Hello, I am Rahul Bhowmick from the Quantum team at Fujitsu Research India Private Limited.
With advancements in software and hardware, quantum computing isn't just theoretical anymore. It’s already showing promise across a range of fields: cryptography, chemistry, materials science, digital communication, and more. Fujitsu is in this race, building our own quantum hardware and developing tools to make this future a reality.
While large-scale, error-free quantum computers are still years away, researchers around the world are actively exploring practical algorithms for today’s Noisy Intermediate-Scale Quantum (NISQ) computers, with relatively small number of qubits and noisy quantum gates. At FRIPL, India, our team is focused on advancing quantum machine learning and variational algorithms. These efforts could enhance fields like quantum chemistry, drug discovery, and artificial intelligence, though their scalability remains uncertain and depends on future hardware improvements.
In this article, I will discuss our recent work on improving variational algorithms by combining classical and quantum methods to reduce quantum resources and improve precision, based on our preprint out on arxiv. This work was also presented at the Fujitsu Quantum Day 2025 held at Kawasaki, Japan.
Introduction
Variational Quantum Algorithms (VQAs) are a promising approach for leveraging quantum computers to solve complex problems. At its core, VQA consists of a quantum circuit with parametrized trainable gates. Depending on the task, these parameters are updated or trained. This is an iterative process where the output from the quantum circuit is measured and then evaluated classically using a loss function. The loss function is dependent on the task and its value decides the quality of solution obtained from the quantum circuit. Until a satisfactory solution is obtained, the parameters are updated iteratively. At the end of the algorithm, we obtain the correct solution from the quantum circuit. Importantly, the core idea is similar to classical machine learning where a model with trainable parameters is also updated based on a loss function. The main difference in VQA is the implementation of this model is done using quantum hardware, as opposed to on a classical computer.
This is a simple idea, however the class of VQAs has many different applications across fields like drug discovery, material exploration, optimization and artificial intelligence among many others. Variational Quantum Algorithms (VQAs) are a promising approach for leveraging quantum computers to solve complex problems.
Current Status and Challenges
For a VQA to reach the correct solution, the parameters are updated using a classical optimizer. However, we also need additional information on which direction to update the parameters in each step. A method of extracting this information is by estimating gradients of the loss function with respect to the trainable parameters of the quantum circuit.
The most employed method for this gradient-estimation step is Parameter Shift Rule (PSR), where the parameters of the quantum circuit are shifted by certain values and the output is measured using quantum hardware. This information is then combined to obtain the gradients. Based on the evaluated gradients, the classical optimizer then decides how to update the parameters, and this process continues iteratively till a good solution is reached.
A limiting aspect of PSR is the high quantum resource cost required to perform it. For L trainable parameter in the quantum circuit, we need 2L unique quantum circuits to obtain the full gradient information. This is in stark contrast to classical neural networks where a single evaluation of the network and some additional memory is enough to obtain the gradient information accurately.
This limitation is illustrated through a specific example. Assuming each circuit iteration takes 1.48 milliseconds (ms) which is equivalent to the latest coherence time of superconducting qubits, one single iteration for a circuit with a billion parameters would take approximately 2 × 109 × 1.48 ms, or 296 million seconds (3 days, 10 hours and 19 minutes). This step will need to be repeated for each iteration in the optimization process until the algorithm finds a good solution. Current classical networks operate in the same scale of parameters but require orders of magnitude less time for each iteration. This poses a major challenge for the scalability of VQAs to the same scale as their classical counterparts in solving real world problems.
Optimization in VQAs becomes challenging when barren plateaus occur, triggered by excessive circuit expressivity, global measurements, high entanglement, or quantum hardware noise. Barren plateaus flatten the optimization landscape, causing gradients to diminish exponentially with the number of qubits. This demands exponentially increasing quantum resources to accurately estimate gradients, significantly prolonging optimization time. Reducing quantum resource requirements remains a key factor to scaling VQAs for real-world applications.
