What Can Be Computed with Indefinite Causal Structure?
Omri Shmueli
AI 速览：在过程矩阵形式下研究不定因果序的计算能力，作者报告在两种设定下给出多项式时间算法，分别覆盖 NP 与唯一汇点定向问题。
Can quantum computation become more powerful when operations do not need to follow a definite causal order? We study this question in the process matrix formalism of Oreshkov et al. (Nature Communications, 2012), which allows quantum operations to be connected without a fixed global order. In the computational model, a quantum circuit succinctly describes a process matrix, and the algorithm is given access to its generated channel, with computational cost measured by the size of the generating circuit. We give polynomial-time algorithms in two settings. First, general process matrix computation suffices to solve every problem in $\textbf{NP}$. The valid processes used by our algorithm satisfy the purification postulate of Araújo et al. (Quantum, 2017), and their generated channels are unitary, although their generating circuits are not. Second, requiring the process matrices themselves to be pure, we give a polynomial-time algorithm for unique-sink orientation, a general problem capturing several longstanding computational problems in optimization. Our unitary generators are constructed explicitly and efficiently. Assuming that unique-sink orientation has no quantum polynomial-time algorithm, our result shows that indefinite causal order can provide computational power beyond $\textbf{BQP}$ within the process matrix formalism, even while preserving unitarity.
AI 速览：作者报告构造出经典色数任意大而量子色数恒为3的图，首次改进了自 Brassard–Cleve–Tapp 以来的量子色数间隙。
We confirm a conjecture of the first author [SODA'26] on a strong form of pseudo-telepathy for Khot's $d$-to-$1$ games: for every $d\geq 4$, every $ε>0$, and every $k\in\mathbb N$, there is a $d$-to-$1$ label-cover game of classical value less than $ε$ that admits a perfect quantum strategy using finite-dimensional shared entanglement with measurement contexts of radius at least $k$. As a consequence, there exist graphs whose quantum chromatic number $χ_q$ is $3$ and whose classical chromatic number $χ$ is arbitrarily large. This is the first improvement on the quantum chromatic gap since the work of Brassard, Cleve, and Tapp [Phys. Rev. Lett.'99], which produced graphs for which $χ$ grows exponentially in $χ_q$. To prove the conjecture, we proceed as follows. We start with a ``stretched'' version of the Mermin--Peres magic square, obtained by spacing out its equations and linking them with additional long equation wires. This operation increases the radius of the initial measurement contexts of a perfect quantum strategy for the game, while preserving classical unsatisfiability. We then apply parallel repetition to make the classical value small, and we turn the system into a pseudo-telepathic multi-layered Kneser hypergraph via a PCP construction of Dinur, Guruswami, Khot, and Regev [STOC'03]. Finally, we feed the resulting hypergraph into the reduction of Fei, Minzer, and Wang from hypergraph colouring to $4$-to-$1$ games, we establish quantum completeness of this reduction, and we show that it preserves context radius.
Ziyi Guan, Yael Tauman Kalai, Ron Rothblum, Zoe Xi
AI 速览：提出可回卷交互证明（RIPs），利用验证者可回卷AI证明者状态的能力，在无需诚实证明者假设且不使用密码学的前提下实现更高效的可扩展监督。
Scalable oversight asks how a weak verifier can check the outputs of computations performed by more powerful AI systems. One proposed approach uses relativizing interactive proofs, in which a weak verifier checks the output of a complex oracle-aided computation by interacting with one or more powerful provers. A prominent example is debate, which is highly efficient but assumes that one of two powerful AI models, or provers, is truthful. For certain classes of oracle-aided computations, this assumption can be removed, but at substantially greater computational cost. In this work, we introduce \emph{rewindable interactive proofs} (RIPs), a new model of relativizing interactive proofs for scalable oversight. In an RIP, the verifier may rewind the prover to an earlier state of the interaction and continue from there. This is possible when the prover is an AI model, since the verifier can save a snapshot of the model's state and later restore it. We construct a relativizing RIP for computations whose outputs are robust to changing a small fraction of oracle answers. Our construction is more efficient than prior relativizing proofs that do not assume a truthful prover, and it uses no cryptography. In our construction the honest prover runs in time linear in that of the original computation, and the verifier, after a one-time preprocessing step, runs in time nearly linear in the input length (which can be much smaller than the computation itself). We implement our protocol, report a preliminary MNIST experiment, and show that an RIP can be faster than a state-of-the-art deployed proof system. RIPs may be of independent interest. Rewinding is possible whenever the verifier controls the prover's machine, for example when the prover runs on trusted hardware or in a virtual machine whose state can be checkpointed.
