Photonic Quantum Information, Computation Experimental & Simulation BosonSampling qt Andrew White lab University of Queensland quantum.info
CENTRE FOR ENGINEERED QUANTUM SYSTEMS : AUSTRALIAN RESEARCH COUNCIL CENTRE OF EXCELLENCE
15,470 km Quantum simulation Quantum foundations & emulation
0.4 0.2 Probability
-8 -6 -4 1 -2 0 2 Nature Physics 11, 249 (2015) 2 4 3 6 4 Scientific Reports 4, 06955 (2014) 8 Physical Review Letters 112, 143603 (2014) Nature Communications 5, 04145 (2014) Physical Review Letters 112, 020401 (2014) Science 339, 794 (2013) Nature Communications 3, 625 (2012) Nature Communications 3, 882 (2012) Physical Review Letters 106, 200402 (2011) New Journal of Physics 13, 075003 (2011) New Journal of Physics 13, 053038 (2011) Physical Review Letters 104, 153602 (2010) Proceedings of the National Nature Chemistry 2, 106 (2010) Academy of Sciences 108, 1256 (2011) Quantum computation Quantum photonics
Physical Review Letters 114, 173603 (2015) Physical Review Letters 114, 090402 (2015) Physical Review Letters 112, 143603 (2014) Physical Review A 89, 042323 (2014) Physical Review Letters 110, 250501 (2013) Physical Review Letters 111, 230504 (2013) Physical Review A 88, 013819 (2013) Physical Review Letters 106, 100401 (2011) Optics Express 21, 13450 (2013) New Journal of Physics 12 083027 (2010) Optics Express 19, 22698 (2011) Nature Physics 5, 134 (2009) Journal of Modern Optics 58, 276 (2011) Physical Review Letters 101, 200501 (2008) Optics Express 19, 55 (2011) Christina Geoff Juan Aleksandrina Markus Martin Azwa The Giarmatzi Gillett Loredo Nikolova Rambach Ringbauer Zakaria Team from July
Marcelo Ivan Jacqui Till Almeida Kassal Romero Weinhold & you ?
Professor, Senior Postdoc,
Heriot-Watt University Oxford University Industry, Scotland England Australia + £1.5M Fellowship
Alessandro Michael Devon Fedrizzi Vanner Biggerstaff
Part 1: Quantum Information: What is it? Why do it? What is information?
Classical: bits ≡ binary digits information that can be stored in a 3-bit register can store 0 1 system is proportional to logb N one number, from 0–7 b is the number of states that can be stored in the unit N is the number of possible states of that system
for three bits, log2 8 = 3
Hartley, Bell System Tech. J. 7, 535–563 (1928) Shannon, Bell System Tech. J. 7, 379–423 (1948) What is information?
Classical: bits ≡ binary digits
3-bit register can store 0 1 one number, from 0–7 N bits: one N-bit number
Quantum: qubits 0 qubit sphere
also known as Poincaré sphere 1 Bloch sphere
Poincaré, Leçons sur la théorie Mathématique de la lumière II,(1897) Hartley, Bell System Tech. J. 7, 535–563 (1928) Bloch, Phys. Rev. 70, 460 (1946) Shannon, Bell System Tech. J. 7, 379–423 (1948) Qubit Sphere
0 also known as Poincaré sphere Bloch sphere 0+1 0+i1 1 Qubit Sphere
0 also known as Poincaré sphere Bloch sphere 0+1 0+i1 1
Pure states are a collective delusion of theorists Qubit Sphere
0 also known as Poincaré sphere Bloch sphere 0+1 0+i1 1
Hermitean
non-negative eigenvalues
unity trace
population Qubit Sphere
0 also known as Poincaré sphere Bloch sphere 0+1 0+i1 1
Hermitean
non-negative eigenvalues
unity trace
population
coherence Qubit Sphere
0 also known as Poincaré sphere Bloch sphere 0+1 0+i1 1 purity Density Matrix
qutrit
ququart or two qubits
quoct or three qubits What is information?
Classical: bits ≡ binary digits Classical gates 3-bit register can store NAND Toffoli 0 1 one number, from 0–7 INPUT OUTPUT INPUT OUTPUT N bits: one N-bit number A B A and B A B C A B C 0 0 1 0 0 0 0 0 0 0 1 1 0 0 1 0 0 1 Quantum: qubits 1 0 1 0 1 0 0 1 0 1 1 0 0 1 1 0 1 1 3-qubit register can 0 1 0 0 1 0 0 irreversible 1 0 1 1 0 1 store eight numbers 1 1 0 1 1 1 1 1 1 1 1 0 a |000 + b |001 reversible + c |010 + d |011 0+1 0+i1 + e |100 + f |101 1 Quantum gates INPUT OUTPUT + g |110 + h |111 A B A B CNOT 0 0 0 0 0 1 0 1 N N qubits: all possible 2 numbers 1 0 1 1 1 1 1 0 0+1 0 00+11 must be reversible 300-qubit register can store more information than number of atoms in universe
Important problems benefit from factoring Physical Review Letters 99, 250505 (2007) this exponential scaling: simulation Nature Chemistry 2, 106 (2010) What is classical computing? quantum.info
insoluble
soluble What is classical computing? quantum.info
insoluble insoluble
hard factoring soluble travelling sales- easy man
Nomenclature easy = “tractable” = “efficiently computable” hard = “intractable” = “not efficiently computable” What is quantum computing? quantum.info
insoluble insoluble
hard factoring soluble travelling sales- easy man
Nomenclature easy = “tractable” = “efficiently computable” hard = “intractable” = “not efficiently computable” What is quantum computing? quantum.info
insoluble insoluble
hard factoring soluble travelling sales- easy man
Nomenclature easy = “tractable” = “efficiently computable” hard = “intractable” = “not efficiently computable” 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 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 Why do quantum computing? quantum.info Aaronson’s Trilemma Shor’s factoring algorithm → Why do quantum computing? quantum.info Aaronson’s • Extended Church-Turing Thesis—foundation of theoretical Trilemma computer science for decades—is wrong Shor’s factoring algorithm →
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
Statement: “All computational problems that are efficiently solvable by realistic physical devices, are efficiently solvable by a probabilistic Turing machine” Why do quantum computing? quantum.info Aaronson’s • Extended Church-Turing Thesis—foundation of theoretical Trilemma computer science for decades—is wrong • Quantum mechanics is wrong Shor’s factoring algorithm →
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
“All computational problems that are efficiently solvable by realistic physical devices, are efficiently solvable by a probabilistic Turing machine” Why do quantum computing? quantum.info Aaronson’s • Extended Church-Turing Thesis—foundation of theoretical Trilemma computer science for decades—is wrong • Quantum mechanics is wrong Shor’s factoring • A fast classical factoring algorithm exists algorithm →
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 Why do quantum computing? quantum.info Aaronson’s • Extended Church-Turing Thesis—foundation of theoretical Trilemma computer science for decades—is wrong • Quantum mechanics is wrong Shor’s factoring • A fast classical factoring algorithm exists algorithm → 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 wrong b. Quantum mechanics is wrong c. Fast factoring exists & QC’s are not necessary d. Two of the above e. All of the above f. Shhh, I’m sleeping.
