Multi-Valued Quantum Neuron (MVQN)

• Multi-Valued Quantum Neuron (MVQN) is a quantum-neuron model that employs roots-of-unity and phase-based threshold logic to extend beyond traditional binary representations.
• It encompasses continuous phase-encoded variants that map real-valued inputs to complex phases, enabling quantum overlap computations and differentiable training.
• MVQN implementations utilize advanced quantum circuits and optical setups, offering fast convergence and improved performance in pattern recognition and classification.

Searching arXiv for papers on MVQN and closely related quantum-neuron formulations. Searching arXiv for "multi-valued quantum neuron", "continuous-valued quantum neuron", and related quantum perceptron papers. Multi-Valued Quantum Neuron (MVQN) denotes a class of quantum-neuron models that replace binary logic with multi-valued representations in the complex plane. In the strict formulation introduced for multiple-valued quantum logic, truth values are represented as unique roots of unity on the unit circle, and the neuron implements multiple-valued threshold logic over the field of complex numbers (AlMasri, 2023). In adjacent literature, the term is also used more broadly for non-binary or continuous-valued quantum neurons that encode real-valued inputs as phases, compute a quantum overlap between input and weight states, and read out an activation through measurement without increasing the number of qubits relative to earlier binary constructions (Mangini et al., 2020, Borba et al., 2024). Across these formulations, the defining shift is from two-valued sign structure to phase- or level-resolved representations, with outputs interpreted either as roots of unity, overlap probabilities, or continuous homodyne observables in optical settings (AlMasri, 2023, Krasimirov-Ivanov et al., 4 Dec 2025).

1. Canonical MVQN as multiple-valued threshold logic

In the canonical formulation, the logic alphabet is generalized from the binary set $E_2=\{-1,1\}$ to the $k$-valued alphabet

$$E_k=\{1=\epsilon_k^0,\epsilon_k,\epsilon_k^2,\dots,\epsilon_k^{k-1}\}, \qquad \epsilon_k^j=(e^{i2\pi/k})^j=e^{i\phi_j}, \qquad \phi_j=\frac{2\pi j}{k}.$$

Each truth value is therefore a point of unit modulus, distinguished by phase, and the neuron is formulated in the complex domain rather than in $\mathbb{R}$ (AlMasri, 2023).

The MVQN itself is defined as

$$f(x_1,x_2,\dots,x_n)=P(\omega_0+\omega_1x_1+\dots+\omega_nx_n),$$

where $x_1,\dots,x_n$ are complex-valued inputs, $\omega_0,\omega_1,\dots,\omega_n$ are complex-valued weights, $\omega_0$ is the bias, and $P$ is a discrete activation function. The activation is a sector quantizer on the complex plane,

$$P(z)=\epsilon_N^n, \qquad \frac{2\pi n}{N}\le \arg z < \frac{2\pi(n+1)}{N},$$

so the output is the root of unity associated with the angular sector containing the weighted sum $k$0 (AlMasri, 2023).

This construction makes the threshold operation intrinsically phase-based. The decision boundary is not a real hyperplane followed by a binary threshold, but a partition of the complex plane into angular sectors. The bias can be associated with the ground state because the ground-state root of unity is $k$1 (AlMasri, 2023). The model is explicitly motivated by the claim that multi-valued logic is more natural for quantum systems with more than two accessible states, including qudits, multi-level oscillators, spin systems, and optical modes (AlMasri, 2023).

The same work links this logic to a classical radix-complexity argument,

$k$2

and notes that differentiating gives a minimum near $k$3, with best integer radix $k$4. Within the paper’s argument, this provides motivation for ternary and more general multi-valued representations instead of binary ones (AlMasri, 2023).

2. Phase-encoded continuous generalizations

A second major line of work generalizes earlier binary quantum neurons into continuous-valued or non-binary phase-encoded neurons. In this formulation, a classical feature vector

$k$5

is mapped to

$k$6

and the weights

$k$7

are mapped to

$k$8

For $k$9, these vectors are embedded into $$E_k=\{1=\epsilon_k^0,\epsilon_k,\epsilon_k^2,\dots,\epsilon_k^{k-1}\}, \qquad \epsilon_k^j=(e^{i2\pi/k})^j=e^{i\phi_j}, \qquad \phi_j=\frac{2\pi j}{k}.$$0 qubits as equally weighted superpositions with feature-dependent phases,

$$E_k=\{1=\epsilon_k^0,\epsilon_k,\epsilon_k^2,\dots,\epsilon_k^{k-1}\}, \qquad \epsilon_k^j=(e^{i2\pi/k})^j=e^{i\phi_j}, \qquad \phi_j=\frac{2\pi j}{k}.$$1

and the neuron computes the overlap

$$E_k=\{1=\epsilon_k^0,\epsilon_k,\epsilon_k^2,\dots,\epsilon_k^{k-1}\}, \qquad \epsilon_k^j=(e^{i2\pi/k})^j=e^{i\phi_j}, \qquad \phi_j=\frac{2\pi j}{k}.$$2

