Neural Quantum States in Many-Body Physics

• Neural quantum states are variational many-body state representations where neural networks parameterize the quantum wave function, enabling efficient modeling of complex entanglement.
• NQS leverage architectures like RBMs and feed-forward networks to capture correlations, though representing volume-law entanglement in certain systems may require exponentially scaling parameters.
• Variational Monte Carlo and natural-gradient descent optimize NQS, advancing applications in quantum error correction, state tomography, and dynamic quantum simulations.

Neural quantum states (NQS) are variational quantum many-body state representations in which the complex amplitude of the wave function in a chosen basis is parameterized by a neural network. NQS provide a scalable and flexible framework for modeling highly entangled quantum systems, enabling the solution of ground states, quantum error-correcting codes, dynamical properties, and quantum state reconstruction. By leveraging neural architectures—such as feed-forward networks, restricted Boltzmann machines, recurrent and convolutional neural networks, and transformer-based models—NQS approaches efficiently encode classes of complex entanglement and correlation structures with polynomial resources in regimes that evade explicit representation or traditional tensor network techniques. Foundationally, NQS have been rigorously analyzed in the context of their representational power, expressivity, limitations, and connections to established methods in quantum information theory and quantum many-body physics.

1. Mathematical Formulation and Expressivity

The central principle of neural quantum states is to represent a quantum many-body state

$$|\psi\rangle = \frac{1}{C} \sum_{i_1,\ldots,i_n} \psi(i_1,\ldots,i_n)\,|i_1\ldots i_n\rangle$$

where $C$ is a normalization constant, and the complex amplitude function $\psi(i_1,\ldots,i_n)$ is computed by a neural network. For a feed-forward neural network with $H$ hidden layers, the recursive computation is

$$\begin{aligned} h^{(1)} &= f_1(W^{(1)} v + b^{(1)}) \ h^{(j)} &= f_j(W^{(j)} h^{(j-1)} + b^{(j)}) \text{ for } j=2,\ldots,H \ o &= W^{(H+1)} h^{(H)} + b^{(H+1)} \end{aligned}$$

The final complex amplitude can be encoded in Cartesian form $\psi(i^{n}) = o_1 + i\, o_2$ or polar form $\psi(i^n) = \exp(o_1 + i\, o_2)$. The parameter count is polynomial in $n$, while the ansatz remains sufficiently expressive to capture nontrivial multipartite entanglement structures in many-body systems (Bausch et al., 2018).

Boltzmann machine (BM) based representations, notably restricted Boltzmann machines (RBMs), are prominent NQS architectures. The RBM ansatz for an $N$-qubit system is

$$\Psi(s) = \exp\left(\sum_i a_i s_i\right) \prod_{j=1}^M 2\cosh\left(b_j + \sum_i W_{ij} s_i\right)$$

where $s_i$ labels spin configurations. The RBM admits analytic marginalization over hidden units due to the absence of intra-layer connections, facilitating efficient sampling and stochastic optimization (Jia et al., 2018). The universal approximation theorems extend to quantum states: with polynomial (often moderate) parameterizations, NQS can approximate any target amplitude distribution to arbitrary accuracy for many relevant state classes.

2. Representation of Entanglement, Limitations, and Volume Law States

The entanglement properties encoded in neural quantum states are determined not simply by the number of parameters, but by the architecture and weight connectivity patterns. RBMs with only local connectivity observe an area law ($S_{\mathcal{A}} \sim |\partial \mathcal{A}|$), whereas RBMs with fully non-local connections or deeper networks can represent states exhibiting volume law entanglement (Jia et al., 2018).

However, the representability of highly entangled states is fundamentally architecture-dependent. For instance, in the Sachdev-Ye-Kitaev (SYK) model—a paradigmatic quantum many-body system with volume-law ground state entanglement—both shallow and deep fully connected feed-forward NQS require an exponential number of parameters in system size to reach fixed energy accuracy, making efficient learning intractable for larger systems (Passetti et al., 2022). This highlights that physical properties such as volume-law entanglement and nonlocality present barriers for efficient representation with generic NQS architectures; the possibility of more specialized architectures that leverage physical structure remains an open question.

