Simulation of Higher-Dimensional Discrete Time Crystals on a Quantum Computer

Abstract

The study of topologically ordered states has given rise to a growing interest in symmetry-protected states in quantum matter. Recently, this theory has been extended to quantum many-body systems, which demonstrate ordered states at low temperatures. An example of this is the discrete time crystal (DTC), which has been demonstrated in a real quantum computer and in driven systems. These states are periodic in time and are protected from disorder to a certain extent. In general, DTCs can be classified into two phases: the stable many-body localization (MBL) state and the disordered thermal state. This work demonstrates the by generalizing DTCs to two dimensions, where there was an decrease in the thermal noise and an increase in the operating range of the MBL range in the presence of disorder.

1. Introduction

An emerging topic in condensed matter physics is the engineering of new states of matter via Floquet driving of the systems. Floquet driving can create new hybrid topological states and ordered phases in matter, which do not exist in normal systems [1,2,3,4]. Crystalline structures are examples of many-body systems, which can give rise to spatially ordered systems under periodic driving. In recent years, the study of discrete time crystals has gained considerable interest as a variation on Floquet systems [5].

Discrete time crystals (DTCs) are periodically driven quantum many-body systems, which break time translation symmetry under certain driving conditions. These systems have been predicted theoretically and have been observed in periodically driven crystal systems [6,7,8,9,10,11,12]. DTCs can be realized into two phases: the strongly disordered thermal phase, which occurs when there is low long-range disorder, and the many-body localization phase, where eigenstates are observable over a long time period and have no dissipation with discrete spectra. Discrete time crystals have been realized experimentally in cold atom systems and in condensed matter systems with a periodic laser pulse [13,14,15,16]. Building on the previous work, these many-body systems have been realized using real quantum computers and in quantum simulations [17,18,19]. Furthermore, there has been a rising theoretical interest in studying DTCs with longer periodic phases and systems, which map to higher dimensions (≥2) [20]. The study of time crystals that exist in a higher-dimensional lattice provides additional degrees of freedom in order to study time translational symmetry with multiple different phases in the same system. Recently, there has been an observation of a 2D discrete time crystal, which was discovered in a 1T-TaS${}_2$ crystal system with periodic driving [21,22,23]. In addition, there has been theoretical modeling of magnetically ordered discrete time crystals, which show that DTCs in two dimensions have a larger protection against thermal disorder in the one dimensional system [24]. When this model is further extended to $N\ge 2$ dimensions, there is further protection against thermal disorder, with diminishing returns with higher dimensions [25,26].

This works presents an extension of 1D discrete time crystals that were realized with quantum algorithms. Using quantum simulation, it was demonstrated that 2D time crystals were more protected from disorder then 1D time crystals. In addition, this theoretical work can be extended into higher-dimensional time crystals, which can also be realized in quantum hardware.

2. Discrete Time Crystals

A 1D DTC can be described as a many-body Bloch–Floquet Hamiltonian with phase-flip and disorder terms [27,28,29,30,31]:

$$\begin{array}{cc}\hfill H=& H_1+H_2+H_3\hfill \\ \hfill H_1=& g(1-ϵ)\sum _i{\sigma }_i^y\hfill \\ \hfill H_2=& \sum _{\langle i,j\rangle }{J}_{i,j}{\sigma }_i^z{\sigma }_j^z\hfill \\ \hfill H_3=& \sum _i{D}_i{\sigma }_i^x.\hfill \end{array}$$

(1)

