Efficient and noise resilient measurements for quantum chemistry
Methods, systems and apparatus for measuring the energy of a quantum chemical system. In one aspect, a method includes obtaining a Hamiltonian describing the chemical system, where the Hamiltonian is expressed in an orthonormal basis; decomposing the Hamiltonian into a sum of terms where each term comprises a respective operator that effects a respective single particle basis rotation, and one or more particle density operators; repeatedly, for each group comprising terms with a same operator that effects a respective single particle basis rotation, measuring expectation values of the terms included in the group, comprising: performing the respective single particle basis rotation on a qubit system encoding a state of the chemical system; and measuring Jordan-Wigner transformations of the one or more particle density operators in the group to obtain a respective measurement result for the group; and determining the energy of the chemical system using the obtained measurement results.
1. A method performed by a classical processor and quantum computing hardware in data communication with the classical processor, the method comprising:
measuring an energy of a chemical system, comprising:
obtaining, as input to the classical processor, a Hamiltonian describing the chemical system, wherein the Hamiltonian is expressed in an orthonormal basis;
implementing, by the classical processor and the quantum computing hardware, a basis rotation grouping measurement strategy of a qubit system included in the quantum computing hardware, the basis rotation grouping measurement strategy comprising:
decomposing, by classical computation, the Hamiltonian describing the chemical system into a sum of terms, wherein each term comprises i) a respective operator that effects a respective single particle basis rotation, and ii) one or more particle density operators;
for each group comprising terms with a same operator that effects a respective single particle basis rotation, measuring expectation values of the terms included in the group, comprising, for each repetition in a plurality of repetitions, wherein a number of repetitions in the plurality of repetitions is dependent on a predetermined accuracy:
encoding, by quantum computation on the quantum computing hardware, a state of the chemical system in a state of the qubit system;
performing, by quantum computation on the quantum computing hardware, the respective single particle basis rotation of the group on the state of the qubit system encoding the state of the chemical system, comprising applying a quantum circuit to the qubit system; and
measuring, by quantum computation on the quantum computing hardware and in a computational basis, the qubit system, comprising measuring Jordan-Wigner transformations of the one or more particle density operators in the group to obtain a respective measurement result for the group; and
determining, by classical computation, the energy of the chemical system using the obtained measurement results.
2. The method of claim 1 , wherein measuring expectation values of the terms included in the group further comprises performing error mitigation by post selection.
3. The method of claim 2 , wherein performing error mitigation by post selection comprises:
computing a total particle number or spin component using the obtained measurement result;
determining whether the computed total particle number or spin component is equal to a respective target value;
in response to determining that the computed total particle number or spin component is equal to a respective target value, providing the measurement result for determining, by classical computation, the energy of the chemical system; and
in response to determining that the computed total particle number or spin component is not equal to a respective target value, discarding the measurement result.
4. The method of claim 1 , wherein the Hamiltonian describing the chemical system comprises a one-electron component and a two-electron component, and wherein decomposing, by classical computation, the Hamiltonian describing the chemical system into a sum of terms comprises:
diagonalizing each scalar coefficient in the two-electron component, comprising representing each scalar coefficient in the two-electron component as a second sum of terms over the single particle bases, each term in the second sum of terms comprising a product of a Hermitian coefficient matrix of one-body operators formed by a first pair of spin orbitals, a matrix, and a Hermitian coefficient matrix of one-body operators formed by a second pair of spin orbitals;
determining, for each term in the sum of terms, a matrix that diagonalizes the one-body operators in the respective Hermitian coefficient matrices; and
determining the respective operators that effect the respective basis rotations using the determined matrices that diagonalize the one-body operators.
5. The method of claim 4 , further comprising discarding finite eigenvalues smaller than a predetermined threshold.
6. The method of claim 1 , further comprising grouping terms of the decomposed Hamiltonian that are diagonal in the same single particle basis, comprising, for each term in the decomposed Hamiltonian:
determining which single particle basis the term diagonalizes; and
assigning the term to a group corresponding to the determined single particle basis.
7. The method of claim 1 , wherein measuring expectation values of the terms included in the group comprises measuring the expectation values of the terms included in the group simultaneously.
8. The method of claim 1 , wherein performing the respective basis rotation comprises applying a respective Givens rotation circuit to the qubit system.
9. The method of claim 1 , wherein determining, by classical computation, the energy of the chemical system using the obtained measurement results comprises:
determining an average measurement result corresponding to each group; and
adding the determined averages.
10. The method of claim 1 , wherein the arbitrary orthonormal basis comprises a Gaussian or molecular orbital basis.
11. The method of claim 1 , wherein the Hamiltonian describing the chemical system comprises multiple terms each comprising products of one or more of i) annihilation operators for respective spin orbitals, ii) creation operators for respective spin orbitals, and ii) scalar coefficients given by one- or two-electron integrals over basis functions in the orthonormal basis.
12. The method of claim 1 , wherein the chemical system comprises a symmetrically stretched Hydrogen chain, symmetrically stretched water molecule, or a stretched Nitrogen dimer.
13. An apparatus comprising:
quantum computing hardware; and
one or more classical processors;
wherein the apparatus is configured to perform operations comprising:
obtaining, as input to the one or more classical processors, a Hamiltonian describing a chemical system, wherein the Hamiltonian is expressed in an orthonormal basis;
implementing, by the one or more classical processors and the quantum computing hardware, a basis rotation grouping measurement strategy of a qubit system included in the quantum computing hardware, the basis rotation grouping measurement strategy comprising:
decomposing, by classical computation, the Hamiltonian describing the chemical system into a sum of terms, wherein each term comprises i) a respective operator that effects a respective single particle basis rotation, and ii) one or more particle density operators;
for each group comprising terms with a same operator that effects a respective single particle basis rotation, measuring expectation values of the terms included in the group, comprising, for each repetition in a plurality of repetitions, wherein a number of repetitions in the plurality of repetitions is dependent on a predetermined accuracy:
encoding, by quantum computation on the quantum computing hardware, a state of the chemical system in a state of the qubit system;
performing, by quantum computation on the quantum computing hardware, the respective single particle basis rotation on the state of the qubit system encoding the state of the chemical system, comprising applying a quantum circuit to the qubit system; and
measuring, by quantum computation on the quantum computing hardware and in a computational basis, the qubit system, comprising measuring Jordan-Wigner transformations of the one or more particle density operators in the group to obtain a respective measurement result for the group; and
determining, by classical computation, the energy of the chemical system using the obtained measurement results.