Quantum Logic with Photons

Yoni Shalibo

By

12/3/2008

OutlineOutline

� Introduction to quantum computation

� Theoretical implementations of quantum gates (KLM)

� Experimental CNOT gates

The QubitThe Qubit

Classical bits can be in one of two states:

0 1Or

Quantum bits (Qubits) can be in a superposition of states:

10 ba +=Ψ , where 122=+ ba

Examples of Qubit states:

Optics:Photon polarization modes, Photon spatial modes

Condensed Matter: States of Josephson junction circuit, two-level atoms (Ion traps), Spins in magnetic field (NMR)

The PoincarThe Poincaréé sphere sphere

A two dimensional mapping of the complex amplitudes a,b into a

unit sphere, where the states are polarization states of photons:

V

H

N =1

2H − V( )

M =1

2H + V( )

VeH

VbHai )2/sin()2/cos( θθ φ+=

+=Ψ

H - Horizontal Polarization

V - Vertical Polarization

• Pure states lie on the sphere, while mixed states lie within it.

φ

θ

H

VM

Qubit GatesQubit Gates

NOT:

Hadamard:

π-Phase shift:

• Represented by a unitary transformation.

• Are thus reversible (unlike some classical gates).

ExamplesExamplesVaHbVbHa +→+ a

b

→

0 1

1 0

a

b

Pauli σX rotation:

VbHaVbHa −→+ Pauli σZ rotation:

H →1

2H + V( ) Hadamard:V → 1

2H − V( )

oo 18090YX → oo 18090 YX →

o180Z

o180X

a

b

→

1 0

0 −1

a

b

a

b

→

1

2

1 1

1 −1

a

b

Multiple Qubit GatesMultiple Qubit Gates• Represented by 2Nx2N unitary transformation.

• Visualized by higher dimensional Poincare spheres

• Can be constructed from the universal C-NOT gate plus single qubit gates

ControlledControlled--NOT (CNOT) GateNOT (CNOT) Gate

=

0100

1000

0010

0001

CNOTUTransformation:

Operation on binary inputs:

A two-qubit operation on the input state: 11100100 dcba +++=Ψ

“Control” qubit

“Target” qubit

1011

1110

0101

0000

→

→

→

→

baaba ⊕→ ,,

2mod)( baba +≡⊕where

Quantum CircuitsQuantum Circuits

The laws of quantum circuits:

• Acyclic (no loops) – one cannot feedback the output into the input

• “Wires” cannot be joined together (reversibility)

• No “cloning” – The circuit cannot make copies of an arbitrary qubit state

Quantum Parallelism?Quantum Parallelism?A quantum circuits can evaluate the outcome of all possible input states simultaneously:

The catch – the output state collapses in the measurement, thus only one outcome can be measured.

One can still interfere different outcomes to exploit this parallelism

� All the values of f(x) are encoded in the output state!

fUx

y

x

yi ⊕ f (x)i

Ψout =1

2Nxi

i

∑ f (xi)

0K 0 H⊗n1

2Nx i

i

∑

0K 0

M

Optical Quantum ComputationOptical Quantum Computation

• Photons exhibit quantum phenomena (as we learned…)• They do not interact strongly with each other and with matter (bad for coupling between them – photon gates usually work with probability smaller than one)

• They can be guided along great distances with low loss (good for communication)

Why photon qubits?

Qubit state representation

The Dual-rail representation:Ψ = a10 + b 01 ≡ a 0 + b 1

01 No photon in the mode 1, One photon in mode 2

Mode 1Mode 2

One photon in mode 1, No photon in the mode 210

Such as polarizationor spatial modes

Phase shifters

Retard a photon mode by passing it through a medium with a higher index of refraction.

Optical elements for quantum gates:Optical elements for quantum gates:

Rz (∆) = exp(−iωZ∆ /2) =1 0

0 e iφ

where :∆ = (n − n0)d /c0

Perform an arbitrary rotation about the Z axis:

Beam splitters (BS)

“Half” silvered mirrors. Reflect a fraction R, and transmit (1-R) of the beam. R is usually parameterized by an angle.

Perform an arbitrary rotation about the X axis:RX (θ) =cos(θ) −sin(θ)sin(θ) cos(θ)

φ

θ

For polarization encoded qubits, a BS transformation can be done by a λ/2 waveplate.