Ziad Ismaili Alaoui, Oded Lachish, Nikhil S. Mande
AI 速览：作者在割查询模型下证明，确定性寻找锦标赛中“国王”的查询复杂度为 Θ(n)。
A king in a tournament is a vertex from which every other vertex is reachable by a directed path of length at most two. We study the deterministic query complexity of finding a king when a query specifies a set of vertices and returns the number of edges directed from that set to its complement. We prove that this complexity is $Θ(n)$ for $n$-vertex tournaments. To show this, we first show that every correct deterministic algorithm in this model must output a maximum-outdegree vertex, even though a king need not have maximum outdegree. We then give an adversary whose answers, after too few queries, remain consistent with a tournament in which the algorithm's output does not have maximum outdegree.
Cubic lower bounds for the permanent in odd characteristic, for Pfaffian representations, and for polynomial entries
Karthik Sheshadri
AI 速览：作者证明了在任意奇特征代数闭域上，永久式的行列式复杂度具有三次下界，并推广到 Pfaffian 表示和多项式元素矩阵。
The determinantal complexity of a polynomial is the order of the smallest matrix whose entries are polynomials of degree at most one and whose determinant equals the polynomial. A lower bound for the permanent in this measure is the algebraic form of the question whether the permanent is harder to compute than the determinant. Over the complex numbers, the best known lower bound grows cubically in the size of the permanent. It holds even when the permanent only has to be a limit of such determinants. I prove that the cubic lower bound holds over every algebraically closed field of odd characteristic $p$, with a constant that depends on $p$. Hence the cubic bound does not depend on characteristic zero. I also prove the bound for Pfaffians, which are square roots of determinants of skew symmetric matrices. Lower bounds for Pfaffians imply the same bounds for determinants. Further, I prove the bound, divided by $a$, for matrices whose entries are polynomials of degree at most $a$. All three results come from one lower bound for smooth hypersurfaces. I prove it in every characteristic by counting intersection points together with their multiplicities.
Random ($\log{n}$)-CNF Formulas Are Hard for Depth-$2$ Frege with Modular Gates
Jonas Conneryd
AI 速览：作者证明随机 k-CNF（k 在 ω_n(1) 到 O(log n) 之间）需要多项式演算度数至少 n/2−O(n/√k)，并借助 Pang 的权衡关系推出深度-2 AC^0[p]-Frege 的指数下界。
We prove that, with high probability, refuting a random $k$-CNF formula on $n$ variables with ${M = n2^{k}}$ clauses for $ω_n(1) \leq k \leq O(\log{n})$ requires polynomial calculus degree at least ${n/2-O(n/\sqrt{k})}$ over every field. Through Pang's recent tradeoff between $\mathsf{AC}^0[p]$-Frege size and polynomial calculus degree [Pang 2026, arXiv:2610.00837v2], this implies a size lower bound of $\exp(Ω_p(\sqrt{k} \log{n}))$ for depth-$2$ $\mathsf{AC}^0[p]$-Frege. In this range of $k$, the bound is superpolynomial in $M$ while $M \leq \mathrm{poly}(n)$. We also prove a nearly matching degree upper bound of $n/2-n\ln(k)/{2k} +O(n/k)$, which shows that other formula families are needed to obtain lower bounds for full $\mathsf{AC}^0[p]$-Frege through Pang's tradeoff.
Half-Optimal Induced Matchings in Random Graphs: Online and Low-Degree Algorithms
Abhishek Dhawan, Eren C. Kızıldağ, Neeladri Maitra, Cheng Mao, Timothy L. H. Wee
AI 速览：作者证明在 Erdős-Rényi 随机图中，寻找大诱导匹配存在尖锐的因子二间隙：简单在线贪心可达约一半最优，而在线算法与低阶多项式算法均无法突破该阈值。
For many optimization problems on random graphs, most prominently the largest independent set problem, natural algorithms reach only half the optimum, and such factor-two gaps are known to be sharp within broad algorithmic classes. We prove such a gap for finding large induced matchings in Erdős-Rényi random graphs. In the worst case, requiring a matching to be induced turns a polynomial-time solvable problem into an NP-hard one; our results reveal an average-case counterpart. In $G(n,p)$ with $C/n\le p\le 1-κ$, the largest induced matching has size $(1+o(1))L$ with high probability, where $L=\log_{1/(1-p)}(np)$. A simple online greedy algorithm finds an induced matching of size $(1/2-ε)L$, while no online algorithm, even with adaptive inspection order and unbounded computation, finds one of size $(1/2+ε)L$ except with probability $\exp(-Ω(L\log(np)))$. In the sparse regime $p=d/n$, low-degree polynomial algorithms exhibit the same sharp threshold. Our guarantees rest on a maximality principle: with high probability, every maximal induced matching is half-optimal. Greedy outputs a maximal induced matching and our low-degree algorithm a near-maximal one. We bypass the delicate coupling arguments classically used to analyze greedy since Karp. For low-degree hardness, we adapt Wein's overlap gap argument, with forbidden structures that track overlaps between endpoint sets rather than edge sets, since a new edge may join two previously used endpoints and thus encounter no fresh randomness. For online hardness, we give a new and significantly simpler argument that avoids the correlated instances of prior OGP-based proofs: at a suitable stopping time, all pairs involving uninspected vertices remain unrevealed, so a direct first-moment bound suffices.
