⚛️ quantum physics

Generalised Quantum Gates for Qudits and their Application in Quantum Fourier Transform

This paper proposes a novel, universally applicable formulation of generalized quantum gates for qudits of any dimension $d$ , demonstrating their validity and utility through the implementation of the Quantum Fourier Transform to broaden the design space for qudit-based quantum algorithms and fault-tolerant architectures.

Original authors: Francesco Pudda, Mario Chizzini, Luca Crippa

Published 2026-08-25

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CC BY 4.0

Original authors: Francesco Pudda, Mario Chizzini, Luca Crippa

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

In the quest to build machines that can solve problems beyond the reach of today's computers, scientists have long focused on the smallest possible unit of information: a two-state switch. Imagine a coin that can be heads, tails, or a blur of both at the same time; this is the qubit, the fundamental building block of current quantum computing. For decades, researchers have treated these two-level systems as the only way forward, building complex circuits to manipulate them. However, nature offers a richer palette. Just as a coin has two sides, many physical systems, such as trapped ions or specific light particles, naturally possess three, four, or even more distinct energy states. These multi-level units, known as qudits, have been largely ignored in mainstream computing designs, despite their potential to hold more information and perform calculations with fewer steps. The challenge has been that the mathematical tools used to control the simple two-state qubits do not easily translate to these more complex, multi-level systems, leaving a gap between the hardware's potential and our ability to use it.

A team of researchers has now bridged this gap by creating a new set of instructions specifically designed for these multi-level quantum units. In their work, they propose a universal method to define and control quantum gates—the operations that change the state of a quantum system—for any number of levels, not just two. Instead of forcing multi-level systems to behave like simple switches, the authors developed a flexible framework that treats the extra levels as a feature rather than a complication. They demonstrated that by using these new definitions, it is possible to construct a complete and functional set of tools for quantum computation using qudits of any size. To prove their theory works, they applied these new rules to build a specific, well-known algorithm called the Quantum Fourier Transform, a critical tool used to analyze patterns in data. By simulating this process on a computer, they showed that their generalized gates could successfully perform the complex calculations required, matching the results of standard mathematical models.

The core of this achievement lies in reimagining how quantum operations work when a system has more than two options. In the familiar two-level world, a basic operation might simply flip a state from zero to one. In the multi-level world of qudits, the researchers defined new versions of these operations that can shift a state up by one level, rotate its phase, or combine two qudits in a way that links their states together. They introduced a specific new operation, a controlled rotation gate, which allows one qudit to influence the rotation of another based on their current levels. This is essential because, unlike simple switches, multi-level systems require a more nuanced way to create the entanglement needed for powerful calculations. The team also defined a method to swap the states of two qudits, a necessary step for rearranging information within a circuit. By combining these new tools, they created a complete toolkit that allows any quantum operation to be broken down into a sequence of these fundamental steps, ensuring that no matter how many levels a qudit has, it can be programmed effectively.

To verify that their new definitions were not just theoretical but actually functional, the researchers turned to the Quantum Fourier Transform. This algorithm is a cornerstone of quantum computing, used to find hidden patterns in data and is a key component in many famous quantum speed-ups. In the past, implementing this algorithm for multi-level systems was difficult because the necessary mathematical definitions for the required gates were missing from the literature. The authors filled this void by extrapolating their new gate definitions to construct the entire algorithm. They simulated the circuit using a standard quantum computing software platform, mapping the multi-level qudits onto groups of simpler two-level qubits to test the logic. The simulation confirmed that their generalized gates produced the exact same outcome as the established mathematical transform, validating that their approach correctly handles the complexity of higher dimensions.

The implications of this work extend beyond a single algorithm. By providing a clear, universal set of rules for controlling qudits, the researchers have opened the door for designing quantum computers that are more efficient and potentially more robust against errors. Because qudits can store more information per unit, they may require fewer physical components to perform the same task, which could simplify the physical construction of these machines. Furthermore, the extra states available in a qudit offer new ways to detect and correct mistakes that occur during computation, a major hurdle in building reliable quantum hardware. While the current study focused on proving the mathematical validity and simulating the performance of these gates, the results suggest a viable path forward for a new generation of quantum architectures. The work does not claim to have solved every problem in the field, but it provides the essential theoretical foundation needed to move from simple two-level experiments to the more powerful, multi-level systems that nature readily provides.

