In this subsection, we will provide a detailed introduction to the MBL Floquet system and the specifics of the model we use.MBL is a phenomenon in quantum many-body systems where the system does not reach thermal equilibrium even over long periods, instead remaining in a localized quantum state. This localization is the result of the combined effects of disorder and interactions within the system, leading to the formation of a series of localized integrals of motion, known as l-bits[24]. These localized integrals of motion are an extensive set of local conservation laws that explain many of the unique features of the MBL phase, including its unusually slow information scramblings[33], the logarithmic entanglement lightcone[34], the area law entropy growth[35, 24] in any initial state, in another word in every eigenstates even in high energy states, the different of orthogonality catastrophe[36] and the possibility of novel forms of "localization-protected" order[37].

Floquet MBL[38] is a special case of MBL involving periodically driven quantum systems. In Floquet systems, the temporal evolution is not generated by a constant Hamiltonian but is described by a discrete time-evolution operator that is repeatedly applied over subsequent time steps. This periodic driving can lead to new physical phenomena, including localization in the absence of energy conservation[39, 25].

There are many ways to construct an Floquet MBL system[25, 30, 39]. In theory, we can construct a unitary that conforms to MBL by sampling random quantum gates on a qubit chain that comply with the GUE (Gaussian Unitary Ensemble), without the need for specific conservation laws[39].

Consider a random time evolution operator composed of a circuit of random unitaries coupling even and odd neighboring spins on a qubit chain in turn. Unitaries constructed from such two-qubit gates can always be decomposed into parameterized circuits of single-qubit and two-qubit gates, thus providing the possibility of constructing them into VQAs.We begin with a general model of a floquet MBL for Eq. (7), it is an XY-Model periodically "Kicked" by a rotation X-gate. The unitary is:

Where the "interacton term" Hintsubscript𝐻intH_{\mathrm{int}}italic_H start_POSTSUBSCRIPT roman_int end_POSTSUBSCRIPT is

In such a model, the disorder strength of the parameter hjzsubscriptsuperscriptℎ𝑧𝑗h^{z}_{j}italic_h start_POSTSUPERSCRIPT italic_z end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_j end_POSTSUBSCRIPT (that is, randomness, in VQA) controls whether the unitary is in the MBL regime. The parameter Jx⁢ysuperscript𝐽𝑥𝑦J^{xy}italic_J start_POSTSUPERSCRIPT italic_x italic_y end_POSTSUPERSCRIPT controls the magnitude of “hopping,” which interferes with the formation of MBL. The parameter JjZsubscriptsuperscript𝐽𝑍𝑗J^{Z}_{j}italic_J start_POSTSUPERSCRIPT italic_Z end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_j end_POSTSUBSCRIPT determines from a fermionic perspective that this is a many-body interaction, and its disorder will determine the stability of the MBL regime; the greater the disorder strength, the more stable it is.

As an example for MBL circuit, we simply sets Jx⁢ysuperscript𝐽𝑥𝑦J^{xy}italic_J start_POSTSUPERSCRIPT italic_x italic_y end_POSTSUPERSCRIPT to zero in this model, with the aim of simplifying the circuit and making it easier to implement on NISQ devices. We only need to set the period T𝑇Titalic_T to 1212\frac{1}{2}divide start_ARG 1 end_ARG start_ARG 2 end_ARG, and these parameters then take the form commonly used in VQA circuits. This model can consider as a disorderd Ising Hamiltonian with on-site longitudinal field, Hc=∑i,i+12⁢Ji⁢σiz⁢σi+1z+∑ihi⁢Zisubscript𝐻𝑐subscript𝑖𝑖12subscript𝐽𝑖subscriptsuperscript𝜎𝑧𝑖subscriptsuperscript𝜎𝑧𝑖1subscript𝑖subscriptℎ𝑖subscript𝑍𝑖H_{c}=\sum_{i,i+1}2J_{i}\sigma^{z}_{i}\sigma^{z}_{i+1}+\sum_{i}h_{i}Z_{i}italic_H start_POSTSUBSCRIPT italic_c end_POSTSUBSCRIPT = ∑ start_POSTSUBSCRIPT italic_i , italic_i + 1 end_POSTSUBSCRIPT 2 italic_J start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT italic_σ start_POSTSUPERSCRIPT italic_z end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT italic_σ start_POSTSUPERSCRIPT italic_z end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_i + 1 end_POSTSUBSCRIPT + ∑ start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT italic_h start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT italic_Z start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT with imperfect qubit flip Hf=∑igi⁢σixsubscript𝐻𝑓subscript𝑖subscript𝑔𝑖subscriptsuperscript𝜎𝑥𝑖H_{f}=\sum_{i}g_{i}\sigma^{x}_{i}italic_H start_POSTSUBSCRIPT italic_f end_POSTSUBSCRIPT = ∑ start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT italic_g start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT italic_σ start_POSTSUPERSCRIPT italic_x end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT specify to dynamicsconsisting of alternating applications of Hcsubscript𝐻𝑐H_{c}italic_H start_POSTSUBSCRIPT italic_c end_POSTSUBSCRIPT and Hfsubscript𝐻𝑓H_{f}italic_H start_POSTSUBSCRIPT italic_f end_POSTSUBSCRIPT for timeT=12𝑇12T=\frac{1}{2}italic_T = divide start_ARG 1 end_ARG start_ARG 2 end_ARG, thus the unitary Operator is

