Neuron Product States in Quantum and Neural Systems

• Neuron Product States (NPS) are a formalism representing wavefunctions as products of neuron-like correlators, capturing long-range correlations in quantum many-body systems.
• They achieve universal approximation via sign-saturating and analytic activation functions, enabling efficient representation of complex quantum states with a controlled number of neurons.
• NPS integrate applications across quantum state representation, reservoir computing, and synthetic quantum-neuromorphic systems, offering new insights for variational algorithms and high-order memory encoding.

Neuron Product States (NPS) are a formal construction appearing in quantum many-body theory, time-series machine learning, and neuromorphic quantum architectures. They denote states or coordinates formed by products of neuron-like factors, where each neuron is a function of weighted sums (or, in some contexts, products) of elementary variables, which can be occupation numbers in Fock space or time-evolved synaptic inputs. The NPS formalism is closely connected to the universal approximation power of neural networks and the capacity of high-order monomial function bases to capture long-range correlations. In quantum settings, NPS provide a conceptually simple, variational class for fermionic wavefunctions in second quantization; in reservoir computing, they realize high-order memory and nonlinear capacity; and in synthetic quantum-neuromorphic systems, multineuron product states play a decisive role in encoding coherent information packets and controlling quantum trajectories.

1. NPS in Quantum Many-Body States: Formal Definition

In second-quantized fermionic systems, NPS are defined on the discrete Fock basis labelled by occupation-number vectors: $\vec n = (n_1, \dots, n_K) \in \{0,1\}^K$ with $K$ spin-orbitals and occupation $n_k$. A general wavefunction is a map: $$\Psi:\{0,1\}^K \longrightarrow \mathbb R\ \text{or}\ \mathbb C$$ The NPS ansatz asserts the amplitude factorizes into products of neuron correlators: $$\boxed{ \Psi_{\rm NPS}(\vec n) = \prod_{\alpha=1}^{N_h} \phi\bigl( b_\alpha + \sum_{k=1}^{K} W_{\alpha k}\,n_k \bigr) }$$ where $N_h$ is the number of neurons, $W_{\alpha k}$ are real weights, $b_\alpha$ are biases, and $\phi$ is an activation function. Alternatively, assuming $\phi(x)$ is nonvanishing,

$$\Psi_{\rm NPS}(\vec n) = \exp\left[ \sum_{\alpha=1}^{N_h} \kappa(b_\alpha + \sum_k W_{\alpha k} n_k) \right]$$

with $\kappa(x) = \ln \phi(x)$.

This construction generalizes restricted Boltzmann machine wavefunctions and is distinct from correlator product states (CPS), as discussed below (Li et al., 7 Nov 2025).

2. Universal Approximation Properties of NPS

A central result is that NPS, with properly chosen neuron activation functions and enough hidden units, can approximate any quantum state on Fock space arbitrarily well. There are two rigorous results:

2.1 Sign-Saturating Activations

If $\phi(x)$ is continuous, sign-saturating ($\phi(x)\in(-1,1)$, $\lim_{x\to+\infty} \phi(x) = 1$), e.g., $\phi(x)=\tanh(x)$, then for any target $\Psi:\{0,1\}^K\to[-1,1]$ and fixed $\epsilon>0$, there exists $N_h \le 2^K$ and parameters $W, b$ achieving

$$|\Psi_{\rm NPS}(\vec n) - \Psi(\vec n)| < \epsilon\ \ \forall\ \vec n$$

The constructive proof utilizes hyperplane separation to isolate each bitstring, then products single-neuron factors to match signs and magnitudes.

2.2 Analytic Non-Polynomial Activations

For more general analytic $\phi$, universality holds if (i) $\phi$ can take positive and negative values, and (ii) $\kappa(x)$ is not a polynomial of degree $<K$. Here, expansion in “spin” variables allows matching multilinear polynomial coefficients recursively; this provides exact control over the Fourier expansion of $\Psi(\vec n)$.

Both proofs guarantee exact universality for $N_h \le 2^K$; in practical scenarios, much fewer neurons may be sufficient.

3. Comparison: NPS vs. Correlator Product States (CPS)

CPS, or Jastrow/entangled-plaquette states, realize wavefunctions as products of full-rank local correlators over orbital clusters: $$\Psi_{\rm CPS}(\vec n) = \prod_{\{i_1,\dots,i_m\}\in\mathcal C} C^{\,n_{i_1}\dots n_{i_m}}$$ where each cluster correlator is a $2^m$-tensor over $m$ sites.

Contrasts between NPS and CPS:

Feature	NPS	CPS
Rank	Low (single function of global sum)	High (local tensor)
Support	Global (combines all sites)	Local (fixed cluster)
Parameter scaling	$O(K 2^K)$ (for universality)	$O(\binom{K}{m} 2^m)$
Expressivity	Universal with $2^K$ units	Exact for full clusters
Entanglement structure	Many simple long-range correlators	Few but high-rank local

NPS is most efficient when global low-rank structure dominates, CPS when local high-rank entanglement is prevalent (Li et al., 7 Nov 2025).

