⚛️ quantum physics

Phase-sensitive representation of Majorana stabilizer states

This paper presents a phase-sensitive representation of Majorana stabilizer states along with efficient algorithms for computing their amplitudes, inner products, and update rules under Majorana Clifford gates.

Original authors: Tomislav Begušić, Garnet Kin-Lic Chan

Published 2026-02-20

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CC BY 4.0

Original authors: Tomislav Begušić, Garnet Kin-Lic Chan

Original paper licensed under CC BY 4.0 (http://creativecommons.org/licenses/by/4.0/). ✨ This is an AI-generated explanation of the paper below. It is not written or endorsed by the authors. For technical accuracy, refer to the original paper. Read full disclaimer

Imagine you are trying to organize a massive, chaotic library. In the world of quantum physics, this library is filled with "quantum states"—complex descriptions of how particles like electrons behave. Usually, describing these states is like trying to write down every single book in the library individually. It's slow, messy, and requires an impossible amount of paper (computing power).

However, some special books in this library have a secret: they follow strict, simple rules. In the quantum world, these are called Stabilizer States. Think of them as books that are perfectly organized by a librarian who only uses a specific set of simple moves (called Clifford gates) to rearrange them. Because they follow these rules, we can describe the whole book with just a tiny cheat sheet instead of writing out every word.

The Problem: The "Fermion" Language Barrier

For a long time, scientists could only use this "cheat sheet" method for simple quantum bits (qubits), like the ones in standard computers. But the real world of chemistry and materials science is made of fermions (like electrons).

To study fermions using the simple cheat sheet, scientists usually had to translate them into "qubit language" first. But this translation is like trying to describe a local neighborhood by mapping it onto a global map of the entire world. The local details get lost, and the map becomes huge and confusing. It's like trying to understand a specific street corner by looking at a satellite image of the whole planet.

The Solution: A New Cheat Sheet for Fermions

This paper introduces a new way to organize the library specifically for fermions, without needing that confusing translation. The authors, Tomislav Begušić and Garnet Kin-Lic Chan, have created a "Phase-Sensitive Representation" for Majorana Stabilizer States.

Here is the breakdown using everyday analogies:

1. The "Majorana" Operators: The Building Blocks

Think of fermions as having two faces: a "creation" face (adding a particle) and an "annihilation" face (removing one). Usually, these are messy. But the authors use Majorana operators, which are like "half-particles."

Analogy: Imagine a coin. A normal coin has a Head and a Tail. A Majorana operator is like cutting the coin in half. You have a "Head-half" and a "Tail-half." These halves are special because they are their own mirror images (they are "Hermitian"). The authors use these halves to build their cheat sheet, keeping the local neighborhood details intact.

2. The "Clifford Group": The Librarian's Rules

In the quantum library, there is a set of allowed moves called Clifford gates.

Analogy: Imagine a librarian who can only move books in specific ways: swapping two shelves, flipping a book upside down, or rotating a whole section. They can't tear pages out or rewrite the text. Because these moves are so predictable, the librarian can keep a simple log (the Stabilizer Tableau) of where every book is, rather than tracking every single page.
The paper defines a new set of rules for fermions (the Majorana Clifford Group) that works just like the librarian's rules but keeps the "local neighborhood" feel of the electrons.

3. The "Phase-Sensitive" Part: The Secret Code

The biggest breakthrough here is the "Phase-Sensitive" part.

The Old Way: Previous cheat sheets for these states were like a map that showed where the books were, but ignored the color of the book covers. For many tasks, this was fine. But if you wanted to combine two libraries or calculate the exact "vibe" (amplitude) of the system, the missing color information made it impossible.
The New Way: This paper adds the "color" back in. It tracks the global phase (a subtle quantum property, like a secret code or a timestamp).
Analogy: Imagine two identical-looking twins. Without a secret code, you can't tell them apart. But if you know one twin has a secret tattoo on their left ankle (the phase), you can perfectly distinguish them and predict exactly how they will react when they meet. This paper provides the algorithm to track that "tattoo" for fermions.

What Can We Do With This?

The authors didn't just write a theory; they built the tools (algorithms) to use this new system. They provide recipes for:

Updating the State: If you apply a quantum gate (a move), how do you update the cheat sheet? (They found the rules are fast, taking roughly the square of the number of particles).
Calculating Amplitudes: What is the probability of finding the system in a specific state? (Like asking, "What are the odds this specific book is on the shelf?").
Comparing States: How similar are two different quantum states? (Like checking if two libraries have the same collection).

Why Does This Matter?

This is a game-changer for Quantum Chemistry and Quantum Computing.

Efficiency: It allows computers to simulate complex molecules (like drugs or new materials) much faster because they don't have to translate everything into the messy "qubit" language first.
Accuracy: By keeping the "phase" (the secret code), scientists can now use these efficient methods to simulate any quantum state, not just the simple ones. It's like upgrading from a sketch of a building to a full 3D blueprint that you can actually build with.

In summary: This paper gives us a new, efficient, and accurate "cheat sheet" for organizing the quantum library of electrons. It keeps the local details clear, tracks the secret codes (phases) that were previously lost, and provides the instructions for how to rearrange the library without getting overwhelmed. This paves the way for simulating complex chemical reactions and materials on classical computers much more effectively.

1. Problem Statement

The classical simulation of fermionic quantum systems is a central challenge in quantum chemistry and many-body physics. While stabilizer states (generated by Clifford gates acting on computational basis states) are well-understood for qubit systems and allow for efficient classical simulation (Gottesman-Knill theorem), their fermionic counterparts have been less explored in a native fermionic framework.

