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.

In 2021, Chen, Liu, and Zhandry presented an efficient quantum algorithm for the average-case ℓ∞-Short Integer Solution (SIS∞) problem, in a parameter range outside the normal range of cryptographic interest, but still with no known efficient classical algorithm. This was particularly exciting since SIS∞ is a simple problem without structure, and their algorithmic techniques were different from those used in prior exponential quantum speedups.