Hello, I am Carlos BISTAFA, a researcher at the Quantum Laboratory, Fujitsu Research. Fujitsu has been exploring quantum computing at all levels, from developing the necessary hardware to creating associated software, as well as investigating applications of this groundbreaking technology.
In this article, I will discuss the results we obtained from our research collaboration with Fujifilm Corporation, which were presented at the 2025 edition of our event, "Fujitsu Quantum Day," held in March here in our home city, Kawasaki.
Introduction
A promising application of quantum computing is the study of molecules and materials. The Schrödinger equation, which is the cornerstone of Quantum Mechanics, has no analytical solution for systems with more than one electron. Consequently, we must rely on alternative methods to study molecules. While there are highly accurate methods available, such as Coupled Cluster Singles and Doubles (CCSD) and Full Configuration Interaction (FCI), their computational cost increases exponentially. This makes it impractical to calculate large molecules even with classical supercomputers.
In a quantum computer, each electron or vacancy in a system can be represented by a single qubit, simplifying calculations significantly. However, the current generation of quantum computers, referred to as NISQ (Noisy Intermediate-Scale Quantum) devices, are still limited. They have a relatively small number of qubits and are affected by quantum noise. As a result, the systems we can study are small, and the results come with inherent errors.
To develop new algorithms and test systems that are currently beyond the capabilities of NISQ devices, we rely on simulators. These are classical supercomputers dedicated to mimicking the behavior of a quantum computer, such as the Fujitsu Quantum Simulator.
Current status of algorithm development and limitation compared to quantum hardware
Currently, the most commonly used algorithm for calculating molecular properties on a NISQ device is the Variational Quantum Eigensolver (VQE). This algorithm represents the wavefunction of a system using a set of parameters. It calculates the expectation values of the system's Hamiltonian with a NISQ device, and then optimizes these parameters using a classical workstation.
The initial set of parameters in the VQE algorithm is known as the ansatz. There are multiple ways to construct an ansatz, but the most precise method is based on the previously mentioned CCSD technique, known as Trotterized Unitary CCSD (UCCSD). However, its high quantum gate count makes it too complex for use in NISQ devices. A quantum gate (a short for quantum logic gate) is a piece of the quantum circuit used to perform a operation over one or two-qubits. Using the UCCSD ansatz, for instance, it would be necessary around 20 thousand quantum gates to calculate the ground state of the water molecule, not feasible using NISQ devices.
There are other options like the Hardware Efficient (HE) ansatz and the Symmetry Preserving (SP) ansatz that require significantly fewer quantum gates. Nonetheless, little is known about their ability to produce chemically accurate results or the conditions necessary to achieve such accuracy, such as the required depth of the quantum circuit. Therefore, we evaluated the accuracy of these ansätze.
Technical points of our study
Two group of molecules were considered: (i) C6H6 (benzene), C10H8 (naphthalene), C14H10 (anthracene); (ii) LiH, BeH2, H2O, CH4, N2. Hamiltonians for the VQE algorithm were constructed from one- and two-electron integrals derived from a Hartree-Fock/STO-3G calculation. Hartree-Fock (HF) is a fundamental method used to study molecular system with multiple electrons, and STO-3G is a basis set employed to represent the wavefunction/orbitals of the system. In fact, STO-3G is the smallest basis set available.
For the molecules in group (i), a strategy known as active space was used. This approach focuses on the most important molecular orbitals needed to describe the molecular system, specifically the π-orbitals, due to the large number of molecular orbitals present. This resulted in using 12, 20, and 28 qubits, respectively. For the molecules in group (ii), which have fewer electrons, all orbitals were included in the calculations.
Calculations were performed on the Fujitsu quantum simulator, the same used in the Fujitsu Quantum Simulator Challenge. The Qulacs program was used, with additional implementations developed by our team. www.fujitsu.com
The backpropagation (BP) method was employed to accelerate the gradient calculations.
Our findings
First, the usage of the BP method for gradient calculation in parameter optimization enabled faster calculations compared to the usual techniques like the Finite Difference Method (FDM) when the depth of the quantum circuit was large [Figure (1)].
Second, with the HE ansatz, the energy was almost equal to or slightly lower than the HF energy for low-depth circuits, while the results approached MP2 (a second-order perturbative correction to HF) energy values at higher depths. In contrast, for the SP ansatz, the energy values gradually decreased as the depth increased and were lower when compared to the HE ansatz results at the same depth. However, these values remained higher than those obtained from accurate classical methods as CCSD, even at high depths [Figure (2) and (3)].
Finally, the calculations using different initial values resulted in a narrow distribution of the obtained energy values [Figure (4)], which is a consequence of the so-called barren plateau problem. The barren plateau problem occurs when the optimization process struggles to move away from initial guesses, leading to similar results regardless of starting points. This issue is commonly observed in calculations involving variational quantum algorithms, where small gradient values cause local optimizations to become trapped in solutions close to the initial values.
Future Perspectives
The SP ansatz is a promising ansatz for practical usage in quantum devices. Our collaboration intends to explore methods to mitigate the barren plateau problem and analyze its applicability to both ground and excited states.
