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1. Long-lived topological time-crystalline order on a quantum processor

   https://doi.org/10.1038/s41467-024-53077-9  

   Xiang, Liang; Jiang, Wenjie; Bao, Zehang; Song, Zixuan; Xu, Shibo; Wang, Ke; Chen, Jiachen; Jin, Feitong; Zhu, Xuhao; Zhu, Zitian; et al (December 2024, Nature Communications)

   Abstract Topologically ordered phases of matter elude Landau’s symmetry-breaking theory, featuring a variety of intriguing properties such as long-range entanglement and intrinsic robustness against local perturbations. Their extension to periodically driven systems gives rise to exotic new phenomena that are forbidden in thermal equilibrium. Here, we report the observation of signatures of such a phenomenon—a prethermal topologically ordered time crystal—with programmable superconducting qubits arranged on a square lattice. By periodically driving the superconducting qubits with a surface code Hamiltonian, we observe discrete time-translation symmetry breaking dynamics that is only manifested in the subharmonic temporal response of nonlocal logical operators. We further connect the observed dynamics to the underlying topological order by measuring a nonzero topological entanglement entropy and studying its subsequent dynamics. Our results demonstrate the potential to explore exotic topologically ordered nonequilibrium phases of matter with noisy intermediate-scale quantum processors. 

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2. Long-lived topological time-crystalline order on a quantum processor

   Xiang, Liang; Jiang, Wenjie; Bao, Zehang; Song, Zixuan; Xu, Shibo; Wang, Ke; Chen, Jiachen; Jin, Feitong; Zhu, Xuhao; Zhu, Zitian; et al (January 2024, arXiv)

   Topologically ordered phases of matter elude Landau's symmetry-breaking theory, featuring a variety of intriguing properties such as long-range entanglement and intrinsic robustness against local perturbations. Their extension to periodically driven systems gives rise to exotic new phenomena that are forbidden in thermal equilibrium. Here, we report the observation of signatures of such a phenomenon -- a prethermal topologically ordered time crystal -- with programmable superconducting qubits arranged on a square lattice. By periodically driving the superconducting qubits with a surface-code Hamiltonian, we observe discrete time-translation symmetry breaking dynamics that is only manifested in the subharmonic temporal response of nonlocal logical operators. We further connect the observed dynamics to the underlying topological order by measuring a nonzero topological entanglement entropy and studying its subsequent dynamics. Our results demonstrate the potential to explore exotic topologically ordered nonequilibrium phases of matter with noisy intermediate-scale quantum processors. 

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3. A quantum processor based on coherent transport of entangled atom arrays

   https://doi.org/10.1038/s41586-022-04592-6  

   Bluvstein, Dolev; Levine, Harry; Semeghini, Giulia; Wang, Tout T.; Ebadi, Sepehr; Kalinowski, Marcin; Keesling, Alexander; Maskara, Nishad; Pichler, Hannes; Greiner, Markus; et al (April 2022, Nature)

   Abstract The ability to engineer parallel, programmable operations between desired qubits within a quantum processor is key for building scalable quantum information systems 1,2 . In most state-of-the-art approaches, qubits interact locally, constrained by the connectivity associated with their fixed spatial layout. Here we demonstrate a quantum processor with dynamic, non-local connectivity, in which entangled qubits are coherently transported in a highly parallel manner across two spatial dimensions, between layers of single- and two-qubit operations. Our approach makes use of neutral atom arrays trapped and transported by optical tweezers; hyperfine states are used for robust quantum information storage, and excitation into Rydberg states is used for entanglement generation 3–5 . We use this architecture to realize programmable generation of entangled graph states, such as cluster states and a seven-qubit Steane code state 6,7 . Furthermore, we shuttle entangled ancilla arrays to realize a surface code state with thirteen data and six ancillary qubits 8 and a toric code state on a torus with sixteen data and eight ancillary qubits 9 . Finally, we use this architecture to realize a hybrid analogue–digital evolution 2 and use it for measuring entanglement entropy in quantum simulations 10–12 , experimentally observing non-monotonic entanglement dynamics associated with quantum many-body scars 13,14 . Realizing a long-standing goal, these results provide a route towards scalable quantum processing and enable applications ranging from simulation to metrology. 

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4. The unitary dependence theory for characterizing quantum circuits and states

   https://doi.org/10.1038/s42005-023-01188-y  

   Hu, Zixuan; Kais, Sabre (April 2023, Communications Physics)

   Abstract Most existing quantum algorithms are discovered accidentally or adapted from classical algorithms, and there is the need for a systematic theory to understand and design quantum circuits. Here we develop a unitary dependence theory to characterize the behaviors of quantum circuits and states in terms of how quantum gates manipulate qubits and determine their measurement probabilities. Compared to the conventional entanglement description of quantum circuits and states, the unitary dependence picture offers more practical information on the measurement and manipulation of qubits, easier generalization to many-qubit systems, and better robustness upon partitioning of the system. The unitary dependence theory can be applied to systematically understand existing quantum circuits and design new quantum algorithms. 

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5. Shannon and von Neumann entropies of multi-qubit Schrödinger's cat states

   https://doi.org/10.1039/d1cp05255a  

   Jansen, Nathan D.; Loucks, Matthew; Gilbert, Scott; Fleming-Dittenber, Corbin; Egbert, Julia; Hunt, Katharine L. (March 2022, Physical Chemistry Chemical Physics)

   Using IBM's publicly accessible quantum computers, we have analyzed the entropies of Schrödinger's cat states, which have the form Ψ = (1/2) 1/2 [|0 0 0⋯0〉 + |1 1 1⋯1〉]. We have obtained the average Shannon entropy S So of the distribution over measurement outcomes from 75 runs of 8192 shots, for each of the numbers of entangled qubits, on each of the quantum computers tested. For the distribution over N fault-free measurements on pure cat states, S So would approach one as N → ∞, independent of the number of qubits; but we have found that S So varies nearly linearly with the number of qubits n . The slope of S So versus the number of qubits differs among computers with the same quantum volumes. We have developed a two-parameter model that reproduces the near-linear dependence of the entropy on the number of qubits, based on the probabilities of observing the output 0 when a qubit is set to |0〉 and 1 when it is set to |1〉. The slope increases as the error rate increases. The slope provides a sensitive measure of the accuracy of a quantum computer, so it serves as a quickly determinable index of performance. We have used tomographic methods with error mitigation as described in the qiskit documentation to find the density matrix ρ and evaluate the von Neumann entropies of the cat states. From the reduced density matrices for individual qubits, we have calculated the entanglement entropies. The reduced density matrices represent mixed states with approximately 50/50 probabilities for states |0〉 and |1〉. The entanglement entropies are very close to one. 

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