Achieving chemical accuracy with shallow quantum circuits is a significant challenge in quantum computational chemistry, particularly for near-term quantum devices. In this work, we present a Clifford-based Hamiltonian engineering algorithm, namely CHEM, that addresses the trade-off between circuit depth and accuracy. Based on a variational quantum eigensolver and hardware-efficient ansatz, our method designs the Clifford-based Hamiltonian transformation that (1) ensures a set of initial circuit parameters corresponding to the Hartree–Fock energy can be generated, (2) effectively maximizes the initial energy gradient with respect to circuit parameters, (3) imposes negligible overhead for classical processing and does not require additional quantum resources, and (4) is compatible with any circuit topology. We demonstrate the efficacy of our approach using a quantum hardware emulator, achieving chemical accuracy for systems as large as 12 qubits with fewer than 30 two-qubit gates. Our Clifford-based Hamiltonian engineering approach offers a promising avenue for practical quantum computational chemistry on near-term quantum devices.
Achieving chemical accuracy with shallow quantum circuits is a significant challenge in quantum computational chemistry, particularly for near-term quantum devices. In this work, we present a Clifford-based Hamiltonian engineering algorithm, namely CHEM, that addresses the trade-off between circuit depth and accuracy. Based on a variational quantum eigensolver and hardware-efficient ansatz, our method designs the Clifford-based Hamiltonian transformation that (1) ensures a set of initial circuit parameters corresponding to the Hartree–Fock energy can be generated, (2) effectively maximizes the initial energy gradient with respect to circuit parameters, (3) imposes negligible overhead for classical processing and does not require additional quantum resources, and (4) is compatible with any circuit topology. We demonstrate the efficacy of our approach using a quantum hardware emulator, achieving chemical accuracy for systems as large as 12 qubits with fewer than 30 two-qubit gates. Our Clifford-based Hamiltonian engineering approach offers a promising avenue for practical quantum computational chemistry on near-term quantum devices.
Quantum computing has emerged as a transformative approach for tackling complex problems in quantum chemistry, particularly in simulating multielectron systems and electron-phonon interactions. However, the current noisy intermediate-scale quantum (NISQ) devices face significant challenges in implementing practical quantum algorithms due to error accumulation caused by increased circuit depth, qubit counts, and gate operations. To address these challenges, we present TenCirChem, an open-source Python package specifically designed for implementing variational quantum algorithms in quantum computational chemistry. TenCirChem demonstrates exceptional performance in simulating unitary coupled-cluster circuits through its innovative use of compact representations for quantum states and excitation operators. This package supports noisy circuit simulation and provides advanced algorithms for variational quantum dynamics, enabling researchers to explore complex chemical phenomena. Its capabilities are exemplified in various applications, including the calculation of potential energy curves and the investigation of quantum gate error impacts on molecular systems. Moreover, TenCirChem’s seamless integration with real quantum hardware makes it a versatile tool for both simulation and experimentation. A key innovation developed within the TenCirChem framework is the Clifford-based Hamiltonian engineering approach for molecules (CHEM). This algorithm addresses the critical challenge of achieving chemical accuracy with shallow quantum circuits, a fundamental requirement for practical applications on NISQ devices. CHEM employs a sophisticated Clifford-based Hamiltonian transformation that operates within the variational quantum eigensolver (VQE) framework using hardware-efficient ansatz. The method ensures four crucial advantages: (1) generation of initial circuit parameters corresponding to Hartree-Fock energy, (2) maximization of initial energy gradients with respect to circuit parameters, (3) minimal classical processing overhead without additional quantum resource requirements, and (4) compatibility with any circuit topology. Through quantum hardware emulator demonstrations, CHEM has achieved chemical accuracy for systems up to 12 qubits with fewer than 30 two-qubit gates, representing a significant advancement in practical quantum computational chemistry. To enhance the efficiency of variational quantum algorithms, we developed the sequential optimization with approximate parabola (SOAP) method, specifically designed for parameter optimization in unitary coupled-cluster ansatz. SOAP