1. Quantum Circuit Decomposition from unitary matrices into elementary gates

2. Prologue • In classical logic synthesis, one may trivially decompose any boolean function into an OR of ANDs (sum of products) • Local optimizations may then be applied to shrink the resulting circuit • Can the same be done in the quantum case?

3. Objectives • Introduce the “controlled-U” gate • Exhibit a decomposition of a controlled-U into CNOT gates and 1-qubit rotation gates • Introduce the QR-decomposition • Use QR to decompose a unitary matrix into controlled-U gates • Conclude that any operator can be built of CNOT gates and 1-qubit rotations

4. References • The A. Barenko et. Al. paper, and how to write a controlled-U gate in elementary gates • U(2) and SU(2) matrices • Controlled-U gates • The Cybenko paper, and how to write an arbitrary unitary matrix in elementary gates • QR decomposition • Making it a circuit

5. Objectives • Introduce the “controlled-U” gate • Exhibit a decomposition of a controlled-U into CNOT gates and 1-qubit rotation gates • Introduce the QR-decomposition • Use QR to decompose a unitary matrix into controlled-U gates • Conclude that any operator can be built of CNOT gates and 1-qubit rotations

6. The “controlled-U” • The block-matrix form of a “controlled-U” gate • These can be decomposed into • CNOT gates • 1-qubit rotations

7. Objectives • Introduce the “controlled-U” gate • Exhibit a decomposition of a controlled-U into CNOT gates and 1-qubit rotation gates • Introduce the QR-decomposition • Use QR to decompose a unitary matrix into controlled-U gates • Conclude that any operator can be built of CNOT gates and 1-qubit rotations

8. One Qubit Rotations • Let U be a SU(2) matrix. U must take the form • Where

9. One Qubit Rotations • Define • So that

10. Some Quick Facts • R takes sums to products (R=Rz or Ry) • R(0)=I. So: • Finally,

11. Circuit Decompositions • The A. Barenko et. Al. paper, and how to write a controlled-U gate in elementary gates • U(2) and SU(2) matrices • Controlled-U gates • The Cybenko paper, and how to write an arbitrary unitary matrix in elementary gates • QR decomposition • Making it a circuit

12. U A B C Controlled-U Gates • Consider the “controlled-U” gate • Claim: this circuit is equivalent

13. A B C Controlled-U Gates • Check this circuit on basis states • One observes

14. A B C Controlled-U Gates • Check this circuit on basis states • One observes

15. A B C Controlled-U Gates • Check this circuit on basis states • One observes

16. A B C Controlled-U Gates • Check this circuit on basis states • One observes

17. A B C Controlled-U Gates • Check this circuit on basis states • One observes • And similarly,

18. A B C Controlled-U Gates • Check this circuit on basis states • One observes • And similarly,

19. A B C Controlled-U Gates • Check this circuit on basis states • One observes • And similarly,

20. A B C Controlled-U Gates • Check this circuit on basis states • One observes • And similarly,

21. A B C Controlled-U Gates • Check this circuit on basis states • One observes • And similarly,

22. A B C Controlled-U Gates • Check this circuit on basis states • By linearity, this circuit performs “controlled-U”

23. D U’ D A B C U Controlled-U Gates • If U’ is in U(2) (as opposed to SU(2)), • write U’=d U, where d2=det U’, U in SU(2) • Then = =

24. = V V* V U Higher Order Controlled-U Gates • Recall (from two weeks ago) • Where V is a square root of U. • This generalizes straight-forwardly to higher numbers of qubits

25. Objectives • Introduce the “controlled-U” gate • Exhibit a decomposition of a controlled-U into CNOT gates and 1-qubit rotation gates • Introduce the QR-decomposition • Use QR to decompose a unitary matrix into controlled-U gates • Conclude that any operator can be built of CNOT gates and 1-qubit rotations

26. QR-Decomposition • Given a vector (a,b), this SU(2) matrix kills the second coordinate

27. QR-Decomposition • The vector (a,b) might be sitting inside a matrix: • Think of this as a rotation of the plane in which the 3rd and 4th coordinates live • Note that this matrix is unitary

28. Making it a Circuit • The matrix used to kill coordinates in the bottom row looks like • This is a (higher order) controlled-U gate!

29. QR-Decomposition • One may iterate this process

30. QR-Decomposition • One may iterate this process

31. QR-Decomposition • One may iterate this process

32. QR-Decomposition • One may iterate this process

33. QR-Decomposition • One may iterate this process

34. QR-Decomposition • One may iterate this process

35. QR-Decomposition • This yields the formula • Where X was the original matrix, the Ui are planar rotations, and R is upper triangular with nonnegative real entries on the diagonal

36. QR-Decomposition • Inverting the Q,

37. QR-Decomposition • If X is unitary, then R is the product of unitary matrices and hence unitary. • A triangular unitary matrix must be diagonal • A diagonal unitary matrix with nonnegative real entries must be the identity

38. Objectives • Introduce the “controlled-U” gate • Exhibit a decomposition of a controlled-U into CNOT gates and 1-qubit rotation gates • Introduce the QR-decomposition • Use QR to decompose a unitary matrix into controlled-U gates • Conclude that any operator can be built of CNOT gates and 1-qubit rotations

39. Making it a Circuit • The matrix used to kill coordinates in the bottom row looks like • This is a (higher order) controlled-U gate!

40. Making it a Circuit • Need to make other planar rotations controlled-U gates • For some j, given an operator Pj • PjUPj-1 is a rotation in the j,j+1 plane. (where U is a rotation in the n-2,n-1 plane)

41. Making it a Circuit • Built the operator out of NOT and CNOT gates • How to do it for the case of 4 qubits, j=5

42. 1 1 1 0 1 1 1 1 1 1 1 0 0 1 1 0 0 1 1 0 Making it a Circuit • Built the operator out of NOT and CNOT gates • How to do it for the case of 4 qubits, j=5

43. 1 1 1 0 1 1 0 0 0 0 1 1 1 1 1 1 1 1 0 1 Making it a Circuit • Built the operator out of NOT and CNOT gates • How to do it for the case of 4 qubits, j=5

44. Making it a Circuit • The general case is not much harder • First, flip all bits that are 0 in both j,j+1 • Then, CNOT every remaining bit that is zero in j+1, controlling by the unique bit that is 1 in j+1 and 0 in j • Finally, switch this unique bit with the low bit

45. Objectives • Introduce the “controlled-U” gate • Exhibit a decomposition of a controlled-U into CNOT gates and 1-qubit rotation gates • Introduce the QR-decomposition • Use QR to decompose a unitary matrix into controlled-U gates • Conclude that any operator can be built of CNOT gates and 1-qubit rotations

46. Conclusion • A unitary matrix can be written as a product of planar rotations • A planar rotation can be written as ZUZ-1, where Z can be decomposed into CNOT and NOT gates, and U is a (higher order) controlled-U gate • A higher order controlled-U gate can be written as a sequence of CNOT gates and singly controlled-U gates • A controlled-U gate can be written as a sequence of CNOT gates and one-qubit rotations

47. Epilogue • The number of gates in this decomposition is exponential in the number of qubits • For certain operators, much smaller circuits are known to exist • Can we automate the process of moving towards these?

48. Reduction • Could try to shrink a long circuit by local optimization techniques • One experimentally observed obstacle: long chains of CNOT gates • These long chains of CNOTs result from certain identities

49. Reduction • Could apply classical techniques…