⚛️ quantum physics

Qudit-ADAPT-VQE: an adaptive variational algorithm with counterdiabatic-inspired improvements for qudits

This paper introduces Qudit-ADAPT-VQE, an adaptive variational algorithm for qudits that utilizes a counterdiabatic-inspired operator pool and a warm-start strategy to construct efficient ansätze for solving Max 3-Cut, thereby achieving higher accuracy, lower gate counts, and improved robustness against barren plateaus compared to fixed-ansatz approaches.

Original authors: Joaquín Molina, Herbert Díaz-Moraga, Dardo Goyeneche, Diego Tancara

Published 2026-08-18

📖 5 min read🧠 Deep dive

CC BY 4.0

Original authors: Joaquín Molina, Herbert Díaz-Moraga, Dardo Goyeneche, Diego Tancara

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 race to build useful quantum computers, scientists have long relied on a basic building block called the qubit. Think of a qubit as a tiny switch that can be off, on, or in a strange superposition of both, much like a coin spinning on a table. This binary nature mirrors the zeros and ones of classical computers, making qubits the standard language of the field. However, just as a single coin can only show two faces, a qubit is limited to two states. Nature, by contrast, offers systems with many more possibilities. A spinning top, for instance, can point in many different directions, not just two. In the quantum world, these multi-state systems are known as qudits. Using qudits instead of qubits could allow researchers to pack more information into fewer particles, potentially making quantum computers smaller, more efficient, and better suited for specific types of complex problems, such as dividing a group of items into three or more categories rather than just two.

The challenge with these advanced systems is that they are notoriously difficult to control. When scientists try to program a quantum computer to solve a problem, they often use a method called a variational algorithm. This process is like tuning a radio: the computer guesses a solution, checks how close it is to the answer, and then adjusts its settings to get better. The problem is that as the system grows larger, the signal often gets lost in static. The adjustments become so tiny that the computer cannot tell which way to turn the dial, a phenomenon known as a barren plateau. Furthermore, if the computer starts with a poor guess, it can get stuck in a local trap, thinking it has found the best solution when it has only found a mediocre one. These obstacles have made it hard to scale up quantum computing, even with the more powerful qudits.

A team of researchers in Chile has proposed a new way to navigate these difficulties, specifically for qudit-based machines. They adapted an existing strategy called ADAPT-VQE, which builds the computer's program step-by-step rather than trying to design the whole thing at once. Instead of guessing the entire structure, the algorithm adds one piece at a time, always choosing the piece that improves the answer the most. To make this even more effective, the researchers borrowed a concept from physics known as counterdiabatic driving. In simple terms, this is a technique used to speed up a process without causing errors, similar to how a skilled driver might steer slightly ahead of a curve to maintain a smooth path. By using this "steering" logic to decide which pieces to add to the program, they created a new algorithm called Qudit-ADAPT.

The team tested their method on a classic puzzle known as the Max 3-Cut problem. Imagine a network of cities connected by roads, where the goal is to divide the cities into three distinct groups so that the number of roads connecting different groups is as high as possible. This is a problem that naturally fits the three-state nature of qudits. The researchers simulated their algorithm on a computer to see how well it performed compared to a standard, fixed-program approach. The results were striking. Their adaptive method found solutions that were significantly more accurate, often reducing the error by more than ten times compared to the fixed approach. Moreover, it achieved this high accuracy using far fewer steps and less complex circuitry, which is crucial for keeping quantum computers stable in the noisy environment of today's technology.

Beyond just finding better answers, the study revealed why the method works so well. The researchers examined the "landscape" of the problem, looking at how the algorithm moved through different possible solutions. They found that the standard fixed approach often got lost in a maze of local traps, where the computer would stop improving because it couldn't see a better path forward. In contrast, the Qudit-ADAPT algorithm, with its step-by-step construction and smart starting points, was able to burrow through these traps. It didn't just get stuck; it kept adding new pieces to its program, reshaping the landscape and finding a way down to the true solution. This suggests that the method is robust against the barren plateau problem, where the signal usually disappears, because it keeps the computer focused on the most promising directions at every step.

The team also explored how the complexity of the "steering" logic affected the results. They tested two versions of their operator pool, one with a simpler set of rules and another with a more detailed, higher-order set. For many of the test cases, both versions worked well, but the more detailed version consistently pushed the accuracy even closer to perfection, especially for the most complex and interconnected networks. This indicates that while the basic method is powerful, adding more sophisticated guidance allows the system to solve harder problems with greater precision. The study did not claim to have solved all quantum computing challenges, nor did it run these tests on a physical quantum machine. Instead, the findings are based on rigorous numerical simulations that model how the algorithm would behave on real hardware.

Ultimately, this work offers a promising roadmap for the future of quantum computing with qudits. By combining an adaptive, step-by-step building process with physics-inspired guidance, the researchers have shown a way to avoid the common pitfalls that have slowed progress in the field. Their approach suggests that we do not need to wait for perfect hardware to start solving complex problems; we can instead design smarter software that works around the limitations of current machines. As the field moves forward, this flexible framework could become a standard tool, helping scientists harness the full potential of multi-state quantum systems to tackle optimization problems that are currently out of reach.