A Unified Complexity Framework for Quantum Property Testing
Qisheng Wang
AI 速览：作者提出一个基于函数型 tr(φ(dρ))/d 的统一框架，用于推导量子性质检验的样本与查询复杂度下界，并给出熵估计与谱估计的（近）紧复杂度。
We develop a unified framework for analyzing the complexity of quantum property testing through functionals of the form $\mathcal{L}_φ(ρ) = \operatorname{tr}(φ(dρ))/d$, where $ρ$ is an unknown $d$-dimensional quantum state and $φ$ is a given function. A master theorem is established that derives sample complexity lower bounds for estimating $\mathcal{L}_φ(ρ)$ from properties of $φ$, combining Haar-random moment encoding with moment matching and best polynomial approximation. Corresponding query complexity lower bounds follow from quantum sample-to-query lifting. The framework yields (nearly) tight bounds for a broad class of problems, including entropy estimation (von Neumann, Rényi, and Tsallis), closeness estimation (trace distance and Uhlmann fidelity), spectrum estimation, rank testing (operator rank, Schmidt rank, and matrix product states). In particular, combined with known upper bounds, we show that the sample complexity of von Neumann entropy estimation is \[ Θ\!\left(\frac{d^2}{\varepsilon\log^2(d)\max\{1, \varepsilon\log^2(d)\}} + \frac{\log^2(d)}{\varepsilon^2}\right), \] and the sample complexity of spectrum estimation is \[ Θ\!\left(\frac{d^2}{\varepsilon^2\log^2(d)\max\{1, \varepsilon^2\log^2(d)\}}\right). \] In summary, our results resolve several open problems and establish the optimality of 37 quantum algorithms since 2015, up to constant or polylogarithmic factors.
In a compact storage system with side access, bins are arranged in adjacent vertical stacks. In each retrieval cycle, bins blocking access are lifted to create a temporary tunnel through which a robot extracts one requested bin from the side. The lifted bins are then lowered again, and the energy expenditure of the cycle equals the number of lifted bins. Given a pick list known in advance, we seek a retrieval order that minimizes total energy expenditure. For every number of stacks, we prove that the total energy separates into an instance constant and a pairwise penalty, and that every energy-optimal order is laminar within each stack. For two stacks, we prove that the retrieval sequencing problem is solvable in polynomial time. For three stacks, we give an exact polynomial-time reduction to a merge problem and characterize which merge-problem instances arise from the storage model. We define a recursive class of retrieval orders and construct a polynomial-time dynamic program that minimizes energy expenditure over this class. Finally, we quantify the additional energy savings from batching, which allows several requested bins to be retrieved in one cycle.
The $T$-count is a dominant cost of fault-tolerant Clifford$+T$ computation. We prove that every Hadamard-free Clifford$+T$ circuit with clean ancillas for a layer of $m$ disjoint CCZ gates, the diagonal core of a parallel Toffoli layer, uses at least $6m+1$ $T$ gates. Campbell and Howard gave a circuit with $6m+1$, and to our knowledge this is the first proof that it is optimal for general $m$. The proof combines Campbell and Howard's self-orthogonality of the columns of the parity matrix with a rank (flattening) bound, sharpened by one. More generally, the argument gives a lower bound on the $T$-count of every diagonal gate in the third level of the Clifford hierarchy, the isotropy floor, computable by Gaussian elimination. It is never below the stabilizer nullity $ν$ and reaches $2ν+1$ on non-Clifford gates whose phase is purely cubic. No bound that also holds for circuits with measurement and feedforward can do this, since such circuits implement CCZ ($ν=3$) with four $T$ gates and the layer with $4m$, where the floor gives seven and $6m+1$. The floor also recovers Campbell and Howard's count $4m+3$ for a fan-out of Toffolis. On the output of PyZX's TODD-based optimizer, relative to its Hadamard placement, it certifies $193$ of $311$ phase-polynomial blocks $T$-optimal once repeated parities are merged (at the best of five optimizer seeds), against $113$ for nullity, including $37$ of the $72$ blocks too large for exhaustive search.
