Papers

Compressed quantum encodings aim to overcome hardware limitations towards tackling challenging problems at scale, with many classical variables mapped onto noncommuting observables of fewer qubits. Classically, relaxations such as the semidefinite program formulation of MaxCut trade solution quality for computational efficiency. By contrast, quantum relaxations based on compression can amplify the worst-case complexity of the problem being solved. We study quantum random access optimization (QRAO), a special case of the Pauli correlation encoding (PCE) framework that assigns up to three binary variables to the Pauli $X$, $Y$, and $Z$ observables of each qubit, with the packing choices determining the compressed Hamiltonian to be optimized. We identify explicit QRAO optimal energy promise problems complete for NP, StoqMA, and QMA, with inverse-polynomial promise gaps for the latter two. Our problem reductions preserve inverse-polynomial promise gaps without requiring gadgets or ancillas. For any prescribed packing, we show that weighted MaxCut instances compress, up to a known shift and rescaling, to arbitrary nonnegative-weight pairwise Pauli couplings allowed by the packing. For QRAO, using one aligned axis gives an NP-complete energy problem. Using two or three positive aligned Pauli axes generally gives QMA-complete problems, with bipartite restrictions in BQP $\cap$ StoqMA. We show that this computational hardness survives compilation and is practically relevant. Notably, this result applies directly to the current QRAO compiler implementation in Qiskit Optimization 0.7.0, confirming our hardness results are not artifacts of artificial or contrived packing rules. Altogether our results identify worst-case complexity barriers arising from quantum compression, while making no broad claims about typical cases or the performance and trainability of algorithm pipelines that use it.

We show that 2-fold Forrelation with inverse-polylogarithmic promise gap can be solved, with bounded error, by polynomial-size QAC$^0$ circuits. Unlike the standard oracle-based Forrelation algorithm, our circuits receive the input explicitly, in the same form as the AC$^0$ circuits against which Forrelation is known to be hard. At constant gap, this yields a natural promise-problem separation between QAC$^0$ and AC$^0$.

In this paper, we characterize resilience for a labeled real-time automaton (LRTA). An LRTA is resilient if whenever a faulty event occurs, after sufficiently many events occur, the LRTA returns to normalcy and the occurrence of the faulty event is not leaked. The notion of resilience reflects the ability of an LRTA recovering from a faulty behavior, and hence can model an intelligent agent. We formulate one definition of resilience for an LRTA and give verification algorithms for the definition based on two basic tools — concurrent composition and observer.

We prove a complexity dichotomy for Boolean Holant problems defined by arbitrary finite sets of algebraic complex-valued signatures. The tractable cases are characterized by an explicit, decidable criterion.

Holographic algorithms solve planar counting problems by encoding each value of a finite domain into several Boolean matchgate wires. Base collapse asks whether every exact representation can be compressed to a number of wires per edge bounded only by the domain size. Chen (STOC 2016) and Xia (STOC 2016) established broad positive collapse theorems under full-rank and related structural hypotheses. Together, these works left open whether a domain-only collapse bound survives when several locally deficient signatures must share one encoding. We resolve this open problem negatively, already on a three-state (qutrit) domain. For every $a\ge4$, we construct a five-label integer-valued qutrit language of maximum arity $a$ with an exact rational matchgate presentation, yet every exactly equivalent label-preserving presentation requires common width $\Omega(2^{a/2}/a)$, even if its domain, base, tensors, and complex weights may all change. The construction avoids the usual local degeneracies, so the obstruction is genuinely simultaneous. Algebraic geometry then drives an exhaustive trichotomy explaining the boundary of collapse: the gadget closure either separates into rays, is Gaussian-mobile and collapses to width at most three, or is confined to a rigid plane-plus-ray flag containing our unbounded family. Pure-spinor geometry forces compression in the mobile case, while dimension bounds for matchgate varieties and Zariski avoidance place exact integer data outside every low-width algebraic image. Thus one geometric framework explains both why collapse occurs and why it can fail without bound.

