Marek Gluza<p>Adrián Pérez-Salinas coordinated a <a href="https://mathstodon.xyz/tags/benasque" class="mention hashtag" rel="tag">#<span>benasque</span></a> session on variational <a href="https://mathstodon.xyz/tags/quantum" class="mention hashtag" rel="tag">#<span>quantum</span></a> algorithms. Here&#39;s the story what I think I learned :)</p><p>People now say various things about the state of the field.<br />It was clear to me from the very beginning that there will be trainability issues and you can verify it because I haven&#39;t written a single paper involving brute force training of circuits. </p><p>But.</p><p>While, I have been avoiding reading papers on VQAs (until I needed to cite variational diagonalization in context of my proposal to use double-bracket flows for diagonalization on quantum computers), now that the field has reached a milestone, here&#39;s a few insights I really like and claim will matter down the line:<br />- statements about <a href="https://mathstodon.xyz/tags/barrenPlateaus" class="mention hashtag" rel="tag">#<span>barrenPlateaus</span></a> are quantitative<br />- appearance of barren plateaus is implied by presence of <a href="https://mathstodon.xyz/tags/t" class="mention hashtag" rel="tag">#<span>t</span></a>-design properties <a href="https://arxiv.org/abs/2101.02138" target="_blank" rel="nofollow noopener noreferrer" translate="no"><span class="invisible">https://</span><span class="">arxiv.org/abs/2101.02138</span><span class="invisible"></span></a><br />- plateaus result from high dimensionality of the training parameter set.</p><p>To paraphrase grandmaster Bronstein, it&#39;s not about what but how. <br />We now know very, very well how VQAs go wrong. Early on it was only clear what the problem will be.</p><p>Each of the points above can guide better <a href="https://mathstodon.xyz/tags/ansatzae" class="mention hashtag" rel="tag">#<span>ansatzae</span></a>:<br />- they need to operate on clumped circuits to reduce the dimensionality<br />- they shouldn&#39;t rotate back and forth but use physics equations to guide the <a href="https://mathstodon.xyz/tags/quantumCompiling" class="mention hashtag" rel="tag">#<span>quantumCompiling</span></a><br />- a restricted ansatz with justified expressibility can be quantitatively tested using the average+variance criteria of regular barren plateaus.</p><p>What the field achieved is to inform a large group of people how to recognize what a good variational ansatz will be once we will encounter it. </p><p>And that it will not be naive? Come on, easy would have been boring.</p>