A universal framework for the quantum simulation of Yang–Mills theory - Nature

Download PDF AbstractQuantum 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. Similar content being viewed by others Simulating two-dimensional lattice gauge theories on a qudit quantum computer Article Open access 25 March 2025 Observable-driven speed-ups in quantum simulations Article Open access 19 August 2025 Long-time error-mitigating simulation of open quantum systems on near term quantum computers Article Open access 21 January 2025 IntroductionRecent advances in quantum error correction have made the prospect of fault-tolerant quantum computing ever more promising. A very exciting prospect of achieving the latter lies in quantum-simulating the holy grail of high-energy physics, QCD1,2,3,4,5,6,7,8,9. For example, this would provide a complementary venue to dedicated particle colliders for the investigation of QCD, aiding in unraveling many of its outstanding mysteries10,11.One of the first steps to realize this potential is to write the QCD Hamiltonian explicitly in a form that can be implemented on digital universal quantum computers. The standard method is to replace the infinite-volume continuum space with a finite-size lattice, in such a way that the continuum and large-volume result can be obtained systematically by sending the lattice size to infinity12. Furthermore, because gluons are bosons, we need to truncate the Hilbert space of the lattice theory by introducing a certain truncation level Λ and taking the limit Λ → ∞. We must write the truncated Hamiltonian explicitly for arbitrary lattice size and arbitrary truncation level, in such a way that the implementation on quantum computers is straightforward.QCD is a Yang–Mills theory with an SU(3) gauge group coupled to fermions in the fundamental representation13. The QCD effects are typically the least well-understood for Standard Model processes and thus limit the theoretical precision reached, especially for time-dependent genuinely non-perturbative processes that cannot be treated by either perturbative QCD (pQCD) or Lattice QCD (LQCD). Currently, phenomenological models using various approximations are used to glean, e.g., information about intermediate-time far-from-equilibrium highly non-perturbative quantum processes underlying the formation of the Quark-gluon plasma. Quantum simulation offers the unique prospect of probing such processes from a first-principles standpoint, providing snapshots of this dynamics that can yield deep insights into outstanding questions14. Furthermore, a major driving force in particle physics is to find and investigate physics beyond the Standard Model, e.g., quantum gravity, for which a plethora of suggestions have been made, typically involving different gauge groups, additional symmetries, or novel interaction terms15.Thus, to fully profit from the possibilities opened up by future, fault-tolerant quantum computing, it will be crucial to develop universal formulations that can easily be adapted to any member of large groups of theories. (In principle, any formulation with the correct continuum limit is eligible.) For example, the large-N limit of SU(N) gauge theories plays a prominent role because it allows us to obtain exact analytic results16,17. Other examples are, e.g., SU(5) and SO(10) candidates for Grand Unified gauge theories. Therefore, the study of SU(N) Yang–Mills theory with N ≥ 