Its activation is the squared modulus

$$E_k=\{1=\epsilon_k^0,\epsilon_k,\epsilon_k^2,\dots,\epsilon_k^{k-1}\}, \qquad \epsilon_k^j=(e^{i2\pi/k})^j=e^{i\phi_j}, \qquad \phi_j=\frac{2\pi j}{k}.$$3

with perfect matching $$E_k=\{1=\epsilon_k^0,\epsilon_k,\epsilon_k^2,\dots,\epsilon_k^{k-1}\}, \qquad \epsilon_k^j=(e^{i2\pi/k})^j=e^{i\phi_j}, \qquad \phi_j=\frac{2\pi j}{k}.$$4 implying $$E_k=\{1=\epsilon_k^0,\epsilon_k,\epsilon_k^2,\dots,\epsilon_k^{k-1}\}, \qquad \epsilon_k^j=(e^{i2\pi/k})^j=e^{i\phi_j}, \qquad \phi_j=\frac{2\pi j}{k}.$$5 (Mangini et al., 2020).

The significance of this formulation is that continuous-valued input vectors are accepted without increasing the number of qubits relative to the binary case. Precision is stored in phases rather than in extra qubits, preserving the qubit efficiency of the earlier binary model while making the neuron compatible with smooth cost functions and optimization methods such as gradient descent or SPSA (Mangini et al., 2020). A related non-binary artificial neuron explicitly frames the same move as replacing the binary $$E_k=\{1=\epsilon_k^0,\epsilon_k,\epsilon_k^2,\dots,\epsilon_k^{k-1}\}, \qquad \epsilon_k^j=(e^{i2\pi/k})^j=e^{i\phi_j}, \qquad \phi_j=\frac{2\pi j}{k}.$$6 restriction with arbitrary phase rotations of complex amplitudes, so that the measurable output becomes the squared modulus of the quantum inner product between an input state and a weight state (Borba et al., 2024).

This literature remains close to the classical perceptron analogy: the weighted sum is replaced by a quantum similarity computation, and the nonlinear effect appears through measurement. The activation in these models is not the sector quantizer of the canonical roots-of-unity MVQN; it is an overlap probability. This suggests a broad but non-uniform usage of the term MVQN: some papers reserve it for discrete multiple-valued threshold logic, while others use it for phase-encoded non-binary quantum neurons that retain a compact qubit register and an overlap-based firing rule (AlMasri, 2023, Mangini et al., 2020, Borba et al., 2024).

3. Training dynamics and learning rules

In the canonical roots-of-unity model, two learning rules are given. The Hebbian rule computes

$$E_k=\{1=\epsilon_k^0,\epsilon_k,\epsilon_k^2,\dots,\epsilon_k^{k-1}\}, \qquad \epsilon_k^j=(e^{i2\pi/k})^j=e^{i\phi_j}, \qquad \phi_j=\frac{2\pi j}{k}.$$7

with normalized form

$$E_k=\{1=\epsilon_k^0,\epsilon_k,\epsilon_k^2,\dots,\epsilon_k^{k-1}\}, \qquad \epsilon_k^j=(e^{i2\pi/k})^j=e^{i\phi_j}, \qquad \phi_j=\frac{2\pi j}{k}.$$8

The main iterative error-correction rule is

$$E_k=\{1=\epsilon_k^0,\epsilon_k,\epsilon_k^2,\dots,\epsilon_k^{k-1}\}, \qquad \epsilon_k^j=(e^{i2\pi/k})^j=e^{i\phi_j}, \qquad \phi_j=\frac{2\pi j}{k}.$$9

where $\mathbb{R}$0, $\mathbb{R}$1, $\mathbb{R}$2 is the learning rate, $\mathbb{R}$3 is the difference between desired and actual outputs, and $\mathbb{R}$4 is the complex conjugate of $\mathbb{R}$5. The corresponding weighted sum updates as $\mathbb{R}$6, which motivates the paper’s description of training as movement along the unit circle: the output phase is moved toward the target phase sector by sector (AlMasri, 2023).

