said sampling hardness result for QCMA, we observe that quantum access to the oracle can be compressed by expressing the problem in terms of bosons -- a novel "second quantization" perspective on compressed oracle techniques, which may be of independent interest. Using this compressed perspective on the sampling problem, we prove the sampling hardness result, completing the proof.

quant-ph↗
arXiv2026

Behavioral Memory under Symmetry in One-Way Quantum Automata

Under compact symmetry, observable behavior reduces to an invariant operator algebra, but its dimension is not yet classical memory: some coordinates are dynamically frozen, some invisible to threshold tests, and some already classical. We develop an operator-algebraic theory that separates these effects through three filters. For one automaton, behavior is the Hilbert--Schmidt pairing between prefix-reachable states and suffix-observable effects, whose rank equals the real Hankel rank without controllability or observability assumptions. Maximizing this invariant over a symmetry-constrained dynamical class gives a structural capacity controlled by the symmetry commutant: its center stores isotypic populations frozen by reversible dynamics, its traceless multiplicity blocks carry movable noncommutative coordinates, dissipation removes the unary spectral loss inside those blocks, and covariant mobility releases relative populations subject to component conservation. Operational realization then determines which surviving coordinates force probabilistic states. For a fixed nontrivial invariant readout, full mobility gives an exact dichotomy in worst-case state cost: a commutative invariant algebra costs exactly its dimension, whereas a noncommutative multiplicity block raises the unrestricted cost by exactly one state. Thus noncommutativity has a one-state worst-case classical price. The known four-letter quadratic-plus-one law at trivial symmetry is the fully mobile endpoint of this principle. Schur--Weyl duality further shows that different preserved symmetries on the same tensor-power Hilbert space can change the worst memory scale from polynomial to exponential, while fixed-weight modules give an exact Catalan law at half filling, with structural capacity equal to the Catalan count minus its central-sector correction.

cs.FL↗
arXiv2026

FIREQ: FPGA Instrumentation for Readout and Qubit control

We present FIREQ (FPGA Instrumentation for Readout and Qubit control), an open-source RFSoC-based framework for the control and readout of superconducting qubits. FIREQ combines a modular AXI-compliant firmware architecture with a PYNQ-based software stack designed to support extensible hardware integration, deterministic experiment timing, and low-overhead execution of repeated calibration and characterization workflows. The firmware implements direct RF synthesis and acquisition, trigger-based sequencing, programmable pulse generation, frequency-multiplexed readout, and memory-efficient acquisition and waveform buffering. The software adopts a client-server architecture with streamed data transfer and dependency-aware configuration updates to reduce host-device and reconfiguration overhead during parameter sweeps. On an AMD Zynq UltraScale+ RFSoC ZCU216, FIREQ generates RF pulses up to 9.3 GHz with a pulse-duration resolution of 107 ps and an event-timing resolution of 1.7 ns. FPGA resource utilization is compared with representative open-source RFSoC control frameworks, showing a low BRAM footprint while retaining full-rate I/Q generation and acquisition. The RF output is characterized in terms of phase noise, noise spectral density, and inter-channel timing skew. End-to-end operation is validated on a superconducting qubit through resonator spectroscopy, Rabi, Ramsey, and relaxation measurements, yielding T1 = 6.94 us and T2* = 13.50 us. FIREQ can therefore be used both as a qubit-control platform and as an experimental environment for evaluating alternative control and readout IP architectures.

