Introduction

Rydberg atom arrays constitute one of the most rapidly developing arenas for quantum simulation and quantum computing. They offer Ising-like interactions between qubits and single site manipulation, and also the ability to arrange hundreds of qubits in arbitrary geometry. This also opens up remarkable opportunities for studying topological phases of matter with fractional excitations. Recent quantum simulation advances have provided remarkable microscopic access to the quantum correlations of a \({{\mathbb{Z}}}_{2}\) quantum spin liquid (QSL)1,2. The \({{\mathbb{Z}}}_{2}\) QSL3,4 is the simplest quantum state in two spatial dimensions with fractionalized excitations and time-reversal symmetry, and has the same anyon content as the toric code5. One of the most interesting directions in quantum simulation is to construct lattice gauge theory (LGT) Hamiltonian and its nontrivial quantum states via Rydberg arrays, superconducting circuits and other platforms1,2,6,7,8. Quantum dimer model as a typical LGT is widely prepared in certain highly frustrated parameter regions of Rydberg arrays9,10,11,12,13,14,15,16. LGT models have Hilbert space fragmentation due to the local constraints, the sub-Hilbert spaces are labeled by different winding numbers and protected by topological defects. The topological defects also induce nontrivial phenomena, such as incommensurate phases and Cantor deconfinement17,18. For example, in the incommensurate phase, each topological sector becomes the ground state in turn while tuning the parameter. In the thermodynamic limit, it means infinite topological sectors one by one arise as the ground state in a finite region of parameter, which is also called ‘devil stairs’19,20. But it is a hardcore problem in experiment to make the state of the system changed from one sector to another while tuning related parameters, because the topology always hinders the jumping between different sectors. At present, there is no control scheme of experiment that can overcome the topological protection and prepare/search quantum states in a certain topological sector accurately. On the other hand, quantum annealing (QA) method is a powerful tool for optimization of Ising-like encoded Hamiltonian21,22,23,24,25,26,27,28, which utilizes the quantum fluctuation to approach the ground state. The degree of quantum acceleration depends very much on the design of QA algorithm29,30,31,32, especially for the lattice gauge model33. Following recent technological advancements in manufacturing coupled qubit systems, the QA algorithm can be embedded into superconducting flux qubits34,35,36. Currently, QA computers, i.e., quantum annealers have also been commercialized, such as the D-wave machine, and it does show higher efficiency than classical computers in certain optimization problems37,38,39,40,41. Recently, Rydberg array simulator can even realize the Ising-like encoded Hamiltonian on a large scale with high tunability1,2, which is thus an ideal QA implementation platform.

The normal optimization problems usually find the ground states of the glassy Hamiltonian as Fig. 1a. In this field, the technology has been developed very well and matured21,22,23,24,25,26,27,28,42. As the fast development of quantum control and simulation, especially recent Rydberg arrays experiments, the optimization problem of LGT Hamiltonian takes another hardcore problems of simulation [Fig. 1b], that is, how to reach the target topological sector containing the ground state. As mentioned in the introduction, for instance, the simulation of ‘devil stairs’ phenomenon requires the system can approach the ground states within different sectors under different parameters. As the complex topological defects that emerged in LGT systems are robust to quantum fluctuations, it is realized that these emergent topological structures could deeply hinder the efficiency of the existing annealing algorithms. Therefore it is the time of finding useful quantum annealing schemes for such systems, hereby, we give a powerful scheme, sweeping quantum annealing (SQA), to solve the problems. Different from the normal QA with local tunneling on each site, the SQA removes the topological defects by additional global annealing on virtual edges to overcome the barrier between the topological sectors. We demonstrate the effectiveness of SQA through some extremely egregious examples which can be realized in Rydberg experiments.

Fig. 1: Energy-Configuration landscape.
figure 1

Conventional optimization problems in glassy systems a and topological optimization problems in lattice gauge theory (LGT) cases b.

