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arxiv: 2606.13481 · v4 · pith:I7W56VDOnew · submitted 2026-06-11 · 🧮 math.OC

Towards a Control interpretation of Quantum Advantage

Dario Pighin This is my paper

Pith reviewed 2026-07-03 23:51 UTC · model grok-4.3

classification 🧮 math.OC
keywords quantum advantagecontrol theorybilinear Schrödinger equationoperator controllabilityminimal time functionquantum Fourier transformmaximum independent setSU(N)
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The pith

Quantum advantage is recast as a polynomial-in-n upper bound on the minimal time to control the bilinear Schrödinger equation on SU(N).

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

The paper develops a control-theoretic framework that treats a target quantum computation as an operator controllability problem on the special unitary group SU(N) governed by the bilinear controlled Schrödinger equation. Quantum advantage is defined precisely as the existence of a polynomial upper bound (in the number of qubits n) on the associated minimal-time function. The framework is illustrated on the quantum Fourier transform, where controllability is established via Lie-algebraic methods and an O(n²) time bound is obtained from the standard circuit decomposition, and on the maximum independent set problem, where the Rydberg-blockade Hamiltonian is treated as a control system and QAOA is rewritten as a continuous-time optimal-control task. A reader would care because this supplies a uniform, time-based criterion for deciding when a quantum algorithm genuinely outperforms classical ones.

Core claim

Quantum advantage is identified with a polynomial-in-n upper bound on the associated minimal-time function for the bilinear controlled Schrödinger equation on SU(N). Operator controllability of the quantum Fourier transform is proved by a Lie-algebraic argument, and an O(n²) upper bound on minimal time is derived via a gate-concatenation lemma. The Rydberg-blockade Hamiltonian for the maximum independent set problem is analyzed as a bilinear control system, controllability is established, and a control-based definition of quantum advantage for that problem is introduced.

What carries the argument

The bilinear controlled Schrödinger equation on SU(N), recast as an operator controllability problem whose minimal-time function supplies the criterion for quantum advantage.

If this is right

  • The QFT on superconducting processors admits an O(n²) upper bound on minimal control time, implying that the transform can be realized in polynomial time under the ideal model.
  • The Rydberg-blockade system for maximum independent set is operator controllable, so the problem can be solved on neutral-atom hardware by steering the continuous-time control.
  • Reformulating QAOA as an optimal-control problem yields a control-theoretic definition of quantum advantage for combinatorial optimization tasks.
  • The same controllability and time-bound analysis can be applied to other quantum algorithms to decide whether they exhibit polynomial scaling.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • Hardware pulse design could be improved by directly optimizing the minimal-time function instead of discretizing into gates.
  • Including realistic decoherence would require modifying the minimal-time function to a minimal-cost function that penalizes error accumulation.
  • The control perspective may link quantum algorithms to classical optimal-control techniques for hybrid classical-quantum solvers.
  • Numerical simulation of the minimal-time function on small-n instances could provide early checks on whether the polynomial bound holds in practice.

Load-bearing premise

The physical hardware is accurately described by the ideal bilinear controlled Schrödinger equation without significant decoherence or control errors.

What would settle it

An experimental measurement showing that the shortest control time required to implement the quantum Fourier transform on a superconducting processor grows exponentially rather than as O(n²) would falsify the proposed identification of quantum advantage with the polynomial bound.

Figures

Figures reproduced from arXiv: 2606.13481 by Dario Pighin.

Figure 1.1
Figure 1.1. Figure 1.1: An instance of the Maximum Independent Set (MIS) problem on a [PITH_FULL_IMAGE:figures/full_fig_p006_1_1.png] view at source ↗
Figure 3.1
Figure 3.1. Figure 3.1: The ibm brisbane coupling map E = (V, E), a heavy-hexagonal (Eagle r3) lattice with |V | = 127 qubits and |E| = 144 two-qubit couplers (maximum vertex degree 3). Each vertex k ∈ V = {0, . . . , 126} is a qubit; each edge (c, t) ∈ E = E carries a cross-resonance/ECR coupler entering the control Hamiltonian of (3.5) through the term P (c,t)∈E vct(t)Zc ⊗ Xt. Ver￾tex color encodes the single-qubit readout (a… view at source ↗
read the original abstract

We develop a control-theoretic framework for understanding Quantum Advantage (QA), providing a systematic route to characterize when and how QA can arise. The bilinear controlled Schr\"odinger equation is the common thread: the target quantum computation is recast as an operator controllability problem on the special unitary group $SU(N)$, and QA is identified with a polynomial-in-$n$ upper bound on the associated minimal-time function. We illustrate the framework on two paradigmatic problems: a) the Quantum Fourier Transform (QFT) on superconducting digital quantum processors (such as IBM's ibm_brisbane), for which we prove operator controllability by a Lie-algebraic argument and derive an $O(n^2)$ upper bound on the minimal time via a gate-concatenation lemma combined with the standard QFT circuit decomposition; b) the Maximum Independent Set (MIS) problem on neutral-atom analog quantum processors (such as Pasqal's hardware), for which we analyze the Rydberg-blockade Hamiltonian as a bilinear control system and reformulate the Quantum Approximate Optimization Algorithm (QAOA) as a continuous-time optimal control problem. By a controllability result, we show how the problem can be solved on Pasqal Quantum Computers and we introduce a control-based definition of Quantum Advantage for MIS. We conclude by outlining several open problems that chart directions for future research at the intersection of Control Theory and Quantum Computing.