We revisit classical diagonalization argument in the setting of semi-Thue systems, viewing computation as nondeterministic rewriting rather than sequential machine execution. Semi-Thue systems naturally generate derivation trees, making existential acceptance, reachability, and infinite derivations fundamental computational notions. We formulate acceptance and decision directly in terms of derivation trees and study the associated assertion, word, and infiniteness problems. We prove that the complements of the assertion and word problems are not recursively enumerable by direct Post-style diagonal arguments. For the infiniteness problem, we give a constructive diagonalization based on a fixed universal semi-Thue system. Unlike the classical approach, the diagonal construction is carried out entirely within the rewriting formalism: the universal system remains an ordinary nondeterministic semi-Thue system while simulating encoded semi-Thue computations, including self-reference. These results show that undecidability in semi-Thue systems arises naturally from universality and self-reference in nondeterministic rewriting, rather than from machine states or control mechanisms. This perspective clarifies the relationship between rewriting systems, recursive enumerability, and classical diagonal arguments originating in the work of Post and Turing.
We construct, for each deterministic Turing machine that halts on all inputs, one polynomial vector field with rational coefficients that performs the same logical computation at unit Euler step, at every smaller or variable step in $(0,1]$ with divergent sum, and along the flow. This unifies discrete-time and continuous-time computation by polynomial dynamics, which earlier characterizations treated separately. For a machine making $T$ transitions in space $H$, a logical scale $h=2^{-s}$ with $s=O(H)$ suffices, with physical time $O(s(T+s))$, drift disturbances up to $h^{2K}$ for a fixed exponent $K$, and $2Ks+3$ fractional bits at unit step. On a fixed compact domain, polynomial time and robust eventual decision characterize P and PSPACE, respectively, with polynomial scale thresholds and no supplied stabilization horizon in the space case. Without sign, floor or conditional primitives, margin amplification at bounded speed, overlapping polynomial gates and interval-product estimates give the common realization, from which we derive shared logical traces, fixed-field PSPACE-complete prediction and a robust safety family with polynomial certificates common to all Euler steps and the flow.
Vector addition systems (VAS) are a fundamental model of concurrent computation in which transitions update nonnegative integer counters by adding integer vectors, provided all counters remain nonnegative. Closely related to Petri nets, VAS are the single-state counterpart of vector addition systems with states (VASS), which use finitely many states to restrict the available transitions. A classical simulation of $d$-VASS by $(d+3)$-VAS connects their reachability problems, but the complexity-theoretic picture of fixed-dimensional VAS remains largely unclear. While reachability in $2$-VASS is PSPACE-complete under binary encoding, the best known upper bound for $5$-VAS was only elementary. We prove that every reachable $5$-VAS instance has a reachability witness of length polynomial in its unary size. Consequently, $5$-VAS reachability is NL-complete under unary encoding and PSPACE-complete under binary encoding. We also establish a polynomial factorization of the reachability relation of every $d$-VAS with $d \ge 3$. For $5$-VAS, this strengthens Hopcroft and Pansiot's effective semilinearity of reachability sets to the full reachability relation. For every fixed $d\ge3$, it also yields an $\mathsf F_d$ upper bound for reachability in $(d+3)$-VAS.
We develop an analytical framework for Boolean promise constraint satisfaction problems (PCSPs) that studies polymorphisms through the notion of influence from Fourier analysis of Boolean functions. Extending the work of Brakensiek, Guruswami, and Sandeep [ICALP'21] on Ordered PCSPs, we identify two general phenomena in Boolean minions indicative of hardness or tractability: (1) preservation of coordinate influence under random 2-to-1 minors and (2) the presence of sharp thresholds. We demonstrate that these phenomena occur in broader settings than previously established, yielding new hardness/tractability results for minions consisting of unate or polynomial threshold functions.
Wen Fang, Xianxian Li, Jun Liu, Jie Luo, Yongxin Tong, Ke Xu
We study self-reference and solution independence in Boolean $K$-SAT as a finite analogue of Gödel incompleteness. Independence is defined through factorial moments of the solution count. Constant-width random formulas fail this criterion due to correlations among overlapping assignments. At a covering threshold, logarithmic-width random CNFs have a Poisson solution count, with positive probabilities of unsatisfiability and unique satisfiability. Replacing a redundant clause by one falsified by the unique solution yields an unsatisfiable companion preserving the unsigned incidence graph. For every permitted local view of a good uniquely satisfiable formula, an unsatisfiable companion preserves that view. Irreducibility defeats deterministic and bounded-error evaluators using $N^c$ clause queries, for fixed $0<c<1$. It establishes a local information obstruction, not an unrestricted time lower bound. For fixed $δ>0$, transcripts of $N^{1-δ}$ queries leave $N-o(N)$ bits of witness entropy. On typical support graphs, expansion yields linear Resolution width, linear PC/PCR and SOS degree, and exponential proof size. Bounds hold for semantic Cutting Planes under a partition condition and for Positivstellensatz with bounded product width, uniformly over unsatisfiable support-preserving signings. Structural evidence and the limits of acceleration strategies motivate the self-referential $K$-SAT exhaustive-search hypothesis (SRK-ESH): no uniform deterministic algorithm deciding the promise family achieves a fixed proportional saving in the exponent. This dual-evidence criterion combines intrinsic hardness mechanisms with algorithmic evidence. We interpret this hypothesis as a quantitative projection of the Gödel-style local--global gap. The information obstruction constrains quantum local observation and statistical learning from bounded local information.
