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Abstract Quantum information science is a rapidly growing interdisciplinary field that is attracting the attention of academics and industry experts alike. It requires talent from a wide variety of traditional fields, including physics, engineering, chemistry, and computer science, to name a few. To prepare students for such opportunities, it is important to give them a strong foundation in the basics of quantum information science, in which quantum computing plays a central role. In this study, we discuss the development, validation, and evaluation of a tutorial on the Bloch sphere, a useful visual tool for developing intuition about single quantum bits (qubits), which are the basic building block of any quantum computer. Students’ understanding was evaluated after they received traditional lecture-based instruction on the requisite topics, and again after engaging with the tutorial. We observe, analyze, and discuss their improvement in performance on concepts covered in the tutorial. 

Abstract One of the primary challenges in realizing large-scale quantum processors is the realization of qubit couplings that balance interaction strength, connectivity, and mode confinement. Moreover, it is very desirable for the device elements to be detachable, allowing components to be built, tested, and replaced independently. In this work, we present a microwave quantum state router, centered on parametrically driven, Josephson-junction based three-wave mixing, that realizes all-to-all couplings among four detachable quantum modules. We demonstrate coherent exchange among all four communication modes, with an average full-iSWAP time of 764 ns and average inferred inter-module exchange fidelity of 0.969, limited by mode coherence. We also demonstrate photon transfer and pairwise entanglement between module qubits, and parallel operation of simultaneousiSWAP exchange across the router. Our router-module architecture serves as a prototype of modular quantum computer that has great potential for enabling flexible, demountable, large-scale quantum networks of superconducting qubits and cavities. 

Topological phases of matter offer a promising platform for quantum computation and quantum error cor- rection. Nevertheless, unlike its counterpart in pure states, descriptions of topological order in mixed states remain relatively underexplored. Our work gives two definitions for replica topological order in mixed states, which involve n copies of density matrices of the mixed state. Our framework categorizes topological orders in mixed states as either quantum, classical, or trivial, depending on the type of information that can be encoded. For the case of the toric code model in the presence of decoherence, we associate for each phase a quantum channel and describes the structure of the code space. We show that in the quantum-topological phase, there exists a postselection-based error correction protocol that recovers the quantum information, while in the classical-topological phase, the quantum information has decohere and cannot be fully recovered. We accomplish this by describing the mixed state as a projected entangled pairs state (PEPS) and identifying the symmetry-protected topological order of its boundary state to the bulk topology. Using this formalism, we enumerate all the possible mixed state phases which result from applying a local decoherence channel to the toric code. In addition to the classical-topological phases that have been previously reported on, we also find mixed states exhibiting chiral topological order. We discuss the extent that our findings can be extrapolated to n → 1 limit.