Nonlocal discrete time crystals in periodically driven surface codes

Discrete time crystals (DTCs) are nonequilibrium phases of matter characterized by robust subharmonic order parameter dynamics. We report a type of DTC in a periodically driven surface code, the subharmonic signature of which is only observable with respect to some nonlocal order parameter. By modifying the commonly used metrics for characterizing ordinary DTCs, i.e., the spectral functions, spin-glass order parameters, and two-point correlators, we further numerically demonstrate the rich phases of the proposed system. Specifically, depending on the system parameters and boundary conditions, we find that the system may fall into either a “surface code” phase, trivial paramagnet phase, nonlocal period-doubling DTC phase, or nonlocal period-quadrupling DTC phase. Our work thus demonstrates the prospect of exploring topological codes for discovering novel phases of matter.

Figure 1

Schematics of $H_{S,\mathrm{surface}}$ on a $6\times 6$ lattice. Physical qubits live on the vertices of the lattice, $X$- and $Z$-type stabilizer generators are highlighted by blue and red, respectively, and the logical operators $\overline{X}$ and $\overline{Z}$ are highlighted by blue- and red-filled circles, respectively.

Figure 2

The stroboscopic dynamics of $\langle \overline{Z}\rangle$ and $\langle \overline{X}\rangle$ with respect to the initial state of ${\prod }_{i,j}{e}^{-i(\pi /8)Y_{i,j}}|00\cdots 0\rangle$ in the surface code of size (a),(c) $2\times 2$ and (b),(d) $3\times 3$. The system parameters are chosen as ${\overline{h}}_{x,1,j}={\overline{h}}_{x,i\ne 1,j}-\pi /2={\overline{h}}_{z,i,1}={\overline{h}}_{z,i,j\ne 1}-\pi /2=0.45\pi , {\overline{J}}_{X,i,j}={\overline{J}}_{Z,i,j}=0.3, \mathrm{\Delta }{h}_{x,i,j}=\mathrm{\Delta }{h}_{z,i,j}=0.005\pi$, and $\mathrm{\Delta }{J}_{X,i,j}=\mathrm{\Delta }{J}_{Z,i,j}=0.5$, and each data point is averaged over 100 disorder realizations.

Figure 3

The stroboscopic dynamics of $\langle \overline{Z}\rangle$ and $\langle \overline{X}\rangle$ with respect to the initial state of ${\prod }_{i,j}{e}^{-i(\pi /8)Y_{i,j}}|00\cdots 0\rangle$ in the surface code of size (a),(c) $2\times 3$ and (b),(d) $2\times 4$. The system parameters are taken to be the same as those of Fig. 2 and each data point is averaged over 100 disorder realizations.

Figure 4

The (a) spectral functions and (b) generalized SG order parameters associated with the periodically driven surface code of size $3\times 3$. Apart from ${\overline{h}}_{x,1,j}={\overline{h}}_{x,i\ne 1,j}-\pi /2={\overline{h}}_{z,i,1}={\overline{h}}_{z,i,j\ne 1}-\pi /2=\overline{h}$, other system parameters are the same as those of Fig. 2 and each data point is averaged over five disorder realizations.

Figure 5

Nonlocal correlators' dynamics of $3\times 3$ periodically driven surface code Hamiltonian over a single period in the (a),(c) nonlocal DTC regime, (b) surface code regime, and (d) trivial regime. The system parameters are taken as $\mathrm{\Delta }{J}_{X,i,j}=\mathrm{\Delta }{J}_{Z,i,j}=0.5, \mathrm{\Delta }{h}_{X,i,j}=\mathrm{\Delta }{h}_{Z,i,j}=0.0125\pi /2,{\overline{J}}_{X,i,j}={\overline{J}}_{Y,i,j}=0.5$; (a) ${\overline{h}}_{X,1,j}={\overline{h}}_{Z,i,1}=0.475\pi , {\overline{h}}_{X,i\ne 1,j}={\overline{h}}_{Z,i,j\ne 1}=0.025\pi$; (b) ${\overline{h}}_{X,1,j}={\overline{h}}_{Z,i,1}=0.025\pi , {\overline{h}}_{X,i\ne 1,j}={\overline{h}}_{Z,i,j\ne 1}=0.475\pi$; (c) ${\overline{h}}_{X,1,j}={\overline{h}}_{Z,i\ne 1,1}=0.475\pi , {\overline{h}}_{X,i\ne 1,j}={\overline{h}}_{Z,1,j\ne 1}=0.025\pi$; and (d) ${\overline{h}}_{X,i,j}={\overline{h}}_{Z,i,j}=0.225\pi$. All data points are averaged over ten disorder realizations.

Figure 6

(a),(b) The stroboscopic dynamics of $\overline{Z}\equiv {\overline{Z}}_1$ and $\overline{X}\equiv {\overline{X}}_1$ with respect to the initial state ${\prod }_{i,j}{e}^{i(\pi /4)X_{i,j}}|00\cdots 0\rangle$ under the periodically driven toric Hamiltonian. The associated (c) spectral functions and (d) generalized SG order parameters as a function of the parameter $\overline{h}$. Other system parameters taken to be the same as those in Fig. 2 are used. The system size is taken to be (a)–(c) $2\times 2$ and (d) $2\times 4$. All data points are averaged over (a),(b) 100 and (c),(d) 5 disorder realizations.

Figure 7

The stroboscopic dynamics of ${\overline{Z}}_2$ with respect to the initial state ${\prod }_{i,j}{e}^{i(\pi /4)X_{i,j}}|00\cdots 0\rangle$ under the periodically driven toric Hamiltonian of size (a) $2\times 2$, (b) $4\times 2$, (c) $2\times 4$, and (d) under the periodically driven surface code (with open boundaries) of size $2\times 4$. We take $\mathrm{\Delta }{J}_{X,i,j}=\mathrm{\Delta }{J}_{Z,i,j}=0.5, \mathrm{\Delta }{h}_{X,i,j}=0.0125\pi , {\overline{J}}_{X,i,j}={\overline{J}}_{Y,i,j}=0.5$, and ${\overline{h}}_{X,1,j}={\overline{h}}_{X,i\ne 1,j}-\pi /2=2{\overline{h}}_{Z,i,j}=0.475\pi$.

Figure 8

(a) Generation of ${\mathcal{S}}_{X,i,j}$ and ${\mathcal{S}}_{Z,i,j}$ by conjugating a single Pauli operator with a series of iSWAP and single qubit gates. (b) The associated quantum circuit implementation of ${\mathcal{S}}_{Z,i,j}$.