One instance of H over a time step $\tau$ is considered to be a Floquet step; any number of iterations can be appended sequentially to form a DTC. $H_1$ indicates the driving part of the Hamiltonian, where the spin rotates are determined by g, the rabi frequency (scaled from 1 to 0), and a small perturbation $ϵ$ for fine tuning. The Pauli matrix ${\sigma }_i^y$ determines which axis of the spin the Rabi driving is coupled to. $H_2$ is the coupling part of the Hamiltonian; $J_{i,j}$ is the Ising coupling between spins i and j, which scales from 0 to $2\pi$. $H_3$ is the disorder part of the Hamiltonian, which is scaled by a disorder parameter that is a random distribution from 0 to $2\pi$. This is used to model noise in the system; however, in order to compare two separate time crystal models, this parameter is set to be static. All Pauli matrices denoted in Equation (1) denote which axis the parameters are couple to (${\sigma }_i^{\gamma }$ or $\gamma =x,y,z$). These matrices and be interchanged or mixed to model different interactions between the parameters and the spin components. For this work, only the spin interactions denoted in Equation (1) were considered. In future work, a mix of states could be considered to model novel interactions with mixed Pauli matrices [32,33,34,35,36,37].

The Ising Hamiltonian, denoted as H, serves as a fundamental theoretical model for defining a system of interacting spins. In this work, these spins correspond to individual qubit orientations. This model can be readily extended to N dimensions by generalizing the hopping Hamiltonians to incorporate the interactions between the next-nearest neighbors. By employing this approach, discrete time crystals can be effectively modeled in the context of quantum computing. This is achieved by applying gates that define the hopping terms to the next-nearest gates within the simulated system [38,39,40,41,42,43,44].

When the Ising Hamiltonian is simulated and realized using quantum computing architectures, it becomes possible to observe the existence of the discrete time crystal (DTC) phase by analyzing the autocorrelation between the state of a qubit at the first time step, denoted as Z${}_i$(0), and its state after a certain number of subsequent time steps, denoted as Z${}_i$(t). This autocorrelation is quantified as the disordered average term $\overline{\langle Z_i\left(0\right)Z_i\left(t\right)\rangle }$. Figure 1A (red line) denotes a 1D time crystal with a near maximum driving (g = 0.9) and minimal disorder in the Hamiltonian (D = 1). This corresponds to the many-body localization (MBL) phase of the DTC; these states are expected to maintain a consistent magnitude and to exhibit oscillations between two distinct states. However, in the thermal regime where disorder is larger (D = 3.14), the magnitude states decay rapidly to “thermal” noise [Figure 1A, blue lines].

When the same 16 qubits are rearranged into a 2D grid where there are more $J_{i,j}$ interactions, the same parameters (g = 0.9, D = 1) lead to a similar stability of the 1D model [Figure 1C, red line]. However, in the thermal region with increased disorder (g = 0.9, D = 3.14), the 2D model shows more resilience to the dispassion caused by thermal noise [Figure 1C, blue lines]. In order to further eluciate the effect of thermal noise on 1D vs. 2D DTCs, a Fourier transform was taken of both time crystals in their respective phases. Figure 1C corresponds to the 1D DTC, and Figure 1D corresponds to the 2D DTC. The red lines in Figure 1C,D show a similar distribution for both the 1D and 2D case; however, when the disorder was increased (3.14), the distribution decreased significantly for the 1D system, while the 2D system remained higher with less states becoming dissipated.

In the event of minimal disorder, the 2D DTC [Figure 1C,D, red] had similar performance to the 1D DTC [Figure 1A,B, red]. However, when D = $\pi$, the 2D DTC [Figure 1C,D, blue] had far less loss, at $\approx 4.3$, than the 1D DTC system, at $\approx 2.6$. This shows that there was a minimal loss of polarization in the 2D DTC when compared to a lower-order system. The difference shows that there was an increased protection against disorder as the order of the discrete time crystal system was increased. Additionally, When D = 1, [Figure 2C, red], there was a minimal loss of polarization after t time steps, and there was a recovery of the polarization of the states as the 2D time crystal developed in time (time steps $t=25-50$), whereas the 1D time crystal had a steadily decaying polarization state in the low-disorder state.