How do we experimentally test this trilemma? Why do quantum computing? quantum.info Aaronson’s • Extended Church-Turing Thesis—foundation of theoretical Trilemma computer science for decades—is wrong • Quantum mechanics is wrong Shor’s factoring • A fast classical factoring algorithm exists algorithm → 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 wrong b. Quantum mechanics is wrong c. Fast factoring exists & QC’s are not necessary d. Two of the above e. All of the above f. Shhh, I’m sleeping. Computer scientists and / or Theoretical physicists and /or Mathematicians & chemists will be badly upset How do we experimentally test this trilemma? Why do quantum computing? quantum.info Aaronson’s • Extended Church-Turing Thesis—foundation of theoretical Trilemma computer science for decades—is wrong • Quantum mechanics is wrong Shor’s factoring • A fast classical factoring algorithm exists algorithm → All three seem crazy. At least one is true!
Church-Turing-Deutsch principle: Any physical process can be efficiently simulated on a quantum computer Why do quantum computing? quantum.info Aaronson’s • Extended Church-Turing Thesis—foundation of theoretical Trilemma computer science for decades—is wrong • Quantum mechanics is wrong Shor’s factoring • A fast classical factoring algorithm exists algorithm → All three seem crazy. At least one is true!
Church-Turing-Deutsch principle: Any physical process can be efficiently simulated on a quantum computer
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
Quantum Simulation Tree of approximation methods for quantum chemistry Why do quantum computing? quantum.info Aaronson’s • Extended Church-Turing Thesis—foundation of theoretical Trilemma computer science for decades—is wrong • Quantum mechanics is wrong Shor’s factoring • A fast classical factoring algorithm exists algorithm → All three seem crazy. At least one is true!
Church-Turing-Deutsch principle: Any physical process can be efficiently simulated on a quantum computer 2-qubit photonic quantum computer: Nuclei movecalculate on the energyelectronic to 20 potential bits energy surface (PES)
Knowledge of PES enables: ▸ Minima: equilibrium structures ▸ Saddle points: transition states experiment & theory ▸ Reaction rates & mechanisms agree to 1 part in 106 ▸ PES characterise most of physical chemistry
Quantum Simulation Tree of approximation methods Lanyon, et al., Nature Chemistry 2, 106 (2010) for quantum chemistry Why do quantum computing? quantum.info Aaronson’s • Extended Church-Turing Thesis—foundation of theoretical Trilemma computer science for decades—is wrong • Quantum mechanics is wrong Shor’s factoring • A fast classical factoring algorithm exists algorithm → All three seem crazy. At least one is true!
Church-Turing-Deutsch principle: Any physical process can be efficiently simulated on a quantum computer
Quantum computers are interesting physical systems in their own right Why do quantum computing? quantum.info Aaronson’s • Extended Church-Turing Thesis—foundation of theoretical Trilemma computer science for decades—is wrong • Quantum mechanics is wrong Shor’s factoring • A fast classical factoring algorithm exists algorithm → All three seem crazy. At least one is true!
Church-Turing-Deutsch principle: Any physical process can be efficiently simulated on a quantum computer
Quantum computers are interesting physical systems in their own right Why do quantum computing? quantum.info
The Economist, Saturday June 20, 2015 Why do quantum computing? quantum.info
The Economist, Saturday June 20, 2015 Why do quantum computing? quantum.info QKD Banks, government
QC Simulation: semiconductors, …
QC Factoring, searching
2005 2010 2015 2020 2025 2030 2035
Courtesy of R. Beausoleil, HP Labs, Palo Alto 38 QKD Banks, government
Few-qubit QIP Distributed algorithms & entanglement, economics, …
QC Simulation: semiconductors, …
QC Factoring, searching
2005 2010 2015 2020 2025 2030 2035
Courtesy of R. Beausoleil, HP Labs, Palo Alto 39 Part 2: Photons and all that quantum.info Photons Classical electromagnetic field
mode functions containing polarisation and spatial phase information chosen so that the Fourier amplitudes, , are dimensionless, i.e. they are solely complex numbers
annihilation operator Quantising the electromagnetic field creation operator
If , , commute, i.e. then the commutator is zero quantum.info Photons 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 Photons
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
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
annihilation operator quantum.info Photons'Number'states,'or,' Annihilation'can'Fock'a'state
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
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
annihilation operator
creation operator quantum.info Oscillators
Consider a mechanical spring … and mass, m
μ k L
m quantum.info Oscillators
Consider a mechanical spring … and mass, m
μ k L
νp m quantum.info Oscillators
Consider a mechanical spring … and mass, m