3 is a promising starting point, covering QCD as well as many models beyond the Standard Model.As an almost trivial but important remark, we note that Yang–Mills theory and QCD are merely a small fraction of many important problems. It is presently hoped that quantum computing will allow us to solve a long list of computational problems for which classical computers are inefficient, and this list is expected to only get longer with time18. To meet all these expectations, one will need versatile codes that allow treating many of these problems without the need to undertake quantum code development for each of them from scratch. The situation will thus be quite different from what it is now, where development focuses on a few, highly specific applications, and invests most work on highly specific resource optimization using, e.g., special properties of the chosen problem that do not generalize to other problems of actual interest. It may be better not to rely on special properties such as the simplicity of the representation theory for U(1) or SU(2), or perhaps, any features specific to Yang–Mills theory, so that we can utilize the power of more generic methods developed by the wide research community.Currently, the most popular choice of lattice Hamiltonian for SU(N) Yang–Mills theory within the high-energy physics community is the Kogut–Susskind formulation19. This is the Hamiltonian version of Wilson’s Lagrangian formulation20 that uses unitary link variables. Specifically, there are unitary link operators \({\hat{U}}_{j,\overrightarrow{x}}\) living on a link connecting lattice site \(\overrightarrow{x}\) and \(\overrightarrow{x}+\hat{j}\), where j = 1, 2, 3 are spatial dimensions and \(\hat{j}\) is the unit vector along the jth direction. In addition, the conjugate momenta \({\hat{E}}_{j,\overrightarrow{x}}\) are introduced. To describe the Hilbert space, one can use the coordinate basis (also called magnetic basis) or the momentum basis (also called electric basis). The coordinate basis uses the coordinate eigenstates \(\left\vert U\right\rangle \) that satisfies \({\hat{U}}_{j,\overrightarrow{x}}\left\vert U\right\rangle ={U}_{j,\overrightarrow{x}}\left\vert U\right\rangle \), where \({U}_{j,\overrightarrow{x}}\) is an N × N special unitary matrix. For a quantum state \(\left\vert \Phi \right\rangle \), the wavefunction \(\Phi (U)=\left\langle U\right\vert \left\vert \Phi \right\rangle \) is defined on \({\prod }_{j,\overrightarrow{x}}{[{{\rm{SU}}}(N)]}_{j,\overrightarrow{x}}\), where \({[{{\rm{SU}}}(N)]}_{j,\overrightarrow{x}}\) is the SU(N) group manifold corresponding to the link between \(\overrightarrow{x}\) and \(\overrightarrow{x}+\hat{j}\). It is a nontrivial task to truncate the SU(N) group manifold systematically so that the truncation effect can be evaluated straightforwardly and at the same time \({\hat{E}}_{j,\overrightarrow{x}}\) takes a simple form. The momentum basis uses the SU(N)-analog of the Fourier transform defined by the Peter–Weyl theorem21,22. This requires complicated group theory, specifically the knowledge of all irreducible representations and their Clebsch–Gordan coefficients. Although the momentum basis allows, in principle, a systematic truncation, as shown in a pioneering paper by Byrnes and Yamamoto23, it is technically complicated except for special cases such as SU(2). Whether we use the coordinate basis or the momentum basis, it is nontrivial