Quantum state preparation via free binary decision diagram

Abstract Quantum state preparation (QSP) is the task of preparing a quantum state from a given classical description. The classical description of an n -qubit quantum state may have exp ⁡ ( O ( n ) ) parameters in general, which makes preparing the corresponding state inherently inefficient in the worst case. However, in many practical cases, we may be able to employ suitable data structures for QSP. Ordered binary decision diagrams (OBDDs) and free BDDs (FBDDs) are data structures that can represent large-scale data in a compressed way. An efficient QSP method for a subclass of OBDDs is known, but requires an O ( 2 n ) -sized quantum circuit in general, while QSP based on FBDDs, which includes OBDDs as a special case, remains unexplored. Here, we construct a state-preparation procedure for QSP when the classical description of a quantum state is given by an FBDD with weighted edges and analyze the space and time complexity of QSP in this setting. We provide a nontrivial example of an n -qubit state that can be represented by a weighted FBDD (WFBDD) with N = O ( poly ( n ) ) nodes rather than exp ( O ( n ) ) . We show that any quantum state represented by a QSP-admissible WFBDD with N nodes can be prepared by an O ( N ) -sized quantum circuit using at most N ancillary qubits, exponentially improving the required circuit size for QSP compared to other BDD-based QSPs. We also provide another example of an n -qubit state that can be represented by a WFBDD with N = O ( n 2 ) nodes and O ( n 2 ) ancillary qubits, for which the known amplitude-amplification-based QSP cost expression is exponential in n . These results provide techniques to employ FBDDs