It can be seen that MBL can be stably maintained over a wide parameter space, which makes it possible for us to perform stable gradient updates on the parameters. The most important two-qubit ZZ-rotation can even cover the entire range of available parameters, which also makes our circuit somewhat resistant to noise. There has already been some discussion of this in earlier theoretical research[31, 29]. In earlier theoretical research[29], it was used to construct discrete time symmetry and double periodicity to create time crystals. Adjusting the size of g𝑔gitalic_g can make the system in the PM phase, DTC phase, or Thermal phase. We retain g𝑔gitalic_g in our circuit to increase non-commutativity, enriching the rotation of the circuit, and allowing each qubit to rotate not only within the z basis. At the same time, we can easily distinguish the variance and entropy dynamics of each regiem by only adjusting g𝑔gitalic_g.

According to previous work[31], to ensure that the evolution conforms to MBL, we need to set the parameters sampled randomly Ji∈[−1.5⁢π,−0.5⁢π]subscript𝐽𝑖1.5𝜋0.5𝜋J_{i}\in[-1.5\pi,-0.5\pi]italic_J start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT ∈ [ - 1.5 italic_π , - 0.5 italic_π ], hi∈[−π,π]subscriptℎ𝑖𝜋𝜋h_{i}\in[-\pi,\pi]italic_h start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT ∈ [ - italic_π , italic_π ],if we repeat this UFsubscript𝑈𝐹U_{F}italic_U start_POSTSUBSCRIPT italic_F end_POSTSUBSCRIPT in D𝐷Ditalic_D times with gi∈(0,0.2⁢π]subscript𝑔𝑖00.2𝜋g_{i}\in(0,0.2\pi]italic_g start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT ∈ ( 0 , 0.2 italic_π ] or gi∈[0.84⁢π,π)subscript𝑔𝑖0.84𝜋𝜋g_{i}\in[0.84\pi,\pi)italic_g start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT ∈ [ 0.84 italic_π , italic_π ), The system will undergo Floquet MBL evolution, particularly, if gi∈[0,0.2⁢π)subscript𝑔𝑖00.2𝜋g_{i}\in[0,0.2\pi)italic_g start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT ∈ [ 0 , 0.2 italic_π ), it will be in the paramagnetic(PM) phase,gi∈[0.84⁢π,π)subscript𝑔𝑖0.84𝜋𝜋g_{i}\in[0.84\pi,\pi)italic_g start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT ∈ [ 0.84 italic_π , italic_π ), it will be in the DTC phase[31].if we repeat UFsubscript𝑈𝐹U_{F}italic_U start_POSTSUBSCRIPT italic_F end_POSTSUBSCRIPT D times and chose this circuit as RPQC, thus the parameters be Jisubscript𝐽𝑖{J_{i}}italic_J start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT hisubscriptℎ𝑖{h_{i}}italic_h start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT and gisubscript𝑔𝑖g_{i}italic_g start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT. We illustrate the gate sequence of this RPQC in Fig. 1 (b).