相关论文: The Garden Hose Complexity for the Equality Functi…

For a simplicial complex X on {1,2, ..., n} we define enriched homology and cohomology modules. They are graded modules over k[x_1, ..., x_n] whose ranks are equal to the dimensions of the reduced homology and cohomology groups. We…

In this addendum to [arXiv:1402.5674] two points are discussed. In the first additional evidence is provided for a dual connection between the geometric length of an Einstein-Rosen bridge and the computational complexity of the quantum…

Kitchloo and Morava give a strikingly simple picture of elliptic cohomology at the Tate curve by studying a completed version of $S^1$-equivariant $K$-theory for spaces. Several authors (cf [ABG],[KM],[L]) have suggested that an equivariant…

In this work, we are interested in the detailed analysis of complexity aspects of both time and space that arises from the implementation of a quantum algorithm on a quantum based hardware. In particular, some steps of the implementation,…

We define a measure of "complexity" of a braid which is natural with respect to both an algebraic and a geometric point of view. Algebraically, we modify the standard notion of the length of a braid by introducing generators $\Delta\_{ij}$,…

Understanding the mechanisms responsible for the equilibration of isolated quantum many-body systems is a long-standing open problem. In this work we obtain a statistical relationship between the equilibration properties of Hamiltonians and…

We consider a subclass of tilings, the tilings obtained by cut and projection. Under somewhat standard assumptions, we show that the natural complexity function has polynomial growth. We compute its exponent \alpha in terms of the ranks of…

Communication complexity, which quantifies the minimum communication required for distributed computation, offers a natural setting for investigating the capabilities and limitations of quantum mechanics in information processing. We…

Alternating quantifier depth is a natural measure of difficulty required to express first order logical sentences. We define a sequence of first order properties on rooted, locally finite trees in a recursive manner, and provide rigorous…

Counting distinct permutations with replacement, especially when involving multiple subwords, is a longstanding challenge in combinatorial analysis, with critical applications in cryptography, bioinformatics, and statistical modeling. This…

We study the round and communication complexities of various cryptographic protocols. We give tight lower bounds on the round and communication complexities of any fully black-box reduction of a statistically hiding commitment scheme from…

For a smooth, closed $n$-manifold $M$, we define an upper semi-continuous integer-valued complexity function on $H^1(M;{\mathbb R})$ using Morse theory. This measures how far an integral class is from being a fiber of a fibration. The fact…

We consider the complexity of the local Hamiltonian problem in the context of fermionic Hamiltonians with $\mathcal N=2 $ supersymmetry and show that the problem remains $\mathsf{QMA}$-complete. Our main motivation for studying this is the…

We give an exposition of the hidden subgroup problem for dihedral groups from the point of view of the standard hidden subgroup quantum algorithm for finite groups. In particular, we recall the obstructions for strong Fourier sampling to…

We prove a general lower bound on the bounded-error entanglement-assisted quantum communication complexity of Boolean functions. The bound is based on the concept that any classical or quantum protocol to evaluate a function on distributed…

The quantum version of communication complexity allows the two communicating parties to exchange qubits and/or to make use of prior entanglement (shared EPR-pairs). Some lower bound techniques are available for qubit communication…

The multiplicative multiple Horn problem is asking to determine possible singular values of the combinations $AB, BC$ and $ABC$ for a triple of invertible matrices $A,B,C$ with given singular values. There are similar problems for…

We introduce a new notion of neighboring databases for coverage problems such as Max Cover and Set Cover under differential privacy. In contrast to the standard privacy notion for these problems, which is analogous to node-privacy in…

Characterizing the express power of the Transformer architecture is critical to understanding its capacity limits and scaling law. Recent works provide the circuit complexity bounds to Transformer-like architecture. On the other hand,…

We present an entirely new geometric and probabilistic approach to synchronization of correspondences across multiple sets of objects or images. In particular, we present two algorithms: (1) Birkhoff-Riemannian L-BFGS for optimizing the…