-level quantum system (qudit) under a time-dependent Hamiltonian. In the present article, we introduce a new general formalism describing the unitary evolution of a qudit ( d ≥ 2 ) in terms of the Bloch-like vector space and specify how, in a general case, this formalism is related to finding time-dependent parameters in the exponential representation of the evolution operator under an arbitrary time-dependent Hamiltonian. Applying this new general formalism to a qubit case ( d = 2 ) , we specify the unitary evolution of a qubit via the evolution of a unit vector in R 4 , and this allows us to derive the precise analytical expression of the qubit unitary evolution operator for a wide class of nonstationary Hamiltonians. This new analytical expression includes the qubit solutions known in the literature only as particular cases.-level quantum system (qudit) under a time-dependent Hamiltonian. In the present article, we introduce a new general formalism describing the unitary evolution of a qudit ( d ≥ 2 ) in terms of the Bloch-like vector space and specify how, in a general case, this formalism is related to finding time-dependent parameters in the exponential representation of the evolution operator under an arbitrary time-dependent Hamiltonian. Applying this new general formalism to a qubit case ( d = 2 ) , we specify the unitary evolution of a qubit via the evolution of a unit vector in R 4 , and this allows us to derive the precise analytical expression of the qubit unitary evolution operator for a wide class of nonstationary Hamiltonians. This new analytical expression includes the qubit solutions known in the literature only as particular cases.

arXiv:2004.09896EntropyMay 3, 20209 citations

Jordan-Wigner formalism for arbitrary 2-input 2-output matchgates and their classical simulation

Richard Jozsa, Akimasa Miyake, Sergii Strelchuk

In Valiant's matchgate theory, 2-input 2-output matchgates are $4\times 4$ matrices that satisfy ten so-called matchgate identities. We prove that the set of all such matchgates (including non-unitary and non-invertible ones) coincides with the topological closure of the set of all matrices obtained as exponentials of linear combinations of the 2-qubit Jordan-Wigner (JW) operators and their quadratic products, extending a previous result of Knill. In Valiant's theory, outputs of matchgate circuits can be classically computed in poly-time. Via the JW formalism, Terhal \& DiVincenzo and Knill established a relation of a unitary class of these circuits to the efficient simulation of non-interacting fermions. We describe how the JW formalism may be used to give an efficient simulation for all cases in Valiant's simulation theorem, which in particular includes the case of non-interacting fermions generalised to allow arbitrary 1-qubit gates on the first line at any stage in the circuit. Finally we give an exposition of how these simulation results can be alternatively understood from some basic Lie algebra theory, in terms of a formalism introduced by Somma et al.In Valiant's matchgate theory, 2-input 2-output matchgates are $4\times 4$ matrices that satisfy ten so-called matchgate identities. We prove that the set of all such matchgates (including non-unitary and non-invertible ones) coincides with the topological closure of the set of all matrices obtained as exponentials of linear combinations of the 2-qubit Jordan-Wigner (JW) operators and their quadratic products, extending a previous result of Knill. In Valiant's theory, outputs of matchgate circuits can be classically computed in poly-time. Via the JW formalism, Terhal \& DiVincenzo and Knill established a relation of a unitary class of these circuits to the efficient simulation of non-interacting fermions. We describe how the JW formalism may be used to give an efficient simulation for all cases in Valiant's simulation theorem, which in particular includes the case of non-interacting fermions generalised to allow arbitrary 1-qubit gates on the first line at any stage in the circuit. Finally we give an exposition of how these simulation results can be alternatively understood from some basic Lie algebra theory, in terms of a formalism introduced by Somma et al.

arXiv:1311.3046Quantum Information and ComputationMay 1, 201510 citations

Efficient Fourier-Based Linear Combination of Unitaries and Applications in Quantum Optimization

Almudena Carrera Vazquez, Daniel J. Egger, Stefan Woerner

We investigate ancilla-free linear combination of unitaries (LCU) as a framework for approximating complex quantum circuits. This is particularly effective for quantum optimization algorithms, where candidate solutions can be evaluated classically and the task is to sample high-quality bitstrings rather than reproduce the full output distribution. We show that Fourier-based LCU constructions efficiently decompose broad classes of diagonal and non-diagonal unitaries, replacing highly connected qubit interactions with single-qubit gate layers or significantly simpler structures at the cost of a polynomial sampling overhead. Applied to algorithms such as QAOA, this yields efficient, hardware-friendly decompositions of, for instance, cardinality-constraint penalties and the fully connected XY-mixer, while maintaining rigorous performance guarantees compared to fully coherent implementations. Furthermore, we establish a formal connection between Fourier-based quantum penalties and Lagrangian relaxation, offering a unified perspective on constraint handling. We validate our approach using exact statevector simulations of 12-qubit circuits and large-scale experiments on 106 superconducting qubits. Our results illustrate how approximate sampling via an LCU systematically trades circuit complexity for sampling overhead, extending the practical reach of near-term quantum optimization.We investigate ancilla-free linear combination of unitaries (LCU) as a framework for approximating complex quantum circuits. This is particularly effective for quantum optimization algorithms, where candidate solutions can be evaluated classically and the task is to sample high-quality bitstrings rather than reproduce the full output distribution. We show that Fourier-based LCU constructions efficiently decompose broad classes of diagonal and non-diagonal unitaries, replacing highly connected qubit interactions with single-qubit gate layers or significantly simpler structures at the cost of a polynomial sampling overhead. Applied to algorithms such as QAOA, this yields efficient, hardware-friendly decompositions of, for instance, cardinality-constraint penalties and the fully connected XY-mixer, while maintaining rigorous performance guarantees compared to fully coherent implementations. Furthermore, we establish a formal connection between Fourier-based quantum penalties and Lagrangian relaxation, offering a unified perspective on constraint handling. We validate our approach using exact statevector simulations of 12-qubit circuits and large-scale experiments on 106 superconducting qubits. Our results illustrate how approximate sampling via an LCU systematically trades circuit complexity for sampling overhead, extending the practical reach of near-term quantum optimization.

quant-pharXiv:2605.18985May 20, 20260 citations

Constructing Quantum Circuits for Unitary Operations

Joanna Wiśniewska

Jan 1, 20110 citations

Models for Quantum Computation

Peter P. Rohde

Cambridge University Press eBooksSep 9, 20210 citations

In Valiant's matchgate theory, 2-input 2-output matchgates are

4\times 4

matrices that satisfy ten so-called matchgate identities. We prove that the set of all such matchgates (including non-unitary and non-invertible ones) coincides with the topological closure of the set of all matrices obtained as exponentials of linear combinations of the 2-qubit Jordan-Wigner (JW) operators and their quadratic products, extending a previous result of Knill. In Valiant's theory, outputs of matchgate circuits can be classically computed in poly-time. Via the JW formalism, Terhal \& DiVincenzo and Knill established a relation of a unitary class of these circuits to the efficient simulation of non-interacting fermions. We describe how the JW formalism may be used to give an efficient simulation for all cases in Valiant's simulation theorem, which in particular includes the case of non-interacting fermions generalised to allow arbitrary 1-qubit gates on the first line at any stage in the circuit. Finally we give an exposition of how these simulation results can be alternatively understood from some basic Lie algebra theory, in terms of a formalism introduced by Somma et al.