scholarly journals Erratum: Localization in Fractonic Random Circuits [Phys. Rev. X 9 , 021003 (2019)]

2019 ◽  
Vol 9 (4) ◽  
Author(s):  
Shriya Pai ◽  
Michael Pretko ◽  
Rahul M. Nandkishore
Keyword(s):  
Random Circuits

scholarly journals Localization in Fractonic Random Circuits

2019 ◽  
Vol 9 (2) ◽  
Author(s):  
Shriya Pai ◽  
Michael Pretko ◽  
Rahul M. Nandkishore
Keyword(s):  
Random Circuits

A generalized Pólya urn and limit laws for the number of outputs in a family of random circuits

Test ◽  
2012 ◽  
Vol 22 (1) ◽  
pp. 46-61 ◽  
Author(s):  
José Moler ◽  
Fernando Plo ◽  
Henar Urmeneta
Keyword(s):  
Limit Laws ◽  
Pólya Urn ◽  
Polya Urn ◽  
Generalized Pólya Urn ◽  
Random Circuits

On the Expected Depth of Random Circuits

1999 ◽  
Vol 8 (3) ◽  
pp. 209-228 ◽  
Author(s):  
SUNIL ARYA ◽  
MORDECAI J. GOLIN ◽  
KURT MEHLHORN
Keyword(s):  
Maximal Depth ◽  
Random Circuits

In this paper we analyse the expected depth of random circuits of fixed fanin f. Such circuits are built a gate at a time, with the f inputs of each new gate being chosen randomly from among the previously added gates. The depth of the new gate is defined to be one more than the maximal depth of its input gates. We show that the expected depth of a random circuit with n gates is bounded from above by ef ln n and from below by 2.04 … f ln n.

scholarly journals On martingale tail sums in affine two-color urn models with multiple drawings

2017 ◽  
Vol 54 (1) ◽  
pp. 96-117 ◽  
Author(s):  
Markus Kuba ◽  
Henning Sulzbach
Keyword(s):  
Urn Model ◽  
Iterated Logarithm ◽  
Large Index ◽  
Pólya Urn ◽  
The Family ◽  
The Standard Model ◽  
Polya Urn ◽  
Random Circuits ◽  
Recursive Trees ◽  
Tail Sums

AbstractIn two recent works, Kuba and Mahmoud (2015a) and (2015b) introduced the family of two-color affine balanced Pólya urn schemes with multiple drawings. We show that, in large-index urns (urn index between ½ and 1) and triangular urns, the martingale tail sum for the number of balls of a given color admits both a Gaussian central limit theorem as well as a law of the iterated logarithm. The laws of the iterated logarithm are new, even in the standard model when only one ball is drawn from the urn in each step (except for the classical Pólya urn model). Finally, we prove that the martingale limits exhibit densities (bounded under suitable assumptions) and exponentially decaying tails. Applications are given in the context of node degrees in random linear recursive trees and random circuits.

scholarly journals Anticoncentration theorems for schemes showing a quantum speedup

Quantum ◽  
2018 ◽  
Vol 2 ◽  
pp. 65 ◽  
Author(s):  
Dominik Hangleiter ◽  
Juan Bermejo-Vega ◽  
Martin Schwarz ◽  
Jens Eisert
Keyword(s):  
Information Science ◽  
Quantum Computers ◽  
Translation Invariant ◽  
Computational Advantage ◽  
Spatial Dimensions ◽  
Universal Quantum ◽  
Closing The Gap ◽  
State Of Affairs ◽  
Random Circuits ◽  
Quantum Speedup

One of the main milestones in quantum information science is to realise quantum devices that exhibit an exponential computational advantage over classical ones without being universal quantum computers, a state of affairs dubbed quantum speedup, or sometimes "quantum computational supremacy". The known schemes heavily rely on mathematical assumptions that are plausible but unproven, prominently results on anticoncentration of random prescriptions. In this work, we aim at closing the gap by proving two anticoncentration theorems and accompanying hardness results, one for circuit-based schemes, the other for quantum quench-type schemes for quantum simulations. Compared to the few other known such results, these results give rise to a number of comparably simple, physically meaningful and resource-economical schemes showing a quantum speedup in one and two spatial dimensions. At the heart of the analysis are tools of unitary designs and random circuits that allow us to conclude that universal random circuits anticoncentrate as well as an embedding of known circuit-based schemes in a 2D translation-invariant architecture.

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