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arXiv:2603.29091

Ether of Orbifolds

Henry Lamm

hep-latquant-phSubmitted Mar 31, 2026• Announced Apr 1, 2026• 0 citations• Revised Apr 6, 2026 (v2)
v1 · Mar 31, 2026v2 · Apr 6, 2026v3 · May 7, 2026
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Abstract

The orbifold lattice has been proposed as a route to practical quantum simulation of Yang--Mills theory, with claims of exponential speedup over all known approaches. Through analytical derivations, Monte Carlo simulation, and explicit circuit construction, we identify compounding costs entirely absent in Kogut--Susskind formulations: a mass-dependent Trotter overhead that scales as $m^4$, non-singlet contamination that grows as $m^2$ and worsens with penalty terms, and a mandatory mass extrapolation. Monte Carlo simulations of SU(3) establish a universal scaling: the continuum limit forces $m^2 \propto 1/a$, binding the Trotter step to the lattice spacing through a cost unique to orbifolds. For a fiducial $10^3$ calculation, the orbifold is $10^4$--$10^{10}$ times more expensive than every published alternative. These results indicate that the claimed computational advantages do not at present survive quantitative scrutiny.

Comments
7+2 pages, 1 figure. Clarified confusion meaning of gauge invariance

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Comments on "Ether of Orbifolds"