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Results

Model

Without losing generality, we use U(1) LGT as an example to test the efficiency of the different QA algorithms. Because 2+1d U(1) LGT has about L2 topological sectors, more than \({{\mathbb{Z}}}_{2}\) LGT which has only four sectors12,43. Moreover, we would design a hard-mode case to demonstrate the power of SQA. The difficult topological optimization problem has three characteristics as shown in Fig. 2a: (1) The minimum energies of many topological sectors are nearly equal. (2) The target topological sector containing the ground state occupies small Hilbert subspace, it is difficult to be found. (3) Larger topological sectors provide enough degree of freedom for quantum fluctuations in the annealing process to compete with the ground energy sector. In other words, annealing tends to pull the system into the wrong sector for a kinetic energy advantage. Therefore, we consider such a frustrated antiferromagnetic (AF) Ising model with small transverse field on a triangular lattice, whose Hamiltonian reads

$${H}_{{{{\rm{A}}}}}={J}_{x}\mathop{\sum}\limits_{{\langle ij\rangle }_{x}}{\sigma }_{i}^{z}{\sigma }_{j}^{z}+{J}_{\wedge }\mathop{\sum}\limits_{{\langle ij\rangle }_{\wedge }}{\sigma }_{i}^{z}{\sigma }_{j}^{z}-\delta \mathop{\sum}\limits_{i}{\sigma }_{i}^{x},$$
(1)

where σz and σx are the Pauli operator, 〈ij〉x and 〈ij〉∧ represent the nearest-neighbor sites on the horizontal bonds and the interchain bonds, respectively, and Jx and J∧ are the corresponding coupling strength. The emergent LGT in this model requires the δ ≪ Jx, J∧. The emergent U(1) gauge fields and topological properties in this model have been well-studied44,45,46,47,48. Due to the antiferromagnetic interaction, every triangle must be composed of two parallel and one antiparallel spins in the low-energy Hilbert space. We dub this local constraint as ‘triangle rule”, and the constraint-satisfying Hilbert space can be exactly mapped to a familiar quantum dimer model10,11,43,49,50,51. Figure 2b shows this mapping between the constrained spin configuration on triangular lattice and the dimer configuration on the dual honeycomb lattice, where the bond with two parallel spins corresponds to a dimer. The dimer density on the honeycomb lattice can be understood as lattice electric field on the dual bond, and the local constraint can be written as the divergenceless condition. There thus emerges an U(1) gauge field in this triangular AF Ising model44,48,52, and the many-body configurations with constraints can be mapped to lattice electromagnetic fields which are naturally classified into different topological sectors44. Recent numerical works48,53, which studied a triangular lattice AFM Ising model with small transverse field (δ ≪ J), have demonstrated that the parameter range of Eq.(1) is very close to the famous Rokhsar-Kivelson point and incommensurate phase with Cantor deconfinement of a general quantum dimer model on honeycomb lattice. To demonstrate the power of our algorithm, we put the Hamiltonian on a periodic boundary lattice, in which case the topological defects are much more robust.

Fig. 2: The emergent topology in the triangular lattice AFM Ising model.
figure 2

a The schematic diagram for energy configurations of different topological sectors (with the three hardcore characteristics discussed in the main text). b The mapping between the constrained spin configuration and the dimers. c The stripe phase with no defect is the ground state of Hamiltonian Eq. (1) in ND = 0 topological sector. d, e are the topological sector with ND = 2 and ND = 4 and the topological defects are denoted by the black lines. As shown in d, spin on corner can be flipped without breaking the 'triangle rule'. The defect before/after spin flipped is labeled by solid/dashed line. f A pair of spinons (triangle with 3 parallel spins) connected with defects. The spinons can be considered as local defects and they can annihilate (with local flipping) when meet.