A tournament solution records a set of winners from a tournament, that is, from the outcomes of all pairwise comparisons among $N$ alternatives. We study the quantum query complexity of computing tournament solutions when the outcome of each comparison has to be learned by a query. Classically, the picture is uniform: the Copeland set, the uncovered set and the top cycle all require $Θ(N^2)$ queries, and so does finding a single vertex of maximum out-degree. We show that the quantum query complexity of the top cycle can be computed with $\tildeΘ(N)$ queries, and a Condorcet winner can be found with $\tilde Θ(\sqrt N)$ queries. Finding a strong king requires $Θ(N^{3/2})$ queries, the same as finding a vertex of maximum out-degree. We also give a self-contained alternative proof of the known $Θ(N^{3/2})$ bound for finding a vertex of maximum degree, established by Gilani, Mande, Patro, and Raja. Our lower bounds reduce counting problems to tournament problems using Ambainis's adversary method. For the top cycle, we find a Hamiltonian path and locate the prefix forming the top cycle by divide and conquer.
In this paper, we study the computational complexity of finding fixed points for set-valued contractions. We first formulate a computational problem for Nadler's fixed-point theorem: Projected-Nadler, and prove that it is $\mathsf{CLS}$-complete by showing its equivalence to Continuous-LocalOpt. We then establish a stronger converse for Nadler's fixed point theorem that can be applied as a tool to analyze the convergence rate of set-valued basic iteration procedure. Finally, we reduce large-margin triplet stationarity problem to Projected-Nadler. Together with its $\mathsf{CLS}$-hardness introduced in [arXiv:2509.16898], this yields $\mathsf{CLS}$-completeness of large-margin triplet stationarity.
Song (2026) proves that, for every fixed $0<ε<1$, the closest codeword problem (CCP) for binary linear codes under Hamming distance is NP-hard to approximate within $N^{1-ε}$, where $N$ is the code length, under deterministic polynomial-time many-one reductions. The code-to-lattice construction gives hardness within $N^{1/2-ε/2}$ for Euclidean CVP, with the same $N$ aslattice rank. This reaches every fixed Euclidean hardness exponent below one half. Aharonov and Regev (2005), with the constant refined by Gao (2026), place GapCVP at factor $cN^{1/2}$ in $\mathrm{NP}\cap\mathrm{coNP}$ for every fixed $c>2/(π\sqrt3)\approx0.3676$. NP-hardness at that scale would imply $\mathrm{NP}=\mathrm{coNP}$. This paper has two goals: to explain Song's theorem and its mathematics for graduate students, and to make the reduction more efficient. The reduction turns satisfiability into a system of quadratic equations and then into CCP for binary linear codes, and recovers assignments from recurrence relations of algebraic moments. We adapt the method to linear codes over any fixed finite field of characteristic $p$. A common recurrence for each block column gives a moment window $L=c_p(κ-1)+1$, where $c_p=\max\{2,p-1\}$ and $κ$ is the rank cutoff. Together with ordinary digit products and flexible sampling, this reduces interpolation degrees and output lengths. For CCP gap $N^{1-ε}$, our explicit schedule gives code length and lattice rank at most $n^{4/ε^2+O(1/ε)}$; Song's prescribed binary schedule gives $n^{8/ε^2+O(1/ε)}$. Here $n$ counts variables in a common quadratic system, equal to the number of original variables plus clauses for 3SAT. These are dimension bounds, not optimality claims or running-time exponents. The limiting hardness exponents remain Song's.
It is $\#\mathrm{P}\text{-hard}$. Even for connected simple undirected unweighted graphs, deciding whether an expected random-walk cover time is at most a given rational threshold is $\#\mathrm{P}\text{-hard}$ under polynomial-time Turing reductions, both from a specified starting vertex and when maximized over starting vertices. Both families have a universal vertex, called the hub, and diameter two. Deleting the hub leaves a planar bipartite graph, and every non-hub vertex has degree at most five. A shared construction encodes an Euler-tour count in rare covering walks. For specified starts, a number of fractional bits linear in the queried graph's vertex count suffices for recovery. For the maximum-start problem, a fixed cube attachment makes the maximizing starts known, and one interpolation of expected cover times recovers the count. Both reductions use polynomial-size graphs and polynomial-bit threshold queries. For general graphs, both comparison problems can be solved in polynomial time with access to an $\mathrm{NP}^{\mathrm{PP}}$ oracle.
