Complexity of Quantum State Verification in the Quantum Linear Systems Problem

We analyze the complexity of quantum state verification in context solving systems linear equations form $A \vec x = b$. show that any operation verifies whether a given is within constant distance from solution problem requires $q=\Omega(\kappa)$ uses unitary prepares $\left| b \right>$, proportional to $\vec b$, and its inverse worst case. Here, $\kappa$ condition number matrix $A$. For typical instances, we $q=\Omega(\sqrt \kappa)$ with high probability. These lower bounds are almost achieved if performed using known algorithms for problem. also copies \right>$ required by procedures prepare measure type. In this case, quadratically worse, being $\Omega(\kappa^2)$ case $\Omega(\kappa)$ instances discuss implications our results variational related approaches problem, where preparation, gate, measurement errors will need decrease rapidly worst-case error correction not used, present some open problems.