Uncovering quantum many-body scars with quantum machine learning

Models

In this work, we consider three different systems harboring QMBSs, as detailed in this section. Our primary focus is the xorX model. In this model, single spin flip (X) occurs when its nearest neighbors satisfy the exclusive or (xor) condition. The xorX model stands out because it allows for the analytical solution of a family of exact scar states, providing a clear testbed for exploring non-thermal phenomena, though it remains an open question whether other types of scar states exist. The xorX model under open boundary condition (OBC) is described by the Hamiltonian^17,30

where \({\sigma }_{i}^{x},{\sigma }_{i}^{y},{\sigma }_{i}^{z}\) are the Pauli-X, Y, Z matrices for the i-th qubit and n is the total number of qubits. In xorX model, the boundary qubits (i = 1, n) are frozen since \(\left[H,{\sigma }_{1}^{z}\right]=\left[H,{\sigma }_{n}^{z}\right]=0\). We focus on the subspace of \(\left\langle {\sigma }_{1}^{z}\right\rangle =\left\langle {\sigma }_{n}^{z}\right\rangle =-1\). A family of exact scar states in the xorX model can be identified as¹⁷

where \({{\mathcal{N}}}_{m}\) is the normalization factor and the operator

with projectors \({P}_{i}^{0}={\left\vert 0\right\rangle }_{i}\left\langle 0\right\vert\) and \({P}_{i}^{1}={\left\vert 1\right\rangle }_{i}\left\langle 1\right\vert\). The domain wall number in the xorX model is conserved as \(\left[H,{\sum }_{i}{\sigma }_{i}^{z}{\sigma }_{i+1}^{z}\right]=0\).

The second model we consider in this work is the PXP model, which is derived from the Rydberg atom system in the Rydberg blockade regime³¹. The Hamiltonian of the PXP model is^31,32

where Ω represents the overall energy scale. Notably, the known scar states exhibit a large overlap with the anti-ferromagnetic (Néel) state \(| {Z}_{2}\left.\right\rangle\), commonly referred to as Z₂ tower states, which accounts for the persistent oscillations observed in experiments¹⁶. Although a few scar states can be analytically solved using matrix product states³², the nature of other scar states remains an active area of research^22,34,35,36.

The far-coupling Ising SSH model is realized on the platform of superconducting circuit³³. The serpentine routing makes it flexible to tune the coupling between different qubits. The Hamiltonian is

where \({\sigma }^{+}=| 1\left.\right\rangle \left\langle \right.0|\) and \({\sigma }^{-}=| 0\left.\right\rangle \left\langle \right.1|\) represent the raising and lowering operators, respectively. J_e and J_o denote the coupling strengths at even and odd positions. J_nn is the next-next-nearest-neighbor coupling strength which breaks integrability. Both numerical simulations and experimental data provide evidence for the existence of scar states in this model, which exhibit a significant overlap with the reference state \({\left\vert {Z}_{1001}\right\rangle = \left\vert 1001\right\rangle }^{\otimes n/4}\).

Non-thermal states

We begin our study with the xorX model in Eq. (1), where there is a family of well-defined exact scar states. Since the exact scar states Eq. (2) are independent of the parameters in the Hamiltonian Eq. (1), the trained QCNN is also parameter independent. Before experimental implementation that will be presented in section Experimental demonstration on quantum device, we first perform numerical simulations on classical computers. After training, the quantum circuit classifies the eigenstates into two types. Interestingly, while the total loss decays during the training, the final converged loss remains near 0.14. Moreover, it successfully recognizes all the exact scar states with an error probability of single-shot measurement less than 1%. This means that the QCNN definitely recognizes all the exact scar state and is expected to do so with sufficient measurement in experiment³⁷. The disparity between the large loss function and the high accuracy in recognizing scar states indicates that the QCNN also classifies some additional states, beyond the known exact scars, as “scar” states. As we detail in section Spin-wave approximation for the marked states in xorX model, we identify a substantial portion of these states as non-thermal states, which bear a significant resemblance to the exact scar states.

