Kagome qubit ice

Topological phases of spin liquids with constrained disorder can host a kinetics of fractionalized excitations. However, spin-liquid phases with distinct kinetic regimes have proven difficult to observe experimentally. Here we present a realization of kagome spin ice in the superconducting qubits of a quantum annealer, and use it to demonstrate a field-induced kinetic crossover between spin-liquid phases. Employing fine control over local magnetic fields, we show evidence of both the Ice-I phase and an unconventional field-induced Ice-II phase. In the latter, a charge-ordered yet spin-disordered topological phase, the kinetics proceeds via pair creation and annihilation of strongly correlated, charge conserving, fractionalized excitations. As these kinetic regimes have resisted characterization in other artificial spin ice realizations, our results demonstrate the utility of quantum-driven kinetics in advancing the study of topological phases of spin liquids.

Topological phases of spin liquids with constrained disorder can host a kinetics of fractionalized excitations. However, spin-liquid phases with distinct kinetic regimes have proven difficult to observe experimentally. Here we present a realization of kagome spin ice in the superconducting qubits of a quantum annealer, and use it to demonstrate a field-induced kinetic crossover between spin-liquid phases. Employing fine control over local magnetic fields, we show evidence of both the Ice-I phase and an unconventional field-induced Ice-II phase. In the latter, a charge-ordered yet spin-disordered topological phase, the kinetics proceeds via pair creation and annihilation of strongly correlated, charge conserving, fractionalized excitations. As these kinetic regimes have resisted characterization in other artificial spin ice realizations, our results demonstrate the utility of quantum-driven kinetics in advancing the study of topological phases of spin liquids.
Dynamics in crystals typically proceed via the motion of topological defects such as dislocation gliding 1 . One might expect the kinetics of disordered systems to be naturally free. But in spin liquids, where disorder is present but constrained, kinetics often also proceeds through defects or excitations endowed with a conserved topological charge 2 . For instance, frustrated spin systems, such as pyrochlore 3,4 or square 5-7 spin ices, remain disordered at low temperatures, leading to a Pauling residual entropy, and their disorder is constrained by the socalled ice rule 8 . There, kinetics consists of creation/annihilation and walks of localized violations of the ice rule, in the form of emergent magnetic monopoles 9 that conserve a topological charge.
Here we present a kagome qubit ice realized in a superconducting quantum annealer. Using this experimental platform, we study its fieldinduced spin-liquid phases and quantum-activated kinetics. We experimentally establish that topological constraints affecting the dynamics proceed via charge-conserving fractionalized excitations. Using thousands of programmable external magnetic fields, we detune the system from its more common ice-rule-obeying "Ice-I" phase into a field-induced "Ice-II" phase, which exhibits charge order while remaining spin-disordered. We then report on the essential differences in quantum-activated kinetics between the two phases.

Kagome spin ice
Kagome spin ice consists of magnetic dipoles as classical binary Ising spins arranged along the edges of a hexagonal lattice and therefore on the sites of a kagome lattice. They point from one triangular "ice vertex" (kagome plaquette) to another (Fig. 1a). We can thus introduce the notion of a charge for a vertex, defined as the number of spins pointing toward the vertex minus those pointing away from it. Because of the odd coordination, a vertex can host only nonzero, odd charges q = −3, −1, 1, 3 (Fig. 1b).
The simplest magnetic kagome model includes interactions only among spins impinging on the same vertex. Since not all pairs of spins at a vertex can simultaneously assume an energy-minimizing head-totail configuration, the system is frustrated. The ground state is therefore an extensively degenerate ensemble of disordered spins obeying the (pseudo-) ice rule: frustration is minimized when each vertex has two spins pointing in and one pointing out, or vice-versa. This ice where each nearest-neighbor spin is coupled antiferromagnetically. We then embed the kagome lattice in the graph of available twobody couplers, as shown in Fig. 1c, by modifying an embedding of a Z 2 lattice gauge theory into the transverse-field Ising model 36 . Each kagome site is represented by a ferromagnetic three-qubit chain, and nearest-neighbor chains are coupled antiferromagnetically with two physical couplers. (Three qubits are needed for each kagome lattice site because it is not possible to directly couple two arbitrarily chosen qubits.) We use h and J (with no index) to denote the total field on a three-qubit chain and the total coupling between two neighboring chains, respectively, obtaining the kagome qubit ice (KQI) Hamiltonian whereΓ = Γ 3 =J 2 FM is an effective transverse field on the three-qubit chains for a ferromagnetic chain coupling J FM 37 ,σ i denotes a logical moment represented by a three-qubit chain, and indices i and j are also over three-qubit chains, rather than individual qubits.

