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

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

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

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

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

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

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

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

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

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

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

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

24. 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 U = V V*

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. 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 1 1 1 1 1

43. 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 1 1 1 1 1

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…