quant-ph↗
arXiv2026

Fast Fault-Tolerant Decoders for Hypergraph Product and Lifted-Product Codes

We design low-complexity, fault-tolerant decoders for quantum low-density parity-check (QLDPC) codes with the goal of reducing decoding latency. We target two major bottlenecks of decoding under the \emph{circuit-level} noise model: (i) post-processing via order-statistics decoding (OSD), and (ii) the large number of auxiliary variable nodes commonly introduced to represent CNOT-induced correlations during syndrome extraction. Our key observation is that propagating CNOT faults (\emph{hook errors}) create \emph{stabilizer-induced} trapping sets (TSs) that are intrinsic to hypergraph-product (HGP) and lifted-product (LP) constructions. Therefore, instead of modeling each such fault with an explicit correlation node and relying on OSD to clean up the resulting failures, we design message-passing decoders that resolve the corresponding \emph{stabilizer-induced} TSs directly. We obtain these decoders by deriving QLDPC decoders from decoders for the parent classical LDPC codes and using them collectively to correct broad families of \emph{stabilizer-induced} TSs. For CNOT faults that manifest primarily as syndrome errors, we show that their effect is equivalent to a data error together with syndrome-bit measurement errors. Consequently, given repeated measurements and a decoding graph that already includes nodes representing syndrome-bit errors, no distinct variable node is needed for each CNOT fault. Using a \emph{phenomenological} Tanner graph with nodes representing only data errors and syndrome-bit errors, simulations on the LP codes show a reduction in, or comparable, logical error rates relative to BP+OSD, at substantially lower decoding complexity.

cs.IT↗
arXiv2026

QILP-0: Constructing Observational Declarative Twins of Quantum Circuits

This paper introduces QXymb, a general framework for constructing observational declarative twins of quantum circuits, and develops QILP-0, its first complete order-0 specialization. QILP-0 constructs a finite multi-valued propositional logic program from observed circuit behaviour within a declared observational scope. The pipeline traverses a declared family of quantum observables incrementally according to a reproducible structural grading and a declared observational reference horizon. Progress is quantified through reference-relative coverage against a fixed target-independent reference. Observable responses are organized through target-independent geometry, while retained latent structure is mapped deterministically back to original observable columns before symbolic processing, preserving observational semantics and provenance. Selected observable profiles are converted into a finite relation through admissible target-independent discretization. The target is used only afterwards to audit twin-admissibility and induce the declarative theory. A theory is certified as an exact observational declarative twin when it completely and correctly reconstructs the resulting finite task-conditioned discrete relation. Logical exactness is therefore separated from numerical, backend, provider, and discretization uncertainty, which is retained as audit metadata. Validation uses two complementary QML settings. Exhaustive Bars & Stripes experiments compare product and grid-CZ embeddings from 16 to 100 qubits and exercise the native-discrete branch. Low-Depth MNIST analyses all 14,708 digit-0/1 instances before and after a trained variational quantum transformation and exercises continuous discretization. In every reported relation, the induced QILP-0 theory achieves complete, conflict-free reconstruction with strict accuracy equal to one.

cs.AI↗
arXiv2026

Quantum MeanFlow: single-shot generative sampling on NISQ hardware

Quantum generative models offer a promising framework for exploring whether quantum computation can enhance generative machine learning. Flow matching is a generative method in which samples are generated by transporting a simple, known distribution to the target data distribution with a learned velocity field. Its quantum counterpart, known as quantum flow matching (QFM), was introduced recently, and, like its classical counterpart, requires integrating an ordinary differential equation over many time steps during inference. As each step requires the output from the previous step, the circuit submission is sequential and a drawback on quantum computers as they have high input/output costs. To alleviate this problem, we introduce Quantum MeanFlow (QMF), the quantum analogue of the MeanFlow formulation, which allows single-step sample generation. While the QFM learns an instantaneous velocity field at each time step, QMF learns the average velocity over a time interval. We use a parameterized quantum circuit to learn these velocity fields and benchmark the two methods on the MNIST dataset. We show that while single-step QMF has lower image quality compared to multi-step QFM, it performs better than the single-step QFM sampling at every shot count. Both of our models are executed on IBM quantum computers and best-of-N rejection sampling recovers most of the accuracy lost to device noise without modifying the circuit. This is especially advantageous for QMF which has only one circuit evaluation per image. Here, We establish QMF as a viable method for single-step quantum generative sampling, saving on quantum circuit evaluations per generated sample.

quant-ph↗
arXiv2026

Projected Riemannian Gradient Descent for the Bures-Wasserstein Barycenter: Dimension-Independent Linear Convergence at Unit Step Size

