Abstract
Quantum manybody systems away from equilibrium host a rich variety of exotic phenomena that are forbidden by equilibrium thermodynamics. A prominent example is that of discrete time crystals^{1,2,3,4,5,6,7,8}, in which timetranslational symmetry is spontaneously broken in periodically driven systems. Pioneering experiments have observed signatures of time crystalline phases with trapped ions^{9,10}, solidstate spin systems^{11,12,13,14,15}, ultracold atoms^{16,17} and superconducting qubits^{18,19,20}. Here we report the observation of a distinct type of nonequilibrium state of matter, Floquet symmetryprotected topological phases, which are implemented through digital quantum simulation with an array of programmable superconducting qubits. We observe robust longlived temporal correlations and subharmonic temporal response for the edge spins over up to 40 driving cycles using a circuit of depth exceeding 240 and acting on 26 qubits. We demonstrate that the subharmonic response is independent of the initial state, and experimentally map out a phase boundary between the Floquet symmetryprotected topological and thermal phases. Our results establish a versatile digital simulation approach to exploring exotic nonequilibrium phases of matter with current noisy intermediatescale quantum processors^{21}.
Main
Symmetryprotected topological (SPT) phases are characterized by nontrivial edge states that are confined near the boundaries of the system and protected by global symmetries^{22,23,24,25,26}. In a clean system without disorder, these edge states typically only occur for the ground states of systems with a bulk energy gap. At finite temperature, they are in general destroyed by mobile thermal excitations. However, adding strong disorder can make the system manybody localized (MBL)^{27,28,29,30,31}, allowing for a sharply defined topological phase and stable edge states even at infinite temperature^{32,33,34,35,36}. Strikingly, the topological phase and corresponding edge states can even survive external periodic driving, as long as the driving frequency is large enough so that the localization persists^{37,38}.
The interplay between symmetry, topology, localization and periodic driving gives rise to various peculiar phases of matter that exist only out of equilibrium^{38}. Understanding and categorizing these unconventional phases poses a wellknown scientific challenge. On the theoretical side, topological classifications of periodically driven (Floquet) systems with^{4,39,40,41,42} and without^{43} interactions have already been obtained through a range of mathematical techniques (such as group cohomology), revealing a number of ‘Floquet SPT’ (FSPT) phases with no equilibrium counterparts^{38}. Yet, we still lack powerful analytical tools or numerical algorithms to thoroughly address these phases and their transitions to other ones. On the experimental side, signatures of discrete time crystals (DTCs)^{1,2,3,4,5,6,7,8}, which are paradigmatic examples of exotic phases beyond equilibrium^{44}, have been reported in a wide range of systems^{9,10,11,12,13,14,15,16,17,18,19,20}. However, none of these experiments encompass topology as a key ingredient. A recent experiment simulating an FSPT phase on a trappedion quantum computer found that the phase was shortlived owing to the presence of coherent errors in the device^{45}. Realizing a longlived FSPT phase, which demands a delicate concurrence of topology, localization and periodic driving, thus still remains a notable experimental challenge.
Here we report the observation of nonequilibrium FSPT phases with a programmable array of 26 superconducting qubits (Fig. 1) with high controllability and long coherence time. We successfully implement the dynamics of prototypical time(quasi)periodic Hamiltonians with \({{\mathbb{Z}}}_{2}\times {{\mathbb{Z}}}_{2}\),\({{\mathbb{Z}}}_{2}\), or no microscopic symmetries, and observe subharmonic temporal responses for the edge spins. In particular, we focus on a onedimensional (1D) timeperiodic Hamiltonian with threebody interactions and \({{\mathbb{Z}}}_{2}\times {{\mathbb{Z}}}_{2}\) symmetry as an example. We digitally simulate this Hamiltonian through a largedepth quantum circuit obtained using a neuroevolution algorithm^{46}. We then measure local spin magnetizations and their temporal correlations and demonstrate that both quantities show a subharmonic response at the boundaries but not in the bulk of the chain. This situation differs drastically from the case of DTCs, which exhibit subharmonic response everywhere in the bulk. This contrast stems from a fundamental distinction between DTC and FSPT phases: the former exhibit conventional longrange order in the bulk intertwined with the spontaneous breaking of discrete timetranslational symmetry^{44,47,48} whereas the latter exhibit SPT order that can only be revealed through boundary effects or nonlocal ‘string operators’ in the bulk^{39,41,49}. The observed boundary subharmonic response persists over an extended range of parameters and is robust to various experimental imperfections, independent of the initial states. We further explore the FSPT phase experimentally from the perspectives of entanglement dynamics, the entanglement spectrum and the dynamics of stabilizer operators that underlies its topological nature. By measuring the variance of the subharmonic peak height in the Fourier spectrum, we experimentally map out the phase boundary between the FSPT and thermal phases.
