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arXiv2026

A quantum let within the lambda calculus

Since the seminal work of Selinger and Valiron, the standard design for quantum lambda calculi has kept the quantum state outside the program: terms manipulate pointers to an external register. This is largely due to the difficulty of eliminating tensor products. For example, the calculus $λ_ρ^\circ$ embeds density matrices directly within terms, where terms carry the entire computation state, a feature particularly appealing for program verification. However, lacking a tensor elimination construct, it can neither access the individual qubits of a compound state nor discard them. Borgna showed that this inability to discard qubits makes the calculus strictly less expressive than the quantum lambda calculus of Selinger and Valiron. In this paper we show that tensor elimination is possible in this setting. The key observation is that the Pauli decomposition, combined with the spectral decomposition of the Pauli matrices, allows any $n$-qubit density matrix to be expressed as a real linear combination of tensor products of single-qubit density matrices. Exploiting this fact, we extend $λ_ρ^\circ$ with a construct $\mathrm{let}\ x^{\otimes n} = ρ \mathrm{in}\ t$, which binds each $x_i$ to a single-qubit density matrix arising from the decomposition of $ρ$. We equip the extended calculus with a rewrite system, a type system, and a denotational semantics, and prove Subject Reduction, Progress, Strong Normalisation, Soundness, and Adequacy. The new construct also recovers the missing ability to discard qubits, thereby restoring expressiveness. Moreover, we show that this is achieved in a physically principled way: a variable unused in $t$ is interpreted exactly as being partial-traced out, as dictated by the no-deleting theorem. We illustrate the resulting compositionality through quantum teleportation and the three-qubit bit-flip code.

cs.LO↗
arXiv2026

Random Garbage Separates XOR from Forward-Only Queries

We give exponential quantum query separations between the standard XOR interface and two forward-only interfaces that supply neither an adjoint nor an inverse oracle. Let $X=\F_2^n$, $N=|X|$, and $f_{h,r}(x)=(h(x),x,r_x)$, where $h:X\to X$ is promised to be either a permutation or a Simon two-to-one function, and $r$ is a fixed table of $n$-bit tags, unrestricted by the promise and reused on every query. The resulting problem is solvable with at most $n+2$ standard XOR queries, but has forward-erasing query complexity $Θ(\sqrt N)$. This answers affirmatively open question 11 in [Scott Aaronson. Open problems related to quantum query complexity. ACM Transactions on Quantum Computing, 2(4):14:1-14:9, 2021] . We also embed these instances into permutations. The detailed construction retains the copy of $x$ in each prescribed output, but for these promises that copy can be replaced by one bit that distinguishes the two inputs in every Simon pair. This gives a permutation domain of size $L=4N^2$ and a permutation problem with the same standard-query upper bound and forward-only in-place query complexity $Θ(\sqrt N)=Θ(L^{1/4})$. Both lower bounds remain valid with a clean coherent bypass. The common lower bound uses an analysis-only recording replacement. In the replacement computation, tracing out the fixed random tag table after $T$ calls gives a sum of positive-semidefinite operator contributions, each depending on $h$ at no more than $T$ addresses. On such a set, the restrictions induced by random permutations and random Simon functions differ only if the set contains a hidden Simon pair, an event of probability $O(T^2/N)$.

cs.CC↗
arXiv2026

Accelerating Fourier--Motzkin elimination: redundancy removal and the choice of variable elimination order

Fourier-Motzkin elimination computes an inequality description of the projection of a polyhedron onto a subset of its coordinates by eliminating one variable at a time. It is used in several areas of optimisation and computer science, and it is a standard way of obtaining the entropic constraints of a causal structure, where the marginalisation over the latent variables produces such a projection. Its limitation is the growth of the intermediate systems of inequalities, which can be doubly exponential in the number of eliminated variables even though the projection itself grows only as a single exponential. In practice the computational overload of the method therefore depends on two choices: how the redundant inequalities are removed after each step, and the order in which the variables are eliminated. We consider both. We first show, by an explicit example, that Imbert's redundancy test cannot be interleaved with redundancy removal by linear programming. We show that the two methods, however, can be combined soundly if the derivation records used by Imbert's test are re-initialised after every step at which linear programming is used. We then propose a rule for choosing the elimination order of the variables that gives a significant computational advantage, however, at the cost of increased resource usage. We demonstrate this advantage on some random polytopes, where the rule reduces the running time by factors of between 6 and 25 compared with the same elimination under a fixed order. For entropic descriptions of causal structures, with more than 250 inequalities and more than 100 variables to eliminate, our rule keeps the number of inequalities handled at each step one to two orders of magnitude lower than a fixed order.