The continuous phase-encoded line replaces discrete sector correction with differentiable losses. For single-pattern learning, one loss is

$\mathbb{R}$7

and for supervised binary classification on labeled examples $\mathbb{R}$8,

$\mathbb{R}$9

The continuous-valued encoding is explicitly presented as crucial because it allows smooth parameter dependence and makes gradient-based learning or automatic differentiation meaningful; the same setup is also used with SPSA (Mangini et al., 2020).

A later non-binary implementation adopts a hybrid training scheme in which the quantum device evaluates the overlap-based output and a classical optimizer updates the weights by gradient descent. Its loss is

$$f(x_1,x_2,\dots,x_n)=P(\omega_0+\omega_1x_1+\dots+\omega_nx_n),$$0

with update rule

$$f(x_1,x_2,\dots,x_n)=P(\omega_0+\omega_1x_1+\dots+\omega_nx_n),$$1

For the simulator, the study uses a 3-input neuron, a training set of 200 random inputs, learning rate $$f(x_1,x_2,\dots,x_n)=P(\omega_0+\omega_1x_1+\dots+\omega_nx_n),$$2, and stopping when the cost drops below $$f(x_1,x_2,\dots,x_n)=P(\omega_0+\omega_1x_1+\dots+\omega_nx_n),$$3 or begins to increase; the reported cost decreases from roughly $$f(x_1,x_2,\dots,x_n)=P(\omega_0+\omega_1x_1+\dots+\omega_nx_n),$$4 to about $$f(x_1,x_2,\dots,x_n)=P(\omega_0+\omega_1x_1+\dots+\omega_nx_n),$$5 after about 4000 steps (Borba et al., 2024).

A separable-state quantum perceptron with N-ary encoding uses a different hybrid quantum-classical training procedure. It updates quaternary weights according to three mismatch cases and repeats until a full cycle occurs without misclassification errors. In one run with $$f(x_1,x_2,\dots,x_n)=P(\omega_0+\omega_1x_1+\dots+\omega_nx_n),$$6 and target quaternary pattern $$f(x_1,x_2,\dots,x_n)=P(\omega_0+\omega_1x_1+\dots+\omega_nx_n),$$7, convergence occurs after 38 steps; averaged over 200 training sessions, the perceptron typically needs less than 60 steps to reach perfect recognition (Cavaletto et al., 2022).

4. Architectural variants and taxonomy

The label MVQN covers models with materially different notions of “multi-valuedness.” Some systems are canonical MVQNs in the roots-of-unity sense; others are better described as MVQN-inspired or MVQN-adjacent.

Model family	Multi-valuedness mechanism	Relation to MVQN
Roots-of-unity threshold neuron (AlMasri, 2023)	Inputs, outputs, and targets are roots of unity; activation is sector quantization	Canonical MVQN
Continuous phase-encoded neuron (Mangini et al., 2020, Borba et al., 2024)	Real-valued features and weights encoded as phases; output is overlap probability	Broad MVQN-style generalization
Separable-state N-ary perceptron (Cavaletto et al., 2022)	Each qubit encodes one multi-valued symbol from a finite alphabet	MVQN-inspired in representation, not canonical
Multi-qubit-potential perceptron (Ban et al., 2021)	Higher-order multi-qubit terms enrich the neural potential	More expressive perceptron, but binary output
Kernel-based constant-depth neuron (Carvalho et al., 2022)	Tensor-product feature mapping and parametrized activation shape	MVQN-relevant, but scalar probabilistic output

The separable-state model encodes each qubit independently as

$$f(x_1,x_2,\dots,x_n)=P(\omega_0+\omega_1x_1+\dots+\omega_nx_n),$$8

with quaternary angles $$f(x_1,x_2,\dots,x_n)=P(\omega_0+\omega_1x_1+\dots+\omega_nx_n),$$9, and compares an input pattern to a target pattern through

$x_1,\dots,x_n$0

The paper explicitly states that its multi-valuedness lives in the representation or encoding, not in a nonlinear quantum activation function, and it is best described as a separable-state quantum perceptron with N-ary encoding rather than a canonical MVQN (Cavaletto et al., 2022).

The multi-qubit-potential model retains a single-output quantum perceptron with excitation-probability readout,

$x_1,\dots,x_n$1

but extends the neural potential from

$x_1,\dots,x_n$2

to

$x_1,\dots,x_n$3

This is presented as an extension of the quantum perceptron, not as a separate MVQN formalism, and its outputs remain binary in the examples (Ban et al., 2021).

