Shallow quantum circuit for generating O(1)-entanged approximate state designs

Abstract: Random quantum states have various applications in quantum information science, including quantum cryptography, quantum simulation, and benchmarking quantum devices. In this work, we discover a new ensemble of quantum states that serve as an $\epsilon$-approximate state $t$-design while possessing extremely low entanglement, magic, and coherence. We show that those resources such quantum states can reach their theoretical lower bounds, $\Omega\left(\log (t/\epsilon)\right)$, which are also proven in this work. This implies that for fixed $t$ and $\epsilon$, those resources do not scale with the system size, i.e., $O(1)$ with respect to the total number of qubits $n$ in the system. Moreover, we explicitly construct an ancilla-free shallow quantum circuit for generating such states. To this end, we develop an algorithm that transforms $k$-qubit approximate state designs into $n$-qubit ones through a sequence of multi-controlled gates, without increasing the support size. The depth of such a quantum circuit is $O\left(t [\log t]^3 \log n \log(1/\epsilon)\right)$, which is the most efficient among existing algorithms without ancilla qubits. A class of shallow quantum circuits proposed in our work offers reduced cost for classical simulation of random quantum states, leading to potential applications in various quantum information processing tasks. As a concrete example for demonstrating utility of our algorithm, we propose classical shadow tomography using an $O(1)$-entangled estimator, which can achieve shorter runtime compared to conventional schemes.