FFromrom F Flippinglipping Q Qubitsubits toto PProgrammable rogrammable QQuantum uantum

PProcessorsrocessors

FFromrom F Flippinglipping Q Qubitsubits toto PProgrammable rogrammable QQuantum uantum

PProcessorsrocessors

Drinking party Budmerice, 1st May 2003

Vladimír Bužek, Mário Ziman, Mark Hillery, Reinhard Werner, Francesco DeMartini

FFlipping a lipping a BBit – it – NNOT OT GGateateFFlipping a lipping a BBit – it – NNOT OT GGateate

0

FFlipping a lipping a BBit – it – NNOT OT GGateateFFlipping a lipping a BBit – it – NNOT OT GGateate

01

Universal NOT GateUniversal NOT Gate

yy^

• NOT gate in a computer basis:

0 1 ; 1 0R R= - =

0 1 0 1y a b y b a^ * *= + ® = -

Poincare sphere – state space

is antipode of

| 0y y^ =

Universal NOT Gate: ProblemUniversal NOT Gate: Problem

yy^is antipode of

- Spin flipping is an inversion of the Poincare sphere- This inversion preserves angels- The Wigner theorem - spin flip is either unitary or anti-unitary operation- Unitary operations are equal to proper rotations of the Poincare sphere- Anti-unitary operations are orthogonal transformations with det=-1- Spin flip operation is anti-unitary and is not CP- In the unitary world the ideal universal NOT gate which would flip a qubit in an arbitrary (unknown) state does not exist

Measurement-based vs Quantum ScenarioMeasurement-based vs Quantum Scenario

Measurement-based scenario: optimally measure and estimate the state then on a level of classical information perform flip and prepare the flipped state of the estimate

Quantum scenario: try to find a unitary operation on the qubit and ancillas that at the output generates the best possible approximation of the spin-flipped state. The fidelity of the operation should be state independent (universality of the U-NOT)

Quantum ClickologyQuantum Clickology

• measurement conditional distribution on a discrete state space of the aparatus A: observables with eigenvalues i

O

System Apparatus

Measurement

• a priori distribution on the state space of the system joint probability distribution

( )0 ˆp r

• Bayesian inversion from distribution on A to distribution on

• – invariant integration measuredW

Quantum Bayesian InferenceQuantum Bayesian Inference

• Reconstructed density operator given the result i

( ) ( )ˆˆ ˆ ˆ, , dest ip Or r J j r l WW

= ò

K.R.W. Jones, Ann. Phys. (N.Y.) 207, 140 (1991)V.Bužek, R.Derka, G.Adam, and P.L.Knight, Annals of Physics (N.Y.), 266, 454 (1998)

Optimal Reconstructions of QubitsOptimal Reconstructions of Qubits

• average fidelity of estimation

12

NF

N+

=+

• Estimated density operator on average

1 ˆˆ ˆ2est

ss I

• Construction of optimal POVM’s – maximize the fidelity F

• POVM via von Neumann projectors – Naimark theorem

• Optimal decoding of information

• Optimal preparation of quantum systems

S.Massar and S.Popescu, Phys. Rev. Lett. 74, 1259 (1995)R.Derka, V.Bužek, and A.K.Ekert, Phys. Rev. Lett 80, 1571 (1998)

2 12

Ns F

N

Quantum Scenario: Universal NOT GateQuantum Scenario: Universal NOT Gate

Theorem

Among all completely positive trace preserving mapsThe measurement-based U-NOT scenario attains the highest possible fidelity, namely

( ) ( ): NT S H S HÄ+ ®

( ) ( )1 2 .F N N= + +

V.Bužek, M.Hillery, and R.F.Werner Phys. Rev. A 60, R2626 (1999)

Quantum Logical Network for U-NOT Quantum Logical Network for U-NOT

( ) 1 13 3

outa Ir r ^= +

V.Bužek, M.Hillery, and R.F.Werner, J. Mod. Opt. 47, 211 (2000)

C-NOT gate:

