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arXiv:2609.20854

Oracle Synthesis Based on X-Map Decision Diagrams

Xin Hong, Kezhen Zhang, Aochu Dai, Sanjiang Li, Shenggang Ying, Mingsheng Ying

quant-phSubmitted Aug 26, 2026• Announced Aug 27, 2026• 0 citations
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Abstract

Quantum oracles act as reversible black-box operators that encode classical Boolean functions into quantum states, enabling efficient function evaluation in quantum superposition. The resource efficiency of oracle implementation is critical to the performance of numerous quantum algorithms. Most state-of-the-art oracle synthesis approaches rely on compact Boolean function representations such as exclusive-sum-of-products (ESOP), yet still suffer from excessive $T$-count and $CX$-count for large-scale functions. In this paper, we propose a novel compact representation named X-Map decision diagram (XMDD) for Boolean functions, which integrates local invertible maps and complement edges to achieve higher compression efficiency. Based on XMDD, we further develop an optimized quantum oracle synthesis algorithm. Extensive experimental results demonstrate that, for Boolean functions with more than seven input variables, our method outperforms the state-of-the-art ESOP-based approach and Qiskit in nearly all test cases, achieving simultaneous reduction in both $T$-count and $CX$-count without an obvious trade-off. Moreover, we show that the performance can be further boosted by employing more optimal variable orderings. The proposed XMDD-based framework provides a scalable and resource-efficient solution for practical oracle synthesis in near-term and fault-tolerant quantum computing.