addresses the critical bottleneck of measurement requirements in VQE by implementing an innovative optimization strategy that approximates the energy landscape as quadratic functions. This approach minimizes the number of energy evaluations while incorporating parameter correlations through the integration of average directions from previous iterations. Benchmark studies demonstrate SOAP’s superior performance, showing faster convergence and enhanced noise robustness compared to traditional optimization methods. The method’s scalability has been validated through numerical simulations of up to 20 qubits, and its practical efficacy has been confirmed through experiments on superconducting quantum computers. For simulating electron-phonon systems, we introduce a variational basis state encoding algorithm that significantly reduces resource requirements compared to conventional unary and binary encoding schemes. Our approach achieves smaller scaling than traditional methods for qubits and gates for systems obeying the area law of entanglement entropy, this remarkable reduction in resource requirements comes at the cost of a constant amount of additional measurements. The algorithm’s effectiveness has been validated through both numerical simulations and quantum hardware experiments, demonstrating that using just one or two qubits per phonon mode can produce quantitatively accurate results across various coupling regimes. The integration of these innovations within the TenCirChem library represents a significant advancement in quantum computational chemistry. The software package provides researchers with a comprehensive toolkit for developing, testing, and implementing quantum algorithms, while the novel methods address fundamental challenges in circuit depth, parameter optimization, and resource efficiency. These developments collectively enhance the practicality of quantum computing for addressing real-world chemical problems in the NISQ era, offering improved accuracy, efficiency, and noise robustness.
Recent breakthroughs have opened the possibility of intermediate-scale quantum computing with tens to hundreds of qubits, and shown the potential for solving classical challenging problems, such as in chemistry and condensed matter physics. However, the high accuracy needed to surpass classical computers poses a critical demand on the circuit depth, which is severely limited by the non-negligible gate infidelity, currently around 0.1%-1%. The limited circuit depth places restrictions on the performance of variational quantum algorithms (VQA) and prevents VQAs from exploring desired nontrivial quantum states. To resolve this problem, we propose a paradigm of Schrödinger-Heisenberg variational quantum algorithms (SHVQA). Using SHVQA, the expectation values of operators on states that require very deep circuits to prepare can now be efficiently measured by rather shallow circuits. The idea is to incorporate a virtual Heisenberg circuit, which acts effectively on the measurement observables, into a real shallow Schrödinger circuit, which is implemented realistically on the quantum hardware. We choose a Clifford virtual circuit, whose effect on the Hamiltonian can be seen as efficient classical processing. Yet, it greatly enlarges the state's expressivity, realizing much larger unitary t designs. Our method enables accurate quantum simulation and computation that otherwise are only achievable with much deeper circuits or more accurate operations conventionally. This has been verified in our numerical experiments for a better approximation of random states, higher-fidelity solutions to the XXZ model, and the electronic structure Hamiltonians of small molecules. Thus, together with effective quantum error mitigation, our work paves the way for realizing accurate quantum computing algorithms with near-term quantum devices.
Quantum simulation of chemical systems is one of the most promising near-term applications of quantum computers. The variational quantum eigensolver, a leading algorithm for molecular simulations on quantum hardware, has a serious limitation in that it typically relies on a pre-selected wavefunction ansatz that results in approximate wavefunctions and energies. Here we present an arbitrarily accurate variational algorithm that, instead of fixing an ansatz upfront, grows it systematically one operator at a time in a way dictated by the molecule being simulated. This generates an ansatz with a small number of parameters, leading to shallow-depth circuits. We present numerical simulations, including for a prototypical strongly correlated molecule, which show that our algorithm performs much better than a unitary coupled cluster approach, in terms of both circuit depth and chemical accuracy. Our results highlight the potential of our adaptive algorithm for exact simulations with present-day and near-term quantum hardware.
