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Inspired by connections to two dimensional quantum theory, we define several models of computation based on permuting distinguishable particles (which we call balls), and characterize their computational complexity. In the quantum setting, we find that the computational power of this model depends on the initial input states. More precisely, with a standard basis input state, we show how to approximate the amplitudes of this model within additive error using the model DQC1 (the class of problems solvable with one clean qubit), providing evidence that the model in this case is weaker than universal quantum computing. However, for specific choices of input states, the model is shown to be universal for BQP in an encoded sense. We use representation theory of the symmetric group to partially classify the computational complexity of this model for arbitrary input states. Interestingly, we find some input states which yield a model intermediate between DQC1 and BQP. Furthermore, we consider a restricted version of this model based on an integrable scattering problem in 1+1 dimensions. We show it is universal under postselection, if we allow intermediate destructive measurements and specific input states. Therefore, the existence of any classical procedure to sample from the output distribution of this model within multiplicative error implies collapse of polynomial hierarchy to its third level. Finally, we define a classical version of this model in which one can probabilistically permute balls. We find this yields a complexity class which is intermediate between L and BPP. Moreover, we find a nondeterministic version of this model is NP-complete.
The classic Ham-Sandwich theorem states that for any $d$ measurable sets in $mathbb{R}^d$, there is a hyperplane that bisects them simultaneously. An extension by Barany, Hubard, and Jeronimo [DCG 2008] states that if the sets are convex and emph{wel
Diffusion-Limited Aggregation (DLA) is a cluster-growth model that consists in a set of particles that are sequentially aggregated over a two-dimensional grid. In this paper, we introduce a biased version of the DLA model, in which particles are limi
The 2008 financial crisis has been attributed to excessive complexity of the financial system due to financial innovation. We employ computational complexity theory to make this notion precise. Specifically, we consider the problem of clearing a fina
We show the problem of counting homomorphisms from the fundamental group of a homology $3$-sphere $M$ to a finite, non-abelian simple group $G$ is #P-complete, in the case that $G$ is fixed and $M$ is the computational input. Similarly, deciding if t
We study the quantum query complexity of finding a certificate for a d-regular, k-level balanced NAND formula. Up to logarithmic factors, we show that the query complexity is Theta(d^{(k+1)/2}) for 0-certificates, and Theta(d^{k/2}) for 1-certificate