We present two elementary constructions of multilayered PCPs that improve upon prior constructions in two ways. Specifically, we give one construction of quasi-linear size, and another one with $2$-to-$2$ constraints. Using these constructions we obtain the following results for the hypergraph vertex cover problem: $\bullet$ For $k=3$, for all $\varepsilon>0$, approximating the minimum vertex cover of a given $3$-uniform hypergraph within factor $1+\sqrt{2}-\varepsilon$ is NP-hard. Previously, the best known result due to [Dinur, Guruswami, Khot, Regev, SICOMP 2005] achieved a factor of $2-\varepsilon$. $\bullet$ For $k\geq 4$, for all $\varepsilon>0$, approximating the minimum vertex cover of a given $k$-uniform hypergraph within factor $k-\varepsilon$ is NP-hard, which is tight. Previous works established this result assuming the Unique-Games Conjecture [Khot, Regev, JCSS 2008], and a weaker factor of $k-1-\varepsilon$ for standard NP-hardness [Dinur, Guruswami, Khot, Regev, SICOMP 2005]. $\bullet$ Assuming the Exponential Time Hypothesis, for all $k\geq 3$ and $\varepsilon>0$ there is $C>0$ such that no $2^{n/\log^C n}$-time algorithm approximates the minimum vertex cover in a $k$-uniform, $n$-vertex hypergraph within factor $k-1-\varepsilon$. The proofs were obtained using ChatGPT 5.6 Pro and subsequently rewritten by the communicators.

We prove that $3\times3$ matrix multiplication over $\mathbb{F}_2$ has bilinear complexity at least $21$, improving the previous lower bound of $20$. Lower bounds for restrictions of the first input constrain how many first factors of a decomposition can lie in each subspace. For a hypothetical $20$-term decomposition, the strengthened restriction bounds force all first factors of matrix rank at least two into a single coset of a three-dimensional rank-one subspace. An exhaustive computation finds no admissible first-factor profile with $20$ terms. We also prove that $2\times3$ by $3\times3$ matrix multiplication over $\mathbb{F}_3$ has rank exactly $15$: three profiles with $14$ terms survive the corresponding computation up to symmetry, and short restriction arguments exclude them. We combine Wang's automated framework for tensor-rank lower bounds with D'Ambrosio's capacity-and-profile strategy. Our computational contributions are rank-one-span searches that strengthen the subspace lower-bound table and a direct, symmetry-reduced profile enumerator that enforces all subspace capacities simultaneously.

Convex optimization on polytopes arises in many areas of science. When the polytope is given implicitly or has exponentially many vertices and facets, standard methods may not apply or be ineffective. This is the case for moment polytopes, such as the entanglement polytopes, which play a foundational role in quantum information and algebraic complexity. They give rise to important entanglement measures and tensor parameters such as the quantum functionals, yet general effective methods for computing these quantities have been elusive. In this paper we address this challenge. We develop a first-order framework called Hadamard mirror descent to optimize suitable convex functions over moment polytopes and, more generally, the gradient sets of geodesically convex functions. It operates locally and does not rely on any explicit description of the polytope. Our framework extends mirror descent, an effective and widely used framework for convex optimization, from the Euclidean setting to Hadamard manifolds, and is motivated by a recent work by Hirai, which we interpret as a Hadamard version of mirror flow. Applying the framework to entanglement polytopes yields the first efficient first-order algorithms to compute the quantum functionals, the symmetric quantum functional, and the G-stable ranks, as well as a new direct algorithm for the non-commutative rank.

In \(d\)-Hitting Set, the input consists of a finite universe \(U\), a family \(\mathcal S\) of subsets of \(U\) with size at most \(d\), and an integer \(k\). The task is to decide whether at most \(k\) elements of \(U\) can intersect every set in \(\mathcal S\). For every fixed \(d\geq3\), we give a one-sided randomized kernel with \(O(k\log^3k)\) elements and a deterministic kernel with \(O(k^2\log k)\) elements for \(d\)-Hitting Set. In the one-sided randomized kernel, every NO-instance is always mapped to a NO-instance, and a YES-instance is mapped to a YES-instance with constant probability. The previously known kernels for \(d\)-Hitting Set contain \(O(k^{d-1})\) elements and \(O(k^d)\) sets. It has been asked in the literature whether \(d\)-Hitting Set allows kernels with \(O(k^{d-1-\varepsilon})\) elements for some constant \(\varepsilon>0\). In this paper, we answer this question affirmatively by giving near-linear element-kernels through a re-encoding of the instance. On the other hand, our kernel may still contain \(k^{O(d)}\) sets and the parameter $k$ may grow polynomially.