Model Hamiltonian and its implementation
We mainly consider a 1D spin\(\frac{1}{2}\) chain governed by the following timeperiodic Hamiltonian (Fig. 1b):
where δ denotes the drive perturbation; \({\hat{\sigma }}_{k}^{x,z}\) is the Pauli matrix acting on the kth spin; J_{k}, V_{k} and h_{k} are random parameters drawn independently from uniform distributions over [J − Δ_{J}, J + Δ_{J}], [V − Δ_{V}, V + Δ_{V}] and [h − Δ_{h}, h + Δ_{h}], respectively. For simplicity, we fix T = 2T′ = 2, which roughly corresponds to 0.3 μs for running the corresponding quantum circuit in our experiment. We note that H(t) has a \({{\mathbb{Z}}}_{2}\times {{\mathbb{Z}}}_{2}\) symmetry. For a suitable parameter regime, it has been shown that H_{2} can be in an MBL phase, in which topological edge states can survive as coherent degrees of freedom at arbitrarily high energies^{34}. The localization and edge states carry over to the case of periodic driving with the Hamiltonian H(t), giving rise to an FSPT phase. In this FSPT phase, the timetranslational symmetry only breaks at the boundary but not in the bulk. The Floquet unitary that fully characterizes the FSPT phase reads U_{F} = U_{2}U_{1}, where \({U}_{1}={{\rm{e}}}^{{\rm{i}}{H}_{1}}\) and \({U}_{2}={{\rm{e}}}^{{\rm{i}}{H}_{2}}\) are the unitary operators generated by the Hamiltonians H_{1} and H_{2}, respectively. The quasienergy spectrum of U_{F} reveals that every eigenstate is twofold degenerate and has a cousin eigenstate separated by the quasienergy π (Fig. 1c). The degenerate eigenstates also exhibit longrange mutual information between the boundary spins; this is essential for the robustness of the subharmonic response of the edge spins against local perturbations, including finite δ and V_{k}, that respect the \({{\mathbb{Z}}}_{2}\times {{\mathbb{Z}}}_{2}\) symmetry (Methods and Supplementary Information I).
To implement H(t) with superconducting qubits, the threebody term in H_{2}, which is crucial for the SPT phase at high energy, poses an apparent challenge because no threebody interaction appears naturally in the superconducting system. We thus use the idea of digital quantum simulation^{50} to implement H(t) with quantum circuits (Fig. 1d). For V_{k} = h_{k} = 0, we find optimal circuits in an analytical fashion that can implement H(t) with arbitrary J_{k} and δ, whereas for nonvanishing V_{k} and h_{k} we use a neuroevolution algorithm^{46} to design suitable quantum circuits (Methods). With the obtained quantum circuits, we perform our experiment on a flipchip superconducting quantum processor (Fig. 1e) with a chain of L = 26 transmon qubits denoted as Q_{1} to Q_{L} (Fig. 1a). See Methods and Supplementary Information for the details of the experimental setup, and for experimental results from another processor with a chain of 14 qubits.
Symmetry breaking at boundaries
The characteristic signature of an FSPT phase is the breaking of the discrete timetranslational symmetry at the boundaries of the chain but not in the bulk. This can be manifested by the persistent oscillation with period 2T of local magnetizations at the boundaries. In Fig. 2, we plot the time evolution of the disorderaveraged local magnetizations \(\overline{\langle {\sigma }_{j}^{z}(t)\rangle }\) for different phases. From Fig. 2a, it is evident that in the FSPT phase, the disorderaveraged magnetizations at the two ends of the chain, namely \(\overline{\langle {\sigma }_{1}^{z}(t)\rangle }\) and \(\overline{\langle {\sigma }_{L}^{z}(t)\rangle }\), oscillate with a 2T periodicity, for up to 40 driving cycles. In stark contrast, the local magnetizations in the bulk of the chain (\(\overline{\langle {\sigma }_{j}^{z}(t)\rangle }\) with \(2\le j\le L1\)) decay quickly to zero and do not show perioddoubled oscillations. This unconventional behaviour is independent of disorder averaging. Even for a single random disorder instance the magnetizations exhibit similar dynamical features, as shown in Fig. 2b. The distinction between the dynamics of boundary and bulk magnetizations can also be clearly seen by examining \(\overline{\langle {\sigma }_{j}^{z}(t)\rangle }\) in the frequency domain. As shown in Fig. 2d, the edge spins lock to the subharmonic frequency of the drive period ω/ω_{0} = 1/2, whereas the bulk spins show no such peak. We stress that the subharmonic response for the edge spins obtained in our experiment is notably robust to various perturbations (including nonzero δ) and experimental imperfections (see Supplementary Information I.B for a more indepth discussion). For comparison, we also experimentally measure the dynamics of the magnetizations in the thermal phase. Our results are shown in Fig. 2c,e, where we see that the magnetizations for both the edge and bulk spins decay quickly to zero and no subharmonic response appears at all.
The breaking of the discrete timetranslational symmetry at the boundaries can also be detected by the disorderaveraged autocorrelators defined as \({\bar{A}}_{j}=\overline{\langle {\Psi }_{0}{\sigma }_{j}^{z}(t){\sigma }_{j}^{z}(0){\Psi }_{0}\rangle }\). Our experimental measurements of autocorrelators for up to 40 driving cycles are plotted in Fig. 2f, again showing the breaking of timetranslational symmetry at the boundaries but not in the bulk. We mention that, in the FSPT phase, the local magnetizations for the edge spins exhibit a gradually decaying envelope, which could be attributed to either external circuit errors (that is, experimental imperfections such as decoherence, pulse distortions and crosstalk effects) or slow internal thermalization (namely, an intrinsic tendency towards thermalization in the model). To distinguish these two mechanisms, we carry out an additional experiment on the echo circuit \({U}_{{\rm{echo}}}\equiv {({U}_{F}^{\dagger })}^{t}{U}_{F}^{t}\), the deviation of which from the identity operator measures the effect of circuit errors^{18}. The square root of the output of U_{echo} (black solid lines shown in Fig. 2f) fits well with the decaying envelope of the results obtained by evolution under U_{F}. This indicates that the decay of the envelope is due to circuit errors rather than thermalization, which corroborates that the system is indeed in the localized phase.
