Introduction

The Hamiltonian formulation, first put forth by Kogut and Susskind1, allows one to perform numerical calculations without encountering a sign problem. Classical computational techniques like exact diagonalization or tensor network calculations constitute a set of nonperturbative methods well suited for calculating real-time dynamics of the gauge theories based on the Hamiltonian framework. Hamiltonian formulations are also expected to provide a natural framework for computation in the upcoming quantum computation era. The Hamiltonian obtained via temporal gauge fixing is defined on a discrete spatial lattice while the time remains continuous. The Hamiltonian dynamics for a gauge theory is constrained by a set of local constraints known as Gauss’ law. The generators of the Gauss law constraints are mutually non-commuting for non-Abelian gauge theories described by an SU(N) gauge group. The SU(3) group is of particular interest, as it forms the basis of Quantum Chromodynamics (QCD)2, the theory that describes the strong nuclear force. The primary ingredients for SU(N) lattice gauge theories are a Hamiltonian \(\hat{H}\) and constraints \({\hat{G}}^{a}(r)\), which satisfy:

$$\left[H,{G}^{a}(r)\right]=0,\,\,\left[{G}^{a}(r),{G}^{b}({r}^{{\prime} })\right]=i{{f}^{ab}}_{c}{G}^{c}(r){\delta }_{r,{r}^{{\prime} }}$$
(1)

∀ r and a = 1, 2, …, N2 − 1. The physical Hilbert space for a gauge theory is generally chosen as the space of states, which are annihilated by all the Gauss’ law operators Ga(r) that generate gauge transformations locally at each site r. In principle, one can choose other gauge symmetry sectors (characterized by an integer quantum number of the Gauss law generator) as the physical Hilbert space, but the Hamiltonian dynamics remains confined to that sector. The conventional choice of gauge invariant sector leads to Wilson loops, strings, and hadrons as the physical degrees of freedom for the gauge theory3,4,5.

Performing Hamiltonian simulation in the full Hilbert space (without imposing Gauss law constraint) involves additional computational costs due to gauge redundancy, and it is often impractical for quantum simulation, where quantum resources are limited and costly. Hamiltonian simulation of gauge theories primarily demands (i) preparing an initial state in the physical Hilbert space, and (ii) guaranteeing protection of gauge invariance throughout the simulated dynamics. Both tasks pose significant challenges due to the choice of the Hamiltonian framework and the quantum hardware. The conventional framework for Hamiltonian lattice gauge theory is given by the well-studied Kogut-Susskind1 formulation. While it is possible to construct a gauge-invariant Hilbert space in this formulation, it is computationally expensive6. Quantum devices in the Noisy-Intermediate-scale Quantum (NISQ) era are prone to error, making state preparation and its subsequent time evolution difficult. The errors will accumulate and drive the dynamics away from the physical Hilbert space, and thus require error correction or error mitigation strategies for a faithful simulation. In the 1 + 1 dimension, one can effectively eliminate all the gauge degrees of freedom for a gauge theory at the cost of introducing non-local interaction among on-site fermionic degrees of freedom. This comes as a solution to the Gauss law and works directly in the gauge-invariant Hilbert space, but the non-locality will be difficult to implement on quantum devices. Over the past decade, there has been a renewed interest in Hamiltonian simulations of lattice gauge theories7,8,9,10,11,12,13. There have been a series of work on addressing this issue in the context of low-dimensional theories, and towards developing novel frameworks particularly 1 + 1 dimensions,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35. On the same note, there have been a multitude of research done into applying one or more of these formalism to describe digital as well as analog simulation protocols in the current NISQ-era16,36,37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,52,53,54,55,56,57,58,59,60,61,62,63,64,65,66,67,68,69,70,71. The noisy nature of current quantum simulation technology demands error mitigation/correction techniques to faithfully simulate gauge-invariant dynamics. This has led to quite a few proposals suggesting different ways to control this error72,73,74,75,76,77,78,79,80. Additionally, there have been progress to incorporate error-correction in the context of quantum simulations of field theories81,82,83,84. The different proposals and error mitigation schemes led to several digital and analog simulations of Abelian and non-Abelian lattice gauge theories over the last few years64,85,86,87,88,89,90,91,92,93,94,95,96,97,98,99,100,101,102,103,104,105,106.

In higher dimensions, the Hamiltonian includes a non-local, gauge-invariant plaquette term that traces a closed path on the spatial lattice. Quantum simulation of these non-local terms while maintaining non-Abelian gauge invariance locally at each site poses significant challenges. The current work addresses how these complications can be avoided by using the Loop-String-Hadron (LSH) framework107,108 towards making concrete progress in quantum simulating QCD.

In this work, we use the LSH formulation of Hamiltonian lattice gauge theory in 1 + 1 dimensions for the SU(3) group and construct a symmetry protection protocol based on two different schemes. We first discuss how noise can be modelled in the context of the Loop-String-Hadron formulation. We then describe two error mitigation schemes, referred to as Scheme I and Scheme II. Scheme I leverages the global symmetries of the LSH Hamiltonian to protect the noisy dynamics, while Scheme II relies on using the local constraints of the LSH formalism to maintain gauge-invariance. Both protection schemes ensure that the dynamics are gauge-invariant. Notably, the first scheme involves only single-body penalty terms, making it relatively simpler to implement, while the second scheme is easily generalizable to higher-dimensional theories. The layout of the paper is as follows. The Results and Discussions section contains the discussion on the LSH basis - its symmetries, possible violation of symmetries in Hamiltonian simulation, the symmetry-protection schemes established by numerical results obtained via exact diagonalization of erroneous LSH dynamics and symmetry protection terms. Discussion on the scheme’s viability in terms of its implementation is also presented. The LSH Hamiltonian governing gauge-invariant dynamics and the possible error models responsible for violating gauge invariance in Hamiltonian dynamics are presented in the Methods section.

Results and Discussions

A suitable framework to study dynamics: LSH framework

As mentioned in the Introduction, we focus on a particular framework, namely the LSH framework developed for SU(2)4 and SU(3)108,109 gauge theories in 1 + 1 dimensions. The SU(2) framework has been studied extensively and found to be advantageous over other available frameworks to describe the same physics. A key feature of the LSH formalism is that it is constructed to be manifestly gauge invariant. The Hilbert space is characterized by on-site quantum numbers, which correspond to the gauge-invariant and physical loop-string-hadron excitations at each site. The dynamics of these excitations are described by a Hamiltonian, which contains combinations of on-site occupation number operators and ladder operators for the associated loop-string-hadron excitations.

