Photonic Quantum Information - Chalmers oc… · Shor’s factoring algorithm → Which of these do...

ExperimentalBosonSampling

Andrew WhiteUniversity of Queensland

qtquantum.info

labCENTRE FORENGINEEREDQUANTUM SYSTEMSAUSTRALIAN RESEARCH COUNCILCENTRE OF EXCELLENCE

:

Photonic Quantum Information, Computation

& Simulation

15,470 km

Nature Physics 11, 249 (2015)Scientific Reports 4, 06955 (2014)Physical Review Letters 112, 143603 (2014)Physical Review Letters 112, 020401 (2014)Nature Communications 3, 625 (2012) Physical Review Letters 106, 200402 (2011)New Journal of Physics 13, 053038 (2011)Proceedings of the National Academy of Sciences 108, 1256 (2011)

Quantum foundations

Physical Review Letters 114, 173603 (2015)Physical Review Letters 112, 143603 (2014)Physical Review Letters 110, 250501 (2013)Physical Review A 88, 013819 (2013) Optics Express 21, 13450 (2013)Optics Express 19, 22698 (2011)Journal of Modern Optics 58, 276 (2011)Optics Express 19, 55 (2011)

Quantum photonics

Physical Review Letters 114, 090402 (2015)Physical Review A 89, 042323 (2014) Physical Review Letters 111, 230504 (2013)Physical Review Letters 106, 100401 (2011)New Journal of Physics 12 083027 (2010)Nature Physics 5, 134 (2009)Physical Review Letters 101, 200501 (2008)

Quantum computation

Nature Communications 5, 04145 (2014) Science 339, 794 (2013) Nature Communications 3, 882 (2012)New Journal of Physics 13, 075003 (2011)Physical Review Letters 104, 153602 (2010)Nature Chemistry 2, 106 (2010)

02

-2-4-6

-8

46

8 43

21

Probab

ility

0.4

0.2

Quantum simulation& emulation

AzwaZakaria

AlessandroFedrizzi

Christina Giarmatzi

AleksandrinaNikolova

MichaelVanner

JuanLoredoThe

TeamfromJuly

MarceloAlmeida

TillWeinhold

IvanKassal

DevonBiggerstaff

GeoffGillett

MartinRingbauer

MarkusRambach

Professor,Heriot-Watt UniversityScotland+ £1.5M Fellowship

JacquiRomero

Senior Postdoc,Oxford UniversityEngland

Industry,Australia

& you ?

Part 1:Quantum Information:

What is it?Why do it?

Classical: bits ≡ binary digits

0 13-bit register can storeone number, from 0–7

What is information?

information that can be stored in a system is proportional to logb N

Hartley, Bell System Tech. J. 7, 535–563 (1928)

Shannon, Bell System Tech. J. 7, 379–423 (1948)

b is the number of states that can be stored in the unitN is the number of possible states of that system

for three bits, log2 8 = 3


0 13-bit register can storeone number, from 0–7N bits: one N-bit number

Quantum: qubits0

1


qubit sphere

also known as Poincaré sphereBloch sphere

Hartley, Bell System Tech. J. 7, 535–563 (1928)

Shannon, Bell System Tech. J. 7, 379–423 (1948)

Poincaré, Leçons sur la théorie Mathématique de la lumière II,(1897)

Bloch, Phys. Rev. 70, 460 (1946)

0

10+1 0+i1


Qubit Sphere

0

10+1 0+i1

Qubit Sphere


Pure states are a collective delusion of theorists

0

10+1 0+i1

Qubit Sphere


Hermitean

non-negative eigenvalues

population

unity trace

0

10+1 0+i1

Qubit Sphere


Hermitean

non-negative eigenvalues

population

coherence

unity trace

0

10+1 0+i1


purity

Qubit Sphere

ququart

qutrit

Density Matrix

two qubits

or

quoct

three qubits

or


0 13-bit register can storeone number, from 0–7N bits: one N-bit number

must be reversible

CNOTINPUT OUTPUT A B A B 0 0 0 0 0 1 0 1 1 0 1 1 1 1 1 0

0+1 0 00+11

Quantum gates

Quantum: qubits0

10+1 0+i1

3-qubit register can store eight numbers a |000 + b |001+ c |010 + d |011+ e |100 + f |101+ g |110 + h |111

N qubits: all possible 2N numbers

300-qubit register can store more information than number of atoms in universe

Classical gates

irreversible

INPUT OUTPUTA B A and B 0 0 1 0 1 1 1 0 1 1 1 0

NAND Toffoli

reversible

INPUT OUTPUT A B C A B C0 0 0 0 0 0 0 0 1 0 0 1 0 1 0 0 1 0 0 1 1 0 1 1 1 0 0 1 0 0 1 0 1 1 0 1 1 1 0 1 1 1 1 1 1 1 1 0

Important problems benefit fromthis exponential scaling:

factoring Physical Review Letters 99, 250505 (2007)

simulation Nature Chemistry 2, 106 (2010)


quantum.infoWhat is classical computing?

insoluble

soluble

quantum.infoWhat is classical computing?

insoluble

soluble

Nomenclature easy = “tractable” = “efficiently computable” hard = “intractable” = “not efficiently computable”

insoluble

easy

factoring

travellingsales-man

hard

quantum.info

insoluble

soluble

insoluble

hard

easy

What is quantum computing?


factoring

travellingsales-man

quantum.info

Software updates, email, online banking, and the entire realm of public-key cryptography and digital signatures rely on just two cryptography schemes to keep them secure—RSA and elliptic-curve cryptography (ECC). They are exceedingly impractical for today’s computers to crack, but if a quantum computer is ever built it would be powerful enough to break both codes. Cryptographers are starting to take the threat seriously, and last fall many of them gathered at the PQCrypto conference, in Cincinnati, to examine the alternatives. —IEEE Spectrum, January 2009

insoluble

soluble

insoluble

hard

easy

What is quantum computing?