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Topological sectors are robust to quantum fluctuations emerged in the LGT systems and can be labeled by the number of topological defects ND48,54,55,56. As examples, we show the spin configurations in different topological sectors in Fig. 2c–e for ND = 0, 2, 4, respectively, if we set the stripe configurations [Fig. 2c] as the ND = 0 reference state. Note that all these topological sectors are degenerate when Jx = J∧ and δ = 0, we thus set Jx = 0.9J∧ and δ ≪ J∧ − Jx in the following to break the degeneracy weakly and make the stripe configuration [Fig. 2c] with ND = 0 has the lowest energy. This setting is for satisfying the hardcore three characteristics mentioned above, as shown in Fig. 2a. In fact, we can also set different configuration with target ND as ground state via changing the related couplings Ji of bond i.

The ground state stripe phase in Fig. 2c is composed of horizontal bonds connecting parallel spins and interchain bonds connecting antiparallel spins. Without breaking the local constraint of the ‘triangle rule’, the number of horizontal antiparallel bonds is conserved in each row, and linking such antiparallel bonds on different chains will construct the one-dimensional global (topological) defect [Fig. 2d, e]. Note that there can be a mass of spin configurations for the same defect number, and the topological sectors thus have extensive nearly degenerate quantum states. In fact, the degeneracy is because flipping the spins at the corners of defects obeys the ‘triangle rule’ with little energy cost, as shown in Fig. 2d. Therefore, the degeneracy (degree of freedom) increases exponentially with the defect number ND12,43,48,51. Changing the defect number requires generating a pair of spinons (constraint-breaking triangles) connected by defects [Fig. 2f], letting them go around the connected boundary to meet and annihilate. Such process changes ND by two. However, exciting spinons and pulling them apart is extremely hard due to the large energy cost and global topology constraints, so it is difficult to achieve the ground-state energy sector from higher energy ones in this way.

Moreover, in the triangle rule limit (δ ≪ Jx, J∧), the J∧ − Jx > 0 make the system favor stripe phase without topological defect while the δΣx term favors more topological defects with flippable corners. The sector with many topological defects, which has much more freedom degree as the ’Sector III’ of Fig. 2a shown, is easy to reach. Of course, we could set the ’Sector III’ as the sector where the ground state in, but it seems trivial for an optimization problem because it is very easy to be found. Thus we set δ ≪ J∧ − Jx in the following (δ = 0 for simplicity), i.e., the ground state is stripe as the ’Sector I’ of Fig. 2a shown. Although the target state is a classical state, it doesn’t lose the generality because the key point here is to jump among different topological sectors while the details of the ground state is not important. Our choice aims increasing the coefficient of difficulty of such optimization.

Sweeping quantum annealing scheme

We first consider the conventional quantum annealing protocol, whose dynamics is generated by a Hamiltonian of the form

$${H}_{{{{\rm{QA}}}}}={H}_{{{{\rm{A}}}}}-h(t)\mathop{\sum}\limits_{i}{\sigma }_{i}^{x},$$
(2)

where σx is the Pauli-x operator. The annealing schedule is controlled by linearly reducing the transverse field h(t) from a sufficiently large initial value h(0) = 5J∧ to the final value h(τ) = 0, so that the original Ising Hamiltonian HA is recovered at the end.

In order to obtain the numerical results with large system size, we adopt the stochastic series expansion (SSE) method within a continuous time frame57,58,59,60,61 to resolve the quantum dynamics, which essentially belongs to the quantum Monte Carlo (QMC) simulation62. The QMC can reveal the scaling behaviors of QA as same as real-time simulations and has been widely used to study the annealing problems63,64,65,66,67,68,69,70. We use the Monte Carlo step (MCS) to label the annealing time. One MCS is defined as visiting all the spins, so it is proportional to size L2. However, as shown in Fig. 3a, the QA protocol can not reach the ground state even for long enough evolution time.