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.

Referee Report

1 major / 2 minor

Summary. The paper develops a control-theoretic framework for quantum advantage (QA) by recasting target quantum computations as operator controllability problems for the bilinear controlled Schrödinger equation on SU(N), with QA identified by definition as the existence of a polynomial-in-n upper bound on the associated minimal-time function. It illustrates the framework via two examples: (a) a Lie-algebraic proof of controllability together with an O(n²) time bound for the Quantum Fourier Transform obtained by gate concatenation applied to the standard circuit decomposition on superconducting hardware, and (b) a reformulation of QAOA for the Maximum Independent Set problem as a continuous-time optimal control problem on the Rydberg-blockade Hamiltonian, together with a controllability result showing solvability on neutral-atom processors.

Significance. If the framework holds, it supplies a systematic route to characterize QA in terms of controllability and minimal times within the ideal bilinear model, potentially bridging quantum information and control theory. The manuscript explicitly invokes standard tools (Lie-algebra rank condition for controllability, gate-concatenation arguments) and states its modeling assumptions (ideal dynamics without decoherence), which are strengths. The two worked examples are internally consistent illustrations rather than independent derivations.

major comments (1)
  1. [Abstract and framework introduction] The central identification of QA with a polynomial upper bound on minimal control time is introduced definitionally in the abstract and framework section; the subsequent examples then verify the bound under precisely the same ideal bilinear modeling assumptions used to define the problem, so the claim holds by construction but does not yet generate predictions that could be falsified outside those assumptions.
minor comments (2)
  1. [Abstract] The abstract refers to 'a control-based definition of Quantum Advantage for MIS' without indicating whether or how this differs from the general polynomial-bound definition introduced earlier.
  2. [QFT example] Notation for the minimal-time function and the precise statement of the gate-concatenation lemma used to obtain the O(n²) bound are not visible in the abstract; adding an explicit equation or lemma number would improve readability.

Simulated Author's Rebuttal

1 responses · 0 unresolved

We thank the referee for their thoughtful review and for highlighting this aspect of the framework. We address the major comment below.

read point-by-point responses

  1. Referee: The central identification of QA with a polynomial upper bound on minimal control time is introduced definitionally in the abstract and framework section; the subsequent examples then verify the bound under precisely the same ideal bilinear modeling assumptions used to define the problem, so the claim holds by construction but does not yet generate predictions that could be falsified outside those assumptions.

    Authors: We appreciate the referee's observation. The identification of quantum advantage with a polynomial-in-n upper bound on the minimal control time is indeed introduced definitionally within the ideal bilinear model, as stated in the abstract and Section 2. This is an intentional modeling choice that recasts QA as an operator controllability problem on SU(N), thereby enabling the direct application of established control-theoretic tools such as the Lie-algebra rank condition and gate-concatenation arguments for time bounds. The two examples are constructed precisely to demonstrate this application under the stated assumptions, yielding the O(n²) bound for QFT and the controllability result for the MIS problem on the Rydberg Hamiltonian. We agree that the framework does not, by itself, produce falsifiable predictions for hardware subject to decoherence or other non-ideal effects; the manuscript explicitly limits its scope to the ideal dynamics and lists extensions to realistic models among the open problems in the conclusion. The contribution lies in supplying a systematic control-theoretic language for QA rather than in empirical validation beyond the model. revision: no

Circularity Check

2 steps flagged

QA defined as polynomial bound on minimal control time; examples confirm by construction

specific steps
  1. self definitional [Abstract]
    "QA is identified with a polynomial-in-n upper bound on the associated minimal-time function."

    The paper defines QA precisely as the existence of the polynomial bound on minimal time for the controlled Schrödinger equation. The subsequent sections then prove controllability and exhibit polynomial bounds for the chosen examples under identical modeling assumptions, so the claimed identification is satisfied by construction rather than derived from independent evidence.

  2. self definitional [Abstract (MIS example)]
    "we introduce a control-based definition of Quantum Advantage for MIS."

    A second, problem-specific definition of QA is introduced inside the same control-theoretic framework; the reformulation of QAOA as an optimal-control problem then satisfies this new definition by construction.

full rationale

The paper's core move is an explicit identification of 'Quantum Advantage' with the existence of a polynomial-in-n upper bound on the minimal-time function of the bilinear Schrödinger system. The two examples then derive exactly such bounds (O(n^2) for QFT via gate concatenation; a control reformulation for MIS) inside the same ideal-dynamics model used to state the definition. This makes the central claim hold tautologically once the modeling assumptions are granted, satisfying the self-definitional pattern. No external benchmark or independent verification is invoked to break the loop. Controllability arguments themselves rely on standard Lie-algebra results and are not circular, but they operate downstream of the definitional step.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

The framework rests on the standard bilinear Schrödinger model and Lie-algebraic controllability criteria from quantum control theory; no new free parameters or invented entities are introduced in the abstract, and the polynomial bound is a definitional choice rather than a fitted quantity.

axioms (2)
  • domain assumption The evolution of the quantum system is governed by the bilinear controlled Schrödinger equation on SU(N).
    Invoked as the common thread for the entire framework and both examples.
  • standard math Operator controllability on SU(N) is equivalent to the Lie algebra generated by the drift and control Hamiltonians being the full su(N).
    Used for the QFT controllability proof.

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