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IEEE Transactions on Computer-Aided Design of Integrated Circuits and Systems

A Divide-And-Conquer Pebbling Strategy for Oracle Synthesis in Quantum Computing

Kezhen Zhang, Riling Li, Mingsheng Ying

Published May 15, 2025• 0 citations
Publisher

Abstract

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.

DOI
10.1109/tcad.2025.3570608

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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.
Hierarchical reversible logic synthesis can find quantum circuits for large combinational functions. The price for a better scalability compared to functional synthesis approaches is the requirement for many additional qubits to store temporary results of the hierarchical input representation. However, implementing a quantum circuit with large number of qubits is a major hurdle. In this paper, we demonstrate and establish how reversible pebble games can be used to reduce the number of stored temporary results, thereby reducing the qubit count. Our proposed algorithm can be constrained with number of qubits, which is aimed to meet. Experimental studies show that the qubit count can be significantly reduced (by up to 63.2%) compared to the state-of-the-art algorithms, at the cost of additional gate count.
Quantum oracles are fundamental building blocks of many quantum algorithms, and their resource consumption directly affects performance, yet structured description and complexity analysis for their composition are still lacking. In this paper, we introduce the Framework for Oracle Recursion Modeling (FORM), a unified formal abstraction of multi-function composition in quantum oracles: it provides a structured description of the composition layer, makes its gate complexity exactly computable, and turns oracle design into an optimizable tree-construction problem. Based on this model, we propose the ShallowGrow algorithm, which constructs an oracle structure under a given ancilla budget and provably minimizes the number of function evaluations. On Boolean quadratic equation systems, ShallowGrow reduces Qiskit-measured circuit depth by 54.1% on average relative to the state-of-the-art W-cycle construction, with consistent reductions on the EPFL and ISCAS85 combinational logic networks under scarce ancilla budgets. Furthermore, pebbling-based syntheses trade space against time within the logic network of a function; ShallowGrow extends this trade-off across functions, and integrated with their published circuits it reduces the ancillary qubits of a complete oracle from one per constraint function to logarithmically many. With half as many ancillas as constraint functions, the T-count falls by a factor of 7.9 to 32 relative to the W-cycle-based construction.
Quantum circuit synthesis translates a classical Boolean function into an equivalent quantum circuit. The syn-thesis is solvable by playing the reversible pebble game on the logic network of the function. However, optimal solutions are im-practical for large networks, affecting the number of qubits and gates of the final quantum circuit. In this work, we improve the solution of the reversible pebble game by leveraging dominance relations on the directed acyclic graph of the classical function, reducing qubits needed for syntheses. The proposed algorithm exposes a tunable tradeoff between the available number of qubits and the circuit size expressed as T-count and T-depth. We experimentally validate our methodology on cryptographic and arithmetic benchmarks, reporting reductions between 37% and 83 % in qubit number with respect to Bennet syntheses algorithms and complete the majority of our syntheses in less than a second, improving on the current running times of state of the art approaches.
One of the main concerns in the era of noisy intermediate-scale quantum (NISQ) computing and fault-tolerant quantum computing is the optimization of circuit implementation for quantum oracles, particularly with limited resources. Synthesizing a satisfiability (SAT) oracle, a crucial component in solving SAT problems, presents a significant challenge. The current state-of-the-art implementation of anmmm-clause SAT-oracle necessitates2m−12m-12m−1ancillary qubits and a linear number of elementary gates. We develop two efficient and ancilla-adjustable synthesis algorithms to reduce the overall quantum resource usage. Our first quantum oracle algorithm achieves quadratic optimization in the number of ancillary qubits with merely eight times increased circuit size. We also show that using only three ancillary qubits with quadratic circuit size expansion is enough. Our second algorithm optimizes the circuit depth of the SAT oracle toO~(log⁡m)\tilde {O}(\log m)O~(logm)usingmmmancillary qubits. By running our algorithms on classical intractable SAT instances featured in SAT competitions, the experiment results show that our required quantum resources align well with our theoretical analysis. Our algorithms highlight the scalability of SAT-oracle-based algorithms in near-term quantum devices, such as Grover’s algorithm.
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.