arxiv: 2604.26945 · v1 · submitted 2026-04-29 · 🪐 quant-ph

Simulating dynamics of RLC circuits with a quantum differential-algebraic equations solver

Arkopal Dutt , Anirban Chowdhury , Kristan Temme , Hari Krovi This is my paper

Pith reviewed 2026-05-07 10:30 UTC · model grok-4.3

classification 🪐 quant-ph

keywords quantum algorithmRLC circuitsdifferential-algebraic equationsBQP-hardnessquantum simulationmodified nodal analysisexponential speedup

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The pith

A quantum algorithm simulates RLC circuit dynamics in polylogarithmic time with exponential speedup over classical methods.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper develops a quantum algorithm that uses a solver for differential-algebraic equations to simulate RLC circuits. Given oracle access to the circuit, it prepares a quantum state with the voltages and currents at a time or over an interval. Under mild assumptions, this takes polylog time in the number of components for exponential speedup. It supports estimating energy and dissipated power in polylog time, which is proven BQP-hard by showing LC circuits include hard oscillator networks. The method uses modified nodal analysis to make the circuit equations compatible with the quantum DAE solver.

Core claim

We introduce a quantum algorithm for simulating the dynamics of electrical circuits consisting of resistors, inductors and capacitors. Given oracle access to the connectivity of the circuit and values of the electrical elements, our algorithm prepares a quantum state that encodes voltages and current values either at a specified time or the history of their evolution over a time-interval. For an RLC circuit with N components, our algorithm runs in time polylog(N) under mild assumptions on the connectivity of the circuit and values of its components. This provides an exponential speed-up over classical algorithms that take poly(N) time in the worst-case. Our algorithm can be used to estimate

What carries the argument

The quantum solver for linear differential-algebraic equations that encodes the solution state given oracle access to the system matrices, applied after modified nodal analysis formulates the RLC equations as a compatible DAE.

Load-bearing premise

The circuit connectivity and component values must satisfy mild assumptions so that the modified-nodal-analysis DAE is compatible with the quantum solver under the oracle access model.

What would settle it

A classical algorithm that estimates dissipated power for general RLC circuits with N components in polylog(N) time under the same oracle model would falsify the BQP-hardness claim.

read the original abstract

We introduce a quantum algorithm for simulating the dynamics of electrical circuits consisting of resistors, inductors and capacitors (aka RLC circuits) along with power sources. Given oracle access to the connectivity of the circuit and values of the electrical elements, our algorithm prepares a quantum state that encodes voltages and current values either at a specified time or the history of their evolution over a time-interval. For an RLC circuit with $N$ components, our algorithm runs in time $\textsf{polylog}(N)$ under mild assumptions on the connectivity of the circuit and values of its components. This provides an exponential speed-up over classical algorithms that take $\textsf{poly}(N)$ time in the worst-case. Our algorithm can be used to estimate energy across a set of components or dissipated power in $\textsf{polylog}(N)$ time, a problem that we prove is BQP-hard and therefore unlikely to be efficiently solved by classical algorithms. The main challenge in simulating the dynamics of RLC circuits is that they are governed by differential-algebraic equations (DAEs), a coupled system of differential equations with hidden algebraic constraints. Consequentially, existing quantum algorithms for ordinary differential equations cannot be directly utilized. We therefore develop a quantum DAE solver for simulating the time-evolution of linear DAEs. For RLC circuits, we employ modified nodal analysis to create a system of DAEs compatible with our quantum algorithm. We establish BQP-hardness by demonstrating that any network of classical harmonic oscillators, for which an energy-estimation problem is known to be BQP-hard, is a special case of an LC circuit. Our work gives theoretical evidence of quantum advantage in simulating RLC circuits and we expect that our quantum DAE solver will find broader use in the simulation of dynamical systems.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Desk Editor's Note private letter to a colleague

Quantum DAE solver for RLC circuits is a solid step but the polylog runtime claim rests on assumptions that may not control the condition number.

read the letter

The paper introduces a quantum solver for linear differential-algebraic equations and applies it to RLC circuit dynamics, plus a BQP-hardness proof for energy or power estimation. That combination is the actual new piece. Prior quantum ODE work does not directly cover the algebraic constraints that appear in circuit equations, so extending the linear-algebra primitives to handle the DAE pencil is a useful technical move. The hardness reduction is straightforward: any network of harmonic oscillators is a special case of an LC circuit, so the known BQP-hard energy estimation problem carries over directly. That part looks clean and does not depend on the runtime details. The claimed polylog(N) runtime under mild assumptions on connectivity and component values is where the soft spot sits. Quantum linear-system solvers scale with the condition number of the effective matrix. Modified nodal analysis for RLC circuits can produce pencils whose condition number grows linearly with N or with the dynamic range of the element values, as in long LC chains. The abstract invokes the assumptions to guarantee both compatibility and the fast runtime, but it is not obvious that those assumptions bound the condition number at polylog(N). If they do not, the exponential speedup over classical poly(N) time does not follow, even though the hardness result remains valid. This work is for people working on quantum simulation of physical systems or on quantum algorithms for engineering models. A reader who already knows quantum linear algebra and DAE theory will get the most out of it. The paper deserves a serious referee to check the full error analysis, the precise statement of the complexity theorem, and whether the mild assumptions actually cap the condition number. I would send it to peer review.

