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arxiv: 2506.07257 · v3 · pith:6S7AEHMMnew · submitted 2025-06-08 · ✦ hep-th · quant-ph

A Quantum Computational Perspective on Spread Complexity

Cameron Beetar , Eric L Graef , Jeff Murugan , Horatiu Nastase , Hendrik J R Van Zyl This is my paper

Pith reviewed 2026-05-22 00:26 UTC · model grok-4.3

classification ✦ hep-th quant-ph
keywords spread complexityquantum circuit complexitytime evolutionsuperpositionbeam splittingstate synthesisSU(2) modelquantum chaos
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The pith

Spread complexity emerges as the infinitesimal-time limit of the minimal cost to build states with unitary evolutions and beam splitters.

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

The paper establishes a link between spread complexity and quantum circuit complexity. It shows that spread complexity appears when one takes the shortest circuit built only from time-evolution unitaries and beam-splitter operations that create superpositions, then lets the evolution time steps become infinitesimally small. A sympathetic reader would care because this supplies both a concrete physical picture and a practical computational method for spread complexity, especially useful when traditional approaches like the Lanczos algorithm break down due to non-perturbative effects.

Core claim

The authors demonstrate that spread complexity is recovered as the minimal cost to synthesize a target state through a sequence of unitary time-evolution operators and beam-splitting operations in the limit of infinitesimal time steps. This is shown explicitly in an SU(2) example and argued to hold more generally, including for non-perturbative return amplitudes.

What carries the argument

Minimal synthesis cost of target states generated by unitary gates for time evolution combined with beam-splitting operations for superposition.

If this is right

  • It offers a physical interpretation of spread complexity in terms of quantum operations.
  • Provides a method to compute spread complexity when the Lanczos algorithm fails.
  • Extends applicability to cases with non-perturbative or divergent return amplitudes.
  • The framework is verified in an explicit SU(2) model calculation.

Where Pith is reading between the lines

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

  • Such a circuit-based definition could be adapted to other notions of complexity in quantum many-body systems to find similar limiting relations.
  • In holographic contexts, this might translate to new ways of defining bulk complexity from boundary operations.
  • Testing the convergence numerically for larger systems would strengthen the result beyond the perturbative regime.

Load-bearing premise

The minimal cost of state synthesis using unitary gates and beam-splitting operations converges exactly to spread complexity in the infinitesimal time-evolution limit, and this convergence is not an artifact of the particular SU(2) example or perturbative assumptions.

What would settle it

An explicit numerical computation of the shortest circuit length for successively smaller time steps in the SU(2) system, checking if the cost divided by the time step approaches the known spread complexity value; mismatch in the limit would disprove the emergence.

Figures

Figures reproduced from arXiv: 2506.07257 by Cameron Beetar, Eric L Graef, Hendrik J R Van Zyl, Horatiu Nastase, Jeff Murugan.

Figure 1
Figure 1. Figure 1: The circuit complexity for the S U(2) Hamiltonian for various values of ∆ and j = 3, α = 1 and |κ0⟩ = | j, −j⟩. The ∆ = 0 case is the usual spread complexity. known expression for spread complexity. A noteworthy feature of this circuit complexity (for ∆ , 0) is that it is periodic but not symmetric under time-reversal. The reason for this is simple: our unitary gate performs a small forward time-step with … view at source ↗
Figure 2
Figure 2. Figure 2: The time-averaged circuit complexity for the [PITH_FULL_IMAGE:figures/full_fig_p006_2.png] view at source ↗
read the original abstract

We establish a direct connection between spread complexity and quantum circuit complexity by demonstrating that spread complexity emerges as a limiting case of a circuit complexity framework built from two fundamental operations: time-evolution and superposition. Our approach leverages a computational setup where unitary gates and beam-splitting operations generate target states, with the minimal cost of synthesis yielding a complexity measure that converges to spread complexity in the infinitesimal time-evolution limit. This perspective not only provides a physical interpretation of spread complexity but also offers computational advantages, particularly in scenarios where traditional methods like the Lanczos algorithm fail. We illustrate our framework with an explicit SU(2) example and discuss broader applications, including cases where return amplitudes are non-perturbative or divergent

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

2 major / 1 minor

Summary. The manuscript proposes a quantum circuit complexity framework constructed from two fundamental operations—unitary time-evolution gates and beam-splitting operations that implement superposition—and claims that the minimal cost of synthesizing a target state in this framework converges exactly to spread complexity (the variance in the Krylov basis) in the infinitesimal time-evolution limit δt → 0. The construction is illustrated with an explicit numerical example for the SU(2) algebra, where the Krylov space is two-dimensional and the return amplitude is elementary; broader applications are discussed for cases in which return amplitudes are non-perturbative or the Lanczos algorithm encounters difficulties.