In a world of quantum computers then we would need to abandon most of the algorithms that are in use today. —Robin Balean, VeriSign Australia, 2008

factoring

travellingsales-man

quantum.infoWhy do quantum computing?

Aaronson’sTrilemma

Shor’s factoring algorithm →


• Extended Church-Turing Thesis—foundation of theoretical computer science for decades—is wrong

The assertion of the Church-Turing thesis might be compared, for example, to Galileo and Newton’s achievement in putting physics on a mathematical basis. By mathematically defining the computable functions they enabled people to reason precisely about those functions in a mathematical manner, opening up a whole new world of investigation. —Michael Nielsen, UQ, 2004



“All computational problems that are efficiently solvable by realistic physical devices,are efficiently solvable by a probabilistic Turing machine”

Statement:

quantum.info

• Extended Church-Turing Thesis—foundation of theoretical computer science for decades—is wrong• Quantum mechanics is wrong

It’s entirely conceivable that quantum computing will turn out to be impossible for a fundamental reason. This would be much more interesting than if it’s possible, since it would overturn our most basic ideas about the physical world. The only real way to find out is to try to build a quantum computer. Such an effort seems to me at least as scientifically important as (say) the search for supersymmetry or the Higgs boson. I have no idea—none—how far it will get in my lifetime. —Scott Aaronson, MIT, 2006

Why do quantum computing?



“All computational problems that are efficiently solvable by realistic physical devices,are efficiently solvable by a probabilistic Turing machine”

quantum.info

• Extended Church-Turing Thesis—foundation of theoretical computer science for decades—is wrong• Quantum mechanics is wrong• A fast classical factoring algorithm exists

I love Shor’s algorithm because it proves to me that a fast classical factoring algorithm must exist. – Peter Sarnak, Princeton, 2012




I believe that there is a polynomial-cost answer to the strong correlation problem. We just haven't found it yet. – Gus Scuseria, Nature Chemistry, 2015

quantum.info

All three seem crazy. At least one is true!





Which of these do you think is the case?

a. The Extended Church-Turing thesis is wrongb. Quantum mechanics is wrongc. Fast factoring exists & QC’s are not necessaryd. Two of the abovee. All of the abovef. Shhh, I’m sleeping.

How do we experimentally test this trilemma?

quantum.info



Computer scientistsand / or

Theoretical physicistsand /or

Mathematicians & chemistswill be

badly upset




Which of these do you think is the case?

a. The Extended Church-Turing thesis is wrongb. Quantum mechanics is wrongc. Fast factoring exists & QC’s are not necessaryd. Two of the abovee. All of the abovef. Shhh, I’m sleeping.

How do we experimentally test this trilemma?

quantum.info



Church-Turing-Deutsch principle: Any physical process can be efficiently simulated on a quantum computer

• Quantum mechanics is wrong• A fast classical factoring algorithm exists




quantum.info





Nuclei move on the electronic potential energy surface (PES)

Knowledge of PES enables:▸ Minima: equilibrium structures▸ Saddle points: transition states ▸ Reaction rates & mechanisms▸ PES characterise most of physical chemistry

Tree of approximation methodsfor quantum chemistry




Quantum Simulation

quantum.info

Lanyon, et al.,Nature Chemistry 2, 106 (2010)





Nuclei move on the electronic potential energy surface (PES)

Knowledge of PES enables:▸ Minima: equilibrium structures▸ Saddle points: transition states ▸ Reaction rates & mechanisms▸ PES characterise most of physical chemistry

Tree of approximation methodsfor quantum chemistry




Quantum Simulation

experiment & theoryagree to 1 part in 106

2-qubit photonic quantum computer:calculate energy to 20 bits

quantum.info




Quantum computers are interesting physical systems in their own right





quantum.info

The Economist, Saturday June 20, 2015


QC Simulation: semiconductors, …

QC Factoring, searching

QKD Banks, government

2005 2010 2015 2020 2025 2030 2035

38Courtesy of R. Beausoleil, HP Labs, Palo Alto

QC Simulation: semiconductors, …

QC Factoring, searching

QKD Banks, government

2005 2010 2015 2020 2025 2030 2035

Few-qubit QIP Distributed algorithms & entanglement, economics, …

39Courtesy of R. Beausoleil, HP Labs, Palo Alto

Part 2:Photons and all that

quantum.infoPhotons

mode functions containing polarisation and spatial phase information

Classical electromagnetic field

chosen so that the Fourier amplitudes, , are dimensionless, i.e. they are solely complex numbers

Quantising the electromagnetic field

If , , commute, i.e.

then the commutator

is zero

annihilation operator

creation operator

quantum.infoPhotons

Energy of field

classically

After canonical transformation, using the boundary conditions of the mode functions:

The Hamiltonian of a collection of quantised simple harmonic oscillators (SHO)

Accordingly, the electromagnetic field can be considered to be a frequency ensemble of modes, each mode represented by a simple harmonic oscillator

quantum.info



The eigenstates of the simple harmonic oscillator Hamiltonian are known as number states or Fock states,

They are eigenstates of the number operator such that

The value of the number state represents the number of photons in that mode, e.g. there are three photons in the mode, , where is the frequency of the mode


Photons

quantum.infoPhotons'Number'states,'or,'Annihilation'can'Fock'a'state



The eigenstates of the simple harmonic oscillator Hamiltonian are known as number states or Fock states,