to write down the truncated Hamiltonian, particularly for N ≥ 3. To get some intuition for the level of complications, see the relatively simple cases of the coordinate basis for SU(2)24,25, Fourier transform for a few discrete subgroups of SU(3)26, the use of q-deformation27,28,29, and some simplifications in the large-N limit30.It is fair to say that it is very challenging to program SU(N) Yang–Mills theory with N ≥ 3 in 2 + 1 or 3 + 1 on a quantum computer using the Kogut–Susskind Hamiltonian; starting already with the simplest task of writing down the explicit Hamiltonian in terms of Pauli strings. Perhaps it is possible to write down the Hamiltonian explicitly either on the momentum basis or coordinate basis by using automated computer algebra systems, but still, there is no clear path to resolve other issues associated with the complicated expressions. Indeed, currently the only known large-scale experimental realizations of lattice gauge theories on quantum hardware are either in 1 + 131,32 or 2 + 1 dimensions33,34,35, with a two-level representation of an Abelian gauge field. This motivates efforts to find an alternative lattice formulation that is straightforward to generalize to any dimension and gauge group. We suggest choosing the orbifold lattice formulation which does not suffer from these technical complications because of its use of non-compact complex link variables \({Z}_{j,\overrightarrow{x}}\) instead of compact unitary link variables \({U}_{j,\overrightarrow{x}}\)36,37.Quantum simulations of a matrix model and a gauge theory using the orbifold lattice formulation are very similar. Matrix models are interesting in their own right for many reasons, most notably as a non-perturbative definition of quantum gravity via gauge/gravity duality38. Therefore, by understanding how matrix models and orbifold lattice gauge theories can be studied on quantum computers, we can approach many important problems including QCD and quantum gravity.In this work, we study SU(N) Yang–Mills theories on orbifold lattices and SU(N) Hermitian matrix models within a universal framework for quantum simulation. We show that both classes of theories can be formulated using standard bosonic variables with simple kinetic and quartic potential terms, so that their Hamiltonians can be written in a compact, universal form,$$\hat{H}=\frac{1}{2}\sum\limits_{a}{\hat{p}}_{a}^{2}+V(\hat{x})\,,$$ (1) where \(V(\hat{x})\) is at most quartic. This enables direct mapping to Pauli strings, efficient implementation on digital quantum computers, and unified resource estimates for qubits and gate counts. Our results establish a broadly applicable approach that simplifies the study of gauge theories and matrix models on quantum hardware, laying the foundation for future exploration of QCD, beyond-the-Standard-Model physics, and quantum gravity.ResultsBasic ideaAs we will see in section “Matrix models” and section “Orbifold lattice”, the Hamiltonians of matrix model and orbifold lattice gauge theory are schematically written as (1), where \(V(\hat{x})\) is at most a fourth-order polynomial. Therefore, we discuss how a Hamiltonian for Nb bosons of the form$$\hat{H}=\sum\limits_{a}\frac{1}{2}{\hat{p}}_{a}^{2}+\sum\limits_{a,b,c,d}{C}_{abcd}\,{\hat{x}}_{a}{\hat{x}}_{b}{\hat{x}}_{c}{\hat{x}}_{d}\,,$$ (2) where Cabcd is an arbitrary real number, can be simulated on a quantum device. In this section, we focus on