In recent years, there have been significant developments in the field of quantum algorithms, especially in applications involving quantum chemistry and representations of molecular Hamiltonians. However, existing methods for simulating quantum chemistry (especially those leveraging simple basis functions like plane waves) are not feasible upon scaling to the continuum limit. The reason being that the majority of quantum computing algorithms for chemistry are based on second quantized simulations where the required number of qubits scales linearly with the number of spin-orbital basis functions.
To overcome this limitation, first quantized simulations of chemistry have been proposed where the idea is to track the basis state each particle is in, instead of storing the occupancy of each basis state. This is comparatively advantageous as the number of qubits needed to represent the state, scale logarithmically with the number of basis functions. Also, such simulations are highly adaptive in cases where the entanglement between the electronic and nuclear subsystems is non-negligible.
In this work, the authors analyze the finite resources required to implement two first quantized quantum algorithms for chemistry; block encodings for the qubitization and interaction picture frameworks. The objective is to compile and optimize these algorithms using a plane wave basis within an error-correctable gate set as well as developing new techniques to reduce the complexity. For qubitization of the Hamiltonian, control registers are used to select the momenta as well as the individual bits to multiply, which significantly decreases the multiplication cost. The results show that the qubitization algorithms requires much less surface code spacetime volume for simulating millions of plane waves as compared to the best second quantized algorithms require for simulating hundreds of Gaussian orbitals. In case of interaction picture based, the cost of computing the total kinetic energy is reduced by computing the sum of squares of momenta at the beginning.
The work shows that the total cost associated with the state preparation cost is reduced by a factor of 3, when one assumes that the state preparation is only needed for the kinetic energy operator, rather than using the standard amplitude amplification for the total energy. The number of bits needed for the discretization of the time intervals is also reduced by a factor of two as indicated by prior well-established methods. The main complexity is therefore focused on selecting the momenta registers. Unlike the previous approaches, only the dynamic degrees of freedom for electrons that participate in reactivity are employed by the algorithm, hence reducing encoding complexity. This approach is particularly more advantageous for a high number of electrons for the qubitization and despite the improved scaling of the interaction picture based method, the qubitization based method proves to be comparatively more practical. Also, the numerical experimentation reveals that these approaches require significantly fewer resources to reach comparable accuracy as compared to second quantized methods. Another interesting finding is that the first quantized approaches developed here, may give lower Toffoli complexities than previous work for realistic simulations of both material Hamiltonians and non-periodic molecules, suggesting more fault tolerant viability than second quantized methods.
This work provides the first explicit circuits and constant factors for simulation of any first quantized quantum algorithm, which can be a promising direction for simulating realistic materials Hamiltonians within quantum error-correction. But perhaps more impressively, the authors have also improved the constant factors; the results demonstrate reduction in circuit complexity by about a thousandfold as compared to naive implementations for modest sized systems. It also provides insights on the resources required to simulate various molecules and materials and gives the first impression of the first quantization-based algorithm for preparation of the eigenstates for phase estimation with required overlap with another eigenstate. This suggests many potential advantages in contrast to the second quantization-based algorithms, as the case in the continuum limit.
In addition to the progress made by the current work, many potential further improvements on the approach have already been proposed by the authors as well, such as modifying the algorithm to encode pseudopotential specific to the problem - relating it to bonds and chemical reactivity. Furthermore, using a non-orthogonal reciprocal lattice and efficient convergence in the thermodynamic limit might be a good direction for future research. The presented algorithms could also be adapted to more natively suit systems of reduced periodicity even thought the underlying representation is based on plane waves. The authors acknowledge relatively little is known about state-preparation in first-quantization as compared to second-quantization. Future work may investigate whether straightforward translations form the latter body of work to the current will be sufficiently efficient. Although this work focuses on preparing eigenstates for energy estimation, it can also be extended for simulation of (non-)Born-Oppenheimer dynamics.
The paper offers an interesting insight and significant progress in designing first-quantization picture quantum chemistry simulations on fault-tolerant hardware. Several strong advantages as compared to second-quantization methods are highlighted, and it will be interesting to see how this field evolves further.