Time-crystalline eigenstate order on a quantum processor
Authors: Mi Xiao , Ippoliti Matteo , Quintana Chris , Greene Ami , Chen Zijun , Gross Jonathan , Arute Frank , Arya Kunal , Atalaya Juan , Babbush Ryan , Bardin Joseph C. , Basso Joao , Bengtsson Andreas , Bilmes Alexander , Bourassa Alexandre , Brill Leon , Broughton Michael , Buckley Bob B. , Buell David A. , Burkett Brian , Bushnell Nicholas , Chiaro Benjamin , Collins Roberto , Courtney William , Debroy Dripto , Demura Sean , Derk Alan R. , Dunsworth Andrew , Eppens Daniel , Erickson Catherine , Farhi Edward , Fowler Austin G. , Foxen Brooks , Gidney Craig , Giustina Marissa , Harrigan Matthew P. , Harrington Sean D. , Hilton Jeremy , Ho Alan , Hong Sabrina , Huang Trent , Huff Ashley , Huggins William J. , Ioffe L. B. , Isakov Sergei V. , Iveland Justin , Jeffrey Evan , Jiang Zhang , Jones Cody , Kafri Dvir , Khattar Tanuj , Kim Seon , Kitaev Alexei , Klimov Paul V. , Korotkov Alexander N. , Kostritsa Fedor , Landhuis David , Laptev Pavel , Lee Joonho , Lee Kenny , Locharla Aditya , Lucero Erik , Martin Orion , McClean Jarrod R. , McCourt Trevor , McEwen Matt , Miao Kevin C. , Mohseni Masoud , Montazeri Shirin , Mruczkiewicz Wojciech , Naaman Ofer , Neeley Matthew , Neill Charles , Newman Michael , Niu Murphy Yuezhen , O’Brien Thomas E. , Opremcak Alex , Ostby Eric , Pato Balint , Petukhov Andre , Rubin Nicholas C. , Sank Daniel , Satzinger Kevin J. , Shvarts Vladimir , Su Yuan , Strain Doug , Szalay Marco , Trevithick Matthew D. , Villalonga Benjamin , White Theodore , Yao Z. Jamie , Yeh Ping , Yoo Juhwan , Zalcman Adam , Neven Hartmut , Boixo Sergio , Smelyanskiy Vadim , Megrant Anthony , Kelly Julian , Chen Yu , Sondhi S. L. , Moessner Roderich , Kechedzhi Kostyantyn , Khemani Vedika , Roushan Pedram 
Published: November 2021

Quantum many-body systems display rich phase structure in their low-temperature equilibrium states1. However, much of nature is not in thermal equilibrium. Remarkably, it was recently predicted that out-of-equilibrium systems can exhibit novel dynamical phases2–8that may otherwise be forbidden by equilibrium thermodynamics, a paradigmatic example being the discrete time crystal (DTC)7,9–15. Concretely, dynamical phases can be defined in periodically driven many-body-localized (MBL) systems via the concept of eigenstate order7,16,17. In eigenstate-ordered MBL phases, the entire many-body spectrum exhibits quantum correlations and long-range order, with characteristic signatures in late-time dynamics from all initial states. It is, however, challenging to experimentally distinguish such stable phases from transient phenomena, or from regimes in which the dynamics of a few select states can mask typical behaviour. Here we implement tunable controlled-phase (CPHASE) gates on an array of superconducting qubits to experimentally observe an MBL-DTC and demonstrate its characteristic spatiotemporal response for generic initial states7,9,10. Our work employs a time-reversal protocol to quantify the impact of external decoherence, and leverages quantum typicality to circumvent the exponential cost of densely sampling the eigenspectrum. Furthermore, we locate the phase transition out of the DTC with an experimental finite-size analysis. These results establish a scalable approach to studying non-equilibrium phases of matter on quantum processors.