Classical-to-Quantum Mappings

Updated 18 December 2025

Classical-to-Quantum Mappings is the process of encoding classical systems into quantum frameworks using techniques like the Koopman–von Neumann formalism on photonic hardware.
It employs continuous-variable photonic representations to translate classical phase-space variables into quantum quadratures via Gaussian and non-Gaussian gate sets.
These mappings enable enhanced simulation and optimization, offering scalable gate decompositions and error correction tailored for quantum-enhanced classical computations.

Classical-to-quantum mappings refer to methodologies that encode, simulate, or embed classical dynamical systems within quantum mechanical frameworks, particularly on continuous-variable (CV) photonic quantum computing architectures. These mappings are foundational in developing quantum algorithms for classical simulation, optimization, and leveraging quantum resources for outperforming classical computational paradigms. Such mappings cover direct Hilbert-space lifts (Koopman–von Neumann formalism), variational encodings for optimization, and framework-level correspondences between classical and quantum gates in the CV regime.

1. Formal Frameworks for Embedding Classical Dynamics

A central approach to mapping classical systems into quantum frameworks is the Koopman–von Neumann (KvN) formalism. KvN reformulates classical mechanics using complex wavefunctions on phase space and linear operators, mimicking quantum mechanics but retaining commutative “position” and “momentum” operators. The KvN evolution is analogous to the quantum Schrödinger equation but evolves classical, non-interferometric distributions. In this mapping:

Every classical phase-space coordinate $(q, p)$ maps to a pair of canonical quadratures $(\hat x, \hat p)$ , where $[\hat x_j, \hat p_k]=i\delta_{jk}$ .
The classical Liouville equation $\partial_t \rho + \{\rho, H\} = 0$ becomes $i\partial_t \Psi(q, p, t) = \hat L \Psi(q, p, t)$ , with Liouvillian $\hat L$ constructed by replacing derivatives with the corresponding momentum operators.
Unitary evolution operators $U(\Delta t) = \exp(-i\Delta t\,\hat L)$ are implemented on photonic quantum hardware as Gaussian and, where necessary, non-Gaussian circuits.

This structure enables direct simulation of classical dynamical systems (e.g., harmonic oscillators, nonlinear PDEs) via quantum circuits, exploiting efficient linear algebra primitives available on quantum hardware (Gao et al., 15 Dec 2025).

2. Continuous-Variable Photonic Quantum Representations

CV photonic quantum computing operates on bosonic modes (qumodes), each represented in an infinite-dimensional Fock space $\mathcal{H} = \mathrm{span}\{\,|n\rangle\,:\,n=0,1,2,\dots\}$ (Choe, 2022, Andersen et al., 2010). Classical phase-space variables map naturally to field quadratures:

$\hat x = (\hat a + \hat a^\dagger)/\sqrt{2}$ (amplitude)
$\hat p = (\hat a - \hat a^\dagger)/(i\sqrt{2})$ (phase)

with $[\hat x, \hat p] = i$ . The basic gate set spans:

Gaussian gates: displacement $D(\alpha)$ , squeezing $S(r)$ , and beam splitter $B(\theta)$
Non-Gaussian gates (e.g., cubic phase $V(\gamma)=\exp(i\gamma \hat x^3)$ )

A pivotal property is that, for many classical-to-quantum mappings and simulation algorithms, only a polynomial number of bosonic modes and gates are required to encode discretized classical systems—often with superior scaling compared to digital qubit mappings (Gao et al., 15 Dec 2025, Abel et al., 15 Mar 2024).

3. Gate-Level Mappings and Circuit Synthesis

For direct simulation of classical time evolution (e.g., for ODEs, PDEs), one implements the time-step evolution operator via a Lie–Trotter decomposition:

The KvN Liouvillian $\hat L$ is split into realizable Hamiltonian terms, commonly a sum of quadratic (Gaussian) and higher-order (non-Gaussian) contributions.
For classical quadratic dynamics (e.g., linear oscillators), the evolution is fully implementable using only Gaussian gates—beam splitters and squeezers.
For nonlinear classical flows, specific non-Gaussian gates (such as cubic phase) are required, which are prepared either via direct ancilla injection or measurement-based quantum circuits (Gao et al., 15 Dec 2025, Abel et al., 15 Mar 2024).

This principle generalizes to more complex systems (e.g., discretized field theories), where each classical variable or spatial point corresponds to a quantum mode. Entangling gates (e.g., $C_Z=\exp(i\hat x_j\hat x_k)$ ) directly encode nearest-neighbor or global couplings present in the classical Hamiltonian.

4. Mapping Classical Optimization and Sampling Problems

Classical-to-quantum mappings are also exploited in variational and optimization frameworks on CV quantum hardware:

Optimization of classical functions $F(\mathbf{x})$ is formulated by encoding $F$ into a problem Hamiltonian $H_p = F(\hat x_1, ..., \hat x_N)$ and engineering a quantum circuit with gates corresponding to $H_p$ and a suitable mixer Hamiltonian.
Counterdiabatic and QAOA-inspired protocols realize gradient-based classical optimization within quantum circuits, mapping continuous variables natively onto field quadratures (Chandarana et al., 2023, Enomoto et al., 2022).
CV quantum approximate optimization algorithms exploit shallow circuits, variational ansatzes constructed from native gates, and measurement in the quadrature basis to directly return classical samples representing optimized solutions.