Phases
When Γ =Γ = 0 and h = 0, the extensively degenerate ground state manifold of H KQI corresponds to that of H I , which is the commonly seen Ice-I phase 19 . But we can go beyond this regime. In nanoscopic realizations, another phase of lower entropy is possible 30,31,33,38 . In such systems, it is driven by the long-range nature of the dipolar interactions 13,14 . It still has disordered ice-rule obeying spins, but with charges ordered in an ionic lattice where A and B vertices have opposite charges. While the spins remain disordered, though at lower entropy 13,14,39 , their disorder is topologically constrained: it can be mapped to a dimer cover model 12,39 and considered a case of classical topological order 2,29,40,41 . This is often called the Ice-II phase, and its topological nature should show topologically protected kinetics. (Note also that Ice-II can also be considered a broken symmetry phase with unsaturated order parameters in the context of magnetic fragmentation 18,42,43 ).
Indeed, the kinetics in the Ice-I phase is not gapped: It is possible to flip a single spin-or indeed an extensive number of single spins-without violating the ice rule and thus without creating an excitation (see also Supplementary Informations). Thus, the system can kinetically explore the phase from within the local low-energy manifold.
Instead, in the Ice-II phase any individual spin flip disrupts the charge balance, thus creating an excitation. Therefore 2 the kinetics of the Ice-II phase must proceed either via pair creation, motion, and annihilation of gapped excitations, or else via cooperative, ungapped flips of entire loops of head-to-tail spins which do not alter the charge distribution. Such kinetics was never probed in previous realizations of kagome ice because the Ice-II phase has proved very hard to reach 30,31,33,38 (see Supplementary Informations). Fortunately, the quantum annealer offers another route: we can induce it by the field h, acting on σ z , and then we can study field-induced Ice-II kinetics.
If we define a staggered charge q s on a vertex such that q s = −q for A vertices and q s = q for B vertices, then the field h determines the vertex energies ε −3 , ε −1 , ε +1 , ε +3 for vertices with q s = −3, −1, 1, and 3, respectively, as shown in Fig. 2 (see also SI).
For 0 < h/J < 4, ε +1 has the lowest energy, leading to the chargeordered, spin-disordered Ice-II phase as the ground state. Within this window, Fig. 2 shows a regime crossover at h/J = 2. The lowest excitations are charge-order violations upsetting the ionic crystals of charges when 0 < h/J < 2, and ice rule violations when 2 < h/J < 4. The two types of excitations are degenerate at h/J = 2 where the excitation gap is highest.
Then, for h/J > 4, the ground state degeneracy vanishes, replaced by an ordered state in which all A and B vertices have charges −3 and 3, respectively.
To estimate pseudo-equilibrium properties of the kagome qubit ice in these different phases, we begin with a random spin state and repeatedly expose the system to quantum fluctuations as described by Eq. (1), by cycling the transverse field Γ on and off. An appropriate magnitude of transverse field drives the kinetics of this kagome qubit ice without erasing the state memory, as previously demonstrated in square ice 7 . After each exposure, we read out a classical spin state. This leads to a sequence of states amenable to statistics (see SI). Increasing h first leads to the ionic ordering of the charge (the Ice-II phase) eventually giving way to a polarized state in which the longitudinal field overcomes the ice rule, forming ionic crystals of ±3 charge, and all spins have value s i = 1. Figure 3b shows the corresponding result in reciprocal space via the Fourier transform of the spins defined as SðqÞ / P ij e iqðr i Àr j Þ ðhs i s j i À hs i ihs j iÞ. Our sign convention for the spins leads to the appearance of peaks only in the Ice-II phase and its proximity, and the formation of pinch points in the topologically protected region when h/J = 2.5. In Fig. 3c, cuts of the Fourier transform through the highsymmetry points in the extended Brillouin zone clearly show growing peaks at K in the proximity of the Ice-II phase. These peaks correspond to the expected logarithmic divergence of the dipolar correlations 12 (see also Fig. 5 in ref. 12 , obtained from a dimer model). They, and the pinch points, follow therefore from the topological properties induced on the phase by the charge ordering. From an implementation point of view, S(q) reveals a highly symmetric system in which the multi-qubit embedding of kagome spins preserves isotropy. This is an important advance over the previous work 7 .  Figure 3d plots the charge order parameter, defined as one-third of the average staggered charge of a vertex. The two broad plateaus at ±1/3 correspond to the Ice-II phases. Figure 3e confirms the high ice-rule obedience throughout the Ice-I and Ice-II phases, which breaks down at h/J > |4| where, from Fig. 2, the lowest energy vertex no longer obeys the ice rule.

