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arXiv:2606.06070

Efficient Quantum Circuit Construction of Controlled Time-Evolution for Arbitrary Pauli-Sum Hamiltonians

Shintaro Fujiwara, Naoki Yamamoto, Naoki Ishikawa

quant-phSubmitted Jun 4, 2026• Announced Jun 5, 2026• 0 citations
arXivHTMLPDF

Abstract

Controlled time-evolution circuits select forward or backward Hamiltonian time evolution according to the state of an ancilla qubit. They are fundamental building blocks in quantum eigenvalue transformation of unitaries, Hamiltonian filtering, and related quantum algorithms. A direct realization adds ancilla control to the elementary gates of the time-evolution circuit and therefore increases the two-qubit gate count, compiled T depth and CX depth. We develop an efficient recursive algorithm that, for an arbitrary Pauli-sum Hamiltonian, partitions the input Pauli terms into groups and assigns to each group a sign-flip Pauli string that anti-commutes with the in-group terms, thereby removing ancilla control from the grouped time-evolution blocks. Numerical benchmarks on random Hamiltonians and structured spin Hamiltonians show reductions in compiled T depth and compiled CX depth. For a Kagome Hamiltonian with 24 spins under full connectivity, the proposed construction reduces the compiled T depth by 85.2% and the compiled CX depth by 68.9%, compared with a conventional implementation that decomposes the Hamiltonian into individual Pauli terms and implements the controlled time evolution of each term by directly adding the ancilla qubit to the corresponding Pauli-rotation circuit.