They are eigenstates of the number operator such that

The value of the number state represents the number of photons in that mode, e.g. there are three photons in the mode, , where is the frequency of the mode


creation operator

quantum.infoOscillators

m

k

L

μ

Consider a mechanical spring … and mass, m


m

k

L

νp

μ



m

k

L

νs

μ



m

k

L

μ

Consider a mechanical spring …

If

Then

and mass, m

phase-matching condition


ω m

k

L

m

μ


Frequency upconversion

m ω2ω

m

k

L

μ



m ω2ω

m

mechanicalsecond-harmonic

generation

k

L

μ

no threshold



m ω2ω

m


generation

k

L

2ω

μ

no threshold



m ω2ω

m


generation

k

L

Frequency downconversion

ω2ω

μ

no threshold



m ω2ω

m


generation

mechanicalparametricoscillation

k

L

Frequency downconversion

ω2ω

μ

no threshold threshold: overcome mechanical losses

quantum.infoPhotons

ωpωs

ωikp

ks

ki

C

= +kp = ks + ki

Optical frequency conversion

quantum.infoPhotons

ωpωs

ωikp

ks

ki

C

Optical Parametric Oscillation

pumpabove threshold

producetwin

bright beams

= +kp = ks + ki

quantum.info

pumpbelow threshold

Photons

ωpωs

ωikp

ks

ki

C

| i = ⌘ |11ia,b + ⌘2 |22ia,b + ⌘3 |33ia,b . . .

Function of crystal properties—including orientation, temperature—and pump power

Spontaneous Parametric Downconversion

producetwin

single photons

= +kp = ks + ki

quantum.infoPhotons

ωpωs

ωikp

ks

ki

C


| i = ⌘ |11ia,b + ⌘2 |22ia,b + ⌘3 |33ia,b . . .

quantum.infoPhotons

ωpωs

ωikp

ks

ki

C


1 2

0.99999999...

P(n)

n0.00000001...

Conditional 1-photon per mode

0.99999...

0.00001...

| i = ⌘ |11ia,b + ⌘2 |22ia,b + ⌘3 |33ia,b . . .

quantum.infoPhotons

ωpωs

ωikp

ks

ki

C


1 2

0.99999999...

P(n)

n0.00000001...

Conditional 1-photon per mode

Brightoutput

normalisedbrightness

Kwiat, Type II 1995 → 0.0044Kwiat, Type I 1999 → 1.98Takeuchi 2001 → 33.3Kurtsiefer 2001 → 13.3White 2003 → 70.4Kim 2006 → 500Jennewein 2007 → ~104

0.99999...

0.00001...

Takeuchi, Optics Letters 26, 843 (2001) Kurtsiefer et al., J. Mod. Opt. 48, 1997 (2001)

Well behaved spatial modes

Coupled into single-mode fibres: true TEM00

quantum.infoPhotons'as'qubits

✓ Single qubits

single rail

low intrinsic decoherence at optical frequencies

quantum.infoPhotons'as'qubits

✓ Single qubits

single rail

dual rail

low intrinsic decoherence at optical frequencies

quantum.infoOptical'qubits'can'be'any'DOF

✓ Single qubits

|0〉 |1〉

H V

Polarisation encoding

Spatial encoding

low intrinsic decoherence at electromagnetic frequencies


✓ Single qubits

|0〉 |1〉

H V


Spatial encoding


useful for qudits & quantum communication

Langford, et al., PRL 93, 053601 (2004)

quantum.info

Frequency encoding

Optical'qubits'can'be'any'DOF

✓ Single qubits

|0〉 |1〉

H V


Spatial encoding




|0〉 |1〉

Temporal encoding

t t

Path encoding

quantum.info

=+iR

=

=

+i

+i

Frequency encoding


✓ Single qubits

|0〉 |1〉

H V


Spatial encoding




|0〉 |1〉

Temporal encoding

t t

Path encoding

quantum.info

✓ Single-qubit gates


Phase shift

quantum.info



can be implemented by:mode seeing a different index of refraction mode seeing a a delay

Phase shift

quantum.info



can be implemented by:mode seeing a different index of refraction mode seeing a a delay

Phase shift

Beamsplitter

θ and φ are rotation angles about two orthogonal axes in the Poincaré sphere

beam splitter and the phase shift suffice to implement any single-qubit operation on a photonic qubit

θ proportional to reflectivityφ gives the phase shift due to reflection

quantum.infoExercise'Problem

If √R = cos θ, φ=0, rewrite these equations:

R


Consider 1 photon in each input:

If √R = cos θ, φ=0, rewrite these equations:

R

Write the output state, , in terms of The probability of seeing a photon in each output mode—a coincidence—is given by:

Calculate PC for indistinguishable photons, i.e. i = j ?

Calculate PC for distinguishable photons, i.e. i ≠ j ?

Calculate the Hong-Ou-Mandel visibility:for R = ½ ?for R = ⅓ ?


When √R = cos θ, φ=0 :

R

The output state,

The probability of seeing a coincidence is:

PC for indistinguishable photons, i.e. i = j :

PC for distinguishable photons, i.e. i ≠ j :

The Hong-Ou-Mandel visibility:

R=½ R=⅓

0

½

1

1/95/9

4/5


✓ Single qubits ✓ Single-qubit gates

? Two-qubit gates

|0〉 |1〉

H V


Spatial encoding

Impossible with bulk nonlinear optics

Milburn, PRL 62, 2124 (1988)



EasyCan use waveplatesAny gate is possible

HardNeed one photon to affectanother

=+iR

=

=

+i

+i

Part 3:Entangling Photonic

Gates

Theorist A physicist that does not do experiments

Experimentalist A physicist that does experiments

Leon Lederman, 1988 Nobel Laureate

quantum.info

StrongNonlinearities

Table of photonic two-qubit gate proposals

Milburn, PRL 62, 2124 (1988)

quantum.info


Linear-optical


Milburn, PRL 62, 2124 (1988)

Knill, Laflamme & Milburn,

Nature 409, 46 (2001)

quantum.info


Interaction-freeMeasurement

Linear-optical


Milburn, PRL 62, 2124 (1988)

Gilchrist, White, & Munro PRA 66,012106 (2002)


Nature 409, 46 (2001)

quantum.info



Linear-optical

Quantum Zeno


Milburn, PRL 62, 2124 (1988)



Nature 409, 46 (2001)