Fig. 3: The sweeping quantum annealing.
figure 3

a The energy is computed for 6 × 6 system and the true ground state energy Eg = − 39.6J∧ is denoted by the gray dashed line. The expectation value of E and its error bars are obtained from averaging over 64 independent annealing runs. The straightforward QA and QA-h have the worst annealing performance, the best among these is the SQA. In the whole paper, we use the statistical error to define the error bar, i.e., \(\sigma /\sqrt{{N}_{{{{\rm{sampling}}}}}}\), where the σ is the standard error and Nsampling is the number of QMC samplings. b Schematic diagram of SQA on a torus square lattice. We perform additional quantum annealing on `virtual edges' of the system to reduce the topological defects. In quantum simulation experiments, the global transverse field term can be controlled with tunable laser or arbitrary waveform generators86,87,88. c The SQA schematic diagram on a triangular lattice with the periodic boundary condition along both x and y directions. The polarized spins lose the Ising interactions which is similar to opening the boundary, as the white region shown.

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We have also tried the quantum annealing with inhomogeneous transverse field, i.e., \({H}_{{{{\rm{QA-h}}}}}={H}_{{{{\rm{A}}}}}+{\sum }_{i}{h}_{i}(t){\sigma }_{i}^{x}\) with site-dependent random field hi(t), which is initially chosen from the uniform distribution [0, 10J∧], and then linearly reduced to 0. We denote such annealing as ‘QA-h’, which is suggested to weaken the effect of second-order phase transition and improve the efficiency of quantum annealing71,72,73,74,75. Nevertheless, as shown in Fig. 3a, it behaves most like the conventional QA protocol and cannot improve efficiency.

The inefficiency of the above two QA protocols can be attributed to the following aspects. If the σx breaks the ‘triangle rule’, it will take a huge energy cost of about 2J∧ which is forbidden at low temperature. Thus, the σx operators always flip the spins at the corners of defects without Ising energy costs [Fig. 2d] and obtain extra kinetic energy advantage of the σx term at the same time. All these low-energy actions can not change the number of topological defects (i.e., the number of topological sectors), but just twist the corner by flipping the spin on it, such as the red spin of Fig. 2d, after flipping the spin the corner on the right side will be turned to left as shown by the dashed line. Therefore, the fluctuations introduced by both QA and QA-h can only deform the defects but cannot create/annihilate them. However, topological optimization problems usually involve non-local excitations and require global operations.

In essence, the optimization problem here is an annealing problem to find the optimal topological sectors in Fig. 2a. In this case, the number of states in a topological sector increases with the defect number ND, so the stripe configuration of the ground state is the smallest sector and hard to be reached in Hilbert space as one needs an annealing algorithm that can both tunnel through different topological sectors and reduce the energy.

In order to obtain non-local excitations to surmount the problem of the topological defects, we develop sweeping quantum annealing to achieve the ground state topological sector. The key point is the topological defects can be moved out continuously through the ’open edge’ by local operators. Thus the system can easily reach the global minimum intuitively if we can change the boundary condition during annealing. The schematic diagram of SQA on a torus lattice taking square lattice as an example is shown in Fig. 3b. The feasible algorithmic scheme on a quantum annealing platform is to do additional annealing with a large transverse field on a virtual edge as the purple circle of Fig. 3b shown. Intuitively, a large transverse field can polarize the spin to the x-axis, acting as if shearing the Ising coupling bond apart. As shown in the right part of Fig. 3b, the central spin has been polarized to reduce its Ising interaction with other spins. A great advantage is that the strength of the transverse field can be easily and continuously adjusted by tunning the Rabi frequency of the laser in the experiment. Similarly, the SQA on the triangular lattice is shown in Fig. 3c, all the spins on the virtual edge are polarized, which is equal to opening the edge effectively.

We then scan all the virtual edges one by one in turn along a certain direction with strongly annealing transverse field, which is expected to achieve the effect across topological sectors. The specific plan is as follows: (1) Keep the conventional quantum annealing process on every site as mentioned above, that is, linearly decreasing the h slowly down from h(0) zero. (2) Divide the whole process into L (system length) parts. In every part, add a extra strong transverse field (as same as h(0)) on virtual edges in order and reduce the field strength slowly to h(t). h(t) is the field strength of the traditional QA at the end of this part. Then move to the next virtual edge and repeat the same process until all the edges (1 ~ L) are visited. From Fig. 3a, we can see an obvious advantage of SQA to arrive at the ground state quickly. The detailed pseudo-code protocol of SQA is shown in Supplemental Note 2.