The constant-depth kernel-based framework likewise does not define a literal multi-valued-output neuron. It encodes input and weight by a feature map $x_1,\dots,x_n$4, computes $x_1,\dots,x_n$5, and outputs a Bernoulli variable equal to 1 with probability $x_1,\dots,x_n$6. Its relevance to MVQN lies in the richer feature-space mappings and in the parametrized activation family of the PCDQN, not in multiple discrete output levels (Carvalho et al., 2022).

A mixed-neuron architecture based on VQC neurons and QuantumFlow neurons is conceptually adjacent but does not mention MVQN explicitly. Its concern is heterogeneous neuron composition, not multi-valued quantum logic (Wang et al., 2021).

5. Circuit constructions and physical implementations

The phase-encoded qubit neuron has a three-stage circuit structure: input state preparation $x_1,\dots,x_n$7, weight-conjugation or overlap extraction $x_1,\dots,x_n$8, and ancilla readout through a multi-controlled NOT. Starting from $x_1,\dots,x_n$9, one applies Hadamards to obtain $\omega_0,\omega_1,\dots,\omega_n$0, then a diagonal phase unitary

$\omega_0,\omega_1,\dots,\omega_n$1

so that

$\omega_0,\omega_1,\dots,\omega_n$2

The overlap is extracted by applying the inverse of the weight-preparation unitary, summarized as

$\omega_0,\omega_1,\dots,\omega_n$3

and the ancilla is flipped conditioned on the final basis state. Measuring the ancilla returns 1 with probability

$\omega_0,\omega_1,\dots,\omega_n$4

which is the neuron activation (Mangini et al., 2020).

A related non-binary implementation introduces two synthesis routes for the phase structure: explicit rotation blocks $\omega_0,\omega_1,\dots,\omega_n$5 and the Hypergraph States Generation Subroutine (HSGS), which organizes phase corrections by Hamming weight using multi-controlled phase rotations $\omega_0,\omega_1,\dots,\omega_n$6. Hardware tests are restricted to $\omega_0,\omega_1,\dots,\omega_n$7 qubits on IBM devices, and the reported discrepancies for continuous-valued inner products are around $\omega_0,\omega_1,\dots,\omega_n$8–$\omega_0,\omega_1,\dots,\omega_n$9, with no statistically meaningful advantage of HSGS over rotation blocks in those experiments (Borba et al., 2024).

Circuit-synthesis work on the same neuron family presents two qubit-based algorithms for realizing diagonal unitaries $\omega_0$0 from $\omega_0$1. Algorithm I uses a matrix constructed from binary and Gray-code strings; Algorithm II uses an unnormalized Hadamard matrix and achieves lower depth. Ignoring multiqubit controlled gates, the reported costs are: circuit size $\omega_0$2 for Algorithm I and $\omega_0$3 for Algorithm II; circuit depth $\omega_0$4 for Algorithm I versus $\omega_0$5 for Algorithm II; and width $\omega_0$6 qubits for both (Mehta et al., 23 Jul 2025).

The same study proposes a quantum optical variant in which the neuron state is encoded in single-photon occupation of spatial modes,

$\omega_0$7

Using only local phase shifters and a multiport device, the probability of detecting the photon in mode 0 reproduces the desired overlap-based output. Strawberry Fields simulations report $\omega_0$8 for a 3D example and $\omega_0$9 for a 4D example, matching the analytical formula. The reported optical circuit size is $P$0, depth is $P$1, and width is $P$2, with the paper emphasizing the absence of expensive multiqubit $P$3 gates (Mehta et al., 23 Jul 2025).

A distinct optical route uses continuous variables rather than qubits or discrete modes. There, coherent-state inputs $P$4 are processed by a Gaussian transformation

$P$5

followed by multi-mode photon subtraction and homodyne readout. In the single-mode case, the activation family is

$P$6

which the paper characterizes as odd, smooth, non-polynomial, and tunable by the squeezing parameter $P$7. In the multi-mode case, each photon-subtracted mode acts like one effective quantum neuron, and correlated multi-mode subtraction produces additional cross-terms; the architecture is claimed to satisfy the Universal Approximation Theorem within a single layer (Krasimirov-Ivanov et al., 4 Dec 2025).

