Quantum algorithm

Quantum mechanics

Algorithm

Quantum computer

Discrete mathematics

Quantum error correction

Implementation

Quantum circuit

Subroutine

Operating system

Computer science

Mathematical optimization

Hamiltonian (control theory)

Electronic circuit

Physics

Mathematics

Exponential function

Mathematical analysis

Quantum

Qubit

Programming language

State (computer science)

Physical Review Letters

Quantum state preparation is an important subroutine for quantum computing. We show that any $n$-qubit quantum state can be prepared with a $\Theta(n)$-depth circuit using only single- and two-qubit gates, although with a cost of an exponential amount of ancillary qubits. On the other hand, for sparse quantum states with $d\geqslant2$ non-zero entries, we can reduce the circuit depth to $\Theta(\log(nd))$ with $O(nd\log d)$ ancillary qubits. The algorithm for sparse states is exponentially faster than best-known results and the number of ancillary qubits is nearly optimal and only increases polynomially with the system size. We discuss applications of the results in different quantum computing tasks, such as Hamiltonian simulation, solving linear systems of equations, and realizing quantum random access memories, and find cases with exponential reductions of the circuit depth for all these three tasks. In particular, using our algorithm, we find a family of linear system solving problems enjoying exponential speedups, even compared to the best-known quantum and classical dequantization algorithms.

Quantum State Preparation with Optimal Circuit Depth: Implementations and Applications