Franson, Jacobs & Pittman PRA 70, 062302 (2004)

quantum.info

Nemoto and Munro, PRL 93, 250502 (2004)



Linear-optical

Quantum Zeno

Amplified weakNonlinearities


Milburn, PRL 62, 2124 (1988)



Nature 409, 46 (2001)

Franson, Jacobs & Pittman PRA 70, 062302 (2004)

quantum.info




Linear-optical

Quantum Zeno


Geometric phasein Hilbert Space


Milburn, PRL 62, 2124 (1988)



Nature 409, 46 (2001)

Franson, Jacobs & Pittman PRA 70, 062302 (2004)Langford & Ramelow,

In prep (2009)

quantum.info

Nemoto and Munro, PRL



Linear-optical

Quantum Zeno




Milburn, PRL

Gilchrist, White, & Munro PRA


Nature 409, 46 (2001)

Franson, Jacobs & Pittman PRALangford & Ramelow,

In prep

quantum.infoLinear optical QC

Knill, Laflamme and Milburn, Nature 409, 46 (2001)

•••

•••

QUBITS QUBITS

a

a LINEAROPTICAL

NETWORK

quantum.infoLinear optical QC


•••

•••

QUBITS QUBITS

a

a LINEAROPTICAL

NETWORK•••

•••

SINGLEPHOTONS

FAST FEEDFORWARD

SINGLE PHOTONDETECTION

quantum.info

Photons have long decoherence times 467 µs (141km) Vienna

Linear optical QC


•••

•••

QUBITS QUBITS

a

a LINEAROPTICAL

NETWORK•••

•••

SINGLEPHOTONS

FAST FEEDFORWARD


Fastest gate times of any architecture 145 ns Vienna

quantum.info

• Teleportation: moving information without measuring it Z

|C〉 BX

|φ〉|C〉

Principles of LOQC

• Non-deterministic gates • Don’t always work, but heralded when they do

NON-DETERMIN

GATE

•••

•••

QUBITS QUBITS

•••

•••

SINGLEPHOTONS


& FEEDFORWARD

quantum.info

• Teleportation: moving information without measuring it Z

|C〉 BX

|φ〉|C〉

X

|T〉 BZ|φ〉

|T〉

Principles of LOQC


NON-DETERMIN

GATE

•••

•••

QUBITS QUBITS

•••

•••

SINGLEPHOTONS


& FEEDFORWARD

quantum.info

X Z

|C〉

X Z

|T〉 B

X

Z|φ〉

|φ〉|CNOT〉

NON-DETERMIN

GATE

B

Measurement-induced nonlinearity

• Teleportation: moving information without measuring it


NON-DETERMIN

GATE

•••

•••

QUBITS QUBITS

•••

•••

SINGLEPHOTONS


& FEEDFORWARD

quantum.info

X Z

|C〉

X Z

|T〉 B

X

Z|φ〉

|φ〉|CNOT〉

NON-DETERMIN

GATE

B

Measurement-induced nonlinearity

• Many non-deterministic gates proposed …

• Teleport non-deterministic gates → deterministic



NON-DETERMIN

GATE

•••

•••

QUBITS QUBITS

•••

•••

SINGLEPHOTONS


& FEEDFORWARD

quantum.info

Internal ancillas

Introduction

Proposed entangling gates

• KLM 4-photon Knill, Laflamme, & Milburn, Nature 409, 46 (2001)

NON-DETERMIN

GATE

➠ 2 ancilla photons ➠ interferometers

quantum.info

Internal ancillas

Introduction



• Simplified 4-photon Ralph, White, Munro, & Milburn, PRA 65, 012314 (2001)

NON-DETERMIN

GATE


➠ 2 ancilla photons

quantum.info

Internal ancillas

Introduction




• Efficient 4-photon Knill, PRA 66, 052306 (2002)

NON-DETERMIN

GATE




quantum.info

Internal ancillas

Introduction



• Entangled ancilla 4-photon Pittman, Jacobs, and Franson, PRL 88, 257902 (2002)



NON-DETERMIN

GATE




® 2 entangled ancilla photons➠

quantum.info

Internal ancillas

Introduction






NON-DETERMIN

GATE




® 2 entangled ancilla photons

➠ 2 degrees of freedom (4-qubit)

• Entangled input 2-photon Sanaka, Kawahara, & Kuga, quant-ph/0108001 (2001)

➠

quantum.info

Internal ancillas

Gasparoni et al., PRL 93, 020504 (2004)Pittman et al., PRA 68, 032316 (2004)Walther et al., Nature 434, 169 (2005)





• Entangled input 2-photon Pittman, Jacobs, and Franson, PRL 88, 257902 (2002)

NON-DETERMIN

GATE


quantum.info

Internal ancillas

• Simplified 2-photon Ralph, Langford, Bell, & White, PRA 65, 062324 (2002)

• Simplified 2-photon Hofmann & Takeuchi, PRA 66, 024308 (2002)

• Linear-optical QND Kok, Lee & Dowling, PRA 66, 063814 (2002)

External ancillas

NON-DETERMIN

GATE

QND

QND






NON-DETERMIN

GATE


quantum.info

+ teleportation + error correction = scalable QC

Internal ancillas

• Simplified 2-photon Ralph, Langford, Bell, & White, PRA 65, 062324 (2002)

• Simplified 2-photon Hofmann & Takeuchi, PRA 66, 024308 (2002)

• Linear-optical QND Kok, Lee & Dowling, PRA 66, 063814 (2002)

External ancillas

NON-DETERMIN

GATE

QND

QND






NON-DETERMIN

GATE


quantum.info

X Z

|C〉

X Z

|T〉 B

X

Z|φ〉

|φ〉|CNOT〉

NON-DETERMIN

GATE

B


• Teleport non-deterministic gates → deterministic

NON-DETERMIN

GATE

•••

•••

QUBITS QUBITS

•••

•••

SINGLEPHOTONS


& FEEDFORWARD


• Linear optical Bell-state measurement of single DOF is non-deterministic → teleporter is non-deterministic

