Time-crystalline eigenstate order on a quantum processor

Abstract

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 phases^2-8 that 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 order^7,16,17. 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 states^7,9,10. 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.

Fig. 1. Order in eigenstates.

a, Equilibrium phases are characterized by long-range order in low-energy eigenstates of time-independent Hamiltonians (for example, an Ising ferromagnet with a pair of degenerate ground states that resemble ‘Schrödinger cats’ of polarized states). b, Thermalizing Floquet systems typically have no ordered states in the spectrum. c, In MBL Floquet systems, every eigenstate can show order. In MBL-DTC, every eigenstate resembles a long-range ordered ‘Schrödinger cat’ of a random configuration of spins and its inversion, with even/odd superpositions split by π.

Fig. 2. Observing an MBL-DTC.

a, The experimental circuit composed of $t$ identical cycles of the unitary ${\hat{U}}_{\mathrm{F}}$. The local polarization of each qubit, ⟨$\hat{Z}\left(t\right)$⟩, is measured at the end. In the following panels, we investigate a number of disorder instances each with a different random bit-string initial state. b, Experimental values of ⟨$\hat{Z}\left(t\right)$⟩ measured at Q₁₁. Data are shown for five representative circuit instances deep in the thermal (g = 0.60; left) and MBL-DTC (g = 0.97; right) phases. c, Autocorrelator $\overline{A}=\overline{⟨\hat{Z}\left(0\right)\hat{Z}\left(t\right)}⟩$ at Q₁₁, obtained from averaging the results of 36 circuit instances. For the same circuit instances, the average autocorrelator at the output of ${\hat{U}}_{\mathrm{ECHO}}={\left({\hat{U}}_{\mathrm{F}}^{†}\right)}^t{\hat{U}}_{\mathrm{F}}^t$ is also measured and its square root, ${\overline{A}}_0$, is shown alongside $\overline{A}$ for comparison. The left (right) panels correspond to g = 0.60 (0.97). d, Top panels: the ratio $\overline{A}/{\overline{A}}_0$ obtained from c. Bottom panels: $\overline{A}/{\overline{A}}_0$ as a function of $t$ and qubit location. The left (right) panels correspond to g = 0.60 (0.97) .

Fig. 3. Distinguishing MBL-DTC from prethermal phenomena.

a, Site- and disorder-averaged autocorrelators $\left[\overline{A}\right]$ measured with $g=0.94$. In the left panel (MBL-DTC), each dataset is averaged over 24 disorder instances of ${\phi }_i$ and $h_i$, with the initial state fixed at one of the following: Néel, $|01{⟩}^{\otimes 10}$; polarized, $|0{⟩}^{\otimes 20}$; random, $|00111000010011001111$⟩. In the right panel (prethermal), the same values of $h_i$ and initial states are used but ${\phi }_i=-0.4$. b, Histograms of$\left|\left[\overline{A}\right]\right|$, from 500 random bit-string initial states, averaged over cycles 30 and 31 and the same disorder instances as in a. The standard deviation (mean) of$\left|\left[\overline{A}\right]\right|$, $\sigma$ (μ), is also listed. Location of the polarized (Néel) state is indicated by a purple (red) arrow. Inset: same collection of$\left|\left[\overline{A}\right]\right|$plotted over the energies of the bit-string states, calculated from the effective Hamiltonian ${\hat{H}}_{\mathrm{eff}}$ approximating the drive (see text). Dashed lines show averaged values within energy windows separated by 0.2. c, ⟨$\hat{Z}\left(t\right)$⟩ for two bit-string initial states that differ only at $Q_{11}$. Top panel shows a single circuit instance with disordered ${\phi }_i$ and bottom panel shows an instance with uniform ${\phi }_i=-0.4$. d, Left and middle panels: relative difference between the two signals ${\overline{\zeta }}_{\mathrm{r}}$ as a function of t and qubit location, averaged over time windows of 10 cycles and over 64 disorder instances for ${\hat{U}}_{\mathrm{F}}$ and 81 instances for ${\hat{U}}_{\mathrm{F}\text{'}}$. Right panel: qubit dependence of ${\overline{\zeta }}_{\mathrm{r}}$, averaged from $t=51$ to $t=60$.

Fig. 4. Probing average spectral response via quantum typicality.

a, Scheme for measuring the autocorrelator, $A_{\psi }=⟨\psi \left|\hat{Z}\left(0\right)\hat{Z}\left(t\right)\right|\psi ⟩$, on $Q_{11}$, of a scrambled quantum state $|\psi$⟩. $|\psi$⟩ is created by scrambling a bit-string state with a circuit ${\hat{U}}_{\mathrm{S}}$. The x-axis projection of an ancilla qubit $Q_{\mathrm{a}}$, ${⟨\hat{X}⟩}_{\mathrm{a}}$, is measured at the end. b, ${\hat{U}}_{\mathrm{S}}$ contains $K$ layers of controlled-Z (CZ) gates interleaved with random single-qubit rotations, $R_{i,k}$, around a random axis along the equatorial plane of the Bloch sphere by an angle $\in [0.4\mathrm{\pi },0.6\mathrm{\pi }]$. c, Upper panel: $A_{\psi }$ for a single disorder instance with $K=20$ cycles in ${\hat{U}}_{\mathrm{S}}$. The square root of the autocorrelator, obtained by replacing ${\hat{U}}_{\mathrm{F}}^t$ with ${\hat{U}}_{\mathrm{ECHO}}$, $A_{\psi ,0}$, is also shown. Bottom panel: normalized autocorrelator, $A_{\psi }/A_{\psi ,0}$, as a function of t. d, Histograms of $\left|A_{\psi }\right|$ from a single disorder instance, averaged over cycles 30 and 31. Each histogram corresponds to a different number of scrambling cycles, $K$, and includes data from 500 random initial bit-string states before ${\hat{U}}_{\mathrm{S}}$.

Fig. 5. Estimating phase transition by varying system size.

Disorder-averaged spin-glass order parameter ${\chi }^{\mathrm{SG}}$ as a function of $g$ for different chain lengths L, measured between $t=51$ and $t=60$. Error bars correspond to statistical errors alone and do not include hardware (for example, gate) errors. Inset shows the size dependence of ${\chi }^{\mathrm{SG}}$ for two different values of g. See Methods for measurement details.