Localizationprotected topological states
In the above discussion, the initial states are random product states. To establish the FSPT phase, additional experiments on other initial states and other local observables are necessary. In this section, we show that the stabilizers in the bulk do not break the discrete timetranslational symmetry, but at the boundaries they do. To understand this, we consider the idealized clusterstate and spinflip limit, that is, V_{k} = h_{k} = 0 and δ = 0. In this limit, H_{2} reduces to a summation of stabilizers: \({H}_{s}=\,{\sum }_{k=2}^{L1}{J}_{k}{S}_{k}\) with \({S}_{k}\equiv {\hat{\sigma }}_{k1}^{z}{\hat{\sigma }}_{k}^{x}{\hat{\sigma }}_{k+1}^{z}\). We choose the initial states to be random eigenstates of H_{s} and evolve the system with the timeperiodic Hamiltonian H(t) to measure the time dependence of local stabilizers.
In Fig. 3a, we show a sketch of the quantum circuit used in our experiment to prepare the desired random eigenstates of H_{s}. To manifest the topological nature of these eigenstates, we study their entanglement spectra^{51}, which are widely used as a crucial diagnostic for universal topological properties of quantum phases^{51,52,53,54}. To show that H(t) preserves the topological nature of the SPT states, we prepare random eigenstates of H_{s} with both open and periodic boundary conditions, evolve the system for one driving period with H(t) and then measure the reduced density matrix ρ_{half} of half of the system through quantumstate tomography. Figure 3b displays the entanglement spectra (eigenvalues of \(\mathrm{ln}({\rho }_{{\rm{half}}})\)) for open and periodic boundary conditions, respectively. From this figure, a clear twofold degeneracy for the lowlying Schmidt states is obtained for the open boundary conditions. This degeneracy corresponds to an effectively decoupled spinhalf degree of freedom at the boundary of the bipartition. For periodic boundary conditions, the spectrum is fourfold degenerate, corresponding to two effectively decoupled spins at the two boundaries of the bipartition. The degeneracy of the entanglement spectrum and its dependence on boundary conditions marks a characteristic feature of the SPT state generated in our experiment. We note that the degeneracy disappears above the entanglement gap. This is due to finitesize effects and experimental imperfections.
In Fig. 3c, we plot the time dependence of local stabilizers in the FSPT phase. We observe that the stabilizers at the boundaries oscillate with a 2T periodicity, indicating again the breaking of discrete timetranslational symmetry at the boundaries. In the bulk, the stabilizers oscillate with a T periodicity and are synchronized with the driving frequency, showing that no symmetry breaking occurs. This is in sharp contrast to the dynamics of bulk magnetizations, which decay rapidly to zero and exhibit no oscillation, as shown in Fig. 2a. In fact, in the FSPT phase, the system is MBL and there exist a set of local integrals of motion, which are the ‘dressed’ versions of the stabilizers with exponentially small tails^{34}. The persistent oscillations of the bulk stabilizers observed in our experiment originate from these local integrals of motion and are a reflection of the fact that the system is indeed in an MBL phase.
Phase transition
We now turn to the phase transition between the FSPT phase and the trivial thermal phase. For simplicity and concreteness, we fix other parameters and vary the drive perturbation δ and the interaction strength V. Theoretically, the system is expected to exhibit an FSPT phase for small δ and V. With increasing δ and V, the strong interaction diminishes localization and eventually thermalizes the system. At some critical values of δ and V, a transition between these two phases occurs. In Fig. 4a, we plot the δ − V phase diagram obtained from numerical simulations, in which the phase boundary, although not very sharp because of finitesize effects (for a small system size L = 8, the coupling between the two edge modes is not negligible and thus will decrease the central subharmonic peak height and result in a blurred boundary), can be located and visualized approximately.
To experimentally examine this phase transition, we further fix the interaction strength V = 0. We probe the transition point by measuring the variance of the subharmonic spectral peak height, that is, the amplitude of the Fourier spectrum of \(\overline{\langle {\sigma }_{1}^{z}(t)\rangle }\) at ω = ω_{0}/2 for the boundary spin. Figure 4b shows the subharmonic peak height as a function of the drive perturbation δ. At small δ, the system is in the FSPT phase, and the peak height remains at a value around 0.5. As we increase δ to a large value, the system transitions out of the topological phase and the peak height vanishes. This is consistent with the theoretical analysis above. The largest variance of the peak height corresponds to the phase transition point. The inset of Fig. 4b shows the measured standard deviation as a function of δ, indicating a phase transition point around δ ≈ 0.30, which is consistent with the numerically predicted value of 0.34. The small deviation between the numerical prediction and experimental result is mainly attributed to finitesize effects, experimental noise and the limited number of disorder instances implemented in the experiment.
Other nonequilibrium SPT phases
The digital simulation approach used in our experiment is generally applicable for quantum simulations of various exotic phases of matter. The model Hamiltonian in equation (1) possesses a \({{\mathbb{Z}}}_{2}\times {{\mathbb{Z}}}_{2}\) symmetry, which can also support robust edge modes in the static equilibrium setting. For driven nonequilibrium systems, however, the edge modes may be stabilized by emergent dynamical symmetries. To demonstrate this and illustrate the general applicability of our approach, we also digitally simulate two other models with our quantum device, namely a periodically driven Ising chain with \({{\mathbb{Z}}}_{2}\) symmetry and a quasiperiodically driven model without any microscopic symmetry (Methods and Supplementary Information VI). Our results are summarized in Extended Data Figs. 1 and 2, in which robust subharmonic edge oscillations are also observed.