μ k L
m νs quantum.info Oscillators
Consider a mechanical spring … and mass, m
μ k L
m If phase-matching condition
Then quantum.info Oscillators
μ k L
m ω m quantum.info Oscillators
Frequency upconversion
μ k L
2ω m ω m quantum.info Oscillators
Frequency upconversion
μ k L
2ω m ω m
mechanical second-harmonic generation
no threshold quantum.info Oscillators
Frequency upconversion
μ k L
2ω m ω m 2ω
mechanical second-harmonic generation
no threshold quantum.info Oscillators
Frequency upconversion Frequency downconversion
μ k L
2ω ω m ω m 2ω
mechanical second-harmonic generation
no threshold quantum.info Oscillators
Frequency upconversion Frequency downconversion
μ k L
2ω ω m ω m 2ω
mechanical mechanical second-harmonic parametric generation oscillation
threshold: overcome no threshold mechanical losses quantum.info Photons
Optical frequency conversion
= + kp = ks + ki ωs ωp k s C ωi kp ki quantum.info Photons
Optical Parametric Oscillation
= + pump above threshold kp = ks + ki ωs ωp k produce s C twin ωi kp bright beams ki quantum.info Photons
Spontaneous Parametric Downconversion
= + pump below threshold kp = ks + ki ωs ωp k produce s C twin ωi kp single photons ki
= ⌘ 11 + ⌘2 22 + ⌘3 33 ... | i | ia,b | ia,b | ia,b
Function of crystal properties—including orientation, temperature—and pump power quantum.info Photons
Spontaneous Parametric Downconversion ωs ωp k s C ωi kp ki
= ⌘ 11 + ⌘2 22 + ⌘3 33 ... | i | ia,b | ia,b | ia,b quantum.info Photons
Spontaneous Parametric Downconversion ωs ωp k s C ωi kp ki
= ⌘ 11 + ⌘2 22 + ⌘3 33 ... | i | ia,b | ia,b | ia,b quantum.info Photons
Spontaneous Parametric Downconversion Conditional 1-photon per mode ωs 0.99999...0.99999999... ωp k s C ωi P(n) kp ki 0.00001...0.00000001... 1 2 n
= ⌘ 11 + ⌘2 22 + ⌘3 33 ... | i | ia,b | ia,b | ia,b quantum.info Photons
Spontaneous Parametric Downconversion Conditional 1-photon per mode ωs 0.99999...0.99999999... ωp k s C ωi P(n) kp ki 0.00001...0.00000001... Well behaved spatial modes 1 2 n Bright output
normalised brightness
Kwiat, Type II 1995 → 0.0044 Kwiat, Type I 1999 → 1.98 Takeuchi 2001 → 33.3 Kurtsiefer 2001 → 13.3 White 2003 → 70.4 Kim 2006 → 500 Coupled into single-mode fibres: true TEM00 Jennewein 2007 → ~104 Takeuchi, Optics Letters 26, 843 (2001) Kurtsiefer et al., J. Mod. Opt. 48, 1997 (2001) quantum.info Photons'as'qubits
✓ Single qubits low intrinsic decoherence at optical frequencies
single rail quantum.info Photons'as'qubits
✓ Single qubits low intrinsic decoherence at optical frequencies
single rail quantum.info Photons'as'qubits
✓ Single qubits low intrinsic decoherence at optical frequencies
single rail
dual rail quantum.info Photons'as'qubits
✓ Single qubits low intrinsic decoherence at optical frequencies
single rail
dual rail quantum.info Optical'qubits'can'be'any'DOF
✓ Single qubits low intrinsic decoherence at electromagnetic frequencies
|0〉 |1〉
Polarisation encoding
H V
Spatial encoding quantum.info Optical'qubits'can'be'any'DOF
✓ Single qubits low intrinsic decoherence at electromagnetic frequencies
|0〉 |1〉
Polarisation encoding
H V
Spatial encoding useful for qudits & quantum communication
Langford, et al., PRL 93, 053601 (2004) quantum.info Optical'qubits'can'be'any'DOF
✓ Single qubits |0〉 |1〉 low intrinsic decoherence at Path encoding electromagnetic frequencies
|0〉 |1〉
Polarisation encoding
Temporal encoding H V
Spatial encoding useful for qudits & quantum communication t t
Frequency encoding
Langford, et al., PRL 93, 053601 (2004) quantum.info Optical'qubits'can'be'any'DOF
✓ Single qubits |0〉 |1〉 low intrinsic decoherence at Path encoding electromagnetic frequencies
|0〉 |1〉
Polarisation encoding +i = Temporal encoding H V R Spatial encoding useful for qudits & quantum communication t t
+i = Frequency encoding
+i =
Langford, et al., PRL 93, 053601 (2004) quantum.info Optical'qubits'can'be'any'DOF
✓ Single-qubit gates
Phase shift quantum.info Optical'qubits'can'be'any'DOF
✓ Single-qubit gates
Phase shift can be implemented by: mode seeing a different index of refraction mode seeing a a delay quantum.info Optical'qubits'can'be'any'DOF
✓ Single-qubit gates
Phase shift can be implemented by: mode seeing a different index of refraction mode seeing a a delay
Beamsplitter
θ proportional to reflectivity φ gives the phase shift due to reflection
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 quantum.info Exercise'Problem
R
If √R = cos θ, φ=0, rewrite these equations: quantum.info Exercise'Problem
R
If √R = cos θ, φ=0, rewrite these equations:
Consider 1 photon in each input:
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 ? for R = ½ ? Calculate the Hong-Ou-Mandel visibility: for R = ⅓ ? quantum.info Exercise'Problem
R
When √R = cos θ, φ=0 :
The output state,
The probability of seeing a coincidence is: R=½ R=⅓
1 PC for indistinguishable photons, i.e. i = j : 0 /9
5 PC for distinguishable photons, i.e. i ≠ j : ½ /9
4 The Hong-Ou-Mandel visibility: 1 /5 quantum.info Optical'qubits'can'be'any'DOF
