50-Site Kitaev Chain — Majorana Zero Modes

BdG exact diagonalisation  ·  Open boundary conditions  ·  Z₂ topological invariant W = −1  ·  Two Majorana bound states

What this benchmark tests

The Kitaev chain is the canonical model for topological quantum computing. It is a 1D p-wave superconductor described by a free-fermion (Bogoliubov–de Gennes) Hamiltonian whose phase diagram is exactly solvable. In the topological phase (|μ| < 2t), two Majorana zero modes emerge at the chain ends — non-Abelian anyons whose quantum state is delocalized and immune to local perturbations.

This benchmark verifies that Qumulator's KLT Kitaev chain engine correctly identifies the topological phase, localizes the zero modes, and reproduces the quantized conductance signature — for a 50-site chain with hopping t = 1, chemical potential μ = 0 (deep topological), and pairing Δ = 1.

Result summary

Phase diagram

The Kitaev chain has an exactly solvable topological phase transition at |μ| = 2t. For this benchmark, t = 1 places the transition at μ = ±2. The μ = 0 operating point is deep in the topological island.

At the boundaries (|μ| = 2t), the bulk gap closes and the Majorana modes delocalize across the entire chain. Inside the topological island, the modes are exponentially localized at the ends with coherence length ξ = 1/ln(t/Δ) in units of the lattice spacing.

Reproduce this result

Submit the Kitaev chain Hamiltonian directly via the Qumulator circuit API using the klt_nexus mode, which maps each site to a KLT Z₂ orbit and resolves the BdG spectrum exactly.

pip install qumulator-sdk

import os
from qumulator import QumulatorClient

client = QumulatorClient(
    api_url=os.environ["QUMULATOR_API_URL"],
    api_key=os.environ["QUMULATOR_API_KEY"],
)

# 50-site Kitaev chain: t=1, mu=0 (deep topological), delta=1
L     = 50
t     = 1.0
mu    = 0.0    # |mu| < 2t → topological phase
delta = 1.0

# Build QASM3 circuit representing the Kitaev Hamiltonian as a sequence
# of nearest-neighbour ZZ + XX + YY gates (Jordan-Wigner Pauli encoding)
instructions = []
for i in range(L):
    # On-site chemical potential: −μ·(c†c − ½) → −μ/2·Z gate
    instructions.append({"gate": "rz", "qubits": i, "params": [-mu]})
for i in range(L - 1):
    # Hopping + pairing: −t(XX+YY)/2 + Δ(XY−YX)/2
    instructions.append({"gate": "cx",  "qubits": [i, i+1]})
    instructions.append({"gate": "ry",  "qubits": i+1, "params": [-t]})
    instructions.append({"gate": "cx",  "qubits": [i, i+1]})
    instructions.append({"gate": "rz",  "qubits": i, "params": [delta]})

result = client.circuit.run(
    n_qubits=L,
    instructions=instructions,
    mode="klt_nexus",
    return_entropy_map=True,
    shots=1,
)

print(f"KLT phase         : {result.klt_phase}")
print(f"Topological index : {result.topological_index}")
print(f"Bulk gap          : {result.bulk_gap:.4f}")
print(f"Majorana modes    : {result.n_majorana_modes}")
print(f"Conductance G     : {result.conductance_norm:.3f} e²/h")

What are Majorana zero modes?

A Majorana fermion is its own antiparticle: γ† = γ. In condensed matter, Majorana bound states emerge at the ends of a topological superconducting wire as quasi-zero-energy excitations. Because they are delocalized across both wire ends, local perturbations (thermal noise, charge fluctuations) cannot shift their energy — making them the leading candidate for fault-tolerant quantum memory.

The KLT connection: in the KLT framework (Paper IV), each Majorana mode corresponds to a nexus birth event where the Frenet–Serret torsion phases satisfy Φ_A + Φ_B = 0 exactly — the same Z₂ linking number that defines the topological invariant W = −1. The two modes at opposite chain ends are the two "half-fermions" of this nexus bond.

Why BdG diagonalisation is tractable

Unlike interacting systems, the Kitaev chain is a free-fermion model. Wick's theorem reduces all N-body correlators to 2×2 blocks, and the BdG Hamiltonian is only 2L×2L even for a chain of L = 10,000 sites. Qumulator's engine diagonalises this exactly via standard LAPACK in O(L³) — no exponential barrier, no variational approximation, no Monte Carlo noise.