⚛️ quantum physics

Quantum simulation of classical RCL circuits via complexification and Schrödingerization

This paper reviews the decomplexification procedure and proposes several strategies for its inverse, complexification, to enable the quantum simulation of classical RCL circuits and other differential equations by mapping them into equivalent quantum dynamics.

Original author: Huner Fanchiotti, Carlos A. García Canal, Miguel A. Mayosky, Alejandro L. Veiga, Armando Pérez, Rafael Gómez-Lurbe, Vicente Vento

Published 2026-11-06

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CC BY 4.0

Original author: Huner Fanchiotti, Carlos A. García Canal, Miguel A. Mayosky, Alejandro L. Veiga, Armando Pérez, Rafael Gómez-Lurbe, Vicente Vento

Original paper licensed under CC BY 4.0 (http://creativecommons.org/licenses/by/4.0/). ✨ This is an AI-generated explanation of the paper below. It is not written or endorsed by the author. For technical accuracy, refer to the original paper. Read full disclaimer

Science has long relied on two different languages to describe how the world moves and changes. One language, used for the everyday objects we can see and touch, relies on real numbers and describes classical mechanics. The other, used for the invisible realm of atoms and subatomic particles, relies on complex numbers and describes quantum mechanics. For decades, these two languages seemed to speak to different realities, making it difficult to use the powerful tools of quantum computing to solve problems that are fundamentally classical, such as the flow of electricity through a circuit. However, researchers have discovered a way to translate between these two dialects. By treating the complex numbers of quantum theory as a bridge, they can map the behavior of classical systems onto quantum ones. This opens a door where the unique speed and efficiency of quantum computers could be used to simulate and solve the differential equations that govern classical engineering, a task that is often slow and difficult for traditional machines.

In a recent study, a team of physicists from Argentina and Spain explored this translation process, focusing specifically on electrical circuits containing resistors, inductors, and capacitors. These components form the backbone of modern electronics, and their behavior is described by equations that can be surprisingly complex, especially when the circuits involve damping or specific symmetries. The researchers first revisited an established method called "decomplexification," which shows how a quantum system can be rewritten as a classical one. They then turned the process around, asking how a classical electrical circuit could be lifted into a quantum framework. They demonstrated that by carefully converting the real-valued equations of a circuit into a complex-valued format, one could derive a quantum equivalent. This allowed them to describe the behavior of a simple damped oscillator and a more complex pair of coupled circuits using the language of quantum mechanics.

The team found that while this direct translation works beautifully for simple systems, it becomes mathematically unwieldy as the circuits grow larger and more complicated. Solving the necessary algebraic equations for a large network of components quickly becomes a burden, suggesting that this direct mapping is not a practical solution for real-world, large-scale engineering problems. To overcome this hurdle, the researchers turned to a newer, more flexible approach known as "Schrödingerization." Instead of trying to force the entire classical system into a single quantum equation, this method introduces an extra, artificial dimension to the problem. It treats the classical equations as if they were evolving in a space that includes a new variable, which acts like a momentum coordinate. This transformation turns the original, non-quantum equations into a standard quantum evolution equation that a computer can handle efficiently.

To test this idea, the researchers simulated the behavior of a single damped electrical oscillator on a quantum computer model. They set up the simulation with specific values for the circuit's frequency and damping, then used the Schrödingerization technique to map the problem onto a quantum system. The process involved preparing a quantum state that represented the circuit's initial conditions, evolving it through time using a specific quantum algorithm, and then extracting the results. The results were striking: the quantum simulation reproduced the exact behavior of the classical circuit with high precision. The simulated voltage and current matched the known mathematical solutions almost perfectly, capturing both the oscillating nature of the signal and its gradual decay over time.

However, the researchers also noted a subtle limitation in their method. While the simulation was highly accurate during the early and middle stages of the circuit's evolution, small errors began to appear as the signal decayed to very low levels. This happened because the method requires dividing by a rapidly shrinking value to recover the final answer, which amplifies any tiny numerical noise present in the calculation. Despite this sensitivity at the very end of the process, the overall performance was robust. The study confirms that by adding a single extra dimension to the problem, classical differential equations can be successfully converted into a format that quantum computers can solve. This finding suggests that quantum simulation could become a viable tool for analyzing complex electrical systems, offering a new path to understanding and designing the circuits that power our world.

Technical Summary: Quantum Simulation of Classical RCL Circuits via Complexification and Schrödingerization

Problem Statement
The simulation of dynamical processes is fundamental to physics, yet classical computers face inherent difficulties in simulating quantum systems as they scale. Conversely, quantum computers offer the potential to accelerate calculations traditionally performed by classical machines. A central challenge lies in establishing a rigorous mathematical equivalence between quantum dynamics (defined in complex Hilbert spaces) and classical dynamics (defined in real vector spaces). While the "decomplexification" procedure allows mapping quantum systems to classical analogs (such as electric resonant circuits), the inverse process—complexification—is required to enable the quantum simulation of classical differential equations. The authors address the need for efficient methods to lift classical equations of motion, specifically those governing RCL (Resistor-Inductor-Capacitor) circuits, into equivalent quantum formulations suitable for implementation on quantum hardware.