3. Variational Optimization and Learning Geometry

Variational Monte Carlo (VMC) provides the main optimization framework for NQS. The variational energy

$$E[\Psi] = \frac{\langle \Psi | \hat{H} | \Psi \rangle}{\langle \Psi | \Psi \rangle}$$

is minimized with respect to the neural parameters, sampling configurations via $|\Psi(x)|^2$. Advanced optimization uses either (stochastic) gradient descent or stochastic reconfiguration (SR). SR leverages the quantum geometric tensor (quantum Fisher matrix) $S_{\alpha\beta}$, performing natural-gradient descent: $S \cdot \delta \theta = \eta f$ with appropriate definitions of $S$ and the force vector $f$ (Park et al., 2019, Song, 3 Jun 2024). The spectrum of the quantum Fisher matrix dynamically encodes learning phases: during ground-state optimization, its evolution reveals transitions between universality, criticality, and product-like phases. Notably, the final values of the neural parameters (“weights”) are highly redundant and not a reliable diagnostic of the quantum state's physical content; instead, the geometry and spectrum of $S$ encode correlation and phase structure (Park et al., 2019).

4. Efficient Encoding of Quantum Codes and Absolutely Maximally Entangled States

One major application is the variational search for quantum codes with high coherent information under noisy channel models. For the generalized amplitude damping (GADC) or dephrasure channels, neural quantum state codes, optimized using global stochastic methods, yield rates that surpass all known repetition codes and even extend the threshold for positive quantum communication capacity beyond regimes accessible to traditional parametrizations (Bausch et al., 2018). The multipartite entanglement patterns discovered by NQS encode superadditivity effects and improved resilience to channel noise.

The same architectures enable efficient variational representations of absolutely maximally entangled (AME) states, i.e., states maximally entangled across every bipartition. The NQS amplitude function can be optimized (e.g., maximizing average marginal entropy) to approximate AME conditions to arbitrary fidelity with a polynomial number of parameters—a task that becomes intractable for explicit amplitude tables in high dimensions (Bausch et al., 2018). Specialized input encoding schemes (e.g., one-hot, scaled, or binary encodings) facilitate the representation of $d$-ary basis states.

5. Classification, Structure, and Generalizations

Neural quantum states possess intrinsic advantages for classification and structural representation tasks in quantum information. Separable neural network states (SNNS) implement structural constraints (blockwise factorization of weights) to variationally witness $K$-separability. Maximizing the fidelity with freely optimized (entangled) NQS versus constrained (product-structured) SNNS allows the construction of entanglement witnesses, and fidelity ratios directly quantify the degree of multipartite entanglement in a given state (Harney et al., 2019). The network transformations thus map directly to quantum separability structure and can be systematically analyzed using combinatorial tools such as Bell numbers.

Architectural advances enable further generalizations and more compact, efficient descriptions. For instance, “compact” NQS representations of Jastrow and stabilizer states achieve exact encoding with only $N-1$ hidden units, as opposed to the $O(N^2)$ required in previous constructions, by exploiting hidden-visible connectivity structures and tensor diagrammatic techniques (Pei et al., 2021). Autoregressive NQS and transformer-based models capture more general correlations, permit perfect sampling, and accommodate symmetries relevant to electronic structure and many-body systems (Malyshev et al., 2023, Cao et al., 23 Aug 2024). Transfer learning, message passing, and subspace expansion schemes further optimize, accelerate, and stabilize NQS optimization in challenging, highly correlated regimes (Kim et al., 2023, Cao et al., 23 Aug 2024).

6. Broader Context and Future Directions

The integration of neural quantum states into quantum many-body theory and quantum information science has broadened the landscape of application domains. These include quantum state tomography (by training a neural output to match measurement statistics over sampled bases), quantum error correction (searching for codes optimized under variational ansatz), quantum dynamics (time-dependent variational methods using dynamically evolving NQS), and simulation of both ground state and dynamical properties in systems ranging from ultra-cold gases to quantum chemistry (Jia et al., 2018, Han et al., 2021, Kim et al., 2023, Joshi et al., 13 Mar 2024).

Several open directions persist:

• Identification of which entanglement and correlation patterns admit efficient neural representations and which do not (especially in regimes with volume-law entanglement or highly nonlocal interactions (Passetti et al., 2022));
• Systematic exploitation and theoretical understanding of complex-valued network architectures for quantum amplitudes;
• Decoding entanglement and symmetry structure directly from trained neural weights and connectivity graphs;
• Efficient scaling of NQS for two- and three-dimensional systems (beyond area-law entanglement), application to codes with topological or holographic structure;
• Integration into quantum-enhanced variational algorithms and hybrid quantum-classical protocols;
• Algorithms to generate physically meaningful initializations by seeding NQS from tensor networks, followed by polynomial-time refinement (Kaneko et al., 30 Jun 2025).

The flexibility and convergence of NQS, relative to traditional direct parametrizations and compact tensor networks, suggest a broad spectrum of future utility for large-scale quantum simulations, quantum error correction, device calibration, state tomography, and the theoretical paper of entanglement in high-dimensional and multipartite systems.