Development of a Large-Scale Energy Gap Calculation Method Using Quantum Computers Toward Industrial and Social Implementation

1. Key points of this research

・Developed ”Tensor-based Quantum Phase Difference Estimation (TQPDE)” by combining quantum phase difference estimation, an energy gap calculation method, with circuit compression technology using tensor networks
・Demonstrated a practical application of quantum chemistry calculations on a superconducting quantum computer with error suppression
・Expectated to enable high-resolution understanding of the physical properties of large-scale molecules through the practical use of quantum computers

2. Challenges in electronic state calculations and expectations for quantum computers

The physical properties of a molecule can be determined by calculating the electronic state of the electrons it contains. However, the computational cost of such calculations increases exponentially with the number of electrons, so approximations are currently used in practice. The electronic ground state* is commonly calculated using DFT*, which approximates electronic correlations. While DFT is widely applied, it often fails to achieve sufficient accuracy for materials with complex electronic structures involving strong Coulomb repulsion.

*Ground State: The lowest-energy electronic state of a system.

*DFT (Density Functional Theory): A computational method that approximates the properties of a many-electron system based on electron density, rather than wavefunctions.

Quantum computers are drawing attention as a potential solution to this challenge, as they can perform computations that are intractable for conventional (classical) computers by leveraging quantum entanglement* and superposition*. One prominent algorithm, quantum phase estimation, is known to exponentially accelerate molecular energy calculations. However, due to the high levels of noise in current quantum hardware, large-scale quantum circuits* cannot yet be executed. As a result, quantum phase estimation has so far only been demonstrated on systems of up to 6 bits.

*Quantum Entanglement: A quantum phenomenon in which the states of multiple qubits become interdependent, such that the state of one qubit cannot be described independently of the others.

*Quantum Superposition: A fundamental property of qubits that allows them to exist in a combination of multiple states simultaneously.

*Quantum Circuit: A structured sequence of quantum gates and measurements designed to manipulate the quantum states of qubits.

3. Result of this work

In this study, we adopted quantum phase difference estimation and tensor networks* to reduce the computational cost of quantum computation. Quantum phase difference estimation enables direct calculation of the energy gap—an important physical property— by using a superposition state of the ground and excited states*. Moreover, comparing the circuits for standard quantum phase estimation [Figure 1(a)] and phase difference estimation [Figure 1(b)] reveals that the latter avoids controlled operation on auxiliary bits during the dynamics computation(-iHt), which is a key factor in reducing computational cost.

*Tensor Network: A tensor is a generalized data structure that extends the concept of scalars (0D), vectors (1D), and matrices (2D) to arbitrary dimensions. A tensor network is a graphical structure in which tensors are connected via contracted indices. In this study, quantum gates and related operations are represented as tensors within the network.

*Excited State: An electronic state with higher energy than the ground state.

Building on the quantum phase difference estimation circuit, the circuit was further compressed using a technique known as a tensor network [Figure 1(c)]. This approach searches for shallow circuits that approximate the target circuit using a classical computation. In addition, implementing the state preparation circuit using a tensor network was found to exponentially suppress quantum noise.


The method was implemented on a 32-system qubit Hubbard model [Figure 2(a)] and a 20-qubit decapentaene molecule [Figure 2(b)] using actual quantum hardware. As a result, the measured distributions were observed to converge toward the target energy gap values, confirming that the algorithm functioned correctly—with an accuracy on the order of 10 milliHartrees. This demonstrates that, with the aid of superconducting quantum hardware and error suppression techniques, quantum phase estimation was successfully executed at a scale exceeding five times that of previous implementations.

Figure 1. Overview of the proposed method TQPDE

(a) Circuit diagram of quantum phase estimation (Bayesian type). The top bit represents the ancillary qubit, while the others correspond to the system qubits. An approximate ground state is prepared in Ug, a phase is added in Phase, the controlled dynamics exp(-iHt) is executed, and finally the ancillary bit is measured. The probability that 0 is measured in the X measurement varies depending on the phase (in the form of a cosine function), and the probability peaks when the phase matches the ground state energy. By varying the evolution time t, the peak width changes, allowing the energy value to be progressively refined through iterative measurements.

(b) Circuit diagram of quantum phase difference estimation. Instead of computing absolute energy values, this circuit directly estimates the energy gap between two levels. Controlled state preparation circuits (Ug, Uex, respectively) are used to prepare the ground and excited states, respectively. Unlike standard phase estimation, the dynamics is no longer controlled, resulting in reduced —since state preparation is generally less complex than controlled time evolution.

(c) Circuit diagram of the proposed method. Using a tensor network, we obtain circuits that approximately compress the state preparation circuit and the dynamics circuit on a classical computer (Uprep and Uevol, respectively). Since classical computation can efficiently simulate only short-term dynamics, a short-time unitary Uevol corresponding to evolution over time Δt is first constructed classically,, and then applied repeatedly t/Δt times on the quantum circuit to achieve long-time evolution. The final measurement is performed in the Z basis on all bits, and the probability of obtaining all zeros is used. This measurement approach is found to exponentially suppress quantum noise.

Figure 2. Calculation target and execution result

(a) Hubbard model. The Hubbard model is a simple model that handles electron correlations and is composed of a kinetic energy term t and a Coulomb energy term U. The lower panel shows the calculation results (32-bit system). The red distribution represents the probability distribution sampled across different phase values, with the peak corresponding to the energy gap. As the number of iterations on the horizontal axis increases, the dynamics simulation time increases, and the distribution (i.e., the period of the cosine function) becomes narrower. It can be observed that the distribution progressively converges toward the reference gap value (dashed line) as the iteration proceed, confirming that the proposed method is executable on a actual hardware.

(b) Linear molecule. Although molecular systems involve more complex interactions than the Hubbard model, by applying g orbital localization, execution on a 20-bit decapentaene system was successfully achieved.

4. Expanding the reach of quantum phase estimation toward practical applications

Quantum phase estimation is one of the most fundamental and widely studied algorithms in quantum computing. It is applicable to a broad range of tasks, not only in quantum chemistry but also in areas such as prime factorization and machine learning. Traditionally, quantum phase estimation has been limited to toy models involving a few bits, which can be easily simulated on classical computers. However, the present study significantly extends the applicable scale to sizes approaching the limits of classical computers. This advancement represents a major step toward the practical application of quantum computers.

5. List of co-authors

・Mitsubishi Chemical Corporation: Shu Kanno, Takao Kobayashi, Qi Gao
・Keio University: Kenji Sugisaki (at the time), Hajime Nakamura, Naoki Yamamoto
・SoftBank Corp.: Hiroshi Yamauchi
・JSR Corporation: Rei Sakuma

6. References

・Demonstration Experiment of a Communication Service Fault Diagnosis System Using Quantum Machine Learning, August 30, 2024
・SoftBank Corp. Launches Joint Research with Keio University with View to Business Utilization of Quantum Computers, July 19, 2023
・A joint paper on “a new computational method for energy calculations of photonic materials on quantum computers” was published in a Nature Research Journal, February 9, 2023
・A Joint Paper on Prediction of Optical Properties of OLED Materials was Published in a Nature Research Journal, May 26, 2021
・Joint paper on “a new computational method for a large-scale and high-accuracy quantum chemistry calculation on quantum computers” published in a Nature Research Journal, June 10, 2024

We would like to express our sincere gratitude to Dr. Shu Kanno of Mitsubishi Chemical Corporation for his valuable contributions to the preparation of this manuscript.