Comments
18 pages, 9 figures

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Controlled time-evolution circuits select forward or backward Hamiltonian time evolution according to the state of an ancilla qubit. They are fundamental building blocks in quantum eigenvalue transformation of unitaries, Hamiltonian filtering, and related quantum algorithms. A direct realization adds ancilla control to the elementary gates of the time-evolution circuit and therefore increases the two-qubit gate count, compiled T depth and CX depth. We develop an efficient recursive algorithm that, for an arbitrary Pauli-sum Hamiltonian, partitions the input Pauli terms into groups and assigns to each group a sign-flip Pauli string that anti-commutes with the in-group terms, thereby removing ancilla control from the grouped time-evolution blocks. Numerical benchmarks on random Hamiltonians and structured spin Hamiltonians show reductions in compiled T depth and compiled CX depth. For a Kagome Hamiltonian with 24 spins under full connectivity, the proposed construction reduces the compiled T depth by 85.2% and the compiled CX depth by 68.9%, compared with a conventional implementation that decomposes the Hamiltonian into individual Pauli terms and implements the controlled time evolution of each term by directly adding the ancilla qubit to the corresponding Pauli-rotation circuit.
Many applications of practical interest rely on time evolution of Hamiltonians that are given by a sum of Pauli operators. Quantum circuits for exact time evolution of single Pauli operators are well known, and can be extended trivially to sums of commuting Paulis by concatenating the circuits of individual terms. In this paper we reduce the circuit complexity of Hamiltonian simulation by partitioning the Pauli operators into mutually commuting clusters and exponentiating the elements within each cluster after applying simultaneous diagonalization. We provide a practical algorithm for partitioning sets of Paulis into commuting subsets, and show that the proposed approach can help to significantly reduce both the number of C N O T operations and circuit depth for Hamiltonians arising in quantum chemistry. The algorithms for simultaneous diagonalization are also applicable in the context of stabilizer states; in particular we provide novel four- and five-stage representations, each containing only a single stage of conditional gates.
Electronic Hamiltonian simulation is commonly formulated by mapping fermionic operators to qubit operators and subsequently expanding the resulting ladder-operator products into Pauli strings. While general, this procedure obscures the higher-level fermionic structure and can hide opportunities for circuit optimization. Building on ladder-string-pair (Lasp) diagonalization originally developed for Hamiltonian simulation of partial differential equations, we construct time-evolution circuits for the second-quantized electronic Hamiltonian without performing a Pauli expansion. For the most general case of complex-valued coefficients, we present a time-evolution circuit for a two-body fermionic Lasp operator that corresponds to 16 Pauli strings in the Pauli-expansion approach but does not suffer from Trotter error at this stage. Expanding the optimization scope from a single operator to a triad of three fermionic Lasp operators sharing the same four spin-orbital indices enables systematic cancellation of CX gates, reducing the CX-gate count from 36 to 12 in the example considered, still without introducing Trotter error at this stage. For nnn spin orbitals, further expanding the optimization scope to a sequence of O(n)O(n)O(n) suitably ordered triads enables a cascade of CX-gate reductions across triad boundaries, reducing the CX-gate count from O(n2)O(n^2)O(n2) to O(n)O(n)O(n). The Lasp-based approach also naturally accommodates controlled time evolution and yields further optimizations for real-valued Hamiltonians. These results demonstrate that the Lasp-based approach enables more efficient time-evolution circuits by preserving high-level circuit structures and thereby expanding the scope of optimization, providing a systematic route toward more efficient electronic Hamiltonian simulation.
We present an implementation of multicontrolled () quantum gates that provides significant reductions of cost compared to state-of-the-art methods. The operator applied on the target qubit is a unitary, special unitary, or the Pauli X operator (multicontrolled Toffoli), and requires one clean ancilla, no ancilla, or one dirty ancilla, respectively. We generalize our methods for any number of target qubits, and provide further cost reductions if additional ancilla qubits are available. For each type of gate, we provide implementations for unrestricted (all-to-all) connectivity and for linear-nearest-neighbor (LNN) connectivity. All of the methods use a linear cost of gates from the Clifford+ T (fault-tolerant) set. In the context of LNN architecture, the cost and depth of our circuits scale linearly irrespective of the position of the qubits on which the gate is applied. Our methods directly improve the compilation process of many quantum algorithms, providing optimized circuits. Given the scale of our improvements, for example, the quadratic-to-linear controlled- count for LNN, they will naturally result in a large reduction of errors.
In quantum computing, the efficient optimization of Pauli string decompositions is a crucial aspect for the compilation of quantum circuits for many applications, such as chemistry simulations and quantum machine learning. In this paper, we propose a novel algorithm for the synthesis of trotterized time-evolution operators that results in circuits with significantly fewer gates than previous solutions. Our synthesis procedure takes the qubit connectivity of a target quantum computer into account. As a result, the generated quantum circuit does not require routing, and no additional CNOT gates are needed to run the resulting circuit on a target device. We compare our algorithm against Paulihedral and TKET, and show a significant improvement for randomized circuits and different molecular ansatzes. We also investigate the Trotter error introduced by our ordering of the terms in the Hamiltonian versus default ordering and the ordering from the baseline methods and conclude that our method on average does not increase the Trotter error.
Improving quantum algorithms run-time performance involves several strategies such as reducing the quantum gate counts, decreasing the number of measurements, advancement in QPU technology for faster gate operations, or optimizing the classical processing. This work focuses on the latter, specifically reducing classical processing and compilation time via hardware-assisted parameterized circuit execution (PCE) for computing dynamical properties of quantum systems. PCE was previously validated for QCVV protocols, which leverages structural circuit equivalencies. We demonstrate the applicability of this approach to computing dynamical properties of quantum many-body systems using structurally equivalent time evolution circuits, specifically calculating correlation functions of spin models using constant-depth circuits generated via Cartan decomposition. Implementing this for spin-spin correlation functions in Transverse field XY (up to 6-sites) and Heisenberg spin models (up to 3-sites), we observed a run-time reduction of up to50%\mathbf{5 0 \%}50%compared to standard compilation methods. This highlights the adaptability of time-evolution circuit with hardware-assisted PCE to potentially mitigate the classical bottlenecks in near-term quantum algorithms.
A variety of quantum algorithms employ Pauli operators as a convenient basis for studying the spectrum or evolution of Hamiltonians or measuring multibody observables. One strategy to reduce circuit depth in such algorithms involves simultaneous diagonalization of Pauli operators generating unitary evolution operators or observables of interest. We propose an algorithm yielding quantum circuits with depths O(nlogr)O(nlogr)O(nlogr) diagonalizing nnn-qubit operators generated by rrr Pauli operators. Moreover, as our algorithm iteratively diagonalizes all operators on at least one qubit per step, it is well suited to maintain low circuit depth even on hardware with limited qubit connectivity. We observe that our algorithm performs favorably in producing quantum circuits diagonalizing randomly generated Hamiltonians as well as molecular Hamiltonians with short depths and low two-qubit gate counts.