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The orbifold lattice has been proposed as a route to practical quantum simulation of Yang--Mills theory, with claims of exponential speedup over all known approaches. Through analytical derivations, Monte Carlo simulation, and explicit circuit construction, we identify compounding costs entirely absent in Kogut--Susskind formulations: a mass-dependent Trotter overhead that scales as m4m^4m4, non-singlet contamination that grows as m2m^2m2 and worsens with penalty terms, and a mandatory mass extrapolation. Monte Carlo simulations of SU(3) establish a universal scaling: the continuum limit forces m2∝1/am^2 \propto 1/am2∝1/a, binding the Trotter step to the lattice spacing through a cost unique to orbifolds. For a fiducial 10310^3103 calculation, the orbifold is 10410^4104--101010^{10}1010 times more expensive than every published alternative. These results indicate that the claimed computational advantages do not at present survive quantitative scrutiny.
We demonstrate that the orbifold lattice Hamiltonian -- an approach known for its efficiency in simulating SU(NNN) Yang-Mills theory and QCD on digital quantum computers -- can reproduce the Kogut-Susskind Hamiltonian in a controlled limit. We show that it emerges naturally as the infinite scalar mass limit of the orbifold lattice formulation, even at finite lattice spacing. Our analysis provides both a general analytical framework applicable to SU(NNN) gauge theories in arbitrary dimensions and specific numerical evidence for (2+1)(2+1)(2+1)-dimensional SU(NNN) Yang-Mills theories (N=2,3N=2,3N=2,3). Using Euclidean path integral methods, we quantify the approach to this limit by comparing the standard Wilson action with the orbifold lattice action, matching lattice parameters, and systematically extrapolating results as the bare scalar mass approaches infinity. This reformulation resolves longstanding technical obstacles and offers a straightforward implementation protocol for digital quantum simulation of the Kogut-Susskind Hamiltonian with exponential speedup compared to classical methods and previously known quantum methods, modulo standard assumptions made also for the original Kogut-Susskind approach. We emphasize that the present work establishes this equivalence using classical Euclidean Monte Carlo; the exponential speedup itself is a property of quantum simulation of the orbifold Hamiltonian established in earlier work, which the equivalence shown here lets one inherit for the Kogut-Susskind Hamiltonian.
Hamiltonian quantum simulation of bosons on digital quantum computers requires truncating the Hilbert space to finite dimensions. The method of truncation and the choice of basis states can significantly impact the complexity of the quantum circuit required to simulate the system. For example, a truncation in the Fock basis where each boson is encoded with a register of QQQ qubits, can result in an exponentially large number of Pauli strings required to decompose the truncated Hamiltonian. This, in turn, can lead to an exponential increase in QQQ in the complexity of the quantum circuit. For lattice quantum field theories such as Yang-Mills theory and QCD, several Hamiltonian formulations and corresponding truncations have been put forward in recent years. There is no exponential increase in QQQ when resorting to the orbifold lattice Hamiltonian, while we do not know how to remove the exponential complexity in QQQ in the commonly used Kogut-Susskind Hamiltonian. Specifically, when using the orbifold lattice Hamiltonian, the continuum limit, or, in other words, the removal of the ultraviolet energy cutoff, is obtained with circuits whose resources scale like QQQ, while they scale like O(exp⁡(Q))\mathcal{O}(\exp(Q))O(exp(Q)) for the Kogut-Susskind Hamiltonian: this can be seen as an exponential speed up in approaching the physical continuum limit for the orbifold lattice Hamiltonian formulation. We show that the universal framework, advocated by three of the authors (M.~H., S.~M., and E.~R.) and collaborators, provides a natural avenue to solve the exponential scaling of circuit complexity with QQQ, and it is the reason why using the orbifold lattice Hamiltonian is advantageous.
We comment on a recent manuscript "Ether of Orbifolds" by Henry Lamm. In the first version, it was mistakenly claimed that the orbifold lattice Hamiltonian is not gauge invariant, and a quantity εgε_gεg​, which has nothing to do with a non-existent "gauge violation", was introduced. The scaling of this εgε_gεg​ was used to claim a huge simulation cost. In fact, εgε_gεg​ characterizes the shift of the effective lattice spacing -- because, in the orbifold lattice formulation, the lattice is generated dynamically from the vacuum expectation value of the complex matrices. In the second version, the claim about the gauge symmetry was partially corrected, based on our comments. However, εgε_gεg​ is still mistakenly interpreted as a measure of "departure from SU(NNN)", inconsistently with the foundational results by Kaplan, Katz, and Ünsal, and also by Arkani-Hamed, Cohen, and Georgi. This interpretation plays a central role in sustaining the argument introduced in the first version.
Quantum computers promise to simulate complex quantum field theories that are intractable for classical computers, potentially revealing new physics in strongly interacting systems. Current approaches for simulating Yang-Mills gauge theories face significant technical barriers due to complex group structures and complicated truncation schemes that vary drastically between different theories. Here we show that the orbifold lattice formulation provides a universal framework for quantum simulation of Yang-Mills theories with arbitrary gauge groups and dimensions. Our approach reduces all theories to the same simple Hamiltonian form, enabling implementation with standard quantum gates regardless of system complexity. We demonstrate explicit quantum circuits using only controlled-NOT and single-qubit operations, with concrete resource estimates for time evolution algorithms. This universal framework simplifies quantum simulation of gauge theories and enables systematic scaling to larger systems on fault-tolerant quantum computers.
Abstract We propose a new framework for simulating U(k) Yang-Mills theory on a universal quantum computer. This construction uses the orbifold lattice formulation proposed by Kaplan, Katz, and Unsal, who originally applied it to supersymmetric gauge theories. Our proposed approach yields a novel perspective on quantum simulation of quantum field theories, carrying certain advantages over the usual Kogut-Susskind formulation. We discuss the application of our constructions to computing static properties and real-time dynamics of Yang-Mills theories, from glueball measurements to AdS/CFT, making use of a variety of quantum information techniques including qubitization, quantum signal processing, Jordan-Lee-Preskill bounds, and shadow tomography. The generalizations to certain supersymmetric Yang-Mills theories appear to be straightforward, providing a path towards the quantum simulation of quantum gravity via holographic duality.
We present a minimal implementation of SU(NNN) pure Yang-Mills theory in 3+13+13+1 dimensions for digital quantum simulation, designed to enable quantum advantage. Building on the orbifold lattice simulation protocol with logarithmic scaling in the local Hilbert-space truncation, we introduce further simplified Hamiltonians. Furthermore, we test simple methods that improve the convergence to the infinite mass limit, thereby removing the requirement of a large scalar mass to obtain the Kogut-Susskind Hamiltonian. For the SU(2) theory, we can cut the resource requirement further by utilizing the embedding of SU(2)≅S3\mathrm{SU}(2)\cong\mathrm{S}^3SU(2)≅S3 into R4\mathbb{R}^4R4. Monte Carlo simulations of the Euclidean path integral were used to benchmark the accuracy of these new analytical improvements to the theory. These results provide further support for the noncompact-variable-based approach as a practical framework for quantum simulation of non-Abelian gauge theories.