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Often algorithms utilize an ansatz $V(a)$ and rely on a classical optimization scheme over a Hamiltonian loss function $L(V(a))$ in order to find the optimal parameters $a$.

Due to many factors such as periodicity of $V(a)$, optimization path and non-convexity of $L$ (others maybe?), two problems with very similar input can result in very different optimal outputs i.e. $\forall~\delta>0 : |x_i - x_j| \leq \delta,~\exists~\epsilon>0$ such that $ | a^{\text{opt}}_i - a^{\text{opt}}_j| > \epsilon$.

I am using non-gradient optimization schemes. I have tried methods such as use constant initial parameters $a^0$, transformations e.g. $\sin(a^{\text{opt}})$, but nothing worked.

Is there a known way to avoid such discontinuities? Maybe e.g. gradient optimization schemes?

EDIT: corrected discontinuity definition

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  • $\begingroup$ The statement $\not \exists~\delta: | a^{\text{opt}}_i - a^{\text{opt}}_j| < \delta$ s.t. $\forall~\epsilon: |x_i - x_j| \leq \epsilon$ does not imply discontinuity at $x_i$ or $x_j$. In fact, according to this definition, it is possible to give a continuous function that satisfies this definition at every point of its domain. One trivial example would be $f(x)=x$. Your statement suggests that the output is unbounded whenever the domain is unbounded. Alternatively, it is possible to give a discontinuous function and show that $\delta$ exists for all $\epsilon$ e.g. Heaviside function. $\endgroup$
    – MonteNero
    Commented Feb 25, 2023 at 0:49
  • $\begingroup$ @MonteNero sorry, my bad. I wanted to say does not imply instead of it doesn't exist, Isn't that more correct? $\endgroup$
    – consthatza
    Commented Feb 26, 2023 at 18:20
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    $\begingroup$ If you want to make statements about discontinuity, then it is better to stick to the standard definition, which says that $f$ is disconti at $x$ if there exists $\epsilon > 0$ such that for all $\delta > 0$, $|x - y| < \delta$ yet $|f(x) - f(y)| > \epsilon$. $\endgroup$
    – MonteNero
    Commented Feb 27, 2023 at 3:12
  • $\begingroup$ Thanks, I will change it $\endgroup$
    – consthatza
    Commented Feb 28, 2023 at 5:42

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Let $C(\theta)$ be an arbitrary quantum circuit parametrized by $\theta \in \mathbb{R}^n$. And let $L(C(\theta))$ be a continuous non-convex objective function we would like to optimize. Given that the pair of initial parameters $\theta_1$ and $\theta_2$ are sufficiently close to a local minimum at $\theta^*$, then any deterministic gradient descent method with a sufficiently small step size will result in convergence to that local minimum for both initial $\theta_1$ and $\theta_2$. Both initial parameters don't even have to be close to each other. The only requirement for $\theta_1$ and $\theta_2$ is being in an $\epsilon$-neighborhood of the local minimum at $\theta^*$.

In practice, the obtained gradients will be noisy because of circuit sampling. So, there is a probability that even if both $\theta_1$ and $\theta_2$ are very close to a local minimum, the final estimates of optimal parameters will be different. However, if you reduce the variance of the gradients, then we will get the convergence to the same local minimum.

A nice example of a quantum natural gradient descent can be found on the Pennylane website. They provide illustrations, mathematical formulation and code.

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