The Solovay-Kitaev theorem shows that we can approximate arbitrary unitary transformations with polynomially many quantum gates. Can we approximate the resulting state vectors in the same way by specifying the quantum states of pairs (or possibly more) of qubits? I.e. if we have $\psi_{i,j}$ be the approximate state of the qubits $i$ and $j$, is this enough to approximate the entire quantum state?
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2$\begingroup$ The Solovay-Kitaev Theorem shows that if you want to approximate a single-qubit gate to accuracy $\epsilon$, you need a sequence of $O(\log(1/\epsilon)^k)$ gates from a finite set such as $H$ and $T$. More modern techniques have show $k=1$. It does not, by itself, talk about many qubits. $\endgroup$– DaftWullieMar 6, 2020 at 11:10
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$\begingroup$ Is this arxiv.org/abs/2203.00811 what you have in mind? $\endgroup$– Jake XuerebMar 7, 2022 at 18:10
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