For polarization encoded qubits, a PS transformation can be done by a λ/2 waveplate, aligned with one of the polarization directions.

Polarizing beam splitters (PBS)

Transmits one mode of polarization and reflects the other. Used to convert from polarization to spatial encoding and vice versa.

Non-linear Kerr media

Induce photon interaction by intensity sensitive change in the refraction index:

InnIn 10)( +=

R(θ,φ) =cos(θ) −eiφ sin(θ)

e−iφ sin(θ) cos(θ)

A generalized beam splitter induces an arbitrary rotation (can be accomplished by combining a regular BS with Phase shifters):

θ ,φ

H + VH

V

H

V

H + V

OutlineOutline

� Introduction to quantum computation

� Theoretical optical implementations of quantum gates

(KLM)

� Experimental optical CNOT gates

E. Knill, R. Laflamme, G.J. Milburn, Nature 409, 46 (2001)

“A Scheme for efficient quantum computation with linear optics”:

Three main results:

• Non-deterministic quantum computing is possible with linear optics

• The probability of success can be increased arbitrarily close to one

• A highly efficient resource for accurately encoded qubits can be achieved using quantum coding

The first two results will be briefly explained hereafter.

NonNon--deterministic Nonlinear Sign flip (with linear optics)deterministic Nonlinear Sign flip (with linear optics)

Performs the following operation :

Ψ = a 0 + b1 + c 2 →a 0 +b1 −c 2

The gate works if and only if detector 2 counts one photon and detector 3 counts no photons. This happens with probability 1/4

The quantum circuit:

θ1 =22.5o;φ1 =0

o;

θ2 =65.5302o;φ2 =0

o;

θ3 =−22.5o;φ3 =0

o;

φ4 =180o;

Single mode state

mode 1

mode 2

mode 3

Construction of unitary transformation of the NS gate:

u =

1 0 0

0 cos(θ3) −sin(θ3)0 sin(θ3) cos(θ3)

cos(θ2) −sin(θ2) 0sin(θ2) cos(θ2) 0

0 0 1

e iφ4 0 0

0 cos(θ1) −sin(θ1)0 sin(θ1) cos(θ1)

=

1− 2 2−1/ 4 3/2− 2

2−1/ 4 1/2 1/2− 1/2

3/2− 2 1/2− 1/2 1/2−1/2

U operates on “tri-rail” states (Fock states of 3 modes occupying 1 photon)!

0

1

0

⊗

1

0

0

⊗

1

0

0

→ u

0

1

0

⊗ u

1

0

0

⊗ u

1

0

0

Input state representation:

( )44 344 21

K444 3444 21

K

n

n

naan

⊗⊗

⊗

≡⊗⊗⊗≡= ++

0

0

1

0

0

1

0

1

0

0,0,10,0,10,1,00,0,00,1, 21

By linearity of the BSs, the U transformation operates on each component:

0

1

0

⊗

1

0

0

⊗

1

0

0

→ u

0

1

0

⊗ u

1

0

0

⊗ u

1

0

0

=

2−1/ 4

1/2

1/2− 1/2

⊗

1− 2

2−1/ 4

3/2− 2

⊗

1− 2

2−1/ 4

3/2− 2

A( 2 → 2 ) =1/2× 1− 2( )2+ 2−1/ 4 × 2−1/ 4 × 1− 2( )+ 2−1/ 4 × 1− 2( )× 2−1/ 4 = −1/2

0

1

0

→ u

0

1

0

=

2−1/ 4

1/2

1/2− 1/2

⇒ A( 0 → 0 ) =1/2

0 photons are in mode 1Case 1:

1 photons are in mode 1Case 2:

2 photons are in mode 1Case 3:

0

1

0

⊗

1

0

0

→ u

0

1

0

⊗ u

1

0

0

=

2−1/ 4

1/2

1/2− 1/2

⊗

1− 2

2−1/ 4

3/2− 2

A(1 → 1 ) =1/2× 1− 2( )+ 2−1/ 4 × 2−1/ 4 =1/2

Conditional Sign flip:Conditional Sign flip:1. Assume qubits Q1 and Q2 are both in state . Modes 1,3 will therefore be in state .