The computation of the Bures-Wasserstein (BW) barycenter of an ensemble of positive definite matrices arises throughout machine learning, optimal transport, and quantum information. Riemannian gradient descent (RGD) at unit step size -- the fixed-point iteration used in practice -- converges rapidly, yet existing analyses present a dichotomy: unit-step guarantees carry worst-case exponential dependence on the dimension, while dimension-independent guarantees require small step sizes that forfeit the empirical speed. We resolve this dichotomy, not by improving the guarantees for unit-step RGD, but by proposing a Projected RGD algorithm that achieves dimension-independent linear convergence at unit step size. The achieved rate, $(1 - κ^{-3/2})$, where $κ$ is the condition number of the ensemble, also polynomially improves on the best small-step guarantee ($κ^{3/2}$ versus $κ^{5/2}$ iteration complexity). The crux is a novel Projection Lemma: clipping the eigenvalues of a positive matrix to an interval $[α, β]$ is the closed-form, non-expansive (1-Lipschitz) BW-metric projection onto the set $\{S : αI \leq S \leq βI\}$ -- a statement which, unlike its known one-sided counterpart, does not follow from convexity. The projection is moreover free: it reuses an eigendecomposition the next iteration must perform in any case, so the projected and unprojected iterations cost the same per step. The same analysis covers the invariant matrix projection problem of Brahmachari et al. (2025), whose fixed-point algorithm we identify as unit-step RGD on a totally geodesic submanifold, thereby extending the dimension-independent guarantee to that setting verbatim.

cs.LG↗
arXiv2026

Quantum Query Algorithms for the Constructive Diagonal Ramsey Theorem

The constructive diagonal Ramsey problem asks, given adjacency-oracle access to an $N$-vertex graph, for a clique or independent set of the order guaranteed by Ramsey's theorem. We give a bounded-error quantum algorithm that, for every $K\ge2$ and $N\ge4^{K-1}$, finds and verifies a homogeneous $K$-set using $O\!\left(2^K K\log\frac Kη\right)$ edge queries with failure probability at most $η$. At the Ramsey scale $N=2^n$, this yields a homogeneous set of order $\lfloor n/2\rfloor+1$ using $O(\sqrt N\log N\log(\log N/η))$ queries, improving on the $O(N)$ queries of the explicit classical recursion and giving, to our knowledge, the first sublinear worst-case algorithm for the Ramsey relation. We also derive an $Ω(N^{1/12})$ quantum lower bound by a reduction from collision finding. The algorithm runs the constructive recursion over implicit candidate sets. Each set is represented by a short conjunction of adjacency constraints and sampled using capped unknown-solution quantum search, and a scale-aware concentration schedule balances estimation accuracy against the increasing cost of sampling deeper sets. We complement the upper bound with an $Ω(N^{1-1/\sqrt2})$ randomized lower bound, transported from the random-Painter analysis of online Ramsey numbers, which holds on the uniform distribution $G(N,1/2)$. On that distribution a greedy quantum search uses only $\widetilde O(N^{1/4})$ queries, giving a provable polynomial quantum speedup for Ramsey search on random graphs. We also give an estimation-free size-biased recursion and extend it to every fixed number of edge colours.

