Guide: the Ising chain on a shared hardware model¶

qsimod.usecases.ising carries an antiferromagnetic Ising chain, ising_magnet, to bose_hubbard, the tilted Bose-Hubbard chain of the Schwinger pipeline, with its namespace, parameters and admissible set of knobs. Two transformations arrive at bose_hubbard, so one knob setting can be judged by both derivations. This page assumes the machinery the Schwinger guide introduces.

The physics follows Simon et al. [1], building on Sachdev, Sengupta and Girvin [2].

The model graph¶

The greyed artifact and arrow are the transformation of the gauge theory into the shared hardware model.

flowchart TB
    subgraph LV1["Application model &nbsp;·&nbsp; <code>models.application</code>"]
        direction LR
        ising_magnet["<b>ising_magnet &nbsp; H<sub>Ising</sub></b><br/>antiferromagnetic<br/>Ising chain<br/><i>Θ = {J<sub>z</sub>, h<sub>z</sub>, h<sub>x</sub>}</i>"]
    end
    subgraph LV2["Intermediate representations &nbsp;·&nbsp; <code>models.intermediate</code>"]
        ising_chain["<b>ising_chain &nbsp; H<sub>Ising</sub></b><br/>Ising chain,<br/>by three energies<br/><i>Θ = {J<sub>z</sub>, Γ, B}</i>"]
        effective_bosonic["<b>effective_bosonic &nbsp; H<sub>IR3</sub></b><br/>boson encoding<br/><i>Θ<sub>IR3</sub> = {m, κ}</i>"]
    end
    subgraph LV3["Hardware model &nbsp;·&nbsp; <code>models.hardware</code>"]
        direction LR
        bose_hubbard["<b>bose_hubbard &nbsp; H<sub>sim</sub></b><br/>tilted, staggered<br/>Bose–Hubbard chain<br/><i>Θ<sub>sim</sub> = {J, U, δ, Δ}</i>"]
    end

    ising_magnet ---> |"<b>field resolution</b><br/><i>exact</i>"| ising_chain
    ising_chain -. "<b>resonant dipole reduction,<br/>solved for the knob settings</b><br/><i>approximate, regime conditions</i>" .-> bose_hubbard

    effective_bosonic -. "<b>degenerate perturbation theory,<br/>solved for the knob settings</b><br/><i>approximate, regime conditions</i>" .-> bose_hubbard

    classDef ham fill:#e0f2ee,stroke:#009371,color:#003e2f
    classDef ctx fill:#f3f3f3,stroke:#919191,color:#545454,stroke-dasharray:4 3
    class ising_magnet,ising_chain,bose_hubbard ham
    class effective_bosonic ctx
    style LV1 fill:#f2f7fa,stroke:#b1d0e5,color:#103c5a
    style LV2 fill:#f2f7fa,stroke:#b1d0e5,color:#103c5a
    style LV3 fill:#f2f7fa,stroke:#b1d0e5,color:#103c5a

from qsimod.usecases.ising import DEVICE_TARGET, shared_graph

graph = shared_graph(3)
for edge in graph.incoming(DEVICE_TARGET):
    print(edge)

The transformations at a glance¶

ising_magnet: the Ising magnet¶

Use case: magnet  ·  library: ising_magnet

One energy \(J_z\) and two dimensionless fields \(h_z\) and \(h_x\); the mapping to the hardware model holds near the multicritical point \((h_z, h_x) = (1, 0)\). The structural type declares no symmetry, since the transverse field does not conserve the total magnetisation. An XXZ chain has the same geometry, algebra and lattice, so the conservation law is what distinguishes the two structural patterns, and the dipole reduction rejects the XXZ chain on this ground:

from qsimod.structure import StructureTypeError
from qsimod.usecases.heisenberg import spin_chain
from qsimod.usecases.ising import dipoles, magnet

print(magnet(3).structure.symmetries or "none declared")
try:
    dipoles().apply(spin_chain(4))  # an XXZ chain: same shape, wrong physics
except StructureTypeError as error:
    print(error)

ising_magnet -> ising_chain: the field resolution¶

Exactness: EXACT and invertible; the same kind of reparametrisation as the anisotropy resolution of the magnet, built by the same helper.

ising_chain: the Ising chain by three energies¶

Use case: ising_chain  ·  library: ising_spin_chain

ising_chain -> bose_hubbard: the resonant dipole reduction¶

A Mott insulator at unit filling is tilted so that the tilt per site cancels the interaction: an atom may move onto its neighbouring site unless that neighbour has moved first. A moved atom is a dipole on a bond, the mutual exclusion of adjacent dipoles is the Ising coupling, and the tunnelling that creates a dipole is the transverse field. The spins live on the bonds, so \(2N-1\) lattice sites carry \(2N-2\) spins, which the transformation declares as its target_sites:

from qsimod.usecases.ising import dipoles, fields

print(dipoles().target_sites(4), fields().target_sites(4))  # 3 4

Exactness: APPROXIMATE. The transverse field is of first order in the tunnelling, since the process is resonant; the leading error is of second order in \(J/U\) and stems from the off-resonant configurations, three atoms on a site and dipoles on adjacent bonds.