Conclusions
In summary, we have experimentally observed signatures of nonequilibrium Floquet SPT phases with a programmable superconducting quantum processor. In contrast to previously reported conventional time crystals, for our observed FSPT phases, the discrete timetranslational symmetry only breaks at the boundaries and not in the bulk. We measured the persistent oscillations of edge spins with a subharmonic frequency and experimentally demonstrated that the FSPT phases are robust to symmetryrespecting perturbations in the drive and imperfections in the experiment. In addition, we also demonstrated that the subharmonic response of boundary observables is independent of the initial state. The digital quantum simulation approach explored in our experiment is generally applicable to the simulation of a wide range of nonequilibrium systems hosting unconventional topological phases, including those with multibody interactions.
Methods
Characterization of the model Hamiltonian
To understand why timetranslational symmetry breaks at the boundary but not in the bulk, we consider the idealized ‘clustermodel’ limit (V_{k} = h_{k} = 0) and set δ = 0. We suppose that the system is initially prepared in a random product state in the computational basis, and we use the dynamics of local magnetization as a diagnostic. In this simple scenario, the topologically nontrivial structure of the cluster states (eigenstates of U_{2}) gives rise to edge modes that behave as free spins. At each driving period, the unitary operator U_{1} flips all spins. As a result, the edge spins are reversed after one period and return to their initial configuration after two, leading to the perioddoubled dynamics of the local magnetization at the boundaries. For spins in the bulk, however, the unitary operator U_{2} plays a part and evolves the random product state to a state with vanishing magnetization, resulting in no period doubling. When V_{k} = 0, the Hamiltonian in equation (1) can be mapped to free Majorana fermions (Supplementary Information I.B and, for example, refs. ^{55,56}). Further setting δ = h_{k} = 0, we find that equation (1) maps onto two decoupled copies of the fixedpoint model of a \({{\mathbb{Z}}}_{2}\) FSPT phase considered in ref. ^{39}. The robustness of the subharmonic responses of the topologically protected edge spins to perturbations respecting the \({{\mathbb{Z}}}_{2}\times {{\mathbb{Z}}}_{2}\) symmetry is discussed in depth in Supplementary Information I.B.
Logarithmic entanglement growth
For a MBL system, the entanglement entropy will feature a logarithmic growth^{57}, which is in sharp contrast to the case of Anderson localization without interactions. For the model Hamiltonian H(t) studied in this Article, we also expect a logarithmic growth of the entanglement entropy inside the FSPT phase with V_{k} ≠ 0. We numerically simulate the entanglement dynamics of the system deep in the FSPT phase with the timeevolving block decimation algorithm up to a system size L = 100 (Supplementary Information II). Our results clearly verify the logarithmic entanglement growth, which again implies that the FSPT phase is indeed MBL with nonvanishing V_{k}. In our experiment, we also study the entanglement dynamics for a small system size (L = 6) through quantum tomography (Supplementary Information V). We find that in the thermal phase the entanglement grows much faster than that in the FSPT phase. However, because of the small system size and experimental imperfections (such as decoherence, pulse distortions and crosstalk effects), we are not able to observe the logarithmic entanglement growth (Supplementary Information VI).
Quantum circuits for implementing H(t)
Direct implementation of the Floquet Hamiltonian H(t) with superconducting qubits faces a notable difficulty: the natural interactions hosted by the superconducting qubits are only twobody, so the threebody terms in H_{2} cannot emerge directly. Fortunately, programmable superconducting qubits are universal for quantum computation; thus we can explore the idea of digital quantum simulation to emulate the dynamics of H(t). However, because of inevitable experimental imperfections, the depth of the quantum circuits is limited. As a result, obtaining wellperforming circuits with an optimal depth that can implement H(t) (or equivalently the Floquet unitary U_{F}) is of crucial importance for the success of our experiment.
To find the desired quantum circuits, we use a neuroevolution method introduced in ref. ^{46}, which outputs a nearoptimal architecture for a family of variational quantum circuits that can implement H(t) with different random disorder instances. For a given instance of J_{k}, V_{k} and h_{k}, we use the gradient decent method to tune the variational parameters of the ansatz circuits to minimize the distance between the unitary represented by the circuit and the unitary generated by H(t) within a small time interval. In the idealized ‘clustermodel’ limit (V_{k} = h_{k} = 0), we can find a simple exact onetoone correspondence between J_{k} and the variational parameters, independent of the system size and the values of J_{k} and δ. Thus, we are able to construct an analytical quantum circuit (see Supplementary Fig. 4c for an explicit illustration of the circuit for L = 6) that can implement H(t) precisely and, at the same time, in a way that is experimentally friendly and practical. The details of how to obtain the desired quantum circuits are given in Supplementary Information III.