quant-ph↗
arXiv2026

A quantum-assisted framework for PDE-based Bayesian inverse problems

Quantum computing offers potential advantages for solving partial differential equations (PDEs). However, most existing quantum PDE solvers primarily focus on preparing quantum states for solutions, while the efficient recovery of classical information from these states remains less explored. Motivated by the readout limitation, we propose a quantum-classical hybrid framework for Bayesian PDE inversion problems: The quantum processor evolves the PDE and evaluate the loss function with sampling noises, while the classical computer tunes the hyper-parameters in the Gaussian Process Regression to explore the next trial candidate. To match the quantum solvers for linear and semi-linear autonomous evolution PDEs, we suggest to use a normalized quantum-state loss as the data-misfit function and evaluate the new misfit by combining quantum PDE solvers with the Hadamard test, thereby allowing us to extract useful classical information using only a limited number of quantum state copies without reconstructing the full solution vector. The analysis of error propagation and overall complexity of loss evaluation under a prescribed accuracy shows that the new data-misfit function outperforms the conventional L2-loss under quantum measurements. Quantum circuit simulations of 1D and 2D linear convection diffusion equations under approximate and finite sampling loss evaluations, together with classical numerical experiments on a nonlinear forced viscous Burgers equation, demonstrate the feasibility of the proposed approach for parameter inversion even when the loss evaluations are affected by sampling noise. This framework may provide a viable quantum-assisted scheme for PDE-based inverse problems and elucidate the potential of quantum PDE algorithms in addressing a complete quantum-to-end optimization stack.

math.NA↗
arXiv2026

Quantum Algorithm for Elliptic Curve Discrete Logarithms with Space-Efficient Point Addition

The Elliptic Curve Discrete Logarithm Problem (ECDLP) is a fundamental problem in cryptography, and reducing the resource requirements of quantum algorithms for solving ECDLP is an important goal. In this work, we present a space-efficient quantum algorithm for solving the ECDLP over prime fields, achieving an implementation with only $3n+6\lfloor \log_2 n \rfloor+O(1)$ logical qubits and $1056n^3/\log_2 n+O(n^2)$ Toffoli gates, where $n$ is the bit-length of the prime. For a 256-bit prime-field curve, our construction requires only 835 logical qubits, reducing the previous best estimates of 1098 and 1175 logical qubits by Chevignard et al. [EUROCRYPT 2026] and Babbush et al. [ArXiv Preprint 2026], respectively. The key to our improvement is a new space-efficient reversible modular inversion circuit, which addresses the dominant space bottleneck in affine-coordinate point addition. Starting from the extended Euclidean algorithm (EEA), we refine the register-sharing technique of Proos and Zalka by introducing length registers and location-controlled arithmetic to compactly store and update intermediate variables. We further optimize the reversible update procedures and construct the corresponding controlled arithmetic circuits, resulting in a modular inversion circuit implemented by only $2n+6\lfloor \log_2 n \rfloor+O(1)$ logical qubits and $229n^2+O(n\log_2 n)$ Toffoli gates. This modular inversion circuit together with mid-circuit measurements and classical feed-forward operations provides a space-efficient controlled affine point-addition circuit and a complete implementation of Shor's algorithm for ECDLP.

quant-ph↗
arXiv2026

Exact quantum splitting and the structure of finite algebras

Berlekamp's algorithm factors a squarefree polynomial $f\in\mathbb{F}_q[x]$ by deterministic linear algebra, reducing the problem to splitting an explicit commutative algebra $B\cong\mathbb{F}_q^r$ into its $r$ simple factors. For large odd $q$, the standard efficient splitting step is randomized, while known derandomizations are conditional on the Extended Riemann Hypothesis. We give an unconditional exact quantum implementation in a circuit model permitting single-qubit rotations through efficiently computable angles. The construction uses an unconditional counting argument. For a block containing $s\ge2$ irreducible factors, a quadratic-character test in odd characteristic and an absolute-trace test in characteristic $2$ yield a nonconstant test element with probability $p_{q,s}\ge\tfrac12$, known exactly in advance and depending only on $q$ and $s$, not on the unknown factorization. Exact amplitude amplification therefore converts each randomized test into a procedure succeeding with certainty after one amplification iteration. The resulting algorithm uses exactly $r-1$ quantum splitting rounds and $O(n^3\log q)$ quantum $\mathbb{F}_q$-operations and $O(n^3)$ classical operations, requiring no primitive root, quadratic non-residue, or distinct-degree preprocessing. The method also splits arbitrary finite-dimensional separable commutative $\\mathbb{F}_q$-algebras given by structure constants. Combined with R'onyai's classical structure theory, which computes the radical deterministically and reduces the remaining tasks deterministically to polynomial factorization, it yields the radical and the Wedderburn decomposition of $A/\mathrm{Rad}(A)$ into minimal two-sided ideals, with certainty, for any $n$-dimensional associative $\mathbb{F}_q$-algebra given by structure constants, using $O(n^4\log q)$ quantum $\mathbb{F}_q$-operations.