A degree-$d$ polynomial source is the output of a polynomial map of degree at most $d$ over $\mathbb{F}_2$ on arbitrarily many uniform random bits. Khodabandeh and Shinkar (FOCS '26) proved that $\mathrm{Ber}(1/3)^{\otimes N}$ has statistical distance $1-o(1)$ from every constant-degree polynomial source and conjectured exponentially small overlap. Independently of Khodabandeh and Shinkar, Byramji, Kane, Morris, and Ostuni (RANDOM '26) asked for an explicit target distribution at distance $1-\exp(-N^{\Omega_d(1)})$. We resolve both questions. For every fixed $d\geq1$, every degree-$d$ polynomial source has overlap at most $\exp(-c_dN)$ with $\mathrm{Ber}(1/3)^{\otimes N}$, where $c_d>0$ is independent of the seed length. For quadratics, $c_2=2^{-26}$ suffices. We amplify Khodabandeh and Shinkar's uniform separation of acceptance probabilities from non-dyadic parameters (numbers not of the form $a/2^b$ for integers $a$ and $b\geq0$). The result extends to other non-dyadic Bernoulli parameters and to coordinates that are Boolean functions of boundedly many bounded-degree polynomials. We also give a uniform deterministic hierarchy between adjacent degrees. Appending the outputs of disjoint AND gates on $d+1$ inputs to uniform seed bits yields flat degree-$(d+1)$ target distributions of entropy $k$ with overlap $\exp(-\Omega_d(\min\{k,N-k\}))$ against every degree-$d$ source, for $\min\{k,N-k\}\geq2(d+1)$. This entropy dependence is optimal up to constants in the exponent among flat target distributions for fixed $d$. The construction has locality $d+1$ and uses $O(N)$ field operations to sample. At $k=\lfloor N/2\rfloor$, it handles $d\leq(1-\varepsilon)\log_2N/3$ with overlap $\exp(-N^{\varepsilon-o(1)})$ for fixed $0<\varepsilon<1$. The proof combines monotonicity of Gowers uniformity norms, pairwise independence of points in a random affine cube, and relative entropy.

For clique complexes $X_1\subseteq X_2$, normalized persistence in degree $d$ is $\operatorname{rank}[H_d(X_1)\to H_d(X_2)]/\dim H_d(X_1)$. Estimating it requires exact endpoint homology and the inclusion-induced map, even with inverse-polynomial endpoint Laplacian gaps. We prove that additive-error $1/24$ estimation is hard for $\mathsf{BQP}_{1}^{G_2}$, the perfect-completeness class over the exact gate set $G_2=\{X,\mathsf{CX},\mathsf{CCX},H\otimes H\}$, and hence for $\mathsf{BQP}_{1}$ over every finite gate set with entries in a cyclotomic field $\mathbb{Q}(\zeta_{2^k})$, even for unweighted clique complexes. The main tool is a finite-certificate kernel-transfer theorem. For a fixed palette of weighted clique gadgets satisfying finitely many exactly checkable local conditions, every unit chain $x$ of the full geometric complex satisfies $\operatorname{dist}(x,K)^2\le C(t\lambda^2+\langle x,\Delta x\rangle/(g\lambda^{26}))$, where $K$ is the embedded kernel of the simulated projector Hamiltonian, $g$ its gap, $t$ the number of gadgets, and $\lambda$ the private vertex weight. Since $\lambda$ is chosen independently of $g$, the geometric Laplacian has exactly $\dim K$ zero modes and a gap linear in $g$ above them. Exact fillings identify the endpoint homology with a quotient $V/W_A$ of the register cycle space, and nested term sets induce the natural quotient epimorphisms, so the persistent rank equals the later kernel dimension without any choice of compatible harmonic representatives. A fixed eight-dimensional label register turns a $\mathsf{BQP}_{1}^{G_2}$ verifier into instances with $\beta_d(X_1)=8$ and normalized persistence $3/4$ or $1/8$, and an established common-copy blow-up transfers everything to unweighted graphs.

