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arXiv:2407.01671 · Journal of Physics A Mathematical and Theoretical

Quantum state preparation via free binary decision diagram

Yu Tanaka, Hayata Yamasaki, Mio Murao

Published Jun 11, 2026• 0 citations

Abstract

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

DOI: 10.1088/1751-8121/ae7c42

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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

Loading classical data into quantum registers is one of the most important primitives of quantum computing. While the complexity of preparing a generic quantum state is exponential in the number of qubits, in many practical tasks the state to prepare has a certain structure that allows for faster preparation. In this paper, we consider quantum states that can be efficiently represented by (reduced) decision diagrams, a versatile data structure for the representation and analysis of Boolean functions. We design an algorithm that utilizes the structure of decision diagrams to prepare their associated quantum states. Our algorithm has a circuit complexity that is linear in the number of paths in the decision diagram. Numerical experiments show that our algorithm reduces the circuit complexity by up to 31.85% compared to the state-of-the-art algorithm, when preparing generic

n

-qubit states with

{n}^{3}

nonzero amplitudes. Additionally, for states with sparse decision diagrams, including the initial state of the quantum Byzantine agreement protocol, our algorithm reduces the number of controlled-nots by 86.61--99.9%.

Quantum state preparation (QSP) is a fundamental task in quantum computing and quantum information processing. It is critical to the execution of many quantum algorithms, including those in quantum machine learning. In this paper, we propose a family of efficient QSP algorithms tailored to different numbers of available ancilla qubits — ranging from no ancilla qubits, to a single ancilla qubit, to a sufficiently large number of ancilla qubits. Our approach exploits the power of Local Invertible Map Tensor Decision Diagrams (LimTDDs) — a highly compact representation of quantum states that combines tensor networks and decision diagrams to reduce quantum circuit complexity. Extensive experiments demonstrate that our methods significantly outperform existing approaches and exhibit better scalability for large-scale quantum states, both in terms of runtime and gate complexity. Furthermore, our method shows exponential improvement in best-case scenarios.

Quantum state preparation (QSP) for a general

n

-qubit state requires

{n}^{3}

CNOT gates and circuit depth, making exact amplitude encoding (EAE) impractical for near-term quantum hardware. We introduce an ancilla-free hybrid classical-quantum strategy that reduces this cost to

O(poly(n))

for a broad class of compressible data. The method first applies a classical compression step to obtain a

d

-sparse representation of the input, loads this sparse vector using a sparse-state preparation routine, and then reconstructs the target state through a polynomial-depth quantum inverse transform. We evaluate the framework on synthetic benchmark signals and real biomedical time series using Fourier and Haar transforms, demonstrating substantial reductions in CNOT counts and circuit depth compared to EAE, together with competitive performance relative to the Fourier Series Loader (FSL). The quantum simulation results show that combining classical data compression with quantum decompression provides a scalable framework for efficient QSP, reducing quantum overhead without requiring variational training or ancillary registers.

An ordered binary decision diagram (OBDD) is a directed acyclic graph that represents a Boolean function.Since OBDDs have many nice properties as data structures,they have been extensively studied for decades in both theoretical and practical fields,such as VLSI (Very Large Scale Integration) design, formal verification, machine learning, and combinatorial problems.Arguably, the most crucial problem in using OBDDs is thatthey may vary exponentially in size depending on their variable ordering (i.e., the order in which the variables are to be read) when they represent the same function.Indeed, it is NP hard to find an optimal variable ordering that minimizes an OBDDfor a given function.Friedman and Supowit provided a clever deterministic algorithm with time/space complexityO*(3n), where

n

is the number of variables of the function,which is much better than the trivial brute-force bound O*(n!2n)$.This paper shows that a further speedup is possible with quantum computersby presenting a quantum algorithmthat produces a minimum OBDD together with the corresponding variable orderingin O*(2.77286n) time and space with an exponentially small error probability. Moreover, this algorithm can be adapted toconstructing other minimum decision diagrams such as zero-suppressed BDDs (ZBDDs or ZDDs).

This paper presents a new data structure, called Weighted Context-Free-Language Ordered BDDs (WCFLOBDDs), which are a hierarchically structured decision diagram, akin to Weighted BDDs (WBDDs) enhanced with a procedure-call mechanism. For some functions, WCFLOBDDs are exponentially more succinct than WBDDs. They are potentially beneficial for representing functions of type B n → D , when a function’s image V ⊆ D has many different values. We apply WCFLOBDDs in quantum-circuit simulation, and find that they perform better than WBDDs on certain benchmarks. With a 15-minute timeout, the number of qubits that can be handled by WCFLOBDDs is 1 − 64 × that of WBDDs (and 1 − 128 × that of CFLOBDDs, which are an unweighted version of WCFLOBDDs). These results support the conclusion that for this application—from the standpoint of problem size, measured as the number of qubits—WCFLOBDDs provide the best of both worlds: performance roughly matches whichever of WBDDs and CFLOBDDs is better. (From the standpoint of running time, the results are more nuanced.)

As a cornerstone for many quantum linear algebraic and quantum machine learning algorithms, controlled quantum state preparation (CQSP) aims to provide the transformation of | i ⟩ | 0 n ⟩ → | i ⟩ | ψ i ⟩ for all i ∈ { 0 , 1 } k for the given n -qubit states | ψ i ⟩ . In this paper, we construct a quantum circuit for implementing CQSP, with depth O ( n + k + 2 n + k n + k + m ) and size O ( 2 n + k ) for any given number m of ancillary qubits. These bounds, which can also be viewed as a time-space tradeoff for the transformation, are optimal for any integer parameters m , k ≥ 0 and n ≥ 1 . When k = 0 , the problem becomes the canonical quantum state preparation (QSP) problem with ancillary qubits, which asks for efficient implementations of the transformation | 0 n ⟩ | 0 m ⟩ → | ψ ⟩ | 0 m ⟩ . This problem has many applications with many investigations, yet its circuit complexity remains open. Our construction completely solves this problem, pinning down its depth complexity to Θ ( n + 2 n / ( n + m ) ) and its size complexity to Θ ( 2 n ) for any