quant-ph↗
arXiv2026

Towards unsupervised representation learning for quantum data: quantum models with inference and generation

With quantum sensors, simulators and networks emerging, a future of quantum technology may produce quantum states as data---that is, coherently rather than as classical measurement records---thus motivating the study of suitable quantum generalisations of modern machine learning, including the automated, unsupervised extraction of useful representations. Two ingredients are central to the latter: inference, mapping observations to latent representations, and generation, mapping latent states back to synthetic data. Both are related to each other and to joint distributions for training models by the chain-rule of classical probability theory. The fact that quantum states however lack such universal, standard factorisation property thus poses a challenge. Here we develop a conceptual and mathematical framework for unsupervised representation learning from quantum data. Models are joint quantum states over visible and latent systems; state-over-time maps provide a notion of factorisation into a marginal state and inference (generation) channel; models with inference (generation) are ambiguous states---states for which such factorisation obtains---subject to a further consistency condition on extended inference maps as data extension. These stipulations are restrictive: we show that non-trivial models must feature non-linear such maps to the extended space. For three representative state-over-time maps, we completely characterise the ambiguous states, uncovering a hierarchy tied to the positive-partial-transpose (PPT) criterion from entanglement theory. Notably, the Leifer-Spekkens construction supports inference and generation exactly for model classes of PPT states, thus allowing genuinely quantum visible-latent correlations. We also formulate quantum counterparts of exact and approximate inference training, explore weaker notions of data extension and sketch a future research programme.

quant-ph↗
arXiv2026

"Train classical, deploy quantum" requires rethinking generalization

Generative models have become central across science and industry, from image and text synthesis to the design of molecules and materials. Quantum generative models are considered one of the most promising applications for quantum computers, since a quantum circuit naturally produces samples from the distribution it encodes, and for suitable circuits that distribution is believed to be hard for any classical computer to reproduce. A leading strategy trains these models on a classical computer and reserves the quantum device for generating samples at deployment. This is possible when the training loss can be evaluated on a classical computer. A prime example is the maximum mean discrepancy (MMD$^2$), a moment-matching loss that compares the model and the data through their Pauli-$Z$ correlations. Research so far has asked whether such models can be trained and whether their sampling is hard; whether minimizing such an objective yields a model that generalizes, rather than one that merely reproduces the training statistics, remains poorly understood. We benchmark a broad set of quantum and classical generative models by direct sampling and show that models trained with a moment-matching loss generally show worse generalization than the likelihood-trained models. We show this on two application-inspired datasets: first a cardinality-constrained dataset at up to $30$ qubits and second a dataset of genomic single-nucleotide variants, whose valid set is the observed data. These results indicate that a converged moment-matching loss is not a reliable measure of generalization, and that train-classical, deploy-quantum workflows will need approaches that target generalization directly, leaving open whether better training objectives suffice or whether the model architectures themselves must change.

quant-ph↗
arXiv2026

Rotational Bell-State Loop-Back Key Establishment and Agreement Certification for Passive Users