The additional non-thermal states have similar energy as the exact scar states, as shown in Fig. 2. These states are situated in the middle of the energy spectrum, distinguishing them from the low-energy integrable modes. Their half-chain entanglement entropy is lower than that of the bulk chaotic states, as shown in Fig. 2a, b, indicating potential deviation from the volume-law entanglement entropy. The QCNN can compensate for deficiencies that the entanglement entropy may fail to distinguish states³⁸. Additionally, the participation ratio (PR), defined as \({\sum }_{i}{\left\vert \langle \psi | i\rangle \right\vert }^{4}\) for a given state \(| \psi \left.\right\rangle\) in the computational basis \(\{| i\left.\right\rangle \}\), is significantly higher than that of the majority of chaotic states, as shown in Fig. 2c, d. They are thus constrained within a smaller Hilbert space compared to chaotic states³⁹.

a, c n_dw = 2, b, d n_dw = 3. Other parameters are λ = J = 10Δ. The red crosses are the eigenstates marked by the QCNN. The prisms are the exact scar states. The number of spins is 12 while two of them at the boundaries are fixed.

In addition to the static metrics presented above, the existence of scars is often demonstrated through revivals of fidelity in quench dynamics. Here we evolve an initial state under the xorX Hamiltonian in Eq. (1). In Fig. 3, we plot the fidelity of the initial state \({\mathcal{F}}={\left\vert \langle {\psi }_{0}| {\psi }_{t}\rangle \right\vert }^{2}\) as a function of time, for three different choices of initial states. As expected, an equal superposition of all known exact scar states shows perfect revivals, as illustrated by the blue dashed curve. The superposition of the additional non-thermal states identified by the enhanced QCNN also exhibits revivals, though with a decaying amplitude and not strictly periodic oscillations, as indicated by the red solid curve. In contrast, the superposition of non-marked states does not exhibit any revival (green dashed), as is expected for generic quantum chaotic dynamics. Additional cases of fidelity oscillations are detailed in Supplementary Note 1. Such revivals in the fidelity of initial states highlight a clear distinction between states marked by the QCNN and generic chaotic states, providing further evidence for the non-thermal characteristics of the former.

The plot shows fidelity as a function of time, with the initial state being an equal-amplitude superposition of exact scar states (blue dashed line), additional states marked by QCNN (red solid line), and non-marked states (green dash-dotted line). The parameters are the same as in Fig. 2.

While the exact scar states of Eq. (2) in the xorX model are parameter independent, we expect that the fraction of these additional non-thermal states can be tuned by varying certain parameters of Hamiltonian (1). For example, upon increasing Δ, pairs of domain walls become more and more confined, which leads to slow thermalization and non-ergodic dynamics⁴⁰. In Fig. 4, we confirm that the ratio of non-thermal states identified by the enhanced QCNN increases with Δ. This further demonstrates the enhanced QCNN’s ability to discern atypical states from the eigenspectrum. On the other hand, when the circuit has more parameters, the criterion becomes stricter. Only states that are close enough to the exact scar states will be marked⁴¹. As a result, the ratio of non-thermal states is smaller, as indicated by the orange dashed line being lower than the blue solid line in Fig. 4.

The results are shown for architectures with 2 (blue solid line) and 3 (orange dashed line) convolutional layers before the pooling layers.

Spin-wave approximation for the marked states in xorX model

The numerical results presented in the previous section suggest a more detailed study on the nature of the additional non-thermal states found by QCNN in the xorX model in Eq. (1). The PR indicates that these states are predominantly localized within a small subregion of the full Hilbert space. In this section, we demonstrate that some of these non-thermal states can be understood in terms of quasiparticles, specifically magnon bound states. We will construct effective tight-binding Hamiltonians that approximately describe the spin-wave modes of these quasiparticles, reproducing key features of the exact many-body eigenstates.

