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Equation 2 <math>C ={1 \over N}\sum_{x}^N{\langle\phi^\text{out}|\rho^\text{out}|\phi^\text{out}\rangle}</math>
=== Barren plateaus ===
[[File:Barren_plateaus_of_VQA.webp|alt=The Barren Plateau problem becomes increasingly serious as the VQA expands|thumb|'''Barren plateaus of VQA'''<ref>{{Cite journal |last1=Wang |first1=Samson |last2=Fontana |first2=Enrico |last3=Cerezo |first3=M. |last4=Sharma |first4=Kunal |last5=Sone |first5=Akira |last6=Cincio |first6=Lukasz |last7=Coles |first7=Patrick J. |date=2021-11-29 |title=Noise-induced barren plateaus in variational quantum algorithms |journal=Nature Communications |language=en |volume=12 |issue=1 |page=6961 |arxiv=2007.14384 |bibcode=2021NatCo..12.6961W |doi=10.1038/s41467-021-27045-6 |issn=2041-1723 |pmc=8630047 |pmid=34845216}}</ref> Figure shows the Barren Plateau problem becomes increasingly serious as the VQA expands.]]
Gradient descent is widely used and successful in classical algorithms. However, although the simplified structure is very similar to neural networks such as CNNs, QNNs perform much worse.
Since the quantum space exponentially expands as the q-bit grows, the observations will concentrate around the mean value at an exponential rate, where also have exponentially small gradients.<ref name=":3">{{Cite journal |last1=McClean |first1=Jarrod R. |last2=Boixo |first2=Sergio |last3=Smelyanskiy |first3=Vadim N. |last4=Babbush |first4=Ryan |last5=Neven |first5=Hartmut |date=2018-11-16 |title=Barren plateaus in quantum neural network training landscapes |journal=Nature Communications |language=en |volume=9 |issue=1 |page=4812 |arxiv=1803.11173 |bibcode=2018NatCo...9.4812M |doi=10.1038/s41467-018-07090-4 |issn=2041-1723 |pmc=6240101 |pmid=30446662}}</ref>
This situation is known as Barren Plateaus, because most of the initial parameters are trapped on a "plateau" of almost zero gradient, which approximates random wandering<ref name=":3" /> rather than gradient descent. This makes the model untrainable.
In fact, not only QNN, but almost all deeper VQA algorithms have this problem. In the present NISQ era, this is one of the problems that have to be solved if more applications are to be made of the various VQA algorithms, including QNN.
==See also==
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