A research study led by Aleksei Ivanov of Riverlane, published on March 23, 2023, proposed a quantum algorithm that uses error-corrected quantum computers to conduct ground-state energy calculations for periodic solids. This innovative algorithm is centred on the sparse quantization approach in second quantization and is specifically created for both Bloch and Wannier basis sets. The study showed a promising first step towards practical algorithms for simulating crystalline solids on error-corrected quantum computers.
According to the research, the Wannier functions require fewer computational resources than Bloch functions for two main reasons. Firstly, the L1 norm of the Hamiltonian is significantly lower for Wannier functions, thereby reducing the computational load. Secondly, the translational symmetry inherent in Wannier functions enables the quantum computer to exploit it, thereby minimizing the amount of classical data that needs to be loaded. This breakthrough development enables more efficient ground-state energy calculations for periodic solids in quantum computing.
For an error rate of 0.1%, quantum error correction necessitates the usage of 4-5 times more physical qubits than logical qubits. In the case of small cells, a few thousand logical qubits are required for simulation, while larger supercells would necessitate around 105 logical qubits.
It should be noted that the enhancement in the physical error rate occasionally decreases the number of logical qubits required. This is because the number of logical qubits is the sum of the computational qubits and the magic state factory qubits. The computational qubit count is a characteristic of the system being studied and is not dependent on the error rate of the quantum computer.
Understanding Wannier and Bloch function
Moreover, Wannier and Bloch’s functions produce the same Hamiltonian spectrum since a unitary transformation relates them. However, the properties of the Hamiltonian vary based on the choice of basis functions. The key objective is to select the functions that minimize the total cost of quantum computation. This study explores how the translational symmetry of the Hamiltonian can be leveraged to reduce the quantum resources needed. Other symmetries, such as point group symmetry, can also be considered.
However, the researchers anticipate that this will not result in a ten-fold reduction of T gates, as qubitization-based algorithms scale with the square root of the number of terms. While using Brillouin zone symmetry can lower the cost of quantum algorithms in the Bloch basis set, they anticipated that Wannier representation would provide better resource estimates for moderate-sized systems. By taking advantage of these symmetries, the study aims to optimize the quantum computational cost for Hamiltonians in various basis sets.
Scaling Properties of the Algorithms
Restricted Hartree-Fock calculations in the Gaussian basis set using the PySCF software were carried out to generate electron repulsion integrals (ERIs). The Fermi-Dirac distribution of occupation numbers was utilized in these calculations.
To investigate the scaling properties of the algorithms, the team conducted calculations on a model hydrogen (H) crystal with a body-centered cubic (BCC) structure. Furthermore, the team performed quantum resource estimations for realistic solids such as lithium hydride (LiH), NiO, and PdO. In the case of H and LiH, all-electron calculations were performed with the STO-3G basis set. On the other hand, the team used GTH pseudopotentials with GTH-SZV basis set for NiO and PdO.
Gaussian density fitting was employed to compute the ERIs. This approach involves approximating the four-center integrals in the Hartree-Fock matrix with a linear combination of products of two one-center functions. Using this technique significantly reduces the computational cost of the ERIs calculation, enabling efficient computation of integrals for large systems. Overall, the team showcases a robust approach for generating ERIs in crystalline solids using restricted Hartree-Fock calculations in the Gaussian basis set.
Their quantum resource estimations demonstrate the potential of this approach for realistic solid-state systems. Additionally, using Gaussian density fitting reduces the computational complexity of the ERIs calculation, further enhancing the efficiency of the overall approach.
Exploring Ground-State energy of Crystalline Solids using qubitization-based quantum phase estimation (QPE)
In this study, the researchers have explored estimating the ground state energy of crystalline solids using qubitization-based quantum phase estimation (QPE) on error-corrected quantum computers. Specifically, the study focused on two challenging systems, NiO and PdO, which have relevance in the field of heterogeneous catalysis and are known to pose significant challenges for classical electronic structure methods.
While this work represents a promising first step towards practical algorithms for simulating crystalline solids on error-corrected quantum computers, it is important to note that there is still much to be done to investigate properties beyond ground state energy calculations. Nonetheless, it made significant progress by adapting the qubitization algorithm to solid-state systems and considering the symmetries of the integrals in the Wannier representation. Additionally, the researchers have developed a generalized sparse qubitization approach for use with complex Hamiltonians that are necessary when employing Bloch functions.
Realistic resource estimations have also been presented for error-corrected quantum computers. Specifically, the simulation of crystalline solids in the minimal basis set using the approach presented in this paper would require an order of 10–100 million physical qubits and 1010–1012 T gates. However, it is anticipated that these numbers could be further reduced by utilizing alternative qubitization techniques, such as Density-Fitting or Tensor Hypercontraction, specifically adapted for solid-state systems.
In summary, the study represents an important contribution toward developing practical quantum algorithms for simulating crystalline solids on error-corrected quantum computers. The study has demonstrated the feasibility of estimating ground-state energy of challenging systems using qubitization-based QPE, highlighting the need for continued research into properties beyond ground-state energy and optimization of resource utilization.
Read the full research article here.