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1. Bosonic Quantum Computational Complexity

   https://doi.org/10.4230/lipics.itcs.2025.33  

   Chabaud, Ulysse; Joseph, Michael; Mehraban, Saeed; Motamedi, Arsalan (January 2025, Schloss Dagstuhl – Leibniz-Zentrum für Informatik)

   Meka, Raghu (Ed.)

   In recent years, quantum computing involving physical systems with continuous degrees of freedom, such as the bosonic quantum states of light, has attracted significant interest. However, a well-defined quantum complexity theory for these bosonic computations over infinite-dimensional Hilbert spaces is missing. In this work, we lay the foundations for such a research program. We introduce natural complexity classes and problems based on bosonic generalizations of BQP, the local Hamiltonian problem, and QMA. We uncover several relationships and subtle differences between standard Boolean classical and discrete-variable quantum complexity classes, and identify outstanding open problems. Our main contributions include the following: 1) Bosonic computations. We show that the power of Gaussian computations up to logspace reductions is equivalent to bounded-error quantum logspace (BQL, characterized by the problem of inverting well-conditioned matrices). More generally, we define classes of continuous-variable quantum polynomial time computations with a bounded probability of error (CVBQP) based on gates generated by polynomial bosonic Hamiltonians and particle-number measurements. Due to the infinite-dimensional Hilbert space, it is not a priori clear whether a decidable upper bound can be obtained for these classes. We identify complete problems for these classes, and we demonstrate a BQP lower bound and an EXPSPACE upper bound by proving bounds on the average energy throughout the computation. We further show that the problem of computing expectation values of polynomial bosonic observables at the output of bosonic quantum circuits using Gaussian and cubic phase gates is in PSPACE. 2) Bosonic ground energy problems. We prove that the problem of deciding whether the spectrum of a bosonic Hamiltonian is bounded from below is co-NP-hard. Furthermore, we show that the problem of finding the minimum energy of a bosonic Hamiltonian critically depends on the non-Gaussian stellar rank of the family of energy-constrained states one optimizes over: for zero stellar rank, i.e., optimizing over Gaussian states, it is NP-complete; for polynomially-bounded stellar rank, it is in QMA; for unbounded stellar rank, it is RE-hard, i.e., undecidable. 

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2. Fréchet Distance for Uncertain Curves

   https://doi.org/10.1145/3597640  

   Buchin, Kevin; Fan, Chenglin; Löffler, Maarten; Popov, Aleksandr; Raichel, Benjamin; Roeloffzen, Marcel (July 2023, ACM Transactions on Algorithms)

   In this article, we study a wide range of variants for computing the (discrete and continuous) Fréchet distance between uncertain curves. An uncertain curve is a sequence of uncertainty regions, where each region is a disk, a line segment, or a set of points. A realisation of a curve is a polyline connecting one point from each region. Given an uncertain curve and a second (certain or uncertain) curve, we seek to compute the lower and upper bound Fréchet distance, which are the minimum and maximum Fréchet distance for any realisations of the curves. We prove that both problems are NP-hard for the Fréchet distance in several uncertainty models, and that the upper bound problem remains hard for the discrete Fréchet distance. In contrast, the lower bound (discrete [ 5 ] and continuous) Fréchet distance can be computed in polynomial time in some models. Furthermore, we show that computing the expected (discrete and continuous) Fréchet distance is #P-hard in some models. On the positive side, we present an FPTAS in constant dimension for the lower bound problem when Δ/δ is polynomially bounded, where δ is the Fréchet distance and Δ bounds the diameter of the regions. We also show a near-linear-time 3-approximation for the decision problem on roughly δ-separated convex regions. Finally, we study the setting with Sakoe–Chiba time bands, where we restrict the alignment between the curves, and give polynomial-time algorithms for the upper bound and expected discrete and continuous Fréchet distance for uncertainty modelled as point sets. 

