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The Quantum State Preparation problem aims to prepare an $n$-qubit quantumstate $|\psi_v\rangle =\sum_{k=0}^{2^n-1}v_k|k\rangle$ from the initial state$|0\rangle^{\otimes n}$, for a given unit vector$v=(v_0,v_1,v_2,\ldots,v_{2^n-1})^T\in \mathbb{C}^{2^n}$ with $\|v\|_2 = 1$.The problem is of fundamental importance in quantum algorithm design,Hamiltonian simulation and quantum machine learning, yet its circuit depth andsize complexity remain open when ancillary qubits are available. In this paper,we study efficient constructions of quantum circuits with $m$ ancillary qubitsthat can prepare $|\psi_v\rangle$ in depth $\tildeO\left(\frac{2^n}{m+n}+n\right)$and size $O(2^n)$, achieving the optimal valuefor both measures simultaneously. These results also imply a depth complexityof $\Theta(4^n/(m+n))$ for quantum circuits implementing a general $n$-qubitunitary using $m = O(2^n/n)$ ancillary qubits. This resolves the depthcomplexity for circuits without ancillary qubits, and for circuits withexponentially many ancillary qubits, this gives a quadratic saving from$O(4^n)$ to $\tilde \Theta(2^n)$. Our circuits are deterministic, prepare thestate and carry out the unitary precisely, utilize the ancillary qubits tightlyand the depths are optimal in a wide range of parameter regime. The results canbe viewed as (optimal) time-space tradeoff bounds, which is not onlytheoretically interesting, but also practically relevant in the current trendthat the number of qubits starts to take off, by showing a way to use a largenumber of qubits to compensate the short qubit lifetime.

Asymptotically Optimal Circuit Depth for Quantum State Preparation and General Unitary Synthesis