Server-aided Bell-state networks with sequential local encoding are an established approach to mediated key distribution. Against this background, we study a Loop-Back architecture in which two source-free, detector-free users transform the same traveling qubit. Alice privately samples $r\in\mathbb{Z}*2^2$, prepares $|β_r\rangle$ over the complete Bell basis, retains one qubit, and uses $r$ as the reference for the returned pair. A deterministic Pauli-composition mode, $U*{\mathrm{eff}}=U_2U_1$, serves as a reference to prior serial-unitary protocols. The main extension instead uses the two same-axis settings $R(\pmα)$ and a reference-versus-complement Bell test. Opposite rotations cancel, whereas equal rotations add, so $P(C\mid\mathrm{disagreement})=0$ and $P(C\mid\mathrm{agreement})=\sin^2(2α)$. At $α=π/8$, agreement is certified with probability $1/2$, and the overall conclusive-key probability is $1/4$ for unbiased choices. The same event supplies one shared raw-key bit. For the Pauli mode, a one-qubit Pauli twirl hides either user's factor from an endpoint mediator that may prepare an arbitrary qubit--ancilla state and choose the final measurement, provided the other Pauli is uniform and the traveler is inaccessible between users. The rotational guarantee is narrower: common-sign privacy holds only for the prescribed Bell source and coarse-grained instrument. A polarization implementation can normalize the private reference to the singlet and use a beam splitter with verified two-photon detection; loss produces a third, discarded outcome. The proposal is theoretical and does not claim composable security or experimental validation.

quant-ph↗
arXiv2026

Towards Scaling Quantum Fine-Tuning of Foundational Time Series Models for Classification

Time-series foundation models produce rich embeddings, but whether quantum models can exploit them, and how far hybrid classical-quantum architectures scale, remains unclear. We address this by fine-tuning Chronos for power-grid event classification (PSML-5) with a quantum head on the model's embeddings. Grouping embeddings by physical sensor type before summarization already surpasses the best published baseline built for this benchmark, and with finer-grained features the quantum head outperforms a larger classical multilayer perceptron on identical inputs by 1.7-2.0 percentage points of balanced accuracy. Yet the gains saturate: past a point, feeding more information to the same fixed-width register yields no improvement. We show the bottleneck is neither the supply of information nor circuit expressiveness, but the bandwidth of the data intake. To overcome this limitation, we introduce the wing module, a self-contained few-qubit circuit that feeds additional information into the core circuit through a sparse, one-way coupling. Under a preregistered four-seed protocol, we attach wings to a fixed 12-qubit core with fixed features. Balanced accuracy increases with each added wing, from 83.6% with no wings (13 qubits, including a post-selection qubit) to 85.2% with two (19 qubits). Ablations establish that a circuit enlarged without new information gains nothing, while a wing fed information from the wrong sample harms accuracy. These results reframe scaling for quantum fine-tuning: added qubits help when they carry added inputs, not merely more parameters. Wings offer a modular and stable route to widening that bandwidth.

quant-ph↗
arXiv2026

Quantum Message Passing Convergence and Vanishing Block-Error Probability for Random LDPC Codes

Belief propagation with quantum messages (BPQM) is a quantum algorithm that decodes classical codes transmitted over classical--quantum channels. It realizes optimal decoding on tree factor graphs over pure-state classical-quantum channels. However, this tree-based analysis does not ensure vanishing block-error probability for LDPC Tanner graphs with cycles. In this work, we construct a two-stage BPQM decoder for random $q$-ary LDPC codes over symmetric $q$-ary pure-state channels, where $q$ is prime, and prove that its ensemble-average block-error probability vanishes as the blocklength $N$ tends to infinity. For regular ensembles with $d_v\geq3$, fidelity bounds yield double-exponential decay of the average symbol-error probability throughout the BPQM success region. We apply depth-$\ell$ BPQM to coordinates with tree neighbourhoods and treat the remaining coordinates as erasures. With a suitable $\ell=Θ(\log\log N)$, a noncommutative union bound controls the BPQM decoding errors, while the minimum-distance property guarantees erasure recovery. We also extend the analysis to finite-support irregular ensembles. These results are relevant to quantum algorithms based on Regev's reduction, where coherent decoding uncomputes a codeword register. Decoded quantum interferometry (DQI) uses a closely related Fourier-based framework that reduces sparse max-LINSAT optimization problems to LDPC decoding problems on pure-state channels. Our results justify the use of BPQM in the decoding step of DQI and of coding-theoretic algorithms based on Regev's reduction whenever the code is drawn from one of the random LDPC ensembles analyzed here and the induced memoryless symmetric pure-state channel lies in the BPQM success region.

quant-ph↗