#QuantumTechnology #QuantumCircuit #ClassiqQmod
Jul 02, 2025
SoftBank Corp.
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Quantum computing has attracted significant attention in recent years as a means of solving quantum many-body problems*1 in fields such as chemistry and materials science. Notably, quantum algorithms offer promising new approaches to estimating ground-state energies of molecules—tasks that are exceedingly difficult for classical computers.
It is important to note that the term quantum circuit used here does not refer to physical wiring, as in conventional electronic circuits. Instead, it denotes a sequence of operations—quantum gates—applied to qubits in a specific order, much like a musical score prescribes the sequence in which notes are played. In this sense, executing a quantum circuit is analogous to plucking the strings of a guitar in a predefined pattern, with gate operations applied to each qubit in turn.
However, each gate operation is associated with a non-negligible probability of error, and these errors accumulate as the number of operations increases. This makes shallow circuits—with fewer gate operations—crucial for practical computation, especially on today’s Noisy Intermediate-Scale Quantum (NISQ) devices, where error rates remain a significant bottleneck.
Among the various quantum algorithms, Quantum Phase Estimation (QPE) is widely recognized as an effective method for estimating ground-state energies. Achieving higher estimation precision with QPE necessitates increasing the number of ancillary (ancilla) qubits. However, this increase leads to a proportional growth in circuit depth and gate count, which in turn makes execution on NISQ hardware increasingly difficult. Thus, there exists a fundamental trade-off between computational accuracy and hardware feasibility.
In light of this challenge, SoftBank's Research Institute of Advanced Technology conducted a study on the estimation of ground-state energies of hydrogen chain molecules, investigating how compression and optimization of QPE circuits affect executability, precision, and computational cost. The experiments were performed on Reimei, an ion-trap quantum computer developed by Quantinuum, and three software development kits (SDKs)—IBM Qiskit, Quantinuum TKET, and Classiq Qmod—were used for circuit generation and optimization. The performance of each SDK was quantitatively evaluated in terms of circuit compression efficiency and job execution cost.
Quantum computers remain an extremely resource-intensive computational platform, and their usage is constrained by the limitations of current hardware performance. Therefore, optimizing quantum circuits to minimize depth is essential for expanding the range of executable computations and enabling high-precision outcomes.
This study demonstrates that circuit compression is a critical factor in determining the accuracy, cost-efficiency, and practical feasibility of quantum computations. Moreover, the choice of SDK plays a decisive role in enabling the practical implementation of quantum algorithms.
This article is based on results obtained from a project commissioned by the New Energy and Industrial Technology Development Organization (NEDO).
*1 Quantum many-body problems involve understanding and predicting the behavior of systems with many interacting quantum particles, whose complexity grows exponentially with system size—making them extremely challenging for classical computers to solve.
In February 2025, Quantinuum’s quantum computer Reimei was installed on-premises at RIKEN’s Wako campus in Japan [1]. This marked the first instance of an on-premises deployment of a Quantinuum quantum system outside the United States. Reimei is a 20-qubit ion-trap quantum computer based on Quantinuum’s System Model H1. It is characterized by full qubit-to-qubit connectivity, long coherence times, and high gate fidelities, offering strong inherent error resilience. In this study, we conducted experimental quantum computations using Reimei, leveraging these advanced hardware capabilities.
The one-dimensional hydrogen chain—a system in which hydrogen atoms are linearly aligned—is structurally simple yet serves as a benchmark model in quantum chemistry and condensed matter physics due to strong electron correlations and quantum phase transition phenomena. In this study, we used this hydrogen chain as a model system, converting it into a form suitable for quantum computation and then estimating its ground-state energy to verify computational accuracy and algorithmic behavior.
To simulate the electronic behavior, we employed the STO-3G*2 basis set, one of the most basic approximation methods. The resulting fermionic Hamiltonian*3 was then transformed into a qubit representation using the Jordan-Wigner transformation*4, enabling implementation on a quantum computer.
When using the STO-3G basis set, one spatial orbital is assigned to each hydrogen atom. Each of these spatial orbitals accommodates two electronic states with opposite spins (spin-up and spin-down), resulting in two spin orbitals per hydrogen atom. On a quantum computer, each spin orbital corresponds to one qubit; therefore, for a system with N spatial orbitals, 2N qubits are required.
Although exploiting system symmetries or introducing active spaces*5 can reduce the number of qubits by retaining only essential electron correlation effects, such simplifications were intentionally avoided in this study. Instead, we adopted the most rigorous approach, equivalent to Full Configuration Interaction (Full-CI)*6, to assess how accurately compressed circuits can estimate energy values on real quantum hardware.
*2 The STO-3G basis set is one of the most fundamental basis sets used in molecular orbital-based quantum chemistry calculations. It approximates a Slater-type orbital (STO) using a linear combination of three Gaussian-type functions.
*3 A fermionic Hamiltonian is an energy operator used to describe systems composed of fermions—particles that obey Fermi-Dirac statistics, such as electrons.
