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arXiv:2505.09320 · IEEE Transactions on Quantum Engineering

Multilevel Gate Set Optimization of Quantum Circuits for Partial Differential Equations

Tamiya Onodera, Yuki Sato, Toshinari Itoko, Naoki Yamamoto

Published Jan 1, 2026• 0 citations
arXivHTMLPDF

Abstract

In this paper, we explore multilevel circuit optimization (MLCO), where we deploy multiple gate sets and progressively lower the source circuit through the gate sets to the target circuit. At each level, we first perform an appropriate set of circuit simplifications and then lower the simplified circuit into the next level, decomposing the gates not supported there. We demonstrate its effectiveness, using as a case study the source circuit for Hamiltonian simulation to solve a partial differential equation, which is densely populated with multi-controlled gates and is transformed by the state-of-the-art circuit compiler to the target circuit with the quadratic number of CX gates in the number of qubits. MLCO makes visible higher-level circuit structures, providing us with insights about how to simplify the circuits and how to decompose the gates. By putting the right circuit structure in place and selecting the right decomposition algorithm, we could cause massive cancellation of entangling gates, thereby having achieved the quadratic reduction in the number of CX gates.

DOI
10.1109/tqe.2026.3706630

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In this paper, we explore multilevel circuit optimization (MLCO), where we deploy multiple gate sets and progressively lower the source circuit through the gate sets to the target circuit. At each level, we first perform an appropriate set of circuit simplifications and then lower the simplified circuit into the next level, decomposing the gates not supported there. We demonstrate its effectiveness, using as a case study the source circuit for Hamiltonian simulation to solve a partial differential equation, which is densely populated with multi-controlled gates and is transformed by the state-of-the-art circuit compiler to the target circuit with the quadratic number of CX gates in the number of qubits. MLCO makes visible higher-level circuit structures, providing us with insights about how to simplify the circuits and how to decompose the gates. By putting the right circuit structure in place and selecting the right decomposition algorithm, we could cause massive cancellation of entangling gates, thereby having achieved the quadratic reduction in the number of CX gates.
Quantum circuits are time-dependent diagrams describing the process of quantum computation. Usually, a quantum algorithm must be mapped into a quantum circuit. Optimal synthesis of quantum circuits is intractable, and heuristic methods must be employed. With the use of heuristics, the optimality of circuits is no longer guaranteed. In this paper, we consider a local optimization technique based on templates to simplify and reduce the depth of nonoptimal quantum circuits. We present and analyze templates in the general case and provide particular details for the circuits composed of NOT, CNOT, and controlled-sqrt-of-NOT gates. We apply templates to optimize various common circuits implementing multiple control Toffoli gates and quantum Boolean arithmetic circuits. We also show how templates can be used to compact the number of levels of a quantum circuit. The runtime of our implementation is small, whereas the reduction in the number of quantum gates and number of levels is significant.
This paper presents novel methods for optimizing multi-controlled quantum gates, which naturally arise in high-level quantum programming. Our primary approach involves rewriting U ( 2 ) gates as S U ( 2 ) gates, utilizing one auxiliary qubit for phase correction. This reduces the number of CNOT gates required to decompose any multi-controlled quantum gate from O ( n 2 ) to at most 32 n . Additionally, we can reduce the number of CNOTs for multi-controlled Pauli gates from 16 n to 12 n and propose an optimization to reduce the number of controlled gates in high-level quantum programming. We have implemented these optimizations in the Ket quantum programming platform and demonstrated significant reductions in the number of gates. For instance, for a Grover's algorithm layer with 114 qubits, we achieved a reduction in the number of CNOTs from 101,252 to 2,684. This reduction in the number of gates significantly impacts the execution time of quantum algorithms, thereby enhancing the feasibility of executing them on NISQ computers.
Abstract With the rapid development of quantum computing, quantum computing devices have entered the Noisy Intermediate-Scale Quantum(NISQ)era. However, current NISQ quantum computing devices still face the challenge of physical limitations. To enable quantum algorithms to be directly executed on quantum computing devices, high-level quantum gates (such as MCT gates) in quantum circuits need to be decomposed and mapped onto quantum computing devices. This paper proposes a quantum device-aware method for decomposing high-level quantum gates. This method selects an optimal sub-topology of the quantum device as the target based on the circuit scale and the goal of reducing error rates. According to the proposed MCT gate decomposition rules, an associated gate-pairs generation algorithm is designed to facilitate subsequent decomposition. The control bits of the MCT gates are partitioned based on the associated gate-pair to better match the selected sub-topology. After eliminating redundant gates through the inverse decomposition of associated gate-pairs, the mapping is performed. Experimental results show that the proposed decomposition method reduces the number of basic quantum gates in quantum circuits, with an average optimization rate of 14.6%. For benchmark circuits decomposed using the method, applying the routing algorithm for mapping requires fewer additional gates, with an average optimization rate of 20.6%. The fidelity of the circuits improved by an average of 14.1% after mapping.
We provide a method for compiling approximate multicontrolled single qubit gates into quantum circuits. Without ancilla qubits, the total number of elementary gates to decompose an n-qubit multicontrolled gate is proportional to 32n elementary operations. The proposed decomposition depends on an optimization technique that minimizes the CNOT gate count for multitarget and multicontrolled CNOT and SU(2) gates. We also provide an approximate decomposition with ancilla qubits with lower-circuit complexity. Computational experiments show the reduction of CNOT gates when multicontrolled U(2) gates are applied. As multicontrolled single-qubit gates serve as fundamental components of quantum algorithms, the proposed decomposition offers a comprehensive solution that can significantly decrease the count of elementary operations employed in quantum computing applications.
We explore the utilization of higher-order discretization techniques in optimizing the gate count needed for quantum computer based solutions of partial differential equations. To accomplish this, we present an efficient approach for decomposing ddd-band diagonal matrices into Pauli strings that are grouped into mutually commuting sets. Using numerical simulations of the one-dimensional wave equation, we show that higher-order methods can reduce the number of qubits necessary for discretization, similar to the classical case, although they do not decrease the number of Trotter steps needed to preserve solution accuracy. This result has important consequences for the practical application of quantum algorithms based on Hamiltonian evolution.
Quantum circuit simulation is paramount to the verification and optimization of quantum algorithms, and considerable research efforts have been made towards efficient simulators. While circuits often contain high-level gates such as oracles and multi-controlled X ( C k X ) gates, existing simulation methods require compilation to a low-level gate-set before simulation. This, however, increases circuit size and incurs a considerable (typically exponential) overhead, even when the number of high-level gates is small. Here we present a gadget-based simulator which simulates high-level gates directly, thereby allowing to avoid or reduce the blowup of compilation. Our simulator uses a stabilizer decomposition of the magic state of non-stabilizer gates, with improvements in the rank of the magic state directly improving performance. We then proceed to establish a small stabilizer rank for a range of high-level gates that are common in various quantum algorithms. Using these bounds in our simulator, we improve both the theoretical complexity of simulating circuits containing such gates, and the practical running time compared to standard simulators found in IBM's Qiskit Aer library. We also derive exponential lower-bounds for the stabilizer rank of some gates under common complexity-theoretic hypotheses. In certain cases, our lower-bounds are asymptotically tight on the exponent.