getting a simple truncated Hamiltonian, postponing an explicit construction of a quantum circuit for the Hamiltonian time evolution until the section “Resource estimate for Suzuki–Trotter time evolution”.The potential part and kinetic part of the Hamiltonian become simple in the coordinate basis and momentum basis, respectively. Unlike in the Kogut–Susskind formulation, the Fourier transform between these two bases is straightforward for a generic gauge group SU(N). Truncation can be performed in a way compatible with the quantum Fourier transform.By setting Nb = 1, the summations in (2) reduce to single terms, and the interaction part can be simplified until we arrive at the simplest nontrivial example, a single quantum anharmonic oscillator,$$\hat{H}=\frac{{\hat{p}}^{2}}{2}+\frac{{\hat{x}}^{4}}{4}\,.$$ (3) We will comment on this example in the section “Discussion”. We are optimistic that, by then, readers will see how this example captures the essence of our approach and serves as an excellent starting point for quantum simulations of more intriguing systems.We begin by explaining truncation in the coordinate basis. Let us use \(\{\left\vert \overrightarrow{x}\right\rangle \}\) to denote all Nb bosons simultaneously. By using this expression, we mean that the coordinate eigenstate of the system is$$\left\vert \overrightarrow{x}\right\rangle ={\otimes }_{a}\left\vert {x}_{a}\right\rangle \,,$$ (4) where each boson has coordinate eigenstate \(\left\vert {x}_{a}\right\rangle \) (a = 1, 2, ⋯, Nb). The coordinate eigenstate \(\left\vert \overrightarrow{x}\right\rangle \) is defined b$$\hat{\overrightarrow{x}}\left\vert \overrightarrow{x}\right\rangle =\overrightarrow{x}\left\vert \overrightarrow{x}\right\rangle .$$ (5) Moreover, we consider the system Hilbert space as a tensor product of the Hilbert spaces of the individual bosons in the coordinate eigenbasis. This Hilbert space \({{\mathcal{H}}}\) corresponds to the extended Hilbert space \({{{\mathcal{H}}}}_{{{\rm{ext}}}}\) introduced in later sections.$${{\mathcal{H}}}={\otimes }_{a}{{{\mathcal{H}}}}_{a}, \quad {{{\mathcal{H}}}}_{a}={{\rm{Span}}}\{\left\vert {x}_{a}\right\rangle | {x}_{a}\in {\mathbb{R}}\}\,.$$ (6) So far, each boson has a wavefunction that can be represented in the basis given by \(\left\vert {x}_{a}\right\rangle \), which lives in an infinite-dimensional Hilbert space. For example, one can imagine a one-dimensional quantum oscillator being in a superposition of many coordinates/positions. In order to reduce the problem to a finite-dimensional Hilbert space, for each boson coordinate xa, we introduce a cutoff,$$-R\le {x}_{a}\le R\,,$$ (7) and discretize xa by introducing Λ≥2 points, with a slight modification of xa,n and δx required when periodic boundary conditions are imposed, as we will see shortly and as depicted in Fig. 1.$${x}_{a,{n}_{a}}=-R+{n}_{a}{\delta }_{x}\,,\qquad {\delta }_{x}=\frac{2R}{\Lambda -1}\,,\qquad {n}_{a}=0,1,\cdots \,,\Lambda -1\,.