Encountering non-local excitations is unavoidable in a gauge theory and remains true for the LSH framework. Non-locality in the LSH framework arises because the gauge-invariant configurations defined on a lattice, analogous to Wilson loops or strings in the theory, are not tensor product states. Instead, all the on-site LSH states are woven via Abelian constraints on each link. The LSH Hamiltonian commutes with this Abelian constraint. This property distinguishes the LSH formulations from other Hamiltonian formulations, and yet the spectrum and dynamics match that of the original Kogut-Susskind Hamiltonian. In this work, we highlight a couple of interesting and useful conclusions regarding using LSH formalism for Hamiltonian simulation, which may be beneficial in suitable contexts.

LSH basis

The LSH basis, defined locally at each lattice site, is characterized by a set of integer-valued quantum numbers, namely the loop and string quantum numbers. Loop quantum numbers are bosonic, implying that the Hilbert space is infinite-dimensional, while the string quantum numbers are fermionic. However, for practical computation, one needs to impose a finite cut-off on the bosonic quantum numbers so that the Hilbert space is of finite dimension.

For SU(2) gauge theory, the LSH basis states at a site r in 1 + 1-d is given by \({\left\vert {n}_{l},{n}_{i},{n}_{o}\right\rangle }_{r}\), where nl ∈ {0, ∞} is the loop quantum number and ni, no ∈ {0, 1} are fermionic quantum numbers. Imposing a cut-off ΛB, one would obtain an onsite Hilbert space of dimension 22(ΛB + 1). For the case of SU(3) gauge theory defined on a one-dimensional spatial lattice, the on-site Hilbert space is characterized by two loop and three string quantum numbers as

$${\vert {n}_{P},{n}_{Q},{\nu }_{\underline{1}},{\nu }_{0},{\nu }_{1}\rangle }_{r},$$
(2)

where, nP, nQ ∈ {0, ∞} and \({\nu }_{\underline{1}},{\nu }_{0},{\nu }_{1}\in \{0,1\}\). Pictorial representations of the LSH states are given in Fig. 1. For a bosonic cut-off ΛB, the dimension of the on-site LSH Hilbert space is \({2}^{3}{({\Lambda }_{B}+1)}^{2}.\)

Fig. 1: On-site SU(2) and SU(3) LSH states in 1 + 1 dimension.
figure 1

(a) For the SU(2) gauge theory, the LSH states are characterized by two fermionic and one bosonic quantum number that corresponds to undirected flux lines. (b) For SU(3) gauge theory, the LSH states are characterized by three fermionic and two bosonic quantum numbers that corresponds to flux lines flowing in two directions. For both SU(2) and SU(3), the fermionic excitations are denoted as circles, which can be either empty or filled, denoting the corresponding fermionic occupation numbers to be 0 and 1, respectively. Within the LSH framework, the fermionic occupation numbers are associated with incoming and outgoing strings in different directions.

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Gauss law constraints in LSH framework

The physical implication of gauge invariance is encoded in non-local structures, most notably in the non-local Wilson loop and string basis, which serve as a gauge-invariant basis for gauge theories. However, as shown in Fig. 1, the LSH framework is an intermediate step in building on-site gauge invariant Hilbert space. Connecting the on-site LSH Hilbert space to the space of Wilson loops and strings requires continuity of flux lines (for each type of directed flux line for SU(3)) between neighboring lattice sites. This property holds in arbitrary dimensions as well.

Analyzing the 1 + 1-dimensional SU(3) theory, let us focus on-site r, which is connected to links along direction 1 and \(\underline{1}\). Careful inspection of directed flux lines in Fig. 1 yields:

$${P}_{1}(r)={n}_{P}(r)+{\nu }_{1}(r)\left(1-{\nu }_{0}(r)\right)$$
(3)
$${Q}_{1}(r)={n}_{Q}(r)+{\nu }_{0}(r)(1-{\nu }_{\underline{1}}(r))$$
(4)

Where P1(r) (and Q1(r)) denote the number of flux lines going from left to right (and right to left) respectively on the link along 1 direction connected to site r.

The same on the link in direction \(\underline{1}\) connected to site r is given by:

$${P}_{\underline{1}}(r)={n}_{P}(r)+{\nu }_{0}(r)\left(1-{\nu }_{1}(r)\right)$$
(5)
$${Q}_{\underline{1}}(r)={n}_{Q}(r)+{\nu }_{\underline{1}}(r)\left(1-{\nu }_{0}(r)\right)$$
(6)

Continuity of flux lines or the local LSH states correspond to Wilson loop and string states of the gauge theory demand the constraints:

$${P}_{1}(r)={P}_{\underline{1}}(r+1)$$
(7)
$${Q}_{1}(r)={Q}_{\underline{1}}(r+1)$$
(8)

This implies that the space of Wilson loops and strings is not a direct product of LSH states but rather a projected subspace that satisfies a pair of local Abelian Gauss-law (AGL) constraints given in ((7),(8)).

Global symmetries of the LSH framework in 1 + 1 dimension

For the SU(3) gauge theory on a one-dimensional staggered lattice, the global symmetry remains SU(3), which can be generated by 8 charges:

$$\begin{array}{rcl}{Q}^{a}&=&\sum\limits_{r=0,2,..}{\Psi }_{\alpha }^{{\dagger} }(r){\left(\frac{{\lambda }^{a}}{2}\right)}_{\beta }^{\alpha }{\Psi }^{\beta }(r)\\ &&-\sum\limits_{r=1,3,..}{\Psi }_{\alpha }(r){\left(\frac{{\lambda }^{* a}}{2}\right)}_{\beta }^{\alpha }{\Psi }^{{\dagger} \beta }(r)\end{array}$$
(9)

where, Ψα(r) are fermionic triplets (anti-triplets) (fundamental irreps of SU(3)) located at even (odd) sites r and λa are Gell-Mann matrices for a = 1, 2, …, 8. A global state defined on the lattice can thus be characterized by global SU(3) Casimirs \(\left\vert p,q,I,M,Y\right\rangle\) of an SU(3) irrep using the standard notation, where p, q is analogous to j quantum number (Casimir) of SU(2) irreps and the set I, M, Y denote the magnetic quantum numbers analogous to m of an SU(2) irrep denoted as \(\left\vert j,m\right\rangle\). Naturally, analyzing the global symmetries of the SU(3) gauge theory is more involved than the SU(2) theory when using the irrep basis or Kogut-Susskind formalism of the theory. Nevertheless, global symmetries play a crucial role by providing a block diagonal structure for the Hamiltonian and help in analyzing the entanglement properties of gauge theories, which is important in the context of quantum computation. The global symmetry structure of the SU(3) theory in the LSH framework is found to be much more intuitive, enabling more efficient computations.

