Motivated by recent observations of exotic hadrons in the near-threshold energy region, the internal structure of near-threshold states has been intensively studied. In various works, a qualitative measure, called the compositeness, has been used to characterize the structure of near-threshold states. The compositeness represents the fraction of the hadronic molecular component in the wavefunction [1]. It is shown that in the limit where the binding energy goes to zero, the compositeness becomes unity as a consequence of the low-energy universality [2]. This indicates that the states exactly at the threshold commonly have a purely molecular structure, independently of the details of the system. Based on this fact, near-threshold states with small but finite eigenenergies are naively expected to be molecular dominant states whose compositeness is close to unity (the threshold energy rule) [3]. However, this rule is empirical, and its theoretical foundation has not yet been established. To understand the nature of near-threshold exotic hadrons, we aim to provide the theoretical foundation of the threshold rule by analyzing the structure of near-threshold states in light of the low-energy universality.

We first focus on near-threshold bound states slightly below the threshold with small and negative eigenenergies. Using an effective field theory model, the model dependence of the compositeness of bound states is examined. We show that the shallow bound states are usually composite dominant without significant fine tuning, which is explained by the emergence of the low-energy universality [4]. This provides the theoretical foundation of the threshold energy rule for bound states. We then consider the structure of the near-threshold resonances, which exist above the threshold with small and positive excitation energy. Using the effective range expansion, we calculate the compositeness of near-threshold resonances, and find that the near-threshold resonances are non-composite dominant [5]. This shows that the structure of near-threshold resonances is completely different from that of bound states, which is another aspect of near-threshold phenomena that deviates from the expectations based on the threshold energy rule. Finally, we also discuss how the Coulomb interaction affects the structure of the near-threshold states.

[1] T. Kinugawa, T. Hyodo, Eur. Phys. J. A 61 , 154 (2025).

[2] T. Hyodo, Phys. Rev. C 90, 055208 (2014).

[3] K. Ikeda, and N. Takigawa, and H. Horiuchi, Prog. Theor. Phys. Suppl. E68, 464-475 (1968).

[4] T. Kinugawa and T. Hyodo, Phys. Rev. C 109 , 045205 (2024).

[5] T. Kinugawa and T. Hyodo, arXiv:2403.12635 [hep-ph].

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Matchgate unitaries are fundamental to quantum computation due to their relation to non-interacting fermions and their utility in benchmarking quantum hardware. In fault-tolerant settings, general unitaries must be decomposed into discrete sets compatible with error-correction primitives, typically the Clifford+T gate set. Here, we propose an alternative paradigm: compiling matchgates using only matchgates. By leveraging the correspondence between n-qubit matchgate circuits and the standard representation of SO(2n), we reduce the compilation task from 2^n \times 2^n unitaries to 2n \times 2n matrices, achieving an exponential reduction in dimensionality. Our first result identifies a discrete gate set that densely generates the matchgate group. We then address approximate synthesis, rigorously showing that approximation errors in the SO(2n) representation propagate only linearly into the full unitary representation. Finally, we characterize exact synthesis, demonstrating that matchgates meeting specific algebraic conditions can be exactly synthesized without ancilla qubits. This allows us to frame optimal synthesis as a Boolean satisfiability (SAT) problem, enabling the construction of circuits with provable guarantees on depth.

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