Integrable states

We begin by considering the simplest scenario. The sequence of exact scar states \(| {{\mathcal{S}}}_{m}\left.\right\rangle\) in Eq. (2) satisfies m ≤ n/2 for a system of n spins. In particular, the state \(| {{\mathcal{S}}}_{\lfloor n/2\rfloor }\left.\right\rangle\) is an anti-ferromagnetic state residing close to the edge of the energy spectrum. For n even, the configuration consistent with the boundary conditions features two domains with different anti-ferromagnetic orders, separated by a single domain wall, as shown in Fig. 5. The Hamiltonian of Eq. (1) acting on this configuration generates a hopping term for the single domain wall and a staggered on-site potential that depends on the sublattice where the domain wall resides. This leads to the following effective single-particle Hamiltonian within the subspace defined by a single domain wall separating two anti-ferromagnetic domains:

where i is the position of the domain wall. In this subspace, the dynamics is fully integrable.

These eigenstates are characterized by distinct eigen-energies with similar entanglement entropy.

The above effective Hamiltonian can be readily diagonalized. Under periodic boundary conditions(PBC), the Hamiltonian in momentum space takes the form:

where we have set the lattice spacing to unity. The eigenenergies are \({E}_{k}=\pm \sqrt{{\Delta }^{2}+4{\lambda }^{2}{\cos }^{2}(k/2)}\), with corresponding eigenstates ϕ_k. Under OBC, the eigenstates approximate standing waves, expressed as superpositions of ϕ_k and ϕ_−k. Specifically, these superpositions take the form of \(\left(|{\phi}_{k}\rangle+|{\phi }_{-k}\rangle\right)/\sqrt{2}\) and \(\left(|{\phi }_{k}\rangle-|{\phi }_{-k}\rangle\right)/\sqrt{2}\). These states exhibit low entanglement entropy, characteristic of integrable systems, as shown in the inset of Fig. 5. The QCNN successfully identifies states within this integrable subspace, marking those with energies similar to the exact scar states.

Approximate quasiparticle states

To understand the nature of the states marked as ‘scar like’ by QCNN, we first calculate the mean and variance of their total z-magnetization \({S}_{z}=\mathop{\sum }\nolimits_{i = 2}^{n-1}{\sigma }_{i}^{z}\), and compare these values with those of typical thermal eigenstates. Figure 6 presents the results for two different domain wall number n_dw sectors. The states marked by QCNN (red crosses) exhibit both a lower average magnetization and a smaller variance compared to typical eigenstates. In the parameter regime with larger Δ, where spin flipping becomes more difficult, the system exhibits increased integrability and better conservation of S_z. Consequently, more eigenstates are marked by QCNN, as shown in Fig. 7.

a, c n_dw = 2; b, d n_dw = 3. The red crosses indicate the eigenstates identified by the QCNN. The green solid curves represent analytical results from the ferromagnetic magnon bound state approximation, while the purple dashed curves correspond to those obtained from the anti-ferromagnetic magnon bound state approximation. Other parameters are λ = J = 10Δ.

a, c n_dw = 2; b, d n_dw = 3. The red crosses indicate the eigenstates identified by the QCNN. The green solid curves represent analytical results from the ferromagnetic magnon bound state approximation, while the purple dashed curves correspond to those obtained from the anti-ferromagnetic magnon bound state approximation. Other parameters are λ = J = 2Δ.

This strongly suggests that these states exhibit a special structure: they can be interpreted as quasiparticles moving within an almost ferromagnetically ordered background (with a negative net magnetization). However, there are two critical differences compared to the quasiparticles in the tower of exact scar states of Eq. (2) and the fully integrable states discussed in the previous section. First, the quasiparticle picture is only approximate. While the exact eigenstates predominantly reside within the quasiparticle subspace, they also have non-negligible components in other configurations within the Hilbert space (see Fig. 9). Second, the quasiparticles in this system are generally more intricate than single magnons or domain walls, often involving longer strings or more complex structures. Moreover, the motion of these quasiparticles typically includes intermediate stages where the size of the quasiparticles first grows and then shrinks (see Fig. 8). In the following, we construct effective Hamiltonians to describe the approximate spin-wave modes of these quasiparticles and demonstrate that they capture similar key features of the exact many-body eigenstates.

On the left side of the dashed line, blue spins form a ferromagnetic background for the propagation of the bound state. On the right side of the dashed line, blue spins are trapped in an antiferromagnetic configuration. a The motion of a ferromagnetic bound state (red spins). b The motion of an anti-ferromagnetic bound state (red spins).