cs.CC↗
arXiv2026

A priori Assessment of Tensor-Network Encoding for Isotropic Turbulent Flows

Tensor networks (TNs), originally developed for simulating many-body quantum systems, provide a systematic framework for approximating high-dimensional fields. This is achieved by factorizing the field into interconnected tensors with small bond dimensions, thereby restricting the correlations captured across field bipartitions. Belonging to the family of TNs, the matrix product state (MPS) ansatz is utilized here as a reduced-order modeling framework to construct truncated representations of isotropic turbulent flow data. Two direct numerical simulation (DNS) datasets are considered: the hydrodynamic field of an incompressible three-dimensional flow, and a conserved Fickian scalar in a similar flow. Each field is encoded as an MPS through a sequence of singular value decompositions (SVDs) in which small singular values are discarded. The truncated representation is contracted back to the full grid, and the resulting reconstructed field is compared against DNS. An interleaved ordering of the spatial tensor indices of the transport variables is applied prior to decomposition in order to localize the dominant inter-tensor correlations. Velocity reconstructions achieve $99.8\%$ fidelity using only $5\%$ of the original DNS memory, while the scalar field reaches the same fidelity at $15\%$ memory usage. A wide range of lower- and higher-order statistics, including velocity gradients, dissipation, and structure functions, are systematically examined. At these compression levels, the total kinetic energy and the scalar energy are both recovered within $0.2\%$ relative error, while the mean dissipation and mean scalar dissipation remain within approximately $10\%$ of the DNS generated values. These findings support the suitability of MPS for scalable reduced-order analysis of complex turbulent datasets and motivate further exploration of TN-based methods in computational turbulence.

physics.flu-dyn↗
arXiv2026

Ordered-Angle Coding for Exact Multiuser Unanimity Testing

We introduce ordered-angle coding for binary-unanimity testing among $n$ transformation-only users in a serial quantum architecture inherited from Loop-Back communication. User $B_i$ fixes one private sign across a logical word and applies $R(s_iα_j)$ in trial $j$. If $w$ users choose the negative sign, serial composition gives $R[(n-2w)α_j]$. We show that every fixed same-axis pulse that is deterministic for both unanimous inputs under the binary Bell readout has $α_j=q_jπ/(2n)$ with integer $q_j$. For a nonadaptive word $\mathbf q=(q_1,\ldots,q_m)$, a mixed Hamming weight imitates the unanimous signature with probability \[ M_{\mathbf q}(w)=\prod_{j=1}^{m}\cos^2\!\left(\frac{πq_jw}{n}\right). \] Within this complete deterministic-unanimity pulse family, a finite perfect word exists if and only if $n$ is a power of two. For $n=2^r$, the dyadic word $(1,2,4,\ldots,2^{r-1})$ is exact and pulse-minimal with $m_{\min}=r=\log_2n$ trials. It replaces the $O(n^2\log(1/\varepsilon))$ worst-case burden of repeated smallest-angle tests by exact $O(\log n)$ verification. Odd primes instead admit balanced statistical words with uniform mixed-weight imitation $2^{-(p-1)}$. For actual operations $R(s_iα_j+δ_{ij})$, an ideal rejecting position is lifted to $\sin^2Δ_j$, where $Δ_j=\sum_iδ_{ij}$. This yields explicit robustness bounds with angular and binary-readout errors. Bell entanglement is not required for the additive algebra, but it keeps the traveler locally maximally mixed in every honest trial. We therefore present the result as a coded multiuser relation primitive with optional raw conference-key-candidate use, not as a composably secure conference-key protocol.

quant-ph↗
arXiv2026

Robust Treasure Hunt in Anonymous Graphs with Quantum Pebbles by Oblivious Agents

