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QFlux: Quantum Circuit Implementations of Molecular Dynamics. Part III - State Initialization and Unitary Decomposition

Alexander V. Soudackov, Delmar G. A. Cabral, Brandon Allen, Xiaohan Dan, Nam P. Vu, Cameron Cianci, Rishab Dutta, Sabre Kais, Eitan Geva, Víctor S. Batista

Published Jan 30, 2026• 0 citations
Publisher

Abstract

This tutorial builds upon the foundations established in Part~II to present a unified, implementation-oriented overview of quantum state initialization and unitary decomposition for n -qubit systems, available in QFlux . We begin with the preparation of arbitrary quantum states and develop two complementary constructions: (i) a recursive multiplexor method (after Shende et al.) that disentangles qubits via multiplexed R y and R z rotations, and (ii) an algebraic scheme based on uniformly controlled rotations (UCRs) (after Möttönen et al.) that realizes the same mapping through analytically defined rotation networks with predictable gate counts. We then extend these state-preparation tools to generic unitary synthesis through Givens-rotation, column-by-column, and recursive cosine-sine (CSD) and quantum Shannon (QSD) decompositions, explicitly linking linear-algebraic factorizations to executable circuits while tracking CNOT complexity. Finally, we introduce the Walsh decomposition for diagonal unitaries and show how Gray-code ordering and local CNOT cancellations yield an O(2^n) entangling-gate cost, providing shallow, NISQ-friendly implementations. Taken together, these techniques form a pedagogical bridge from matrix analysis to hardware-efficient circuit constructions, offering clear design rules, closed-form parameters, and scalable synthesis pathways for simulation and experiment.