✓ Single qubits ✓ Single-qubit gates low intrinsic decoherence at electromagnetic frequencies Easy |0〉 |1〉 Can use waveplates Polarisation encoding Any gate is possible +i = H V R ? Two-qubit gates Spatial encoding Hard Need one photon to affect +i = another Impossible with bulk +i = nonlinear optics
Milburn, PRL 62, 2124 (1988) Langford, et al., PRL 93, 053601 (2004) Part 3: Entangling Photonic Gates Leon Lederman, 1988 Nobel Laureate
Experimentalist A physicist that does experiments
Theorist A physicist that does not do experiments Table of photonic two-qubit gate proposals quantum.info
Strong Nonlinearities Milburn, PRL 62, 2124 (1988) Table of photonic two-qubit gate proposals quantum.info
Linear-optical Knill, Laflamme & Milburn, Strong Nature 409, Nonlinearities 46 (2001) Milburn, PRL 62, 2124 (1988) Table of photonic two-qubit gate proposals quantum.info
Linear-optical Knill, Laflamme & Milburn, Strong Nature 409, Interaction-free Nonlinearities 46 (2001) Measurement Milburn, Gilchrist, White, & Munro PRL 62, 2124 (1988) PRA 66,012106 (2002) Table of photonic two-qubit gate proposals quantum.info
Linear-optical Knill, Laflamme & Milburn, Strong Nature 409, Interaction-free Nonlinearities 46 (2001) Measurement Milburn, Gilchrist, White, & Munro PRL 62, 2124 (1988) PRA 66,012106 (2002)
Quantum Zeno Franson, Jacobs & Pittman PRA 70, 062302 (2004) Table of photonic two-qubit gate proposals quantum.info
Linear-optical Knill, Laflamme & Milburn, Strong Nature 409, Interaction-free Nonlinearities 46 (2001) Measurement Milburn, Gilchrist, White, & Munro PRL 62, 2124 (1988) PRA 66,012106 (2002)
Quantum Zeno Franson, Jacobs & Pittman PRA 70, 062302 (2004)
Amplified weak Nonlinearities Nemoto and Munro, PRL 93, 250502 (2004) Table of photonic two-qubit gate proposals quantum.info
Linear-optical Knill, Laflamme & Milburn, Strong Nature 409, Interaction-free Nonlinearities 46 (2001) Measurement Milburn, Gilchrist, White, & Munro PRL 62, 2124 (1988) PRA 66,012106 (2002)
Geometric phase Quantum Zeno in Hilbert Space Franson, Jacobs & Pittman Langford & Ramelow, PRA 70, 062302 (2004) In prep (2009) Amplified weak Nonlinearities Nemoto and Munro, PRL 93, 250502 (2004) Table of photonic two-qubit gate proposals quantum.info
Linear-optical Knill, Laflamme & Milburn, Strong Nature 409, Interaction-free Nonlinearities 46 (2001) Measurement Milburn, Gilchrist, White, & Munro PRL PRA
Geometric phase Quantum Zeno in Hilbert Space Franson, Jacobs & Pittman Langford & Ramelow, PRA In prep Amplified weak Nonlinearities Nemoto and Munro, PRL Linear optical QC quantum.info
Knill, Laflamme and Milburn, Nature 409, 46 (2001)
a
• • QUBITS • • • • QUBITS
a LINEAR OPTICAL NETWORK Linear optical QC quantum.info
Knill, Laflamme and Milburn, Nature 409, 46 (2001)
a
• • QUBITS • • • • QUBITS
a LINEAR OPTICAL NETWORK FAST SINGLE • • FEEDFORWARD PHOTONS • • • •
SINGLE PHOTON DETECTION Linear optical QC quantum.info
Knill, Laflamme and Milburn, Nature 409, 46 (2001)
a
• • QUBITS • • • • QUBITS
a LINEAR OPTICAL NETWORK FAST SINGLE • • FEEDFORWARD PHOTONS • • • •
SINGLE PHOTON DETECTION Fastest gate times of any architecture 145 ns Vienna
Photons have long decoherence times 467 µs (141km) Vienna Principles of LOQC quantum.info
• • QUBITS • • QUBITS • • • Non-deterministic gates NON- DETERMIN SINGLE PHOTON • Don’t always work, but heralded when they do SINGLE GATE • • DETECTION PHOTONS • • • • & FEEDFORWARD
|C〉 • Teleportation: B moving information |φ〉 without measuring it X Z |C〉 Principles of LOQC quantum.info
• • QUBITS • • QUBITS • • • Non-deterministic gates NON- DETERMIN SINGLE PHOTON • Don’t always work, but heralded when they do SINGLE GATE • • DETECTION PHOTONS • • • • & FEEDFORWARD
|C〉 • Teleportation: B moving information |φ〉 without measuring it X Z |C〉 X Z |T〉 |φ〉 B |T〉 Measurement-induced nonlinearity quantum.info
• • QUBITS • • QUBITS • • • Non-deterministic gates NON- DETERMIN SINGLE PHOTON • Don’t always work, but heralded when they do SINGLE GATE • • DETECTION PHOTONS • • • • & FEEDFORWARD
|C〉 • Teleportation: B moving information |φ〉 without measuring it NON- X X Z DETERMIN |CNOT〉 GATE X Z Z |φ〉 B |T〉 Measurement-induced nonlinearity quantum.info
• • QUBITS • • QUBITS • • • Non-deterministic gates NON- DETERMIN SINGLE PHOTON • Don’t always work, but heralded when they do SINGLE GATE • • DETECTION PHOTONS • • • • & FEEDFORWARD
|C〉 • Teleportation: B moving information |φ〉 without measuring it NON- X X Z DETERMIN |CNOT〉 GATE X Z Z |φ〉 • Teleport non-deterministic B gates → deterministic |T〉
• Many non-deterministic gates proposed … Introduction quantum.info Proposed entangling gates Internal ancillas
NON- DETERMIN GATE
• KLM 4-photon ➠ 2 ancilla photons Knill, Laflamme, & Milburn, Nature 409, 46 (2001) ➠ interferometers Introduction quantum.info Proposed entangling gates Internal ancillas
NON- DETERMIN GATE
• KLM 4-photon ➠ 2 ancilla photons Knill, Laflamme, & Milburn, Nature 409, 46 (2001) ➠ interferometers
