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arXiv:2502.05716 · International Symposium on Lattice Field Theory (LATTICE)

Qubit Regularization of Quantum Field Theories

Shailesh Chandrasekharan A

Published Feb 20, 2025• 4 citations
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Abstract

To study quantum field theories on a quantum computer, we must begin with Hamiltonians defined on a finite-dimensional Hilbert space and then take appropriate limits. This approach can be seen as a new type of regularization for quantum field theories, which we refer to as qubit regularization. A related finite-dimensional regularization, known as the D-theory approach, was proposed long ago as a general framework for all quantum field theories. In this framework, the dimensionality of the local Hilbert space at each spatial point can increase as needed through an additional flavor index. To reproduce asymptotically free QFTs, most studies assume that qubit-regularized theories require extending the local Hilbert space to infinity. However, contrary to this common belief, recent discoveries in (1+1) dimensions have revealed two examples where asymptotic freedom appears to emerge within a strictly finite-dimensional local Hilbert space through a novel renormalization group (RG) flow. These findings motivate further investigation into whether asymptotically free gauge theories could also emerge within a strictly finite-dimensional local Hilbert space. To support these explorations, we propose an orthonormal basis called the monomer-dimer-tensor-network (MDTN) basis and use it to construct new types of qubit-regularized lattice gauge theories.

DOI
10.22323/1.466.0001

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To study quantum field theories on a quantum computer, we must begin with Hamiltonians defined on a finite-dimensional Hilbert space and then take appropriate limits. This approach can be seen as a new type of regularization for quantum field theories, which we refer to as qubit regularization. A related finite-dimensional regularization, known as the D-theory approach, was proposed long ago as a general framework for all quantum field theories. In this framework, the dimensionality of the local Hilbert space at each spatial point can increase as needed through an additional flavor index. To reproduce asymptotically free QFTs, most studies assume that qubit-regularized theories require extending the local Hilbert space to infinity. However, contrary to this common belief, recent discoveries in (1+1) dimensions have revealed two examples where asymptotic freedom appears to emerge within a strictly finite-dimensional local Hilbert space through a novel renormalization group (RG) flow. These findings motivate further investigation into whether asymptotically free gauge theories could also emerge within a strictly finite-dimensional local Hilbert space. To support these explorations, we propose an orthonormal basis called the monomer-dimer-tensor-network (MDTN) basis and use it to construct new types of qubit-regularized lattice gauge theories.
Traditional SU ( N ) lattice gauge theories (LGTs) can be formulated using an orthonormal basis constructed from the irreducible representations (irreps) V λ of the SU ( N ) gauge symmetry. On a lattice, the elements of this basis are tensor networks comprising dimer tensors on the links labeled by a set of irreps { λ ℓ } and monomer tensors on sites labeled by { λ s } . These tensors naturally define a local site Hilbert space, H s g , on which gauge transformations act. Gauss’s law introduces an additional index α s = 1 , 2 , … , D ( H s g ) that labels an orthonormal basis of the gauge-invariant subspace of H s g . This monomer-dimer tensor-network (MDTN) basis, | { λ s } , { λ ℓ } , { α s } ⟩ , of the physical Hilbert space enables the construction of new qubit-regularized SU ( N ) gauge theories that are free of sign problems while preserving key features of traditional LGTs. Here, we investigate finite-temperature confinement-deconfinement transitions in a simple qubit-regularized SU(2) and SU(3) gauge theory in d = 2 and d = 3 spatial dimensions, formulated using the MDTN basis, and show that they reproduce the universal results of traditional LGTs at these transitions. Additionally, in d = 1 , we demonstrate using a plaquette chain that the string tension at zero temperature can be continuously tuned to zero by adjusting a model parameter that plays the role of the gauge coupling in traditional LGTs.
Other than the commonly used Wilson's regularization of quantum field theories (QFTs), there is a growing interest in regularizations that explore lattice models with a strictly finite local Hilbert space, in anticipation of the upcoming era of quantum simulations of QFTs. A notable example is Euclidean qubit regularization, which provides a natural way to recover continuum QFTs that emerge via infrared fixed points of lattice theories. Can such regularizations also capture the physics of ultraviolet fixed points? We present a novel regularization of the asymptotically free massive continuum QFT that emerges at the Berezenski-Kosterlitz-Thouless (BKT) transition through a hard core loop-gas model, discussing the advantages this model provides compared to traditional regularizations. In particular, we demonstrate that without the need for fine-tuning, it can reproduce the universal step-scaling function of the classical lattice XY model in the massive phase as we approach the phase transition.
We construct simple qubit-regularized Hamiltonian lattice gauge theories formulated in the monomer--dimer--tensor-network (MDTN) basis that are free of sign problems in the pure gauge sector. These models naturally realize both confined and deconfined phases. Using classical Monte Carlo methods, we investigate the associated finite-temperature phase transitions and show that they exhibit the expected universality classes of conventional SU(N) lattice gauge theories in various spacetime dimensions. Furthermore, we argue that second-order quantum phase transitions separating the confined and deconfined phases are likely to exist. Such critical points would provide a nonperturbative route to defining continuum limits of qubit-regularized gauge theories, potentially allowing Yang--Mills theory and related continuum gauge theories to emerge from finite-dimensional lattice constructions.
Other than the commonly used Wilson’s regularization of quantum field theories (QFTs), there is a growing interest in regularizations that explore lattice models with a strictly finite local Hilbert space, in anticipation of the upcoming era of quantum simulations of QFTs. A notable example is Euclidean qubit regularization, which provides a natural way to recover continuum QFTs that emerge via infrared fixed points of lattice theories. Can such regularizations also capture the physics of ultraviolet fixed points? We present a novel regularization of the asymptotically free massive continuum QFT that emerges at the Berezenski-Kosterlitz-Thouless (BKT) transition through a hard core loop-gas model, discussing the advantages this model provides compared to traditional regularizations. In particular, we demonstrate that without the need for fine-tuning, it can reproduce the universal step-scaling function of the classical lattice XY model in the massive phase as we approach the phase transition.
We construct simple qubit-regularized Hamiltonian lattice gauge theories formulated in the monomer--dimer--tensor-network (MDTN) basis that are free of sign problems in the pure gauge sector. These models naturally realize both confined and deconfined phases. Using classical Monte Carlo methods, we investigate the associated finite-temperature phase transitions and show that they exhibit the expected universality classes of conventional SU(N) lattice gauge theories in various spacetime dimensions. Furthermore, we argue that second-order quantum phase transitions separating the confined and deconfined phases are likely to exist. Such critical points would provide a nonperturbative route to defining continuum limits of qubit-regularized gauge theories, potentially allowing Yang--Mills theory and related continuum gauge theories to emerge from finite-dimensional lattice constructions.
We explore if spacetime symmetric lattice field theory models with a finite Hilbert space per lattice site can reproduce asymptotic freedom in the two-dimensional O(4)O(4)O(4) model. We focus on a simple class of such models with a five-dimensional local Hilbert space. We demonstrate how even the simplest model reproduces asymptotic freedom within the D-theory formalism but at the cost of increasing the size of the Hilbert space through coupling several layers of a two-dimensional lattice. We then argue that qubit regularization can be viewed as an effective field theory (EFT) even if the continuum limit cannot be reached, as long as we can tune the model close enough to the continuum limit where perturbation theory, or other analytical techniques, become viable. We construct a simple lattice model on a single layer with a four-dimensional local Hilbert space that acts like an excellent EFT of the original theory.