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Quantum oracles act as reversible black-box operators that encode classical Boolean functions into quantum states, enabling efficient function evaluation in quantum superposition. The resource efficiency of oracle implementation is critical to the performance of numerous quantum algorithms. Most state-of-the-art oracle synthesis approaches rely on compact Boolean function representations such as exclusive-sum-of-products (ESOP), yet still suffer from excessive TTT-count and CXCXCX-count for large-scale functions.
In this paper, we propose a novel compact representation named X-Map decision diagram (XMDD) for Boolean functions, which integrates local invertible maps and complement edges to achieve higher compression efficiency. Based on XMDD, we further develop an optimized quantum oracle synthesis algorithm. Extensive experimental results demonstrate that, for Boolean functions with more than seven input variables, our method outperforms the state-of-the-art ESOP-based approach and Qiskit in nearly all test cases, achieving simultaneous reduction in both TTT-count and CXCXCX-count without an obvious trade-off. Moreover, we show that the performance can be further boosted by employing more optimal variable orderings. The proposed XMDD-based framework provides a scalable and resource-efficient solution for practical oracle synthesis in near-term and fault-tolerant quantum computing.
A quantum oracle is a reversible quantum subroutine that implements a classical Boolean function within a quantum algorithm. Classical synthesis methods for quantum oracles typically involve two steps: reversible logic synthesis and quantum compilation. In the reversible logic synthesis phase, producing a compact reversible circuit is crucial to minimizing the quantum cost of the final quantum circuit. In this paper, we investigate Boolean functions that simultaneously exhibit two distinct XOR-based regularities: autosymmetry and D-reducibility. These regularities can be effectively utilized for efficient reversible circuit synthesis of Boolean functions. In particular, we propose and implement a new method for the quantum synthesis of autosymmetric and D-reducible Boolean functions. The experimental results demonstrate the effectiveness of our approach on the considered benchmarks, showing reductions in T-count and CNOT-count after Clifford+T quantum compilation.
Quantum oracle synthesis involves compiling arbitrary Boolean functions into quantum circuits using specific quantum gates supported by the target quantum computer. The Clifford+T gate library is particularly common in fault-tolerant quantum computing systems. Utilizing XOR-AND-inverter graphs (XAGs) as the logic representation for the target Boolean functions has received extensive attention due to the observed direct correlation between the number of AND nodes in an XAG and the T count and the helper qubit count of the quantum oracle optimally compiled from it. However, to be deployed onto fault-tolerant quantum hardware, quantum gates must be further re-expressed by logical quantum error correction (QEC) code operations, a process known as back-end compilation. This paper enhances the current XAG-based oracle synthesis techniques by establishing a link between the properties of XAGs and quality measures of back-end-compiled quantum oracles. This link unlocks more optimization opportunities---experimental results demonstrate average reductions of 4.49% in T count, 7.00% in logical time steps, and 14.89% in helper qubit count, respectively, on benchmarks optimized by the proposed back-end-aware XAG optimization approaches.
Quantum oracles are quantum circuits that implement classical Boolean functions, frequently used as independent black boxes in quantum algorithm design. The hierarchical reversible logic synthesis approach, a scalable method for large-scale oracle synthesis, relies on ancilla qubits to store intermediate results. The sequence in which these results are computed and uncomputed, governed by the reversible pebble game, greatly affects both qubit usage and circuit gate count. This paper introduces a novel divide-and-conquer pebbling strategy that reduces qubit count while maintaining a reasonable circuit gate count. Experimental results show up to a 90% reduction in qubits compared to the most commonly-used strategy, with an average increase of 3 to 4 times in quantum operations, all achieved within a reasonable runtime. In qubit-constrained tasks, our method achieves over 90% reduction in T-count. Additionally, we have incorporated our strategy into the hierarchical synthesis method and implemented it as a potential synthesizer for Qiskit, with experiments demonstrating that it outperforms the default synthesizer. The proposed approach shows promise in optimizing both qubit usage and circuit gate count for large-scale oracle synthesis problems.
The Boolean Satisfiability (SAT) problem is a canonical NP-complete problem and a natural candidate for quantum acceleration via search-based algorithms. In Grover-based quantum SAT solvers, the dominant computational cost stems from the construction of a reversible oracle that evaluates the Boolean formula, rendering the choice of SAT encoding crucial for overall quantum resource efficiency. Although SAT instances are conventionally expressed in Conjunctive Normal Form (CNF), such encodings typically translate into quantum circuits with significant qubit overhead and high non-Clifford gate complexity. In this work, we investigate an Exclusive-Sum-of-Products (ESOP)-based CNF (e-CNF) representation tailored for quantum SAT solving and analyze its impact on oracle construction. We derive tighter upper bounds on qubit requirements and Clifford+TTT gate counts for Grover-based SAT solvers when e-CNF encodings are employed in place of standard CNF. In addition, we propose a scalable transformation from Boolean formulas to e-CNF and present a systematic procedure for interpreting e-CNF representations as reversible quantum circuits suitable for oracle implementation. Experimental evaluation on representative SAT benchmarks demonstrates that the proposed e-CNF-based approach yields substantial and consistent reductions in quantum resources, including qubit count, T-gate complexity, and circuit depth, when compared to CNF-based oracle constructions. These results establish e-CNF as an effective quantum-aware SAT encoding that significantly improves the practicality of oracle-based quantum SAT solving.
Tensor networks serve as a powerful tool for efficiently representing and manipulating high-dimensional data in applications such as quantum physics, machine learning, and data compression. Tensor Decision Diagrams (TDDs) offer an efficient framework for tensor representation by leveraging decision diagram techniques. However, the current implementation of TDDs and other decision diagrams fail to exploit tensor isomorphisms, limiting their compression potential. This paper introduces Local Invertible Map Tensor Decision Diagrams (LimTDDs), an extension of TDDs that incorporates local invertible maps (LIMs) to achieve more compact representations. Unlike LIMDD, which uses Pauli operators for quantum states, LimTDD employs the XPXPXP-stabilizer group, enabling broader applicability across tensor-based tasks. We present efficient algorithms for normalization, slicing, addition, and contraction, critical for tensor network applications. Theoretical analysis demonstrates that LimTDDs achieve greater compactness than TDDs and, in best-case scenarios and for quantum state representations, offer exponential compression advantages over both TDDs and LIMDDs. Experimental results in quantum circuit tensor computation and simulation confirm LimTDD's superior efficiency. Open-source code is available at https://github.com/Veriqc/LimTDD.
The Exclusive-OR Sum-of-Product (ESOP) minimization problem has long been of interest to the research community because of its importance in classical logic design (including low-power design and design for test), reversible logic synthesis, and knowledge discovery, among other applications. However, no exact minimal minimization method has been presented for more than seven variables on arbitrary functions. This paper presents a novel quantum-classical hybrid algorithm for the exact minimal ESOP minimization of incompletely specified Boolean functions. This algorithm constructs oracles from sets of constraints and leverages the quantum speedup offered by Grover's algorithm to find solutions to these oracles, thereby improving over classical algorithms. Improved encoding of ESOP expressions results in substantially fewer decision variables compared to many existing algorithms for many classes of Boolean functions. This paper also extends the idea of exact minimal ESOP minimization to additionally minimize the cost of realizing an ESOP expression as a quantum circuit. To the extent of the authors' knowledge, such a method has never been published. This algorithm was tested on completely and incompletely specified Boolean functions via quantum simulation.