The LSH framework, being a solution to all the non-Abelian constraints, is still constrained by the local Abelian Gauss law. As a consequence, the global symmetries for LSH also become Abelian. The LSH dynamics is governed by the Hamiltonian consisting of an electric field term HE, a staggered mass term HM, and a matter gauge interaction term HI, yielding:

$$H={H}_{E}+{H}_{M}+{H}_{I}.$$
(10)

The representation of these terms in terms of LSH operators is presented in the Methods section (see subsection The Hamiltonian governing LSH dynamics). For 1 + 1-d theory coupled to staggered matter fields, as depicted in (32), the interaction term HI manifestly preserves the global occupation numbers of the three individual gauge-invariant fermionic or string modes:

$${q}_{f}=\sum\limits_{r}{\nu }_{f}(r)$$
(11)

for \(f=\underline{1},0,1\). Alternately, one can also define the linear combinations of \({q}_{\underline{1}},{q}_{0},{q}_{1}\) as108:

$$\begin{array}{rcl}{{\mathcal{F}}}&=&{q}_{\underline{1}}+{q}_{0}+{q}_{1}\\ {{\mathcal{P}}}&=&{q}_{1}-{q}_{0}\\ {{\mathcal{Q}}}&=&{q}_{0}-{q}_{\underline{1}}\end{array}$$
(12)

Note that \({{\mathcal{F}}}\) gives the total fermionic occupation number for the system, while \({{\mathcal{P}}}\) and \({{\mathcal{Q}}}\) give the imbalance between the incoming and outgoing flux of the lattice individually for leftward and rightward flux lines. The Hamiltonian matrix written in the LSH basis is block diagonal, where each block is characterized by a unique set of global quantum numbers \(({{\mathcal{F}}},{{\mathcal{P}}},{{\mathcal{Q}}})\). An ideal Hamiltonian simulation is expected to respect the block diagonal structure of the theory.

A definite benefit of using the LSH framework is the link it provides between these global symmetries and the remnant local Abelian constraints in 1+1 dimensions. As manifested for the SU(2) theory in110, the particular structure of the interaction Hamiltonian for a 1 + 1d theory that conserves global symmetries as a consequence of the interaction Hamiltonian being invariant under the AGL constraint. However, a similar connection is nontrivial to demonstrate for the case of SU(3) via an analytic expression. The string end operators are now of three types and involve an intermediate or 0th mode rather than corresponding only to incoming and outgoing strings. However, the current work establishes the connection between AGL and global symmetries for SU(3), demonstrated numerically in the next subsection.

Erroneous dynamics in quantum simulation

In this subsection, we discuss various types of errors that may occur in a quantum simulation and lead to deviation from ideal gauge-invariant LSH dynamics. Later in this article, we provide a unified solution for gauge protection and numerically establish its validity. These solutions are primarily useful in tensor network simulation and analog simulation protocols and may also be useful in digital quantum circuits.

Because the SU(3) Hilbert space has a much larger local dimension, there are several ways errors can arise when simulating such a theory. The physical states in the LSH picture are only the particular combinations of string-out-string-in states satisfying AGL (i.e., with continuity of flux lines in each direction), and any other combination, as represented in Fig. 1, will correspond to an AGL-violating state. With this in mind, we classify the possible types of errors that can appear in the SU(3) LSH Hamiltonian.

  • Error Model I: We can have nearest-neighbour hopping, which can cause transitions between different states on neighbouring sites in an AGL-violating manner.

  • Error Model II: There can be uncontrolled transitions between these states at a single site, resulting in a violation of the AGL. One can model such transitions by considering a two-body operator acting locally at a single site. These types of errors can be more detrimental to the simulation as they are ultra-local on the lattice. This two-body operator will violate the two symmetry operators \({{\mathcal{P}}}\) and \({{\mathcal{Q}}}\)

The interaction Hamiltonian modelling these errors is given by \({H}_{I}^{{\prime} }\) and \({H}_{I}^{{\prime}\prime}\). The LSH representation of both the erroneous matter gauge interactions is provided in the Methods section (see subsection Error Models) in equations (39) and (40).

We rely on the diagrammatic description of the LSH basis states defined in108 to understand how each hopping term violates the AGL constraints. Let us consider a state lying in the strong-coupling vacuum sector. This corresponds to an LSH state in the global symmetry sectors of \({{\mathcal{F}}},{{\mathcal{P}}},{{\mathcal{Q}}}=(6,0,0)\). For simplicity, we consider a four-site lattice. In the diagrammatic language, an example of such a state can be defined as shown in Fig. 2(a). The action of one of the terms in \({H}_{I}^{{\prime} }\), as given in equation (39) on the strong-coupling vacuum is shown in Fig. 2(b). There is a spontaneous violation of AGLs across every link on the 4-site example, with each link having disconnected P, Q fluxes. This corresponds to static charges on the lattice, which contradicts the initial zero-static charge configuration. The second kind of erroneous hopping generated by \({H}_{I}^{{\prime}\prime}\), equation (40), also creates similar disconnected P, Q fluxes, but their action on the strong-coupling vacuum is zero. This is straightforward to see since any linear/bilinear annihilation/creation operator combinations that are strictly onsite will end up annihilating the state. But this can have AGL violations for string-like states. A single action of the AGL invariant hopping will produce an extended string-like state. The on-site error terms will break AGLs for these string-like states. As a result, the on-site error-hoppings will be more prominent at an intermediate time scale.

