The SU(2) lattice model and the LSH basis

Part 3 of the series From quarks to quantum advantage

Quarks and color flux form one locally constrained system in the previous part. This immediately raises the computational problem: how do we preserve the physical states without simulating a huge collection of redundant and unphysical field configurations?

The paper uses two steps for this. First, the continuous theory is written on a spatial grid using the Kogut-Susskind-Hamiltonian. Then that Hamiltonian is rewritten in the Loop-String-Hadron basis. The LSH basis solves the three non-Abelian Gauss laws locally and describes the remaining physical information with a through flux and two binary string ends.

The three parts of the Hamiltonian

For SU(2) gauge fields coupled to staggered fermions, the paper writes the Hamiltonian as

\[
H=\frac{g^2}{2a}H_E+mH_M+\frac{1}{2a}H_I.
\]

Here \(a\) is the lattice spacing, \(g\) is the gauge coupling and \(m\) is the fermion mass. The three operators have different roles:

  • \(H_E\) measures the electric energy on the links.
  • \(H_M\) gives the staggered matter a mass energy.
  • \(H_I\) moves matter between sites while adjusting the gauge field.

The interaction term is essential. A quark cannot simply hop to the next site while the flux remains unchanged. Matter motion and flux change together must continue to respect the local Gauss law.

It is practical to make all quantities dimensionless. The paper defines

\[
W=\frac{2a}{g^2}H=H_E+\mu H_M+xH_I,
\]

with \(x=1/(g^2a^2)\) and \(\mu=2(m/g)\sqrt{x}\). For a fixed mass ratio, large \(x\) corresponds to small lattice spacing and weak coupling. The continuum direction therefore requires not only \(x\to\infty\), but also a sufficiently large number of sites \(N\to\infty\), so that the physical volume does not disappear.

The main instance uses \(N=60\), \(x=100\), and \(m/g=1\), with a Trotter step of 0.0015. That is not a continuum limit. It is a large, weakly coupled lattice point at which the classical references are still partially verifiable.

The original local variables

In the Kogut-Susskind formulation, each link contains a SU(2) matrix \(U(r)\). Electric field operators \(E_L^a(r)\) and \(E_R^a(r)\) belong to both ends. The sites contain two-color staggered fermion fields. The generator of a local gauge transformation is

\[
G^a(r)=E_L^a(r)+E_R^a(r-1)
+\psi^\dagger(r)\frac{\sigma^a}{2}\psi(r).
\]

All physical states must be canceled by each \(G^a(r)\). That condition is exact, but not useful as hardware encoding. A registry that stores all matter and link variables independently contains many states that do not respect the constraint. Moreover, the three SU(2) Gauss laws cannot be trivially reduced to single classical bits.

Solve locally instead of filtering afterwards

The LSH formulation starts with prepotentials: harmonic oscillator variables at the left ends. By combining them locally with the fermion fields to form SU(2) singlets, gauge-invariant operators and basic states are immediately created.

A local LSH state is written as

\[
|n_l,n_i,n_o\rangle_r.
\]

The three numbers mean:

  • \(n_l\ge 0\): the number of continuous flux lines or walking flux on the site;
  • \(n_i\in\{0,1\}\): presence of an incoming string end;
  • \(n_o\in\{0,1\}\): presence of an outgoing string end.

When both string ends are present, the local combination can be interpreted as a hadronic singlet. Because \(n_i\) and \(n_o\) are fermionic, they are binary. \(n_l\) is bosonic and in principle has no upper limit; classical calculations must choose a cutoff for this.

flowchart LR
    A["matter and SU(2) link fields"] --> B["solve local Gauss laws"]
    B --> C["LSH state |n_l,n_i,n_o>"]
    C --> D["map n_i and n_o to two qubits"]
    C --> E["reconstruct n_l from the link constraint"]

The LSH basis solves the non-Abelian Gauss laws at each site by construction. A simpler link condition remains: the flux leaving a site must match the flux entering the next site. The paper calls this the Abelian Gauss Law:

\[
\left[n_l+n_o(1-n_i)\right]_r
=
\left[n_l+n_i(1-n_o)\right]_{r+1}.
\]

Abelian does not mean here that the original theory has suddenly become an Abelian gauge theory. The local SU(2) constraints are already basically solved; what remains is a numerical matching of flux over the link.

The full LSH Hamiltonian

The same physics is retained in this basis. The full LSH Hamiltonian

\[
W^{(\mathrm{LSH})}
=H_E^{(\mathrm{LSH})}+\mu H_M^{(\mathrm{LSH})}
+xH_I^{(\mathrm{LSH})}
\]

is exactly equivalent to the Kogut-Susskind Hamiltonian within the same bosonic cutoff. The electric and mass terms are diagonal in the local LSH numbers. The interaction term contains ladder operators that change string ends and flux together. The square root factors in that term ensure that the amplitude depends on the flux present.