• Simplified 4-photon ➠ Ralph, White, Munro, & Milburn, 2 ancilla photons PRA 65, 012314 (2001) Introduction quantum.info Proposed entangling gates Internal ancillas
NON- DETERMIN GATE
• KLM 4-photon ➠ 2 ancilla photons Knill, Laflamme, & Milburn, Nature 409, 46 (2001) ➠ interferometers
• Simplified 4-photon ➠ Ralph, White, Munro, & Milburn, 2 ancilla photons PRA 65, 012314 (2001)
• Efficient 4-photon ➠ 2 ancilla photons Knill, PRA 66, 052306 (2002) Introduction quantum.info Proposed entangling gates Internal ancillas
NON- DETERMIN GATE
• KLM 4-photon ➠ 2 ancilla photons Knill, Laflamme, & Milburn, Nature 409, 46 (2001) ➠ interferometers
• Simplified 4-photon ➠ Ralph, White, Munro, & Milburn, 2 ancilla photons PRA 65, 012314 (2001)
• Efficient 4-photon ➠ 2 ancilla photons Knill, PRA 66, 052306 (2002)
• Entangled ancilla 4-photon ➠® 2 entangled ancilla Pittman, Jacobs, and Franson, photons PRL 88, 257902 (2002) Introduction quantum.info Proposed entangling gates Internal ancillas
NON- DETERMIN GATE
• KLM 4-photon ➠ 2 ancilla photons Knill, Laflamme, & Milburn, Nature 409, 46 (2001) ➠ interferometers
• Simplified 4-photon ➠ Ralph, White, Munro, & Milburn, 2 ancilla photons PRA 65, 012314 (2001)
• Efficient 4-photon ➠ 2 ancilla photons Knill, PRA 66, 052306 (2002)
• Entangled ancilla 4-photon ➠® 2 entangled ancilla Pittman, Jacobs, and Franson, photons PRL 88, 257902 (2002)
• Entangled input 2-photon ➠ 2 degrees of Sanaka, Kawahara, & Kuga, quant-ph/0108001 (2001) freedom (4-qubit) Proposed entangling gates quantum.info
Internal ancillas
NON- DETERMIN GATE
• KLM 4-photon Knill, Laflamme, & Milburn, Nature 409, 46 (2001)
• Simplified 4-photon Ralph, White, Munro, & Milburn, PRA 65, 012314 (2001)
• Efficient 4-photon Knill, PRA 66, 052306 (2002)
• Entangled ancilla 4-photon Pittman, Jacobs, and Franson, PRL 88, 257902 (2002) Gasparoni et al., PRL 93, 020504 (2004) • Entangled input 2-photon Pittman et al., PRA 68, 032316 (2004) Pittman, Jacobs, and Franson, Walther et al., Nature 434, 169 (2005) PRL 88, 257902 (2002) Proposed entangling gates quantum.info External ancillas Internal ancillas
NON- NON- QND DETERMIN DETERMIN GATE GATE QND
• KLM 4-photon Knill, Laflamme, & Milburn, Nature 409, 46 (2001) • Simplified 2-photon Ralph, Langford, Bell, & White, PRA 65, 062324 (2002) • Simplified 4-photon Ralph, White, Munro, & Milburn, • Simplified 2-photon PRA 65, 012314 (2001) Hofmann & Takeuchi, PRA 66, 024308 (2002) • Efficient 4-photon Knill, PRA 66, 052306 (2002) • Linear-optical QND Kok, Lee & Dowling, PRA 66, • Entangled ancilla 4-photon 063814 (2002) Pittman, Jacobs, and Franson, PRL 88, 257902 (2002)
• Entangled input 2-photon Pittman, Jacobs, and Franson, PRL 88, 257902 (2002) Proposed entangling gates quantum.info External ancillas Internal ancillas
NON- NON- QND DETERMIN DETERMIN GATE GATE QND
• KLM 4-photon Knill, Laflamme, & Milburn, Nature 409, 46 (2001) • Simplified 2-photon Ralph, Langford, Bell, & White, PRA 65, 062324 (2002) + teleportation • Simplified 4-photon Ralph, White, Munro, & Milburn, • Simplified 2-photon + error correction PRA 65, 012314 (2001) Hofmann & Takeuchi, PRA 66, 024308 (2002) = scalable QC • Efficient 4-photon Knill, PRA 66, 052306 (2002) • Linear-optical QND Kok, Lee & Dowling, PRA 66, • Entangled ancilla 4-photon 063814 (2002) Pittman, Jacobs, and Franson, PRL 88, 257902 (2002)
• Entangled input 2-photon Pittman, Jacobs, and Franson, PRL 88, 257902 (2002) The Scaling Problem quantum.info
• • QUBITS • • QUBITS • • • Non-deterministic gates NON- DETERMIN SINGLE PHOTON • Don’t always work, but heralded when they do SINGLE GATE • • DETECTION PHOTONS • • • • & FEEDFORWARD
|C〉 • Teleportation: B moving information |φ〉 without measuring it NON- X X Z DETERMIN |CNOT〉 GATE X Z Z |φ〉 • Teleport non-deterministic B gates → deterministic |T〉
• Linear optical Bell-state measurement of single DOF is non-deterministic → teleporter is non-deterministic 1 1 PB = /2 → PCNOT = /4 Fixing the Teleporter quantum.info Fixing the Teleporter quantum.info Fixing the Teleporter quantum.info
• 2001 - KLM’s solution
1. Increase the entangled resources
2. Concatenate the physical gates Fixing the Teleporter quantum.info
• 2001 - KLM’s solution
1. Increase the entangled resources |C〉 B e.g. “Level 1” encoding |φ〉 U
NON- DETERMIN |CNOTCNOT GATE U1 ... Un | 〉 P = (2/ )2 = 4/ |φ〉 CNOT 3 9 U |T〉 B
2. Concatenate the physical gates Fixing the Teleporter quantum.info
• 2001 - KLM’s solution
1. Increase the entangled resources |C〉 B e.g. “Level 1” encoding |φ〉 U
NON- DETERMIN |CNOTCNOT GATE U1 ... Un | 〉 P = (2/ )2 = 4/ |φ〉 CNOT 3 9 U |T〉 B
2. Concatenate the physical gates e.g. to achieve a single entangling operation with success probability 95%
9 ~300 successful “Level 2” encoded gate operations, each P = /16 10,000’s of optical elements Fixing the Teleporter quantum.info
• 2001 - KLM’s solution
1. Increase the entangled resources
e.g. “Level 1” encoding
2 2 4 PCNOT = ( /3) = /9
2. Concatenate the physical gates e.g. to achieve a single entangling operation with success probability 95%
