2

Initializing a parameterized quantum circuit with a unitary sampled from a uniform distribution (so sampling from the Haar distribution) or in a weaker form: a unitary which is a unitary 2-design, brings the danger of starting in a barren plateau.

So why do we sample from the uniform? Because we have no prior knowledge?

The author in this paper without title argues, that to sample measurements from a distribution that is hard using classical computing, we need unitaries which are unitary 2-designs. Is this the reason we consider the unitary 2-design?

Actually, this topic with the 2-design is quite confusing for me, why isn't it possible to use non-2-design unitaries?

glS
  • 24,708
  • 5
  • 34
  • 108
nuemlouno
  • 177
  • 5
  • Unitary two designs can be used to compute expectation values, i.e. quadratic (2)-forms. This is because when forming the inner product you get a quadratic polynomial in terms of the state vector. – Cuhrazatee May 21 '22 at 02:54
  • Just a note, here is a recent paper on a similar topic that has a title http://arxiv.org/abs/2205.05056! I do not think it answers your question about initialization though. As far as I remember (can't find a reference now) initializing parameters near zero values was in fact recommended to avoid barren plateaus. – Nikita Nemkov May 21 '22 at 14:56

1 Answers1

1

Here is my view: The reason why we use the unitary 2-design concept is based on the fact that random quantum circuits are approximate unitary 2-design (I do not know how to add citation here, you can search it ). Moreover, to prove barren plateaus with numerical evidence that the gradient vector or the single partial derivative has a zero-average and exponentially suppressed variance, those integrations like $\langle \partial_{\theta_j}C\rangle=\int d\theta \partial_{\theta_j}C$ can be evaluated using equations 1 and 2 in your linked pdf.

Using a non-2-design circuit is one method to address the problem. People are intended to use the problem-inspired ansatz for such a topic, but little theoretical work has been provided. known results show that quantum convolutional neural networks do not establish BP

Another method is finding a "good" set of initial parameters. As it was shown that solid expressibility can lead to poor trainability, reducing the expressibility, like correlating parameters in the ansatz can improve the trainability (arXiv:2101.02138). And several parameter initialization methods have been proposed.

刘环宇
  • 89
  • 3
  • But from my linked paper: We need a unitary 2-design for showing quantum advantage. This is what I get from the first page – nuemlouno May 24 '22 at 12:48
  • Indeed lots of works showing quantum advantages are based on random quantum circuits sampling, where quantum computers provide a huge speedup. – 刘环宇 May 25 '22 at 01:49
  • However, I think the unitary 2-design property of parameterized quantum circuits, especially in the hardware-efficient ansatz, is applied for convenience in the analysis of gradients. The proposition of the HEA is due to the fact that the gates in the ansatz are easily implementable. – 刘环宇 May 25 '22 at 01:54
  • As for quantum advantages in parameterized quantum circuits, whether it can be realized is still not clear. While unitary 2-design circuit leads to BP, we prefer to use less-expressive circuits as long as they can reach the optimal points, which is challenging. – 刘环宇 May 25 '22 at 02:00