Classical simulation constants and ordering for quantum chemistry simulation

1. A computer executable method for designing a quantum circuit for computation using a plurality of qubits, the method comprising:

defining a first set of quantum circuits based on one-body Hamiltonian coefficients, the set including one or more gates that apply control-Z rotations based on corresponding one-body Hamiltonian coefficients of the form h pp , wherein p is an integer denoting a spin orbital;

defining a second set of quantum circuits based on two-body Hamiltonian coefficients, the set including one or more gates that apply control-Z rotations based on corresponding two-body Hamiltonian coefficients of the form h prrp , wherein r is an integer denoting a spin orbital; and

arranging first and second sets of quantum circuits based on the definitions of the first and second sets of quantum circuits so that a plurality of qubits is coupled to the first set of quantum circuits prior to coupling to the second set of quantum circuits.

2. The method of claim 1 , further comprising defining a third set of quantum circuits based on a one-body Hamiltonian coefficients, the set including one or more gates that apply control-Z rotations based on corresponding one-body Hamiltonian coefficients of the form h pq , wherein q is an integer denoting a spin orbital and p≠q, and situating a third set of quantum circuits based on the definition of the third set of quantum circuits so as to be coupled to the qubits after the first and second sets of gates.

3. The method of claim 1 , further comprising:

defining a third set of quantum circuits based on one-body Hamiltonian coefficients and two-body Hamiltonian coefficients, the set including one or more gates that apply control-Z rotations based on corresponding one-body and two-body Hamiltonian coefficients, wherein gates that apply control-Z rotations associated with two-body Hamiltonian coefficients of the form h prrq , wherein q is an integer denoting a spin orbital, p≠q, follow gates that apply control-Z rotations based on corresponding one-body Hamiltonian coefficients of the form h pq ; and

situating a third set of quantum circuits based on the definition of the third set of quantum circuits so as to be coupled to the qubits after the second set of quantum circuits.

4. The method of claim 3 , further comprising:

defining a fourth set of quantum circuits based on two-body Hamiltonian coefficients, the set including one or more gates that apply control-Z rotations based on corresponding two-body Hamiltonian coefficients, wherein the one or more gates apply control-Z rotations associated with two-body Hamiltonian coefficients of the form h prsq , wherein s is an integer denoting a spin orbital and r≠s; and

arranging a fourth set of quantum circuits based on the definition of the fourth set of quantum circuits so as to follow the third set of quantum circuits.

5. The method of claim 3 , wherein the first set of quantum circuits includes gates corresponding to one or more of a number operator, an excitation operator, a Coulomb and exchange operator, a number-excitation operator, and a double-excitation operator.

6. The method of claim 1 , further comprising defining a first series of entanglement gates associated with the first set of quantum circuits and a second series of entanglement gates associated with the second set of quantum circuits, wherein the entanglement gates of the first series and the second series are situated so as to be applied to the qubits prior to application of the first set of quantum circuits and the second set of quantum circuits, respectively.

7. The method of claim 1 , further comprising defining at least a first series of basis change gates associated with the first set of quantum circuits and at least a second series of basis change gates associated with the second series of quantum circuits, wherein the basis change gates of the first series and the second series are situated so as to be applied to the qubits prior to application of the first set of quantum circuits and the second set of quantum circuits, respectively.

8. The method of claim 7 , wherein the basis change gates are Hadamard gates, Pauli-Y gates, and Pauli-X gates, or combinations thereof.

9. The method of claim 1 , further comprising:

selecting at last one Hamiltonian coefficient of the form h pq , p≠q or h pqrs , q≠r, wherein q and s are integers denoting spin orbitals; and

adjusting the value of the selected at least one Hamiltonian coefficient.

10. The method of claim 9 , wherein the selected at least one Hamiltonian coefficient is adjusted based on one or more Hamiltonian coefficients of the form h pp and h prrq .

11. The method of claim 9 , wherein the value is adjusted based on a number of Trotter-Suzuki steps.

12. An apparatus for specifying a quantum computing circuit, comprising:

a memory device storing one-body and two-body Hamiltonian coefficients associated with a material of interest; and

a processor that:

receives the stored one-body and two-body Hamiltonian coefficients;

selects an order in which quantum gates associated with the stored one-body and two-body Hamiltonian coefficients h pq , h pqrs , respectively are to be applied to a plurality of qubits, wherein p, q, r, s are integers identifying orbitals, such that coefficients are applied in the order h pp , for all p, h prrp , for all p and r; and

defines a sequence of quantum gates based on the order.

13. The apparatus of claim 12 , wherein the order is selected such that gates associated with Hamiltonian coefficients h pq and h prrq , wherein p q, follow gates associated with the Hamiltonian coefficients h pp , for all p and h prrp , for all r.

14. The apparatus of claim 12 , further wherein the selected order includes the Hamiltonian coefficients h pp , for all p and h prrp , for all r, followed by alternating pairs of Hamiltonian coefficients of the form h pq , h prrq .

15. The apparatus of claim 14 , wherein the selected order includes Hamiltonian coefficients of the form h pq , wherein p≠s and q≠r.

16. The apparatus of claim 12 , wherein the sequence of quantum gates includes entanglement gates and basis change gates.

17. The apparatus of claim 16 , wherein the entanglement gates are CNOT gates, and the basis change gates are Hadamard gates, Pauli-X gates, and Pauli-Y gates, or combinations thereof.

18. The apparatus of claim 12 , wherein the processor adjusts values of at least some Hamiltonian coefficient associated with non-commuting operators.

19. The apparatus of claim 18 , where the adjusted values are based on a number of Trotter-Suzuki steps.

20. A quantum circuit, comprising:

a first series of quantum gates defined based on one-body Hamiltonian coefficients associated with a material of interest, the Hamiltonian coefficients corresponding to commuting operators, wherein the first series of quantum gates is defined based on one-body Hamiltonian coefficients of the form h pp , wherein p is an integer denoting an orbital; and

an intermediate series of quantum gates defined based on one-body and two-body Hamiltonian coefficients of the forms h pq , h prrq , respectively, wherein p, q, r, s are integers denoting orbitals and p≠q, and the series includes alternating pairs of gates h pq , h prrq for at least some values of p, q, r; and

a second series of quantum gates defined based on one-body or two-body Hamiltonian coefficients associated with the material of interest, wherein the Hamiltonian coefficients are non-commutative, wherein the second series of quantum gates is defined based on two-body Hamiltonian coefficients of the form h pqrs , wherein p, q, r, s are integers denoting orbitals and p≠s and q≠r; and

wherein the first series of quantum gates, the intermediate series of quantum gates, and the second series of quantum gates are situated so that the first series precedes the intermediate series and the intermediate series precedes the second series.