WCCT→Lattice Gauge Technical Note (v0.1)
Author: QuantumCore Technologies (Dustin Thornton, PI)
Date: 2025‑08‑28
Scope: Practical mapping from WaveCore Continuum Theory (WCCT) to lattice gauge models (U(1), Z₂), with device‑ready circuit sketches, parameter dictionaries, and analysis scripts suitable for superconducting or trapped‑ion stacks.
1) Executive Summary
This note provides a concrete, falsifiable bridge between WCCT’s scalar‑wave framework and standard lattice gauge Hamiltonians used in near‑term quantum simulations. We present two encodings:
Compact U(1) with staggered matter (continuous, truncated to finite dimension per link)
Z₂ gauge theory with dynamical strings (binary links; hardware‑efficient)
We supply: parameter dictionaries, state‑prep routines, Trotterized circuit sketches, measurable observables, scaling predictions unique to WCCT, and minimal Python analysis stubs.
2) WCCT → Lattice Discretization
Let sites be i = 0…N−1 and links l ≡ (i,i+1). WCCT variables:
Site scalar potential: φ_i
Link gradient: Δφ_l ≡ φ_{i+1} − φ_i
Canonical momentum on link: Π_l (conjugate to Δφ_l)
Flux on link: P_l = −Δφ_l
Source density on site: ρ_i
WCCT energy (discrete effective form):
H_WCCT = Σ_l [ μ/2 · Π_l² + κ/2 · (Δφ_l)² ] + Σ_i [ (m_φ²/2) φ_i² + λ φ_i⁴ ] − J Σ_l cos(Δφ_l − θ_l)
Constraint (local conservation): (∇·P)_i − ρ_i = 0, with P_l = −Δφ_l.
3) Target Gauge Models
3.1 Compact U(1), 1+1D with staggered matter
H_U(1) = m Σ_i (−1)^i n_i + w Σ_i (ψ_i† U_{i,i+1} ψ_{i+1} + h.c.) + (g²/2) Σ_l E_l²
Gauss’s law: (∇·E)_i − n_i = 0
Link: U_l = e^{i A_l}, Electric field: E_l
Dictionary (minimal):
A_l ↔ α · Δφ_l
U_l = e^{iA_l} ↔ e^{i α Δφ_l}
E_l ↔ β · Π_l
n_i ↔ ρ_i
g² ↔ μ/β², string tension σ ↔ κ α², m ↔ m_φ, nonlinearity ↔ λ
3.2 Z₂ Gauge Theory with dynamical matter
H_Z2 = m Σ_i Z_i − h Σ_{⟨i,i+1⟩} ( X_{i,i+1} Z_i Z_{i+1} ) − Γ Σ_l X_l
Gauss’s law: G_i = X_{i−1,i} X_{i,i+1} Z_i = +1
Dictionary (minimal):
Link string bit (present/absent) ↔ X_l eigenvalue
Local scalar phase parity ↔ Z_i
String motion/creation ↔ X_l flips (parity jump in Δφ)
Γ ↔ string fluctuation rate; h ↔ J; m ↔ m_φ
4) WCCT‑Specific, Falsifiable Predictions
P1. Nonlinear sharpening: At fixed σ = κ α², increasing λ narrows flux‑tube width; break length L_break grows sublinearly with λ.
P2. Observer‑phase renormalization: Adding observer modulation term α_obs cos(kx) renormalizes J → J_eff = J [1 − c α_obs²], lengthening pre‑break plateaus in Loschmidt echo.
P3. Toroidal harmonic sidebands: Weak drives at {3,6,9,…} harmonic ratios induce discrete sidebands in string oscillations absent in linear gauge models.
5) State Preparation
Static charge pair (distance L):
Prepare gauge‑invariant vacuum |Ω⟩.
Create charges ±q at sites (i₀, i₀+L) via local matter flips respecting Gauss’s law.
Apply a Wilson line W = ∏_{l∈path} U_l to initialize a straight flux tube.
Z₂ hardware‑efficient recipe:
Initialize all matter Z_i = +1, links X_l = +1.
Flip matter at endpoints to Z = −1, then set X_l = −1 along the chosen path (straight string).
6) Time Evolution (Trotter Sketches)
6.1 U(1) (truncated dimension d per link)
One Trotter step U(δt):
Field energy: exp[−i (g²/2) Σ_l E_l² δt] → diagonal in E basis (phase rotations per link).
Mass term: exp[−i m Σ_i (−1)^i n_i δt] → Z‑rotations on matter qubits.
Hopping: exp[−i w Σ_i (ψ_i† U_i ψ_{i+1} + h.c.) δt] → three‑body gate compiled as two‑qubit + controlled‑phase on link register.
6.2 Z₂ (qubit‑native)
One Trotter step U_Z2(δt):
Mass: ∏_i R_Z_i(2 m δt)
Link flip: ∏_l R_X_l(2 Γ δt)
Matter‑link interaction: For each ⟨i,i+1⟩, implement e^{+i h δt X_{i,i+1} Z_i Z_{i+1}} via two CNOTs from Z_i,Z_{i+1} onto link, followed by R_X on link, then uncompute.