We study how to find a hidden treasure in anonymous graphs using an agent that has no persistent memory. The nodes are indistinguishable, and only edges have local port numbers. Classical pebbles placed by an oracle cannot guide an oblivious agent to the treasure. We introduce \emph{quantum pebbles}, which are sources that emit qubits in a fixed (unknown) state, encoding at every node the outgoing port on the shortest path to the treasure. By measuring in several non-orthogonal bases, an oblivious agent recovers the port and can reach the treasure in $D$ steps using $D$ quantum pebbles. This requires $O(Δ^{3}(\log D + \log Δ))$ measurements per node, where $Δ$ is the maximum degree. We further establish \emph{error robustness}, distinguishing two models of state preparation error. Under \emph{per-node persistent} error, where a device returns the same faulty encoding on every read, a single mislabelled coloured pebble can trap an oblivious agent in an infinite loop and every randomized strategy decays exponentially in $D$. Quantum pebbles inherit the same exponential decay. Under \emph{per-emission} error, the intended encoding is correct, but each emitted qubit independently changes state as $ρ= (1-e)\,\lvertψ\rangle\langleψ\rvert + e\,σ$ for an arbitrary noise matrix $σ$. Here the quantum protocol is provably robust. A threshold decoding rule with $O((\log D + \log Δ)/γ^{2})$ measurements per basis, where $γ= (1-e) - δ_e$ and $δ_e = (1-e)δ+ e$, has success probability close to $1$ as $D \to \infty$, provided $e < e^{*} = \sin^2(π/2Δ)/(1+\sin^2(π/2Δ))$. The separation that we establish is thus among quantum pebbles with per-emission error and a persistent marker.

quant-ph↗
arXiv2026

Generalized quantum asymptotic equipartition

The asymptotic equipartition property (AEP) states that in the limit of a large number of independent and identically distributed (i.i.d.) random experiments, the output sequence is virtually certain to come from the typical set, each member of which is almost equally likely. This property is a form of the law of large numbers and lies at the heart of information theory. In this work, we prove a generalized quantum AEP beyond the i.i.d. framework where the random samples are drawn from two sets of quantum states. In particular, under suitable assumptions on the sets, we prove that all operationally relevant divergences converge to the quantum relative entropy between the sets. More specifically, both the quantum hypothesis testing relative entropy and the smoothed max-relative entropy approach the regularized relative entropy between the sets. Notably, the asymptotic limit has explicit convergence guarantees and can be efficiently estimated through convex optimization programs, despite the regularization, provided that the sets have efficient descriptions. The generalized AEP directly implies a new quantum Stein's lemma for conducting quantum hypothesis testing between two sets of quantum states. Combined with the accompanying work [IEEE TIT 72(4):2330-2342 (2026)], this addresses open questions raised by Brandao et al. [IEEE TIT 66(8):5037-5054 (2020)] and Mosonyi et al. [IEEE TIT 68(2):1032-1067 (2022)], which seek a Stein's lemma with computational efficiency. Moreover, we propose a new framework for quantum resource theory in which state transformations are performed without requiring precise characterization of the states being manipulated, making it more robust to imperfections. We demonstrate the reversibility (also referred to as the second law) of such a theory and identify the regularized relative entropy as the unique measure of the resource in this new framework.

