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arXiv:2506.00755

Exponential speedup in quantum simulation of Kogut-Susskind Hamiltonian via orbifold lattice

Georg Bergner, Masanori Hanada, Mendicelli, Emanuele

Published May 31, 2025• 0 citations
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Abstract

We demonstrate that the orbifold lattice Hamiltonian -- an approach known for its efficiency in simulating SU($N$) 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($N$) gauge theories in arbitrary dimensions and specific numerical evidence for $(2+1)$-dimensional SU($N$) Yang-Mills theories ($N=2,3$). 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.

DOI
10.48550/arxiv.2506.00755

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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.
We present a quantum simulation framework universally applicable to a wide class of quantum systems, including quantum field theories such as quantum chromodynamics (QCD). Specifically, we generalize an efficient quantum simulation protocol developed for bosonic theories in [Halimeh et al., arXiv:2411.13161] which, when applied to Yang-Mills theory, demonstrated an exponential resource advantage with respect to the truncation level of the bosonic modes, to systems with both bosons and fermions using the Jordan-Wigner transform and also the Verstraete-Cirac transform. We apply this framework to QCD using the orbifold lattice formulation and achieve an exponential speedup compared to previous proposals. As a by-product, exponential speedup is achieved in the quantum simulation of the Kogut-Susskind Hamiltonian, the latter being a special limit of the orbifold lattice Hamiltonian. In the case of Hamiltonian time evolution of a theory on an LdL^dLd spatial lattice via Trotterization, one Trotter step can be realized using O(Ld)\mathcal{O}(L^d)O(Ld) numbers of CNOT gates, Hadamard gates, phase gates, and one-qubit rotations. We show this analytically for any matter content and SU(N)\mathrm{SU}(N)SU(N) gauge group with any NNN. Even when we use the Jordan-Wigner transform, we can utilize the cancellation of quantum gates to significantly simplify the quantum circuit. We also discuss a block encoding of the Hamiltonian as a linear combination of unitaries using the Verstraete-Cirac transform. Our protocols do not assume oracles, but rather present explicit constructions with rigorous resource estimations without a hidden cost, and are thus readily implementable on a quantum computer.
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Simulating lattice gauge theories on quantum computers presents unique challenges that drive the development of novel theoretical frameworks. The orbifold lattice approach offers a scalable method for simulating SU(NNN) gauge theories in arbitrary dimensions. In this work, we present three improvements: (i) two new simplified Hamiltonians, (ii) an encoding of the SU(2) theory with smaller number of qubits, and (iii) a reduction in the requirement for large scalar masses to reach the Kogut-Susskind limit, achieved via the inclusion of an additional term in the Hamiltonian. These advancements significantly reduce circuit depth and qubit requirements for quantum simulations. We benchmarked these improvements using Monte Carlo simulations of SU(2) in (2+1) dimensions. Preliminary results demonstrate the effectiveness of these developments and further validate the use of noncompact variables as a promising framework for scalable quantum simulations of gauge theories.
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.