Network theory classification of quantum matter based on wave function snapshots

TL;DR

This work develops an interpretable framework to classify quantum phases directly from stochastic wave-function snapshots. It combines an Occam-inspired search for a minimal-complexity measurement basis with wave function networks (WFNs) built from Parisi two-replica overlaps to quantify correlations. In 1D spin models, the approach yields distinct data-structure fingerprints: paramagnets and SSB phases show low intrinsic dimension with structured networks, while SPT and critical phases exhibit high complexity and Erdős-Rényi or scale-free networks, respectively; a Kadanoff decimation procedure further differentiates symmetry breaking from topological order. The method is experimentally relevant, model-agnostic, and extensible to advanced network and topological analyses, offering a new route to characterize quantum matter from limited wave-function samples with potential impact on quantum simulations and information tasks.

Abstract

Quantum computers and simulators offer unparalleled capabilities of probing quantum many-body states, by obtaining snapshots of the many-body wave function via collective projective measurements. The probability distribution obtained by such snapshots (which are fundamentally limited to a negligible fraction of the Hilbert space) is of fundamental importance to determine the power of quantum computations. However, its relation to many-body collective properties is poorly understood. Here, we develop a theoretical framework to link quantum phases of matter to their snapshots, based on a combination of data complexity and network theory analyses. The first step in our scheme consists of applying Occam's razor principle to quantum sampling: given snapshots of a wave function, we identify a minimal-complexity measurement basis by analyzing the information compressibility of snapshots over different measurement bases. The second step consists of analyzing arbitrary correlations using network theory, building a wave-function network from the minimal-complexity basis data. This approach allows us to stochastically classify the output of quantum computers and simulations, with no assumptions on the underlying dynamics, and in a fully interpretable manner. We apply this method to quantum states of matter in one-dimensional translational invariant systems, where such classification is exhaustive, and where it reveals an interesting interplay between algorithmic and computational complexity for many-body states. Our framework is of immediate experimental relevance, and can be further extended both in terms of more advanced network mathematics, including discrete homology, as well as in terms of applications to physical phenomena, such as time-dependent dynamics and gauge theories.

Figures (8)

• Figure 1: Schematic representation the wave function network analysis: (a) A given quantum state is produced either in experimental quantum simulations or in numerical experiments. (b) Snapshot datasets are built in different bases. This can be done experimentally via global projective measurements, or by following some sampling scheme in numerical simulations. (c) Networks in different bases are built from datasets: each node corresponds to a snapshot and connections between nodes are established based on a cutoff of their affinity [see Sec. \ref{['sec:Construction_WFN']}]. Here we depict wave function networks (WFNs) obtained for the cluster Ising model [Eq. \ref{['eq:ClusterIsing']}], projected on the plane spanned by the two principal components of the data matrix. Color scale and nodes' sizes refer to their degrees. Top panel is an Erdős–Rényi network (ERN) in the symmetry-protected topological phase, namely at $h=0.5$ in the $x$-basis. Bottom panel is a scale-free network (SFN) at the critical point $h=1.0$ in the minimal-complexity basis $y$. In the latter, we note the presence of highly connected nodes, called hubs, characteristic of scale-free networks. Moreover, at this point two clusters start forming, due to the two-fold degeneracy of the ground state in the antiferromagnetic phase. Such clusters become more and more evident as $h$ increases. (d) From the WFNs, the quantities of interest are computed, namely the intrinsic dimension $I_d$ and the degree distribution $P_k$. Top panel shows the $I_d$ of the cluster Ising model in the $x$-, $y$-, and $z$-basis as a function of $h$. Bottom panel shows the $P_k$ of the networks displayed in (c). Note that at the critical point, in the minimal-complexity basis $y$, the degree distribution is a power law, characterizing a scale-free network.
• Figure 2: Scheme of the stochastic classification. After a filtering (a) through intrinsic dimension ($I_d$) to select the minimal-complexity basis, the network structure is studied (b). From top to bottom: Phases that show probability networks can be distinguished between paramagnetic and spontaneously symmetry-broken (SSB) ones by the presence of clusterization in data space (c), which reflects the ground state degeneracy typical of SSB phases. Erdős-Rényi networks can signal either SSB or symmetry-protected topological phases. The two can be distinguished by computing $I_d$ after sampling on increasingly larger supports (d): In the former case, it decreases abruptly as soon as the support of the order parameter is reached, while in the latter it is left unchanged. Lastly, scale-free networks are associated to gapless phases and points
• Figure 3: Schematic representation of intrinsic dimension and of the wave function networks.(a) Data points in a high-dimensional data space are constrained by correlations to live on a lower-dimensional manifold, with minimal loss of information. In the toy example shown here, the dimension of the data space, i.e. the embedded dimension, is 3, while the intrinsic dimension is 1. (b) Construction of a wave function network: (i) Given a state $\ket{\Psi}$, snapshots are taken as explained in section \ref{['sec:datasets']}. (ii) Configurations in the dataset are points in the data space and represent the nodes of the networks. (iii) A metric is chosen on the data space, and a cutoff distance $R$ is fixed. Links are then drawn to connect pairs of nodes whose distance is below $R$. The basis label $\alpha$ has been dropped for the sake of clarity.
• Figure 4: Connection between sampling support and complexity. (a) Decimated sampling procedure: pairs of neighboring sites in the MPS are merged and sampled with respect to their product as spin-1/2 variables. Performing this procedure $\ell$ times is equivalent to sampling observables with support on $2^\ell$ sites. To compare results for different values of $\ell$, we consider systems of different sizes $L$ so that $L/\ell$ is constant. (b)$I_d$ of SSH model [Eq. \ref{['eq:SSHmodel']}] for samplings with different supports. In the dimerized ordered phase, $J_A/J_B>1$, the complexity drops significantly once the support of the sampling exceeds 2. On the other hand, it doesn't decrease in the SPT phase, $J_A/J_B<1$. Note that the sampling with support bigger than one site breaks the $U(1)$ invariance of the model, while preserving the $\mathbb{Z}_2$ symmetry. For this reason, as some constraints are dropped, the complexity of the state increases from $\ell=0$ to $\ell=1$. (c)$I_d$ as a function of $\ell$. In the dimerized ordered phase of SSH model, as soon as the support of the sampling exceeds 2, the $I_d$ drops significantly. On the other hand, in the SPT phases of both Cluster Ising and SSH models, shown here respectively at $h=0.25$ and $J_A/J_B=0.1$, the $I_d$ never decreases, as there's no way to reduce the complexity of the state by sampling with respect to non-global operators.
• Figure 5: Intrinsic dimension of the ground states of 1D quantum spin model.(a-c) Intrinsic dimension of the ground states of the 1D quantum spin chains under examination, with $L=80$, sampled in $x$, $y$ and $z$ bases. We observe low value of the $I_d$ in the minimal-complexity basis (highlighted in pink) deep in the ferromagnetic and paramagnetic phases, and high values in SPT phases and inside critical phases and points. All values are computed for a number of samples $N_s=10^3$. (d) Phase diagrams of the three considered models. (e) Scaling of $I_d$ with number of samples $N_s$. The computed values are stable if $N_s$ is increased by one order of magnitude. (f) Scaling of $I_d$ with system size $L$. In ordered and disordered phases, $I_d$ has a weak dependence on $L$. In critical and SPT phases, the $I_d$ grows faster with $L$.