This approach circumvents the need for digital encoding of real variables into binary qubit strings and enables direct quantum-enhanced optimization for both continuous and integer-valued classical cost functions.

5. Resource Scaling, Fidelity, and Error Models

The physical feasibility of classical-to-quantum mappings critically depends on:

Squeezing levels: High squeezing ( $r$ in dB) increases simulation precision and the range over which classical variables can be mapped with low noise. For fault-tolerant encoding (e.g., GKP codes), squeezing thresholds of $10$–$13$ dB are required (Renault et al., 17 Dec 2024).
Gate fidelity: Each step in a classical-to-quantum mapping introduces noise proportional to finite squeezing and optical loss. For cluster-state and circuit-based simulation, the accumulated error is additive and scales as $e^{-2r}$ per gate (Renault et al., 17 Dec 2024, Gao et al., 15 Dec 2025, Abel et al., 15 Mar 2024).
Quadrature basis truncation: Simulating continuous spectra necessitates a Fock-space cutoff ( $N_{\text{cut}}$ ) such that population in $|n\geq N_{\text{cut}}\rangle$ is negligible.
Error correction: Embedding classical variables in CV hardware at scale requires bosonic codes (e.g., GKP) to correct for displacement and photon loss errors (Clark et al., 5 Jun 2025, Renault et al., 17 Dec 2024).

The quantum resource requirements scale linearly with the number of encoded classical variables (qumodes), significantly better than qubit-based logarithmic encodings when all-to-all or dense Hamiltonians are involved (Gao et al., 15 Dec 2025, Abel et al., 15 Mar 2024).

6. Physical Implementations and Architectures

Integrated photonic platforms, particularly those based on LiNbO $_3$ and Si $_3$ N $_4$ , provide the high-fidelity Gaussian and non-Gaussian gate set required for classical-to-quantum mappings (Clark et al., 5 Jun 2025, Lenzini et al., 2018). Specific features include:

On-chip squeezed state generation via $\chi^{(2)}$ or $\chi^{(3)}$ nonlinearities, with squeezing levels exceeding $10$ dB now achievable.
Programmable optical interferometers and electro-optic phase shifters enabling arbitrary symplectic transformations for Gaussian operations (Lenzini et al., 2018).
Homodyne detection for reading out the continuous quadrature statistics directly corresponding to mapped classical variables (Masada et al., 2015).
Non-Gaussian state synthesis via photon subtraction, cat-state breeding, or photon-number-resolving detectors, enabling universal computation and classical simulation of nonlinear systems (Renault et al., 17 Dec 2024, Abel et al., 15 Mar 2024).
Large-scale architectures utilizing frequency and time multiplexing to realize multi-dimensional cluster states for the measurement-based model (Wu et al., 2019, Pfister, 2019).

A critical implication is that existing and near-term integrated photonic hardware can support all requisite operations for classical-to-quantum mapping protocols at experimentally relevant scales (Clark et al., 5 Jun 2025).

7. Applications and Theoretical Implications

Classical-to-quantum mappings on CV quantum computers have yielded practical advances in:

Quantum simulation of both quantum and classical Hamiltonian dynamics, including nontrivial PDEs and quantum field theories, with demonstrated algorithms preserving the continuous spectra of original classical models (Gao et al., 15 Dec 2025, Abel et al., 15 Mar 2024).
Variational and machine-learning approaches to optimization and generative modeling, with OpticalGAN demonstrating quantum GANs in the CV regime (Shrivastava et al., 2019).
Enhanced classical sampling and estimation, e.g., the frequency-Hermite–Gauss mapping for single-photon processing (Fabre et al., 10 Feb 2024).
Unification of classical and quantum probabilistic flows in a single Hilbert-space framework, enabling deployment of quantum-native spectral and linear algebraic tools for classical problems (Gao et al., 15 Dec 2025).

Broader implications include the prospect of hybrid quantum-classical variational algorithms that optimize over quantum representations of classical control and fluid dynamics, as well as new directions in error-corrected simulation of classically intractable systems (Gao et al., 15 Dec 2025, Abel et al., 15 Mar 2024).

References:

Implementing the Koopman-von Neumann approach on continuous-variable photonic quantum computers (Gao et al., 15 Dec 2025)
Simulating quantum field theories on continuous-variable quantum computers (Abel et al., 15 Mar 2024)
End-to-end switchless architecture for fault-tolerant photonic quantum computing (Renault et al., 17 Dec 2024)
Integrated photonics for continuous-variable quantum optics (Clark et al., 5 Jun 2025)
Quantum computing overview: discrete vs. continuous variable models (Choe, 2022)
Continuous Variable Quantum Information Processing (Andersen et al., 2010)
Photonic counterdiabatic quantum optimization algorithm (Chandarana et al., 2023)
Continuous-variable quantum approximate optimization on a programmable photonic quantum processor (Enomoto et al., 2022)
OpticalGAN: Generative Adversarial Networks for Continuous Variable Quantum Computation (Shrivastava et al., 2019)
Continuous-variable quantum computing in the quantum optical frequency comb (Pfister, 2019)
Quantum-Computing Architecture based on Large-Scale Multi-Dimensional Continuous-Variable Cluster States in a Scalable Photonic Platform (Wu et al., 2019)
Integrated photonic platform for quantum information with continuous variables (Lenzini et al., 2018)