# Quantum Circuit Decomposition

## Presentation on theme: "Quantum Circuit Decomposition"— Presentation transcript:

Quantum Circuit Decomposition
from unitary matrices into elementary gates

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?

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

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

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

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

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

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

One Qubit Rotations Define So that

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

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

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

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

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

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

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

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

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

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

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

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

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

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’ = =

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*

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

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

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

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

QR-Decomposition One may iterate this process

QR-Decomposition One may iterate this process

QR-Decomposition One may iterate this process

QR-Decomposition One may iterate this process

QR-Decomposition One may iterate this process

QR-Decomposition One may iterate this process

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

QR-Decomposition Inverting the Q,

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

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

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

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)

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

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

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

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

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

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

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?

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

Reduction Could apply classical techniques…