Validity conditions. One domain and four regime conditions:

The superlattice \(\delta\) acts as a staggered longitudinal field on the spins and must be small on the scale of the transverse field.

from qsimod.usecases.ising import ParameterNames as P
from qsimod.usecases.ising import coupling_map, dipoles, transverse_map, validity

step = dipoles()
print(step.forward_relation_kind)  # CLOSED_FORM
knobs = {P.TUNNELLING: 0.0177, P.INTERACTION: 1.0, P.SUPERLATTICE: 0.0025, P.TILT: 1.033}
print(
    round(coupling_map().evaluate_real(knobs), 4), round(transverse_map().evaluate_real(knobs), 4)
)
print(
    validity().report(
        {**knobs, P.COUPLING_ISING_CHAIN: 1.0, P.TRANSVERSE: 0.05, P.LONGITUDINAL: 0.967}
    )
)

Convention and boundary term. The constraint that forbids two dipoles on adjacent bonds is represented as a finite penalty, which the source gives as "of order \(U\)"; the package takes it equal to \(U\), and the quoted \((h_z, h_x)\) values do not depend on this choice. Rewriting the penalty as a coupling plus a uniform field is exact in the bulk only: the two end spins see a field that is short by half a coupling each, the same identity as in the superexchange of the magnet. dipole_boundary_field gives its strength and end_magnetisation_term builds the term; manifold_check in examples/simon2011.py shows the spectrum agreeing with it and deviating without it.

The hardware model, read twice¶

bose_hubbard is unchanged: the same builder, knobs and Hamiltonian. The two theories need different corners of its admissible set and pose their requests against different boxes:

from qsimod.parameters import ConstraintOrigin
from qsimod.usecases import ising, schwinger

box = schwinger.device_limits()
for constraint in box.constraints:
    print(constraint)
print(len(box.constraints_from(ConstraintOrigin.APPARATUS)), "of them are the apparatus")
print(len(ising.device_limits().constraints), "left once a second theory poses its request")

Every coupled constraint declares an origin. The two coupled constraints of the Schwinger box have the origin DERIVATION: they are requirements of the superlattice derivation, not of the apparatus, and at the dipole resonance \(\Delta \approx U\) one of them cannot hold. The Ising use case therefore poses its request against apparatus_only and leaves the validity conditions to each derivation.

What a shared hardware model lets the framework decide¶

One knob setting is judged by both derivations:

from qsimod.usecases import ising, schwinger

knobs = {
    "bose_hubbard.J": 0.0177,
    "bose_hubbard.U": 1.0,
    "bose_hubbard.delta": 0.0025,
    "bose_hubbard.Delta": 1.033,
}
coupling = ising.coupling_map().evaluate_real(knobs)
ising_report = ising.validity().report(
    {
        **knobs,
        "ising_chain.Jz": coupling,
        "ising_chain.Gamma": ising.transverse_map().evaluate_real(knobs),
        "ising_chain.B": ising.longitudinal_map().evaluate_real(
            {**knobs, "ising_chain.Jz": coupling}
        ),
    }
)
gauge_report = schwinger.validity().report(
    {
        **knobs,
        "effective_bosonic.m": schwinger.mass_map().evaluate_real(knobs),
        "effective_bosonic.kappa": schwinger.coupling_map().evaluate_real(knobs),
    }
)
print("Ising:", ising_report.is_valid, round(ising_report.weakest_margin, 2))
print("gauge:", gauge_report.is_valid, round(gauge_report.weakest_margin, 2))

The superlattice derivation requires \(\Delta \ll \delta\), the dipole derivation \(\delta \ll \Gamma\). The two declared regimes are disjoint: no setting of this lattice realises both theories, and the margins report this before any operator is built.

References¶

1. J. Simon, W. S. Bakr, R. Ma, M. E. Tai, P. M. Preiss and M. Greiner, Quantum simulation of antiferromagnetic spin chains in an optical lattice, Nature 472, 307-312 (2011). doi:10.1038/nature09994
2. S. Sachdev, K. Sengupta and S. M. Girvin, Mott insulators in strong electric fields, Phys. Rev. B 66, 075128 (2002). doi:10.1103/PhysRevB.66.075128