9 ~300 successful “Level 2” encoded gate operations, each P = /16 10,000’s of optical elements
Infinite entangled modes required for PCNOT = 1 Two paths to scalability quantum.info
conventional circuit
flow of quantum information
Uz(α1) Uz(α2) Uz(α3) Uz(α4)
Uz(β1) Uz(β2) Uz(β3) Uz(β4)
Uz(γ1) Uz(γ2) Uz(γ3) Uz(γ4)
two-qubit single (entangling) qubit gates gates
Raussendorf and Briegel, PRL 86, 5188 (2001) Nielsen, PRL 93, 040503 (2004) Two paths to scalability quantum.info
conventional circuit cluster/graph circuit
flow of quantum information flow of classical information
α1 ±α2 ±α3 ±α4 Uz(α1) Uz(α2) Uz(α3) Uz(α4)
Uz(β1) Uz(β2) Uz(β3) Uz(β4) β1 ±β2 ±β3 ±β4
Uz(γ1) Uz(γ2) Uz(γ3) Uz(γ4) γ1 ±γ2 ±γ3 ±γ4
two-qubit single entanglement qubit measurement (entangling) qubit cos + sin gates θ σx θ σy gates
Raussendorf and Briegel, PRL 86, 5188 (2001) Nielsen, PRL 93, 040503 (2004) Two paths to scalability quantum.info
conventional circuit cluster/graph circuit
flow of quantum information flow of classical information
α1 ±α2 ±α3 ±α4 Uz(α1) Uz(α2) Uz(α3) Uz(α4)
Uz(β1) Uz(β2) Uz(β3) Uz(β4) β1 ±β2 ±β3 ±β4
Uz(γ1) Uz(γ2) Uz(γ3) Uz(γ4) γ1 ±γ2 ±γ3 ±γ4
two-qubit single entanglement qubit measurement (entangling) qubit cos + sin gates θ σx θ σy gates CZ
qubit qubit
Raussendorf and Briegel, PRL 86, 5188 (2001) Nielsen, PRL 93, 040503 (2004) Two paths to scalability quantum.info
conventional circuit cluster/graph circuit
flow of quantum information flow of classical information
α1 ±α2 ±α3 ±α4 Uz(α1) Uz(α2) Uz(α3) Uz(α4)
Uz(β1) Uz(β2) Uz(β3) Uz(β4) β1 ±β2 ±β3 ±β4
Uz(γ1) Uz(γ2) Uz(γ3) Uz(γ4) γ1 ±γ2 ±γ3 ±γ4
two-qubit single entanglement qubit measurement (entangling) qubit cos + sin gates θ σx θ σy gates
CZ
qubit qubit qubit
Raussendorf and Briegel, PRL 86, 5188 (2001) Nielsen, PRL 93, 040503 (2004) Two paths to scalability quantum.info
conventional circuit cluster/graph circuit
flow of quantum information flow of classical information
α1 ±α2 ±α3 ±α4 Uz(α1) Uz(α2) Uz(α3) Uz(α4)
Uz(β1) Uz(β2) Uz(β3) Uz(β4) β1 ±β2 ±β3 ±β4
Uz(γ1) Uz(γ2) Uz(γ3) Uz(γ4) γ1 ±γ2 ±γ3 ±γ4
two-qubit single entanglement qubit measurement (entangling) qubit cos + sin gates θ σx θ σy gates
qubit
Raussendorf and Briegel, PRL 86, 5188 (2001) Nielsen, PRL 93, 040503 (2004) Two paths to scalability quantum.info
conventional circuit cluster/graph circuit
flow of quantum information flow of classical information
α1 ±α2 ±α3 ±α4 Uz(α1) Uz(α2) Uz(α3) Uz(α4)
Uz(β1) Uz(β2) Uz(β3) Uz(β4) β1 ±β2 ±β3 ±β4
Uz(γ1) Uz(γ2) Uz(γ3) Uz(γ4) γ1 ±γ2 ±γ3 ±γ4
two-qubit single entanglement qubit measurement (entangling) qubit cos + sin gates θ σx θ σy gates CZ
qubit qubit
Raussendorf and Briegel, PRL 86, 5188 (2001) Nielsen, PRL 93, 040503 (2004) Two paths to scalability quantum.info
conventional circuit cluster/graph circuit
flow of quantum information flow of classical information
α1 ±α2 ±α3 ±α4 Uz(α1) Uz(α2) Uz(α3) Uz(α4)
Uz(β1) Uz(β2) Uz(β3) Uz(β4) β1 ±β2 ±β3 ±β4
Uz(γ1) Uz(γ2) Uz(γ3) Uz(γ4) γ1 ±γ2 ±γ3 ±γ4
two-qubit single entanglement qubit measurement (entangling) qubit cos + sin gates θ σx θ σy gates
CZ
qubit qubit qubit
Raussendorf and Briegel, PRL 86, 5188 (2001) Nielsen, PRL 93, 040503 (2004) Graph-state computation quantum.info 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: 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 1 3 Acceptable loss for a scalable architecture “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. But, in practice, there is.” – Jan L.A. van de Snepscheut (1953-1994) quantum.info Two?qubit'gate
CSIGN gate
C0 C1 180˚ phase T0 shift T1
Ralph, Langford, Bell & White, PRA 65, 062324 (2002) Hofmann & Takeuchi, PRA 66, 024308 (2002) quantum.info Two?qubit'gate
CSIGN gate
-1 /3 C0 C1
1 /3 T0 T1
1 /3
Ralph, Langford, Bell & White, PRA 65, 062324 (2002) Hofmann & Takeuchi, PRA 66, 024308 (2002) quantum.info Two?qubit'gate
CSIGN gate
-1 /3 C0 C1
1 /3 T0 T1
1 /3
Ralph, Langford, Bell & White, PRA 65, 062324 (2002) Hofmann & Takeuchi, PRA 66, 024308 (2002) quantum.info Two?qubit'gate
CSIGN gate
-1 /3 C0 C1
1 /3 T0 T1
1 /3
Ralph, Langford, Bell & White, PRA 65, 062324 (2002) Hofmann & Takeuchi, PRA 66, 024308 (2002) quantum.info Two?qubit'gate
CSIGN gate
-1 /3 C0 C1
1 /3 T0 T1
1 /3
Ralph, Langford, Bell & White, PRA 65, 062324 (2002) Hofmann & Takeuchi, PRA 66, 024308 (2002) quantum.info Two?qubit'gate
CSIGN gate
-1 /3 C0 C1
1 /3 T0 T1
1/ 3 both transmitted
Ralph, Langford, Bell & White, PRA 65, 062324 (2002) Hofmann & Takeuchi, PRA 66, 024308 (2002) quantum.info Two?qubit'gate
CSIGN gate
-1 /3 C0 C1
1 /3 T0 T1
1/ 3 both reflected
Ralph, Langford, Bell & White, PRA 65, 062324 (2002) Hofmann & Takeuchi, PRA 66, 024308 (2002) quantum.info Two?qubit'gate
CSIGN gate
-1 /3 C0 C1
1 /3 T0 T1
1 /3
Ralph, Langford, Bell & White, PRA 65, 062324 (2002) Hofmann & Takeuchi, PRA 66, 024308 (2002) quantum.info Controlled?NOT'gate
CNOT gate
-1 /3 C0 Control in Control out C1
1 /3 T0 T1 Target in Target out HWP 1 HWP /3
one photon can flip the polarisation of another!
