Quantum State Preparation with Optimal T-Count

How many $T$ gates are needed to approximate an arbitrary $n$-qubit quantum state to within error $\varepsilon$? Improving prior work of Low, Kliuchnikov, and Schaeffer, we show that the optimal asymptotic scaling is $\Theta\left(\sqrt{2^n \log(1/\varepsilon)} + \log(1/\varepsilon)\right)$ if we allow ancilla qubits. We also show that this is the optimal $T$-count for implementing an arbitrary diagonal $n$-qubit unitary to within error $\varepsilon$. We describe applications in which a tensor product of many single-qubit unitaries can be synthesized in parallel for the price of one.