A computer implemented method for compiling a quantum circuit for quantum computations using a plurality of qubits, the method comprising: receiving stored one-body and two-body Hamiltonian coefficients wherein the stored one-body and two-body Hamiltonian coefficients are associated with a Trotter-Suzuki expansion of a second quantized form of a Hamiltonian simulating a material of interest; compiling a first set of quantum circuits based on one-body Hamiltonian coefficients, the first 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; compiling a second set of quantum circuits based on two-body Hamiltonian coefficients, the second 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; arranging the first and second sets of quantum circuits so that the plurality of qubits is coupled to the first set of quantum circuits prior to coupling to the second set of quantum circuits; compiling a third set of quantum circuits based on one-body Hamiltonian coefficients and two-body Hamiltonian coefficients, the third 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 p ≠ q, follow gates that apply control-Z rotations based on corresponding one-body Hamiltonian coefficients of the form h pq; and situating the third set of quantum circuits so as to be coupled to the qubits after the second set of quantum circuits; compiling (308, 310) a fourth set of quantum circuits based on two-body Hamiltonian coefficients, the fourth 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 r ≠ s; and arranging the fourth set of quantum circuits so as to follow the third set of quantum circuits; and outputting a compiled circuit corresponding to the arranging of the first, second, third and fourth sets of quantum circuits, whereby errors introduced by non-commutating operators are reduced based on the arrangement. receiving stored one-body and two-body Hamiltonian coefficients wherein the stored one-body and two-body Hamiltonian coefficients are associated with a Trotter-Suzuki expansion of a second quantized form of a Hamiltonian simulating a material of interest; compiling a first set of quantum circuits based on one-body Hamiltonian coefficients, the first 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; compiling a second set of quantum circuits based on two-body Hamiltonian coefficients, the second 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; arranging the first and second sets of quantum circuits so that the plurality of qubits is coupled to the first set of quantum circuits prior to coupling to the second set of quantum circuits; compiling a third set of quantum circuits based on one-body Hamiltonian coefficients and two-body Hamiltonian coefficients, the third 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 p ≠ q, follow gates that apply control-Z rotations based on corresponding one-body Hamiltonian coefficients of the form h pq; and situating the third set of quantum circuits so as to be coupled to the qubits after the second set of quantum circuits; compiling (308, 310) a fourth set of quantum circuits based on two-body Hamiltonian coefficients, the fourth 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 r ≠ s; and arranging the fourth set of quantum circuits so as to follow the third set of quantum circuits; and outputting a compiled circuit corresponding to the arranging of the first, second, third and fourth sets of quantum circuits, whereby errors introduced by non-commutating operators are reduced based on the arrangement.