In this paper, we investigate the low-depth circuit complexity of Group Isomorphism in the multiplication (Cayley) table model. We prove the first circuit lower bounds for Group Isomorphism: namely, we show that every family of depth-$2$ Boolean circuits deciding Group Isomorphism requires quasipolynomial-size. (We also extend this to a limited class of depth-3 circuits, namely $\textsf{CNF}$-of-$\textsf{XOR}$s.) We complement this with upper bounds of uniform depth-$2\frac{1}{2}$ circuits of quasipolynomial-size: depth 3 circuits with poly-logarithmic bottom fan-in. A sequence of previous results from 1970-2025 progressively reduced the circuit depth from polynomial to $3\frac{1}{2}$; all of these results relied on the generator-enumerator strategy and, in fact, applied more generally to quasigroups. In contrast, our depth-$2\frac{1}{2}$ construction follows a fundamentally different strategy that exploits structure more specific to groups. We guess a composition series for each group, together with generators for its terms and the isomorphism types of its composition factors. We then inductively verify that the corresponding extensions at each level of the two composition series are compatible. A central part in this approach brings to bear the extensive work on the Short Presentation Conjecture, in tandem with the algorithmic theory of group extensions and cohomology.
Optical and wave-based analog computing has re-emerged as a route to accelerating linear algebra, with metamaterials and photonic circuits shown to invert matrices and solve integral equations. However, it has remained unclear which problems such passive structures can represent, and at what cost, once encoding, precision, conditioning and readout are accounted for. Here, we show that every polynomially conditioned sparse linear system can be compiled into a three-dimensional dielectric -- a cavity filled with lossless, isotropic material of bounded permittivity -- whose time-harmonic response encodes its solution. For $m$ nonzero coefficients, the volume of the structure and the cost of writing it down and reading out the answer scale as $\tilde O(m^{3/2})$, coefficients require only $O(\log m)$ bits, and every field that approximately satisfies Maxwell's equations decodes to an approximate solution. We prove the construction for the continuum equations and give a computer-assisted version for the Yee discretization underlying finite-difference solvers. Read in reverse, it implies that electromagnetic simulation is, up to this overhead, as hard as general sparse linear algebra: a sufficiently fast general-purpose Maxwell solver would yield a faster sparse linear solver.
We study fault-tolerant quantum computation using a fixed, local error-correction rule and a geometrically simple code: the cubical $(d,d')$-Toric code with $d',d-d'\geq 2$, including the 4D Toric code. Under sufficiently weak local-stochastic Pauli noise, our construction uses noiseless, bounded-depth classical processing per physical time step. Its correction cycle acts only on the current syndrome, has constant depth, and is repeated unchanged during idle logical computation; it requires neither multiscale decoder states nor scale-dependent communication delays. Our proof treats errors and syndromes as chains in a cubical complex, combining geometric shrinking under the sweep rule with a space--time explanation-tree argument inspired by Toom's theorem. It does not rely on expansion. We establish three results: 1.Explicit higher-dimensional sweep rule. We give a local algorithm for choosing sweep corrections on cubical $(d,d')$-Toric complexes, resolving the local-existence question of Kubica and Preskill. 2. Fault-tolerant computation. We prove a circuit-level threshold theorem for noisy sweep cycles interleaved with logical gates. In particular, the 4D Toric code admits a fixed, constant-depth syndrome-based correction cycle without a multiscale decoder or expansion-based encoding. We do not claim constant total-depth overhead. 3. Ideal decoding under local-stochastic noise. For an initial local-stochastic Pauli error of rate $p$ below threshold, $n^{d+1}$ noiseless sweep cycles recover the encoded state on a side-length-$n$ $(d,d')$-Toric code with probability at least $1-O(\operatorname{poly}(n))p^{Ω(n)}$, extending earlier guarantees for independent errors.
Foivos Fioravantes, Dušan Knop, Jan Matyáš Křišťan, Nikolaos Melissinos, Michal Opler
Consider the scenario where multiple agents have to move in an optimal way through a network, each one towards their ending position while avoiding collisions. By optimal, we mean as fast as possible, which is evaluated by a measure known as the makespan of the proposed solution. This is the setting studied in the Multiagent Path Finding problem. In this work, we additionally provide the agents with a way to communicate with each other. Due to size constraints, it is reasonable to assume that the range of communication of each agent will be limited. What should be the trajectories of the agents to, additionally, maintain a backbone of communication? In this work, we study the Multiagent Path Finding with Communication Constraint problem under the parameterized complexity framework. Our main contribution is three exact algorithms that are efficient when considering particular structures for the input network. We provide such algorithms for the case when the communication range and the number of agents (the makespan resp.) are provided in the input and the network has a tree topology, or bounded maximum degree (has a tree-like topology, i.e., bounded treewidth resp.). We complement these results by showing that it is highly unlikely to construct efficient algorithms when considering the number of agents as part of the input, even if the makespan is $3$ and the communication range is $1$.