Fig. 2: Cartoon depicting AGL preserving and violating LSH states.
figure 2

(a) A graphical representation of the strong-coupling vacuum (odd sites completely filled and even sites empty) for a 4-site lattice using the LSH quantum numbers that satisfy AGLs on all the links. The kets corresponds to the SU(3) LSH basis defined as \({\vert {n}_{P},{n}_{Q},{\nu }_{\underline{1}},{\nu }_{0},{\nu }_{1}\rangle }_{r}\). (b) A graphical representation of another state, charaterized by on-site LSH quantum numbers. This state do not satisfy on-link AGLs as depicted by discontinuity of directed flux lines across the links. The erroneous hopping terms \({H}_{I}^{{\prime} }\) in (13) would cause a non-zero matrix element between the states given in (a) and (b). (LSH representation of \({H}_{I}^{{\prime} }\) is given in equation (39)). As depicted in the cartoon, the total number of fermions on the lattice is still conserved (the filled black circles correspond to fully-filled occupation numbers, as shown in Fig. 1). The discontinuity of directed flux lines are connected to violation of individual global charges qf defined in (11).

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Numerical demonstration of erroneous dynamics

Let us consider the Error model I for investigating the erroneous dynamics, where the Hamiltonian contains erroneous interaction \({H}_{I}^{{\prime} }\) (replacing the original matter gauge interaction HI) as

$${H}_{err}={H}_{E}+{H}_{M}+{H}_{I}^{{\prime} }$$
(13)

The erroneous matter gauge interaction term \({H}_{I}^{{\prime} }\) is defined in equation (39) in terms of LSH operators. It is important to note that the first global symmetry characterized by \({{\mathcal{F}}}\) is easy to preserve, as that refers to only a perfect state preparation with a fixed total number of qubits given by \({{\mathcal{F}}}\). Even while preserving the value of \({{\mathcal{F}}}\), the hopping term \({H}_{I}^{{\prime} }\) violates \({{\mathcal{P}}},{{\mathcal{Q}}}\), related to an erroneous 2-qubit gate operation and propagates exponentially affecting the entire simulation.

We numerically demonstrate in Fig. 3 that time evolution with the erroneous Hamiltonian is not confined to the AGL satisfying sector of the LSH Hilbert space.

Fig. 3: Numerical demonstration of violation of global and local symmetries.
figure 3

The first row contains the dynamics of expectation values of the global charges \({{\mathcal{P}}}\) and \({{\mathcal{Q}}}\), and the second row contains the same for the six AGLs across three links evolving under the erroneous Hamiltonian Herr given in equation (13). Each column corresponds to the evolution of an initial state in a particular global symmetry sector denoted by \(({{\mathcal{P}}},{{\mathcal{Q}}})\). Here, \({\langle AG{L}^{1}\rangle }_{r,r+1},{\langle AG{L}^{2}\rangle }_{r,r+1}\) corresponds to equation (7),(8) respectively. The red-dotted line corresponds to the ideal expectation value of the 6 AGLs (0 for any gauge-invariant state) and the global charges \({{\mathcal{P}}}\) and \({{\mathcal{Q}}}\) (0 or 1 depending on the symmetry sector of the initial state) under gauge-invariant dynamics. The plots demonstrate that none of the global and local symmetries are preserved by Herr as the dynamics do not coincide with reference lines (red dotted lines) for each of the panels above.

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We also impose a cutoff on the bosonic quantum numbers nP, nQ≤2. The QuSpin Package111,112 is used to construct the Hamiltonian and its subsequent time evolution. The initial state is chosen to be the strong-coupling vacuum (unless otherwise specified), which satisfies the AGLs across all the links and lies in the global symmetry sector \(({{\mathcal{F}}},{{\mathcal{P}}},{{\mathcal{Q}}})=(6,0,0)\). In the LSH quantum numbers given in (2), the initial state is given by :

$$\begin{array}{rcl}{\left\vert \Psi \right\rangle }_{i}&=&{\left\vert 0,0,0,0,0\right\rangle }_{0}\otimes {\left\vert 0,0,1,1,1\right\rangle }_{1}\\ &\otimes &{\left\vert 0,0,0,0,0\right\rangle }_{2}\otimes {\left\vert 0,0,1,1,1\right\rangle }_{3}\end{array}$$
(14)

The above-defined state is time-evolved using the erroneous Hamiltonian defined in equation (13) as \(\left\vert \Psi (t)\right\rangle ={e}^{-i{H}_{err}t}{\left\vert \Psi \right\rangle }_{i}\). Fig. 3 shows the averages of the AGLs across each link along with the average of the global symmetry operators \({{\mathcal{P}}},{{\mathcal{Q}}}\) as a function of time. The averages move away from the ideal values and signify mixing with non-AGL satisfying states. Next, we propose a symmetry protection protocol and numerically demonstrate its capability for each error model we considered. With proper protection, the expectation values of the constraint operator fluctuate at its zero value (while the precision depends upon protection strength) as given in Supplementary Note 2.

Symmetry protection scheme I: via global symmetry protection

Protecting erroneous nearest-neighbour hopping Hamiltonian

In this section, we present a scheme for dynamically protecting global symmetries of SU(3) gauge theory coupled to dynamical matter fields in 1 + 1 dimensions. This method is a generalization of110, and can be understood as projecting the dynamics in different superselection sectors. As discussed previously, \({H}_{I}^{{\prime} }\) violates the superselection rules, i.e, the global symmetries of the LSH Hamiltonian. To counteract this phenomenon, we introduce a pair of protection terms that preserve the global symmetries of the SU(3) LSH Hamiltonian, namely the \({{\mathcal{P}}}\) and \({{\mathcal{Q}}}\) operators. We propose adding the following set of operators to the Hamiltonian:

$${H}_{{{\mathcal{P}}}}=\Lambda \left[{{\mathcal{P}}}-{\sum }_{r=0}^{N-1}\left[{\hat{\nu }}_{1}-{\hat{\nu }}_{0}\right]\right]$$
(15)
$${H}_{{{\mathcal{Q}}}}=\Lambda \left[{{\mathcal{Q}}}-{\sum }_{r=0}^{N-1}\left[{\hat{\nu }}_{0}-{\hat{\nu }}_{\underline{1}}\right]\right]$$
(16)

Here, \({{\mathcal{P}}}\) and \({{\mathcal{Q}}}\) can take integer values depending on the symmetry sector of interest. Adding these terms to the total Hamiltonian will ensure that an initially gauge-invariant state can be protected with, at most, a single body protection term. The total Hamiltonian will now read as:

$$H={H}_{E}+{H}_{M}+{H}_{I}^{{\prime} }+{H}_{{{\mathcal{P}}}}+{H}_{{{\mathcal{Q}}}}$$
(17)

where Λ is a constant term kept large enough to ensure a sufficient gap between the symmetry-protected sector and the other part of the spectrum. The scheme’s validity in confining the dynamics in each super-selection sector is demonstrated in Fig. 4.