Quantum computing is emerging as a new computational paradigm with the potential to transform several research fields, including quantum chemistry. However, current hardware limitations (including limited coherence times, gate infidelities, and limited connectivity) hamper the straightforward implementation of most quantum algorithms and call for more noise-resilient solutions. In quantum chemistry, the limited number of available qubits and gate operations is particularly restrictive since, for each molecular orbital, one needs, in general, two qubits. In this study, we propose an explicitly correlated Ansatz based on the transcorrelated (TC) approach, which transfers -- without any approximation -- correlation from the wavefunction directly into the Hamiltonian, thus reducing the number of resources needed to achieve accurate results with noisy, near-term quantum devices. In particular, we show that the exact transcorrelated approach not only allows for more shallow circuits but also improves the convergence towards the so-called basis set limit, providing energies within chemical accuracy to experiment with smaller basis sets and, therefore, fewer qubits. We demonstrate our method by computing bond lengths, dissociation energies, and vibrational frequencies close to experimental results for the hydrogen dimer and lithium hydride using just 4 and 6 qubits, respectively. Conventional methods require at least ten times more qubits for the same accuracy.
One of the most significant challenges in quantum chemistry simulations is the accurate and efficient treatment of strongly correlated systems using classical computers. Quantum computational chemistry is an emerging field that has grown alongside the development of quantum computers, aiming to tackle challenges beyond the reach of traditional computational chemistry methods. Given the current limitations of quantum hardware, hybrid quantum-classical algorithms have become the dominant approach in quantum computational chemistry, with the Variational Quantum Eigensolver (VQE) serving as a representative example. In the realm of variational quantum algorithms, the design of the ansatz plays a pivotal role, as it directly determines both the accuracy of the results and the optimization efficiency of the algorithm, making it a central component in achieving success in quantum computational chemistry. However, existing ansätze often fails to strike a balance between precision and hardware efficiency, posing a major bottleneck for practical applications. In our previous work, we proposed a novel strategy for designing hardware-efficient ansätze with built-in physical constraints (PCHEA). This approach demonstrated outstanding performance in simulating one-dimensional Heisenberg models and several typical molecular systems. By embedding physical insights into the design, PCHEA not only improved computational accuracy but also significantly enhanced hardware compatibility, making it a promising candidate for near-term quantum devices. Nevertheless, challenges remain. When applying the one-dimensional PCHEA to two-dimensional systems, the circuit costs increase significantly. To address this challenge, in this work, we simplify the implementation of the physics-constrained hardware-efficient ansatz, using fSim gates arranged in hardware-efficient linear and brickwall layouts to construct one-dimensional fSim linear and fSim brickwall ansätze. This modification substantially reduces both the number of two-qubit gates and the circuit depth. These ansätze ensure universality, systematic improvement, and size consistency within a particle-number-conserving Hilbert space, with significant improvements observed in the simulation of the one-dimensional Heisenberg model. Furthermore, building on the advancements in quantum hardware connectivity, we extend the one-dimensional PCHEA to a two-dimensional version by arranging it in a grid pattern. The two-dimensional PCHEA not only retains the physical constraints of the one-dimensional PCHEA but also exploits hardware connectivity to further reduce circuit costs. Simulations on two-dimensional Heisenberg models show that the two-dimensional PCHEA achieves chemical accuracy with considerably lower circuit depth, fewer parameters, and fewer two-qubit gates compared to the one-dimensional PCHEA. This work reinforces the effectiveness of designing hardware-efficient ansätze with embedded physical constraints, presenting a promising approach to advancing the practical application of quantum computers for simulating two-dimensional systems. Such progress brings us closer to achieving more efficient and scalable quantum simulations while addressing key challenges in quantum computational chemistry.
Recently (Physica Scripta, 100(10):105401, 2025), an algorithm was introduced that deterministically generates a Clifford transformation from the Qubit Coupled Cluster (QCC) algorithm which we call Q-Cliff (QCC+Clifford). There, it was shown that Q-Cliff could be utilized to generate a hardware efficient version of the QCC ansatz. Here, we examine and refine these techniques and show that Q-Cliff can be utilized to generate efficient classical and quantum approximations to the ground states of chemical systems. The algorithm generates an efficient variational method that generally has accuracy between MP2 and CISD with O(N6). Furthermore, we show through DMRG calculations that the entanglement between qubits is reduced significantly and therefore the accuracy for a given bond dimension can be vastly improved (up to an order of magnitude). Finally, we refine the previously reported algorithm to generate low-depth and CNOT efficient circuits that can be optimized with a comparable number of energy evaluations to state-of-the-art VQE algorithms. All these results show that this Hamiltonian derived Clifford transformation should be a tool used for many classical and quantum algorithms.