The physical reason for the superior behavior of SQA in the topological frustrated optimization problem is that it effectively removes topological defects in the systematic sweeping. As demonstrated in Fig. 4a, different from the QA and QA-h, the proportion of ND = 0 sectors increases with the annealing time of SQA. Furthermore, the proportion of different topological sectors is shown in Fig. 4b. The conventional QA and QA-h are stuck in ND = 4 sectors, which reveals the reason for their inefficiency. Clearly, the best of all annealing schemes is the SQA which can straightforwardly change the topology of the system at any position. In order to strengthen the evidence of the effectiveness and universality of SQA, we simulate another difficult topological optimization problem of a fully frustrated Ising system on the square lattice (corresponding to the square lattice quantum dimer model) in Supplementary Note 4, where we see that our SQA protocol is also efficient.

Fig. 4: The topological sector changes only by SQA.
figure 4

a The proportion of ground state sector as a function of the annealing time with different methods. b The proportion of all topological sectors as a function of the annealing time with different methods.

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Remarkably, our SQA protocol can prepare the target topological state at a time that scales linearly with the system size (N = L2). Figure 5 shows the energy per site E/N as a function of MCS, in which every MCS is defined with scaling as L2, i.e., 1MCS ~ L2. We see that the lines almost coincide with each other for different system size, which clearly demonstrates a linear increase of annealing time with the system size, since the annealing time is proportional to MCS67,68,69,70. The power cost for the algorithm is a strong advantage.

Fig. 5: Time complexity.
figure 5

The energy per site E/N is computed for 4 × 4, 6 × 6 and 8 × 8 systems and the true ground state energy per site is denoted by the dashed line. The expectation value of E/N and its error bars are obtained from averaging over 64 independent annealing runs. It shows a good convergence for all sizes. One MCS is defined as visiting all the spins, so MCS scales as L2.

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In addition, we have also discussed the effect of thermal annealing (TA)76,77 in Supplementary Note 6 although it is almost impossible to realize in the recent cold atom experiments. Hopefully, the result may inspire related experiments in the future.

Experimental candidates

The experimental candidates for implementing the SQA protocol are many, for example in the superconducting circuits1,78,79,80. We find SQA is in particular applicable to the Rydberg arrays, which have recently been utilized to simulate the \({{\mathbb{Z}}}_{2}\) quantum spin liquid2.

Below we discuss in detail the Rydberg arrays systems are the idea platform to simulate the U(1) LGT model [see Eq. (1)] and implement the SQA protocol. The effective Hamiltonian of Rydberg arrays is81,82

$${H}_{{{{\rm{R}}}}}=h\mathop{\sum}\limits_{i}{\sigma }_{i}^{x}-\mu \mathop{\sum}\limits_{i}{n}_{i}+V\mathop{\sum}\limits_{i > j}\frac{{n}_{i}{n}_{j}}{| i-j{| }^{6}},$$
(3)

where i and j are the site labels, ni = 0, 1 is the density operator to probe the ground state or Rydberg state, respectively, and σx is the tunneling term (Pauli-x operator) to connect the two states. We can use the spin-1/2 language to describe the qubit within the two-level Rydberg system, since there is a common mapping between the hardcore boson and spin, that is, n = (σz + 1)/2 and ground/Rydberg state corresponds to spin down/up state.