Topologically protected quasi-classical kinetics
These measurements validate the annealer's effectiveness as an experimental platform for probing phases of the Ising kagome spin ice system near a low-temperature equilibrium. By employing consecutive output states separated dynamically by a relatively short exposure to a relatively weak transverse field Γ (compared to J), we can also probe the quasi-classical kinetics.
As mentioned above, in the Ice-II ground state a single spin flip always corresponds to fractionalized excitations, as either violation of the Ice-II charge-order constraint, or a violation of the kagome ice rule (Fig. 2). We can define a topological charge (or t-charge) as q t = q +1, q t = q−1 for A and B vertices, respectively. In the Ice-II charge-ordered ground state, the topological charge is zero on all vertices. Instead, excitations of the Ice-II phase are topologically charged. Their t-charge  is conserved: flipping a spin creates a pair of fractional excitations of t-charges ± 2 and zero net t-charge. Further flips can separate the tcharges, which can then be annihilated when meeting other, opposite ones. This situation of paired fractional excitations is very reminiscent of square and pyrochlore ice 7,44 , although here the topological charge is not the magnetic charge, as it can describe both a violation of the ice rule and a violation of the charge order.
To probe the thermal and quantum-activated kinetics of the Ice-I and Ice-II phases, we compare QA output samples. Between consecutive samples, the qubits are exposed to the transverse field for 1 μs, and at the same time J is dropped. This protocol is depicted in Fig. 4a (see also Fig. 5). Since the system is in a thermal bath at 12 mK, this allows both quantum and thermal fluctuations to drive dynamics 7 .
In agreement with the description above, our results show a kinetics of fractionalized excitations, that can be created and annihilated in pairs of opposite topological charge, and more rarely a kinetics consisting of flips of entire loops of spins-which can always be construed mathematically as creations followed by the annihilation of topologically charged pairs. Figure 4b shows two representative samples from each of h/J = 0.5, 2.5, 4, corresponding roughly to the boundaries and the middle of the field-induced Ice-II phase. Ice-rule and charge-order violations are shown as triangles. Between the two samples, we highlight the spins that flip during the exposure to fluctuations, as well as the motion of fractional excitations.
At h/J = 0.5 the charge order is fragile and we are close to the Ice-I phase (see also Fig. 3d). We see many excitations popping up erratically, and they are charge order violations, due to their small energy cost (see also Fig. 2).
At h/J = 2.5 we see far fewer excitations, and the kinetics consists of their wandering. We also see the flipping of closed loops of spins. One fractional excitation escapes off the boundary, one appears from the boundary, and one moves to another location through a chain of flipped spins. This picture is consistent with the large energy gap shown in Fig. 2, which suppresses the pair creation of excitations.
At h/J = 4, we again see a regime in which excitations can appear at low cost; these cheap excitations are now ice-rule violations, in contrast to the charge-order violations seen near the Ice-I phase, consistent with the energetics (see also Fig. 2).
To quantify the creation/annihilation and motion of fractional excitations, we consider the subgraph of the honeycomb lattice whose edges correspond to flipped spins (Fig. 4c, d) between consecutive states. We measure the degrees (valencies) of honeycomb sites in this graph. A closed loop of flipped spins results in only degree-two honeycomb sites. Conversely, an open chain of flipped spins will have degree two in the interior, and degree one on the ends. This can involve the motion of a fractional excitation, with or without creation/ annihilation. In general, degree-two spins correspond to the motion of excitations, while degree-one spins correspond to creation/annihilation. Figure 4e shows that the system is overall most active around h/J = 0 and h/J = 4, which corresponds to points of degeneracy (see also Fig. 2) where excitations are cheapest. The plot of the relative frequency of excitation motion over pair creation/annihilation shows a maximum around h/J = 2, the point of the maximum gap, where excitations are most expensive and kinetics consists mostly of their random walk, much like monopoles in square or pyrochlore spin ice at low temperature.
The non-monotonicity of the curves in Fig. 3e shows that in kagome qubit ice, by tuning the gap of the phase, the topological protection of the kinetics can be controlled, from a hard-todistinguish soup of excitations at h/J = 0, 4, to a clear picture of creation/annihilation and motion of fractionalized excitations around the value h/J = 2.