Ralph, Langford, Bell & White, PRA 65, 062324 (2002) Hofmann & Takeuchi, PRA 66, 024308 (2002) Building entangling gates C0 quantum.info Interferometric gate C1 T0 T1 Jamin-Lebedeff interferometer HWP HWP Very stable: insensitive to x-y-z translation
V C H
O’Brien, Pryde, White, Ralph, and Branning, Nature 426, 264 (2003) O’Brien, Pryde, Gilchrist, James, Langford, Ralph and White, PRL 93, 080502 (2004) Building entangling gates C0 quantum.info Interferometric gate C1 T0 T1 Jamin-Lebedeff interferometer HWP HWP Very stable: insensitive to x-y-z translation
V H
T
O’Brien, Pryde, White, Ralph, and Branning, Nature 426, 264 (2003) O’Brien, Pryde, Gilchrist, James, Langford, Ralph and White, PRL 93, 080502 (2004) Building entangling gates C0 quantum.info Interferometric gate C1 T0 T1 Jamin-Lebedeff interferometer HWP HWP Very stable: insensitive to x-y-z translation
V C V,H H
T
O’Brien, Pryde, White, Ralph, and Branning, Nature 426, 264 (2003) O’Brien, Pryde, Gilchrist, James, Langford, Ralph and White, PRL 93, 080502 (2004) Building entangling gates C0 quantum.info Interferometric gate C1 T0 T1 Jamin-Lebedeff interferometer HWP HWP Very stable: insensitive to x-y-z translation
V C V,H H
T Non-classical interference
O’Brien, Pryde, White, Ralph, and Branning, Nature 426, 264 (2003) O’Brien, Pryde, Gilchrist, James, Langford, Ralph and White, PRL 93, 080502 (2004) Building entangling gates C0 quantum.info Interferometric gate C1 T0 T1 Jamin-Lebedeff interferometer HWP HWP Very stable: insensitive to x-y-z translation
V C V,H H
T Non-classical interference ✓ Tunable beamsplitters ✗ Dual-path interferometers limit performance
O’Brien, Pryde, White, Ralph, and Branning, Nature 426, 264 (2003) O’Brien, Pryde, Gilchrist, James, Langford, Ralph and White, PRL 93, 080502 (2004) Interferometric'controlled?NOT'gate quantum.info
Control in Automated Target in CNOT tomography gate
Control out Target out
O’Brien, Pryde, et al., Nature 426, 264 (2003) Interferometric'controlled?NOT'gate quantum.info
Control in Automated Target in CNOT tomography gate
Control out Target out
O’Brien, Pryde, et al., Nature 426, 264 (2003) Interferometric'controlled?NOT'gate
From Mick Withford Marshall, et al., Optics Express 17, 12546 (2009) Beamsplitter'controlled?NOT'gate quantum.info
C0 C1
T0 T1
HWP HWP
C
R = 1/ Non-classical interference H 3 RV = 1 T
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
C0 C1
T0 T1
HWP HWP
C
R = 1/ Non-classical interference H 3 RV = 1 T
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
C0 C1
T0 T1
HWP HWP
C
R = 1/ Non-classical interference H 3 RV = 1 T ✓ No dual-path interferometers ✗ No adjustment if wrong splitting ratio
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) Building'graph'states quantum.info
1 2 2 photon cluster
1
2
Non-classical interference between |H1 and |H2 Building'graph'states quantum.info
1 2 3 3 photon cluster
1
2 3
Non-classical interference Non-classical interference between |H1 and |H2 between |V2 and |H3 Building'graph'states quantum.info
1 2 3 3 photon cluster
Repeat unit 1
2 3 Exponential blow-out
Non-classical interference Non-classical interference between |H1 and |H2 between |V2 and |H3 Table of photonic two-qubit gate proposals quantum.info
Linear-optical Knill, Laflamme & Milburn, Strong Nature Interaction-free Nonlinearities 46 (2001) Measurement Milburn, Gilchrist, White, & Munro PRL PRA
Geometric phase Quantum Zeno in Hilbert Space Franson, Jacobs & Pittman Langford & Ramelow, PRA 70, 062302 (2004) In prep Amplified weak Nonlinearities Nemoto and Munro, PRL Strong Nonlinearities quantum.info
• 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
Franson, Jacobs, and Pittman, quant-ph/0401113 (2004) quant-ph/0408097 (2004) Strong Nonlinearities quantum.info
• 2004 - Franson’s solution: deterministic entangling gate by combining linear optics gates, quantum Zeno effect and nonlinear optics 1. Linear optics 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
Franson, Jacobs, and Pittman, quant-ph/0401113 (2004) quant-ph/0408097 (2004) Strong Nonlinearities quantum.info
• 2004 - Franson’s solution: deterministic entangling gate by combining linear optics gates, quantum Zeno effect and nonlinear optics 1. Linear optics 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
effects of loss? c.f. “interaction-free” measurements Franson, Jacobs, and Pittman, quant-ph/0401113 (2004) Kwiat, et al., PRL 83, 4725 (1999) Gilchrist, et al., PRA 66, 012106 (2002) quant-ph/0408097 (2004) Quantum Zeno quantum.info
induced by 20 photons
Hendrickson, Lai, Pittman and Franson, PRL 105, 173602 (2010) Getting photon-photon nonlinearities quantum.info
Luiten, Adelaide (2009) Table of photonic two-qubit gate proposals quantum.info
Linear-optical Knill, Laflamme & Milburn, Strong Nature Interaction-free Nonlinearities 46 (2001) Measurement Milburn, Gilchrist, White, & Munro PRL PRA