PB = 1/2 → PCNOT = 1/4

The Scaling Problem

quantum.infoFixing the Teleporter

quantum.info

• 2001 - KLM’s solution

1. Increase the entangled resources

2. Concatenate the physical gates

Fixing the Teleporter

quantum.info

|C〉

|T〉

|φ〉|CNOT〉NON-

DETERMINGATE

U|φ〉

B

UB

U1 ... Un




e.g. “Level 1” encoding

PCNOT = (2/3)2 = 4/9


|CNOT

quantum.info

|C〉

|T〉

|φ〉|CNOT〉NON-

DETERMINGATE

U|φ〉

B

UB

U1 ... Un




e.g. to achieve a single entangling operation with success probability 95%


PCNOT = (2/3)2 = 4/9

~300 successful “Level 2” encoded gate operations, each P = 9/16

10,000’s of optical elements


|CNOT

quantum.info


Infinite entangled modes required for PCNOT = 1



e.g. to achieve a single entangling operation with success probability 95%


PCNOT = (2/3)2 = 4/9

~300 successful “Level 2” encoded gate operations, each P = 9/16

10,000’s of optical elements


quantum.info

Nielsen, PRL 93, 040503 (2004)

conventional circuit

Raussendorf and Briegel, PRL 86, 5188 (2001)

Uz(α1) Uz(α2) Uz(α3) Uz(α4)

Uz(β1) Uz(β2) Uz(β3) Uz(β4)

Uz(γ1) Uz(γ2) Uz(γ3) Uz(γ4)

singlequbitgates

two-qubit(entangling)

gates

flow of quantum information

Two paths to scalability

quantum.info

Nielsen, PRL 93, 040503 (2004)






singlequbitgates


gates


α1 ±α2 ±α3 ±α4

β1 ±β2 ±β3 ±β4

γ1 ±γ2 ±γ3 ±γ4

cluster/graph circuitflow of classical information

entanglement qubit measurementcosθ σx + sinθ σy


quantum.info

Nielsen, PRL 93, 040503 (2004)



qubit qubit

CZ




singlequbitgates


gates


α1 ±α2 ±α3 ±α4

β1 ±β2 ±β3 ±β4

γ1 ±γ2 ±γ3 ±γ4




quantum.info

CZ

qubit

Nielsen, PRL 93, 040503 (2004)



qubit qubit




singlequbitgates


gates


α1 ±α2 ±α3 ±α4

β1 ±β2 ±β3 ±β4

γ1 ±γ2 ±γ3 ±γ4




quantum.info

Nielsen, PRL 93, 040503 (2004)



qubit




singlequbitgates


gates


α1 ±α2 ±α3 ±α4

β1 ±β2 ±β3 ±β4

γ1 ±γ2 ±γ3 ±γ4




quantum.info

Nielsen, PRL 93, 040503 (2004)



qubit




singlequbitgates


gates


α1 ±α2 ±α3 ±α4

β1 ±β2 ±β3 ±β4

γ1 ±γ2 ±γ3 ±γ4



qubit

CZ


quantum.info

Nielsen, PRL 93, 040503 (2004)



qubit




singlequbitgates


gates


α1 ±α2 ±α3 ±α4

β1 ±β2 ±β3 ±β4

γ1 ±γ2 ±γ3 ±γ4



qubit

CZ

qubit


quantum.infoGraph-state computation

quantum.info

Graph states (clusters and parity-encoding techniques) have greatly reduced the required resources and the loss-tolerance threshold for LOQC:

OQC Anti-Moore’s Law

quantum.info

Graph states (clusters and parity-encoding techniques) have greatly reduced the required resources and the loss-tolerance threshold for LOQC:

Resources (Bell states, operations, etc.) for a reliable entangling gate

Acceptable loss for a scalable architecture

OQC Anti-Moore’s Law

13

“In theory, there is no difference between theory and practice.

– Jan L.A. van de Snepscheut (1953-1994)

“In theory, there is no difference between theory and practice.

– Jan L.A. van de Snepscheut (1953-1994)But, in practice, there is.”

quantum.info

C0

C1

T0

T1

C0

C1

T0

T1

Two?qubit'gate

Ralph, Langford, Bell & White, PRA 65, 062324 (2002) Hofmann & Takeuchi, PRA 66, 024308 (2002)

180˚phase shift

CSIGN gate

quantum.info

C0

C1

T0

T1


-1/3

1/3

1/3

CSIGN gate

Two?qubit'gate

quantum.info

bothtransmitted

C0

C1

T0

T1


-1/3

1/3

1/3

CSIGN gate

Two?qubit'gate

quantum.info

bothreflected

C0

C1

T0

T1


-1/3

1/3

1/3

CSIGN gate

Two?qubit'gate

quantum.info

C0

C1

T0

T1


-1/3

1/3

1/3

CSIGN gate

Two?qubit'gate

quantum.info

C0

C1

T0

T1

HWP HWP


-1/3

1/3

1/3

Control in Control out

Target in Target out

CNOT gate

one photon can flip the polarisation of another!