$$ (8) Fig. 1: Discretization of a bosonic coordinate.The coordinate operator \(\hat{x}a\) acts on a limited number of different states \(\left\vert {n}_{a}\right\rangle \) with discretized eigenvalues \({x}_{a,{n}_{a}}\), labeled by integers na. The gradient shading represents a possible wavefunction realization for this single bosonic degree of freedom.Full size imageWe consider Λ, δx, and R as truncation parameters that can and need to be adjusted. In particular, they each have limiting values that should be reached in order to recover the original infinite-dimensional Hilbert space: Λ should be sent to ∞, together with R, while δx goes to 0.By using \(\left\vert {n}_{a}\right\rangle \) to denote \(\left\vert {x}_{a,{n}_{a}}\right\rangle \), we can write the operator \({\hat{x}}_{a}\) acting diagonally on \({{{\mathcal{H}}}}_{a}\) as$${\hat{x}}_{a}={\sum }_{{n}_{a}=0}^{\Lambda -1}{x}_{a,{n}_{a}}\left\vert {n}_{a}\right\rangle \left\langle {n}_{a}\right\vert =-R\cdot {{\bf{1}}}+{\delta }_{x}\cdot {\hat{n}}_{a}\,,$$ (9) where$${\hat{n}}_{a}\equiv \sum\limits_{{n}_{a}}{n}_{a}\left\vert {n}_{a}\right\rangle \left\langle {n}_{a}\right\vert $$ (10) is the bosonic number operator. This operator can be extended to the operator acting on \({{\mathcal{H}}}\), assuming that it acts as the identity on \({{{\mathcal{H}}}}_{{a}^{{\prime} }}\) for \({a}^{{\prime} }\ne a\).We then write \({\hat{n}}_{a}\) as a sum of Pauli operators acting on Q qubits representing a number of states equal to the number of points Λ = 2Q. Using the binary form with ba,i = {0, 1},$$\left\vert {n}_{a}\right\rangle =\left\vert {b}_{a,1}\right\rangle \left\vert {b}_{a,2}\right\rangle \cdots \left\vert {b}_{a,Q}\right\rangle \,,\qquad {n}_{a}={b}_{a,1}+2{b}_{a,2}\cdots +{2}^{Q-1}{b}_{a,Q}\,,$$ (11) the number operator can be written by using Pauli σz gates,$${\hat{n}}_{a} = - \frac{{\hat{\sigma }}_{z;a,1}-{{\bf{1}}}}{2}-2\cdot \frac{{\hat{\sigma }}_{z;a,2}-{{\bf{1}}}}{2}-\cdots -{2}^{Q-1}\cdot \frac{{\hat{\sigma }}_{z;a,Q}-{{\bf{1}}}}{2} \\ = - \frac{{\hat{\sigma }}_{z;a,1}}{2}-2\cdot \frac{{\hat{\sigma }}_{z;a,2}}{2}-\cdots -{2}^{Q-1}\cdot \frac{{\hat{\sigma }}_{z;a,Q}}{2}+\frac{\Lambda -1}{2}\cdot {{\bf{1}}},$$ (12) where \({\hat{\sigma }}_{z;a,i}\) is the Pauli \({\hat{\sigma }}_{z}\) operator acting on \(\left\vert {b}_{a,i}\right\rangle \). Note that our convention is$${\sigma }_{z}\equiv \left\vert 0\right\rangle \left\langle 0\right\vert -\left\vert 1\right\rangle \left\langle 1\right\vert \,.$$ (13) Therefore,$${\hat{x}}_{a}=-{\delta }_{x}\cdot \left(\frac{{\hat{\sigma }}_{z;a,1}}{2}+2\cdot \frac{{\hat{\sigma }}_{z;a,2}}{2}+\cdots +{2}^{Q-1}\cdot \frac{{\hat{\sigma }}_{z;a,Q}}{2}\right)\,.$$ (14) There are many four-boson couplings of the form \({\hat{x}}_{a}\otimes {\hat{x}}_{b}\otimes {\hat{x}}_{c}\otimes {\hat{x}}_{d}\). Each \(\hat{x}\) is a sum of \({\hat{\sigma }}_{z,1}\),..., \({\hat{\sigma }}_{z,Q}\). Therefore, each four-boson coupling consists of Q4 terms, each of which is a tensor product of four \({\hat{\sigma }}_{z}\)’s. If the same boson appears more than once in a given coupling, e.g., \({\hat{x}}_{a}^{2}{\hat{x}}_{b}^{2}\), then terms with fewer than four \({\hat{\sigma }}_{z}\)’s also appear. The same structure is already present for the harmonic oscillator (3) and was explicitly studied in ref. 39, and much earlier in ref. 40.Next, we discuss the implementation of the periodic boundary condition. Technically, it is convenient to use the periodic boundary condition x + 2R ~ x. In this case, a convenient convention is to take$${\delta }_{x}=\frac{2R}{\Lambda }$$ (15) and$${x}_{a,{n}_{a}}=-\frac{\Lambda -1}{\Lambda }R+{n}_{a}{\delta }_{x}=\left({n}_{a}-\frac{\Lambda -1}{2}\right){\delta }_{x}\,.