We study the approximability of rank aggregation under the Ulam metric. In the Ulam median problem, the goal is to find a ranking (permutation) minimizing the sum of its Ulam distances to the input rankings, while in the Ulam center problem, the objective is to minimize the maximum such distance. We prove that, for every $0<\varepsilon< 1/34$, it is $\mathrm{NP}$-hard to approximate either Ulam median or Ulam center within a factor of $35/34-\varepsilon$, even when the input consists of only four rankings. We further show that unless P = NP, neither problem admits a polynomial-time additive approximation scheme. Prior to our work, only the exact versions of both problems were known to be $\mathrm{NP}$-hard, and that too only when the number of input rankings is unbounded [Fischer et al., ESA'25 and Bachmaier et al., J. of Discrete Algorithms'15]. Furthermore, our inapproximability results are optimal in terms of the number of input rankings since for three inputs it is already known to be polynomial-time solvable [Chakraborty, Das, Krauthgamer, SODA'21]. En route, we introduce a new general framework for reducing Boolean constraint satisfaction problems (CSP) to the Ulam median with only four inputs. As a specific instantiation of the reduction framework, we obtain our hardness-of-approximation results. The corresponding hardness for the Ulam center follows from a reduction from the Ulam median.

We formulate the generalized hidden shift problem over arbitrary finite groups and obtain upper and lower bounds for the sample complexity in terms of group theoretic data.

Hao, Huang, and Liu (STOC'26) recently showed that optimal quantum query complexity may require large workspace even for short-output problems, and asked whether a quantum query advantage over classical computation can itself require space. We resolve this question by exhibiting an explicit total Boolean function with quantum query complexity $Q=\widetilde{\Theta}(M)$, and randomized query complexity $R=\Theta(M^{21/20})$, whereas, for every fixed $0<\eta<1/100$ and $S\le O(M^{1/100-\eta})$, its $S$-space quantum query complexity satisfies $Q_S=\omega(M^{21/20})$. Consequently, $$ Q<R<Q_S, $$ so the unrestricted quantum query advantage disappears under sufficiently small workspace. The separation is obtained through a one-bit filtered-parity construction and a space-sensitive quantum lower bound based on compressed-oracle capacity and a new parity-to-capacity inequality, which may be of independent interest.

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).

Algebraic combinatorics often seeks bijections that explain identities between distributions object by object. Encoding combinatorial objects as words lets automata theory study such a bijection as a word-to-word computation and measure its memory, input access, and control of output order. This refines existence questions by asking which computational mechanisms a bijection requires. We develop this viewpoint for Dyck paths. Our motivating example is the $q,t$-Catalan polynomial. Let $D_n$ be the set of Dyck paths of semilength $n$, let $D=\bigcup_{n\ge 0}D_n$, and let $area, dinv, bounce \colon D\to\mathbb{N}$ be the standard statistics. Then, \[ C_n(q,t)=\sum_{P\in D_n}q^{area(P)}t^{bounce(P)} =\sum_{P\in D_n}q^{dinv(P)}t^{area(P)}. \] Haglund's zeta map $\zeta\colon D\to D$ gives a bijective proof: it preserves semilength and sends $(dinv,area)$ to $(area,bounce)$. By contrast, the full symmetry $C_n(q,t)=C_n(t,q)$ still lacks a direct explanation: no explicit, uniform, semilength-preserving bijection is known that swaps area and dinv on every Dyck path. Polyregular maps from automata theory provide a natural computational starting point, but we prove that neither $\zeta$ nor the classical height-sweep bijection witnessing Narayana symmetry is polyregular. The missing mechanism is global ordering by numerical levels whose range grows with the input. We call this a rank sort and introduce weighted-rank polyregular maps (WRP), extending polyregular maps by one such sort and containing both bijections. Nevertheless, WRP is a proper subclass of deterministic logspace. We prove that $\zeta^{-1}$ lies outside WRP and that no WRP map can realise a semilength-preserving area-dinv swap. Thus the rank-sorting strategy behind $\zeta$ cannot be extended within WRP to exchange the two statistics.

