 Publication Date:
 NSFPAR ID:
 10298626
 Journal Name:
 Electronics
 Volume:
 10
 Issue:
 14
 Page Range or eLocationID:
 1690
 ISSN:
 20799292
 Sponsoring Org:
 National Science Foundation
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Recent work has proposed and explored using coreset techniques for quantum algorithms that operate on classical data sets to accelerate the applicability of these algorithms on nearterm quantum devices. We apply these ideas to Quantum Boltzmann Machines (QBM) where gradientbased steps which require Gibbs state sampling are the main computational bottleneck during training. By using a coreset in place of the full data set, we try to minimize the number of steps needed and accelerate the overall training time. In a regime where computational time on quantum computers is a precious resource, we propose this might lead to substantial practical savings. We evaluate this approach on 6x6 binary images from an augmented bars and stripes data set using a QBM with 36 visible units and 8 hidden units. Using an Inception score inspired metric, we compare QBM training times with and without using coresets.

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Quantum computational supremacy arguments, which describe a way for a quantum computer to perform a task that cannot also be done by a classical computer, typically require some sort of computational assumption related to the limitations of classical computation. One common assumption is that the polynomial hierarchy ( P H ) does not collapse, a stronger version of the statement that P ≠ N P , which leads to the conclusion that any classical simulation of certain families of quantum circuits requires time scaling worse than any polynomial in the size of the circuits. However, the asymptotic nature of this conclusion prevents us from calculating exactly how many qubits these quantum circuits must have for their classical simulation to be intractable on modern classical supercomputers. We refine these quantum computational supremacy arguments and perform such a calculation by imposing finegrained versions of the noncollapse conjecture. Our first two conjectures poly3NSETH( a ) and perintNSETH( b ) take specific classical counting problems related to the number of zeros of a degree3 polynomial in n variables over F 2 or the permanent of an n × n integervalued matrix, and assert that any nondeterministic algorithm that solves them requires 2 c nmore »

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Abstract We consider the problem of covering multiple submodular constraints. Given a finite ground set
N , a weight function ,$$w: N \rightarrow \mathbb {R}_+$$ $w:N\to {R}_{+}$r monotone submodular functions over$$f_1,f_2,\ldots ,f_r$$ ${f}_{1},{f}_{2},\dots ,{f}_{r}$N and requirements the goal is to find a minimum weight subset$$k_1,k_2,\ldots ,k_r$$ ${k}_{1},{k}_{2},\dots ,{k}_{r}$ such that$$S \subseteq N$$ $S\subseteq N$ for$$f_i(S) \ge k_i$$ ${f}_{i}\left(S\right)\ge {k}_{i}$ . We refer to this problem as$$1 \le i \le r$$ $1\le i\le r$MultiSubmodCover and it was recently considered by HarPeled and Jones (Few cuts meet many point sets. CoRR.arxiv:abs1808.03260 HarPeled and Jones 2018) who were motivated by an application in geometry. Even with$$r=1$$ $r=1$MultiSubmodCover generalizes the wellknown Submodular Set Cover problem (SubmodSC ), and it can also be easily reduced toSubmodSC . A simple greedy algorithm gives an approximation where$$O(\log (kr))$$ $O(log(kr\left)\right)$ and this ratio cannot be improved in the general case. In this paper, motivated by several concrete applications, we consider two ways to improve upon the approximation given by the greedy algorithm. First, we give a bicriteria approximation algorithm for$$k = \sum _i k_i$$ $k={\sum}_{i}{k}_{i}$MultiSubmodCover that covers each constraint to within a factor of while incurring an approximation of$$(11/e\varepsilon )$$ $(11/e\epsilon )$ in the cost. Second, we consider the special case when each$$O(\frac{1}{\epsilon }\log r)$$ $O(\frac{1}{\u03f5}logr)$ is a obtained from a truncated coverage function and obtain an algorithm that generalizes previous work on partial set cover ($$f_i$$ ${f}_{i}$PartialSC ), covering integer programs (CIPs ) and multiple vertex cover constraintsmore »