Quantum state preparation with optimal T-count

TL;DR

This paper establishes the optimal asymptotic T-count for approximating arbitrary n-qubit states and diagonal unitaries within error ε, demonstrating improvements over prior bounds and enabling efficient parallel synthesis of unitaries.

Contribution

It proves the optimal asymptotic T-count scaling for state and diagonal unitary synthesis with ancillas, advancing quantum circuit complexity understanding.

Findings

Optimal T-count scales as Θ(√(2^n log(1/ε)) + log(1/ε)) with ancillas.

This T-count is proven to be optimal for approximating arbitrary diagonal unitaries.

Parallel synthesis of tensor products of single-qubit unitaries is achievable at the cost of one unitary.

Abstract

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.