The widespread adoption of artificial intelligence (AI) within real-world applications has raised a lot of concerns regarding their trustworthiness, especially in critical applications. The field of eXplainable AI (XAI) has emerged with the objective of providing explanations to the users about the decisions made by AI systems. Several explanations for boolean classifiers have been introduced in the literature, including abductive and contrastive explanations, each giving a different insight on the decision of the classifier. However, computing an explanation for a decision of a boolean classifier is a hard problem in general. One way to deal with this complexity is to rely on a compiled representation of the classifier for which each explanation can be computed efficiently. Unfortunately, we prove in this paper that several classes of abductive explanations, remain hard to compute even for Ordered Binary Decision Diagrams, one of the most tractable subsets of the knowledge compilation map. Included in such classes are shorter abductive explanations or abductive explanations that include the explainee's preferences. To recover the benefits of working with compiled representations, we show that a proper representation of the dual-rail encoding of the classifier can be used to compute efficiently these classes of explanations.

We study exact single-source shortest paths when the output is only the materialized labeled distance vector ($\mathrm{DIST}$), rather than a distance order. In the full deterministic comparison-addition model, the minimum worst-case number of additions on every fixed directed topology is exactly the maximum number $\rho_{\mathrm{fwd}}$ of forward nonsource endpoint classes over rooted vertex orders; the lower bound permits adaptive control, literals, and arbitrary mixed sums. This arithmetic law aligns with the comparison optimum on DAGs, where the full resource region is an exact rectangle. Cycles destroy that alignment: a two-spoke shared-hub graph has coordinatewise optima $(4,2)$ but requires five comparisons at the two-addition budget. Its $k$-spoke extension forces $k\log_2 k+O(k)$ comparisons at the addition optimum and has an entropy-tight deterministic tradeoff $C_{k+r}^*(H_k)=\Theta(k+\Lambda_{k,r})$, where $\Lambda_{k,r}=\log_2(k!/[r!(r+1)^{k-r}])$, with leading constant one when $\Lambda_{k,r}/k\to\infty$. Because the two coordinatewise minima need not belong to one program, these conflicts lead to the same-program benchmark $\operatorname{OPT}_{\mathrm{DIST}}=\inf_A\sup_w(C_A(w)+P_A(w))$. An exact transcript-cone game yields one uniform interpreter whose charged addition-comparison cost equals $\operatorname{OPT}_{\mathrm{DIST}}$ on every topology; its optimal actions are synthesizable in polynomial space but may require exponential time. Finally, an active-core reduction and the current deterministic directed-SSSP bound give an efficient uniform $O\!\bigl(\operatorname{OPT}_{\mathrm{DIST}}\sqrt{\log(2+\operatorname{OPT}_{\mathrm{DIST}})\log\log(4+\operatorname{OPT}_{\mathrm{DIST}})}\bigr)$ charged-operation bound. Thus optimal numerical policies exist uniformly, while efficient constant-competitive navigation remains open.

Exact CNOT synthesis seeks a minimum-size circuit implementing a given invertible binary linear transformation. We study its vanilla formulation: synthesis from the identity on fixed labeled qubits, without ancillas and with all-to-all connectivity. We prove that this problem is NP-hard and that its bounded decision version is NP-complete. Hardness persists even for unitriangular targets that are low-rank perturbations of the identity and under near-linear gate budgets. Our proof gives a polynomial-time reduction from the NP-complete Hamiltonian path problem on grid graphs. A unary hypercube embedding translates graph vertices into circuit parities and edge traversals into CNOT updates. The central challenge is that Hamiltonian paths require intermediate vertex visits, whereas exact synthesis constrains only the final transformation. We bridge this gap with a replicated recorder qubits construction that encodes the required parities in the final outputs, together with a tight gate budget that forces every sufficiently short implementation to trace a Hamiltonian path. Crucially, this path structure is enforced by the construction rather than imposed as a restriction on the circuit. The result directly implies NP-hardness for shortest word problem and Cayley-graph distance computation over $\mathrm{GL}(n,2)$ with elementary transvections as generators, as well as fixed-width sequential XOR program minimization. For CNOT-minimal exact phase polynomial synthesis, it yields NP-hardness with a zero phase polynomial; extending the reduction to grid-graph Hamiltonian cycle problem establishes hardness even when the final linear transformation is the identity. These complementary results show that the linear and phase components each independently suffice for NP-hardness of exact phase polynomial synthesis.