This is the Hamiltonian that approximates the tensor network baseline of the paper. A local tensor index contains the full state \(|n_l,n_i,n_o\rangle\), with a cutoff \(j_{\max}=5/2\). As a result, TN simulates more local physical information than the final two-qubit circuit.

Cutoff and elimination are different approaches

The bosonic flux \(n_l\) is unbounded in the exact LSH basis. A classical TN calculation keeps track of that variable explicitly, but truncates the local basis at a chosen \(j_{\max}\). The error question then is whether the time evolution develops a relevant amplitude at flux levels above that cutoff. Convergence can be tested by increasing \(j_{\max}\), as long as computation time allows.

The quantum circuit does something different. It does not store a finite flux register that can be made larger later, but eliminates \(n_l\) with the link constraint and the large-boundary-flux approximation. The error therefore does not become an ordinary Hilbert space cutoff, but a model approximation in the amplitudes and electric phases.

This difference explains why 120 qubits does not directly mean that TN and QPU use the same local Hilbert space. TN stores more LSH information per site; the QPU uses fewer degrees of freedom thanks to a physically motivated approach. The comparison between the two methods is therefore also a test of that reduction.

The strong-coupling vacuum state

The simple product state used as a reference is called the strong-coupling vacuum, or SCV. In the LSH variables it is

n_l(r) = 0                         for all r
n_i(r) = n_o(r) = 0               for even r
n_i(r) = n_o(r) = 1               for odd r

With the staggered definition of \(n_f\), this state has no local particle excitations. It has global baryon charge \(B=0\), net flux \(q=0\) and \(Q=B+N=N=60\).

The name can be misleading. The SCV is the simple strong coupling product state, but the paper uses it as the initial state for dynamics at \(x=100\), i.e. in a weakly coupled regime. Strong-coupling vacuum here describes the chosen basic reference, not the coupling under which the circuit subsequently evolves.

The meson is placed on top of the same background by naming the two central sites \(n_f=1\). Both initial states lie in the same global \((B,q)=(0,0)\) sector. Therefore, their difference can isolate a local meson signal without comparing two different global charge sectors.

Why n_l does not need to be on a qubit register

An exact local encoding should also store the unbounded flux variable \(n_l\). The paper avoids that with an approach that specifically uses one space dimension, open borders and the weak coupling regime choice.

Allow a large incoming edge flux \(l_i\gg1\) to enter the lattice. By the Abelian Gauss Law, the flux at site \(r\) is then determined by the edge flux and all previous string ends:

\[
n_l(r)=l_i-n_i(r)[1-n_o(r)]
+\sum_{r'<r}[n_o(r’)-n_i(r’)].
\]

In the large flux approximation used, the non-local sum in the leading local interaction factors is neglected and flux-dependent ratios such as \(n_l/(n_l+1)\) are approximated by one. The dominant interaction term then becomes local to the binary string ends. Some of the large background flux contributes only a global phase and does not need to be stored as a dynamic register.

This specifies each physical configuration for the circuit by all \(n_i(r)\) and \(n_o(r)\). Because each number is zero or one, exactly two qubits are needed per site.

What is gained and what is given up?

The benefit is substantial. The local non-Abelian Gauss laws have already been solved, the unbounded flux registers disappear from the hardware circuit and the interactions can be built with local two-qubit blocks. For 60 sites, the encoding remains at 120 qubits.

The price is that the quantum circuit does not perform the full LSH Hamiltonian. The large edge flux approximation, simplified square root factors and treatment of the electric background define an approximate weak coupling model. Trotter error will be added in the next step.

That is precisely why the tensor network calculation of the full LSH Hamiltonian is necessary. Agreement between TN and the ideal circuit is the empirical evidence that the approximation does not destroy the relevant hadron dynamics in the early time window examined.

What has been demonstrated here?

The LSH formulation provides a local, gauge-invariant basis that can describe exactly the same SU(2) physics as the Kogut-Susskind formulation. For the hardware implementation, an additional weak-link approach reduces the dynamic information to two binary string ends per site.

What has not yet been proven?

The two-qubit encoding is not exactly equal to the full LSH Hamiltonian for random coupling, flux and time. Its area of ​​validity must be determined by small, precise calculations and by comparison with the entire TN route.

In part 4 we take the two binary string ends as a starting point. We show how Jordan-Wigner strings, Trotterization, interaction gates, phase gates, and SWAP layers combine to form the final 120-qubit QASM.

Sources

  1. Fran Ilčić et al., paper v3, especially Methods, equations 4-28.
  2. I. Raychowdhury and J. R. Stryker, LSH formulation for SU(2).
  3. Z. Davoudi, A. F. Shaw and J. R. Stryker, General quantum algorithms for Hamiltonian simulation with applications to a non-Abelian lattice gauge theory.
  4. R. Dasgupta and I. Raychowdhury, Weak coupling approach to string and hadron dynamics.
  5. Hadron repository and reproducibility notes.