Engineering of arbitrary U(N) transformation by quantum Householder reflections

P. A. Ivanov, E. S. Kyoseva, and N. V. Vitanov

Plan of the talk

• Standard, generalized and coupled Householder Reflections

• Physical Implementations of standard and generalized HRs

• Decomposition of U(N) group by standard and generalized HRs

• Navigation in Hilbert space by HRs

• Physical Implementation of coupled HR

• Decomposition of U(N) group by coupled HR

For an arbitrary vector we wish to find another vector , which is the reflection of on the other side of the plane P

REFLECTIONREFLECTION

STANDARD HOUSEHOLDER REFLECTIONSTANDARD HOUSEHOLDER REFLECTION

1 2

2

M I

M

2

P

A. S. Householder, J. ACM 5, 339 (1958)

21

1

1

PROPERTIESPROPERTIES

REFLECTIONREFLECTION

1. If we reflect a vector twice through the same plane, we get the same vector again

2

1 1 M M M I

1 M M

; 1ie M I

2. M is equal to its inverse

GENERALIZED HRGENERALIZED HR

The standard HR is a special case of the generalized HR for , ; M M

COUPLED HRCOUPLED HR

1

1 1 1 11

, , ; , , ( 1)n

Ni

N N n nn

e

M I=

APPLICATION of HRAPPLICATION of HR

• Householder decomposition of matricesHouseholder decomposition of matrices• Solving systems of linear equationsSolving systems of linear equations• Finding eigenvalues of high-dimensional Finding eigenvalues of high-dimensional

matricesmatrices• Least-square optimizationLeast-square optimization• Grover algorithmGrover algorithm

Physical Implementations of the standard and generalized Householder Reflections

1

.2

N

nn

H t e e t n e H c

ni

n nt e f t

We consider (N+1)-state system with N degenerate ground states, which represent the qunit, coupled coherently via a common excited state by pulsed external fields of the same time dependence but possible different amplitudes and phases. Such an N-pod system can be formed, e.g., by coupling the magnetic sublevels of several J=1 levels to a single J=0 level; for a qutrit only one J=1 level suffices

Phys. Rev. A 73 023420 (2006)

HAMILTONIAN OF THE SYSTEM Rabi frequency

1 2 N

e

N21

t

Morris- Shore transformation

The coupled (N+1)-state system can be decomposed into a set of N-1 dark ground states, and a two-state system, consisting of a bright state and the excited state

Morris- Shoretransformation

2d1db

e

N321

e

1Nd

2

1

N

nn

t

21

3 N

Phys. Rev. A 27 906 (1983)

112 1

212 2

1 2

1 2

211 1 2 1

2 2 2

221 2 2 2

2 2 2 2

1

21 2

2 2 2

* * *1 2

1 1 1 1

1 1 1 1

,

1 1 1 1

N

N

N N N

N

ii iN

ii iN

N i

i i iN N N N

ii iN

ee ea a a ib

ee ea a a ib

t t

e e ea a a ib

ee eib ib ib a

U

*

1 1, ,N i N i

di t t t t t

dt U H U

Schrödinger Equation

The exact solution for the propagator

2 21a b

Cayley-Klein parameters

STANDARD QHR: Exact resonance

cos / 2

sin / 2

a A

b i A

2 2 1 0,1,2,...

f

i

t

t

A f t dt

A k k

1

0

0

0 0 1

N

UU

0

nin nt f t e

Pulse Area

The propagator reduces to:The propagator reduces to:U is an N-dimensional unitary matrix, which represents the propagator within the N- state degenerate manifold, which has exactly the standard QHR form:

2 0U M I

The components of the vector are the normalized Rabi frequencies

1 21 2

1, ,..., N

Tii iNe e e

The propagator U of the N- pod system driven by Hamiltonian represents a compact physical realization of standard QHR in a single interaction step.

Generalized QHR: Rosen- Zener model

0

cosht

t

0

2 ( 1,2,...)A l l The Rozen - Zener propagatorThe Rozen - Zener propagator

1

0

0

0 0

RZ

N

ie

UU

Physical realization of the generalized QHRPhysical realization of the generalized QHR

The Rozen- Zener model can be seen as an extensionof the resonance solution to nonzero detuning for aspecial pulse shape (hyperbolic secant)

The phase depends on the detuning and for an arbitrary integerl we find

; 1RZ ie 0U M I

1

00

2arg 2 1l

k

T i k

The use of nonresonant interaction, besidesan additional phase parameter, has anotherimportant advantage over resonant pulses:lower transient population of the intermediatestate. This can be crucial if the lifetime ofthis state is short compared to the interactionduration