Technical Summary: Qudit-ADAPT-VQE

Problem Statement
Variational Quantum Algorithms (VQAs) based on qudits (d-level quantum systems) offer potential advantages over qubit-based approaches, including reduced circuit complexity for high-dimensional problems and natural encoding for variables with dimensionality greater than two (e.g., Max $k$ -Cut). However, qudit-based VQAs face significant challenges similar to their qubit counterparts: the design of efficient, expressive ansätze and the "barren plateau" problem, where cost-function gradients vanish exponentially with system size, hindering trainability. Furthermore, increasing the local Hilbert space dimension ( $d > 2$ ) can amplify barren plateau effects. Fixed-ansatz approaches, such as the qudit extension of the Quantum Approximate Optimization Algorithm (QAOA), struggle to balance expressibility with circuit depth, often requiring deep circuits that exacerbate noise and optimization difficulties.

Methodology
The authors propose Qudit-ADAPT, an adaptive variational algorithm tailored for qudit systems. The core of the method is an iterative ansatz construction that selects operators based on energy gradients, rather than using a fixed circuit structure.

Operator Formalism: The algorithm utilizes angular-momentum operators ( $L_x, L_y, L_z$ ) to encode ternary variables ( $\ell=1$ , $d=3$ ). The cost Hamiltonian ( $H_C$ ) for the Max 3-Cut problem is formulated using these operators, which possess closed commutation relations suitable for nested-commutator expansions.
Counterdiabatic-Inspired Operator Pool: The operator pool is derived from an approximate Counterdiabatic (CD) Hamiltonian. Inspired by adiabatic evolution enhanced with counterdiabatic driving, the method constructs an Adiabatic Gauge Potential (AGP) using a nested-commutator expansion.
- The AGP is approximated as $A^{(l)}_\lambda = \sum a_j(t) D_j$ , where $D_j$ are strings of angular-momentum operators.
- A critical technical adaptation for qudits is addressed: unlike Pauli strings in qubit systems, products of angular-momentum operators in qudits may not be Hermitian. The authors define a Hermitian operator pool $V_j = (D_j + D_j^\dagger)/2$ to ensure physical validity.
Adaptive Construction (CD-ADAPT): The algorithm initializes the state as the ground state of a mixer Hamiltonian ( $H_M$ $H_{M}$ ). In each iteration:
- Energy gradients for all operators in the pool are evaluated.
- The operator with the largest gradient magnitude is selected and appended to the ansatz.
- A Variational Quantum Eigensolver (VQE) optimization is performed over all accumulated parameters using a warm-start strategy (reusing optimized parameters from the previous iteration).
- The process repeats until the gradient norm falls below a convergence threshold.

Key Contributions

First Qudit-ADAPT Implementation: This work presents the first application of the ADAPT-VQE strategy to qudit-based quantum computing.
Hermitian Pool Construction: The authors resolve the non-Hermiticity issue inherent in qudit operator strings by constructing a symmetrized operator pool derived from the approximate AGP.
Counterdiabatic Integration: The method integrates counterdiabatic driving concepts into the adaptive pool selection, theoretically guiding the ansatz toward directions relevant for suppressing diabatic transitions.
Scalability Analysis: The paper investigates the trainability of the algorithm in the context of barren plateaus and local minima, specifically analyzing the impact of the warm-start strategy in high-dimensional Hilbert spaces.

Results
The algorithm was numerically tested on the Max 3-Cut problem using various graph instances (irregular and regular) with six vertices.

Accuracy vs. QAOA: Qudit-ADAPT consistently outperformed qudit QAOA (fixed ansatz). For irregular graph instances, Qudit-ADAPT achieved relative errors at least one order of magnitude lower than QAOA. In many cases, it reached high approximation ratios ( $r > 0.99$ ) with significantly fewer variational parameters and native gates.
Operator Pool Order: Comparing first-order ( $l=1$ ) and second-order ( $l=2$ ) approximations of the AGP, the $l=2$ pool provided superior performance, particularly for highly connected graphs (e.g., $K_6$ ), reaching machine precision with fewer parameters than the $l=1$ pool or QAOA.
Robustness and Trainability:
- Local Minima: The study analyzed the optimization landscape by comparing warm-start, cold-start, and random restarts. The warm-start strategy consistently converged to lower energy states than random initializations, often by orders of magnitude.
- Burrowing Mechanism: The results support the "burrowing" mechanism, where the iterative addition of operators allows the algorithm to escape local traps that would stall a fixed-ansatz optimizer.
- Barren Plateaus: The adaptive construction, starting with shallow circuits, avoids the initial barren plateau regime. The warm-start strategy ensures the optimization remains in regions with resolvable gradients, suggesting robustness against the barren plateau effects that typically worsen with increasing qudit dimension.

Significance and Claims
The paper claims that Qudit-ADAPT provides a scalable and robust framework for variational quantum algorithms on qudit architectures. By combining a counterdiabatic-inspired operator pool with an adaptive, warm-start construction, the method addresses two central challenges of fixed-ansatz approaches: the difficulty of designing efficient ansätze and the susceptibility to unfavorable optimization landscapes (local minima and barren plateaus).

The authors assert that this approach achieves higher accuracy with lower resource requirements (fewer parameters and native gates) compared to fixed-ansatz methods like QAOA. They conclude that the combination of iterative ansatz growth and warm-start initialization allows the algorithm to navigate rough optimization landscapes effectively, making it a promising candidate for NISQ-era qudit processors. The work suggests that strategies developed for qubit-based ADAPT-VQE, such as hardware-aware penalties and commuting operator grouping, can be directly transferred to this qudit framework.

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