Joshua Brakensiek, David Conlon, Venkatesan Guruswami, Xiaoyu He, Jiaxi Nie, Maya Sankar, Hung-Hsun Hans Yu
In the study of constraint satisfaction problems (CSPs), the non-redundancy of a relation measures the maximum number of clauses in a CSP instance which can be logically independent. In the last few years, CSP non-redundancy has found a number of applications including in query complexity, sparsification, kernelization and streaming. In this paper, we present a novel technique for upper-bounding the non-redundancy of some relations using locally decodable codes (LDCs) and locally correctable codes (LCCs). More precisely, known limitations on the structure of 2-query LDCs and LCCs can be translated into limitations on the size of certain non-redundant CSP instances. This technique builds on a technique used by Brakensiek and Guruswami [STOC 2025] to bound non-redundancy via techniques from matching vector codes. Using LDCs and LCCs, we give improved upper bounds on non-redundancy of the BCK predicate pioneered by Bessiere, Carbonnel, and Katsirelos [AAAI 2020] as well as its well-studied Boolean variant. We also give the first example of a Boolean predicate whose non-redundancy is provably not of the form $n^{k\pm o(1)}$ for some positive integer $k$.
The low-degree method and its associated lower bounds are widely used to guide algorithm design and to provide evidence of computational hardness in average-case inference, high-dimensional statistics, random optimization, and related problems. This led to the low-degree conjecture, which predicts that when the low-degree advantage between a planted distribution and a uniform null distribution remains bounded, no efficient distinguisher can succeed after independent noise, provided that the planted distribution has permutation symmetry. Several works have produced counterexamples to variants of this conjecture or to versions for algorithms with higher time complexity, but the conjecture remained open in its standard binary, polynomial-time formulation. We disprove the polynomial-time low-degree conjecture by giving a family of examples. For every fixed integer $r\geq3$, we construct a permutation-invariant distribution $\mathbb{P}_n$ on simple graphs, with $\mathbb{Q}_n=G(n,1/2)$, such that every marginal of $\mathbb{P}_n$ on at most $D_n=Θ((\log n)^{r-1})$ edges is uniform. Therefore, the low-degree advantage is zero through degree $D_n$. Nevertheless, after every edge is independently resampled at a fixed positive rate, a deterministic rank test strongly distinguishes the resulting distribution from $\mathbb{Q}_n$ in linear time in the graph input size. The construction chooses a subspace of a Reed--Muller code whose nonzero polynomials have small absolute bias, selects points whose evaluation vectors have no short linear dependencies, and evaluates a random alternating bilinear form on pairs of these vectors. Our result shows that low-degree indistinguishability, a uniform null distribution, permutation invariance, and independent resampling do not by themselves imply polynomial-time hardness, and suggests that a valid general conjecture must impose an additional condition.
A depth lower bound for ReLU networks asserts that every shallower network errs on some input; we ask when a weak theory can prove such a bound so that an error can be computed. For the exact computation of $\max(x_1,\ldots,x_n)$, recent work shows two hidden layers suffice with unrestricted rational weights. By contrast, Haase, Hertrich and Loho (ICLR 2023) proved that integer weights require depth logarithmic in $n$, and Averkov, Hojny and Merkert (ICLR 2025) extended the bound to nonbias weights whose denominators are powers of a fixed integer not divisible by a prime $q$. With integer weights the bound is sharp: the classical balanced construction has depth $\lceil\log_2 n\rceil$. For each fixed $q$ and depth cap $K$, we prove this known bound in $\mathbf{VTC}^0$, the bounded-arithmetic theory of uniform constant-depth threshold computation, together with a refuter that is itself such a computation and whose correctness the theory proves. Given a network with at most $K$ hidden layers, nonbias weights whose reduced denominators avoid $q$, arbitrary rational biases and $n>q^K$ inputs, the refuter returns a rational input of polynomial length on which the network errs. The proof makes the obstruction finite: an iterated intersection product of piecewise-linear functions is divisible by $q$ for the network but not for the maximum, so one of finitely many test points exposes the error. By witnessing, provable refutation corresponds to a provably correct refuter, and provable existence of a certified network to a provably correct constructor. On the construction side, we introduce certificates that label each maximum gadget by the inputs it aggregates; with them the theory verifies the balanced construction, which meets the refuter at the sharp threshold $n\le2^K$, and a rational family of polynomial description length and depth $O(\log n/\log\log n)$ built from known depth-two blocks.