$$ (16) Hence, \({x}_{a,{n}_{a}}\) takes values \(\pm \frac{{\delta }_{x}}{2}\), \(\pm \frac{3{\delta }_{x}}{2}\), ..., \(\pm \frac{(\Lambda -1){\delta }_{x}}{2}\). To reduce the truncation effect, we must take R large enough for x ~ ± R not to be significantly excited, such that the boundary condition does not affect the physics of interest. In other words: if the boundary condition matters, R is not large enough. The study of truncation effects is usually specific for the system under study, including its parameters, such as the coupling constant. It is often the case that these systematic effects need to be studied numerically. An example targeting expectation values computed via classical sampling methods was reported in ref. 39.Next, we introduce the quantum Fourier transform and the corresponding momentum basis. With the periodic boundary condition, the shift operator \({\hat{S}}_{a}\equiv {\sum }_{{n}_{a}}\left\vert {n}_{a}+1\right\rangle \left\langle {n}_{a}\right\vert \) is identified with \({e}^{{{\rm{i}}}{\delta }_{X}{\hat{p}}_{a}}\). Therefore, we can approximate \({\hat{p}}_{a}\) by \(\frac{{\hat{S}}_{a}^{1/2}-{\hat{S}}_{a}^{-1/2}}{{{\rm{i}}}{\delta }_{X}}\) up to corrections of order δX. Then,$${\hat{p}}_{a}^{2} = \frac{2\cdot {{\bf{1}}}-{\hat{S}}_{a}-{\hat{S}}_{a}^{-1}}{{\delta }_{X}^{2}}=\frac{1}{{\delta }_{X}^{2}}{\sum }_{{n}_{a}=0}^{\Lambda -1}\left\{2\left\vert {n}_{a}\right\rangle \left\langle {n}_{a}\right\vert -\left\vert {n}_{a}+1\right\rangle \left\langle {n}_{a}\right\vert -\left\vert {n}_{a}\right\rangle \left\langle {n}_{a}+1\right\vert \right\}\,.$$ (17) Because \({\sum }_{{n}_{a}}\left\vert {n}_{a}\right\rangle \left\langle {n}_{a}\right\vert \) is the identity, the nontrivial parts of \({\hat{p}}_{a}^{2}\) are \({\hat{S}}_{a}={\sum }_{{n}_{a}}\left\vert {n}_{a}+1\right\rangle \left\langle {n}_{a}\right\vert \) and \({\hat{S}}_{a}^{-1}\).By applying the quantum Fourier transform, we can switch to the momentum eigenstates \(\left\vert {\tilde{n}}_{a}\right\rangle \) (\({\tilde{n}}_{a}=0,1,\cdots \,,\Lambda -1\)):$$\left\vert {\tilde{n}}_{a}\right\rangle =\frac{1}{\sqrt{\Lambda }}\sum\limits_{{n}_{a}}{e}^{2\pi {{\rm{i}}}{\tilde{n}}_{a}({n}_{a}+1/2)/\Lambda }\left\vert {n}_{a}\right\rangle \,.$$ (18) The shift operator becomes diagonal in the momentum basis:$$\hat{S}\left\vert {\tilde{n}}_{a}\right\rangle ={e}^{2\pi {{\rm{i}}}{\tilde{n}}_{a}/\Lambda }\left\vert {\tilde{n}}_{a}\right\rangle \,.$$ (19) Therefore, \({\hat{p}}_{a}\) is diagonal, too:$${\hat{p}}_{a}\left\vert {\tilde{n}}_{a}\right\rangle =\frac{2}{{\delta }_{X}}\sin \left(\frac{\pi {\tilde{n}}_{a}}{\Lambda }\right)\left\vert {\tilde{n}}_{a}\right\rangle \,.$$ (20) Alternatively, we can define \({\hat{p}}_{a}\) as$${\hat{p}}_{a}\left\vert {\tilde{n}}_{a}\right\rangle =\frac{2\pi }{{\delta }_{X}\Lambda }\left({\tilde{n}}_{a}+\frac{1}{2}\right)\left\vert {\tilde{n}}_{a}\right\rangle = \frac{\pi }{R}\left({\tilde{n}}_{a}+\frac{1}{2}\right)\left\vert {\tilde{n}}_{a}\right\rangle ,$$ (21) restricting the range of \({\tilde{n}}_{a}\) from \(-\frac{\Lambda }{2}\) to \(+\frac{\Lambda }{2}-1\) instead of (20). Here, we used \({\tilde{n}}_{a}+1/2\) rather than \({\tilde{n}}_{a}\) to respect the symmetry under \({\hat{p}}_{a}\to -{\hat{p}}_{a}\). Associated with this, the Fourier transform is modified to$$\left\vert {\tilde{n}}_{a}\right\rangle =\frac{1}{\sqrt{\Lambda }}\sum\limits_{{n}_{a}}{e}^{2\pi {{\rm{i}}}({\tilde{n}}_{a}+1/2)({n}_{a}+1/2)/\Lambda }\left\vert {n}_{a}\right\rangle \,.