Fig. 4: Numerical demonstration of symmetry protection for the Error Model I under Scheme I.
figure 4

a Illustrates the time-averaged values of the global symmetry operators \({{\mathcal{P}}},{{\mathcal{Q}}}\). The inset shows the quantum fluctuations of the two symmetry operators as a function of \(\frac{{x}^{{\prime} }}{\Lambda }\). b–d The time-averaged Abelian Gauss laws for the three links are plotted as a function of increasing Λ. Each of the insets for (b–d) corresponds to the quantum fluctuations of the AGLs as a function of \(\frac{{x}^{{\prime} }}{\Lambda }\). We have taken x = 1 while \({x}^{{\prime} }=0.05\), and the Λ → ∞ limit ensures that \({{\mathcal{P}}},{{\mathcal{Q}}}\) tend to the sector the initial state belonged to along with the Abelian Gauss laws across the three links converging to zero. The time-averaging has been done by looking at the time-scale regime where the error suppression is maximum.

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The numerical study presented in Fig. 3 demonstrated that the modified interaction Hamiltonian \({H}_{I}^{{\prime} }\) does, in effect, violate all of the Abelian Gauss laws for the lattice and drive the dynamics into the non-AGL satisfying sector of the LSH Hilbert space. However, the numerical study performed in this work establishes that protection of all of the global symmetries of this theory, in effect, protects each individual Abelian Gauss law originally present in the theory.

Suitable protection strength Λ, with respect to the coupling constant \({x}^{{\prime} }\) (associated with \({H}_{I}^{{\prime} }\)), is also capable of ensuring a sufficient gap between the AGL-satisfying and AGL-violating states in the Hilbert space. As a result, the AGL-violating states are suppressed during time evolution, and the dominant contribution comes from the AGL-satisfying states. We time evolve an initial state, which we choose as the strong coupling vacuum defined in equation (14), under the new Hamiltonian defined in equation (17). In the limit of Λ → ∞, this procedure ensures that the AGLs and global symmetries are restored. Fig. 4 shows how the local and global symmetries are restored. The insets correspond to the quantum fluctuations of the operators of interest, mainly \(\{AG{L}_{r,r+1}^{\{1,2\}},{{\mathcal{P}}},{{\mathcal{Q}}}\}\). These operators are temporally averaged for a short time of 0.5T, where T is the total number of time steps, which is taken to be 103 with dt = 0.5. This is because the size of the Hilbert space for the 4-site lattice is of the order 107, resulting in the norm of the Hamiltonian defined in equation (17) becoming very large. This renders it difficult to simulate for larger time scales.

If one prepares an initial state in the correct global symmetry sector that also satisfies the AGLs across each link, the symmetry protection protocol will ensure that in the large Λ limit, the global symmetries and the local constraints are approximately conserved. With such an initial state, we can compute the dynamics of relevant observables of the theory, namely the staggered fermion density and the electric Hamiltonian density across the lattice, and compare them with the gauge-invariant dynamics. This scheme also applies to initial states belonging to different global symmetry sectors, and Supplementary Note 1 contains the results for such a choice of initial state.

Protecting error due to on-site hopping

A similar numerical experiment was considered for the Error Model II that involves on-site AGL violating interactions \({H}_{I}^{{\prime}\prime}\). The explicit form of \({H}_{I}^{{\prime}\prime}\) is defined in equation (40) in terms of LSH operators. The erroneous Hamiltonian can contain both the on-site term and the nearest-neighbour violation. The on-site term plays a stronger role in erroneous dynamics as the SU(3) states admit a larger local Hilbert space. This can lead to random transitions between the different levels at a single site. The total Hamiltonian now reads as

$$H={H}_{E}+{H}_{M}+{H}_{I}^{{\prime} }+{H}_{I}^{{\prime}\prime}+{H}_{{{\mathcal{P}}}}+{H}_{{{\mathcal{Q}}}}$$
(18)

Here, H includes the AGL preserving terms of the original Hamiltonian, the on-site violations \({H}_{I}^{{\prime}\prime}\), the nearest neighbour violations \({H}_{I}^{{\prime} }\), as well as the protection terms. To demonstrate the effect of the two protection terms we introduced in equation (15) and (16), we consider the same initial state defined in equation (14). The dimensionless coupling constants associated with \({H}_{I}^{{\prime} },{H}_{I}^{{\prime}\prime}\) are respectively \({x}^{{\prime} },{x}^{{\prime}\prime}\). We choose \({x}^{{\prime} } > {x}^{{\prime}\prime}\), where x″ corresponds to the coupling of the on-site hopping, \({x}^{{\prime} }\) corresponds to the coupling of the nearest-neighbour hopping, and x to the AGL invariant hopping term.

The coupling is chosen in such a way that the influence of the on-site violations is minimal. We have also done simulations with \({x}^{{\prime} } < {x}^{{\prime}\prime}\) and see very little difference between these two choices. The results are quoted in Supplementary Note 1. With the addition of the on-site hopping term \({H}_{I}^{{\prime}\prime}\), a qualitatively comparable result is observed. Fig. 5(a) shows the time average symmetry operator values converging towards the (0, 0) sector of the initial state. Fig. 5 (b), (c), and (d) show the time-averaged AGLs. We observe the suppression decreases in the large Λ limit, and the time-series plots indicate (see Supplementary Note 2) that the dynamics are truly suppressed on the short time scale. This is primarily due to the two AGL-violating terms, \({H}_{I}^{{\prime} }\) and \({H}_{I}^{{\prime}\prime}\), in the simulated Hamiltonian.

Fig. 5: Numerical demonstration of symmetry protection for the Error Model II under Scheme I.
figure 5

a Global symmetry operators time-averaged as a function of the protection strength for the error model described in equation (40). The inset shows the quantum fluctuations of the two symmetry operators. b–d Correspond to the time-averaged AGLs across the three links. The x-axis for all the main plots and insets is \(\frac{{x}^{{\prime}\prime}}{\Lambda }\).

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For both the error models, we have averaged the operators for only the initial short time scales, where the error suppression is higher. The time-series plots corresponding to both error models are given in Supplementary Note 2.

