Shallow Quantum Circuit for Generating Extremely Low-Entangled Approximate State Designs

Random quantum states have various applications in quantum information science. We discover a new ensemble of quantum states that serve as an ε-approximate state t-design while possessing extremely low entanglement, magic, and coherence. These resources can reach their theoretical lower bounds, Ω(log(t/ε)), which are also proven in this Letter. This implies that, for fixed t and ε, entanglement, magic, and coherence do not scale with the system size, i.e., O(1) with respect to the total number of qubits n. Moreover, we explicitly construct an ancilla-free shallow quantum circuit for generating such states by transforming k-qubit approximate state designs into n-qubit ones without increasing the support size. The depth of such a quantum circuit, O(t[logt]^{3}log n log[1/ε]), is the most efficient among existing algorithms without ancilla qubits. A class of quantum circuits proposed in our Letter offers reduced cost for classical simulation of random quantum states, leading to potential applications in quantum information processing. As a concrete example, we propose classical shadow tomography using an estimator with superpositions between only two states, from which almost all quantum states can be efficiently certified by requiring only O(1) measurements and classical postprocessing time.