What can a single layer of self-attention compute? We study head complexity: the minimum number of attention heads required to compute a function in a one-layer attention-only model. We establish an exact hierarchy under this measure: $k$ heads compute $k$-bit parity but cannot compute $(k+1)$-bit parity. The lower bound is unconditional in the two resources a transformer might otherwise exploit; it holds at unbounded embedding dimension and unbounded numerical precision. The proof rests on an alternating-sum obstruction: after clearing the softmax denominators, every monomial in the resulting decision polynomial omits at least one of the $k+1$ input bits, forcing its correlation with parity to vanish. The same obstruction yields lower bounds for related tasks, including the well-studied multi-hop induction-head task. We also establish compactness bounds for embedding dimension and numerical precision. Specifically, a compactness theorem shows that any function computable at all can be computed with embedding dimension and precision bounded by the discrete data of the task, namely, head count, alphabet size, and length. Thus, potentially unbounded dimension or precision provably cannot substitute for heads. Finally, we derive nearly matching universal bounds for general binary functions: $2^n$ heads suffice to compute every $n$-bit binary function, with one head per monomial in its multilinear expansion, while a counting argument shows almost all such functions require $\Omega(2^n/n^2)$ heads. This lower bound matches the upper bound to within a $\operatorname{poly}(n)$ factor, even when dimension and precision are unbounded. Together, these results characterize head requirements for Boolean computation in this model.

The Knaster-Tarski fixed-point theorem states that every monotone function over a complete lattice has a fixed point. Beyond its fundamental role in order theory, the theorem and its algorithmic variants have found broad applications in areas such as economics, game theory, and programming languages. While the query complexity of finding a Tarski fixed point has been extensively studied in classical models, comparatively little is known in the quantum setting. We prove an $\Omega(k\log n)$ quantum query lower bound for finding a fixed point of a monotone function on $[n]^k$, using the nonnegative spectral adversary method. In the two extremal regimes $n = 2$ and $k = 1$, our quantum lower bound matches the previous classical lower bounds $\Omega(k)$ and $\Omega(\log n)$, respectively. For $n, k\geq 2$, our bound improves the best previous classical lower bound when $n < k$ and is within a factor of $\log n / \log k$ compared to the known classical lower bound when $n \geq k$. To construct the adversary matrix, we develop the Tree–Filtration Adversary Method. Besides yielding our lower bound, the method offers a more transparent combinatorial interpretation of the nonnegative spectral adversary method. When the hard instances of a problem admit a tree-like organization and suggest an intuition analogous to classical decision-tree lower bounds, our method provide a promising approach to establishing quantum complexity lower bounds.

We study non-adaptive selection of a feasible set $S$ that maximizes the expected sum of the $k$ largest realized values among independent nonnegative discrete random variables. The same objective arises in team hiring and as VCG welfare in an $\ell$-unit auction. The main setting is a fixed-dimensional nonnegative packing family, whose natural LP has $d=O(1)$ packing inequalities with binary coefficients. We give a PTAS for every $k\ge 1$ on every such family, including binary one- and two-dimensional knapsack, by approximating the occupancy functional $p\mapsto\mathbb{E}[\min(k,N(p))]$ and realizing the resulting signatures in the packing LP. As a generic guarantee the scheme is essentially optimal: there is no FPTAS that works for every such $\mathcal{F}$ unless $P=NP$, and no EPTAS unless $W[1]=FPT$. An incomparable sufficient condition is a query-weight exact-sum oracle (DAG paths, matchings), which likewise yields a PTAS for every $k$. The same signatures give a PTAS for $\min_{S\in\mathcal{F}}\mathbb{E}[\mathrm{Top}_k(S)]$ on every fixed-$d$ covering family; two-dimensional covering knapsack rules out a generic FPTAS on that class.