quant-ph↗
arXiv2026

Well-conditioned iterative methods for large open quantum systems

Markovian open quantum systems are well modeled by the Lindblad Master Equation (ME) $\frac{\mathrm{d}}{\mathrm{d} t} ρ_t = \mathcal{L} ρ_t$, where $\mathcal{L}$ is a linear (super-)operator and $ρ_t$ is the system state, a positive matrix. When designing or characterizing a quantum system, one is usually interested in the steady state $ρ_\infty$ (such that $\mathcal{L} ρ_\infty = 0$), the first few excited states, and trajectories $t\mapsto ρ_t$. In finite dimension, $ρ_t$ is an $n\times n$ matrix, $\mathcal{L}$ thus typically costs $n^4$ to store explicitly as a dense matrix, and $O(n^6)$ to diagonalize or invert exactly, making standard linear algebraic techniques expensive for large systems. However, $\mathcal{L}$ usually costs only $O(n^3)$ to apply. This makes iterative methods appealing, but they do not work without a good preconditioner. In this article, our main observation is that a part of the Lindblad equation, corresponding to the so-called no-jump evolution $\mathcal{S}$, can be inverted efficiently. Using this inverse map, we introduce an auxiliary completely positive trace-preserving (CPTP) map $Φ$ whose fixed point is directly related to $ρ_\infty$, all the other eigenvalues having smaller magnitude. The map $Φ$ is thus well suited to iterative methods, and $ρ_\infty$ can be found in a few Arnoldi iterations. Using the same inverse map $\mathcal{S}^{-1}$ as preconditioner, we compute the low-lying spectrum efficiently via shift-invert Arnoldi, and, as a proof of concept, build an implicit time integrator that is competitive on stiff systems in the low-precision regime. For the steady-state and low excited states problems, our methods scale like $O(n^3)$ per iteration and offer state-of-the-art performance on CPU and GPU.

quant-ph↗
arXiv2026

Quantum Bicycle LDPC Codes with High $kd^2/n$ from Divisor-Driven Search

Bicycle (two-block circulant) quantum low-density parity-check (LDPC) codes include some of the best known small quantum codes, yet their design has relied on group-algebra formulations in which the dimension and distance are accessible only through matrix computation. We show that in the cyclic case the construction collapses into the polynomial ring $\F_2[x]/(x^{l}-1)$: self-orthogonality is automatic, the quantum dimension is read off from a polynomial gcd, and the minimum distance is certified exactly through the Calderbank correspondence to additive codes over $\F_4$, turning code search into an algebraically pre-filtered enumeration that reaches parameter regimes poorly covered by existing tables. A computer search based on this framework recovers the short codes $[[42,12,4]]_2$ and $[[62,12,4]]_2$ and produces a family of codes with competitive figure of merit $kd^2/n$, including $[[66,20,7]]_2$ with $kd^2/n=14.85$, above the bivariate bicycle code $[[144,12,12]]_2$ ($kd^2/n=12$) at less than half the block length, together with $[[46,2,8]]_2$, $[[66,2,9]]_2$, $[[66,4,8]]_2$, $[[66,6,8]]_2$ and, at $n=90$, $[[90,16,6]]_2$, $[[90,18,6]]_2$, $[[90,20,6]]_2$. An exhaustive census at $n=48$ delineates the boundary of this picture: we exhibit a $[[48,10,6]]_2$ code from a minimal $48$-element group (the Aydin--Tamo--Barg realization uses $72$ elements), and prove that distance $5$ forces a stabilizer-rank loss, which excludes $[[48,10,5]]_2$ from the weight-$8$ symmetric coset family. The framework thus opens a systematic route to bicycle-type quantum LDPC codes beyond the reach of group-theoretic searches, and identifies exactly where genuinely coset-theoretic phenomena begin.

cs.IT↗
arXiv2026

Quantum Query Complexity of Persistence Statistics in Graph Zigzags

We study the query complexity of estimating scalar summaries of zigzag bar lifetimes from snapshot-adjacency bits. For graphs $G_1,\ldots,G_m$ on $n$ labeled vertices, let $\ell_b$ be the snapshot lifetime of a degree-one bar $b$ of the intersection zigzag. For a probability generating function $ϕ(x)=\mathbb{E}[x^R]$, the statistic $F_ϕ=\sum_bϕ(\ell_b/m)$ includes normalized degree-$r$ total persistence and the mean generalized rank over a uniform time window. An exact identity underlies our algorithm: sample $R$ uniform times; the expected generalized rank between their minimum and maximum equals $F_ϕ$. For graphs that rank is the circuit rank of an intersection graph, so a nonlinear barcode functional becomes an average of edge and component counts, and no barcode is computed. Without spectral-gap, homology-state, or QRAM assumptions, this gives a quantum estimator with additive error $\varepsilon n$ and $\widetilde O(\sqrt{m(K+n)}/\varepsilon)$ queries when a bound $K\ge F_ϕ$ is supplied, against $\widetilde O(m\min\{n^2,(K+n)/\varepsilon^2\})$ classically, and an adaptive quantum variant with the same instance dependence. These estimators are optimal in two regimes. For every fixed power weight $x^r$, $r\ge2$, and for the uniform-window mean, the worst-case complexities are $\widetildeΘ(n\sqrt m/\varepsilon)$ quantum and $Θ(n^2m)$ classical. On sparse instances, under an explicit split-leakage promise met by power and binomial weights of logarithmic degree and the promise $F_ϕ\le K$, they are $\widetildeΘ(\sqrt{mK}/\varepsilon)$ and $\widetildeΘ(m\min\{n^2,K/\varepsilon^2\})$. The classical lower bounds hold against fully adaptive algorithms, and fewer than $m$ such statistics cannot determine the positive-lifetime histogram. All bounds concern snapshot access; with an explicit update stream, near-linear full-barcode algorithms are known.