Engineering of arbitrary U(N) transformation by rotations

,

1 0 0 0

0 sin cos 0

0 cos sin 0

0 0 0 1

i i

i j

e e

iT9

1

, 1 , 2 2,1N N N N

U T T T F

The matrix a beam-splitter with a phase shifter The matrix a beam-splitter with a phase shifter

Any N-dimensional unitary matrix can be represented as product N(N-1)/2 rotations and one phase transformation

Phys. Lett. 73 58 (1994)

Householder Decomposition

1

1 21

, , ,N

n Nn

U M F

Standard QHR decompositionStandard QHR decomposition

Any N-dimensional unitary matrix U can be expressed as a product of N-1 standard QHRs and one phase gate

Generalized QHR decompositionGeneralized QHR decomposition

Any N-dimensional unitary matrix U can be expressed as a product of N generalized QHRs

1

;N

n nn

U M

Phys. Rev. A 74 022323 (2006)

Proof of the Standard QHR decomposition

1. First we define the normalized vector

11 1 1

1 1 0, ( 2,3, )

i

n

u e e

e u n N

M

M

1

1

1 11

112 1 Re

i

i

u e e

u e

1

1

1

0 0

0

0

i

N

e

M U

U

It is easy to see

Hence the action of M upon Unullifies the first row and the first column except for the first element

1 11arg u

nu

1 [1,0, ,0]Te

nth column of U

2. We repeat the same procedure on MU and construct the vector

1

2

2 1

2

0 0

0 0

0 0

i

i

N

e

e

M M U

U

1 2 1 1 2, , ,N N M M M U F

2

2

'2 2

2'222 1 Re

i

i

u e e

u e

The corresponding QHR applied to MU, has the following effects:(i) it nullifies the second row and second column of MU except for thediagonal element, and (ii) does not change the first row and first column.

After N-1 interaction steps

By repeating the same procedure N-1 times, we construct N-1 consecutive Householder reflections, which nullify all off-diagonal elements, to produce a diagonal matrix comprising N phase factors

Proof of the Generalized QHR decomposition

1.We first define the normalized vector

1 1 1 1

1 1 1

,

, 0, ( 2,3, )n

u e

e u n N

M

M

It is readily seen that

1 11

11

1 11

2(1 Re )

2arg 1

u e

u

u

Therefore, the action of the firs generalizedHR upon U nullifies the first row and first column except for the first element, which is turned into unit

1 11

1 0 0

0,

0N

M U

U

2.We repeat the same procedure

1

;n nn N

M U=I=

'2 2

2 '22

'2 22

2(1 Re )

2arg 1

u e

u

u

2 2 1 12

1 0 0

0 1 0; ;

0 0N

M M U

U

The corresponding QHR applied to MU, has the following effects :(i) it nullifies the second row and second column of MU except for the diagonal element which is turned into unit, and (ii) does not change the first row and first column.

By repeating the same procedure N times, we construct N consecutiveHouseholder reflection, which nullify all off-diagonal elements, to producethe identity matrix

COMMENT

2. The QHR decomposition of the U(N) group provides a simple and efficient physical realization of a general transformation of a qunit by only N interaction steps; this is a significant advance compared to O(N2) operations in existing recipes.

3. Each QHR vectors is N-dimensional, but the nonzero elements decrease

4. The decomposition is also of mathematical interest because it provide a very natural parametrization of the U(N) group.

1. The choice of the QHRs is not unique: for example, the first QHR vector can beconstructed from first row, instead of the first column. Furthermore, the final diagonal matrix occurs due to the unitarity of U;QHR sequence produces a triangular matrix in general.

2

2

2 1N

n

n N N

from N in the first step to just 1 in the final step

Quantum Fourier Transform

2 1 1 /

1

1 Ni n k NF

k

n e kN

U

e1 11

1 12

12 2 , 2 2

2

F

T

U

For Qubit, the QFT can be written as a single Householder reflection

Physical Implementation

2

1 2

1

0T

2A

1 2 2T

1 2 2T

1 2

EXAMPLES

For Qutrit, the QFT can be written as a product of two standard QHRs and one phase gate, or as a product of two generalized QHRs

, 1

( ) ( )N

FN jk N jk

j k

U U

2 /3 2 /3

2 /3 2 /3

1 1 11

13

1

F i i

i i

e e

e e

U

1 2

1 2; ; / 2

F

F

U M M F

U M M

e

2 31

sec [( 5) / ]f t h t T sec [( 5) / ]f t h t T

1T

2A

1 0T

2,3 2T

0T

2A

1 0.919T

2,3 1.256T

1 32

1J

0J

NAVIGATION IN HILBERT SPACE

Given the initial state and final state of the qunit, we wish to find propagator such that

