Modeling superconducting quantum circuit systems

1. A computer-implemented simulation method comprising:

generating, by operation of a graph generating unit executed by one or more processors, a graph representing a quantum circuit system, wherein the graph includes vertices and edges, the edges representing circuit elements of the quantum circuit system and the vertices representing physical connections between the circuit elements of the quantum circuit system;

storing the graph in first memory;

assigning, to the graph stored in the first memory, inverse inductances, conductances, capacitances, and junction inverse inductances to respective edges of the graph based on a lumped-element approximation of the quantum circuit system;

providing a system of equations of motion representing current in the quantum circuit system;

generating, by operation of a tree generating unit executed by the one or more processors, a spanning tree based on the graph;

storing the spanning tree in second memory;

generating a matrix representation of the system of equations of motion based on the spanning tree; and

simulating the quantum circuit system based on the matrix representation;

wherein the quantum circuit system is comprised of a plurality of modular units with a cascading construction.

2. The method of claim 1 , wherein the matrix representation is specific to the plurality of modular units within the quantum circuit system, the modular units being embedded in circuits of the quantum circuit system, the circuits being identified at specific ports.

3. The method of claim 1 , wherein the plurality of modular units comprises a lumped element resonator.

4. The method of claim 1 , wherein the plurality of modular units comprises a transmission line.

5. The method of claim 1 , wherein the plurality of modular units comprises a transmon qubit.

6. The method of claim 1 , further comprising determining a transfer function of the quantum circuit system based on the simulation of the quantum circuit system.

7. The method of claim 1 , further comprising determining quantized energy levels of the quantum circuit system based on the simulation of the quantum circuit system.

8. The method of claim 1 , further comprising determining a relaxation rate of the quantum circuit system based on the simulation of the quantum circuit system.

9. A non-transitory computer-readable medium storing computer-executable instructions which, when executed by one or more processors, cause the one or more processors to perform operations comprising:

generating a graph representing a quantum circuit system, wherein the graph includes vertices and edges, the edges representing circuit elements of the quantum circuit system and the vertices representing physical connections between the circuit elements of the quantum circuit system;

assigning, to the graph, inverse inductances, conductances, capacitances, and junction inverse inductances to respective edges of the graph based on a lumped-element approximation of the quantum circuit system;

determining a coordinate system based on the graph;

generating a matrix representation of the system based on the coordinate system;

determining a Hamiltonian for the quantum circuit system using the matrix representation; and

simulating the quantum circuit system based on the Hamiltonian;

wherein the matrix representation is specific to a plurality of sub-circuits within the quantum circuit system, the sub-circuits being identified at specific ports.

10. The medium of claim 9 , wherein the quantum circuit system is comprised of a plurality of modular units with a cascading construction.

11. The medium of claim 10 , wherein the plurality of modular units comprises a lumped element resonator.

12. The medium of claim 10 , wherein the plurality of modular units comprises a transmission line.

13. The medium of claim 10 , wherein the plurality of modular units comprises a transmon qubit.

14. The medium of claim 9 , wherein the operations further comprise determining a transfer function of the quantum circuit system based on the simulation of the quantum circuit system.

15. The medium of claim 9 , wherein the operations further comprise determining quantized energy levels of the quantum circuit system based on the simulation of the quantum circuit system.

16. The medium of claim 9 , wherein the operations further comprise determining a relaxation rate of the quantum circuit system based on the simulation of the quantum circuit system.

17. A computer system comprising:

memory that stores computer-executable instructions; and

one or more processors configured to access the memory, wherein the one or more processors are configured to execute the computer-executable instructions to:

generate a graph representing a quantum circuit system, wherein the graph includes vertices and edges, the edges representing circuit elements of the quantum circuit system and the vertices representing physical connections between the circuit elements of the quantum circuit system;

assign, to the graph, inverse inductances, conductances, capacitances, and junction inverse inductances to respective edges of the graph based on a lumped-element approximation of the quantum circuit system;

determine a coordinate system based on the graph;

generate a matrix representation of the system based on the coordinate system;

determine a Hamiltonian for the quantum circuit system using the matrix representation; and

simulate the quantum circuit system based on the Hamiltonian;

wherein the matrix representation is specific to a plurality of sub-circuits within the quantum circuit system, the sub-circuits being identified at specific ports.

18. The system of claim 17 , wherein the quantum circuit system is comprised of a plurality of modular units with a cascading construction.

19. The system of claim 18 , wherein the plurality of modular units comprises a lumped element resonator.

20. The system of claim 18 , wherein the plurality of modular units comprises a transmission line.

21. The system of claim 18 , wherein the plurality of modular units comprises a transmon qubit.

22. The system of claim 17 , wherein the one or more processors are configured to execute the computer-executable instructions to determine a transfer function of the quantum circuit system based on the simulation of the quantum circuit system.

