Systems and methods for calculating the ground state of non-diagonal Hamiltonians
A hybrid computing system comprising a digital and an analog processor calculates the ground energy state of a non-diagonal Hamiltonian via diagonalization of the Hamiltonian in different bases and reverse annealing. A first basis is rotated to render part of the Hamiltonian diagonal, then the quantum processor evolves backwards until a value s* of the normalized evolution coefficient. Another basis is rotated to render another part of the Hamiltonian diagonal and the quantum processor evolves backwards again until s*. The bases can be rotated via discrete Fourier transform. The quantum processor may pause for a time t after each backward evolution. The ground state energy is calculated using the final ground states.
1. A method of operating a hybrid computing system to calculate the ground state of a non-diagonal Hamiltonian, the hybrid computing system comprising a quantum processor and a digital processor, wherein the non-diagonal Hamiltonian comprises a first term and a second term, the first term diagonal in a first basis and the second term diagonal in a second basis, the second basis different from the first basis, the method comprising:
programming the quantum processor to evolve forwards with the first term in the first basis to a first ground state;
applying a first transformation to the first basis;
programming the quantum processor to evolve backwards with the second term in the second basis until reaching a value s 1 * of the normalized evolution coefficient s;
causing the evolution of the quantum processor to pause for a first period of time during the evolving backwards with the second term in the second basis;
applying a second transformation to the second basis;
programming the quantum processor to evolve backwards with the first terms in the first basis until reaching a value s 2 * of the normalized evolution coefficient s;
causing the evolution of the quantum processor to pause for a second period of time during the evolving backwards with the first term in the first basis;
programming the quantum processor to evolve forwards with the first term in the first basis to a second ground state;
reading out the second ground state; and
using the second ground state to calculate the ground energy of the non-diagonal Hamiltonian.
2. The method of claim 1 wherein applying a second transformation to the second basis includes applying to the second basis a second transformation that is the inverse of the first transformation.
3. The method of claim 2 wherein applying a first transformation to the first basis includes applying a discrete Fourier transform to the first basis.
4. The method of claim 1 wherein applying a first transformation to the first basis includes applying a first transformation to the first basis of a non-diagonal Hamiltonian that includes a plurality of terms and applying a first transformation to the first basis includes applying a first transformation to the first basis so that more than one term of the non-diagonal Hamiltonian becomes diagonal in the first basis.
5. The method of claim 1 further comprising programming the quantum processor to evolve forward with the second term in the second basis to a third ground state after pausing the evolution of the quantum processor for a first period of time during the evolving backwards with the second term in the second basis; and reading out the third ground state.
6. The method of claim 1 wherein programming the quantum processor to evolve forwards with the first term in the first basis to a first ground state includes programming the quantum processor to evolve forwards with the first term in the first basis to a first set of ground states; and further comprising programming the quantum processor to evolve forwards with the first term in the first basis to a second set of ground states before reading out the second set of ground states.
7. The method of claim 1 wherein programming the quantum processor to evolve backwards with the first term in the first basis until a value s 2 * of the normalized evolution coefficient s includes programming the quantum processor to evolve backwards with the first term in the first basis until a value s 2 * of the normalized evolution coefficient s is equal to s 1 *.
8. The method of claim 1 wherein calculating the ground energy of the non-diagonal Hamiltonian includes calculating the ground energy of a non-diagonal Hamiltonian describing the electronic structure of a period lattice.
9. The method of claim 8 further comprising determining whether an exit condition has been met.
10. The method of claim 9 wherein determining whether an exit condition has been met includes determining at least one of: whether an error condition is below an error threshold and whether a number of evolution iterations has been completed.
11. The method of claim 1 further comprising until an exit condition is met, iteratively repeating:
applying a first transformation to the first basis;
programming the quantum processor to evolve backwards with the second term in the second basis until reaching a value s 1 * of the normalized evolution coefficient s;
causing the evolution of the quantum processor to pause for a first period of time during the evolving backwards with the second term in the second basis;
applying a second transformation to the second basis;
programming the quantum processor to evolve backwards with the first term in the first basis until reaching a value s 2 * of the normalized evolution coefficient s;
causing the evolution of the quantum processor to pause for a second period of time during the evolving backwards with the first term in the first basis;
programming the quantum processor to evolve forwards with the first term in the first basis to a n th ground state; and
reading out the n th ground state.
12. A hybrid computing system, the hybrid computing system comprising a quantum processor and a digital processor, the digital processor operable to:
program the quantum processor with a first term of a non-diagonal Hamiltonian, the non-diagonal Hamiltonian comprising the first term and a second term, the first term diagonal in a first basis and the second term diagonal in a second basis different from the first basis, to evolve forwards in the first basis to a first ground state;
apply a first transformation to the first basis;
program the quantum processor to evolve backwards with the second term in the second basis until reaching a value s 1 * of the normalized evolution coefficient s;
pause the evolution of the quantum processor for a first period of time during the evolving backwards with the second term in the second basis;
apply a second transformation to the second basis;
program the quantum processor to evolve backwards with the first term in the first basis until reaching a value s 2 * of the normalized evolution coefficient s;
pause the evolution of the quantum processor for a second period of time during the evolving backwards with the first term in the first basis;
program the quantum processor to evolve forwards with the first term in the first basis to a second ground state;
read out the second ground state; and
use the second ground state to calculate the ground energy of the non-diagonal Hamiltonian.
13. The hybrid computing system of claim 12 wherein the second transformation is the inverse of the first transformation.
14. The hybrid computing system of claim 13 wherein the first transformation is a discrete Fourier transform.
15. The hybrid computing system of claim 12 wherein the non-diagonal Hamiltonian includes a plurality of terms and the digital processor is operable to apply a first transformation to the first basis so that more than one term of the non-diagonal Hamiltonian becomes diagonal.
16. The hybrid computing system of claim 12 wherein the digital processor is further operable to program the quantum processor to evolve forward with the second term in the second basis to a third ground state after pausing the evolution of the quantum processor for a first period of time during the evolving backwards with the second term in the second basis; and to read out the third ground state.
17. The hybrid computing system of claim 12 wherein the digital processor is further operable to program the quantum processor to evolve forwards in the first basis to a first set of ground states; and to cause the quantum processor to evolve forwards in the first basis to a second set of ground states before reading out the second set of ground states.
18. The hybrid computing system of claim 12 wherein s 2 * is equal to s 1 *.
19. The hybrid computing system of claim 12 wherein the non-diagonal Hamiltonian describes the electronic structure of a period lattice.
20. The hybrid computing system of claim 12 wherein the digital processor is further operable to, until an exit condition is met, iteratively repeatedly:
apply a first transformation to the first basis;
program the quantum processor to evolve backwards with the second term in the second basis until reaching a value s 1 * of the normalized evolution coefficient;
pause the evolution of the quantum processor for a first period of time during the evolving backwards with the second term in the second basis;
apply a second transformation to the second basis;
program the quantum processor to evolve backwards with the first term in the first basis until reaching a value s 2 * of the normalized evolution coefficient;
pause the evolution of the quantum processor for a second period of time during the evolving backwards with the first term in the first basis;
program the quantum processor to evolve forwards with the first term in the first basis to a n th ground state; and
read out the n th ground state.