The Rydberg arrays Hamiltonian [Eq. (3)] on the triangular lattice can then be cast into the form of [Eq. (2)] with emergent U(1) LGT, it has been realized in experiment in fact83. Firstly, since the repulsive interaction decays very fast, i.e., 1/r6, the strength of the second nearest neighbors on triangular lattice is about 0.037 compared with that of the first neighbor, so the model can be further simplify as \({H}_{{{{\rm{R}}}}}=h{\sum }_{i}{\sigma }_{i}^{x}-\mu {\sum }_{i}{n}_{i}+V{\sum }_{\langle ij\rangle }{n}_{i}{n}_{j}\) with only nearest neighbors. Then, we set μ = V and use \({n}_{i}=2{\sigma }_{i}^{z}+1\) to simplify the Hamiltonian in σ operator form as \({H}_{R}=h{\sum }_{i}{\sigma }_{i}^{x}+\frac{V}{4}{\sum }_{\langle ij\rangle }{\sigma }_{i}^{z}{\sigma }_{j}^{z}.\) Lastly, set the distances of Rydberg atoms along x-axis a little farther than along other directions, and then the anisotropy Hamiltonian in Eq. (2), \({H}_{R}=h{\sum }_{i}{\sigma }_{i}^{x}+\frac{{V}_{\wedge }}{4}{\sum }_{{\langle ij\rangle }_{\wedge }}{\sigma }_{i}^{z}{\sigma }_{j}^{z}+\frac{{V}_{x}}{4}{\sum }_{{\langle ij\rangle }_{x}}{\sigma }_{i}^{z}{\sigma }_{j}^{z}\), is obtained. In this way, the superior behavior of SQA we have discussed for Eq. (2), can be readily applied to the Rydberg arrays, and all the parameters can be tuned by adjusting laser detuning or Rabi frequency. For the experimental sources of error, the main decay channels of the Rydberg states include blackbody radiation-induced errors and spontaneous radiative decay transitions to lower-lying states84. In order to mitigate them, several hardware-efficient and fault-tolerant quantum computation schemes has been proposed85. We therefore think it has well experimental feasibility to implement our SQA protocol on the present Rydberg atom array platform.

Discussion

With the fast development of Rydberg arrays simulation, precise regulation of the quantum state of LGT within the topological sector is required. For such topological optimization problems, there is still a lack of systematic quantum simulation protocol. In this work, we find that the emergent topological properties in frustrated Ising systems greatly reduce the efficiency of both conventional thermal and quantum annealing. Borrowing the idea of changing topology by cutting and gluing the system, we invent such a generalized algorithm—the sweeping quantum annealing method to solve these problems with huge quantum speedup. Moreover, the SQA can be easily implemented in realistic quantum simulation experimental platforms because the global transverse field term can be controlled finely by tunable laser or arbitrary waveform generators78,79,80,86,87,88,89,90,91. After comparing with conventional quantum and sweeping quantum annealing algorithm, we find that the SQA presents high efficiency and validity while other annealing schemes fail. The sweeping quantum annealing therefore opens up an effective and innovative way for controlling the quantum states of LGT. It will also be of interest to study the effectiveness of SQA under noises and dissipations92,93 which commonly exist in the quantum simulator.

It’s worth noting that the topological defects are in one direction in this work. Topological defects along the other direction can also be removed under the SQA along that direction. Therefore, the system with defects in both directions can be optimized by SQA mixed in both directions in principle. In future works, more general systems with both nearly degenerate topological sectors in two directions and glassy minimums will be discussed.

Methods

Quantum Monte Carlo simulation

We use the continuous-time quantum Monte Carlo method for the quantum annealing in this paper. Because the discrete-time QMC may take the opposite result due to the error67. The temperature T = 1/β we set is J∧/20, which is much smaller than the energy scale of spinon excitation ~ 2J∧. The temperature can keep the configurations obeying the ‘triangle rule’ with emergent LGT. The dynamic of annealing can be simulated via QMC because the time scaling of QMC is the same as real-time evolution, these simulations have been widely used to study the quantum annealing behaviors67,68,69,70.

The QMC we used is a stochastic series expansion (SSE) algorithm57,58,59. The partition function Z is dealt with a Taylor expansion as:

$$Z=\mathop{\sum}\limits_{\alpha }\mathop{\sum }\limits_{n=0}^{\infty }\frac{{\beta }^{n}}{n!}\langle \alpha | {(-H)}^{n}| \alpha \rangle$$
(4)

Then we extract the wanted information via sampling these configurations of Taylor expansion. More details of quantum Monte Carlo simulations for QA, QA-h, and SQA have been explained in Supplementary Note 1.