Adiabatic quantum simulation of quantum chemistry

Abstract

We show how to apply the quantum adiabatic algorithm directly to the quantum computation of molecular properties. We describe a procedure to map electronic structure Hamiltonians to 2-body qubit Hamiltonians with a small set of physically realizable couplings. By combining the Bravyi-Kitaev construction to map fermions to qubits with perturbative gadgets to reduce the Hamiltonian to 2-body, we obtain precision requirements on the coupling strengths and a number of ancilla qubits that scale polynomially in the problem size. Hence our mapping is efficient. The required set of controllable interactions includes only two types of interaction beyond the Ising interactions required to apply the quantum adiabatic algorithm to combinatorial optimization problems. Our mapping may also be of interest to chemists directly as it defines a dictionary from electronic structure to spin Hamiltonians with physical interactions.

Figure 1. A diagram relating several different approaches to the quantum simulation of quantum chemistry with the procedures and approximations implicit in each approach.

Some of these approaches have been demonstrated experimentally using quantum information processors. References ,,, and ,,,,,, are cited in the figure above.

Figure 2. Numerics comparing the minimum spectral gaps required to reduce the term αX₁Y₂Z₃ to 2-local with an error in the eigenspectrum of at most .

On the left, is fixed at 0.001 and gaps are plotted as a function of α. On the right, α is fixed at 0.1 and gaps are plotted as a function of . Here we compare the bit-flip construction, the Oliveira and Terhal construction and an improved variant on Oliveira and Terhal by Cao et al..

Figure 3. The six equivalent bit-flip processes at third order which produce the effective interaction A · B · C.

Each of these diagrams also occurs backwards on the part of the ground state in |111〉.

Figure 4. Diagrams showing an example of each of the four processes at fourth order.

In the upper left is the process B₁ (H_else + Λ)² B₁. In the upper right is the process . In the lower left is the process D₂B₁D₂B₁. In the lower right is the process A₂B₂C₂D₂.

Figure 5. The three bit-flip processes at second order.

These occur for each term. Note that each of these diagrams occurs in reverse for the part of the ground state in |111〉.

Figure 6. Diagrams for the competing process encountered at third order.

Note that each of these diagrams can also occur backwards if the system starts in |111〉.

Figure 7. Interaction graph for embedded molecular hydrogen Hamiltonian.

Each node represents a qubit. The solid, black edges represent ZZ terms and the black loops represent local Z terms. The dashed, red edges represent XX terms and the red loops represent local X terms. The dotted, blue edges represent XZ terms. It is easy to see the unperturbed Hamiltonians corresponding to the six 3-operator terms (the black triangles).