Journal of the ACM (JACM)

Aharonov, Dorit; Irani, Sandy

doi: 10.1145/3769108pmid: N/A

Despite immense progress in quantum Hamiltonian complexity in the past decade, little is known about the computational complexity of quantum physics at the thermodynamic limit. In fact, even defining the problem properly is not straight forward. We study the complexity of estimating the ground energy of a fixed, translationally-invariant (TI) Hamiltonian in the thermodynamic limit, to within a given precision; this precision (given by n the number of bits of the approximation) is the sole input to the problem. Understanding the complexity of this problem captures how difficult it is for a physicist to measure or compute another digit in the approximation of a physical quantity in the thermodynamic limit. We show that this problem is contained in FEXPQMA-EXP and is hard for FEXPNEXP. This means that the problem is doubly exponentially hard in the size of the input.As an ingredient in our construction, we study the problem of computing the ground energy of translationally invariant finite 1D chains. A single Hamiltonian term, which is a fixed parameter of the problem, is applied to every pair of particles in a finite chain. In the finite case, the length of the chain is the sole input to the problem and the task is to compute an approximation of the ground energy. No thresholds are provided as in the standard formulation of the local Hamiltonian problem. We show that this problem is contained in FPQMA-EXP and is hard for FPNEXP. Our techniques employ a circular clock structure in which the ground energy is calibrated by the length of the cycle. This requires more precise expressions for the ground energies of the resulting matrices than were required for previous QMA-completeness constructions and even exact analytical bounds for the infinite case which we derive using techniques from spectral graph theory. To our knowledge, this is the first use of the circuit-to-Hamiltonian construction that shows hardness for a function class.

Koutavas, Vasileios; Lin, Yu-Yang; Tzevelekos, Nikos

doi: 10.1145/3765737pmid: N/A

We present the first fully abstract normal form bisimulation for call-by-value PCF (PCFv). Our model is based on a labelled transition system (LTS) that combines elements from applicative bisimulation, environmental bisimulation and game semantics. In order to obtain completeness while avoiding the use of semantic quotienting, the LTS constructs traces corresponding to interactions with possible functional contexts. The model gives rise to a sound and complete technique for checking of PCFv program equivalence, which we implement in a bounded bisimulation checking tool. We test our tool on known equivalences from the literature and new examples.

Peng, Richard; Vempala, Santosh

doi: 10.1145/3767721pmid: N/A

Can linear systems be solved faster than matrix multiplication? Although there has been much progress on systems with additional structures, in the general setting, the complexity of solving an n × n linear system Ax=b is at least Ω (nω) bit operations, where ω < 2.372 is the matrix multiplication exponent. Improving on this bound of n< has been an open problem even for sparse linear systems with poly(n) condition number.In this article, we present an algorithm that solves linear systems in sparse matrices asymptotically faster than matrix multiplication or any ω > 2. This speedup holds for any input matrix A with o(nω-1/log (κ (A))) nonzeros, where κ (A) is the condition number of A. For poly(n)-conditioned matrices with Õ(n) nonzeros, and the current value of ω, the bit complexity of our algorithm to solve to within any 1/poly(n) error is O(n2.331).Our algorithm can be viewed as an efficient, randomized implementation of the block Krylov method via recursive low displacement rank factorizations. It is inspired by the algorithm of [Eberly et al. ISSAC ‘06 ‘07] for inverting matrices over finite fields. In our analysis of numerical stability, we use matrix anti-concentration techniques to bound the smallest eigenvalue and the smallest gap in eigenvalues of semi-random matrices. Incorporating the matrix anti-concentration bounds by [Nie STOC‘22] gives an improved bit complexity of O(n2.271).

Platzer, André; Qian, Long

doi: 10.1145/3763228pmid: N/A

This article proves the completeness of an axiomatization for initial value problems (IVPs) with compact initial conditions and compact time horizons for bounded open safety, open liveness and existence properties. Completeness systematically reduces the proofs of these properties to a complete axiomatization for differential equation invariants. This result unifies symbolic logic and numerical analysis by a computable procedure that generates symbolic proofs with differential invariants for rigorous error bounds of numerical solutions to polynomial initial value problems. The procedure is modular and works for all polynomial IVPs with rational coefficients and initial conditions and symbolic parameters constrained to compact sets. Furthermore, this article discusses generalizations to IVPs with initial conditions/symbolic parameters that are not necessarily constrained to compact sets, achieved through the derivation of fully symbolic axioms/proof-rules based on the axiomatization.

Kisfaludi-Bak, Sándor; Nederlof, Jesper; Węgrzycki, Karol

doi: 10.1145/3766548pmid: N/A

We revisit the classic task of finding the shortest tour of n points in d-dimensional Euclidean space, for any fixed constant d ⩾ 2. We determine the optimal dependence on ɛ in the running time of an algorithm that computes a (1 + ɛ)-approximate tour, under a plausible assumption. Specifically, we give an algorithm that runs in \(2^{\mathcal {O}(1/\varepsilon ^{d-1})} n\log n\) time. This improves the previously smallest dependence on ɛ in the running time \((1/\varepsilon)^{\mathcal {O}(1/\varepsilon ^{d-1})}n \log n\) of the algorithm by Rao and Smith (STOC 1998). We also show that a \(2^{o(1/\varepsilon ^{d-1})}\mathrm{poly}(n)\) algorithm would violate the Gap-Exponential Time Hypothesis (Gap-ETH).Our new algorithm builds upon the celebrated quadtree-based methods initially proposed by Arora (J. ACM 1998), but it adds a new idea that we call sparsity-sensitive patching. On a high level this lets the granularity with which we simplify the tour depend on how sparse it is locally. We demonstrate that our technique extends to other problems, by showing that for Steiner Tree and Rectilinear Steiner Tree it yields the same running time. We complement our results with a matching Gap-ETH lower bound for Rectilinear Steiner Tree.