4. NPS in Reservoir Computing: Product-Unit Architectures

In reservoir computing, notably “Product Reservoir Computing” (Goudarzi et al., 2015), NPS denote reservoir coordinates formed by multiplicative neurons. For a scalar input $u(t)$ and vector state $\mathbf{x}(t) \in \mathbb R^N$,

$$x_i(t) = \left( \prod_{j=1}^N x_j(t-1)^{\Omega_{i,j}} \right) u(t-1)^{\omega_i}$$

which yields

$$\mathbf{x}(t) = \exp[ \Omega\,\log \mathbf{x}(t-1) + \boldsymbol\omega\,\log u(t-1) ]$$

These product-unit reservoirs encode exponentially many monomials of the input history, i.e. high-order time correlations, realizable as NPS. When combined with a linear readout,

$y(t) = \Psi[ \mathbf{x}(t); 1 ]$

arbitrary nonlinear functionals can be approximated.

Product reservoirs match or surpass standard tanh-ESNs for nonlinear memory retention and prediction benchmarks (Mackey-Glass, Lorenz), and are analytically tractable due to linear dynamics in the log-domain.

5. NPS in Synthetic Quantum-Neuromorphic Networks

In synthetic neuron networks with memristive qubit architectures (Nayfeh et al., 22 Jul 2025), “Neuron Product States” refer to joint product states of several neuron-qubits, each with Hamiltonian: $$H_{\rm neuron\;qubit}(t) = - g_s \sin(\theta_k(t)) (\sigma_+^{(k)} + \sigma_-^{(k)}) - A_{\rm co} g_e (\sigma_+^{(k)} + \sigma_-^{(k)}) + (\hbar \omega_{q}/2) \sigma_z^{(k)}$$ Product states across multiple neurons: $$|i_1, i_2, ..., i_M\rangle = |i_1\rangle \otimes |i_2\rangle \otimes ... \otimes |i_M\rangle$$ are generated by initializing each qubit to the ground state ($|0\rangle$), with coupling bias set to zero. Burst-mode spikes control rotations and entanglement by varying the coupling strengths ($g_{12}(t)$). Sufficiently weak coupling maintains separable product states for non-Markovian memory timescales, quantified by purity and entanglement negativity.

Algorithmic protocols include calibrated burst initialization, selective rotations, controlled entangling gates via bias adjustment, readout via membrane conductance, and coherence/entanglement measurement for packet generation. Table-I logic maps entanglement and non-Markovianity to packet routing and decision outcomes.

6. Computational and Representational Considerations

Universality proofs for NPS, as with feedforward NNs and neural-network backflow (NNBF), necessitate exponential scaling in the number of hidden units ($N_h \sim 2^K$). Each neuron requires $O(K)$ parameters; thus, total complexity for exact universality is $O(K 2^K)$. In realistic systems, substantial dimensional reduction is expected via physical or structural priors; typical applications deploy $N_h \ll 2^K$.

Comparatively, CPS with fixed local cluster size $m$ scales as $O(\binom{K}{m} 2^m)$, and is efficient for area-law entanglement. FNN and NNBF require similar exponential resources for formal universality; in practice, the target wavefunction $\Psi$ should exhibit compressibility for scalable deployment.

In product reservoir computing, simulations proceed via the $\exp/\log$ transformation and matrix algebra, affording analytic tractability and efficient memory/capacity calculations. For synthetic quantum networks, separability and memory retention hinge critically on maintaining weak coupling and tuning burst-mode spikes.

7. Connections to Universal Neural Network Quantum States

The theoretical framework underlying NPS closely mirrors universal approximation theorems in classical and quantum neural networks. Rigorous proofs leverage hyperplane separation, multilinear expansions, and analytic control of activation functions. For FNN: $$\Psi_{\rm FNN}(\vec n) = \sum_{\alpha=1}^{N_h} c_\alpha\,\sigma(b_\alpha + \vec w^T_\alpha \vec n)$$ and for NNBF: $$\Psi_{\rm NNBF}(\vec n) = \det[\phi_{p_k m}(\vec n)] = \det[\vec c_{pm}^T \sigma(b + W \vec n)]$$ both achieve universality for $N_h=2^K$ with appropriate activation tailoring.

The “sanity check” established by these results demonstrates that NPS, FNN, NNBF, and CPS all saturate the representational capacity of the many-body Hilbert space for exponential resources, but offer fundamentally distinct architectures for correlation generation and entanglement control.

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Neuron Product States unify product-form neuron correlations in quantum, classical, and neuromorphic architectures, establishing a versatile platform for universal representation, high-order memory, and controlled quantum dynamics. The formal and computational properties have direct implications for variational quantum algorithms, reservoir computing, and quantum information packet processing.