6. Applications, performance claims, and limitations

The canonical MVQN paper proposes applications in analyzing the energy spectrum of quantum systems, orbital angular momentum of light, and multi-level systems such as molecular spin qudits, explicitly mentioning $P$8 as a possible platform (AlMasri, 2023). The broader phase-encoded neuron literature emphasizes grayscale image recognition, pattern recognition, and classification of real-valued datasets. In one formulation, grayscale images are normalized into phase angles in $P$9 to avoid ambiguities due to periodicity and sign inversion, and a bias-like parameter can be introduced through phase encoding, for example with

$$P(z)=\epsilon_N^n, \qquad \frac{2\pi n}{N}\le \arg z < \frac{2\pi(n+1)}{N},$$0

to classify nonlinearly separable datasets such as concentric circles (Mangini et al., 2020).

Several papers report specific performance or expressivity claims. The roots-of-unity MVQN states that such networks enjoy fast convergence and higher functionalities compared with quantum neural networks based on binary input with the same number of neurons and layers (AlMasri, 2023). The separable-state N-ary perceptron reports recognition heatmaps with diagonal entries near 1 and off-diagonal entries $$P(z)=\epsilon_N^n, \qquad \frac{2\pi n}{N}\le \arg z < \frac{2\pi(n+1)}{N},$$1 for $$P(z)=\epsilon_N^n, \qquad \frac{2\pi n}{N}\le \arg z < \frac{2\pi(n+1)}{N},$$2 and $$P(z)=\epsilon_N^n, \qquad \frac{2\pi n}{N}\le \arg z < \frac{2\pi(n+1)}{N},$$3, based on 1024 shots, and describes itself as superior to its classical counterpart while acknowledging lower encoding power than entanglement-based alternatives (Cavaletto et al., 2022). The parametrized constant-depth quantum neuron reports that PCDQN achieved 1.0 AUC ROC on all toy tasks considered and, for digit 1 in one-vs-all handwritten-digit recognition, attained AUC ROC $$P(z)=\epsilon_N^n, \qquad \frac{2\pi n}{N}\le \arg z < \frac{2\pi(n+1)}{N},$$4 versus $$P(z)=\epsilon_N^n, \qquad \frac{2\pi n}{N}\le \arg z < \frac{2\pi(n+1)}{N},$$5 for CDQN, $$P(z)=\epsilon_N^n, \qquad \frac{2\pi n}{N}\le \arg z < \frac{2\pi(n+1)}{N},$$6 for CVQN, and $$P(z)=\epsilon_N^n, \qquad \frac{2\pi n}{N}\le \arg z < \frac{2\pi(n+1)}{N},$$7 for RBDQN (Carvalho et al., 2022). Mixed-neuron architecture work, while not MVQN-specific, reports 90.62% accuracy on full MNIST for QF-MixNN, compared with 52.77% for VQC and 69.92% for QuantumFlow (Wang et al., 2021). The continuous-variable optical neural network reports perfect accuracy in the stated one-neuron and two-neuron classification setups for “moons” and “circles,” as well as applications to multi-class MNIST with continuous latent encodings and to cubic-phase-gate synthesis (Krasimirov-Ivanov et al., 4 Dec 2025).

The limitations are equally heterogeneous. The continuous phase-encoded qubit neuron notes that circuit construction can require $$P(z)=\epsilon_N^n, \qquad \frac{2\pi n}{N}\le \arg z < \frac{2\pi(n+1)}{N},$$8 operations to imprint all phases, that the model is not yet a full multilayer feedforward network, and that more efficient compilation or approximate variational preparation methods may be needed for scalability (Mangini et al., 2020). The later non-binary implementation reports that simulator behavior does not transfer cleanly to actual IBM hardware because of noise and limited access time, and that HSGS does not show a strong experimental advantage on present devices (Borba et al., 2024). The separable-state N-ary perceptron trades encoding power for reduced multi-qubit-gate count and fault-tolerance compatibility (Cavaletto et al., 2022). The continuous-variable optical architecture argues that deeper stacks may suffer from vanishing gradients and low heralding rates when non-Gaussian resources are fixed (Krasimirov-Ivanov et al., 4 Dec 2025).

A recurrent misconception is that every non-binary quantum neuron is an MVQN in the same sense. The literature does not support that simplification. In the strict sense, MVQN denotes a roots-of-unity neuron with sector-based discrete activation (AlMasri, 2023). In broader usage, it can refer to phase-encoded non-binary neurons, N-ary separable perceptrons, or photonic neuron analogs with continuous outputs (Mangini et al., 2020, Cavaletto et al., 2022, Krasimirov-Ivanov et al., 4 Dec 2025). This suggests that MVQN is presently better understood as a family of related quantum-neuron design principles centered on moving beyond binary encoding, rather than as a single standardized architecture.