Ferromagnetic magnon bound state

We begin by analyzing the motion of a single magnon in a background of down spins, as depicted in Fig. 8a. In Fig. 9a, b, we present the total weight of each eigenstate within the single-magnon configuration subspace. The data reveal that certain marked states exhibit significantly larger weights in this subspace. The motion of this single magnon will necessarily involve intermediate configurations where the magnon first grows into longer strings and then shrinks. For instance, consider the following intermediate configurations (totaling four configurations): { ⋯ 00111100 ⋯ , ⋯ 0011100 ⋯ , ⋯ 001100 ⋯ , and ⋯ 00100 ⋯ }. The effective Hamiltonian within this subspace has the following form in momentum space (assuming PBC):

The analytical solution of the ground state energy is \({E}_{k}=-\Delta \sqrt{10+3{u}^{2}+\sqrt{64+48{u}^{2}+5{u}^{4}}}/\sqrt{2}\), where \(u=\left\vert \lambda +\lambda {e}^{ik}\right\vert /\Delta\). Under OBC, the system approximately forms standing waves as a superposition of states with momenta ±k. By varying k, we calculate E_k and the corresponding S_z. The relationship between these quantities is illustrated by the green solid curves in Figs. 6 and 7. Most of the marked states closely align with these curves, suggesting that they can indeed be interpreted as spin-wave modes. Furthermore, as Δ increases, the agreement between the marked states and the analytical approximation improves, as shown in Fig. 7.

The weights of the ferromagnetic magnon bound state are shown in a and b while the weights of the anti-ferromagnetic magnon bound states are shown in c and d. The special states marked by QCNN have anomalously large weight on the subspace of a particular type of quasiparticle, compared to typical eigenstates. Other parameters are λ = J = 10Δ, a, c n_dw = 2, b, d n_dw = 3.

Anti-ferromagnetic magnon bound state

We identify another component of special states, recognized by QCNN, which can be understood as quasiparticles of a short anti-ferromagnetic string, as depicted in Fig. 8b. In Fig. 9c, d, we plot the total weight of each eigenstate in the subspace of the shortest anti-ferromagnetic string. The results confirm that some marked states exhibit unusually large weights in these configurations compared to typical eigenstates. Restricting to the subspace of the four configurations shown in Fig. 8b, we can similarly write down an effective Hamiltonian:

The energy of the ground state and the first excited state are \({E}_{k}=-\sqrt{{\Delta }^{2}+2{\lambda }^{2}(1\pm \cos (k/2))}\). Since the energy of the first excited state is closer to the exact scar state, we present it with purple dashed curves in Figs. 6 and 7. The deviation of the antiferromagnetic magnon bound states from the exact eigenstates is greater than that of the ferromagnetic magnon bound states, leading to a smaller component contribution, as shown in Fig. 9.

It is worth emphasizing that there are also some integrable local modes found in this model in the low-energy regime⁴². They can be approximated by oscillators in a linear potential, which give rise to spatially localized modes. Such states with localized modes have a rather distinct nature compared to the tower of states in Eq. (2) that we use as training set. Indeed, these trivial states are not marked by the QCNN. The non-thermal states discussed in this section, in contrast, are situated in the middle of the energy spectrum.

Experimental demonstration on quantum device

We demonstrate the performance of our QCNN on IBM’s quantum hardware, with the training process carried out classically via noise-free simulations. We then prepare the exact scar state \(| {{\mathcal{S}}}_{1}\left.\right\rangle\) using a shallow circuit, which is fed into the trained QCNN on the quantum device to evaluate its performance. To combat noise, we introduce a shallow general layer as a preprocessing step to enhance hardware efficiency. Furthermore, the learning circuit is optimized by reducing the number of two-qubit gates, improving overall implementation.

The circuit used to prepare the \(| {{\mathcal{S}}}_{1}\left.\right\rangle\) state is depicted in Supplementary Note 2. The trained QCNN successfully identifies this state, achieving a success rate of over 99% in a noiseless classical simulation. The success rate observed on quantum hardware is presented in Fig. 10, which improves as the number of iterations in the learning process increases. However, due to the inherent noise in real-world quantum devices, the overall success rate is lower than that achieved in noiseless simulations.