Controlled?NOT'gate

quantum.info

VH

C

Building entangling gates

Interferometric gate

O’Brien, Pryde, White, Ralph, and Branning, Nature 426, 264 (2003)

Jamin-Lebedeff interferometer Very stable: insensitive to x-y-z translation

O’Brien, Pryde, Gilchrist, James, Langford, Ralph and White, PRL 93, 080502 (2004)

C0

C1

T0

T1

HWP HWP

quantum.info

VH

T






C0

C1

T0

T1

HWP HWP

quantum.info

VV,HH

C

T






C0

C1

T0

T1

HWP HWP

quantum.info

VV,HH

C

T



Non-classical interference




C0

C1

T0

T1

HWP HWP

quantum.info

VV,HH

C

T






✓ Tunable beamsplitters✗ Dual-path interferometers limit performance


C0

C1

T0

T1

HWP HWP

quantum.info

Control in

Target in CNOT gate

Control out Target out

Automated tomography

Interferometric'controlled?NOT'gate

O’Brien, Pryde, et al., Nature 426, 264 (2003)

Interferometric'controlled?NOT'gate

Marshall, et al., Optics Express 17, 12546 (2009)From Mick Withford

quantum.info

C

T

Non-classical interference RH = 1/3 RV = 1

C0

C1

T0

T1

HWP HWP

Langford, Weinhold, Prevedel, Pryde, O’Brien, Gilchrist and White, PRL 95, 210504 (2005) Okamoto, Hofmann, Takeuchi, and Sasaki PRL 95, 210505 (2005) Kiesel, Schmid, Weber, Ursin, and Weinfurter, PRL 95, 210506 (2005)

Beamsplitter'controlled?NOT'gate

quantum.info

C

T


✓ No dual-path interferometers✗ No adjustment if wrong splitting ratio

RH = 1/3 RV = 1

C0

C1

T0

T1

HWP HWP

Langford, Weinhold, Prevedel, Pryde, O’Brien, Gilchrist and White, PRL 95, 210504 (2005) Okamoto, Hofmann, Takeuchi, and Sasaki PRL 95, 210505 (2005) Kiesel, Schmid, Weber, Ursin, and Weinfurter, PRL 95, 210506 (2005)

Beamsplitter'controlled?NOT'gate

quantum.info

2 photon cluster

1

2

Non-classical interference between |H1 and |H2

1 2

Building'graph'states

quantum.info

1

2


Non-classical interference between |V2 and |H3

3 photon cluster3

3

1 2


quantum.info

1

2


Non-classical interference between |V2 and |H3

3 photon cluster3

3

Repeatunit

1 2


Exponential blow-out

quantum.info

Nemoto and Munro, PRL



Linear-optical

Quantum Zeno




Milburn, PRL



Nature46 (2001)

Franson, Jacobs & Pittman PRA 70, 062302 (2004)Langford & Ramelow,

In prep

quantum.info

Franson, Jacobs, and Pittman, quant-ph/0401113 (2004) quant-ph/0408097 (2004)

• 2004 - Franson’s solution: deterministic entangling gate by combining linear optics gates, quantum Zeno effect and nonlinear optics

1. Linear-optical gates fail by emitting 2 photons into one mode 2. Quantum-Zeno effect can suppress 2 photon events

3. Requires nonlinear interaction

e.g. universal gate: √SWAP

Strong Nonlinearities

quantum.info


1. Linear optics gates fail by emitting 2 photons into one mode 2. Quantum Zeno effect can suppress 2 photon events





quantum.info


1. Linear optics gates fail by emitting 2 photons into one mode 2. Quantum Zeno effect can suppress 2 photon events




effects of loss? c.f. “interaction-free” measurements

Kwiat, et al., PRL 83, 4725 (1999) Gilchrist, et al., PRA 66, 012106 (2002)


quantum.info

Hendrickson, Lai, Pittman and Franson, PRL 105, 173602 (2010)

induced by20 photons

Quantum Zeno

quantum.infoGetting photon-photon nonlinearities

Luiten, Adelaide (2009)

quantum.info




Linear-optical

Quantum Zeno




Milburn, PRL



Nature46 (2001)

Franson, Jacobs & Pittman PRALangford & Ramelow,

In prep

quantum.info

Entangler

Homodyneclassical

feed-forward

Measurement & nonlinearities

• 2004 Barrett, et al.: amplify weak optical nonlinearities using measurement

Barrett et al., PRA 71, 060302R (2005)

quantum.info

Entangler

Homodyneclassical

feed-forward




H

V

quantum.info

Entangler

Homodyneclassical

feed-forward




VH

quantum.info

Entangler

Homodyneclassical

feed-forward

H

HV

V




quantum.info

Entangler

Homodyneclassical

feed-forward




• 2004 Nemoto & Munro: deterministic CNOT gate


quantum.info

85Rb

10µm

Perrella, Light, Anstie, Benabid, Stace, White, & Luiten, PRA 88, 013819 (2013)

π rad at Pm=150 nW0.7 µrad / photon

phase Psig25 µW

Nonlinear photonicsGetting photon-photon nonlinearities

quantum.info

85Rb

10µm


π rad at Pm=150 nW0.7 µrad / photon80% absorption

phaseabs

Psig25 µW


quantum.info

85Rb

10µm



phaseabs

Psig25 µW

0.13 µrad / photon0.5% absorption


quantum.info

85Rb

10µm



phaseabs

Psig25 µW

0.13 µrad / photon0.5% absorption

Nonlinear photonics

Venkataraman, Saha, Gaeta,Nature Photonics 7, 138 (2013)

0.3 mrad6 µm hole

Getting photon-photon nonlinearities

Using quantum memories

recall efficiency of 69% quantum-noise limited

Using quantum memories

recall efficiency of 87%Using quantum memories

recall efficiency of 87%Using quantum memories

fidelity of 98%

Takea break

Part 4:Measuring

entangling gates

quantum.info

quantum.infoSingle'gate'performance

quantum.info

• State tomography: measure combinations of basis states

0 1 0 + 1 0 + i1 H V D R

Quantum'Tomography

H

V

DR

Poincaré Sphere

AL

quantum.info


0 1 0 + 1 0 + i1 H V D R

Quantum'Tomography

degree of polarisation

discrimination

H

V

DR

Poincaré Sphere

AL P

quantum.info


0 1 0 + 1 0 + i1 H V D R

Quantum'Tomography

degree of polarisation

discrimination

H

V

DR

Poincaré Sphere

AL P

DH,V 0.006 ± 0.010 0.992±0.012 0.003±0.003 DD,A 0.994 ± 0.010 0.004±0.009 0.001±0.002 DR,L 0.000 ± 0.000 0.004±0.009 0.996±0.003

quantum.infoQuantum'Tomography

Kleinlogel and White, PLoS ONE 3, e2190 (2008)