DOI
10.26434/chemrxiv.10001767/v1

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This tutorial builds upon the foundations established in Part~II to present a unified, implementation-oriented overview of quantum state initialization and unitary decomposition for n -qubit systems, available in QFlux . We begin with the preparation of arbitrary quantum states and develop two complementary constructions: (i) a recursive multiplexor method (after Shende et al.) that disentangles qubits via multiplexed R y and R z rotations, and (ii) an algebraic scheme based on uniformly controlled rotations (UCRs) (after Möttönen et al.) that realizes the same mapping through analytically defined rotation networks with predictable gate counts. We then extend these state-preparation tools to generic unitary synthesis through Givens-rotation, column-by-column, and recursive cosine-sine (CSD) and quantum Shannon (QSD) decompositions, explicitly linking linear-algebraic factorizations to executable circuits while tracking CNOT complexity. Finally, we introduce the Walsh decomposition for diagonal unitaries and show how Gray-code ordering and local CNOT cancellations yield an O(2^n) entangling-gate cost, providing shallow, NISQ-friendly implementations. Taken together, these techniques form a pedagogical bridge from matrix analysis to hardware-efficient circuit constructions, offering clear design rules, closed-form parameters, and scalable synthesis pathways for simulation and experiment.
Quantum computers offer a powerful platform for simulating real-time molecular and materials dynamics, where coherence, entanglement, and many-body correlations govern processes such as charge transport, tunneling, and spin exchange. Building on the foundations introduced in Part I, this tutorial presents practical algorithms for quantum dynamical simulation on qubit-based hardware using the open-source QFlux framework. We formulate molecular and spin Hamiltonians in the Pauli basis and map their propagators to quantum circuits using parity-based phase constructions and local basis rotations. Expectation values and state overlaps are obtained with the Hadamard test, or by direct measurement of Pauli strings, providing a unified protocol for extracting complex observables and variational quantities. We introduce the Quantum Split-Operator Fourier Transform ( Q-SOFT ), the quantum analog of the classical SOFT method for quantum dynamics simulations, which applies kinetic and potential phase operators in conjugate bases connected by the Quantum Fourier Transform to implement second-order Trotterized evolution. A simulation of proton transfer in an asymmetric double-well potential modeling a DNA base pair reproduces benchmark simulations. We also present a variational time-evolution scheme based on the McLachlan principle, implemented in QFlux’s VarQRTE driver. Applied to a two-spin Heisenberg model, it captures coherent spin exchange with shallow, noise-resilient circuits. Together, these methods establish a reproducible workflow—from Hamiltonian encoding to circuit execution and measurement—bridging classical and quantum simulation and providing a practical toolkit for real-time quantum dynamics on near-term hardware.
Quantum computers offer a powerful platform for simulating real-time molecular and materials dynamics, where coherence, entanglement, and many-body correlations govern processes such as charge transport, tunneling, and spin exchange. Building on the foundations introduced in Part I, this tutorial presents practical algorithms for quantum dynamical simulation on qubit-based hardware using the open-source QFlux framework. We formulate molecular and spin Hamiltonians in the Pauli basis and map their propagators to quantum circuits using parity-based phase constructions and local basis rotations. Expectation values and state overlaps are obtained with the Hadamard test, or by direct measurement of Pauli strings, providing a unified protocol for extracting complex observables and variational quantities. We introduce the Quantum Split-Operator Fourier Transform ( Q-SOFT ), the quantum analog of the classical SOFT method for quantum dynamics simulations, which applies kinetic and potential phase operators in conjugate bases connected by the Quantum Fourier Transform to implement second-order Trotterized evolution. A simulation of proton transfer in an asymmetric double-well potential modeling a DNA base pair reproduces benchmark simulations. We also present a variational time-evolution scheme based on the McLachlan principle, implemented in QFlux’s VarQRTE driver. Applied to a two-spin Heisenberg model, it captures coherent spin exchange with shallow, noise-resilient circuits. Together, these methods establish a reproducible workflow—from Hamiltonian encoding to circuit execution and measurement—bridging classical and quantum simulation and providing a practical toolkit for real-time quantum dynamics on near-term hardware.
We introduce QFlux, an open-source Python package for simulating quantum dynamics in chemical systems across multiple levels of theory using a unified classical and quantum computational framework. QFlux integrates deterministic wavefunction propagation, Lindblad master equations, generalized quantum master equations (GQMEs), operator-splitting techniques, and variational quantum algorithms within a modular architecture designed for systematic benchmarking and hybrid classical-quantum execution. The framework supports both closed-and open-system dynamics, including Markovian and non-Markovian regimes, and provides explicit mappings between classical propagators and quantum circuit constructions executable on NISQ hardware. Built atop established scientific and quantum software ecosystems, QFlux emphasizes reproducibility, interoperability, and controlled cross-validation between numerical approximations and hardware-oriented algorithms. Part I in this series of papers presents the theoretical foundations, software design principles, and methodological scope of QFlux, and positions it within the broader landscape of classical, tensor-network, and quantum-circuit-based simulation platforms. The series is designed to serve as a teaching reference for graduate students, a practical guide for researchers implementing custom quantum simulations, and a foundational reference for the broader QFlux ecosystem.
We introduce the flag decomposition as a central tool for unitary synthesis. It lets us carve out a diagonal unitary with 2n2^n2n degrees of freedom in such a way that the remaining flag circuit is parametrized by the optimal number of 4n−2n4^n-2^n4n−2n rotations. This enables us to produce parameter-optimal quantum circuits for generic unitaries and matrix product state preparation. Our approach improves upon the state of the art, both when compiling down to the {Clifford + Rot} gate set with what we call selective de-multiplexing, and when using phase gradient resource states together with quantum arithmetic to implement multiplexed rotations. All of our synthesis methods are efficiently implementable in terms of recursive Cartan decompositions realized by standard linear algebra routines, making them applicable to all practically relevant system sizes.
In quantum computation every unitary operation can be decomposed into quantum circuits---a series of single-qubit rotations and a single type entangling two-qubit gates, such as controlled-not(cnot) gates. Two measures are important when judging the complexity of the circuit: the total number of cnot gates needed to implement it and the depth of the circuit, measured by the minimal number of computation steps needed to perform it. Here we give an explicit and simple quantum circuit scheme for preparation of arbitrary quantum states, which can directly utilize any decomposition scheme for arbitrary full quantum gates, thus connecting the two problems. Our circuit reduces the depth of the best currently known circuit by a factor of 222. It also reduces the total number of cnot gates from 2n{2}^{n}2n to 23242n\frac{23}{24}{2}^{n}2423​2n in the leading order for even number of qubits. Specifically, the scheme allows us to decrease the upper bound from 111111 cnot gates to 999 and the depth from 111111 to 555 steps for four qubits. Our results are expected to help in designing and building small-scale quantum circuits using present technologies.
This paper proposes a new optimized quantum block-ZXZ decomposition method [7,8,10] that results in more optimal quantum circuits than the quantum Shannon decomposition (QSD)[27], which was introduced in 2006 by Shende et al. The decomposition is applied recursively to generic quantum gates, and can take advantage of existing and future small-circuit optimizations. Because our method uses only one-qubit gates and uniformly controlled rotation-Z gates, it can easily be adapted to use other types of multi-qubit gates. With the proposed decomposition, a general 3-qubit gate can be decomposed using 19 CNOT gates (rather than 20). For general nnn-qubit gates, the proposed decomposition generates circuits that have 22484n−322n+53\frac{22}{48}4^n - \frac{3}{2}2^n +\frac{5}{3}4822​4n−23​2n+35​ CNOT gates, which is less that the best known exact decomposition algorithm by (4n−2−1)/3(4^{n-2} -1)/3(4n−2−1)/3 CNOT gates.