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

U

IN0

02

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

2/

2/

2/

2/

2/2/

2/)(2/

2221

1211

0

0

2/cos2/sin

2/sin2/cos

0

0

2/cos2/sin

2/sin2/cos

i

i

i

i

ii

ii

e

e

e

e

ee

ee

uu

uuU

21

11

2111

2111

arctan

argarg

argarg

q

q

qq

qq

One Qubit Rotations

Define

So that

2/cos2/sin

2/sin2/cos)(

yR

2/

2/

0

0)(

i

i

ze

eR

01

10X

)()()( zyz RRRU

2/

2/

2/

2/

0

0

2/cos2/sin

2/sin2/cos

0

0

i

i

i

i

e

e

e

eU

Some Quick Facts

R takes sums to products (R=Rz or Ry)

R(0)=I. So:

Finally,

XRXR )()(

)()()( RRR

)()( 1 RR

2/cos2/sin

2/sin2/cos)(

yR

2/

2/

0

0)(

i

i

ze

eR

01

10X

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

)(1010 bUaba U

2

22/

2/

z

zy

yz

RC

RRB

RRA

)()()( zyz RRRU

BA C

2/cos2/sin

2/sin2/cos)(

yR

2/

2/

0

0)(

i

i

ze

eR

01

10X

Controlled-U Gates

Check this circuit on basis states

One observes

ABCaa 00

2222

zzyyz RRRRRABC

AXBXCbb 11

BA C

2

22/

2/

z

zy

yz

RC

RRB

RRA

Controlled-U Gates

Check this circuit on basis states

One observes

ABCaa 00

22

zzz RRRABC

AXBXCbb 11

BA C

2

22/

2/

z

zy

yz

RC

RRB

RRA

Controlled-U Gates

Check this circuit on basis states

One observes

ABCaa 00

)( zz RRABC

AXBXCbb 11

BA C

2

22/

2/

z

zy

yz

RC

RRB

RRA

Controlled-U Gates

Check this circuit on basis states

One observes

ABCaa 00

IABC

AXBXCbb 11

BA C

2

22/

2/

z

zy

yz

RC

RRB

RRA

Controlled-U Gates

Check this circuit on basis states

One observes

And similarly,

ABCaa 00

IABC

AXBXCbb 11

2222

zzyyz XRRXRRRAXBXC

BA C

2

22/

2/

z

zy

yz

RC

RRB

RRA

Controlled-U Gates

Check this circuit on basis states

One observes

And similarly,

ABCaa 00

IABC

AXBXCbb 11

2222

zzyyz XRXRRRRAXBXC

BA C

2

22/

2/

z

zy

yz

RC

RRB

RRA

Controlled-U Gates

Check this circuit on basis states

One observes

And similarly,

ABCaa 00

IABC

AXBXCbb 11

2222

zzyyz XXRRRRRAXBXC

BA C

2

22/

2/

z

zy

yz

RC

RRB

RRA

Controlled-U Gates

Check this circuit on basis states

One observes

And similarly,

ABCaa 00

IABC

AXBXCbb 11

zyz RRRAXBXC

BA C

2

22/

2/

z

zy

yz

RC

RRB

RRA

Controlled-U Gates

Check this circuit on basis states

One observes

And similarly,

ABCaa 00

IABC

AXBXCbb 11

UAXBXC

BA C

2

22/

2/

z

zy

yz

RC

RRB

RRA

Controlled-U Gates

Check this circuit on basis states

By linearity, this circuit performs “controlled-U”

aa 00 )(11 bUb

BA C

2

22/

2/

z

zy

yz

RC

RRB

RRA

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

U’ U

D= =

dD

0

01

BA C

D

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* 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

0

1 22

22

bab

a

ab

ba

ba

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

*'*'*'0

*'*'*'

****

****

***

***

****

****

//00

//00

0010

0001

C

b

a

CaCb

CbCa22

baC

0

1 22

22

bab

a

ab

ba

ba

Making it a Circuit

The matrix used to kill coordinates in the bottom row looks like

This is a (higher order) controlled-U gate!

U

IN0

02

QR-Decomposition

One may iterate this process

***0

****

****

****

****

****

****

****

0

1 22

22

bab

a

ab

ba

ba

QR-Decomposition

One may iterate this process

***0

***0

****

****

***0

****

****

****

0

1 22

22

bab

a

ab

ba

ba

QR-Decomposition

One may iterate this process

***0

***0

***0

****

***0

***0

****

****

0

1 22

22

bab

a

ab

ba

ba

QR-Decomposition

One may iterate this process

**00

***0

***0

****

***0

***0

***0

****

0

1 22

22

bab

a

ab

ba

ba

QR-Decomposition

One may iterate this process

**00

**00

***0

****

**00

***0

***0

****

0

1 22

22

bab

a

ab

ba

ba

QR-Decomposition

One may iterate this process

*000

**00

***0

****

**00

**00

***0

****

0

1 22

22

bab

a

ab

ba

ba

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

RXUU n ...1

0

1 22

22

bab

a

ab

ba

ba

QR-Decomposition

Inverting the Q,

RXUU n ...1

RUUX n1

11...

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

RUUX n1

11...

RXUU n ...1

11

1... UUX n

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!

U

IN0

02

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)

11

2

Nj

Nj

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

11

2

Nj

Nj

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

11

2

Nj

Nj

1

0

0

0

1

0

0

1

1

0

1

1

1

1

1

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

11

2

Nj

Nj

0

1

1

0

0

1

1

1

0

1

1

1

0

1

1

1

1

0

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

11

2

Nj

Nj

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…