23. The system of claim 17 , wherein the one or more processors are configured to execute the computer-executable instructions to determine quantized energy levels of the quantum circuit system based on the simulation of the quantum circuit system.

24. The system of claim 17 , wherein the one or more processors are configured to execute the computer-executable instructions to determine a relaxation rate of the quantum circuit system based on the simulation of the quantum circuit system.

25. A non-transitory computer-readable medium storing computer-executable instructions which, when executed by one or more processors, cause the one or more processors to perform operations comprising:

generating a graph representing a quantum circuit system, wherein the graph includes vertices and edges, the edges representing circuit elements of the quantum circuit system and the vertices representing physical connections between the circuit elements of the quantum circuit system;

assigning, to the graph, inverse inductances, conductances, capacitances, and junction inverse inductances to respective edges of the graph based on a lumped-element approximation of the quantum circuit system;

providing a system of equations of motion representing current in the quantum circuit system;

generating a spanning tree based on the graph;

generating a matrix representation of the system of equations of motion based on the spanning tree; and

simulating the quantum circuit system based on the matrix representation;

wherein the quantum circuit system is comprised of a plurality of modular units with a cascading construction.

26. The medium of claim 25 , wherein the plurality of modular units comprises a lumped element resonator.

27. The medium of claim 25 , wherein the plurality of modular units comprises a transmission line.

28. The medium of claim 25 , wherein the plurality of modular units comprises a transmon qubit.

29. The medium of claim 25 , wherein the operations further comprise determining a transfer function of the quantum circuit system based on the simulation of the quantum circuit system.

30. The medium of claim 25 , wherein the operations further comprise determining quantized energy levels of the quantum circuit system based on the simulation of the quantum circuit system.

31. The medium of claim 25 , wherein the operations further comprise determining a relaxation rate of the quantum circuit system based on the simulation of the quantum circuit system.

32. A computer system comprising:

memory that stores computer-executable instructions; and

one or more processors configured to access the memory, wherein the one or more processors are configured to execute the computer-executable instructions to:

generate a graph representing a quantum circuit system, wherein the graph includes vertices and edges, the edges representing circuit elements of the quantum circuit system and the vertices representing physical connections between the circuit elements of the quantum circuit system;

assign inverse inductances, conductances, capacitances, and junction inverse inductances to respective edges of the graph based on a lumped-element approximation of the quantum circuit system;

provide a system of equations of motion representing current in the quantum circuit system;

generate a spanning tree based on the graph;

generate a matrix representation of the system of equations of motion based on the spanning tree; and

simulate the quantum circuit system based on the matrix representation;

wherein the quantum circuit system is comprised of a plurality of modular units with a cascading construction.

33. The system of claim 32 , wherein the plurality of modular units comprises a lumped element resonator.

34. The system of claim 32 , wherein the plurality of modular units comprises a transmission line.

35. The system of claim 32 , wherein the plurality of modular units comprises a transmon qubit.

36. The system of claim 32 , wherein the one or more processors are configured to execute the computer-executable instructions to determine a transfer function of the quantum circuit system based on the simulation of the quantum circuit system.

37. The system of claim 32 , wherein the one or more processors are configured to execute the computer-executable instructions to determine a quantized energy levels of the quantum circuit system based on the simulation of the quantum circuit system.

38. The system of claim 32 , wherein the one or more processors are configured to execute the computer-executable instructions to determine a relaxation rate of the quantum circuit system based on the simulation of the quantum circuit system.

39. A computer-implemented simulation method comprising:

generating a graph representing a quantum circuit system, wherein the graph includes vertices and edges, the edges representing circuit elements of the quantum circuit system and the vertices representing physical connections between the circuit elements of the quantum circuit system;

assigning, to the graph, inverse inductances, conductances, capacitances, and junction inverse inductances to respective edges of the graph based on a lumped-element approximation of the quantum circuit system;

determining a coordinate system based on the graph;

generating a matrix representation of the system based on the coordinate system;

determining a Hamiltonian for the quantum circuit system using the matrix representation; and

simulating the quantum circuit system based on the Hamiltonian;

wherein the matrix representation is specific to a plurality of sub-circuits within the quantum circuit system, the sub-circuits being identified at specific ports.

40. The method of claim 39 , wherein the quantum circuit system is comprised of a plurality of modular units with a cascading construction.

41. The method of claim 40 , wherein the plurality of modular units comprises a lumped element resonator.

42. The method of claim 40 , wherein the plurality of modular units comprises a transmission line.

43. The method of claim 40 , wherein the plurality of modular units comprises a transmon qubit.

44. The method of claim 39 , further comprising determining a transfer function of the quantum circuit system based on the simulation of the quantum circuit system.

45. The method of claim 39 , further comprising determining quantized energy levels of the quantum circuit system based on the simulation of the quantum circuit system.

46. The method of claim 39 , further comprising determining a relaxation rate of the quantum circuit system based on the simulation of the quantum circuit system.