Our Technical Contribution
Due to the limited scalability of PSR, we aim for a hybrid approach. Instead of relying fully on quantum hardware for gradient evaluation, we delegate the task to both classical and quantum hardware. Our approach is based on the following:
- A new quantum ansatz called Hardware Efficient and dynamical LIe algebra supported Ansatz (HELIA), which is made up of two blocks of quantum gates. For one of the blocks, gradient is evaluated using standard PSR, while for the other block the same is done using a classical simulation algorithm called g-sim. By cleverly choosing the gates in the second block, based on a group-theoretic structure called dynamical Lie algebra, we can efficiently calculate gradients for it.
Our proposed HELIA is composed of two blocks of quantum gates: Orange block (Uq) trained using PSR, and Blue block (Ug) trained using g-sim - Several training schemes tailored to HELIA, namely: Alternate and Simultaneous, as well as a combination of the two we refer to as Alt+Sim.
By using the above ideas, we can significantly reduce the quantum resource for training (up to 60%) and reduce the error (up to an order of magnitude reduction) in finding the ground state energy of a Hamiltonian. We are also able to mitigate the Barren Plateau phenomenon. We will discuss these findings in detail in the next section.
Our Key Findings
We apply these ideas to the task of finding ground state or lowest energy state of a Hamiltonian. VQA used for this task is known as Variational Quantum Eigensolver, which searches over quantum states iteratively to find the ground state of the chosen Hamiltonian. For this task we employ the 16 qubit XY Hamiltonian. We test our idea for 4 different configurations of HELIA, by choosing 1,3 6 and 9 layers of YZ linear gates. Our results are described below:
- Reduction in Quantum Processing Unit (QPU) usage: Comparing the proposed protocols Alternate and Alt+Sim, with standard PSR we see significant reduction in the number of QPU calls required for finding the correct solution. For Layer 1, reduction of 60% can be seen.
QPU resources required by Alternate (blue) and Alt+Sim (purple) as compared to standard PSR (orange). QPU calls are reduced in all cases using our proposed Alternate and Alt+Sim protocol. Especially in Layer 1 configuration we find a huge 60% reduction in quantum resources - Reduction in error: We compare the error between the solution obtained from the quantum circuit to the exact solution, for standard PSR, Alternate and Alt+Sim. Although the Alternate scheme struggles in some cases, the Alt+Sim scheme can reach significantly lower errors than standard PSR. For Layer 1, up to an order of magnitude reduction can be seen.
Error values reached by Alternate (blue) and Alt+Sim (purple) as compared to standard PSR (orange). Alternate method struggles in some cases, while Alt+Sim always gives lower error than standard PSR. Especially in Layer 1 configuration we find an order of magnitude reduction using Alt+Sim compared to PSR - Mitigation of Barren Plateaus (BP): A major challenge of training VQAs is the exponentially decreasing magnitude of gradients at higher qubits, known as the Barren Plateau phenomenon. Using our proposed quantum ansatz HELIA, we can mitigate this issue and maintain large magnitude of gradients even at high qubits. This is demonstrated by comparing variance of gradients for the two blocks of HELIA using Alternate and standard PSR, with a deep Hardware Efficient Ansatz where BP is known to occur from literature.
Variance of gradient is compared for our proposed HELIA using PSR (dotted line) and Alternate (solid line) protocol, with a deep circuit (green line with label HEA50) when BP is known to occur. The variance for the orange and blue block of HELIA are plotted in their corresponding colors. They are able to maintain larger values consistently at high qubits compared to HEA50 in green line, demonstrating the mitigation of BP phenomenon.
Conclusion
We proposed quantum circuit ansatz and training protocols for VQAs. In a benchmark of finding ground state energy, we are able to
- Reduce quantum cost for training (up to 60% reduction)
- Improve precision (up to an order of magnitude reduction in error)
- Mitigate Barren Plateau phenomenon
VQAs generally face challenges related to quantum resource demands in training. However, variational methods have a proven track record in classical computing and are widely used in applications like machine learning and quantum chemistry. Given the heuristic nature of VQAs, their full potential may unfold with future quantum devices. Moreover, VQAs remain relevant beyond the NISQ era, serving as approximation for non-variational methods like quantum phase estimation. At the current stage, our work does not resolve all current challenges of VQAs to bring them to the same scalability as training classical variational models. However, our method paves the potential way for their practical applications in both near-term and fault-tolerant quantum computing, much like the prominence of variational methods in classical computing