*4 The Jordan-Wigner transformation is a mathematical technique used to simulate fermionic systems on quantum computers. It maps fermionic creation and annihilation operators to strings of Pauli operators on qubits, allowing direct implementation within quantum circuits.
*5 An active space is a simplification strategy in electronic structure calculations in which only the orbitals and electrons most relevant to the system’s physical behavior are included, thereby reducing the complexity of the system.
*6 The Full Configuration Interaction (Full-CI) method is one of the most accurate approaches in quantum chemistry for accounting for electron–electron interactions. Within a specified basis set, it considers all possible electron configurations (Slater determinants) to compute the system’s energy and wavefunction exactly.
Quantum Phase Estimation (QPE)[2] is a quantum algorithm that computes the eigenvalues of a unitary operator. Given a unitary operator U and one of its eigenstates |ψ⟩, the relationship
holds. QPE expresses as a binary fraction:
The Quantum Phase Estimation (QPE) subroutine involves operations analogous to those used in the Hadamard test. Specifically, it encodes the eigenvalues of powers of a unitary operator U2k into the phases of n ancillary qubits. These phase-encoded qubits are then processed via the Inverse Quantum Fourier Transform (IQFT)*7 to extract the phase information. This procedure generates a quantum state of the form |φ1 φ2 ・・・ φn⟩, where each qubit encodes a binary digit of the eigenvalue phase. By measuring the ancilla qubits, one can recover the eigenvalue of the unitary operator.
As the number of ancilla qubits increases, the number of binary digits representing the phase also increases, improving the precision of the phase estimation. However, this comes at the cost of exponentially increasing circuit depth, making the computation more susceptible to gate-induced errors. This illustrates a fundamental trade-off between estimation accuracy and robustness to noise. The quantum circuit for this subroutine is shown in Figure 1.
Figure 1: Quantum circuit for the Quantum Phase Estimation (QPE) subroutine.
*7 The Quantum Fourier Transform (QFT) is an operation that performs a Fourier transform on numerical data encoded in a register of quantum bits. Analogous to the classical Fast Fourier Transform (FFT), the QFT is distinguished by its ability to act in parallel on quantum bits in superposition. The Inverse Quantum Fourier Transform (IQFT) is the reverse operation of the QFT.
The primary objective of this study was to estimate the ground-state energy of the hydrogen chain using QPE. Quantum circuits were generated and optimized using SDKs provided by IBM, Quantinuum, and Classiq. We first compared the compression performance of these tools—i.e., how compactly they could represent the same algorithm.
Next, we executed the circuits on the Quantinuum Reimei quantum computer to evaluate the precision of the resulting energy estimates.
QPE employs auxiliary qubits known as ancilla qubits. Increasing the number of these qubits allows for a greater number of significant digits in the phase estimation, thereby enhancing its precision. In this study, we conducted experiments by varying the number of ancilla qubits from two to six. To avoid excessive use of quantum computational resources, the number of ancilla qubits and measurement shots was adjusted based on the size of the hydrogen chain. Specific settings are detailed in Section C on job execution cost.
For execution on actual hardware, circuits were compiled using Quantinuum’s TKET and Classiq’s Qmod tools. Evaluation was based on two main criteria: (1) job execution cost and (2) accuracy of energy estimation.
For each hydrogen chain configuration, QPE circuits were constructed using 2–6 ancilla qubits. Three SDKs were used: IBM Qiskit, Quantinuum TKET, and Classiq Qmod.
Each generated circuit was optimized using the respective SDK’s compiler. The backend is configured to use a simulator. The number of CX (CNOT) gates was counted and shown in Figures 2–4. Missing data points indicate unsuccessful compilation under specific conditions.
Figure 2: Number of CX gates for H2.
Figure 3: Number of CX gates for H3.
Figure 4: Number of CX gates for H4.
QPE circuits contain iterative gate operations (i.e., power-of-gate structures). To address these efficiently, we also tested a “flexible” circuit architecture optimized for such repetitions, indicated accordingly in the figures.
As a representative example, Figures 5-7 present the full QPE circuits constructed using four ancilla qubits for the H2molecule.
Overall, circuits generated with TKET and Qmod were more compact than those from Qiskit. Notably, Qmod in its flexible configuration achieved significantly better compression, especially as the number of ancilla qubits—and thus circuit depth—increased.
Figure 5: QPE circuit for the H2 molecule with 4 ancilla qubits compiled using Qiskit.
Figure 6: QPE circuit for the H2 molecule with 4 ancilla qubits compiled using TKET.
Figure 7: QPE circuit for the H2 molecule with 4 ancilla qubits compiled using Qmod(flexible).
Job execution cost on the Quantinuum quantum computer is measured in Hardware Quantum Credits (HQC), computed as follows[3]:
where N1q, N2q, and Nm denote the number of single-qubit gates, two-qubit gates, and state preparation and measurement (SPAM) operations, respectively, and C is the number of measurement shots.
