Many-Body Physics in the NISQ Era: Quantum Programming a Discrete Time Crystal

Recent progress in the realm of noisy intermediate-scale quantum (NISQ) devices [J. Preskill, Quantum 2, 79 (2018)] represents an exciting opportunity for many-body physics by introducing new laboratory platforms with unprecedented control and measurement capabilities. We explore the implications of NISQ platforms for many-body physics in a practical sense: we ask which physical phenomena, in the domain of quantum statistical mechanics, they may realize more readily than traditional experimental platforms. While a universal quantum computer can simulate any system, the eponymous noise inherent to NISQ devices practically favors certain simulation tasks over others in the near term. As a particularly well-suited target, we identify discrete time crystals (DTCs), novel nonequilibrium states of matter that break time translation symmetry. These can only be realized in the intrinsically out-of-equilibrium setting of periodically driven quantum systems stabilized by disorder-induced many-body localization. While promising precursors of the DTC have been observed across a variety of experimental platforms—ranging from trapped ions to nitrogen-vacancy centers to NMR crystals—none have all the necessary ingredients for realizing a fully fledged incarnation of this phase, and for detecting its signature long-range spatiotemporal order. We show that a new generation of quantum simulators can be programmed to realize the DTC phase and to experimentally detect its dynamical properties, a task requiring extensive capabilities for programmability, initialization, and readout. Specifically, the architecture of Google’s Sycamore processor is a remarkably close match for the task at hand. We also discuss the effects of environmental decoherence, and how they can be distinguished from ‘internal’ decoherence coming from closed-system thermalization dynamics. Already with existing technology and noise levels, we find that DTC spatiotemporal order would be observable over hundreds of periods, with parametric improvements to come as the hardware advances.

Popular Summary

The advent of noisy intermediate-scale quantum (NISQ) devices promises new possibilities not only for computer science, but also for many-body quantum physics. Indeed, NISQ platforms possess unprecedented capabilities for control and readout, which in turn allow the realization and detection of elusive quantum phenomena. But how can many-body physicists take advantage of these novel capabilities with present-day noisy devices? In this work, we argue that nonequilibrium phases of matter in periodically driven (Floquet) systems constitute an ideal first step in this direction, and we formulate a detailed proposal for the realization of a “discrete time crystal” (DTC) phase on a NISQ processor.

By reviewing past experiments on various quantum simulator platforms, we distill a checklist of technical requirements for the realization of a DTC phase. We show that these requirements, not fully achieved in past experiments, are instead an excellent fit for the capabilities of digital gate-based quantum simulators. Focusing on the Sycamore processor for concreteness, we propose an experimental blueprint for a first observation of the DTC phase, emphasizing in particular how to distinguish a stable DTC phase from various transient phenomena even within the limited coherence time that characterizes NISQ devices.

While we focus on the DTC for specificity, our work applies more broadly to nonequilibrium phases of matter in periodically driven systems, and sets the stage for their observation on present-day and near-term quantum processors within realistic constraints.

Figure 1

Illustrative sketch for distinguishing MBL and prethermal DTCs, even with access to only finite experimental coherence times (vertical dashed line, $n_{\exp }$). (a) In a MBL DTC, local autocorrelators remain large for all initial product states in the $Z$ basis. (b) In a prethermal DTC, low-temperature initial states have long-lived autocorrelators whose lifetime may exceed the experimental coherence time. However, generic bitstring initial states (high temperature) decay quickly: the dependence on initial states—visible within experimental time scales $n_{\exp }$—is a signature of prethermalization. (c) In a $U(1)$ prethermal DTC, the total magnetization $Z_{\mathrm{tot}}=\sum _i{Z}_i$ is nearly conserved (for all initial states, including high-temperature ones). However, local operators $Z_i$ decay quickly: the $U(1)$-prethermal behavior is revealed through site-resolved measurements.

Figure 2

Simulating a 1D Floquet DTC on the Sycamore chip. (a) Modified gate $\tilde{G}$ in terms of the native gate $G$ and single-qubit $Z$ rotations. (b) Circuit for the DTC Floquet unitary: each Floquet cycle acts with $\tilde{G}$ on each pair of neighboring qubits, followed by single-qubit $X$ rotations, as depicted. (c) A closed loop through the Sycamore chip, simulating a 1D system. During each cycle, $\tilde{G}$ gates act first on the blue bonds, then on the black bonds. All other bonds remain idle during the dynamics.

Figure 3

Phase diagram of the circuit, Eq. (7), as a function of the pulse parameter $g$ and the average iswap angle $\overline{\theta }$. The phase boundary is based on finite-size crossing points (black dots) of the level spacing ratio, computed numerically for systems of $8\le L\le 12$ qubits. Inset: level spacing ratio $⟨r⟩$, Eq. (9), versus $g$ on the $\overline{\theta }=0$ cut. The value of $⟨r⟩$ is averaged over eigenstates and over between 400 and 4000 realizations of disorder (depending on $L$).

