Quantum Computational Chemistry Algorithms For Solving Eigenstates Of Molecular Systems

Due to the direct product structure of the state space for quantum systems with multiple particles,accurately describing an arbitrary many-body quantum state requires exponentially increasing storage space as the size of the system grows.In the field of quantum chemistry,precise calculation of eigenstates is often limited to molecular systems with very few atoms.For slightly larger molecules,approximations are usually necessary,but these approximations are not applicable to all systems.For strongly correlated systems,the calculation error caused by such approximations are often unacceptable.So far,the exact calculation of the eigenstates of quantum many-body systems based on classical computation has been a very challenging problem.However,with the gradual development of quantum computing,researchers have started to explore new computing models to tackle problems that are difficult or impossible to solve in using classical computing.Based on the superposition and entanglement effects of qubits,quantum registers are naturally suitable for representing quantum many-body states.Therefore,solving quantum chemistry problems is considered as one of the most promising applications of quantum computing.This dissertation focuses on quantum chemistry methods based on quantum computing,with the core task being to solve the ground and excited states of molecular systems using the framework of quantum computing.The specific research work includes the following aspects:Firstly,this thesis proposes a new and more compact quantum circuit called the kQUpCCGSD ansatz,which is applicable to existing VQE(Variational Quantum Eigensolver)methods.The proposed quantum circuit is a multi-layered quantum network based on generalized single-particle exchange operators and paired double-particle exchange operators.Each operator discards parity checking and only retains the particle exchange property.The corresponding quantum circuit has a complexity of O(1),and the number of exchange operators in each layer is O(M2),where M represents the number of qubits.Therefore,the complexity of the ansatz circuit is only O(kM2),which is smaller than other existing types of ansatz circuits.Numerical simulations show that the newly proposed circuit structure can achieve VQE ground state calculations within the chemical accuracy range with smaller computational costs.As the current technology is still immature and limits the maximum circuit depth that practical quantum processors can support,developing quantum algorithms with lower circuit depth will help demonstrate the practical superiority of quantum computing.This work is believed to bring new value to the practical application of VQE methods on quantum devices.Secondly,the original VQE methods are only applicable to ground state calculations.To broaden the application of the variational principle in VQE to excited state calculations,this thesis proposes a new type of variational quantum algorithm:OSRVE(orthogonal state reduction variational eigensolver).Based on rigorous theoretical derivation,this dissertation proves that by introducing auxiliary qubits and combining the low-order eigenstates calculation circuit parameters,the OSRVE method can remove the contribution of all low-order eigenstates in the output state energy.Based on this,optimizing the quantum circuit can only obtain the target excited state.This method is also applicable to energy degeneracy cases,and numerical simulations have shown its reliability.Compared with other existing quantum computing methods for excited states,the proposed OSRVE method has significant advantages in computing low-order excited states.Its circuit depth increases only linearly as the computation order increases,the number of samples is also acceptable,and it does not require any hyperparameters.This work extends the theoretical framework of VQE to excited state calculations,expanding the boundaries of variational quantum algorithms in addressing quantum chemistry problems.Thirdly,all variational quantum algorithms encounter limitations in the state representation of quantum circuits and face the challenges of complex nonlinear optimization,potentially leading to inaccurate calculation results for VQE and OSRVE methods.To enhance the accuracy of variational quantum algorithms,this thesis introduces the VQE/OSRVE-FNFCIQMC method.This approach employs the output of the VQE or OSRVE method as a guiding wave function and carries out fixed-node full configuration interaction quantum Monte Carlo to achieve more accurate results.Numerical simulations have demonstrated that this method can boost the accuracy of VQE or OSRVE calculations by 30%to 60%.Moreover,the VQE/OSRVE-FN-FCIQMC method offers additional possibilities for selecting guiding wave functions for quantum Monte Carlo methods.Although corresponding guide wave functions can be constructed using classical computers,qubits possess superior representation capabilities,particularly for excited state calculations.Presently,most wave functions built on classical computers are only applicable to the ground state,with few effective forms for excited state wave functions.Nevertheless,the OSRVE method in quantum computing effectively addresses this issue,significantly enriching and improving the performance of quantum Monte Carlo methods.Fourthly,this thesis introduces a novel non-variational classical-quantum hybrid algorithm,the PSHO(power of sine Hamiltonian operator)method,which tackles some of the challenges inherent in variational quantum algorithms.The PSHO method can compute the normalized energy of the sinn(Hτ)|ψ0>state for any Hamiltonian H and initial reference state |ψ0>.As the power n increases with different τ values,the sinn(Hτ)|ψ0>state converges to different eigenstates of the Hamiltonian.More specifically,it converges to the eigenstate with the maximum value of |sin(Eiτ)| and non-zero overlap with the initial reference state,enabling the calculation of eigenstate energies for molecular systems.Compared to the currently popular variational quantum algorithms,the PSHO method has a universal and standard process for designing the corresponding quantum circuit,which can be applied to any molecular system.In contrast,algorithms like VQE lack a universal method to ensure the accurate preparation of target eigenstates for any molecular system,and any type of quantum circuit structure may only be applicable to specific molecular systems,lacking generalizability.Moreover,variational quantum algorithms face complex nonlinear optimization problems,and finding the global optimal solution to the optimization problem is often difficult,which also affects the accuracy of results.However,the PSHO method effectively avoids these problems,which provides a beneficial perspective and direction for the development of quantum computational chemistry.In conclusion,this dissertation proposes a series of novel quantum computing algorithms that significantly broaden the scope and potential of existing quantum chemistry methods.These advancements offer valuable insights for the practical application of quantum computing in the field of quantum chemistry.