1,625 research outputs found
Strong ETH Breaks With Merlin and Arthur: Short Non-Interactive Proofs of Batch Evaluation
We present an efficient proof system for Multipoint Arithmetic Circuit
Evaluation: for every arithmetic circuit of size and
degree over a field , and any inputs ,
the Prover sends the Verifier the values and a proof of length, and
the Verifier tosses coins and can check the proof in about time, with probability of error less than .
For small degree , this "Merlin-Arthur" proof system (a.k.a. MA-proof
system) runs in nearly-linear time, and has many applications. For example, we
obtain MA-proof systems that run in time (for various ) for the
Permanent, Circuit-SAT for all sublinear-depth circuits, counting
Hamiltonian cycles, and infeasibility of - linear programs. In general,
the value of any polynomial in Valiant's class can be certified
faster than "exhaustive summation" over all possible assignments. These results
strongly refute a Merlin-Arthur Strong ETH and Arthur-Merlin Strong ETH posed
by Russell Impagliazzo and others.
We also give a three-round (AMA) proof system for quantified Boolean formulas
running in time, nearly-linear time MA-proof systems for
counting orthogonal vectors in a collection and finding Closest Pairs in the
Hamming metric, and a MA-proof system running in -time for
counting -cliques in graphs.
We point to some potential future directions for refuting the
Nondeterministic Strong ETH.Comment: 17 page
On quantum backpropagation, information reuse, and cheating measurement collapse
The success of modern deep learning hinges on the ability to train neural
networks at scale. Through clever reuse of intermediate information,
backpropagation facilitates training through gradient computation at a total
cost roughly proportional to running the function, rather than incurring an
additional factor proportional to the number of parameters - which can now be
in the trillions. Naively, one expects that quantum measurement collapse
entirely rules out the reuse of quantum information as in backpropagation. But
recent developments in shadow tomography, which assumes access to multiple
copies of a quantum state, have challenged that notion. Here, we investigate
whether parameterized quantum models can train as efficiently as classical
neural networks. We show that achieving backpropagation scaling is impossible
without access to multiple copies of a state. With this added ability, we
introduce an algorithm with foundations in shadow tomography that matches
backpropagation scaling in quantum resources while reducing classical auxiliary
computational costs to open problems in shadow tomography. These results
highlight the nuance of reusing quantum information for practical purposes and
clarify the unique difficulties in training large quantum models, which could
alter the course of quantum machine learning.Comment: 29 pages, 2 figure
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