quant-ph↗
arXiv2026

Near-Optimal Quantum Lower Bounds for Convex Optimization via Fourier Rank

We establish a near-linear quantum query lower bound for high-accuracy convex optimization over an explicit family of $n$-dimensional ellipsoids. We focus on linear optimization with an explicitly given objective, where the feasible set is accessed through a membership oracle. We show that any algorithm that, for every unit linear objective, returns an exactly feasible point with additive objective error $Θ(n^{-2})$ requires $Ω\!\left(\frac{n}{\log n\,\log\log n}\right)$ membership queries. The same lower bound can be shown to hold if the returned point is only required to be approximately feasible, within $Θ(n^{-2})$ distance from the feasible set. This resolves, up to logarithmic factors, an open question posed by Chakrabarti, Childs, Li, and Wu~(\textit{Quantum}, 2020) and by van Apeldoorn, Gilyén, Gribling, and de Wolf~(\textit{Quantum}, 2020). Coupled with the upper bounds in these papers, the query complexity of high-accuracy convex optimization is characterized tightly up to logarithmic factors. The proof is built around a lower bound for determinant computation that is derived via a novel polynomial method based on Fourier-rank. In the continuous matrix phase-query model, computing the determinant of a real $n\times n$ matrix requires at least $n/2$ matrix-vector product queries. The construction also yields an $Ω(n)$ phase-query lower bound for estimating the minimum eigenvalue of a real symmetric $n\times n$ matrix to additive accuracy $Θ(n^{-2})$. These results extend the determinant and minimum-eigenvalue lower bounds of Childs, Hung, and Li~(ICALP 2021) from finite fields to the real-valued setting. Based on the same constructions, we also prove a near-optimal gradient-query lower bound for constant-accuracy optimization of smooth and strongly convex functions.

quant-ph↗
arXiv2026

Distributed Quantum Hypothesis Testing under Zero-rate Communication Constraints

The trade-offs between error probabilities in quantum hypothesis testing are by now well-understood in the centralized setting, but much less is known for distributed settings. Here, we study a distributed binary hypothesis testing problem to infer a bipartite quantum state shared between two remote parties, where one of these parties communicates to the tester at (asymptotic) zero-rate, while the other party communicates to the tester at zero-rate or higher. As our main contribution, we derive an efficiently computable single-letter formula for the Stein's exponent of this problem, when the state under the alternative is the product of their marginals. For proving the converse direction of our result, we utilize a novel technique based on reverse hypercontractivity of a quantum markov semigroup combined with the pinching method. For the general case with vanishing type I error probability, we show that the Stein's exponent when (at least) one of the parties communicates classically at zero-rate is given by a multi-letter expression involving regularized measured relative entropy maximized over a sub-class of binary outcome separable measurements. When the state under the alternative commutes with a product eigenbasis of the marginal states under the null and has a larger support, we show that the exponent is characterized as a max-min optimization of regularized measured relative entropy over a sub-class of local binary outcome projective measurements. While this expression becomes single-letter for the fully classical case, we further prove that this already does not happen in the same way for classical-quantum states in general. The converse proof of the max-min characterization relies on an extension of the classical blowing-up lemma to bipartite quantum states satisfying the aforementioned commutativity condition, which could be of independent interest.

quant-ph↗
arXiv2026

Optimal inequalities for completely bounded polynomials and the limitations of quantum query algorithms