$$ (22) With this option, the kinetic term is nonlocal in the coordinate basis. These two options are the same up to truncation effects.Note that the Fourier transform can be performed for each boson in parallel, and hence the depth of the circuit depends only on the truncation level Λ and not on the number of bosons. Therefore, if we can perform a Fourier transform on a one-boson system, like the anharmonic oscillator (3), in principle, we only have to add more qubits that can describe more bosons and add the same circuits for the other bosons.Scalar quantum field theoryAn important example of quantum field theory is a scalar ϕ4 theory in 3 + 1 spacetime dimensions. Despite its simplicity, this theory contains many important features of quantum field theory, and it is often used to demonstrate new concepts or new techniques. See e.g., the famous textbooks by Peskin and Schröder41 and by Fradkin42. Naturally, the seminal paper on quantum computing by ref. 43 studies this theory, too.We regularize this theory on a cubic lattice with equal lattice spacing a in all three directions. Following the notations in ref. 39, we write the lattice Hamiltonian as$$\hat{H}=\sum\limits_{\overrightarrow{n}}\left(\frac{1}{2}{\hat{\pi }}_{\overrightarrow{n}}^{2}+\frac{1}{2}{\sum }_{j=1}^{3}{\left({\hat{\phi }}_{\overrightarrow{n}+\hat{j}}-{\hat{\phi }}_{\overrightarrow{n}}\right)}^{2}+\frac{{m}^{2}}{2}{\hat{\phi }}_{\overrightarrow{n}}^{2}+\frac{\lambda }{4}{\hat{\phi }}_{\overrightarrow{n}}^{4}\right)\,.$$ (23) The scalar field \(\hat{\phi }\) and its conjugate momentum \(\hat{\pi }\) are dimensionless. They correspond to fields in the continuum theory according to \(\hat{\phi }=a{\hat{\phi }}_{{{\rm{cont.}}}}\) and \(\hat{\pi }={a}^{2}{\hat{\pi }}_{{{\rm{cont.}}}}\) (where a is the dimensionfull lattice spacing). The Hamiltonian and the mass parameter are also made dimensionless, i.e., \(\hat{H}=a\times {\hat{H}}_{{{\rm{cont.}}}}\), m = a × mcont.. The lattice sites are labeled by \(\overrightarrow{n}\in {{\mathbb{Z}}}^{d}\); and \(\hat{j}\) is the unit vector along the jth dimension of the spatial lattice (j = 1, 2, 3).The canonical commutation relation is imposed, i.e.,$$[{\hat{\phi }}_{\overrightarrow{n}},{\hat{\pi }}_{{\overrightarrow{n}}^{{\prime} }}]={{\rm{i}}}{\delta }_{\overrightarrow{n},{\overrightarrow{n}}^{{\prime} }}\,.$$ (24) These operators \(\hat{\phi }\) and \(\hat{\pi }\) are the same as \(\hat{x}\) and \(\hat{p}\) in the previous sections, and the Hamiltonian takes the universal form (1).As a minor comment on terminology, we note that the quadratic term in \(\hat{\phi }\) in the lattice Hamiltonian (23) corresponds to the spatial derivative term \({({\partial }_{j}\hat{\phi })}^{2}\) in the continuum theory, which is usually called a kinetic term. However, in the context of the universal form (1), we regard it as a quadratic part of the potential \(V(\hat{x})\), because \(\hat{\phi }\) is playing the role of a (bosonic) coordinate.Next, we define the Hilbert space describing the lattice system. A convenient way to define the Hilbert space is to use coordinate eigenstates \(\left\vert \phi \right\rangle \) that satisfy \({\hat{\phi }}_{\overrightarrow{n}}\left\vert \phi \right\rangle ={\phi }_{\overrightarrow{n}}\left\vert \phi \right\rangle \)$${{\mathcal{H}}}=\left\{\left\vert \Psi \right\rangle \equiv \int{{{\rm{d}}}}^{{V}_{{{\rm{lattice}}}}}\phi \,\Psi (\phi )\left\vert \phi \right\rangle | \int{{{\rm{d}}}}^{{V}_{{{\rm{lattice}}}}}\phi \,| \Psi (\ph