Referee Report

2 major / 1 minor

Summary. The paper introduces a quantum algorithm for simulating the dynamics of RLC circuits by formulating them as linear differential-algebraic equations (DAEs) via modified nodal analysis and developing a dedicated quantum DAE solver. It claims that, given oracle access to circuit connectivity and component values, the algorithm prepares a quantum state encoding voltages and currents (at a fixed time or over an interval) in polylog(N) time for an N-component circuit under mild assumptions on connectivity and component values. This is asserted to yield an exponential speedup over classical poly(N) methods. The paper further proves that estimating energy across components or dissipated power is BQP-hard by reduction from networks of classical harmonic oscillators (a special case of LC circuits).

Significance. If the runtime bound holds, the work supplies concrete theoretical evidence of quantum advantage for a practical simulation task in electrical engineering and circuit design. The quantum DAE solver construction is a reusable technical contribution that could extend to other linear dynamical systems with algebraic constraints. The BQP-hardness reduction is a clear strength, as it directly leverages a previously established hard problem without circularity.

major comments (2)

[Abstract and complexity analysis (Theorem on runtime)] The central polylog(N) runtime claim (abstract and complexity theorem) requires that the condition number κ of the effective linear system obtained from the MNA DAE (Eẋ = Ax + b) remains polylog(N) under the stated mild assumptions. No explicit bound on κ or on the DAE index is provided; ladder or chain topologies with O(1) component values can produce κ = Θ(N), which would make the QLSA-based runtime superpolylogarithmic and eliminate the claimed exponential speedup while leaving the BQP-hardness intact.
[DAE solver construction and MNA formulation] The reduction of the DAE to a form amenable to quantum linear-system primitives (block-encoding or QLSA) is described at a high level but lacks a detailed error analysis or explicit handling of the algebraic constraints; without this, it is impossible to verify that the overall simulation error remains controlled in polylog(N) time.

minor comments (1)

[Modified nodal analysis section] The notation for the MNA matrices E and A is introduced without an equation number, making later references to their spectral properties harder to follow.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for their careful reading and constructive feedback. The comments identify areas where the manuscript would benefit from greater precision and detail. We address each major comment below and commit to revisions that strengthen the presentation without altering the core claims.

read point-by-point responses

Referee: [Abstract and complexity analysis (Theorem on runtime)] The central polylog(N) runtime claim (abstract and complexity theorem) requires that the condition number κ of the effective linear system obtained from the MNA DAE (Eẋ = Ax + b) remains polylog(N) under the stated mild assumptions. No explicit bound on κ or on the DAE index is provided; ladder or chain topologies with O(1) component values can produce κ = Θ(N), which would make the QLSA-based runtime superpolylogarithmic and eliminate the claimed exponential speedup while leaving the BQP-hardness intact.

Authors: We agree that an explicit bound on the condition number κ (and DAE index) is required to rigorously support the polylog(N) runtime. The manuscript invokes 'mild assumptions on connectivity and component values' with the intention that these suffice to keep κ polylogarithmic, but this is not stated or proved explicitly. In the revised version we will add a dedicated lemma (or subsection) that either (i) proves κ = O(polylog(N)) under the stated assumptions or (ii) augments the assumptions with an explicit polylogarithmic bound on κ. If the latter is necessary, we will also qualify the runtime statement accordingly. The BQP-hardness reduction remains unaffected, as it does not rely on the runtime bound. revision: yes
Referee: [DAE solver construction and MNA formulation] The reduction of the DAE to a form amenable to quantum linear-system primitives (block-encoding or QLSA) is described at a high level but lacks a detailed error analysis or explicit handling of the algebraic constraints; without this, it is impossible to verify that the overall simulation error remains controlled in polylog(N) time.

Authors: We acknowledge that the current description of the quantum DAE solver and its embedding of the modified-nodal-analysis formulation is high-level. The revised manuscript will expand the relevant section with a complete error analysis. This will include: (a) how algebraic constraints are incorporated into the block-encoding construction, (b) propagation of truncation and approximation errors through the DAE index, and (c) an end-to-end bound showing that the total simulation error can be made inverse-polylogarithmic while preserving the overall polylog(N) runtime under the (possibly augmented) assumptions on κ. revision: yes

Circularity Check

0 steps flagged

No significant circularity detected

full rationale

The paper establishes BQP-hardness via an explicit reduction showing that networks of classical harmonic oscillators (a previously known hard problem) are a special case of LC circuits. The algorithm is constructed from standard quantum linear-algebra primitives (block-encoding or QLSA applied to the modified-nodal-analysis DAE) rather than any self-referential fitting or redefinition of inputs as outputs. The 'mild assumptions' on connectivity and component values are external conditions for compatibility and are not defined in terms of the claimed polylog runtime or condition number; they function as stated hypotheses rather than circularly derived quantities. No load-bearing self-citations, ansatzes smuggled via prior work, or renamings of known results appear in the derivation chain. The overall argument remains self-contained against external benchmarks.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

The central claim rests on standard quantum-computing oracles and the applicability of modified nodal analysis to produce linear DAEs; no new free parameters or invented entities are introduced.

axioms (2)

domain assumption Quantum computers can implement linear-algebra operations on oracles in polylog time
Invoked for the runtime claim and standard in quantum simulation literature.
domain assumption Modified nodal analysis yields a linear DAE compatible with the quantum solver
Stated in the abstract as the modeling step that enables the algorithm.

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discussion (0)

Reference graph

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