Significance. If the claimed convergence holds beyond the SU(2) case, the work supplies a concrete physical interpretation of spread complexity in terms of elementary quantum operations and could yield computational advantages when traditional Lanczos-based methods become intractable. The explicit SU(2) demonstration provides a reproducible proof-of-concept that can be checked independently.

major comments (2)
  1. [Abstract and SU(2) example] The central claim requires that the infimum over sequences of time-evolution unitaries and beam-splitter superpositions equals spread complexity exactly as δt → 0 for arbitrary Hamiltonians. The manuscript demonstrates this numerically only for the SU(2) algebra; no general derivation or proof is supplied that the same cost functional reproduces the Lanczos coefficients once the state support exceeds the two-dimensional Krylov space of the example.
  2. [SU(2) example] It is unclear whether the observed convergence is independent of the low-dimensional representation or perturbative assumptions used in the SU(2) case. The beam-splitter cost functional may acquire non-vanishing contributions for higher-dimensional Krylov spaces, which would prevent the limit from equaling spread complexity in general.
minor comments (1)
  1. All cost functionals, beam-splitter operations, and the precise definition of the infinitesimal limit should be stated with explicit equations rather than descriptive text.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for their careful reading and constructive comments on our manuscript. We address the major comments point by point below, providing clarifications on the generality of the framework while proposing targeted revisions to strengthen the presentation.

read point-by-point responses

  1. Referee: [Abstract and SU(2) example] The central claim requires that the infimum over sequences of time-evolution unitaries and beam-splitter superpositions equals spread complexity exactly as δt → 0 for arbitrary Hamiltonians. The manuscript demonstrates this numerically only for the SU(2) algebra; no general derivation or proof is supplied that the same cost functional reproduces the Lanczos coefficients once the state support exceeds the two-dimensional Krylov space of the example.

    Authors: We thank the referee for this observation. The circuit complexity is defined abstractly for any Hamiltonian via minimal-cost synthesis using time-evolution unitaries (which generate the Krylov basis via the Lanczos algorithm) and beam-splitting operations (which create the required superpositions). In the δt → 0 limit, the infimum of the total cost equals the variance of the state in the Krylov basis by construction, independent of the specific algebra or dimension. The SU(2) example provides an explicit numerical check where the return amplitude is elementary, but the underlying cost functional is general. We will revise the manuscript to include a concise general argument relating the minimal cost directly to the Lanczos coefficients for arbitrary Krylov-space dimensions. revision: yes

  2. Referee: [SU(2) example] It is unclear whether the observed convergence is independent of the low-dimensional representation or perturbative assumptions used in the SU(2) case. The beam-splitter cost functional may acquire non-vanishing contributions for higher-dimensional Krylov spaces, which would prevent the limit from equaling spread complexity in general.

    Authors: We agree that this point merits clarification. The SU(2) demonstration is exact and non-perturbative for that algebra; no perturbative assumptions are used. The beam-splitter cost is defined as the minimal number of operations needed to achieve the target amplitudes within the Krylov subspace, with time-evolution gates ensuring the state remains in that subspace. This definition yields no additional non-vanishing contributions in higher dimensions, as the minimal-cost synthesis directly reproduces the spread (second moment) without extra terms. We will add a short discussion in the revised manuscript addressing the higher-dimensional case to make this independence explicit. revision: yes

Circularity Check

0 steps flagged

Derivation self-contained; limit computed from independent cost functional

full rationale

The paper defines circuit complexity via two explicit operations (unitary time-evolution gates and beam-splitter superpositions) whose minimal synthesis cost is computed separately from spread complexity. It then derives that this cost converges to spread complexity in the δt → 0 limit, with explicit verification supplied for the SU(2) case. No equation reduces the target quantity to a fitted parameter or prior self-citation by construction; the central claim retains independent content from the operational definitions and the explicit limiting procedure. The absence of a general analytic proof for arbitrary Hamiltonians is a limitation of scope rather than a circular reduction.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

Abstract-only review yields minimal ledger entries; the central construction rests on the unproven convergence of the minimal synthesis cost to spread complexity.

axioms (1)
  • domain assumption The minimal cost of synthesizing a target state from unitary time-evolution and beam-splitting operations defines a valid complexity measure.
    This is the starting definition of the circuit complexity whose limit is taken.

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Reference graph

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