---
title: "Protecting the Gauge Subspace Interactive Lab | Sandbox Physics"
description: "Does a large Gauss penalty keep an imperfect simulator physical?"
canonical_url: "https://sandboxphysics.com/labs/gauge-subspace/"
language: "en"
page_type: "lab"
alternate_languages:
  zh-CN: "https://sandboxphysics.com/zh/labs/gauge-subspace/"
---

FR04-C · An energy cost can suppress a wrong hop

# Protecting the Gauge Subspace

**Does a large Gauss penalty keep an imperfect simulator physical?** A large energy penalty makes gauge-violating transitions costly and can reduce leakage caused by an imperfect hopping term. Only selecting the physical subspace enforces Gauss law exactly; finite penalties retain small violations and can alter the motion.

Compare exact selection of physical states with an imperfect full-space evolution. Turn up the energy penalty for violating Gauss law and see how much unwanted motion survives.

> Interactive simulation: open this page in a web browser to change the parameters and run the model.

Model assumptions

The comparison uses a four-site open chain with finite electric cutoff. The physical calculation projects onto the Gauss sector. The full-space calculations add an explicit matter-hop error that leaves its electric link unchanged; the protected case also adds the sum of squared Gauss generators as an energy penalty. A finite penalty suppresses leakage but does not impose an exact constraint. The plotted records are model simulations.

Deterministic model calculation

## Probability outside the physical sector

Numerical diagnostics and full record

$\lVert\psi\rVert^2=1$

$\Delta\langle E_1\rangle=\langle E_1\rangle_\lambda-\langle E_1\rangle_{\rm phys}$ compares the central link in the penalty branch with exact physical-subspace evolution at the same time. A small leakage probability can coexist with a visible field shift.

The record saves the frozen Hamiltonian parameters, initial state, sampled numerical trajectory and conservation diagnostics. Curves join exact finite-matrix samples; all data are simulated.

[Continue the course: Gauss law, strings and gauge protection](https://sandboxphysics.com/stories/lattice-gauge/)

Physics tutorial

## Protecting the Gauge Subspace

**Background** An ideal gauge-invariant hop moves a fermion and changes the crossed electric link. An imperfect simulator can also move the fermion while leaving that link untouched.

**Why it matters** A physical-state basis removes forbidden configurations entirely. An energy penalty offers another way to discourage them when hardware cannot remove them.

Start with the essentials

- **Focus question**: Does a large Gauss penalty keep an imperfect simulator physical?
- **One-sentence intuition**: A finite energy penalty can suppress a gauge-breaking error, but exact selection of the physical sector is a different operation.

### Core mathematical model

#### A controlled gauge-breaking error

$$
H_{\rm err}=-\epsilon\sum_n(c_n^\dagger c_{n+1}+\mathrm{h.c.})
$$

This term deliberately omits the link operator. It preserves total fermion number but changes two neighboring Gauss generators.

#### The energy penalty

$$
H_{\lambda}=H+H_{\rm err}+\lambda\sum_nG_n^2
$$

Every violating configuration gains a positive energy cost. The strength of suppression depends on the error and on the other energy scales; finite penalties need not improve every observable monotonically.

#### Probability outside the physical sector

$$
\mathcal L(t)=1-\langle P_{\rm phys}\rangle_t
$$

The data plot measures probability in any Gauss-violating basis state. A vanishing mean Gauss generator is insufficient because positive and negative violations can cancel.

### Common difficulties

#### Suppression is not projection

**Typical misconception** A sufficiently large finite penalty makes the computation identical to deleting every forbidden state.

**Better mental model** Forbidden states remain in the full basis and can contribute through virtual transitions. Compare leakage and physical observables as the penalty grows.

### Run the experiment

1. 01

   #### Let the error act

   Choose Errors spread and watch the three chains at the same observation time.

   **What to observe:** The physical-basis calculation stays in its selected sector. The other two agree when the penalty is zero, and the wrong-hop term spreads probability into forbidden states.

2. 02

   #### Turn up the penalty

   Move Gauss penalty continuously, then compare Weak protection and Strong protection.

   **What to observe:** The gold trace usually drops as the energy separation grows. It stays finite at the shown settings, and its physical observables can differ from the exact physical-basis trace.

3. 03

   #### Remove the cause

   Choose No hopping error. Vary the penalty again.

   **What to observe:** Without the explicit gauge-breaking term, all three evolutions preserve the initial Gauss sector. A gauge-invariant Hamiltonian does not need a penalty to enforce its own symmetry.

Physical conclusion

Compare exact projection, unprotected errors and a finite penalty at the same time and parameters.

## Related experiments and articles

Lattice gauge fields and string breaking · Reading path

### Gauss constraints, screening and numerical control

Articles on this path that use this experiment, followed by the other experiments they suggest.

1. [Article **Projecting out violations or penalizing them**](https://sandboxphysics.com/stories/lattice-gauge/physical-subspace/)

Quantum & fields · Starter path

## Start from a single measurement

Predict an outcome, then count what the measurements actually give. The path starts with one qubit and moves on to wave packets, bound states and two-photon interference.

[Start here → **Quantum State & Born Measurement** Predict a probability, turn the analyzer, then compare your prediction with the measured frequency over many shots. Observe · 15 min](https://sandboxphysics.com/labs/qubit-state/)

1. [1 Quantum State & Born Measurement](https://sandboxphysics.com/labs/qubit-state/)
2. [2 Stern–Gerlach Sequence](https://sandboxphysics.com/labs/stern-gerlach-sequence/)
3. [3 Wave Packet & Uncertainty](https://sandboxphysics.com/labs/wave-packet-uncertainty/)
4. [4 Quantum Harmonic Oscillator](https://sandboxphysics.com/labs/quantum-harmonic-oscillator/)
5. [5 Which Path & Quantum Eraser](https://sandboxphysics.com/labs/quantum-eraser/)
6. [6 Hong–Ou–Mandel Two-Photon Interference](https://sandboxphysics.com/labs/hong-ou-mandel/)

[View the full path →](https://sandboxphysics.com/topics/quantum/#starter-path)