Phys. Rev. A 74 012323 (2007)

f i U

12 1 Re{( ) }

ik

ik

ii

ii

i

e k

e

Transition by standard QHRs

1. First we define the normalized vector

The corresponding QHR acting uponthe initial state reflects it onto single qunit state

ikii i e k M

NAVIGATION IN HILBERT SPACE

12 1 Re{( ) }

fk

fk

i

f

fi

f

e k

e

2. Next, we define the vector

The action of corresponding standard QHR upon the single qunit state reflects it onto the final state

fk

ik fk

i

f f

i

f i i f

k e

e

M

M M

Physical realization of the propagator U requires only two standard Householder reflections

1. Transfer from an arbitrary single state to an arbitrary superposition state

1

2f i

( )fU M

2 1

f i

f i

2. Transfer from an arbitrary superposition state to an arbitrary single state

3. Transfer between orthogonal states

4. Transfer between states with real coefficients

( )iU M

In several important special cases only a single standard QHR is neededfor a pure-to pure transition

NAVIGATION IN HILBERT SPACE

Transition by generalized QHR

( ; ) U M

A generalized QHR is ideally for pure-to-pure transition because only onegeneralized QHR is sufficient to reflect initial state into the final state

2 1 Re

2arg 1

f i

f i

f i

In this case the solution is unique; there is no arbitrariness in thechoice of QHR vector (up to an unimportant global phase) and the phase

1

1.732T

e

321

32

1 1.839T

2 2.512T

3 2.512T

4A

sec [ / ]f t h t T

1

e

32

321

M

11 2 3

3f 1i

1J

0J

1

( / ) sec [ 5 / ]f t T h t T

1 1.531T

2 3 4 5

e

( / ) sec [ 5 / ]f t T h t T

2 3.696T

3,4,5 0T

1.732T

4A

4A

1.732T

1 2.103T

1 1.701n T

11 2 3 4 5

5f i 1

1 22

i

1 2 M M

1 2 3 4 5

e

1 2 3

e

3 2 1

( ) sec [ / ]f t h t T

0.7T

1 10.239 0.693T

2 21.678 0.910T

3 31.061 0.308T

2A

1 2 3

e

3 2 1

11 2

2i 1

1 2 33

f i

; M

COHERENT NAVIGATION BETWEEN MIXED STATES

† 1f i f i U U U R R

The transfer between two mixed states requires a general U(N) propagator. Thelatter can be can be expressed as a product of N-1 standard QHRs and one phase gate, or by N generalized QHRs

Synthesis of arbitrary preselected mixed state

Mixed states with different invariants cannot be connected by Hermitian evolutionBecause these invariants are constant of motion 1. Using dephasing

2. Using spontaneous emission

Physical Implementation of the coupled Householder reflection

21

†

0 ( )

( ) ( )

Vf tH

V f t D t

2nm nmt V f t

N3

1 2 M t

HAMILTONIAN OF THE SYSTEM Rabi frequency

Morris-Shore Transformation

The system is transformed by Morris-Shore transformation into a set ofM independent two-state systems, and a set of K=N-M decoupled dark state

1d

2e Me

Mb1b2b

Kd

t1e

M21

The propagator in the original basis

N NM

MN M

U UU

U U

1

( 1)M

N n n nn

a b b

U I=

1

*

1

M

NM n n nn

Mi

MN n n nn

b b e

e b e b

U

U

*

1

( 1)M

iM n n n

n

e a e e

U I=

Connects states within the lower set

Connects states within the upper set

Mix states from the lower and upper sets

COUPLED HOUSEHOLDER REFLECTIONCOUPLED HOUSEHOLDER REFLECTION

0

1,2, ,n

n

in

b

a e n M

Of particular significance is the special case when the Cayley-Klein parameters bn

(n=1,2…,M) are all equal to zero

1

( 1)n

Mii

M n nn

e e e e

U I=

1

( 1)n

Mi

N n nn

e b b

U I=

0NM MN U U

All transition probabilities in the MS basis, as well as these in the originalbasis from the lower set to the upper set, vanish

The propagator in the lower setsThe propagator in the lower sets The propagator in the upper setsThe propagator in the upper sets

Decomposition of U(N) transformations by Decomposition of U(N) transformations by Coupled HRCoupled HR

Any N-dimensional unitary matrix U can be expressed as a product of one coupled HR and one phase gate

1 1 1 1 1 1, , ; , , , , ,N N U M F

THANK YOU