Experimental setup
Our experiment is performed on a flipchip superconducting quantum processor designed to encapsulate a square array of 6 × 6 transmon qubits with adjustable nearestneighbour couplings (Fig. 1e), on which a chain of up to L = 26 qubits, denoted as Q_{1} to Q_{L}, that alternate with L − 1 couplers, denoted as C_{1} to C_{L − 1}, are selected to observe the FSPT phase (Fig. 1a). All L qubits can be individually tuned in frequency with flux biases, excited by microwaves, and measured using onchip readout resonators; all couplers are also of transmon type with characteristic transition frequencies higher than those of the qubits, which can be controlled with flux biases to tune the effective nearestneighbour couplings. During an experimental sequence (Fig. 1d), we first initialize each qubit, Q_{J}, in \(0\rangle \) at its idle frequency ω_{j}, following which we alternate the singlequbit gates at ω_{j} with the twoqubit controlledπ (CZ) gates realized by biasing Q_{J} and its neighbouring qubit to the pairwise frequencies of group A(B) listed in \(({\omega }_{j}^{{\rm{A}}({\rm{B}})},{\omega }_{j+1}^{{\rm{A}}({\rm{B}})})\) for a fixed interaction time (Supplementary Information III.C). Meanwhile, each coupler is dynamically switched between two frequencies^{58,59,60,61,62,63}: one is to turn off the effective coupling where the neighbouring two qubits can be initialized and operated with singlequbit gates; the other one is to turn on the nearestneighbour coupling to around 11 MHz for a CZ gate. After n layers of the alternating single and twoqubit gates, we finally tune all qubits to their respective \({\omega }_{j}^{{\rm{m}}}\) (here, the superscript 'm' stands for 'measurement') for simultaneous quantumstate measurement. Qubit energy relaxation times measured around ω_{j} are in the range of 7–41 μs, averaging above 30 μs. More characteristic qubit parameters, including the above mentioned frequencies, anharmonicities and readout fidelities, can be found in Supplementary Table 1. The parameters for another processor with 14 qubits used are displayed in the Supplementary Table 2.
We explore a quantum digital simulation scheme to implement the dynamics of the system under the driven Hamiltonian H(t). More specifically, we decompose the evolution operators into the experimentally feasible singlequbit gates (X(θ), Y(θ) and Z(θ)) and twoqubit gates (CR_{z}(±π)), where X(θ), Y(θ) and Z(θ) are rotations around the x, y and z axes by the angle θ, respectively, and CR_{z}(±π) are the zaxis rotations of the target qubit by ±π conditioned on the state of the control qubit (Fig. 1d and Supplementary Information III.A for the ansatz that generates the gate sequences). Here X(θ) and Y(θ) are realized by applying 50nslong microwave pulses with a fullwidth halfmaximum of 25 ns, for which the quadrature correction terms are optimized to minimize state leakages to higher levels^{64}. Simultaneous randomized benchmarkings indicate that the singlequbit gates used in this experiment have reasonably high fidelities, averaging above 0.99 (Supplementary Table 1). Then Z(θ) is realized using the virtualZ gate, which encodes the information θ in the rotation axes of all subsequent gates^{65}, and is combined with CZ to assemble CR_{z}(±π). Here we adopt the strategy reported elsewhere^{62,66} to realize the CZ gate, that is, we diabatically tune the coupler frequency while keeping \(11\rangle \) and \(02\rangle \) (or \(20\rangle \)) for the subspace of the two neighbouring qubits in near resonance. When simultaneously running the 40nslong CZ gates for multiple pairs of neighbouring qubits as required in the experimental sequence, the average CZ gate fidelities can be above 0.98, as obtained by simultaneous randomized benchmarking (Supplementary Table 1).
Further experiments on nonequilibrium SPT phases
The digital simulation strategies of our experiments are capable of simulating a wide range of models hosting unconventional nonequilibrium topological phases. To illustrate this, we also implement two other dynamical SPT phases with our superconducting quantum processor: an FSPT phase in a periodically driven random Ising chain^{4} and an emergent dynamical SPT (EDSPT) phase in a quasiperiodically driven chain^{67}.
The first model has a \({{\mathbb{Z}}}_{2}\) (Ising) symmetry. For the FSPT phase (ref. ^{4} and Supplementary Information VI.A), the evolution is realized by applying two unitaries in an alternating fashion (Extended Data Fig. 1a) to random initial states. For the parameters chosen in our experiments, the corresponding Floquet unitary \({U}_{{\rm{F}}}={{\rm{e}}}^{{\rm{i}}{H}_{{\rm{Ising}}}}{{\rm{e}}}^{{\rm{i}}{H}_{{\rm{single}}}}\), where \({H}_{{\rm{single}}}=\sum _{k}{g}_{k}{\hat{\sigma }}_{k}^{x}\) and \({H}_{{\rm{Ising}}}=\sum _{k}{J}_{k}{\hat{\sigma }}_{k}^{z}{\hat{\sigma }}_{k+1}^{z}\) with g_{k} and J_{k} being coupling parameters respectively, maintains a \({{\mathbb{Z}}}_{2}\times {\mathbb{Z}}\) symmetry (where \({\mathbb{Z}}\) describes discrete timetranslation symmetry), despite the fact that the original static Hamiltonian only possesses a \({{\mathbb{Z}}}_{2}\) symmetry. This enlarged dynamical symmetry protects the edge modes of this phase, one at quasienergy 0 and the other at quasienergy π. This leads to unusual dynamics of the edge spins. If one applies this evolution to a product state in the x basis, the edge spins will return to their initial states only at even periods. In our experiments, we measure the random disorderaveraged local magnetization \(\overline{\langle {\sigma }_{k}^{x}\rangle }\) during the evolution (Extended Data Fig. 1b). Persistent subharmonic oscillations are observed for the edge spins, whereas the averaged magnetizaiton in the bulk is synchronized with the driving frequency and shows no breaking of the discrete timetranslational symmetry.
