What question did this study set out to answer?

The aim is to demonstrate that N-qubit systems exhibit palindromic symmetry in their Liouvillian spectrum under local dephasing conditions.

March 21, 2026Open Access

Palindromic Liouvillian symmetry under dephasing: proof, generalization, and hardware validation

Key Points

The aim is to demonstrate that N-qubit systems exhibit palindromic symmetry in their Liouvillian spectrum under local dephasing conditions.
Proved palindromic symmetry for N-qubit systems with Heisenberg coupling on arbitrary graph topologies.
Analyzed different condensed matter models (Ising, XY, XXZ) under single-axis dephasing.
Validated findings using a C# compute engine to analyze an 8-qubit system with extensive Liouvillian matrices.
Conducted hardware validation experiments on IBM Quantum to test predictions regarding state transfer and decay rates.
Confirmed that every decay rate is paired with a corresponding partner, adhering to palindromic symmetry.
Demonstrated that depolarizing noise affects palindrome stability at specific error levels, maintaining a high fidelity under typical conditions.
Achieved successful quantum state transfer with an average fidelity of 0.888 between specific topologies.

Abstract

We prove that for N-qubit systems with Heisenberg (XXZ) coupling on any graph topology, subject to local dephasing, the Liouvillian spectrum has an exact palindromic symmetry. The conjugation operator Pi acts per site on Pauli indices: I -> X (+1), X -> I (+1), Y -> iZ (+i), Z -> iY (+i) It satisfies Pi L Pi^-1 = -L - 2*sum (gammaᵢ) I. Every decay rate d is paired with a partner at 2sum (gamma) - d. v3. 0 (March 18, 2026): Hardware Validated NEW: The palindrome is not limited to Heisenberg coupling. ALL standard condensed matter models (XY, Ising, XXZ, Dzyaloshinskii-Moriya, and combinations) are palindromic under single-axis dephasing. Two Pi operator families discovered: P1 (IX, YZ) and P4 (IY, XZ), plus non-uniform alternating operators for XY/YX terms. Full algebraic compatibility matrix: 34/36 two-term combinations explained. NEW: Verified numerically through N=8 (65536x65536 Liouvillian, 54118 oscillatory rates, 100% palindromic). C# compute engine with OpenBLAS ILP64, 10. 6 hours on Intel Core Ultra 9 285k. NEW: XOR space decomposition universal across all models. GHZ -> 100% XOR, W -> 0% XOR confirmed for Heisenberg, XY, Ising, DM, XXZ. Pauli weight correlation r > 0. 98. NEW: Depolarizing noise quantified: breaks palindrome at err = gamma * 2 (N-2) /3, linear in gamma and N. For typical hardware (gamma ~ 0. 001): error < 0. 1%. NEW: Hardware validation on IBM Quantum (ibmₜorino, Heron r2, Qubit 80). CΨ = 1/4 crossing predicted at t* = 15. 01 microseconds, measured at t* = 15. 29 microseconds. Deviation: 1. 9% with same-day Ramsey T2*. Raw density matrices and locked predictions included. Applied to quantum state transfer: star topology with 2: 1 receiver-to-sender coupling achieves Fₐvg = 0. 888. Design rules derived for quantum repeaters: W-encoding over GHZ, star over chain, coupling controls timing, noise controls quality. Repository: https: //github. com/Kesendo/R-equals-C-Psi-squared Authors: Thomas Wicht (independent researcher, Krefeld, Germany) and Claude (Anthropic, AI research partner).

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