DH,V 0.006 ± 0.010 0.992±0.012 0.003±0.003 DD,A 0.994 ± 0.010 0.004±0.009 0.001±0.002 DR,L 0.000 ± 0.000 0.004±0.009 0.996±0.003



row 2

row 3row 4

row 5

row 6

mid-banddorsal

ventral

D

V

L M

row 1

1mm

D, A

R, L

H, V

DH,V 0.006 ± 0.010 0.992±0.012 0.003±0.003 DD,A 0.994 ± 0.010 0.004±0.009 0.001±0.002 DR,L 0.000 ± 0.000 0.004±0.009 0.996±0.003



Mantis shrimp, Gonodactylus Smithii, hasoptimal polarisation vision

quantum.infoMeasuring CNOT gate operation

Quantum state tomography

• For 1 photon states: measure Stokes parameters H V D R 0 1 0 + 1 0 + i1

• For 2 photon states: use bi-photon Stokes parameters HH HV HD HR

VH VV VD VR

DH DV DD DR

RH RV RD RR

quantum.info

0˚ 180˚45˚ 90˚ 135˚

1

0

PAB

H HV

1 0 0 00 0 0 00 0 0 00 0 0 0

HH HV VH VV

HH HV VH VV

Measuring CNOT gate operation




VH VV VD VR

DH DV DD DR

RH RV RD RR

White, James, Eberhard & Kwiat, PRL 83, 3103 (1999) James, Kwiat, Munro & White, PRA 64, 052312 (2001)

quantum.info

0˚ 180˚45˚ 90˚ 135˚

1

0

PAB

H HV0˚ 180˚45˚ 90˚ 135˚

1

0

PAB

H HV

1 0 0 00 0 0 00 0 0 00 0 0 0

HH HV VH VV

HH HV VH VV

14

1 1 1 11 1 1 11 1 1 11 1 1 1

HH HV VH VV

HH HV VH VV





VH VV VD VR

DH DV DD DR

RH RV RD RR


quantum.info

0˚ 180˚45˚ 90˚ 135˚

1

0

PAB

H HV

12

1 0 0 10 0 0 00 0 0 01 0 0 1

HH HV VH VV

HH HV VH VV



• For 1 photon states: measure Stokes parameters

• n qubit states require 4n measurements 2 qubit states require 16 measurements

H V D R 0 1 0 + 1 0 + i1



VH VV VD VR

DH DV DD DR

RH RV RD RR

• “impossible” set of correlations |HH〉+|VV〉|DD〉+|AA〉|RL〉+|LR〉 ➠ entanglement

quantum.info

• Process tomography: measure combinations of basis processes

|0〉= |H〉

|0+1〉=|D〉

|0+i1〉=|R〉


I

n qubit state requires 22n measurements

rotations on Poincare sphere

0 1 0 + 1 0 + i1 H V D R

for 2 photon states, bi-photon Stokes parametersHH HV HD HR

VH VV VD VR

DH DV DD DR

RH RV RD RR

White, Gilchrist, Pryde, O’Brien, Bremner, & Langford, JOSA B, 24, 172-183 (2007)

Quantum'Tomography

quantum.info


|0〉= |H〉

|0+1〉=|D〉

|0+i1〉=|R〉


I X



0 1 0 + 1 0 + i1 H V D R


VH VV VD VR

DH DV DD DR

RH RV RD RR


Quantum'Tomography

quantum.info


|0〉= |H〉

|0+1〉=|D〉

|0+i1〉=|R〉


I X Z



0 1 0 + 1 0 + i1 H V D R


VH VV VD VR

DH DV DD DR

RH RV RD RR


Quantum'Tomography

quantum.info


|0〉= |H〉

|0+1〉=|D〉

|0+i1〉=|R〉


I X Y Z



0 1 0 + 1 0 + i1 H V D R


VH VV VD VR

DH DV DD DR

RH RV RD RR


Quantum'Tomography

quantum.info


|0〉= |H〉

|0+1〉=|D〉

|0+i1〉=|R〉


I X Y Z

n qubit gate requires 24n measurements



• Reconstructed states and processes are unphysical: effect of uncertainties Maximum likelihood or Bayesian analysis required

0 1 0 + 1 0 + i1 H V D R

for 2 photon states, bi-photon Stokes parameters

I I I X I Y I Z

X I XX XY XZ

Y I YX YY YZ

Z I ZX ZY ZR

HH HV HD HR

VH VV VD VR

DH DV DD DR

RH RV RD RR


Quantum'Tomography

quantum.info Quantum process tomography

Measuring a state is not measuring a gate … normally



General recipe: Nielsen & Chuang, `Quantum Information´ (2003)

Standard n2n input states

1 qubitgate

ρH

^

ρV

^

ρD

^

ρR

^

ρH

^

ρV

^

ρD

^

ρR

^

´´´´





1 qubitgate

ρH

^

ρV

^

ρD

^

ρR

^

ρH

^

ρV

^

ρD

^

ρR

^

´´´´

Ancilla assisted

Altepeter, Branning, Jeffrey, Wei, Kwiat, Thew, O’Brien, Nielsen, & White, PRL 90, 193601 (2003)

2n-qubit input state

´1 qubitgate

ρ2

^ ρ2

^


n qubit gate requires 42n measurements

2-qubit gate recipes: White, Gilchrist, Pryde, O’Brien, Bremner & Langford, quant-ph/0308115 (2003)


2 qubit gate (CNOT) requires 256 measurements



1 qubitgate

ρH

^

ρV

^

ρD

^

ρR

^

ρH

^

ρV

^

ρD

^

ρR

^

´´´´

Ancilla assisted

Altepeter, Branning, Jeffrey, Wei, Kwiat, Thew, O’Brien, Nielsen, & White, PRL 90, 193601 (2003)