The EDSPT model has no microscopic symmetry (see refs. ^{45,67} and Supplementary Information VI.B). The evolution of an initial state is realized by applying on it a sequence of evolution unitaries at Fibonacci times, \({U}^{(\nu )}=U({t}_{\nu }={F}_{\nu })\), with F_{v} being the vth element of the Fibonacci sequence. Although the underlying Hamiltonian of this model includes random fields breaking all microscopic symmetries, the evolution unitary possesses a locally dressed \({{\mathbb{Z}}}_{2}\times {{\mathbb{Z}}}_{2}\) symmetry emergent from the quasiperiodic drive^{45,67}. The emergent symmetry hosts two nontrivial edge modes, which can be manifested by the distinct dynamics of the edge spins. In particular, the edge spins would exhibit 3vperiodic oscillations when measured at Fibonacci times t_{v} = F_{v}, whereas the magnetization of the bulk spins will decay to zero rapidly. In our experiment, we prepare random initial states and use the circuits shown in Extended Data Fig. 1a,b to implement the quasiperiodic driving of the system. We measure the random disorderaveraged magnetizations \(\overline{\langle {\sigma }_{j}^{z}\rangle }\) and \(\overline{\langle {\sigma }_{j}^{x}\rangle }\) at Fibonacci times. Our experimental results are summarized in Extended Data Fig. 2c, in which persistent quasiperiodic oscillations for edge spins are indeed observed. The results shown in Extended Data Figs. 1 and 2 were obtained using 12 qubits on a third device with slightly improved performance. We note that an experimental implementation of the EDSPT model with ten trappedion qubits has recently also been reported^{45}.
Data availability
The data presented in the figures and that support the other findings of this study are available for download at https://doi.org/10.5281/zenodo.6510867.
Code availability
The data analysis and numerical simulation codes are available at https://doi.org/10.5281/zenodo.6510867.
References
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. Phase structure of driven quantum systems. Phys. Rev. Lett. 116, 250401 (2016).
Sacha, K. & Zakrzewski, J. Time crystals: a review. 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 driven systems. Phys. Today 71, 40 (2018).
Khemani, V., Moessner, R. & Sondhi, S. A brief history of time crystals. Preprint at https://arxiv.org/abs/1910.10745 (2019).
Zhang, J. et al. Observation of a discrete time crystal. Nature 543, 217–220 (2017).
Kyprianidis, A. et al. Observation of a prethermal discrete time crystal. Science 372, 1192–1196 (2021).
Choi, S. et al. Observation of discrete timecrystalline order in a disordered dipolar manybody 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. Manybodylocalized discrete time crystal with a programmable spinbased quantum simulator. Science 374, 1474–1478 (2021).
Rovny, J., Blum, R. L. & Barrett, S. E. Observation of discretetimecrystal signatures in an ordered dipolar manybody system. Phys. Rev. Lett. 120, 180603 (2018).
Pal, S., Nishad, N., Mahesh, T. S. & Sreejith, G. J. Temporal order in periodically driven spins in starshaped clusters. Phys. Rev. Lett. 120, 180602 (2018).
Smits, J., Liao, L., Stoof, H. T. C. & van der Straten, P. Observation of a spacetime crystal in a superfluid quantum gas. Phys. Rev. Lett. 121, 185301 (2018).
Autti, S., Eltsov, V. B. & Volovik, G. E. Observation of a time quasicrystal and its transition to a superfluid time crystal. Phys. Rev. Lett. 120, 215301 (2018).
Mi, X. et al. Timecrystalline eigenstate order on a quantum processor. Nature 601, 531–536 (2022).
Ying, C. et al. Floquet prethermal phase 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 onedimensional superconducting qubit chain. Preprint at https://arxiv.org/abs/2108.00942 (2021).
Preskill, J. Quantum computing in the NISQ era and beyond. Quantum 2, 79 (2018).
Pollmann, F., Berg, E., Turner, A. M. & Oshikawa, M. Symmetry protection of topological phases in onedimensional quantum spin systems. Phys. Rev. B 85, 075125 (2012).
Chen, X., Gu, Z.C., Liu, Z.X. & Wen, X.G. Symmetryprotected topological orders in interacting bosonic systems. 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. Symmetryprotected topological phases of quantum matter. Annu. Rev. Condens. Matter Phys. 6, 299–324 (2015).
Chiu, C.K., 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. Manybody localization and thermalization in quantum statistical mechanics. Annu. Rev. Condens. Matter Phys. 6, 15–38 (2015).
Schreiber, M. et al. Observation of manybody localization of interacting fermions in a quasirandom optical lattice. Science 349, 842–845 (2015).
Smith, J. et al. Manybody localization in a quantum simulator with programmable random disorder. Nat. Physics 12, 907–911 (2016).
Xu, K. et al. Emulating manybody localization with a superconducting quantum processor. Phys. Rev. Lett. 120, 050507 (2018).
Abanin, D. A., Altman, E., Bloch, I. & Serbyn, M. Colloquium: Manybody localization, thermalization, and entanglement. Rev. Mod. Phys. 91, 021001 (2019).
Huse, D. A., Nandkishore, R., Oganesyan, V., Pal, A. & Sondhi, S. L. Localizationprotected quantum order. Phys. Rev. B 88, 014206 (2013).
