Quantum-Inspired Circuits

• Quantum-inspired circuits are engineered systems that integrate quantum design principles like entangling gates and tensor decompositions into classical, analog, or hybrid architectures.
• They employ methodologies such as tensor network decompositions, circuit Laplacian mappings, and teleportation-inspired rewiring to emulate quantum dynamics on accessible hardware.
• This approach enables scalable simulations, controlled error in signal processing, and the emulation of quantum phenomena including topological phases and state discrimination.

Quantum-inspired circuits are engineered systems—digital, analog, or hybrid—which borrow quantum circuit design principles and phenomena to deliver functionalities or informational capabilities traditionally associated with quantum mechanics, but using platforms that may be classical, hybrid, or deploy non-unitary operations. The field encompasses a diverse array of architectures, including tensor-network circuits for quantum-inspired algorithms, mapped Schrödinger-type dynamics in electrical networks, quantum-inspired unitary and non-unitary circuits in classical or hybrid computation, and adaptations of quantum logical, combinatorial, or computational protocols for classically-tractable simulation and direct information processing.

1. Definitions and Theoretical Foundations

Quantum-inspired circuits are not constrained to actual quantum processing units (QPUs); rather, they combine quantum design principles—such as unitaries, entangling gates, projective operations, or tensor decompositions—with alternative hardware or algorithmic implementations. Quantum-inspired approaches may exploit:

• Classical tensor network algorithms that structurally emulate quantum circuits for tasks such as large-scale simulation, signal processing, and data transformation, often leveraging the formal equivalence of tensor contractions and quantum circuit evolution (Jaseem et al., 25 Jan 2026).
• Circuit Laplacian–Schrödinger correspondences where the node-voltage dynamics of classical RLC/RC networks map directly to lattice Hamiltonians or tight-binding models, enabling simulation of quantum transport and topological phases (Chen et al., 2024).
• Pseudo-fermionic factorization and PT-symmetric analogues, where classical dissipative systems (e.g., RLC circuits) are cast in a non-Hermitian quantum dynamical framework, revealing spectral properties and dualities such as loss–gain symmetry (Alicata et al., 2018).
• Unitary and non-unitary transformations embedded in classical hardware, e.g., via MPO-decomposition of (possibly non-unitary) transforms or classical implementations of quantum walks/search formats (Jaseem et al., 25 Jan 2026, Chen et al., 2024).
• Logical wire-redirections and teleportation-inspired mappings which reduce the simulation complexity of random circuits, efficiently repackaging quantum computational processes within classical algorithms (Chen et al., 2019).

2. Mathematical Structure and Key Examples

a. Quantum-Inspired Algorithms: Tensor-Network Circuits

Tensor-network circuits, notably matrix product operators (MPOs), serve as a structural backbone for efficiently realizing both unitary and non-unitary linear maps derived from quantum algorithms. For example, the quantum-inspired Laplace transform algorithm realizes the signal map

$$\chi_{k,\ell} = \sum_{j=0}^{N-1} x_j \, z_{k,\ell}^{j}$$

using a composite circuit $L = F_N \circ D$, where $D$ is a non-unitary exponential damping transform and $F_N$ is the (unitary) quantum Fourier transform (Jaseem et al., 25 Jan 2026). The resulting MPO can be compressed to logarithmic bond dimension, ensuring both controlled error and near-linear runtime in $N$.

b. Quantum-Inspired State Discrimination

Deutschian closed-timelike-curve (D-CTC)-inspired circuits allow, in the limit of infinite copies, for discrimination of non-orthogonal pure quantum states with error exponents exactly matching the multiple Chernoff bound: $$\xi = -\ln\left(\max_{i < j} |\langle \psi_i | \psi_j \rangle|^2\right)$$ The circuit is constructed from a fixed set of unitaries $\{U_i\}$, classical feedforward, and local projective measurements. It features an adaptive structure equivalent to a single-qubit memory plus a fresh test state per iteration (Vairogs et al., 2021).

c. Quantum Teleportation–Inspired Algorithms

Circuit simulation methods leveraging teleportation-like rewiring of logical flows permit low-depth but wide quantum circuits (e.g., 1000-qubit, 42-layer random circuits) to be classically simulated by recasting them into circuits on $L \ll N$ logical qubits: $$L \sim D \quad \text{(1D)}, \qquad L \sim N_{\text{cols}} \times (D/8) \quad \text{(2D grids)}$$ where $D$ is circuit depth (Chen et al., 2019). This mapping preserves output distributions and is crucial for scaling classical simulation up to the "memory barrier" of roughly $L = F_N \circ D$0 qubits.

d. Quantum-Inspired Classical Circuits and Topological Emulation

RLC, RC, and more general impedance networks, via the mapping $L = F_N \circ D$1, directly replicate a broad range of quantum lattice models. Such mappings are used to realize:

• Emulations of SSH chains, Kitaev chains (hosting Majorana-like end modes), and 2D/3D/4D topological insulators,
• Implementation of quantum combinational logic, quantum walks, and Grover-like search by leveraging voltage-dynamical analogues of quantum evolution (Chen et al., 2024).

