Phase-sensitive representation of Majorana stabilizer states

Stabilizer states hold a special place in quantum information science due to their connection with quantum error correction and quantum circuit simulation. In the context of classical simulations of many-body physics, they are an example of states that can be both highly entangled and efficiently represented and transformed under Clifford operators. Recently, Clifford operators have been discussed in the context of fermionic quantum computation through their extension, the Majorana Clifford group. Here, we document the phase-sensitive form of the corresponding Majorana stabilizer states, as well as the algorithms for computing their amplitudes, their inner products, and update rules for transforming Majorana stabilizer states under Majorana Clifford gates.

💡 Research Summary

The paper “Phase‑sensitive representation of Majorana stabilizer states” develops a complete, phase‑aware framework for representing and manipulating stabilizer‑type states in fermionic systems using Majorana operators. Traditional stabilizer theory, which is central to quantum error correction and Clifford‑gate simulation, has been largely confined to qubit systems. When applied to fermionic problems, the usual Jordan‑Wigner mapping turns local fermionic operators into non‑local Pauli strings, obscuring physical locality and making classical simulation inefficient. The authors avoid this by working directly with the two Majorana operators per site, (c_k = a_k + a_k^\dagger) and (\tilde c_k = i(a_k^\dagger - a_k)), which are both Hermitian, unitary, and retain locality.

A key technical contribution is the introduction of a compact binary representation ((\phi, z, x)) for any Majorana string (\Gamma). Here (z) and (x) are binary vectors indicating the presence of the parity operator (p_k = i\tilde c_k c_k) and the Majorana operator (c_k) respectively, while (\phi\in{0,1,2,3}) encodes the overall phase (i^\phi). This mirrors the symplectic representation of Pauli operators and enables O((n^2)) algorithms for multiplication, commutation testing, and action on computational basis states. Explicit formulas are given for the product (\Gamma\Gamma’) and the commutation condition, both expressed in terms of simple binary dot‑products and parity functions.

The authors define Majorana Clifford gates as rotations (U(\Gamma,\theta)=\exp(-i\theta\Gamma/2)). Because (\Gamma^2=I), the exponential reduces to (\cos(\theta/2)-i\sin(\theta/2)\Gamma). When (\theta=j\pi/2) (integer (j)), the rotation maps any Majorana operator to another Majorana operator, thus belonging to the Majorana Clifford group. They focus on the parity‑preserving subgroup (“p‑Cliffords”) generated by three families of gates:

• (\eta_{jk}=e^{\pi/4,c_jc_k}) (two‑index braids),
• (\eta_j=e^{-i\pi/4,p_j}) (single‑site phase),
• (W_{jk}=e^{-i\pi/4,p_jp_k}) (parity‑parity rotations).

These generators are sufficient to build any p‑Clifford gate, and the paper provides O((n)) or O((n^2)) update rules for each.

A Majorana stabilizer state is defined as (|\psi\rangle = U|0\rangle) where (U) is a product of Majorana Clifford gates and (|0\rangle) is the fermionic vacuum. The authors adopt a phase‑sensitive CH‑form analogue: \