Symmetry protection scheme II: via local symmetry protection

Since it is clear that \({H}_{I}^{{\prime} }\) violates each individual AGL, the most straightforward way to protect the AGLs is by adding protection terms proportional to the AGLs themselves to the erroneous Hamiltonian. In this section, we demonstrate that this approach works for the erroneous Hamiltonian given in (13). In contrast with the previous section, where we used a single body protection term to simulate an approximate gauge theory, here the focus is on two body protection terms, which are just equivalent to imposing the square of the two AGLs across each link. As before, we consider equation (39) to be the erroneous interaction term.

The full Hamiltonian will now be modified to include the following two terms:

$${H}_{P}=\Lambda {\sum }_{r=0}^{N-1}{\left[{P}_{1}(r)-{P}_{\underline{1}}(r+1)\right]}^{2}$$
(19)
$${H}_{Q}=\Lambda {\sum }_{r=0}^{N-1}{\left[{Q}_{1}(r)-{Q}_{\underline{1}}(r+1)\right]}^{2}$$
(20)

where \({P}_{1},{P}_{\underline{1}},{Q}_{1},{Q}_{\underline{1}}\) are defined according to equation (3)–(6). The newly modified Hamiltonian can now be written out as follows:

$$H={H}_{E}+{H}_{M}+{H}_{I}^{{\prime} }+{H}_{P}+{H}_{Q}$$
(21)

Once again, we repeat the numerical experiment of starting with the state defined in equation (14) and measure the AGLs as a function on Λ → ∞. In Fig. 6, we demonstrate that this scheme, with a suitable choice of protection strength Λ, allows the dynamics to remain confined in the AGL satisfying subspace of the Hilbert space.

Fig. 6: Numerical demonstration of symmetry protection for local symmetries under Scheme II.
figure 6

Here, we constrain the dynamics by imposing the AGLs as an energy penalty. The average of the 6 AGLs across the three links is plotted as a function of the increasing Λ. The inset shows the quantum fluctuation, ΔAGL, of each of these AGLs. In both plots, we see a clear suppression of AGL violation.

Full size image

The Λ → ∞ limit is where an approximate gauge theory is expected to be recovered, and as per the numerical simulation, we see that the operator averages tend to the desired values, with the expectation value of the AGL to be vanishing. Additionally, the temporally average operator fluctuations are also of the order 10−5, indicating that the non-AGL satisfying states’ contributions are becoming negligible.

Such a local Abelian protection can be applied to higher dimensions, where it is essential. In this work, we do not present calculations for higher dimensions as they become computationally heavy, and the one-dimensional study is sufficient to demonstrate the proof of principle.

A key finding in this one-dimensional demonstration of the local protection scheme is that each successful protection of local symmetry guarantees that the dynamics are confined to the original global symmetry sectors (as per the initial state prepared) of the LSH Hamiltonian. This is also demonstrated in Fig. 7. Additionally, this protection scheme is also expected to work for the error model described by \({H}_{I}^{{\prime}\prime}\) as given in (40), as the interplay between the Abelian Gauss laws and global symmetries of the LSH Hamiltonian remains the same.

Fig. 7: Numerical demonstration of symmetry protection for global symmetries under Scheme II.
figure 7

The global symmetry operators \({{\mathcal{P}}}\) and \({{\mathcal{Q}}}\) are plotted as a function of increasing Λ. The inset shows the quantum fluctuation of these two operators.

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Conclusions

Quantum simulation of non-Abelian gauge theories must preserve gauge invariance, but this task is undoubtedly difficult. Using conventional techniques, detecting the same for SU(3) gauge theories is yet to be demonstrated113. In contrast, the loop string hadron framework enables easier protocols for verification of gauge symmetry by digital circuit solution61 and for implementing symmetry protection while using analog/digital hardware110. These have been demonstrated for continuous and non-Abelian gauge group SU(2) in recent years. The current work demonstrates the feasibility and limitations of a simple scheme for protecting all the symmetries of SU(3) theories during dynamical simulations.

The challenges in simulating non-Abelian gauge theories are twofold. The first non-trivial task is preparing the initial state, which is a global state, yet gauge singlet at each and every lattice site. As the Hamiltonian is expected to commute with the generators of all gauge transformations, the dynamics ideally should remain confined in a subspace of the entire gauge theory Hilbert space. However, in terms of erroneous quantum devices, this is not expected to be the reality, and hence, the simulation scheme demands a symmetry protection protocol for any available platform. Following the warm-up exercises towards quantum simulating simpler gauge theories carried out over more than a decade now, the community is now focusing on the SU(3) gauge theories28,90,101,106 in 1 + 1 dimension, with the final aim of performing simulations in higher dimensions. The major restriction in achieving this goal is imposing the complicated SU(3) symmetries and protecting the same. The loop-string-hadron framework for SU(3) in108,109 provides a convenient framework for Hamiltonian simulation, where the basis states are inherently SU(3) singlets, leading to no need to impose any SU(3) symmetry at any point of calculating or simulating dynamics. The current protocol takes advantage of this fact and provides a viable scheme for symmetry protection, demonstrating the advantage of using the LSH framework.

1 + 1 dimensional gauge theories, although they offer the first test-bed for novel simulation protocols, are often a lot easier to analyze as the gauge field is not dynamic. However, the global symmetry still possesses a rich non-Abelian structure, implying the superselection rules to be governed by SU(3) symmetry and its representations. The LSH framework, being fully Abelianised, leads to simpler Abelian global symmetry sectors, which are far more intuitive. The current protocol utilizes the same for protecting all the symmetries in 1 + 1 dimensions. This is the major advantage of enabling one to simulate SU(3) gauge theory without imposing either any non-abelian or any local symmetries. Work is in progress in utilizing this scheme for developing analog quantum simulation protocols, which will be reported shortly. This scheme also remains useful for multiple directions of ongoing research efforts in developing tensor network algorithms for calculating dynamics114,115,116,117, as well as quantum simulation of scattering of wave-packets118,119, or studying thermalization dynamics of gauge theories103,120,121 using quantum simulators. Works are progressing towards generalising these schemes to deliver building blocks for the ultimate goal of quantum-simulating QCD, and this current work is one of those. An actual quantum device will interact with its environment and hence require a more sophisticated gauge symmetry protection protocol based upon the behaviour of the theory as an open quantum system. Future work is planned in this direction.