The design can incorporate negative impedance converters (INICs) or operational amplifiers to instantiate gain, nonreciprocity, or synthetic gauge fields.

3. Decomposition, Simulation, and Optimization Protocols

Quantum-inspired circuit design and simulation benefit from tensor decomposition and hybrid optimization strategies. For decomposing arbitrary matrix product states (MPS) to shallow quantum circuits:

• Layer-wise analytic disentangling: Iteratively truncate the MPS bond dimension, convert to two-qubit layers, and compose.
• Constrained circuit optimization: Parameterize and globally optimize all two-qubit gates to maximize circuit fidelity to a target MPS.
• Hybrid sequential protocols: Combine analytic layer addition with periodic global optimizations, achieving order-of-magnitude gains in fidelity per added depth, highly favorable in resource-constrained regimes (Rudolph et al., 2022).

These protocols provide a formal link between tensor-network-based classical methods and parametrized quantum circuits.

4. Quantum-Inspired Information Processing in Classical Networks

Classical circuits explicitly engineered to mimic quantum information tasks include:

• Topological quantum computation analogues: By emulating Majorana braiding with switchable bond-grounded T-junctions in Kitaev-chain circuits, one can generate non-Abelian gates (Chen et al., 2024).
• Quantum walks and Grover search: Continuous-time quantum walks (CTQW) on circuit networks reproduce quantum search dynamics, including quadratic hitting-time scaling and analogues of amplitude amplification.
• Unitary transforms via RLC bridges: Classical beamsplitter circuits instantiate Hadamard/DFT and arbitrary unitaries. Mesh architectures allow for synthesis of arbitrary $L = F_N \circ D$2 transforms by matrix-factorization sequences.
• Quantum-inspired activation functions: High-order cosine product functions and quantum Chebyshev-polynomial networks realize function classes and convergence rates outperforming standard Tanh or ReLU activations, with provable three-layer universal approximation theorems (Li et al., 2024).

5. Circuit Laplacian Techniques, Non-Hermitian Extensions, and Exceptional Points

Circuit Laplacian-based approaches unlock further formal territory:

• Spectral and biorthogonal analysis: Series RLC circuits can be recast in a Liouville (non-Hermitian) operator form, generating dual biorthogonal eigenbases. The gain–loss duality is encapsulated algebraically via adjoint Liouville matrices, and exceptional points associated with PT symmetry breaking are observed at critical parameter values (Alicata et al., 2018).
• Pseudo-fermionic structure: First-order RLC dynamics can be factorized using pseudo-fermionic operators satisfying $L = F_N \circ D$3, $L = F_N \circ D$4, reflecting underlying algebraic symmetries analogously to canonical quantum modes.

6. Experimental Scalability, Error Control, and Applications

Modern quantum-inspired circuits have realized practical scalability and controlled error schemes:

• MPO truncation via SVD: Operator-norm errors in MPO-based algorithms are constrained by exponential decay of singular values. Retaining $L = F_N \circ D$5 singular values ensures an error of at most $L = F_N \circ D$6, typically scaling as $L = F_N \circ D$7 (Jaseem et al., 25 Jan 2026).
• Topological and metrological applications: Realized circuits serve as robust physical simulators for topological phases, filters, and sensors. Topological modes (edge or corner, and higher-order) retain functionality under $L = F_N \circ D$8 component disorder, and non-Hermitian sensors achieve exponential sensitivity via topolectrical circuit design (Chen et al., 2024).
• Hardware requirements: Classical RLC/INIC circuits operate across bands from kHz (Schrödinger emulation) to GHz (microwave graph states), with standard passive/active components (Chen et al., 2024).

7. Outlook and Future Directions

Research on quantum-inspired circuits advances along several major fronts:

• Signal processing and analog computing: MPO-based decompositions permit direct realization of transforms (e.g., Laplace, Fourier) at scales ($L = F_N \circ D$9) tractable on classical hardware, outperforming dense implementations in $D$0 time (Jaseem et al., 25 Jan 2026).
• Quantum Chebyshev and hybrid neural architectures: QCPN approaches point to shallower, more expressive networks for classical and quantum machine learning, with potential for integration into large-scale, high-dimensional models (Li et al., 2024).
• Integrated circuit and metrology platforms: Prospects for fully CMOS-integrated topological circuits, active nonlinear topolectrical networks, and fractal/hyperbolic lattice implementations are being explored (Chen et al., 2024).
• Theory-experiment feedback: As quantum-inspired models continue to incorporate exceptional-point physics, non-Hermitian algebra, and deep connections to quantum information theory, the dovetailing of classical–quantum simulation and engineering advances is poised to further bridge quantum and classical processing paradigms.

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Quantum-inspired circuits thus constitute a mathematically rigorous, experimentally accessible, and rapidly expanding framework translating quantum computation, control, and information theory into new algorithmic and physical architectures (Vairogs et al., 2021, Jaseem et al., 25 Jan 2026, Alicata et al., 2018, Chen et al., 2024, Li et al., 2024, Rudolph et al., 2022, Chen et al., 2019).