Chandran, A., Khemani, V., Laumann, C. R. & Sondhi, S. L. Manybody localization and symmetryprotected topological order. Phys. Rev. B 89, 144201 (2014).
Bahri, Y., Vosk, R., Altman, E. & Vishwanath, A. Localization and topology protected quantum coherence at the edge of hot matter. Nat. Commun. 6, 7341 (2015).
Parameswaran, S. A., Potter, A. C. & Vasseur, R. Eigenstate phase transitions and the emergence of universal dynamics in highly excited states. Ann. Phys. (Berl.) 529, 1600302 (2017).
Parameswaran, S. A. & Vasseur, R. Manybody localization, symmetry and topology. Rep. Prog. Phys. 81, 082501 (2018).
Ponte, P., Chandran, A., Papić, Z. & Abanin, D. A. Periodically driven ergodic and manybody localized quantum systems. Ann. Phys. (N.Y.) 353, 196–204 (2015).
Harper, F., Roy, R., Rudner, M. S. & Sondhi, S. Topology and broken symmetry in floquet systems. Annu. Rev. Condens. Matter Phys. 11, 345–368 (2020).
von Keyserlingk, C. W. & Sondhi, S. L. Phase structure of onedimensional interacting floquet systems. i. abelian symmetryprotected topological phases. Phys. Rev. B 93, 245145 (2016).
Else, D. V. & Nayak, C. Classification of topological phases in periodically driven interacting systems. Phys. Rev. B 93, 201103 (2016).
Potter, A. C., Morimoto, T. & Vishwanath, A. Classification of interacting topological floquet phases in one dimension. Phys. Rev. X 6, 041001 (2016).
Potirniche, I.D., Potter, A. C., SchleierSmith, M., Vishwanath, A. & Yao, N. Y. Floquet SymmetryProtected Topological Phases in ColdAtom Systems. Phys. Rev. Lett. 119, 123601 (2017).
Roy, R. & Harper, F. Periodic table 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 phase in a trappedion quantum simulator. Preprint at https://arxiv.org/abs/2107.09676 (2021).
Lu, Z., Shen, P.X. & Deng, D.L. Markovian quantum neuroevolution for machine learning. Phys. Rev. Appl. 16, 044039 (2021).
von Keyserlingk, C. W., Khemani, V. & Sondhi, S. L. Absolute stability and spatiotemporal longrange order in floquet systems. Phys. Rev. B 94, 085112 (2016).
Khemani, V., von Keyserlingk, C. W. & Sondhi, S. L. Defining time crystals via representation theory. Phys. Rev. B 96, 115127 (2017).
Kumar, A., Dumitrescu, P. T. & Potter, A. C. String order parameters for onedimensional 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 nonabelian fractional quantum hall effect 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. Boundarylocality and perturbative structure of entanglement spectra in gapped systems. Phys. Rev. Lett. 108, 227201 (2012).
Fendley, P. Parafermionic edge zero modes in Z_{n}invariant spin chains. J. Stat. Mech. Theory Exp. 2012, P11020 (2012).
Iadecola, T., Santos, L. H. & Chamon, C. Stroboscopic symmetryprotected topological phases. Phys. Rev. B 92, 125107 (2015).
Bardarson, J. H., Pollmann, F. & Moore, J. E. Unbounded growth of entanglement in models of manybody localization. Phys. Rev. Lett. 109, 017202 (2012).
Yan, F. et al. Tunable coupling scheme for implementing highfidelity twoqubit gates. Phys. Rev. Appl. 10, 054062 (2018).
Arute, F. et al. Quantum supremacy using a programmable superconducting processor. Nature 574, 505–510 (2019).
Xu, Y. et al. Highfidelity, highscalability twoqubit gate scheme for superconducting qubits. Phys. Rev. Lett. 125, 240503 (2020).
Collodo, M. C. et al. Implementation of conditional phase gates based on tunable zz interactions. Phys. Rev. Lett. 125, 240502 (2020).
Sung, Y. et al. Realization of highfidelity cz and zzfree iswap gates with a tunable coupler. Phys. Rev. X 11, 021058 (2021).
Wu, Y. et al. Strong quantum computational advantage using a superconducting quantum processor. Phys. Rev. Lett. 127, 180501 (2021).
Song, C. et al. 10qubit entanglement and parallel logic operations with a superconducting circuit. Phys. Rev. Lett. 119, 180511 (2017).
McKay, D. C., Wood, C. J., Sheldon, S., Chow, J. M. & Gambetta, J. M. Efficient z gates for quantum computing. Phys. Rev. A 96, 022330 (2017).
Foxen, B. et al. Demonstrating a continuous set of twoqubit gates for nearterm quantum algorithms. Phys. Rev. Lett. 125, 120504 (2020).
Friedman, A. J., Ware, B., Vasseur, R. & Potter, A. C. Topological edge modes without symmetry in quasiperiodically driven spin chains. Phys. Rev. B 105, 115117 (2022).