2. The two photons emerge “bunched” from BS1, i.e. :

3. The NS gates will induce a phase shift to the above state with probability 1/16.

4. will emerge from BS2. In all other cases NS do nothing, and the state will not change.

111

13

ΨBS1

= 2013

− 0213

−1113

ab → −1( )ab ab

Bunching in BSsBunching in BSs

BS(45o) =1

2

1 −1

1 1

11ab

≡1

0

a

⊗0

1

b

→ uBS1

0

a

⊗ uBS0

1

b

=1

2

1

1

a

⊗1

−1

b

≡ 2 ×1

220

ab− 02

ab( )+1

211

ab− 11

ab( )=1

220

ab− 02

ab( )

Bunching!

NearNear--deterministic conditional sign flip via deterministic conditional sign flip via TeleportationTeleportation

Bell statepreparation

Bell measurement

• The NS gates operate on qubits state only if it works.

• Conditional sign flip will work only if teleportation works

• Teleportation of each qubit state works with probability 1/2

OutlineOutline

� Introduction to quantum computation

� Theoretical optical implementations of quantum gates (KLM)

� Experimental optical CNOT gates

J.L O’Brien, G.J. Pride, A.G. White, T.C. Ralph, D.Branning, Nature 426, 264 (2003)

“Demonstration of an all-optical quantum controlled-NOT gate”

1/2 1/2

1/3

1/3

1/3

Cin

Tin

C0

C1

T0

T1

Cout

Tout

The following circuit performs a CNOT gate on qubits C, T using linear optics with success probability 1/9:

• Numbers indicate reflection probability• Blue side of BS reflects with a π phase shift

Controlqubit

Targetqubit

• The gate works only when a single photon is detected on each qubit.• When one photon exists in the upper mode and zero photons in the lower mode, the qubit is in the state . Otherwise it is in state .0 1

Calculation of the CNOT gate operation.Calculation of the CNOT gate operation.

C0,out+ =

1

3C0,in

+

C1,out+ =

1

3−C1,in

+ + T0,in+ + T1,in

+( )

Ψout

= aC0,out+T0,out

+ + bC0,out+T1,out

+ + cC1,out+T0,out

+ + dC1,out+T1,out

+( )0000Plugging output operators and taking only terms with single photon in each qubit:

Ψout

= a1010 + b1001 + c 0110 + d 0101

Fock basis

Output modes operators:

T0,out+ =

1

3C1,in

+ + T0,in+( )

T1,out+ =

1

3C1,in

+ + T1,in+( )

Ψin

= a1010 + b1001 + c 0101 + d 0110

≡ aC0,in+T0,in

+ + bC0,in+T1,in

+ + cC1,in+T0,in

+ + dC1,in+T1,in

+( )0000

Input state:

Output state:

C T

Experimental setup for equivalent CNOT circuitExperimental setup for equivalent CNOT circuit

- HWP

CinTin

Automated Quantum State Tomography

Cout

Tout

“1/3”OA: 62.5O

“1/2”OA: 22.5O

“1/2”OA: 22.5O

State Preparation

- QWP- PBS - SPCM

1. The qubits are prepared in the polarization basis.

2. First and last HWPs perform Hadamard transformation, which is equivalent to a balanced BS.

3. The 1st PBS separates the two polarization modes into two spatial modes, thus allowing modes from the two qubits to interfere non classically at the 1/3 BS.

4. The 2nd HWP performs a polarization rotation, which is equivalent to a 1/3 BS in the polarization basis.

5. The 2nd PBS recombines different spatial modes into the same spatial mode.

Experimental VS Ideal output:Experimental VS Ideal output:

The probability of getting the correct result is low in the case of non-classical interference (i.e., more than one photon): P≈75%

In the case of single photon, the probability of success is ≈95%

Experimental: Ideal:

“Probabilistic quantum logic using polarizing beam splitters”

T.B. Pittman, B.C. Jacobs, J.D. Franson, Phys. Rev. A 64, 062311 (2001)

destructive CNOT + Quantum Encoder = nondestructive CNOT + Quantum Encoder = non--destructive CNOTdestructive CNOT

in2'

Control:

Dc

2’

c

a

2 out2

φab+

b

φab+ =

1

2H

aH

b+ V

aV

b( )

in3'

Target:

Dd

3’

d

3 out3

The following gate performs CNOT with probability ¼:

Destructive CNOT gateDestructive CNOT gate

in3'

Target:

Control: φb

Dd

3’

b

d

3 out3

45o rotated PBS

Performs the following operation with a probability of ½: 00 → 0

01 → 1

10 → 1

11 → 0

targettargetcontrol

The gate works if Dd detects 1AO1 photon!

