Wilczek, F. Quantum time crystals. Phys. Rev. Lett. 109, 160401 (2012).
Else, D. V., Bauer, B. & Nayak, C. Floquet time crystals. Phys. Rev. Lett. 117, 090402 (2016).
Yao, N. Y., Potter, A. C., Potirniche, I.-D. & Vishwanath, A. Discrete time crystals: rigidity, criticality, and realizations. Phys. Rev. Lett. 118, 030401 (2017).
Khemani, V., Lazarides, A., Moessner, R. & Sondhi, S. L. Section construction of pushed quantum techniques. Phys. Rev. Lett. 116, 250401 (2016).
Sacha, Ok. & Zakrzewski, J. Time crystals: a overview. Rep. Prog. Phys. 81, 016401 (2017).
Else, D. V., Monroe, C., Nayak, C. & Yao, N. Y. Discrete time crystals. Annu. Rev. Condens. Matter Phys. 11, 467–499 (2020).
Yao, N. Y. & Nayak, C. Time crystals in periodically pushed techniques. Phys. In the present day 71, 40 (2018).
Khemani, V., Moessner, R. & Sondhi, S. A quick historical past of time crystals. Preprint at https://arxiv.org/abs/1910.10745 (2019).
Zhang, J. et al. Statement of a discrete time crystal. Nature 543, 217–220 (2017).
Kyprianidis, A. et al. Statement of a prethermal discrete time crystal. Science 372, 1192–1196 (2021).
Choi, S. et al. Statement of discrete time-crystalline order in a disordered dipolar many-body system. Nature 543, 221–225 (2017).
O’Sullivan, J. et al. Signatures of discrete time crystalline order in dissipative spin ensembles. New J. Phys. 22, 085001 (2020).
Randall, J. et al. Many-body-localized discrete time crystal with a programmable spin-based quantum simulator. Science 374, 1474–1478 (2021).
Rovny, J., Blum, R. L. & Barrett, S. E. Statement of discrete-time-crystal signatures in an ordered dipolar many-body system. Phys. Rev. Lett. 120, 180603 (2018).
Pal, S., Nishad, N., Mahesh, T. S. & Sreejith, G. J. Temporal order in periodically pushed spins in star-shaped clusters. Phys. Rev. Lett. 120, 180602 (2018).
Smits, J., Liao, L., Stoof, H. T. C. & van der Straten, P. Statement of a space-time crystal in a superfluid quantum fuel. Phys. Rev. Lett. 121, 185301 (2018).
Autti, S., Eltsov, V. B. & Volovik, G. E. Statement of a time quasicrystal and its transition to a superfluid time crystal. Phys. Rev. Lett. 120, 215301 (2018).
Mi, X. et al. Time-crystalline eigenstate order on a quantum processor. Nature 601, 531–536 (2022).
Ying, C. et al. Floquet prethermal section protected by u(1) symmetry on a superconducting quantum processor. Phys. Rev. A 105, 012418 (2022).
Xu, H. et al. Realizing discrete time crystal in an one-dimensional superconducting qubit chain. Preprint at https://arxiv.org/abs/2108.00942 (2021).
Preskill, J. Quantum computing within the NISQ period and past. Quantum 2, 79 (2018).
Pollmann, F., Berg, E., Turner, A. M. & Oshikawa, M. Symmetry safety of topological phases in one-dimensional quantum spin techniques. Phys. Rev. B 85, 075125 (2012).
Chen, X., Gu, Z.-C., Liu, Z.-X. & Wen, X.-G. Symmetry-protected topological orders in interacting bosonic techniques. Science 338, 1604–1606 (2012).
Chen, X., Gu, Z.-C., Liu, Z.-X. & Wen, X.-G. Symmetry protected topological orders and the group cohomology of their symmetry group. Phys. Rev. B 87, 155114 (2013).
Senthil, T. Symmetry-protected topological phases of quantum matter. Annu. Rev. Condens. Matter Phys. 6, 299–324 (2015).
Chiu, C.-Ok., Teo, J. C. Y., Schnyder, A. P. & Ryu, S. Classification of topological quantum matter with symmetries. Rev. Mod. Phys. 88, 035005 (2016).
Nandkishore, R. & Huse, D. A. Many-body localization and thermalization in quantum statistical mechanics. Annu. Rev. Condens. Matter Phys. 6, 15–38 (2015).
Schreiber, M. et al. Statement of many-body localization of interacting fermions in a quasi-random optical lattice. Science 349, 842–845 (2015).
Smith, J. et al. Many-body localization in a quantum simulator with programmable random dysfunction. Nat. Physics 12, 907–911 (2016).
Xu, Ok. et al. Emulating many-body localization with a superconducting quantum processor. Phys. Rev. Lett. 120, 050507 (2018).
Abanin, D. A., Altman, E., Bloch, I. & Serbyn, M. Colloquium: Many-body localization, thermalization, and entanglement. Rev. Mod. Phys. 91, 021001 (2019).
Huse, D. A., Nandkishore, R., Oganesyan, V., Pal, A. & Sondhi, S. L. Localization-protected quantum order. Phys. Rev. B 88, 014206 (2013).
Chandran, A., Khemani, V., Laumann, C. R. & Sondhi, S. L. Many-body localization and symmetry-protected topological order. Phys. Rev. B 89, 144201 (2014).
Bahri, Y., Vosk, R., Altman, E. & Vishwanath, A. Localization and topology protected quantum coherence on the fringe of scorching matter. Nat. Commun. 6, 7341 (2015).