Enforce Gauss’s law either by compilation (gauge‑invariant gates only) or a penalty e^{−i Λ (G_i−1)² δt} with Λ ≫ {m,h,Γ}.
7) Observables and Readout
Electric/flux profile: U(1): ⟨E_l⟩; Z₂: ⟨X_l⟩ parity string along path.
Wilson loop: W(C) = ⟨∏_{l∈C} U_l⟩. For Z₂, loop parity product of X_l.
String breaking: Identify L_break where potential σ L exceeds 2m (or effective 2 m_φ). Extract from parity decay and endpoint correlators.
Entanglement growth: Bipartition entropy vs time after quench; WCCT predicts slower growth as λ increases at fixed σ.
Loschmidt echo: E(t) = |⟨ψ₀| e^{iHt} e^{−i(H+εV)t} |ψ₀⟩|²; test observer‑phase prediction P2.
8) Parameter Calibration (Z₂ reference)
Choose N ∈ [8, 24], open boundaries.
Initialize L ∈ [2, N/2].
Sweeps: Γ ∈ [0.02, 0.3], h ∈ [0.05, 0.5], m ∈ [0.05, 0.5], penalty Λ ≥ 5 × max{m,h,Γ}.
WCCT mapping: J ≈ h, μ ↔ 1/Γ scale, m_φ ≈ m, σ ≈ κ α² set by effective h/Γ ratio.
9) Minimal Circuit Sketches
9.1 Z₂ three‑body term e^{+i h δt X_l Z_i Z_{i+1}}
For each triple (i,l,i+1):
CNOT(Z_i → anc)
CNOT(Z_{i+1} → anc)
CNOT(anc → l)
R_X(l, 2 h δt)
Uncompute CNOTs (reverse order)
Notes: anc may be the link qubit if compiled as parity‑controlled rotation using two ZZ entanglers.
9.2 U(1) truncated link (d = 3 or 4)
Encode link in qudit→qubit register (e.g., 2 qubits for d=4).
Field term: phase rotations conditioned on |E⟩ levels.
Hopping: controlled‑phase on link register between neighboring matter qubits.
10) Data Analysis Stubs (Python)
Below are minimal, implementation‑agnostic pseudocode snippets. Replace backend hooks with qiskit, cirq, pytket, or Braket as needed.
10.1 Flux‑tube profiling (Z₂)
import numpy as np
def flux_profile(bitstrings, link_indices):
# bitstrings: list of measured bitstrings; link_indices: positions of link qubits
# Return ⟨X⟩ per link via parity from computational-basis readout with pre‑measurement Hadamards on links
vals = np.zeros(len(link_indices))
for s in bitstrings:
for j, idx in enumerate(link_indices):
vals[j] += 1 if s[idx] == '0' else -1
return vals / len(bitstrings)
10.2 Wilson loop estimator (Z₂)
def wilson_loop(bitstrings, loop_links):
total = 0
for s in bitstrings:
parity = 1
for idx in loop_links:
parity *= (1 if s[idx] == '0' else -1)
total += parity
return total / len(bitstrings)
10.3 Break‑length extraction
from scipy.optimize import curve_fit
# Model: plateau up to L_break then exponential decay
def model(L, Lb, a, b):
return np.where(L < Lb, a, a*np.exp(-b*(L-Lb)))
# lengths: array of separations; obs: measured loop/parity observable vs L
def fit_break_length(lengths, obs):
p0 = [np.median(lengths), obs.max(), 0.5/np.median(lengths)]
pars, _ = curve_fit(model, lengths, obs, p0=p0, maxfev=10000)
Lb, a, b = pars
return {"L_break": Lb, "plateau": a, "decay": b}
10.4 Entanglement proxy via randomized measurements
# Given bitstrings over two random local bases, estimate 2nd Renyi entropy growth
# See Elben et al. protocols; plug in your backend’s randomized compile.
def renyi2_from_randomized(collisions):
# collisions: probability that two random outcomes coincide
return -np.log(collisions)
11) Experimental Checklist
Choose model (Z₂ to start)
Layout qubits: interleave matter–link–matter…
Gauge‑invariant compilation or strong Λ penalty
Prepare endpoints (charges) and initial string
Trotter depth scan with echo variants
Measure: link X parity, loops, matter correlators, randomized entanglement
Parameter sweeps to test P1–P3; record L_break vs {h,Γ,m,λ_eff}
Compare against WCCT scaling fits and extract λ‑dependent sharpening
12) Reporting Template
Include: device, connectivity, native gates, error rates, calibration times; N, depth, shots; parameter table; raw bitstring archives; analysis notebooks; best‑fit values with confidence intervals; WCCT mapping values (σ, μ, J, m_φ, λ, α, β).
13) Next Extensions
Move from Z₂ to truncated U(1) (d=3/4) to probe compactness effects
Add weak global phase drive to test observer‑renormalization P2
Multi‑string interactions and reconnections (WCCT interference of flux tubes)
Cross‑hardware replication (superconducting ↔ trapped‑ion) to isolate compilation artifacts
End of v0.1. Ready for circulation to collaborators and device teams.