Methods

The Hamiltonian governing LSH dynamics

Following the original Kogut-Susskind construction of the Hamiltonian1, the Hamiltonian in the LSH framework also consists of electric, mass, fermionic hopping, and magnetic field terms. Considering the simplest case of one spatial dimension, the magnetic term is not present, but the Hamiltonian (scaled to make it dimensionless) is written as in (10)H = HE + HM + HI. With proper rescaling to make all of the couplings dimensionless, one can associate μ and x as dimensionless parameters with HM and HI terms; the coupling with HE is set to 1. The values of the parameters μ, x depend upon the lattice spacing, coupling, and mass of the fermions. The strong coupling limit of lattice gauge theory is always interesting as the theory is confined in this regime, defined by x → 0, keeping μ to be finite. The LSH basis described before is a strong coupling eigenbasis, i.e., in the LSH basis, the matrix representation of HE and Hm are both diagonal. Also note that both the terms HE and Hm are ultra-local in the strong coupling basis, so contributions from all the lattice sites are added to make the electric energy and mass of the entire system.

In the LSH framework, one can define on-site occupation number operators for the LSH basis both for the SU(2) and SU(3) case, and the onsite contribution of the electric and mass term is obtained as a function of occupation numbers of the LSH excitations at that site:

$${\hat{n}}_{l}\vert \{{n}_{l}\},\{{\nu }_{f}\}\rangle ={n}_{l}\vert \{{n}_{l}\},\{{\nu }_{f}\}\rangle$$
(22)
$${\hat{\nu }}_{f}\vert \{{n}_{l}\},\{{\nu }_{f}\}\rangle ={\nu }_{f}\vert \{{n}_{l}\},\{{\nu }_{f}\}\rangle$$
(23)

where {nl} denotes the set of bosonic quantum numbers, i.e., nl for SU(2) and nP&nQ for SU(3). Interestingly, the mass part of the Hamiltonian is effectively proportional to the total occupation number of fermions at each site. The electric part, on the other hand, counts the flux lines that originate or enter a site. A closer inspection of Fig. 1 reveals that non-zero occupation of the fermionic LSH modes often comes with flux lines attached to it, and hence, the fermionic LSH excitations also contribute to the on-site electric energy. The full expressions for the mass and the electric part can be written down as follows:

$${H}_{M}=\mu \sum\limits_{r}{(-1)}^{r}({\hat{\nu }}_{\underline{1}}(r)+{\hat{\nu }}_{0}(r)+{\hat{\nu }}_{1}(r))$$
(24)
$$\begin{array}{rcl}{H}_{E}&=&\sum\limits_{r}\left[\frac{1}{3}\left({\hat{P}}^{2}(1,r)+{\hat{Q}}^{2}(1,r)+\hat{P}(1,r)\hat{Q}(1,r)\right)\right.\\ &&\left.+\hat{P}(1,r)+\hat{Q}(1,r)\right]\end{array}$$
(25)

where \(\hat{P}\) and \(\hat{Q}\) are defined as:

$$\hat{P}(1,r)={\hat{n}}_{P}(r)+{\hat{\nu }}_{1}(r)(1-{\hat{\nu }}_{0}(r))$$
(26)
$$\hat{Q}(1,r)={\hat{n}}_{Q}(r)+{\hat{\nu }}_{0}(r)(1-{\hat{\nu }}_{\underline{1}}(r))$$
(27)

For a 1 + 1 dimensional theory, the only term responsible for dynamics is the hopping or ‘matter-gauge interaction’ term HI. The construction of the interaction Hamiltonian HI is nontrivial, compared to its SU(2) counterpart, as evident in the detailed construction given in108. This is due to the fact that matter content at a SU(3) site contains much more detailed inner structure than just being present as string end operators for the SU(2) case. This can be qualitatively appreciated from Fig. 1. To finalize a representation of HI in LSH basis, we consider the following set of normalized ladder operators, which, acting on an on-site LSH state, changes its excitations by  ± 1 unit.

$${\hat{\Gamma }}_{L}^{{\dagger} }\vert \{{n}_{l}\},\{{\nu }_{f}\}\rangle \, = \, \vert \{{n}_{l}| {n}_{L}+1\},\{{\nu }_{f}\}\rangle$$
(28)
$${\hat{\chi }}_{F}^{{\dagger} }\vert \{{n}_{l}\},\{{\nu }_{f}\}\rangle \, = \, (1-{\delta }_{1,{\nu }_{F}})\vert \{{n}_{l}\},\{{\nu }_{f}| {\nu }_{F}+1\}\rangle$$
(29)
$${\hat{\Gamma }}_{L}\vert \{{n}_{l}\},\{{\nu }_{f}\}\rangle \, = \, \vert \{{n}_{l}| {n}_{L}-1\},\{{\nu }_{f}\}\rangle$$
(30)
$${\hat{\chi }}_{F}\vert \{{n}_{l}\},\{{\nu }_{f}\}\rangle \, = \, (1-{\delta }_{0,{\nu }_{F}})\vert \{{n}_{l}\},\{{\nu }_{f}| {\nu }_{F}-1\}\rangle$$
(31)

The notation {nl∣nL ± 1} and {νf∣νF ± 1} within the ket denotes that only the specific bosonic (L) or fermionic (F) LSH quantum number has changed, while keeping all other in the set fixed.

The following definition captures the inner structure of gauge-invariant dynamics for this theory:

$${H}_{I} = \, x\sum\limits_{r}\left[\sum\limits_{F=\underline{1},0,1}{\hat{\chi }}_{F}^{{\dagger} }(r)\left[{\hat{O}}_{1}^{(F)}(\{{\hat{n}}_{l}(r),{\hat{\nu }}_{f}(r)\})\times {\hat{O}}_{2}^{(F)}(\{{\hat{n}}_{l}(r+1), \\ {\hat{\nu }}_{f}(r+1)\})\right]\right.\\ \left.{\hat{\chi }}_{F}(r+1)+{{\rm{h.c.}}}\right]$$
(32)