3. 1D vs. 2D DTC

The 1D DTC was modeled with the Cirq quantum simulator for 50 Floquet time steps with 16 qubits and $g=0.9$. In order to add thermal noise to the Hamiltonian in this simulation, a disorder parameter D was added. When mapped to each qubit, the autocorrelation $\overline{\langle Z\left(0\right)Z\left(t\right)\rangle }$ showed the relation to the qubit at that time step as it related to each respective qubit’s initial state. Since all qubits were initialized to zero in this work, the yellow states corresponded to the |0〉 state, while the dark blue states corresponded to the |1〉 state. Any mix between these two states was a superposition between the two states in the computational basis, thus corresponding to a decrease in the intenisty of the mapping [Figure 2].

When $D=1$, [Figure 2A] the autocorrelation $\overline{\langle Z\left(0\right)Z\left(t\right)\rangle }$ showed that the system was in the MBL phase, since there was a lower loss of magnitude and minimal depolarization in the system. Since $g=0.9$, the system was in the strong driving phase and should be in the MBL phase with minimal disorder. When D was increased to $\pi$, the disorder began to have a larger effect on the total coherence of the system. The polarization measurement of the 1D DTC showed an ordered state until $t=10$; then, it transitioned to a disordered state with increasing time steps. [Figure 2B] shows an eventual evolution into the thermal regime.

The 2D discrete time crystal is a stronger interaction form of the 1D discrete time crystal. There was an increased amount of spin interaction terms ($R_{ZZ}$ gates or $J_{i,j}$) between the nearest neighbors when the qubits were arranged in a “virtual” 2D crystal, which corresponded to the 2D grid layout of the quantum computer. For comparison, the same disorder parameter was used for the 2D case as for the 1D DTC simulation. When (D = 1), there was a similar peformance to the 1D DTC when examining the polarization measurement (D = 1) [Figure 2C]. However, when D = $\pi$, the 2D DTC retained its MBL phase with less thermal dissipation. Although there were errors that developed in the high-disorderd state, there was less dissipation [Figure 2D] than there was in the 1D DTC [Figure 2D].

4. Simulated Noise

It is important to consider the impact of thermal noise in NISQ (noisy intermediate-scale quantum) computers on the stability of the qubits and the resulting dynamics of the DTC. Over time, thermal noise can depolarize the qubits in the quantum computer, thus leading to a gradual decay of the states in the MBL phase of the DTC. To ensure reliable and robust simulations, all reported simulations set the initial polarization of the qubits to be fully polarized, as this configuration has been determined to yield the most stable qubit states over an extended period of time.

In real quantum hardware, there will always be some percentage of noise, which will effect the measurement of the qubits. In order to ensure that the simulation presented in this work could perform under large noise conditions, a simulation of the noise was performed. For the first noise simulation, the depolarization error rate was set to 0.5%, the phase damping error was set to 0.5%, and the amplitude damping error rate was set to 0.5%. These parameters were set in order to simulate real noise conditions in quantum computers, which have a fidelity rate of about 99.5% in modern architectures. For the 1D DTC model, 0.5% noise caused significant error rates in the 1D DTC, whose interactions could not overcome the noise [Figure 3A]. In the 2D model with the same number of qubits rearranged into a 2D lattice of ZZ interactions, there was a better protection from the noise, and the DTC model stayed relatively stable in the presence of noise [Figure 3C]. In order to model higher noise conditions, beyond those at which error-corrected quantum computers operate at, a 1% error rate was added for depolarization, phase damping, and amplitude damping. For the 1D DTC, there was still not a recovery of the DTC phase; however, the DTC did not depolarize at the same rate as the 0.5% error rate model [Figure 3B]. For the 2D DTC model, it was again able to outperform the 1D model and was able to maintain a recognizable time crystal model, even in the presence of 1% noise. Additionally, qubits that did become out of phase or damped recovered the DTC phase where some qubits may have been out of phase with the main DTC phase. In this case, qubits 5–10 formed their own DTC phase, which was 180 degrees out of phase with the main DTC phase [Figure 3D].

