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Abstract We study quantum circuits constructed from $\sqrt{\mathrm{i}\text{SWAP}}$gates and, more generally, from the entangling gates that can be realized with theXX + YYinteraction alone. Such gates preserve the Hamming weight of states in the computational basis, which means they respect the global U(1) symmetry corresponding to rotations around thezaxis. Equivalently, assuming that the intrinsic Hamiltonian of each qubit in the system is the PauliZoperator, they conserve the total energy of the system. We develop efficient methods for synthesizing circuits realizing any desired energy-conserving unitary usingXX + YYinteraction with or without single-qubit rotations around thezaxis. Interestingly, implementing generic energy-conserving unitaries, such as CCZ and Fredkin gates, with two-local energy-conserving gates requires the use of ancilla qubits. When single-qubit rotations around thez-axis are permitted, our scheme requires only a single ancilla qubit, whereas with theXX+YYinteraction alone, it requires two ancilla qubits. In addition to exact realizations, we also consider approximate realizations and show how a general energy-conserving unitary can be synthesized using only a sequence of $\sqrt{\mathrm{i}\text{SWAP}}$gates and two ancillary qubits, with arbitrarily small error, which can be bounded via the Solovay–Kitaev theorem. Our methods are also applicable for synthesizing energy-conserving unitaries when, rather than theXX + YYinteraction, one has access to any other energy-conserving two-body interaction that is not diagonal in the computational basis, such as the Heisenberg exchange interaction. We briefly discuss the applications of these circuits in the context of quantum computing, quantum thermodynamics, and quantum clocks.