2n-qubit input state

´1 qubitgate

ρ2

^ ρ2

^

quantum.info

χ matrix: Table of process measurement probabilities, and coherences between themFor a CZ gate:

Single'gate'performance

quantum.info


χ matrix: Table of process measurement probabilities, and coherences between them

Ideal Measured

For a CZ gate:


quantum.info

2005, F = 89.3 ± 0.1 %2004, F = 94 ± 2 %


χ matrix: Table of process measurement probabilities, and coherences between them

Ideal Measured

O’Brien, Pryde, et al., Nature 426, 264 (2003)

For a CZ gate:

average gate fidelity:

1 gate works 90-95% of time At best, 2 gates should work 80-90% of time

Fp is process fidelity: overlap between ideal & exp processes


quantum.info

(II, IX, IY, IZ, XI, XY, XZ, YI, YX, YY, YZ, ZI, ZX, ZY, ZZ)

• Physical interpretation? Change basis

CNOT ⊗ ⊗ CNOT

Measuring entangling gates

Quantum process tomography

quantum.info

Measured (Re) Measured (Im)



CNOT ⊗ ⊗ CNOT



quantum.info

Measured (Re) Measured (Im)



CNOT ⊗ ⊗ CNOT

= 87%




quantum.info

F = (89.3 ± 0.1) %

Weinhold, et al., arXiv:0808.0794 (2008)

dependent photons: not scalable

CZ

• Gate performance depends on: photon source, circuit quality, detectorsSingle'gate'performance

quantum.info

F = (89.3 ± 0.1) %circuit & mode-

matching is same, source is different



CZ

F = (82.5 ± 1.5) %

independent photons: scalable

CZ


quantum.info

2.Beamsplitterreflectivities, Δ=1-2%

3.Photon loss,90-97%

1. Independentdownconversion,1.2, 3.2% higherorder terms

F = (89.3 ± 0.1) %

• Model source & gate based on measured parameters

circuit & mode-matching is same, source is different



CZ

F = (82.5 ± 1.5) %

independent photons: scalable

CZ


quantum.info

ideal 0 %1-Fp



quantum.info

ideal srce (2+3) 2.8 %all (1+2+3) 18.6 %

photon source Δ ε=15.8%ideal 0 %

1-Fp



quantum.info

21.8% ±1.5%experiment

ideal srce (2+3) 2.8 %all (1+2+3) 18.6 %

photon source Δ ε=15.8%mode mismatch Δ ε= 3.2%

ideal 0 %1-Fp



quantum.info


ideal srce (2+3) 2.8 %all (1+2+3) 18.6 %

model predicts that with good photon source & optics experimental gate error is: εg ~ 3.2 ± 1.5 %c.f. Knill’s tolerance threshold: ε0 ≤ 6 %


ideal 0 %1-Fp



quantum.info


ideal srce (2+3) 2.8 %all (1+2+3) 18.6 %

model predicts that with good photon source & optics experimental gate error is: εg ~ 3.2 ± 1.5 %c.f. Knill’s tolerance threshold: ε0 ≤ 6 %


ideal 0 %1-Fp


photonics QC: scalability requires sources, detectors are rate limiting factor


COMPRESSED SENSING

Candès, Romberg, and Tao, IEEE Transactions on Information Theory, 52 489 - 509 (2006)

5000-pt signal

COMPRESSED SENSING

500 random samplesfrom 5000-pt signal


COMPRESSED SENSING


Least-squares reconstruction


COMPRESSED SENSING


Compressed sensing(Sparse matrix)


quantum.info

d-dimensional quantum system needs

COMPRESSED TOMOGRAPHY

for qubits d=2n

how many measurement configurations?

quantum.info



for qubits d=2n

Full QT

O(d2)


process

state

O(d4)

quantum.info



for qubits d=2n

Full QT Compressed QT

O(d2)


process

state

O(d4)

O(d r log2 d) quadratically fasterblind, r is matrix rank

Gross, Liu, Flammia, Becker, & Eisert,PRL 105, 150401 (2010).

quantum.info


Shabani, Kosut, Mohseni, Rabitz, Broome, Almeida, Fedrizzi, & White,PRL 106, 100401 (2011)


for qubits d=2n


O(d2)


process

state

O(d4)

O(d r log2 d)

O(s log d)

quadratically fasterblind, r is matrix rank


exponentially fasterneeds prior, s is matrix sparsity

quantum.info




for qubits d=2n


O(d2)


process

state

O(d4)

O(d r log2 d)

O(s log d)




engineered quantum systems aim to implement a unitary process which is maximally-sparse in its eigenbasis

quantum.info




for qubits d=2n


O(d2)


process

state

O(d4)

O(d r log2 d)

O(s log d)




in practice—as observed in QPT experiments in liquid-state NMR,photonics, ion traps, and superconducting circuits—nearly-sparse, still compressible!

engineered quantum systems aim to implement a unitary process which is maximally-sparse in its eigenbasis

COMPRESSED TOMOGRAPHYtwo-photon CZ gate low-noise, purity 91%

Full tomo

Shabani, Kosut, Mohseni, Rabitz,Broome, Almeida, Fedrizzi, and White,Physical Review Letters 106, 100401 (2011)

COMPRESSED TOMOGRAPHYtwo-photon CZ gate low-noise, purity 91%

Full tomoCompressed tomo


COMPRESSED TOMOGRAPHYtwo-photon CZ gate

F=95%

low-noise, purity 91%


2 measurement combinations ofRI and IR

16 input combinationsof H,V,D,R

32 combinations


COMPRESSED TOMOGRAPHYfour-photon CZ gate

F=85%

high-noise, purity 62%


2 measurement combinations ofRI and IR

16 input combinationsof H,V,D,R

32 combinations


Photonic Quantum Information - Chalmers oc… · Shor’s factoring algorithm → Which of these do...

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