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 $\Theta(\sqrt N)$. 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 $\Theta(\sqrt N)=\Theta(L^{1/4})$. 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 $O(T^2/N)$.

The standard semidefinite programming (SDP) relaxation of Max-Cut is exact when its optimum equals the maximum cut value. Delorme and Poljak resolved NP-completeness of recognizing exactness for weighted graphs and left the unweighted case open. We show that recognition is NP-complete even for connected simple unweighted graphs, and hence strongly NP-complete for nonnegative integer edge weights. The reduction provides an explicit SDP optimum and makes the additive integrality gap equal to the minimum number of unsatisfied clauses in the source formula. Recognition remains NP-complete even when an exact rational optimal primal–dual pair is supplied. We also establish strong NP-hardness of recognizing exactness of the Frieze–Jerrum Max-$k$-Cut relaxation for every fixed $k\ge3$, even for connected graphs with nonnegative integer edge weights. An independent bounded-weight construction gives a second proof for Max-Cut. Finally, reductions preserving the additive gap up to explicit factors establish strong NP-completeness of exactness recognition for a basic Max-DiCut SDP and NP-hardness for a Max-Bisection SDP.

We study the computational complexity of solving promise systems of equations over finite algebras. Given two algebras $\mathbf{A}$ and $\mathbf{B}$ with a homomorphism from $\mathbf{A}$ to $\mathbf{B}$, the promise system of equations problem is to determine if an input system of equations has a solution in $\mathbf{A}$ or not even in $\mathbf{B}$. We generalize the results of Larrauri, Mottet, and Živný [ACM ToCL'26] to obtain a $\mathbf{P}-\mathbf{NP}$-hard dichotomy result for promise systems of equations over a class of algebras which contains all monoids, and a dichotomy result for promise systems of equations over algebras in a congruence modular variety. We then consider the metaproblem for promise systems of equations over algebras in a congruence modular variety: given finite algebras $\mathbf{A}$ and $\mathbf{B}$ such that $\mathbf{A}$ is in a congruence modular variety, we show there is a quasi-polynomial time algorithm for determining whether or not the associated promise system of equations problem is in $\mathbf{P}$.

Pseudorandomness is increasingly recognized as a key property of ensembles in quantum information theory, statistical mechanics, and quantum many-body physics. Yet it appears in two conceptually different forms: statistical pseudorandomness, embodied by unitary designs, and computational pseudorandomness captured by pseudorandom unitaries (PRUs). The relationship between these two forms of pseudorandomness remains surprisingly poorly understood. Existing PRU constructions reveal this interplay where a statistically randomizing ingredient, a unitary design, is combined with classical cryptographic primitives to produce computational pseudorandomness. We show that statistical pseudorandomness is not necessary for computationally pseudorandom unitaries. We do this by replacing the unitary $2$-design layer in the existing constructions with ensembles that are not even state $1$-designs, yet are sufficiently distinct, a property we identify to be necessary for any PRU. This yields new non-adaptively secure PRU ensembles whose computational pseudorandomness is obtained without an underlying statistically pseudorandom quantum ensemble, such as a $2$-design. We characterize distinctness via an entangled analogue of anticoncentration and use it to show that distinctness already captures constraints on coherence and imaginarity of PRUs, while identifying broad classes of inputs for which the latter obstruction disappears, enabling real-valued PRUs even for certain (maximally) entangled states. As an application, we use lack of distinctness to constrain the conjectured pseudorandomness of the random phase-Hadamard ensemble to form a PRU.

