Abstract: A fault-tolerant quantum computing scheme implements quantum computing based on three-dimensional cluster states using just a small number of physical components. The technique may be implemented using a single actively controlled qubit [100], e.g., a quantum emitter, and a pair of delay line loops [106, 108], which can be physically realized using existing technologies in quantum photonic and phononic systems. A three-dimensional cluster state is prepared by using the actively con-trolled qubit to generate data qubits that propagate through the two delay line loops, interact with the actively controlled qubit, and then are measured by a detector [110].

Abstract: Methods, systems, and apparatus for enhancing classical shadows using matchgate quantum circuits. In one aspect, a method for computing a classical shadow of an n-qubit quantum state includes repeatedly sampling, by a classical computer, a unitary operator from an ensemble of random unitaries, wherein the ensemble of random unitaries comprises a generalized matchgate group; for each sampled unitary operator: applying, by a quantum computer, a quantum circuit to the n-qubit quantum state to obtain an evolved quantum state, wherein the quantum circuit implements the sampled unitary operator, measuring, by the quantum computer, the evolved quantum state to obtain a respective bit string, and storing, by the classical computer, a record of the respective bit string and the sampled unitary operator; and providing the records as a classical shadow of the quantum state.