Quantum Phase Estimation Enables Advanced Molecular Simulation in Future Quantum Computers

The article discusses a proof of concept for ab initio molecular simulation using a hybrid approach that could be realized by future fault-tolerant quantum computers. The method involves approximating basic equations by polynomials, transforming these to a specific form, and determining unknown variables as eigenvalues of transformation matrices via quantum phase estimation. The authors also detail a method of quantum simulation of materials using numeric, symbolic, and quantum algorithms. The quantum phase estimation computes the eigenvalues of transformation matrices, with results encoded in a quantum state. The authors demonstrate how to apply this computational scheme.

What is the Feasibility of First Principles Molecular Dynamics in Fault-Tolerant Quantum Computer by Quantum Phase Estimation?

The article presents a proof of concept regarding the feasibility of ab initio molecular simulation, where the wavefunctions and the positions of nuclei are simultaneously determined by the quantum algorithm. This is realized by the so-called Car-Parrinello method of classical computing. The approach used in this article is of a hybrid style, which shall be realized by future fault-tolerant quantum computers.

The basic equations are approximated by polynomials. These polynomials are then transformed to a specific form wherein all variables representing the wavefunctions and the atomic coordinates are given by the transformations acting on a linear space of monomials with finite dimension. The unknown variables could be determined as the eigenvalues of those transformation matrices. The eigenvalues are determined by quantum phase estimation. Following these three steps, namely symbolic, numeric, and quantum steps, we can determine the optimized electronic and atomic structures of molecules.

How Does the Quantum Simulation of Materials Work?

The authors of the article have developed a method of quantum simulation of materials, wherein numeric, symbolic, and quantum algorithms are employed. This approach uses several steps. The molecular integrals are given by analytic functions generated from analytic atomic bases such as GTO or STO through the standard method of computational chemistry. The energy is given by an analytic function. It is a multivariate polynomial where the variables represent the coefficients of LCAO.

A system of polynomial equations is derived according to the minimum condition of the energy functional. Those polynomials are represented by the coefficients of LCAO, the orbital energies, and other variables, say the atomic coordinates. The polynomials in the system of equations compose an ideal I in a suitable commutative ring RCx1 x2 x n where C is the number field. Then the Gröbner basis G for I can be computed. G and I are equivalent in such a way that the zeros of these two systems depict the same geometric object in the Cartesian coordinate space.

What is the Role of the Quantum Phase Estimation?

Quantum phase estimation (QPE) computes the eigenvalues of those transformation matrices. The common eigenvector vj of those matrices is encoded in the quantum states, and the eigenvalues of the matrices ξj ll are successively recorded in the ancillary component. Namely, the computation’s result shall be encoded in a quantum state.

As the transformation matrices are not Hermitian, their time evolution should be executed by special quantum circuits wherein a matrix A is embedded in the leading principal block of a larger unitary matrix U acting on the full Hilbert space. This sort of unitary operation is realized in a quantum circuit by the trick of block encoding, as is illustrated in Figure 1. The block encoding of A in U enables us to put an arbitrary state vector in the input and execute the matrix-vector multiplication Av. Consequently, the time-evolution exp 1A for the non-Hermitian matrix A, which is required in the QPE, is implemented in a quantum circuit.

How is the Computational Scheme Applied in Determination?

In the next section, the authors show how to apply this computational scheme in the determination. The quantum circuit generates Aψ/Aψ. Note that the block encoding is applied only for the matrix that satisfies aij ≤ 1 for all entries. If the matrix A does not satisfy this condition, it should be multiplied by a suitable factor beforehand.

In conclusion, the article presents a method of quantum simulation of materials, which uses numeric, symbolic, and quantum algorithms. The quantum phase estimation computes the eigenvalues of transformation matrices, and the results are encoded in a quantum state. The transformation matrices are not Hermitian, so their time evolution should be executed by special quantum circuits. The authors show how to apply this computational scheme in the determination.