To conserve quantum resources, job settings for each molecule were as follows:
・H2：1000 shots; 2-6 ancilla qubits
・H3：1000 shots; 2-4 ancilla qubits
・H4：500 shots; 2-4 ancilla qubits
Table 1 presents HQC usage across different molecules. Qmod yielded lower job execution costs by reducing circuit depth.
Table 1: HQC usage for each hydrogen chain molecule. H2 and H3: 1000 shots; H4 : 500 shots.
In this study, three software development kits (SDKs)—Qiskit, TKET, and Qmod—were used for comparing quantum circuit compilation performance. However, for execution on actual hardware, we limited the scope to TKET and Qmod. This decision was based on the fact that Qiskit is primarily designed for IBM quantum processors and lacks direct execution compatibility with Quantinuum hardware.
From a technical standpoint, it is possible to export quantum circuits created in Qiskit into the OpenQASM format and subsequently convert and recompile them into a Quantinuum-compatible form via TKET or similar tools. However, circuits constructed in Qiskit are not designed with the native gate set or full qubit connectivity of Quantinuum hardware in mind. As a result, such compiled circuits are likely to be suboptimal, leading to inefficient execution and unrepresentative benchmarking results.
For these reasons, Qiskit was used exclusively for logical circuit construction and compilation performance comparison. Actual hardware execution was conducted only with TKET and Qmod, which are specifically designed to accommodate the architectural characteristics of Quantinuum quantum devices.
Figures 8-10 show energy estimation results from both simulator (aer-simulator) and the Reimei quantum computer, using circuits generated by Qmod and TKET. Black horizontal lines represent reference (true) energy values.
For example, in Figure 8, Qmod’s circuits showed improved estimation accuracy with increasing ancilla qubits. TKET's circuits also achieved accurate estimates up to five ancilla qubits, but estimation significantly deteriorated at six.
This discrepancy is attributed to the two-qubit gate error rate of the actual quantum hardware. For the Reimei quantum computer used in this study, the two-qubit gate error rate is reported as p2=1.41×10-3, from which the practical upper limit for executable two-qubit gates is estimated to be approximately 700. As shown in Figure 2, Qmod’s circuit with six ancilla qubits stayed within this limit, whereas TKET’s exceeded it, resulting in cumulative errors and degraded accuracy.
Thus, Qmod enabled high-fidelity energy estimation by compressing circuits sufficiently to remain within the hardware's gate limit.
Figures 9 and 10 reveal that, for larger molecules (H3, H4), both Qmod and TKET circuits exceeded the 700 two-qubit gate threshold. Consequently, estimation results varied widely and became unstable, as evidenced in the figures.
Furthermore, Figures 3 and 4 indicate that, for the H3 and H4 molecules, the number of two-qubit gates exceeds 700 in both types of circuits. Such large-scale circuits are difficult to execute accurately on current quantum hardware, and as shown in Figures 9 and 10, the estimated energy values exhibit significant variability and instability.
Figure 8: Estimated energy values for H2.
Figure 9: Estimated energy values for H3.
Figure 10: Estimated energy values for H4.
In this study, we evaluated the performance of quantum circuit compression and execution through the task of estimating the ground-state energy of hydrogen chain molecules using the Quantum Phase Estimation (QPE) algorithm. Quantum circuits were generated and optimized using three software development kits (SDKs): IBM Qiskit, Quantinuum TKET, and Classiq Qmod. These circuits were then executed on the Quantinuum ion-trap quantum computer Reimei, enabling a quantitative comparison of differences among the SDKs.
The results revealed that, in particular, Classiq Qmod was capable of maintaining a manageable number of gates even as the number of ancilla qubits increased. This enabled a favorable balance between high-precision energy estimation and low execution cost. It was shown that the choice of SDK significantly affects circuit compression performance, computational resource consumption, and the accuracy of the results. Therefore, SDK selection is a critical factor directly influencing the feasibility and reliability of quantum computations.
The advantages of circuit compression observed in this study are not limited to QPE but are expected to be broadly applicable to other quantum algorithms as well. Future work will involve applying similar methodologies to different algorithms, with the aim of analyzing how characteristics of circuit design and gate arrangement impact compression performance.
Furthermore, while the present investigation targeted circuits intended for NISQ devices without quantum error correction, future studies will expand the scope to include error-corrected circuits. This will enable us to evaluate the effectiveness of circuit compression under fault-tolerant conditions, thereby contributing to the development of practically viable quantum algorithms for real-world deployment.
[1] Quantinuum.Quantinuum’s “Reimei”Quantum Computer Now Fully Operational at RIKEN, Ushering in a New Era of Hybrid Quantum High-Performance Computing, 2025.
[2] Michael A. Nielsen and Isaac L. Chuang. Quantum Computation and Quantum Information: 10th Anniversary Edition. Cambridge University Press, 2010.
[3] Microsoft. Pricing plans for Azure Quantum providers.
Rei Nishimura