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3. On the partial convexification for low-rank spectral optimization: rank bounds and algorithms

   https://doi.org/10.1007/s10107-025-02316-y  

   Li, Yongchun; Xie, Weijun (December 2025, Mathematical Programming)

   Abstract A Low-rank Spectral Optimization Problem (LSOP) minimizes a linear objective function subject to multiple two-sided linear inequalities intersected with a low-rank and spectral constrained domain. Although solving LSOP is generally NP-hard, its partial convexification (i.e., replacing the domain with its convex hull), termed “LSOP-R, is often tractable and yields a high-quality solution. This motivates us to study the strength of LSOP-R. Specifically, we derive rank bounds for any extreme point of LSOP-R in different matrix spaces and prove their tightness. The proposed rank bounds recover two well-known results in the literature from a fresh angle and allow us to derive sufficient conditions under which the relaxation LSOP-R is equivalent to LSOP. To effectively solve LSOP-R, we develop a column generation algorithm with a vector-based convex pricing oracle and a rank-reduction algorithm, which ensures that the output solution always satisfies the theoretical rank bound. Finally, we numerically verify the strength of LSOP-R and the efficacy of the proposed algorithms. 

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4. Sparse representation of vectors in lattices and semigroups

   https://doi.org/10.1007/s10107-021-01657-8  

   Aliev, Iskander; Averkov, Gennadiy; De Loera, Jesús A.; Oertel, Timm (June 2021, Mathematical Programming)

   null (Ed.)

   Abstract We study the sparsity of the solutions to systems of linear Diophantine equations with and without non-negativity constraints. The sparsity of a solution vector is the number of its nonzero entries, which is referred to as the $$\ell _0$$ ℓ 0 -norm of the vector. Our main results are new improved bounds on the minimal $$\ell _0$$ ℓ 0 -norm of solutions to systems $$A\varvec{x}=\varvec{b}$$ A x = b , where $$A\in \mathbb {Z}^{m\times n}$$ A ∈ Z m × n , $${\varvec{b}}\in \mathbb {Z}^m$$ b ∈ Z m and $$\varvec{x}$$ x is either a general integer vector (lattice case) or a non-negative integer vector (semigroup case). In certain cases, we give polynomial time algorithms for computing solutions with $$\ell _0$$ ℓ 0 -norm satisfying the obtained bounds. We show that our bounds are tight. Our bounds can be seen as functions naturally generalizing the rank of a matrix over $$\mathbb {R}$$ R , to other subdomains such as $$\mathbb {Z}$$ Z . We show that these new rank-like functions are all NP-hard to compute in general, but polynomial-time computable for fixed number of variables. 

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5. The Replenishment Schedule to Minimize Peak Storage Problem: The Gap Between the Continuous and Discrete Versions of the Problem

   https://doi.org/10.1287/opre.2018.1839  

   Hochbaum, Dorit S.; Rao, Xu (September 2019, Operations Research)

   The replenishment storage problem (RSP) is to minimize the storage capacity requirement for a deterministic demand, multi-item inventory system, where each item has a given reorder size and cycle length. We consider the discrete RSP, where reorders can only take place at an integer time unit within the cycle. Discrete RSP was shown to be NP-hard for constant joint cycle length (the least common multiple of the length of all individual cycles). We show here that discrete RSP is weakly NP-hard for constant joint cycle length and prove that it is strongly NP-hard for nonconstant joint cycle length. For constant joint cycle-length discrete RSP, we further present a pseudopolynomial time algorithm that solves the problem optimally and the first known fully polynomial time approximation scheme (FPTAS) for the single-cycle RSP. The scheme is utilizing a new integer programming formulation of the problem that is introduced here. For the strongly NP-hard RSP with nonconstant joint cycle length, we provide a polynomial time approximation scheme (PTAS), which for any fixed [Formula: see text], provides a linear time [Formula: see text] approximate solution. The continuous RSP, where reorders can take place at any time within a cycle, seems (with our results) to be easier than the respective discrete problem. We narrow the previously known complexity gap between the continuous and discrete versions of RSP for the multi-cycle RSP (with either constant or nonconstant cycle length) and the single-cycle RSP with constant cycle length and widen the gap for single-cycle RSP with nonconstant cycle length. For the multi-cycle case and constant joint cycle length, the complexity status of continuous RSP is open, whereas it is proved here that the discrete RSP is weakly NP-hard. Under our conjecture that the continuous RSP is easier than the discrete one, this implies that continuous RSP on multi-cycle and constant joint cycle length (currently of unknown complexity status) is at most weakly NP-hard. 

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