No-Cloning Theorem & U-QCM No-Cloning Theorem & U-QCM

( ) 2 13 6

outa Ir r= +

W.Wootters and W.H.Zurek, Nature 299, 802 (1982)V.Bužek and M.Hillery, Phys. Rev. A 54, 1844 (1996)S.L.Braunstein, V.Bužek, M.Hillery, and D.Bruss, Phys. Rev. A 56, 2153 (1997)

( ) 2 13 6

outb Ir r= +

( ) 1 13 3

outc Ir r ^= +

U-NOT via OPA U-NOT via OPA

† † † † . .IH i a b a b hcy yy yc ^ ^

é ù= - +ê úë ûh

• Original qubit is encoded in a polarization state of photon

• This photon is injected into an OPA excited by mode-locked UV laser

• Spatial modes and are described by the operators and 1k 2k a b

• Initial state of a qubit is 1 1

† 0,0 1,0k ka

• The other mode is in a vacuum 20,0 k

• Under given conditions OPA is SU(2) invariant

• Evolution – stimulated emission

21 2 1 2 1 211,0 0,0 1,0 0,0 ( 0,12,02 1,1 1,0 )k k k kk k k kU gÄ Ä + Ä + Ä;

• Evolution – spontanous emission

1 2 1 2 1 2 1 20,0 0,0 0,0 0,0 ( 1,0 0,1 0,1 1,0 )k k k k k k k kU gÄ Ä + Ä + Ä;

2 2

† 0,0 0,1k kb

Optical Parametric AmplifierOptical Parametric Amplifier

Z

BS1

M

D2*

D2

Da

MP

k2k1

Q Q

QWP

2

PBS2

*

PBS2

nm

kP

BBOType II

WP2*

Q

Db

A.Lamas-Linares, C.Simon, J.C.Howell, and D.Bouwmeester, Science 296, 712 (2002)F.DeMartini, V.Bužek, F.Sciarino, and C.Sias, Nature 419, 815 (2002)

Optimal Universal-NOT GateOptimal Universal-NOT Gate

1R

FR

=+

There is Something in This Network There is Something in This Network

S.L.Braunstein, V.Bužek, and M.Hillery, Phys. Rev. A 63, 052313 (2001)

Quantum Information Distributor Quantum Information Distributor

2( ) 21

2out IN Nab b

r a ræ ö÷ç= + +÷ç ÷çè ø

S.L.Braunstein, V.Bužek, and M.Hillery, Phys. Rev. A 63, 052313 (2001)

- Covariant device with respect to SU(2) operations- POVM measurements eavesdropping- programmable beamsplitter

2( ) 22

2out IN Nab a

r b ræ ö÷ç= + +÷ç ÷çè ø

( )3 2

2 ( 2 )out T NI

N Nab ab

r r-

= +

POVM Measurement POVM Measurement

V.Bužek, M.Roško, and M.Hillery, unpublished

Model of Classical ProcessorModel of Classical Processor

Classical processor

0010110111

1110010110

1101110110

Heat

inrdata register

program register

output register

[ ]out inTr r=

outr

Quantum Processor Quantum Processor

Quantum processor

dr

px

dr¢

px¢

Quantum processor – fixed unitary transformation Udp

d – data system, S(d) – data statesp – program system, S(p) – program states

data register

program register

output data register

Quantum processor

Two ScenariosTwo Scenarios

• Measurement-based strategy - estimate the state of program

• Quantum strategy – use the quantum program register

conditional (probabilistic) processors

unconditional processors

1NF

N d+

=+

C-NOT as Unconditional Quantum ProcessorC-NOT as Unconditional Quantum Processor

d

| p

dCNOT 0 0

CNOT 1 1x

• program state

• program state

• general pure state

• unital operation, since

• program state is 2-d and we can apply 2 unitary operations

220 1 d d xp dp xp d ra b r a r b s r s¢= +X = + Þ a

[ ] 22x xa b s sF = + =1 1 1 1

0 implemented, i.e. d d d1 r r r¢® =Þ

1 implemented, i.e. dx d x d xr rs s r s¢® =Þ

Question Question

Is it possible to build a universal programmable quantum gate array which take as input a quantum state specifying a quantum program and a data register to which the unitary operation is applied ?