Existing approaches often map fermions to qubits (e.g., via Jordan-Wigner transformation) to utilize qubit-based stabilizer formalisms. However, this mapping introduces non-locality, obscuring the physical locality of fermionic operators and complicating the interpretation of results. Furthermore, while Majorana Clifford operators (which preserve the Majorana algebra) have been identified as a tool for fermionic quantum computation, a phase-sensitive representation of the resulting Majorana stabilizer states has been missing. Without phase information, one cannot:

Represent general fermionic states as superpositions of stabilizer states.
Compute probability amplitudes or inner products between states.
Accurately simulate quantum circuits dominated by Clifford gates in the fermionic picture.

2. Methodology

The authors develop a formalism analogous to the CH form (Hadamard-Clifford) used for qubit stabilizer states but adapted for the Majorana algebra.

A. Majorana Operator Representation

The paper defines a compact $(\phi, z, x)$ representation for Majorana strings (operators):
$\Gamma = i^\phi \prod_{k=0}^{n-1} p_k^{z_k} c_k^{x_k}$
where $c_k$ and $\tilde{c}_k$ are local Majorana operators, $p_k = i\tilde{c}_k c_k$ is the parity operator, and $\phi, z, x$ are binary/integer parameters. This allows for efficient storage and manipulation (commutation checks, products) of Majorana operators.

B. Phase-Sensitive State Representation

A Majorana stabilizer state $|\psi\rangle$ is defined as $U|0\rangle$ , where $U$ is a product of Majorana Clifford operators. The authors represent $|\psi\rangle$ as:
$|\psi\rangle = e^{i \frac{\pi}{4} \phi} U_C U_B |s\rangle$

$|s\rangle$ : A computational basis state (bitstring).
$U_B$ : A product of commuting braiding operators $B_k = e^{-i \frac{\pi}{4} c_k \tilde{c}_{k+1}}$ , parameterized by a binary vector $b$ .
$U_C$ : A "control-type" Clifford operator that preserves the vacuum and maps $p$ -type operators to $p$ -type operators. It is stored using a stabilizer tableau (matrices $E, F, G$ and phase vector $\omega$ ) analogous to the qubit stabilizer tableau.
$\phi$ : A global phase integer.

C. Algorithmic Updates

The core of the methodology involves deriving update rules for the parameters $(\phi, U_C, b, s)$ when acted upon by generators of the parity-preserving Majorana Clifford group (p-Cliffords), specifically:

$\eta_j$ and $W_{jk}$ (Diagonal in the computational basis).
$\eta_{jk}$ (Non-diagonal, analogous to Hadamard gates).
General Majorana Clifford rotations.

The update for $\eta_{jk}$ is the most complex, following a strategy similar to the CH-form update for qubits:

Decompose the action into a superposition of basis states.
Apply a control-type operator to align the basis states so they differ only at two neighboring sites.
Update the tableau and phase accordingly.

3. Key Contributions

Formalism Definition: The paper establishes the first phase-sensitive representation for Majorana stabilizer states, enabling the description of general fermionic states as superpositions of these states.
Efficient Algorithms: The authors provide explicit algorithms for:
- State Updates: Applying Majorana Clifford gates ( $O(n)$ to $O(n^2)$ complexity).
- Amplitude Calculation: Computing $\langle x | \psi \rangle$ in $O((|x|+c)n)$ time.
- Inner Products: Computing $\langle \phi | \psi \rangle$ in $O(n^3)$ time.
- Expectation Values: Computing $\langle \psi | \Gamma | \psi \rangle$ efficiently.
Tableau Identities: The derivation of specific matrix identities (e.g., $E \cdot G^T = I$ ) required to maintain the consistency of the stabilizer tableau under Majorana transformations.
Implementation: A Python implementation is provided (linked in the paper) and verified against Qiskit's statevector simulator using Jordan-Wigner mapping, confirming the correctness of the native fermionic approach.

4. Results and Complexity Analysis

The paper summarizes the computational complexity of operations in Table I:

Operation	Complexity	Notes
Apply $\eta_j$ or $W_{jk}$	$O(n)$	Diagonal gates; update phase and tableau only.
Apply $\eta_{jk}$	$O(n^2)$	Non-diagonal; requires tableau updates similar to Hadamard.
General Majorana Clifford	$O(n^2)$	General rotation.
Probability Amplitude $\langle x\|\psi\rangle$	$O((\|x\|+c)n)$	Depends on the weight of the basis state.
Inner Product $\langle \phi\|\psi\rangle$	$O(n^3)$	Dominated by matrix multiplication in tableau construction.

The authors demonstrate that these algorithms allow for the efficient manipulation of highly entangled fermionic states without mapping to qubits, preserving the locality of the original fermionic operators.

5. Significance

This work bridges a critical gap between quantum information theory and fermionic many-body physics:

Native Fermionic Simulation: It enables low-rank stabilizer methods to work directly in the fermionic picture. This avoids the non-locality issues inherent in qubit mappings (like Jordan-Wigner), potentially leading to more efficient simulations for quantum chemistry and condensed matter systems.
Quantum Error Correction & Computation: The formalism supports the development of Majorana-based quantum error correction codes and fault-tolerant quantum computation schemes, where understanding the phase and structure of stabilizer states is essential.
Magic State Distillation: By providing a way to compute overlaps and amplitudes, the method facilitates the quantification of "non-stabilizerness" (magic) in fermionic states, a key resource metric for quantum advantage.
Scalability: The polynomial scaling ( $O(n^2)$ to $O(n^3)$ ) suggests that large-scale fermionic systems dominated by Clifford-like dynamics can be simulated classically, serving as a benchmark for quantum hardware or a component in hybrid quantum-classical algorithms.

In summary, Begušić and Chan provide the necessary mathematical toolkit to treat Majorana stabilizer states as a first-class resource for classical simulation, extending the power of stabilizer formalisms from qubits to fermions.