The experiment was conducted in 12 groups, each spaced hours apart to assess drift error. Each group involved 10⁴ measurement repeats to determine the success probability. Error bars indicated by the shaded region represent the standard error across different groups.

To further enhance the performance of QCNN, we use error mitigation techniques to extrapolate to the noiseless limit. In particular, there is error from the state preparation circuit for \(| {{\mathcal{S}}}_{1}\left.\right\rangle\), due to the imperfect two-qubit gates that implement non-local swaps. Our error mitigation technique uses two different methods to boost error, which yields a relation between error rate and the overall performance that can be used for extrapolation. The first method involves randomly adding single-qubit Pauli gates to multi-qubit gates, shown in Fig. 11a. The effect of such single-qubit errors can be simulated by Monte Carlo sampling. The second method uses the transformation \(U\to U{\left({U}^{\dagger }U\right)}^{r}\), as shown in Fig. 11b. In the noiseless situation, this is equivalent to applying a single unitary and hence has no effect on the result. On a noisy device, however, appending additional layers of U^†U increases the net error accumulated during the circuit evolution, and the performance becomes worse as the number of appended layers increase. Moreover, it can be directly implemented on the quantum device.

a, b Two different methods for boosting errors in the circuit: a adding single-qubit Pauli errors; b replacing U with \(U{\left({U}^{\dagger }U\right)}^{r}\). c Error mitigation according to input fidelity. Linear regression suggests P₁ ≈ 67 ± 1% in the limit of a perfect input state. The red and blue dots correspond to Monte Carlo sampling of single-qubit Pauli errors (averaged over 10³ noise realizations) and repeating U^†U, respectively. d Error mitigation according to error rate in the second approach. Linear regression using data prior to saturation suggests P₁ ≈ 63 ± 1% in the limit of a perfect input state.

In Fig. 11c, d, we show results obtained using the two error mitigation methods discussed above. In Fig. 11c, we plot the success rate P₁ against the fidelity of the input state \({{\mathcal{F}}}_{{\rm{in}}}\), for different error rates via either adjusting the rate of single-qubit errors, or adjusting the number of appended layers r in \({({U}^{\dagger }U)}^{r}\). The method for estimating \({{\mathcal{F}}}_{{\rm{in}}}\) is detailed in Supplementary Note 2. Regression analysis in the log-log scale suggests that the success rate of the QCNN, in the limit of a perfect input state \(| {{\mathcal{S}}}_{1}\left.\right\rangle\), reaches 67%. In Fig. 11d, we instead plot the success rate P₁ against the number of appended layers r in the second approach. By repeating the gates using \(U\to U{\left({U}^{\dagger }U\right)}^{r}\), the error is assumed to be 1 + 2r times that of U. The success probability decreases with increasing r and eventually saturates due to the finite size of the Hilbert space. Regression analysis before saturation suggests that the success rate of the QCNN, in the absence of input errors, reaches 63%.

Although experimental errors weaken the performance of QCNN, the classification signal is not completely drowned out by the noise. Our experimental results demonstrate that QCNN can still achieve a good success rate in the presence of noise, which we recover by using error mitigation. In this work we mitigate the state preparation noise, while the QCNN circuit error mitigation will be deferred to future works dedicated to full experimentation.

Generalizations to other models

In this section, we extend our QCNN-based approach to two additional models that host QMBSs. We begin with the PXP model, utilizing training data that include both analytically solvable and numerically identified scar states. Specifically, we incorporate the four exact scar states reported in ref. ³² and consider the states with the largest overlap with the \(| {Z}_{2}\left.\right\rangle\) across various energy windows as scar states³¹. Notably, the QCNN identifies many additional states as potentially non-thermal states, as shown in Fig. 12a. Some of these states exhibit smaller overlaps with \(| {Z}_{2}\left.\right\rangle\), remaining hidden within the chaotic spectrum. Inspired by ref. ⁴³, a symmetric subspace \({\mathcal{K}}\) can be constructed, which exhibits regular motion. The quasimodes within this subspace can be viewed as approximations of certain eigenstates. The subspace \({\mathcal{K}}\) is spanned by the basis states