on a qubit an Ą number of operations can be performed

No-go Theorem No-go Theorem

( ) ( ) ,dp U Ud pP U yy y ¢Ä X = Ä X

dy

U pX

ddUy y¢ =

,U pY¢X

Pdp

• no universal deterministic quantum array of finite extent can be realized

• on the other hand – a program register with d dimensions can be used to implement d unitary operations by performing an appropriate sequence of controlled unitary operations

M.A.Nielsen & I.L.Chuang, Phys. Rev. Lett 79, 321 (1997)

C-NOT as Probabilistic Quantum Processor C-NOT as Probabilistic Quantum Processor

0

1

0

1

y

0

'y

j

( )i / 2 i / 210 1

2e ej jj -= +

i2 zU ejs

j-

=

•Vidal & Cirac – probabilistic implementation of

G.Vidal and J.I.Cirac, Los Alamos arXiv quant-ph/0012067 (2000)G.Vidal, L.Mesanes, and J.I.Cirac, Los Alamos arXiv quant-ph/0102037 (2001).

dyU j

C-NOT as Probabilistic Quantum Processor C-NOT as Probabilistic Quantum Processor

0

1

0

1

y

1

j-

( )i / 2 i / 210 1

2e ej jj -= +

Correction of the error – new run of the processor with |2

dyU j-

“universal” processor

projective yes/no measurement

probability of success:

2

1

1,

D

dp k k k k l k l klk D

U U TrU U

2

1

1,

D

kpk

yes no ID

M

success 2

1

DP

Universal Probabilistic ProcessorUniversal Probabilistic Processor

- Quantum processor Udp

- Data register d, dim Hd = D

- Quantum programs Uk = program register p, dim Hp =

• Nielsen & Chuang: - N programs N orthogonal states

- Universal quantum processors do not

• BHZ:- Probabilistic implementation

- {Uk} operator basis,

- program state

k

kkkk DUUUU Tr

1,

k

kk U

Example: Data register = qudit, program register = 2 qudits

1

0

2exp

D

s

mnk sns

N

ismUU

1

0

1 2exp

D

k mns

isms s n

ND

2N D

Set of operations

Implementation of Maps via Unconditional Quantum Processors

Implementation of Maps via Unconditional Quantum Processors

U

( )PS H

P

†' [ ] K Km mm

r r r= F = å

( )HS ( )HS

G

Description of Quantum ProcessorsDescription of Quantum Processors

• definition of Udp via “Kraus operators”

• normalization condition

• induced quantum operation

• general pure program state

• can be generalized for mixed program states

:kl dpp pA l U k

dp kld p d pl

U k A l

1 2 1 2

† k l k l k k dl

A A 1

†d d k d kl d kl

l

A A

kp pk

k † d d d l d l

l

A A

l dp k klp pk

A l U A

Inverse Problem: Quantum SimulatorsInverse Problem: Quantum Simulators

“Given a set x of quantum operations . Is it possible to design a processor that performs all these operations?”

1. Continous set of unitaries = question of universality

2. Phase damping channel = model of decoherence

3. Amplitude damping channel = model exponetial decay

NO

YES

NO

Quantum LoopsQuantum Loops

Quantum processor

data

program

Analogy of “for-to” cycles in classical programming

• Introducing loops control system – “quantum clocks”• Halting problem – how (when) to stop the computation process

Conclusions & Open QuestionsConclusions & Open Questions

• programmable quantum computer – programs via quantum states programs can be outputs of another QC

• some CP maps via unconditional quantum processors

• arbitrary CP maps via probabilistic programming

• controlled information distribution (eavesdropping)

• simulation of quantum dynamics of open systems

• set of maps induced by a given processor (loops)

• quantum processor for a given set of maps

• quantum multi-meters

M.Hillery, V.Buzek, and M.Ziman: Phys. Rev. A 65, 022301 (2002).

M.Dusek and V.Buzek: Phys. Rev. A 66, 022112 (2002).

M.Hillery, M.Ziman, and V.Buzek: Phys. Rev. A 66, 042302 (2002)