Figure 4

Dynamics of the ideal (noise-free) circuit in the MBL DTC ($g/\pi =39/80$), thermal ($g/\pi =19/80$), and MBL paramagnetic ($g/\pi =1/80$) phases ($\overline{\theta }=0$). (a)–(c) Position- and disorder-averaged temporal autocorrelator $C(n)$ starting from various initial bitstring states for $L=20$ qubits. In the DTC phase the envelopes at even and odd times are highlighted; the signal oscillates stroboscopically between these two envelopes (lighter curves). Insets: space-time color plots of expectation values $⟨Z_x(n)⟩$ for $L=16$ qubits. (d)–(f) Disorder-averaged probability distribution of the Hamming distance $d$ from the initial bitstring in two consecutive Floquet cycles at late time, $n={10}^4$, for $L=20$ qubits.

Figure 5

Other diagnostics of the DTC order. (a) Fourier transform of the temporal autocorrelator, $C(\omega )$, averaged over position and disorder, for several values of $g$ spanning the DTC and thermal phases. Data from dynamics simulations of $L=18$ qubits starting from a fixed bitstring state and evolving to $n_{\max }={10}^4$ Floquet cycles. Inset: height of the $\omega =\pi$ peak as a function of $g$. (b),(c) Spectral function $\mathcal{C}(\pi ,\delta \omega )$ [see Eq. (12)] from exact diagonalization of $U_F$ on small sizes, averaged over disorder. The DTC phase develops a plateau for $\delta \omega \to 0$ corresponding to a delta-function $\pi$ peak in the Fourier response, while in the thermal and MBL PM phases we find that $\mathcal{C}(\pi ,\delta \omega )\sim \delta \omega$. (d) Spin glass order parameter ${\chi }^{\mathrm{SG}}$ evaluated at late times, $n_{\max }/2\le n\le n_{\max }$, from dynamics simulations as in (a). A crossing for increasing system size indicates a transition consistent with the phase boundary in Fig. 3 at $\overline{\theta }=0$.

Figure 6

Noisy dynamics. (a) Time evolution of the spatially averaged correlator $C(n)$ for a circuit with $L=20$ qubits in the presence of depolarizing noise ($p={10}^{-2}$) for the MBL DTC, MBL paramagnetic, and thermal phases, starting from a fixed bitstring state. The dashed line is the noise limit $e^{-\gamma t}$ (see the main text). Inset: Fourier transforms of the signals show (broadened) peaks at $\omega =\pi$ for the MBL DTC and $\omega =0$ for the MBL PM. (b) DTC signal for different system sizes $L$ and error rates $p$. All curves in the main panel overlap within statistical error, showing that the signal does not depend on $L$, and depends on $p$ only through the product $pn$, proportional to the number of accumulated errors per site. Inset: same data versus the number of Floquet cycles $n$. Different system sizes are indistinguishable.

Figure 7

Temporal autocorrelator $C(n)$ for $L=16$ qubits, averaged over position, 100 disorder realizations, and 100 initial states, with maximal disorder in the Ising-odd $h$ fields ($h\in [0,2\pi ]$), and with disorder $W$ in the Ising-even $\varphi$ angles.

Figure 8

(a) Temporal autocorrelator $C(n)$ for $L=20$ qubits in the presence of noise (rate $p={10}^{-2}$) in the MBL DTC phase ($g/\pi =39/80$, disorder in the controlled-phase angles $W=\pi /2$) for the four initial bitstring states indicated in the legend. For each state, the data are averaged over position and at least ${10}^3$ combined realizations of disorder and noise (i.e., quantum trajectories). The dashed line indicates the decoherence bound $e^{-\gamma n}$, also present in Fig. 6, with $\gamma$ defined in Eq. (17). The bitstring indicated by $|\mathsf{r}\mathsf{a}\mathsf{n}\mathsf{d}\mathsf{o}\mathsf{m}⟩$ is $|00101010100110111001⟩$. (b) Same simulation but without disorder in the controlled-phase angles, $W=0$ [with $\overline{\varphi }=\pi$ as in (a)]. Other model parameters are $\mathrm{\Delta }\theta =\overline{\theta }=\mathrm{\Delta }h=\pi /50$.

Figure 9

Effect of different noise models on the DTC signal [autocorrelator $C(n)$, averaged over position and ${10}^3$ disorder realizations] for $L=20$ qubits at $g/\pi =39/80$. All noise channels are single qubit and have rate $p=3\times {10}^{-3}$ (see the text).

Figure 10

Numerical simulation of $L=16$ qubits in the DTC phase with depolarizing noise (as in Fig. 6, with $p=0.01$) acting only on a single bond (qubits $i=0,1$; boundary conditions are periodic), averaged over ${10}^4$ realizations of disorder. The dashed line represents an arbitrary threshold (0.8) used to extract a “decay time” $\tau$ for each site. Inset: DTC signal’s decay time, $\tau (d)$, diverges exponentially in the distance $d$ from the noisy bond, consistent with the presence of exponentially localized integrals of motion in the MBL DTC phase.