Decoding errors using quantum subspace expansion

1. A method for correcting a result of a quantum computation, the method comprising:

selecting, by one or more classical processors, a quantum error correcting code to perform the quantum computation, wherein the quantum error correcting code is defined by a respective set of stabilizer generators;

determining, by the one or more classical processors, a set of symmetry operators, comprising:

selecting a subset of the set of stabilizer generators, the subset comprising a plurality of stabilizer generators;

determining, for each stabilizer generator in the selected subset, a sum between an identity operator and the stabilizer generator; and

multiplying the determined sums together to form a summation of terms, wherein each term in the summation is equal to a respective symmetry operator;

measuring, using quantum computing hardware, a projective correction of a physical observable over an output quantum state of the quantum computation using the determined set of symmetry operators, wherein the physical observable corresponds to the result of the quantum computation; and

determining, by the one or more classical processors, a corrected result of the quantum computation using the measured projective correction of the physical observable.

2. The method of claim 1 , wherein measuring the projective correction of a physical observable over an output quantum state of the quantum computation using the determined set of symmetry operators comprises measuring P † Γ P where P represents a summation of the symmetry operators in the determined set of symmetry operators with uniform coefficients, and Γ represents the physical observable.

3. The method of claim 1 , wherein measuring a projective correction of a physical observable over an output quantum state of the quantum computation using the determined set of symmetry operators comprises:

selecting one or more pairs of operators, wherein each pair comprises i) a respective component of the physical observable, and ii) a respective symmetry operator from the determined set of symmetry operators;

for each selected pair of operators:

performing the quantum computation on an initial quantum state to obtain the output quantum state, and

measuring the selected pair of operators over the output quantum state to obtain a respective measurement result.

4. The method of claim 3 , wherein determining a corrected result of the output of the quantum computation comprises determining a corrected result of the output of the quantum computation using the obtained measurement results.

5. The method of claim 4 , wherein determining a corrected result of the output of the quantum computation using the obtained measurement results comprises computing a linear combination of the obtained measurement results.

6. The method of claim 3 , wherein selecting one or more pairs of operators comprises randomly sampling one or more pairs of operators according to a stochastic sampling scheme.

7. The method of claim 3 , wherein the physical observable comprises a weighted sum of Pauli operators, and wherein components of the physical observable comprise Pauli operators in the sum of Pauli operators.

8. The method of claim 3 , wherein measuring a selected pair of operators over the output quantum state to obtain a respective measurement result comprises measuring Γ j M k over the output quantum state, wherein Γ j represents the component of the physical observable in the selected pair and M k represents the symmetry operator in the selected pair.

9. The method of claim 1 , wherein the result of the quantum computation comprises an expectation value of the physical observable.

10. The method of claim 1 , wherein the size of the selected subset of the set of stabilizer generators is dependent on one or more of i) a target computation accuracy, or ii) a target computation cost.

11. An apparatus comprising:

quantum hardware; and

one or more classical processors;

wherein the apparatus is configured to perform operations for correcting a result of a quantum computation, the operations comprising:

selecting, by the one or more classical processors, a quantum error correcting code to perform the quantum computation, wherein the quantum error correcting code is defined by a respective set of stabilizer generators;

determining, by the one or more classical processors, a set of symmetry operators, comprising:

selecting a subset of the set of stabilizer generators, the subset comprising a plurality of stabilizer generators;

determining, for each stabilizer generator in the selected subset, a sum between an identity operator and the stabilizer generator; and

multiplying the determined sums together to form a summation of terms, wherein each term in the summation is equal to a respective symmetry operator;

measuring, using the quantum hardware, a projective correction of a physical observable over an output quantum state of the quantum computation using the determined set of symmetry operators, wherein the physical observable corresponds to the result of the quantum computation; and

determining, by the one or more classical processors, a corrected result of the quantum computation using the measured projective correction of the physical observable.

12. The apparatus of claim 11 , wherein measuring the projective correction of a physical observable over an output quantum state of the quantum computation using the determined set of symmetry operators comprises measuring P † Γ P where P represents a summation of the symmetry operators in the determined set of symmetry operators with uniform coefficients, and Γ represents the physical observable.

13. The apparatus of claim 11 , wherein measuring a projective correction of a physical observable over an output quantum state of the quantum computation using the determined set of symmetry operators comprises:

selecting one or more pairs of operators, wherein each pair comprises i) a respective component of the physical observable, and ii) a respective symmetry operator from the determined set of symmetry operators;

for each selected pair of operators:

performing the quantum computation on an initial quantum state to obtain the output quantum state, and

measuring the selected pair of operators over the output quantum state to obtain a respective measurement result.

14. The apparatus of claim 13 , wherein determining a corrected result of the output of the quantum computation comprises determining a corrected result of the output of the quantum computation using the obtained measurement results.

15. The apparatus of claim 14 , wherein determining a corrected result of the output of the quantum computation using the obtained measurement results comprises computing a linear combination of the obtained measurement results.

16. The apparatus of claim 13 , wherein selecting one or more pairs of operators comprises randomly sampling one or more pairs of operators according to a stochastic sampling scheme.

17. The apparatus of claim 13 , wherein the physical observable comprises a weighted sum of Pauli operators, and wherein components of the physical observable comprise Pauli operators in the sum of Pauli operators.

18. The apparatus of claim 13 , wherein measuring a selected pair of operators over the output quantum state to obtain a respective measurement result comprises measuring Γ j M k over the output quantum state, wherein Γ j represents the component of the physical observable in the selected pair and M k represents the symmetry operator in the selected pair.

19. The apparatus of claim 11 , wherein the result of the quantum computation comprises an expectation value of the physical observable.

20. The apparatus of claim 11 , wherein the size of the selected subset of the set of stabilizer generators is dependent on one or more of i) a target computation accuracy, or ii) a target computation cost.