5. Discussion

The increased protection from disorder can be explained by a stronger interaction between qubits, because there are more nearest-neighbor qubits interacting with each other. The Ising interaction $J_{i,j}$ scales as $J_0/{|i-j|}^{\alpha }$. This means that the nearest-neighbor bits contribute the most to the protection against disorder, and the corner or edge states are the most prone to disorder. By reformulating the qubits from a 1D chain to a 2D lattice, it is possible to have more interaction terms with nearest neighbors. For the first nearest-neighbor term, this went from two for the 1D case to four for the 2D case with a square lattice. This effect can be scaled to N dimensions with an increase in nearest-neighbor protections to the qubits.

6. Methods

6.1. Simulation

All calculations were performed in python v 3.9.1 with the Google quantum AI Cirq package v1.0.0. GPU acceleration was enabled with the Nvidia cuQuantum 22.07.1 SDK. Within the Cirq library, the FSIM gates were used with constant perimeters $\theta =\pi$, $\varphi =\pi$, $\alpha =\pi$, $\beta =\pi$. Qubits were simulated with a grid layout similar to Google quantum machines. All qubits are typically measured in the $\widehat{Z}$ basis in a quantum machine. However, for this simulation, qubits were measured using CIRQ’s vector state simulator; the polarization measurement was then performed in the $\widehat{Z}$ basis in order to obtain the qubit rotation with respect to the $\widehat{Z}$ axis.

6.2. Edge State

When computing the average of the qubit states, all qubits were taken into consideration. Edge qubits and their effects were considered to be part of the system and were integral to the results.

6.3. Circuits

The time crystal algorithm can be split into 3 main stages: the rotations, the mixing parameters, and the long-range interactions; these stages correspond to $H_1$, $H_2$, an $H_3$, respectively, of Equation (1). The Bloch–Floquet time step can be translated into a quantum algorithm by 3 simple gates. The first stage of gates were $R_y$ single-gate rotations; for this work, all gates were rotated by $\pi$ since this is the most stable and optimal rotation for a real quantum algorithm. Next were the “spin” interactions, which can be modeled ideally with an 2-qubit $R_{ZZ}$ gate. This gate belonged to the more generalized FSIM gates, which have been used in previous works to fine-tune the circuit; however, the $R_{ZZ}$ variation was used in this work. In order for form a 2D state, the $R_{ZZ}$ gates were applied to the nearest neighbors (NN) with corresponding qubits that were theoretically arranged in a square. Since Google quantum computers are arranged in a square pattern, this algorithm could be realized with an NN algorithm. Finally, the phased XZ gate with the exponent $\alpha =0$ resulted in a rotation about the X axis with a phase determined by $Z\left(D\right)$ [Figure 4].

7. Conclusions

In conclusion, a method was developed to create two dimensional time crystals utilizing cutting-edge quantum computing architecture and novel quantum algorithms. This work shows that 2D time crystals possess superior stability and resilience compared to their one-dimensional counterparts, thereby making them highly resistant to disorder within their respective systems. However, it has been observed that the approach of incorporating additional dimensions exhibited a diminishing return in terms of reducing the noise in the system, as it required a progressively larger number of qubits to define a system with the same edge lengths. Furthermore, the implementation of time crystals in dimensions greater than or equal to three ($N\ge 3$) necessitates the existence of adjacent qubits to facilitate rapid and low-noise swapping is a requirement that poses significant challenges for quantum hardware that limited to two-dimensional or three-dimensional architectures, which must be worked out in the foreseeable future. Taking these limitations into account, there remains a keen interest in exploring the behavior of higher-dimensional time crystals using real quantum hardware, as it promises invaluable insights into the potential of these intriguing quantum phenomena. In future work, there may be a way to extend the higher-dimensional model in order to create exotic time crystals.