PBS as twoPBS as two--qubit gate:qubit gate:

in3'

φb

Dd

3’

b

d

3 out3

Ψ3'b

= a H3'+ b V

3'( )⊗1

2H

b+ V

b( )=1

2a H

3'H

b+ a H

3'V

b+ b V

3'H

b+ b V

3'V

b( )→ PBS →

1

2a H

3H

d+ a H

3V

3+ b V

dH

d+ b V

3V

d( )

≡1

2a H

3H

d+ b V

3V

d+ ΨII( )

Note: given that Dd detects 1AO1 photon, if the two inputs are the same, modes dand 3 will be the same (called Quantum Parity Check)

Ψ3'b

= a H3'+ b V

3'( )⊗ Hb

Case 1:

a

2F

dF

3+ S

dS

3( )+b

2F

dF

3− S

dS

3( )

+

1

2ΨII

H-V Basis: F-S Basis:

a

2F

3'− S

3'( )+b

2F

3'+ S

3'( )

⊗

1

2F

b− S

b( )

PBS PBS

12

Hd

a H3+ b V

3( )+ V d a V 3 + b H 3( )[ ]+1

2ΨII

Ψ3'b

= a H3'+ b V

3'( )⊗Vb

Case 2:

a

2F

dF

3− S

dS

3( )+b

2F

dF

3+ S

dS

3( )

+

1

2ΨII

H-V Basis: F-S Basis:

a

2F

3'− S

3'( )+b

2F

3'+ S

3'( )

⊗

1

2F

b+ S

b( )

PBS PBS

12

Hd

a V3+ b H

3( )+ V d a H 3 + b V 3( )[ ]+1

2ΨII

Up to a single qubit NOT gate (case Dd measures a V photon), the gate works with a probability of ½.

H

VFS

Quantum EncoderQuantum EncoderGoal: encode the state of the control into an additional photon (no cloning going on here)

in2'

Control:

Dc

2’

c

a

2 out2

φab+

outb

b

Ψ2'b

= a H2'

+ b V2'

→ a H2

Hb+ b V

2V

b

φab+ =

1

2H

aH

b+ V

aV

b( )

The following gate performs encoding (via quantum parity check) with probability 1/2:

(After 1AO1 photon post selection)

“Experimental controlled-NOT logic gate for single photons in the coincidence basis”

T.B. Pittman, M.J. Fitch, B.C. Jacobs, J.D. Franson, Phys. Rev. A 68, 032316 (2003)

Scheme:Scheme:

• It is very difficult to produce heralded (on demand) entangled pairs, but easy to produce heralded single photons (parametric down conversion)• The new scheme also contains a quantum encoder, but there is a drawback: In order for the encoder to work, 1AO1 photon must be detected in each output –demolition measurement.

Quantum demolition measurement

Experimental setup:Experimental setup:

Coherent state with an average of zero photons (single photon source)

Fiber polarization rotator

ResultsResults

The counts were acquired during 1500s.

BibliographyBibliography

M.A. Nielsen, I.L Chuang, “Quantum computation and quantum information”, Cambridge university press, Cambridge (2000)

E. Knill, R. Laflamme, G.J. Milburn, “A scheme for efficient quantum computation with linear optics”, Nature 409, 46 (2001)

J.L O’Brien, G.J. Pride, A.G. White, T.C. Ralph, D.Branning, “Demonstration of an all-optical quantum controlled-NOT gate”, Nature 426, 264 (2003)

T.B. Pittman, M.J. Fitch, B.C. Jacobs, J.D. Franson, “Experimental controlled-NOT logic gate for single photons in the coincidence basis”, Phys. Rev. A 68, 032316 (2003)

T.B. Pittman, B.C. Jacobs, J.D. Franson, “Probabilistic quantum logic using polarizing beam splitters”, Phys. Rev. A 64, 062311 (2001)