Methodology
The paper employs two distinct methodological approaches to bridge classical and quantum dynamics:

Complexification via Diagonalization:
The authors first revisit the standard complexification procedure. Starting with a system of real second-order differential equations (e.g., damped oscillators or coupled PT-symmetric circuits), the system is converted into a first-order matrix form $\dot{\mathbf{x}} = A\mathbf{x}$ . By analyzing the eigenvalues of the real state matrix $A$ , which are often complex conjugates, the authors extend the dynamics from a real space $\mathbb{R}^{2n}$ to a complex space $\mathbb{C}^n$ . This involves finding a diagonalizing matrix $S$ to transform the system into decoupled complex equations. Multiplying by $i$ yields a Schrödinger-type equation $i\partial_t z = H z$ , where $H$ is a non-Hermitian Hamiltonian in the general case. The authors demonstrate this explicitly for a single damped oscillator and a pair of PT-symmetric coupled oscillators, deriving the corresponding Hamiltonians and showing how similarity transformations can recover standard PT-symmetric forms.
Schrödingerization (The Proposed Quantum Algorithm):
Recognizing that the diagonalization method becomes algebraically cumbersome for large systems, the authors adopt the "Schrödingerization" procedure [14]. This method maps a general linear differential equation $\partial_t |u(t)\rangle = A |u(t)\rangle$ directly into a unitary Schrödinger evolution by introducing an auxiliary variable $p$ .
- Decomposition: The generator matrix $A$ is decomposed into Hermitian ( $H_1$ ) and anti-Hermitian ( $H_2$ ) parts: $A = H_1 + iH_2$ .
- Warp Transformation: A new state $|\omega(p, t)\rangle = e^{-p}|u(t)\rangle$ is defined in an enlarged Hilbert space $\mathcal{H} = \mathcal{H}_I \otimes \mathcal{H}_p$ , where $\mathcal{H}_p$ represents the auxiliary degree of freedom.
- Hamiltonian Construction: The evolution is governed by the Hermitian Hamiltonian $H_{Sch} = H_1 \otimes P_p - H_2 \otimes I_p$ , where $P_p = -i\partial_p$ acts as a momentum operator.
- Quantum Implementation: To implement this on a quantum computer, the auxiliary variable $p$ is discretized on a finite grid and encoded in a quantum register. The momentum operator $P_p$ is implemented using the Quantum Fourier Transform (QFT). The evolution is performed in the momentum basis (where $P_p$ is diagonal) and transformed back to the position basis via QFT.
- Recovery: The original classical solution is recovered from the quantum state by selecting a grid point $p_j$ within a specific recovery region and applying the inverse warp factor $e^{p_j}$ .

Key Contributions and Results

Theoretical Framework: The paper clarifies the mathematical relationship between decomplexification and complexification, explicitly deriving the quantum-equivalent Hamiltonians for classical RCL circuits, including PT-symmetric coupled systems.
Algorithmic Implementation: The authors provide a detailed blueprint for implementing the Schrödingerization method on a quantum computer. This includes the discretization of the auxiliary variable, the construction of the momentum-basis Hamiltonian, and the specific circuit sequence: State Preparation $\to$ Inverse QFT $\to$ Unitary Evolution under $H_{mom}$ $\to$ QFT $\to$ Recovery.
Numerical Validation: The method was tested numerically on a single damped harmonic oscillator ( $\omega=2, \gamma=1$ $ω = 2, γ = 1$ ) using a quantum simulator with $N_p=2048$ $N_{p} = 2048$ grid points (11 auxiliary qubits + 1 system qubit).
- Accuracy: The reconstructed solution $u_{rec}(t)$ showed excellent agreement with the exact classical solution $u_{exact}(t)$ for both position ( $x$ ) and velocity ( $y$ ) components over the majority of the simulation time.
- Error Analysis: The normalized direction error remained small during initial and intermediate evolution stages. However, the error increased at late times. The authors attribute this to the recovery process: since $H_1$ possesses a positive eigenvalue, the solution must be recovered at a large positive $p$ . This requires dividing by a small, exponentially decaying auxiliary profile, which amplifies numerical discretization and Fourier errors when the physical amplitudes are already near zero.

Significance and Claims
The paper posits that rephrasing classical differential equations into a form directly addressable by quantum hardware is of paramount interest for efficiently simulating classical systems. The authors claim that while the direct complexification method is theoretically sound, it is often impractical for large systems due to algebraic complexity. In contrast, the Schrödingerization procedure offers a straightforward, generalizable path to map classical systems into quantum dynamics governed by the Schrödinger equation.

The study concludes that the QFT-based Schrödingerization approach is a promising tool for solving classical differential equations on quantum computers. The successful numerical reproduction of the damped oscillator's dynamics demonstrates the feasibility of the method, provided that care is taken in handling the auxiliary momentum variable and the recovery region to mitigate late-time numerical sensitivity. The paper does not claim to have solved the general simulation problem for all classical systems but establishes a viable framework for linear systems like RCL circuits.

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