We consider the problem of establishing limitations on the power of quantum query algorithms via the completely bounded polynomial method. In particular, we prove several optimal functional inequalities involving different notions of completely bounded polynomials. These inequalities lead to limiting theorems for the power of quantum query algorithms that improve on prior works. 1. An optimal root-influence bound for block-multilinear polynomials. Prior work showed that block-multilinear polynomials $p$ of degree $t$ satisfy a root-influence bound, $\|p\|_{\text{cb}}\geq \sum_i \sqrt{\mathrm{Inf}_i[p]}/t^2$, which is stronger than the bound appearing in the Aaronson-Ambainis conjecture. We find the optimal constant in that inequality: $\|p\|_{\text{cb}}\geq \sum_i \sqrt{\mathrm{Inf}_i[p]}/t$. Since the amplitudes of quantum algorithms that query disjoint blocks of inputs-such as $t$-fold forrelation- are block-multilinear polynomials with $\|p\|_{\text{cb}}\leq 1,$ our inequality shows that they satisfy $t\geq \sum_i\sqrt{\mathrm{Inf}_i[p]}$. We prove that this inequality yields both a more efficient classical simulation than prior results based on the Aaronson-Ambainis argument, and a qualitative improvement: all classical queries are nonadaptive. 2. Optimal Fourier growth of the highest level of quantum query algorithms. We show that for every polynomial $p$ defined on $\{-1,1\}^n$ of degree $2t$, the Fourier Growth at the level $2t,$ namely $\|\widehat p_{2t}\|_{\ell_1}$, satisfies $\|\widehat p_{2t}\|_{\ell_1}\leq (en/(2t-1))^{\frac{2t-1}{2}}\|p\|_{\text{cb}}$. This is optimal up to the factor $e$, as witnessed by $2t$-fold forrelation. As quantum query algorithms that make $t$ queries (to the whole input) satisfy $\|p\|_{\text{cb}}\leq 1$, this yields a Fourier growth bound for these algorithms, partially resolving a question by Girish (STOC, 2026).

quant-ph↗
arXiv2026

Python in the front, party in the Backline: compiling quantum workloads across CPUs, GPUs, and FPGAs

Moving from quantum research and development to production-grade, fault-tolerant quantum workload execution remains one of the most significant challenges facing quantum platform builders. While Python frameworks have enabled an easy entry point for quantum algorithm design, the low-latency requirements for real-time quantum error correction (QEC) demand performance that traditional interpreted environments cannot provide. FPGAs and ASICs play a central role at these layers, but their specialized programming models make development rigid and time-consuming. CPUs, GPUs, and other accelerators introduce a different challenge: as infrastructure becomes increasingly heterogeneous, programming across different devices and their associated abstractions becomes more complex. Allowing researchers to write workloads in high-level languages that map to low-latency execution across diverse distributed target platforms will enable the development of key infrastructure for utility-scale quantum systems. For this, we introduce $\textit{Backline}$, a heterogeneous compilation and runtime framework built within PennyLane and Catalyst. Backline allows us to design and build quantum-classical workloads for high-performance and low-latency devices, with compilation directly from a Python interface through MLIR. We demonstrate the compilation and execution of several quantum workloads with low-latency data movement across a mix of CPUs, GPUs, and FPGAs, for both local and distributed remote hardware targets, all from a vendor-agnostic Python frontend. With an AMD VPK120 FPGA board as the controller, issuing each round from its hardware-handshake engine, we measured median steady-state round-trip latencies over RoCE v2 of $2.305~μ$s to an AMD Ryzen Threadripper PRO CPU and $4.5~μ$s to an AMD Instinct MI210 GPU across $10^6-1$ rounds per path, demonstrating microsecond-scale synchronous co-processing.

quant-ph↗
arXiv2026

Quantum Private Distributed Matrix Multiplication: Extending the Classical Codes and Limitations