We study the question of answering linear queries with differential privacy using few (expected) random bits. We provide a randomness-efficient analog of the $\| \cdot \|_K$-norm mechanism of Hardt and Talwar [HT10]. For the $\ell_\infty$-error, our algorithm can answer $d$ linear queries with $O(d / \varepsilon)$ error using $O(\log d)$ random bits, improving upon algorithms of Canonne et al. and Ghentiyala [CSV25, Ghe26]; this is optimal when $\varepsilon \le 1/d$. We also provide a computationally efficient version of our algorithm, albeit with an $O(\log d)$ multiplicative increase in the error.

We study online maximization of nonnegative, non-monotone DR-submodular functions over compact convex down-closed subsets of the $d$-dimensional unit cube. The best known constructive offline approximation factor is $0.401$ under the corresponding meta-solvability assumptions, whereas comparable adversarial online guarantees had remained at $1/e$. We show that this factor is also achievable online. In the post-decision full-information value-oracle model, our algorithm attains factor $0.401$ with sublinear approximate regret when oracle feedback is conditionally unbiased and bounded. The online algorithm does not run the offline construction on a changing objective. Instead, it replaces the offline objective-dependent box step by a weighted online learner that controls the required residual terms cumulatively. An exact asymmetric balance theorem preserves the offline coefficients despite adversarial variation. The direct implementation has $O(T^{3/4})$ regret and uses $O(dT^{1/4})$ oracle calls per round. More generally, for every $\delta\in[0,1/4]$, batching gives $O(T^\delta)$ calls per round and $O(T^{4/5-\delta/5})$ regret, including a one-call $O(T^{4/5})$ endpoint. Under a positive-anchor condition, randomized blocking retains factor $0.401$ with $O(T^{5/6})$ one-point bandit regret.

Classical lower bounds for $\mathrm{AC^0}$ circuits proceed bottom-up by simplifying or approximating gates beginning at the input layer. We introduce a complementary top-down model called the Chopping Game, played by adversaries Spoiler and Duplicator on the sets of $0$- and $1$-inputs of a Boolean function. In each round, Spoiler keeps at least a $1/m$-fraction of one side, and Duplicator arbitrarily restricts the other; Spoiler seeks to minimize (and Duplicator to maximize) the number of rounds until some coordinate separates the two remaining sets. Every depth-$d$, fan-in-$m$ circuit induces a $d$-round winning strategy for Spoiler, while Duplicator strategies that survive $d$ rounds formalize top-down lower-bound arguments. Through the Chopping Game and using the polynomial-approximation method, we first obtain the classical lower bound for depth-$d$ $\mathrm{AC^0}$ circuits in a top-down fashion. We then consider a $k$-local variant of the Chopping Game, which relaxes Spoiler's win condition by requiring a separating coordinate within each Hamming ball of radius $k$, rather than a single coordinate globally. We put forward a conjecture that the $d$-round $k$-local Chopping Game for $\mathrm{PARITY}$ requires $m = n^{\omega(1)}$ in the regime $d \ll k \ll n$. We prove such a lower bound $m \ge n^{\Omega(k^{1/d}/d)}$ when Spoiler is restricted to so-called affine strategies, a class of strategies that achieves the best known upper bounds. Finally, we formulate a version of the $k$-local Chopping Game on $n$-regular graphs of girth $>2k$, and we conjecture a graph-theoretic analogue of “$\mathrm{PARITY} \notin \mathrm{AC^0}$”.

We construct a constant-degree and constant-gap quantum expander on $n$ qubits where each unitary can be implemented by a depth-$1$ and 1D circuit of Pauli or CNOT gates. We provide two applications of this expander. First, we use it to construct a family of frustration-free 1D Hamiltonians whose ground states obey the entanglement-gap relation $S = \Theta(\Delta^{-1/2})$; this is believed to be optimal, but achieving it had been open. Second, we use it to provide a streaming protocol that tests for closeness to a class of 1D volume-law entangled states. Moreover, we extend our quantum expander to a constant-degree and constant-gap expander on the unitary group where each unitary is a single $T$ gate, a single $T^{\dagger}$ gate, or a depth-$1$ Clifford circuit. This implies that a random sequence of unitaries from the expander yields a gapped walk on a dense subgroup of the unitary group. This improves upon previous work by Bourgain and Gamburd which did not control the dependence of the gap on the dimension.