Friedl and Löh (2021, Confl. Math.) prove that testing whether or not there is an epimorphism from a finitely presented group to the direct product of an abelian and a finite group, or to a virtually cyclic group, is decidable. Here we prove that these problems are $\mathsf{NP}$-complete. In addition, we show that testing epimorphism is $\mathsf{NP}$-complete when the target is is the semi-direct product of a finitely generated free-abelian group and a finite group which factors through $\{\pm\text{Id}\}$, thus extending the class of virtually abelian target groups for which decidability of epimorphism is known.
In fault-tolerant Clifford+T circuits, a Boolean oracle costs T gates and extra qubits, and on four or more qubits every exact Toffoli gate needs an extra qubit. Gosset, Kothari and Zhang showed that sampling a fresh random circuit for each call makes the T-count of the Toffoli gate independent of its size, but a direct implementation of their sampler uses about one extra qubit per bit of precision. We show that computing the sampled parities in place, on the gate's own controls, removes every extra qubit once the gate is large enough: each random circuit then uses about 8 T gates per bit of precision and borrows only the gate's own idle inputs. This is one T gate more per circuit than the best measurement-free variant that keeps one clean ancilla. The same holds for exclusive-ors of indicators of affine subspaces, such as equality tests and checks of linear equations. For randomized circuits that do not measure, this is within a factor of eight of optimal up to an additive constant, even with extra qubits, and when each random circuit computes a Boolean function exactly, closing the factor asymptotically is equivalent to determining the leading constant in the T-count of exact multi-controlled Toffoli gates. Finally, the oracle of the indicator of a random set of fixed density needs exponentially many T gates at constant error, while that of an affine subspace of the same density needs a constant number, and extra qubits reduce the exponential cost only quadratically.
We give complexity-theoretic separations between $\#\mathrm{BIS}$ and $\#\mathrm{IS}$. Given a graph $G$, and a fugacity $λ>0$, the hard-core model is given by $Z_G(λ)=\sum_{I\in\mathcal I(G)}λ^{|I|}$, where $\mathcal I(G)$ denotes the independent sets of $G$. For $F\ge1$, an $F$-factor approximation to $Z_G(λ)$ is an estimate $\widehat Z$ satisfying $F^{-1}Z_G(λ)\le\widehat Z\le FZ_G(λ)$. Let $\mathcal G_Δ$ denote the class of $Δ$-regular graphs and $\mathcal G_{Δ,Δ}$ the class of $Δ$-regular bipartite graphs. We give two separation results. (1) There exist an integer $Δ\ge3$, a sufficiently large fugacity $λ$, and a constant $c>0$ so that the following hold: (a) For every $δ\in(0,1)$, deterministic $(1+δ)$-factor approximation on $\mathcal G_{Δ,Δ}$ is possible in time $\exp(cn)\operatorname{poly}(n,1/δ)$. (b) Under the Gap-Exponential Time Hypothesis, there exists no deterministic algorithm running in time $\exp(2cn)\operatorname{poly}(n)$ on $\mathcal G_Δ$ that gives an $\frac{11}{10}$-factor approximation. (2) For every fixed $Δ\ge3$ and $\varepsilon\in(0,1)$, for all sufficiently large fugacities $λ$, the following hold: (a) There exists an algorithm which outputs a deterministic $(1+\varepsilon)^n$-factor approximation to $Z_G(λ)$ on $\mathcal G_{Δ,Δ}$ in polynomial time. (b) Under the assumption $\mathrm P\ne\mathrm{NP}$, there exists no deterministic polynomial-time algorithm on $\mathcal G_Δ$ that gives a $(1+\varepsilon)^n$-factor approximation. Both our algorithmic results follow from a new spanning-tree encoding of independent sets that enables an efficient representation of large independent sets.
We give a deterministic quasipolynomial-time construction of Reed-Solomon codes that are list-decodable beyond the Johnson radius. Specifically, for every $ε\in(0,1]$ and all sufficiently large $n$, we construct an $[n,k]$ Reed-Solomon code of rate $R=Ω(ε/(1+\log(1/ε)))$ over a field of size $(nk)^{quasi-poly(nk)}$ that is $(1-ε,O(1/ε))$ list-decodable. The construction runs in quasi-poly(nk) time. We also give a deterministic construction of $[n,k]$-MDS$(3)$ Reed-Solomon codes for all integers $1\leq k<n$, with the same asymptotic bounds on field size and construction time. In the constant-rate regime, our field-size bound substantially improves the bound of $O(n^{k^{2k}})$ achieved by Roth's construction (Roth, IEEE TIT 2022). Our constructions exploit a connection to polynomial identity testing for read-once determinants, together with the quasipolynomial isolating-weight construction of Gurjar and Thierauf for linear matroid intersection (Gurjar and Thierauf, STOC 2017 and Computational Complexity 2020).