In this paper, we explore how quantum resources can be used to increase the rate of private distributed matrix multiplication (PDMM). In PDMM, a user who has two high-dimensional matrices, A and B, and lacks the computational capabilities to apply matrix multiplication locally, divides the matrices A and B into K and L sub-blocks, respectively. Then, the user sends them to N servers to apply the required multiplication \emph{privately}, i.e., any $T$ colluding servers cannot get any information about the user's matrices. The goal is to reduce the number of servers needed to perform the required matrix multiplication, thereby decreasing the communication cost. First, in the high-privacy regime, the state-of-the-art classical code is called the gap additive secure polynomial (GASP) code. We define a feasibility requirement in the quantum setting for the GASP code such that the highest performance is achieved when the requirement is satisfied. Thus, super-dense coding gain is achieved when the feasibility condition is satisfied. We show that when $T \geq KL-K+1$, the feasibility condition is always satisfied and the GASP code can be extended to the quantum version. In the case of $T < KL-K+1$, the feasibility can still be satisfied. To further examine this behavior, we numerically study how the minimum privacy requirement depends on the matrix dimensions and provide a quadratic estimate for this relation. The results suggest that feasibility can be achieved when $T \sim 0.5 KL$. Second, in the low-privacy regime, the recently developed cyclic-addition degree tables (CAT) and discretely optimized GASP (DOG) codes are among the most efficient known classical constructions for PDMM. We show that the feasibility condition developed for GASP can be adopted for both CAT and DOG codes as well, thus unifying the feasibility framework for multiple classical PDMM coding schemes.

cs.IT↗
arXiv2026

How to Build a Quantum Supercomputer: Scaling from Hundreds to Millions of Qubits

In the span of four decades, quantum computation has evolved from an intellectual curiosity to a potentially realizable technology. Today, small-scale demonstrations have become possible for quantum algorithmic primitives on hundreds of physical qubits. Nevertheless, there are significant outstanding challenges in quantum hardware, fabrication, software architecture, and algorithms on the path towards a full-stack scalable quantum computing technology. Here, we provide a comprehensive review of these scaling challenges. We show how to facilitate scaling by adopting existing semiconductor technology to build much higher-quality qubits, employing systems engineering approaches, and performing distributed heterogeneous quantum-classical computing. We provide a detailed resource and sensitivity analysis for quantum applications on surface-code error-corrected quantum computers given current, target, and desired hardware specifications based on superconducting qubits, accounting for a realistic distribution of errors. We provide comprehensive resource estimates for several utility-scale applications including quantum chemistry calculations, catalyst design, NMR spectroscopy, and Fermi-Hubbard simulation. We show that orders of magnitude enhancement in performance could be obtained by a combination of hardware improvements and tight quantum-HPC integration. Furthermore, we introduce high-performance architectures for quantum-probabilistic computing with custom-designed accelerators to tackle today's industry-scale classical optimization, machine learning, and quantum simulation tasks in a cost-effective manner.

quant-ph↗
arXiv2026

AutoScientist-Quant: Self-Evolving Coding Agents for Automatic Research in Quantitative Investment

Large language model agents can discover alphas, yet current methods have three weaknesses. The search cannot adapt during the run, automation usually ends at alpha generation while library selection and model choice stay manual, and alpha discovery can read the test window through loop feedback or code problems. We present AutoScientist-Quant, a self evolving search process that regards quantitative research as one budgeted search problem. A single controller conditions every decision on the remaining budget, choosing at each round whether to improve, combine, pivot, or stop, which node to expand, how many alphas to generate, and how to retrieve past trajectories from the shared memory. The same core then selects from the library and tunes the model, closing the loop from hypothesis to deployable strategy. We also review the evaluation pipeline reused from prior work, fix two lookahead problems, and keep the feedback window disjoint from the held out test window, so every comparison tests true generalization. On CSI universes, the framework attains the best value of nearly every metric in every setting, and these conclusions hold across several backbones and markets.

cs.AI↗
arXiv2026

Geometric integrators for adiabatically closed simple thermodynamic systems

A variational formulation for non-equilibrium thermodynamics was developed by Gay-Balmaz and Yoshimura. In a recent article, the first two authors of the present paper introduced partially cosymplectic structures as a geometric framework for thermodynamic systems, recovering the evolution equations obtained variationally. In this paper, we develop a discrete variational principle for adiabatically closed simple thermodynamic systems, which can be utilised to construct numerical integrators for the dynamics of such systems. The effectiveness of our method is illustrated with several examples.

math-ph↗