Acknowledgements
We thank L.M. Duan and S.L. Xu for helpful discussion. The device was fabricated at the MicroNano Fabrication Centre of Zhejiang University. X. Zhang, J.D., K.W., J.C., P.Z., W.R., H.D., S.X., Y.G., F.J., X. Zhu, Q.G., H.L., C.S., Z.W. and H.W. acknowledge the support of the National Natural Science Foundation of China (grant nos. 11725419, U20A2076 and 92065204), the National Basic Research Programme of China (grant no. 2017YFA0304300), the Zhejiang Province Key Research and Development Programme (grant no. 2020C01019) and the KeyArea Research and Development Programme of Guangdong Province (grant no. 2020B0303030001). W.J. and D.L.D. are supported by the National Natural Science Foundation of China (grant no. 12075128), Tsinghua University and the Shanghai Qi Zhi Institute. A.V.G. and F.L. acknowledge funding by the AFOSR, DoE QSA and NSF QLCI (award no. OMA2120757).
Author information
Authors and Affiliations
Contributions
X. Zhang and J.D. carried out the experiments under the supervision of Z.W. and H.W. H.L., Z.W. and J.C. fabricated the device supervised by H.W. K.W., P.Z., W.R., H.D., S.X, Y.G., F.J., X. Zhu, Q.G. and C.S. characterized and calibrated the device. W.J. performed the numerical simulations under the supervision of D.L.D. A.V.G., T.I., F.L., Z.X.G. and D.L.D. conducted the theoretical analysis. All authors contributed to the experimental setup, the discussions of the results and the writing of the manuscript.
Corresponding authors
Ethics declarations
Competing interests
The authors declare no competing interests.
Peer review
Peer review information
Nature thanks the anonymous reviewers for their contribution to the peer review of this work. Peer reviewer reports are available.
Additional information
Publisher’s note Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.
Extended data figures and tables
Extended Data Fig. 1 FSPT phase protected by a \({{\mathbb{Z}}}_{2}\) symmetry.
a Schematic of the experimental circuits for the Floquet unitary, where the first layer represents the evolution under the onebody Hamiltonian H_{single}, and the following layers represents the evolution under the Ising Hamiltonian H_{Ising}. b The dynamics of the edge and bulk magnetizations. Here, the expectation values of \({\hat{\sigma }}_{j}^{x}\) are multiplied by the signs of those of the random initial product states prepared in the \({\hat{\sigma }}^{x}\) basis. The edge magnetizations are averaged over 12 random disorder realizations, and the bulk magnetizations are averaged over 12 random realizations and all bulk sites. The dashed lines show numerical simulations with experimental noises. Here, we set \({J}_{k}=\frac{\pi }{2}0.1\), and choose g_{k} uniformly from \(\left[\frac{\pi }{4}\frac{\pi }{3},\frac{\pi }{4}+\frac{\pi }{3}\right]\) (Supplementary Information VI.A).
Extended Data Fig. 2 EDSPT phase protected by emergent dynamical symmetry.
a Schematic of the experimental circuits for implementing U_{x} and U_{z} respectively, which are the building blocks of the quasiperiodicallydriven EDSPT model. b The circuit implementations of the evolution unitary \({U}^{(\nu +1)}={U}^{(\nu 1)}{U}^{(\nu )}\) that defines the EDSPT model. c, d The dynamics of the edge and bulk magnetizations. Here, the expectation values of \({\hat{\sigma }}_{j}^{z}\), \({\hat{\sigma }}_{j}^{x}\) are multiplied by the signs of those of the random initial states respectively. The edge magnetizations are averaged over 12 random disorder realizations and 10 random initial states, and the bulk magnetizations are averaged over 12 random disorder realizations, 10 random initial states, and all bulk sites. The dashed lines show numerical simulations taking into account experimental noise. The imperfect pulse is set as J = 0.99π. The coupling parameters \({K}_{k}^{x},\,{K}_{k}^{z}\) are uniformly chosen from [0, 4π]. The norms of the fields \({{\bf{B}}}_{k}^{x},\,{{\bf{B}}}_{k}^{z}\) are uniformly chosen from [0, 0.3], and their random directions are also chosen uniformly (Supplementary Information VI.B).
Supplementary information
Supplementary Information
This Supplementary Information file contains the following seven sections: I. Theoretical understanding; II. Details of the TEBD method; III. Experimental details; IV. Numerical simulations considering experimental imperfections; V. Quantumstate tomography; VI. Dynamics of entanglement; VII. Extended models. It includes Supplementary Tables 1 and 2, Figs. 1–10 and additional references.
Rights and permissions
Open Access This article is licensed under a Creative Commons Attribution 4.0 International License, which permits use, sharing, adaptation, distribution and reproduction in any medium or format, as long as you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made. The images or other third party material in this article are included in the article’s Creative Commons license, unless indicated otherwise in a credit line to the material. If material is not included in the article’s Creative Commons license and your intended use is not permitted by statutory regulation or exceeds the permitted use, you will need to obtain permission directly from the copyright holder. To view a copy of this license, visit http://creativecommons.org/licenses/by/4.0/.
About this article
Cite this article
Zhang, X., Jiang, W., Deng, J. et al. Digital quantum simulation of Floquet symmetryprotected topological phases. Nature 607, 468–473 (2022). https://doi.org/10.1038/s41586022048543
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1038/s41586022048543
This article is cited by

Observing and braiding topological Majorana modes on programmable quantum simulators
Nature Communications (2023)

Noisy intermediatescale quantum computers
Frontiers of Physics (2023)

Quantum simulation of Hofstadter butterfly with synthetic gauge fields on twodimensional superconductingqubit lattices
Frontiers of Physics (2023)
Comments
By submitting a comment you agree to abide by our Terms and Community Guidelines. If you find something abusive or that does not comply with our terms or guidelines please flag it as inappropriate.