Technical Summary: Generalised Quantum Gates for Qudits and Their Application in Quantum Fourier Transform

Problem Statement
While quantum computing research has predominantly focused on qubits (two-level systems), there is a growing recognition of the potential benefits offered by qudits (quantum systems with $d > 2$ levels). Qudits promise expanded computational capacity, reduced circuit complexity, improved noise resilience, and the potential for embedded error correction. However, the development of a comprehensive, general theory for qudit quantum gates has not been thoroughly explored. Specifically, existing literature often lacks explicit, universally applicable definitions for qudit gates across arbitrary dimensions $d$ , and crucial matrix definitions for controlled rotations in the context of the Quantum Fourier Transform (QFT) have been absent.

Methodology
The authors propose a novel formulation of quantum gates applicable to qudits of any dimension $d$ , extending standard quantum gate theory without dimensional restrictions. The methodology involves:

Generalizing Fundamental Gates: The authors adopt Schwinger's canonical notations to define generalized Pauli ( $X_d$ , $Z_d$ ) and Hadamard ( $H_d$ ) gates for $d$ -dimensional systems. They establish the relationship $X = HZH^\dagger$ and define the action of these gates on basis states $|j\rangle$ .
Defining Rotation Gates: To achieve universality beyond the Clifford group (as required by the Gottesman-Knill theorem), the authors define generalized $S$ -gates (half $Z$ -rotations) and $T$ -gates (quarter $Z$ -rotations). They introduce a generalized phase gate, $P_d(\theta)$ , which rotates the phase of a qudit state $|j\rangle$ by an amount proportional to $j \cdot \theta$ .
Constructing Controlled Gates: To enable entanglement and universal computation, the authors define:
- The SUMX-gate (generalized controlled-NOT), which shifts the target qudit state cyclically by the value of the control qudit state: $|j\rangle|k\rangle \to |j\rangle|(j+k) \mod d\rangle$ .
- The SUMP-gate (generalized controlled-phase), which applies a phase rotation to the target qudit proportional to the product of the control and target states: $|j\rangle|k\rangle \to \omega^{j \cdot k \cdot \theta / \pi} |j\rangle|k\rangle$ .
- The K-gate (Complement operator), defined as $K_d |j\rangle = |-j\rangle \mod d$ , which is utilized alongside the SUMX-gate to construct a SWAP-gate for $d > 2$ .
QFT Implementation: The authors apply these generalized gates to construct the Quantum Fourier Transform (QFT) for arbitrary $d$ . They adapt the standard QFT algorithm, replacing standard controlled rotations with the proposed SUMP-gates and utilizing the generalized Hadamard gates.

Key Contributions

Universal Gate Set Formulation: The paper provides explicit matrix definitions and operational rules for a universal set of qudit gates (Pauli, Hadamard, $P$ , $S$ , $T$ , SUMX, SUMP, K, and SWAP) that are valid for any dimension $d$ .
The SUMP-Gate: A novel definition of the controlled-phase gate (SUMP) is introduced, filling a gap in the literature regarding matrix definitions for controlled rotations in general qudit scenarios.
QFT Verification: The authors demonstrate the validity of their generalized gate set by successfully implementing the QFT algorithm for arbitrary $d$ .
Simulation and Validation: The proposed circuits were simulated using Qiskit by mapping $d$ -dimensional qudits to $n$ qubits (where $d=2^n$ ). The results confirmed that the overall circuit operator, constructed using the proposed gates, perfectly matches the conjugate of the Discrete Fourier Transform (DFT) as computed by SciPy.

Results
The study successfully derived explicit gate operations that form a universal set for qudit quantum computation. The implementation of the QFT using these gates yielded a circuit operator that aligns precisely with the theoretical DFT $^\dagger$ . The authors note that simulations were limited to $n \le 4$ (i.e., $d \le 16$ ) due to RAM constraints, but the mathematical consistency and simulation results for these cases verify the correctness of the approach.

Significance
The paper claims that this work offers an alternative foundation for designing qudit-based algorithms and architectures. By providing a complete set of theoretical definitions and demonstrations, the authors aim to broaden the design space for quantum algorithms and fault-tolerant architectures. The methodology paves the way for advancements in qudit-based quantum computing, specifically facilitating the development of optimized error correction and more efficient quantum circuits that leverage the higher-dimensional state spaces of qudits. The work is presented as a necessary step toward realizing the practical potential of qudit systems in both theoretical research and practical applications.

Technical Summary: Generalised Quantum Gates for Qudits and Their Application in Quantum Fourier Transform

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