As a general-purpose data structure, the decision diagram was originally born in the field of representation and analysis of classical Boolean functions. It is essentially a directed acyclic graph that compactly encodes the truth table of a function through variable nodes, 0/1 branch edges, and terminal nodes, avoiding exponential storage explosion. In its simplified or reduced form, the decision diagram achieves extremely high compression rates by merging identical subgraphs, eliminating redundant nodes, and sharing paths. HOLO extends this classical tool to the quantum domain and proposes a class of quantum states that can be efficiently represented using reduced Algebraic Decision Diagrams (ADD). Specifically, for a quantum state |ϕ⟩ = ∑ α_s |s⟩, where s is an n-bit basis state string, the non-zero amplitudes α_s correspond to paths in the decision diagram. Each internal node represents a qubit variable, solid edges represent the 1-branch, dashed edges represent the 0-branch, and terminal nodes store the normalized amplitude values. Through reduction rules — merging identical terminals, eliminating single-child nodes, and sharing identical subgraphs — a state that might originally have m non-zero amplitudes is compressed into a decision diagram with only k paths, where k is often much smaller than m or even 2^n. This representation captures the sparsity and repetitive patterns of the quantum state, such as certain subsets of basis states sharing the same amplitude or substructure, thereby providing a roadmap for subsequent circuit construction.
Based on this decision diagram representation, the core of the algorithm developed by HOLO lies in directly utilizing the structure of the graph to construct quantum circuits, rather than blindly enumerating all basis states. The algorithm employs a single auxiliary qubit (initialized to |1⟩, acting as an unprocessed flag), with data qubits initialized to |0⟩. The entire process is completely deterministic, requiring no measurements or random post-selection. The algorithm first traverses the decision diagram in post-order, computes the local probability p_0 for each node (based on the sum of squared amplitudes of child nodes, with reduced nodes multiplied by a 2^e factor), and precomputes the corresponding Ry rotation gate G(p_0), which rotates |0⟩ to √p_0|0⟩ + √(1-p_0)|1⟩, encoding the branch probability entering that subtree. It then proceeds to pre-order traversal, recursively constructing the circuit starting from the root node: for branch nodes, it applies a doubly-controlled G(p_0) gate (controlled by the auxiliary qubit and the nearest |1⟩ in the path) to the current qubit; for single-child nodes, it inserts a doubly-controlled X gate (CNOT) and handles half-probability rotations between reduced nodes; when reaching a terminal, it first applies a phase gate e^{i arg(α)} to adjust the amplitude phase, and finally uses a multi-controlled X gate (MCX, controlled by all branch nodes in the path) to flip the auxiliary qubit, marking it as |0⟩ (processed). This design ensures that subsequent path preparation does not interfere with previously completed paths, as the auxiliary qubit acts as a protection switch on the processed subspace.
The key to the technical implementation logic lies in the sharing and sequential processing of decision diagram paths. HOLO cleverly sorts the paths in descending binary value order (p1 ≻ p2 ≻ ... ≻ pk), so that each time construction continues from the last common prefix node of the previous path, avoiding redundant operations. The number of gates contributed by each path is at most O(n), including n doubly-controlled gates (decomposed into 4-6 CNOTs) and one MCX (also decomposed into O(n) CNOTs). However, due to prefix sharing and node reduction, the actual total circuit complexity is only O(kn), i.e., linearly related to the number of paths k in the decision diagram, rather than to the number of non-zero amplitudes m or 2^n. This forms a sharp contrast with traditional methods: general preparation often requires O(m n) or even more gates and struggles to exploit structure; although early decision diagram-based methods offered some compression, they did not fully exploit path marking and reduction, resulting in higher gate counts. HOLO’s algorithm ensures a clean circuit and theoretically achieves a fidelity of 1 (under ideal noiseless conditions) through the auxiliary qubit’s path-locking mechanism and precise probability/phase injection, truly realizing deterministic preparation.
It is particularly worth mentioning that the algorithm demonstrates extreme performance on the initial state of the quantum Byzantine agreement protocol. The quantum Byzantine agreement protocol is a key protocol for achieving consensus in distributed quantum computing, and its initial quantum state often exhibits a highly sparse decision diagram with specific shared substructures. HOLO’s experiments show that for the initial state of this protocol, the reduction in the number of CNOT gates ranges from 86.61% to 99.9%. This means that in actual multi-party quantum networks, the resource overhead in the protocol initialization phase is significantly reduced, with higher fidelity, thereby improving the overall reliability and scalability of the protocol. This application directly proves the practical value of the technology.
Looking back at the development history of quantum computing, from Shor’s algorithm to Grover’s search, and then to variational quantum eigensolvers (VQE), every advancement has relied on efficient fundamental primitives. As the data entry point, the optimization of quantum state preparation directly affects the performance of the entire stack. HOLO’s algorithm not only fills the gap in structured quantum state preparation but also lays the foundation for broader applications of decision diagrams in the quantum field — which may expand to areas such as quantum circuit synthesis and simulation acceleration in the future. As the scale of qubits advances toward hundreds or even thousands, the exponential barrier of general state preparation will become increasingly prominent. HOLO’s algorithm provides a clear path: as long as the problem has a structure that can be represented by a decision diagram, efficient injection can be achieved.
This is not only a demonstration of HOLO’s R&D strength but also a milestone event in the quantum computing ecosystem. In today’s fiercely competitive global quantum race, achieving a quantum breakthrough in the classical field of decision diagrams demonstrates a powerful capability to transform from theory to engineering.
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