Discussion
We have realized kagome qubit spin ice in 2742 superconducting flux qubits of a quantum annealing processor and explored its fieldinduced spin-liquid ice phases. We have studied the quantum-activated, topologically protected kinetics of the Ice-II phase and shown that it proceeds via creation/annihilation and propagation of chargeconserving fractionalized excitations. We emphasize that quantum fluctuations are used here only to drive kinetics, but can be employed in the future to study entangled states. Furthermore, the kagome antiferromagnet in a transverse field Γ has a rich ground-state phase diagram 45 arising from high-order perturbations in Γ, which may be probed in future work. Our results demonstrate that quantum annealers are capable of implementing exotic programmable phases of frustrated spin liquids, whose gap and topologically protected kinetic regimes can be finely tuned.

Quantum annealing methods
The QA processor used in this work was a D-Wave Advantage QPU (Fig. 1d) housed in Burnaby, BC, Canada, operating at T = 12 mK and accessed remotely. The QPU contains 5627 operable superconducting flux qubits of which we used 2739 to implement our kagome qubit spin ice. The architecture is discussed in ref. 46 .
In quantum annealing, the Hamiltonian (1) in the Main Text is controlled by an annealing parameter s ranging from 0 to 1: where Γð0Þ≫J ð0Þ and Γð1Þ ≈ 0 ≪ J ð1Þ. Thus a typical "forward anneal", in which s is ramped linearly for the duration of anneal time t a (s = t/t a ) begins in an easily prepared superposition ground state and ends in a low-energy state of a classical Ising Hamiltonian. For simulating spin systems, it has proven useful 7,47 to employ a "quantum evolution Monte Carlo" method, in which a chain of classical samples S 0 , …S k is generated (see Fig. 5). To generate S i , the system is initialized in state S 0 at the end of the anneal (s = 1), then "reverse annealed" back to some intermediate s * , paused at s * to allow equilibration for some time t p , then quickly quenched back to s = 1. Although this method can be used to estimate observables from a transverse field Ising model at s *47 , here we just use quantum fluctuations as a driver of mixing dynamics between low energy states in the kagome ice system.
In this work, we generate chains of k = 128 samples, starting with a random initial state S 0 . To estimate equilibrium properties (Fig. 3 of Main Text) we use t p = 256 μs and discard the first 64 samples of each chain (and the random initial state) as Monte Carlo burn-in. For dynamics inquiries (Fig. 4 of Main Text and Fig. 5) we use t p = 1 μs. In both cases, we interrogate the Hamiltonian of Eq. (4) using s * = 0.32, which was chosen to give an appropriate amount of mixing in one microsecond (smaller s leads to faster mixing since both Γ=J and T=J are larger 7 ). When statistical quantities are estimated, we take the average of 200 repeated experimental iterations; each iteration includes a call to the QPU for each value of h probed.

Graph embedding
The qubits in the QA processor are intercoupled in a "Pegasus" layout 48 , in which a qubit is coupled to up to 15 other qubits. From these available couplers, we select a geometry that represents a kagome graph using three qubits per kagome spin as depicted in Fig. 1c of the Main Text. We show the full embedded lattice in Fig. 6. The kagome embedding does not require the use of all qubits, and it is possible to embed a defect-free lattice with no site vacancies, despite the existence of some inoperable qubits (empty circle in Fig. 6, panel c).
Since ferromagnetic chains are sometimes broken, they are majority-voted to provide an unambiguous mapping from the qubit system to the kagome system. We run all experiments presented herein with J ij = 0.9 for AFM couplers and J ij = −1.5 for FM couplers. This choice of ferromagnetic coupling is sufficient to guarantee that chains are almost never broken in QPU output, despite the frustration in the system.

Disorder suppression
In this application, we perform many experiments on a single programmed lattice, whose classical ground state is highly degenerate. Under such conditions, it is appropriate to refine the general-purpose QA calibration by exploiting symmetries in the system. For example, when h = 0 each qubit should have average magnetization s i = 0. Thus we tune per-qubit flux offsets to balance qubits at zero for h = 0, then use the same flux offsets when h ≠ 0. In this experiment, we are not interested in probing boundary conditions. Rather, we want to simulate the thermodynamic limit of an infinite system. In an infinite system, for any fixed h, the correlation of two neighboring kagome sites hs i s j i is the same. Thus we fine-tune the AFM couplers to promote this property. Since the three-qubit FM chains are almost never broken, we do not fine-tune the FM couplers. Similarly, for any fixed h ≠ 0, the magnetization of each qubit should be equal; we fine-tune the per-qubit fields h i to promote this property (maintaining the property that the average 1 N P i h i does not change from the nominal value h). These calibration refinements are performed before collecting the analyzed data. Figure 7 shows an example of this refinement for J = 0.9, h = 0.6, with the magnetizations and correlations achieved, and the programmed values that achieve them.

Data availability
All data are available upon request.
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