where, the operators \({\hat{O}}_{1/2}^{F}\) are defined as

$${\hat{O}}_{1}^{(\underline{1})}=\left[{\left({\hat{\Gamma }}_{Q}\right)}^{1-{\hat{\nu }}_{0}}\sqrt{1+{\hat{\nu }}_{0}/\left({\hat{n}}_{Q}+1\right)}\sqrt{1+{\hat{\nu }}_{1}/\left({\hat{n}}_{P}+{\hat{n}}_{Q}+2\right)}\right]$$
(33)
$${\hat{O}}_{2}^{(\underline{1})}=\left[\sqrt{1-{\hat{\nu }}_{0}/\left({\hat{n}}_{Q}+2\right)}\sqrt{1-{\hat{\nu }}_{1}/\left({\hat{n}}_{P}+{\hat{n}}_{Q}+3\right)}{\left({\hat{\Gamma }}_{Q}\right)}^{{\hat{\nu }}_{0}}\right]$$
(34)
$${\hat{O}}_{1}^{(0)}=\left.\left[{\left({\hat{\Gamma }}_{P}\right)}^{1-{\hat{\nu }}_{1}}{\left({\hat{\Gamma }}_{Q}^{{\dagger} }\right)}^{{\hat{\nu }}_{1}}\sqrt{1+{\hat{\nu }}_{1}/\left({\hat{n}}_{P}+1\right.}\right)\sqrt{1-{\hat{\nu }}_{\underline{1}}/\left({\hat{n}}_{Q}+2\right)}\right]$$
(35)
$${\hat{O}}_{2}^{(0)}=\left.\left[\sqrt{1-{\hat{\nu }}_{1}/\left({\hat{n}}_{P}+2\right.}\right)\sqrt{1+{\hat{\nu }}_{\underline{1}}/\left({\hat{n}}_{Q}+1\right)}{\hat{\chi }}_{0}{\left({\hat{\Gamma }}_{P}\right)}^{{\hat{\nu }}_{1}}{\left({\hat{\Gamma }}_{Q}^{{\dagger} }\right)}^{1-{\hat{\nu }}_{1}}\right]$$
(36)
$${\hat{O}}_{1}^{(1)}=\left[{\left({\hat{\Gamma }}_{P}^{{\dagger} }\right)}^{{\hat{\nu }}_{0}}\sqrt{1-{\hat{\nu }}_{0}/\left({\hat{n}}_{P}+2\right)}\sqrt{1-{\hat{\nu }}_{\underline{1}}/\left({\hat{n}}_{P}+{\hat{n}}_{Q}+3\right)}\right]$$
(37)
$${\hat{O}}_{2}^{(1)}=\left[\sqrt{1+{\hat{\nu }}_{0}/\left({\hat{n}}_{P}+1\right)}\sqrt{1+{\hat{\nu }}_{\underline{1}}/\left({\hat{n}}_{P}+{\hat{n}}_{Q}+2\right)}{\left({\hat{\Gamma }}_{P}^{{\dagger} }\right)}^{1-{\hat{\nu }}_{0}}\right]$$
(38)

Error models

As outlined in the Results and Discussions section, we consider two possible routes for protecting erroneous dynamics, dubbed as Scheme I and Scheme II, respectively, as given in (13), (17), (18). The explicit mathematical form for modelling these two errors is given by Hamiltonian interaction \({H}_{I}^{{\prime} }\) and \({H}_{I}^{{\prime}\prime}\) given by:

  • Error Model I: \({H}_{I}^{{\prime} }=\)

    $$\begin{array}{rcl}&&\sum\limits_{r}\sum\limits_{F,{F}^{{\prime} }=\underline{1},0,1}\left[{x}^{{\prime} }(1-{\delta }_{F,{F}^{{\prime} }})\left[{\hat{O}}_{1}^{(F)}(\{{\hat{n}}_{l}(r),{\hat{\nu }}_{f}(r)\})\right]{\hat{\chi }}_{F}^{{\dagger} }(r)\right.\\ &&\left.\left[{\hat{O}}_{2}^{({F}^{{\prime} })}(\{{\hat{n}}_{l}(r+1),{\hat{\nu }}_{f}(r+1)\})\right]{\hat{\chi }}_{{F}^{{\prime} }}(r+1)\right]\\ &&+{{\rm{h.c.}}}\end{array}$$
    (39)

  • Error Model II: \({H}_{I}^{{\prime}\prime}=\)

    $$\begin{array}{rcl}&&\sum\limits_{r}\sum\limits_{F,{F}^{{\prime} }=\underline{1},0,1}\left[{x}^{{\prime}\prime}(1-{\delta }_{F,{F}^{{\prime} }})\left[{\hat{O}}_{1}^{(F)}(\{{\hat{n}}_{l}(r),{\hat{\nu }}_{f}(r)\})\right]{\hat{\chi }}_{F}^{{\dagger} }(r)\right.\\ &&\left.\left[{\hat{O}}_{2}^{({F}^{{\prime} })}(\{{\hat{n}}_{l}(r),{\hat{\nu }}_{f}(r)\})\right]{\hat{\chi }}_{{F}^{{\prime} }}(r)\right]\\ &&+{{\rm{h.c.}}}\end{array}$$
    (40)

The inclusion of any of these two operators breaks the Abelian Gauss Law (AGL) as shown in Fig. 2(b). Note that only a set of sub-terms of (39), i.e., only for \(F={F}^{{\prime} }\) cases in the sum, gives the original interaction Hamiltonian (32). All our numerical calculations presented in Fig. 3 are performed for a 4-staggered site lattice with \(x={x}^{{\prime} }=1\) to ensure maximal mixing of the three fermionic modes and the mass parameter μ = 1.

Protection schemes

The error protection scheme under the scheme I, i.e., the global scheme, requires adding protection terms presented in (15) and (16). These two term individually protect the global quantum numbers \({{\mathcal{P}}}\) and \({{\mathcal{Q}}}\), numerical results demonstrate that protecting these two global symmetries guarantee protection of each and every local symmetries of the theory. The testbed for the numerical demonstrations is a 4-staggered site lattice with \(x=1,{x}^{{\prime} }=0.05\), μ = 1, and the strong coupling vacuum as our initial state defined in equation (14). The scheme is explained in detail in the subsection on Symmetry protection scheme I: via global symmetry protection within Results and Discussions section. The numerical results are presented in 4. For the Error Model II, the numerical calculations presented in Fig. 5 are performed with \(x=1,{x}^{{\prime} }=0.05,{x}^{{\prime}\prime}=0.005\).

The scheme II of symmetry protection requires protecting each and every local constraint, which are still Abelian, even for the SU(3) gauge theory in any dimension. The corresponding protection Hamiltonian is given in ((19), (20)) along with numerical evidence. Numerical results obtained in Figs. 6 and 7 are based on calculations using the parameters \(x=1,{x}^{{\prime} }=1,\mu =1\).