Parameswaran, S. A., Potter, A. C. & Vasseur, R. Eigenstate section transitions and the emergence of common dynamics in extremely excited states. Ann. Phys. (Berl.) 529, 1600302 (2017).
Parameswaran, S. A. & Vasseur, R. Many-body localization, symmetry and topology. Rep. Prog. Phys. 81, 082501 (2018).
Ponte, P., Chandran, A., Papić, Z. & Abanin, D. A. Periodically pushed ergodic and many-body localized quantum techniques. Ann. Phys. (N.Y.) 353, 196–204 (2015).
Harper, F., Roy, R., Rudner, M. S. & Sondhi, S. Topology and damaged symmetry in floquet techniques. Annu. Rev. Condens. Matter Phys. 11, 345–368 (2020).
von Keyserlingk, C. W. & Sondhi, S. L. Section construction of one-dimensional interacting floquet techniques. i. abelian symmetry-protected topological phases. Phys. Rev. B 93, 245145 (2016).
Else, D. V. & Nayak, C. Classification of topological phases in periodically pushed interacting techniques. Phys. Rev. B 93, 201103 (2016).
Potter, A. C., Morimoto, T. & Vishwanath, A. Classification of interacting topological floquet phases in a single dimension. Phys. Rev. X 6, 041001 (2016).
Potirniche, I.-D., Potter, A. C., Schleier-Smith, M., Vishwanath, A. & Yao, N. Y. Floquet Symmetry-Protected Topological Phases in Chilly-Atom Programs. Phys. Rev. Lett. 119, 123601 (2017).
Roy, R. & Harper, F. Periodic desk for floquet topological insulators. Phys. Rev. B 96, 155118 (2017).
Watanabe, H. & Oshikawa, M. Absence of quantum time crystals. Phys. Rev. Lett. 114, 251603 (2015).
Dumitrescu, P. T. et al. Realizing a dynamical topological section in a trapped-ion quantum simulator. Preprint at https://arxiv.org/abs/2107.09676 (2021).
Lu, Z., Shen, P.-X. & Deng, D.-L. Markovian quantum neuroevolution for machine studying. Phys. Rev. Appl. 16, 044039 (2021).
von Keyserlingk, C. W., Khemani, V. & Sondhi, S. L. Absolute stability and spatiotemporal long-range order in floquet techniques. Phys. Rev. B 94, 085112 (2016).
Khemani, V., von Keyserlingk, C. W. & Sondhi, S. L. Defining time crystals through illustration idea. Phys. Rev. B 96, 115127 (2017).
Kumar, A., Dumitrescu, P. T. & Potter, A. C. String order parameters for one-dimensional floquet symmetry protected topological phases. Phys. Rev. B 97, 224302 (2018).
Georgescu, I. M., Ashhab, S. & Nori, F. Quantum simulation. Rev. Mod. Phys. 86, 153–185 (2014).
Li, H. & Haldane, F. D. M. Entanglement spectrum as a generalization of entanglement entropy: Identification of topological order in non-abelian fractional quantum corridor impact states. Phys. Rev. Lett. 101, 010504 (2008).
Swingle, B. & Senthil, T. Geometric proof of the equality between entanglement and edge spectra. Phys. Rev. B 86, 045117 (2012).
Fidkowski, L. Entanglement spectrum of topological insulators and superconductors. Phys. Rev. Lett. 104, 130502 (2010).
Alba, V., Haque, M. & Läuchli, A. M. Boundary-locality and perturbative construction of entanglement spectra in gapped techniques. Phys. Rev. Lett. 108, 227201 (2012).
Fendley, P. Parafermionic edge zero modes in Zn-invariant spin chains. J. Stat. Mech. Idea Exp. 2012, P11020 (2012).
Iadecola, T., Santos, L. H. & Chamon, C. Stroboscopic symmetry-protected topological phases. Phys. Rev. B 92, 125107 (2015).
Bardarson, J. H., Pollmann, F. & Moore, J. E. Unbounded progress of entanglement in fashions of many-body localization. Phys. Rev. Lett. 109, 017202 (2012).
Yan, F. et al. Tunable coupling scheme for implementing high-fidelity two-qubit gates. Phys. Rev. Appl. 10, 054062 (2018).
Arute, F. et al. Quantum supremacy utilizing a programmable superconducting processor. Nature 574, 505–510 (2019).
Xu, Y. et al. Excessive-fidelity, high-scalability two-qubit gate scheme for superconducting qubits. Phys. Rev. Lett. 125, 240503 (2020).
Collodo, M. C. et al. Implementation of conditional section gates primarily based on tunable zz interactions. Phys. Rev. Lett. 125, 240502 (2020).
Sung, Y. et al. Realization of high-fidelity cz and zz-free iswap gates with a tunable coupler. Phys. Rev. X 11, 021058 (2021).
Wu, Y. et al. Robust quantum computational benefit utilizing a superconducting quantum processor. Phys. Rev. Lett. 127, 180501 (2021).
Track, C. et al. 10-qubit entanglement and parallel logic operations with a superconducting circuit. Phys. Rev. Lett. 119, 180511 (2017).
McKay, D. C., Wooden, C. J., Sheldon, S., Chow, J. M. & Gambetta, J. M. Environment friendly z gates for quantum computing. Phys. Rev. A 96, 022330 (2017).
Foxen, B. et al. Demonstrating a steady set of two-qubit gates for near-term quantum algorithms. Phys. Rev. Lett. 125, 120504 (2020).
Friedman, A. J., Ware, B., Vasseur, R. & Potter, A. C. Topological edge modes with out symmetry in quasiperiodically pushed spin chains. Phys. Rev. B 105, 115117 (2022).