where x represents the binary configuration of the spin chain, with the constraint that neighboring spins cannot both be in the “1″ state. Here, n₁ and n₂ denote the total number of “1”s at odd and even positions, respectively, and \({{\mathcal{N}}}_{{n}_{1},{n}_{2}}\) is the normalization factor. The Hamiltonian in this subspace is expressed as \(\langle {n}_{1},{n}_{2}| {H}_{{\rm{xorX}}}| {n}_{1}^{{\prime} },{n}_{2}^{{\prime} }\rangle\). The eigenstates of quasimodes in this subspace are shown as yellow squares in Fig. 12a, and these quasimodes closely align with some of the marked states. In Fig. 12b, we plot the probability distribution of eigenstates within the subspace \({\mathcal{K}}\). Notably, the marked states have significant components within this subspace, demonstrating that the QCNN effectively learns hidden properties of the quasimodes without prior knowledge. Additionally, the QCNN not only identifies the top band in Fig. 12a, which is near the quasimodes, but also marks a second, lower band. This band has attracted considerable interest, though it still lacks a theoretical explanation.

a Overlap between the eigenstates and the \(| {Z}_{2}\left.\right\rangle\) state, with red crosses marking the states identified by the QCNN. The yellow squares are the eigenstates of quasimodes in the symmetric subspace \({\mathcal{K}}\). b Weights of each eigenstates within the symmetric subspace \({\mathcal{K}}\). c Density of QCNN-identified states based in (a). d Density of energy differences between the QCNN-identified states in (a). In both (c) and (d), only states with non-zero overlap with the \(| {Z}_{2}\left.\right\rangle\) state are counted, with smoothing applied using Gaussian broadening of each points set to 0.1Ω/2.

The dynamics of these states are governed by their energy spectrum. The marked states exhibit energies similar to those of the Z₂ tower states. These states form distinct energy towers, as depicted in Fig. 12c. Their energies are approximately equally spaced, as shown in Fig. 12d, indicating that their linear superposition can lead to stable oscillations.

The perturbations in the PXP model exhibit distinct behaviors. In cases where perturbations enhance quantum many-body scars^44,45,46, we observe a reduction in the number of non-thermal states identified by the QCNN. Under a uniform magnetic field perturbation, the QCNN additionally marks the second-highest energy band, alongside the top band, which is generally recognized as hosting scar states. Results for various perturbations are presented in Supplementary Note 3.

We also train the QCNN using the far-coupling Ising SSH model. The training data includes numerically solved scar states that exhibit significant overlaps with the Z₁₀₀₁ state³³. Due to the smaller number of scar states in this model compared to others with the same number of qubits, the accuracy of the QCNN is reduced. Nonetheless, the QCNN identifies several additional states, as shown in Fig. 13a. Some of these states have smaller overlaps with the Z₁₀₀₁ state. Although the peaks in the energy spectrum shown in Fig. 13b appear mixed and unclear, we can clarify their behavior by examining the energy spectrum in Fig. 13c. In Fig. 13c, the towers near ΔE = ±2.9J_e are primarily contributed by the scar states, whereas the towers near ΔE = ±1.2J_e are contributed by the additional states identified by QCNN. This suggests that the newly found non-thermal states exhibit a different oscillation frequency compared to the scar states.

a Overlap between the eigenstates and the Z₁₀₀₁ state. The red crosses are states marked by QCNN. b Density of QCNN-identified states based in (a). c Density of energy differences between the QCNN-identified states in (a). In both (b) and (c), only states with non-zero overlap with the \(| {Z}_{1001}\left.\right\rangle\) state are counted, with smoothing applied using Gaussian broadening of each points set to 0.25J_e.

We extend the QCNN approach to other models and discover additional states. The energy difference spectrum reveals that these identified states are non-thermal with relatively small dispersion. This demonstrates the QCNN’s capability to identify non-thermal states across various models. Analytical understanding of these non-thermal states, similar to that presented for the xorX model in this work, is an interesting open problem.