Geometric phase Quantum Zeno in Hilbert Space Franson, Jacobs & Pittman Langford & Ramelow, PRA In prep Amplified weak Nonlinearities Nemoto and Munro, PRL 93, 250502 (2004) quantum.info Measurement & nonlinearities
• 2004 Barrett, et al.: amplify weak optical nonlinearities using measurement
Homodyne classical feed-forward
Entangler
Barrett et al., PRA 71, 060302R (2005) quantum.info Measurement & nonlinearities
• 2004 Barrett, et al.: amplify weak optical nonlinearities using measurement
Homodyne classical feed-forward H
V Entangler
Barrett et al., PRA 71, 060302R (2005) quantum.info Measurement & nonlinearities
• 2004 Barrett, et al.: amplify weak optical nonlinearities using measurement
Homodyne classical feed-forward
V H
Entangler
Barrett et al., PRA 71, 060302R (2005) quantum.info Measurement & nonlinearities
• 2004 Barrett, et al.: amplify weak optical nonlinearities using measurement
Homodyne classical feed-forward H V H V Entangler
Barrett et al., PRA 71, 060302R (2005) quantum.info Measurement & nonlinearities
• 2004 Barrett, et al.: amplify weak optical nonlinearities using measurement
Homodyne classical feed-forward
Entangler
• 2004 Nemoto & Munro: deterministic CNOT gate
Barrett et al., PRA 71, 060302R (2005) Nemoto and Munro, PRL 93, 250502 (2004) Getting photon-photon nonlinearities quantum.info Nonlinear photonics
10µm
85 Rb phase Psig 25 µW π rad at Pm=150 nW 0.7 µrad / photon
Perrella, Light, Anstie, Benabid, Stace, White, & Luiten, PRA 88, 013819 (2013) Getting photon-photon nonlinearities quantum.info Nonlinear photonics
10µm
85 Rb phase Psig 25 µW π rad at Pm=150 nW abs 0.7 µrad / photon 80% absorption
Perrella, Light, Anstie, Benabid, Stace, White, & Luiten, PRA 88, 013819 (2013) Getting photon-photon nonlinearities quantum.info Nonlinear photonics
10µm
85 Rb phase Psig 25 µW π rad at Pm=150 nW abs 0.7 µrad / photon 80% absorption 0.13 µrad / photon 0.5% absorption
Perrella, Light, Anstie, Benabid, Stace, White, & Luiten, PRA 88, 013819 (2013) Getting photon-photon nonlinearities quantum.info Nonlinear photonics
10µm
85 Rb phase Psig 25 µW π rad at Pm=150 nW abs 0.7 µrad / photon 80% absorption 0.13 µrad / photon 0.5% absorption Venkataraman, Saha, Gaeta, 0.3 mrad Nature Photonics 7, 138 (2013) 6 µm hole Perrella, Light, Anstie, Benabid, Stace, White, & Luiten, PRA 88, 013819 (2013) Using quantum memories Using quantum memories
recall efficiency of 69% quantum-noise limited Using quantum memories recall efficiency of 87% Using quantum memories recall efficiency of 87%
fidelity of 98% Take a break Part 4: Measuring entangling gates quantum.info quantum.info Single'gate'performance Quantum'Tomography quantum.info
• State tomography: measure combinations 0 1 0 + 1 0 + i1 of basis states H V D R H
L D A R
V Poincaré Sphere Quantum'Tomography quantum.info
• State tomography: measure combinations 0 1 0 + 1 0 + i1 of basis states H V D R H
L P degree of polarisation D A R discrimination V Poincaré Sphere Quantum'Tomography quantum.info
• State tomography: measure combinations 0 1 0 + 1 0 + i1 of basis states H V D R H
L P degree of polarisation D A R discrimination V Poincaré Sphere
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'Tomography quantum.info
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 Kleinlogel and White, PLoS ONE 3, e2190 (2008) Quantum'Tomography quantum.info
D
L M
V D, A
dorsal mid-band R, L row 1 row 2 row 3 row 4 row 5 row 6 H, V ventral
1mm
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 Kleinlogel and White, PLoS ONE 3, e2190 (2008) Quantum'Tomography quantum.info
Mantis shrimp, Gonodactylus Smithii, has optimal polarisation vision Kleinlogel and White, PLoS ONE 3, e2190 (2008) Measuring CNOT gate operation quantum.info 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 VH DH RH HV VV DV RV HD VD DD RD HR VR DR RR Measuring CNOT gate operation quantum.info 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 VH DH RH HV VV DV RV 1 HD VD DD RD HR VR DR RR PAB
0 0˚ 45˚ 90˚ 135˚ 180˚ H V H
HH HV VH VV HH 1 0 0 0 HV 0 0 0 0 VH 0 0 0 0 VV 0 0 0 0
White, James, Eberhard & Kwiat, PRL 83, 3103 (1999) James, Kwiat, Munro & White, PRA 64, 052312 (2001) Measuring CNOT gate operation quantum.info 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 VH DH RH HV VV DV RV 1 1 HD VD DD RD HR VR DR RR PAB PAB
0 0 0˚ 45˚ 90˚ 135˚ 180˚ 0˚ 45˚ 90˚ 135˚ 180˚ H V H H V H
HH HV VH VV HH HV VH VV HH 1 0 0 0 HH 1 1 1 1 HV 0 0 0 0 1 HV 1 1 1 1 VH 0 0 0 0 4 VH 1 1 1 1 VV 0 0 0 0 VV 1 1 1 1
White, James, Eberhard & Kwiat, PRL 83, 3103 (1999) James, Kwiat, Munro & White, PRA 64, 052312 (2001) Measuring CNOT gate operation quantum.info 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 VH DH RH HV VV DV RV 1 HD VD DD RD HR VR DR RR PAB
0 • “impossible” set of correlations 0˚ 45˚ 90˚ 135˚ 180˚ H V H |HH〉+|VV〉 |DD〉+|AA〉 HH HV VH VV |RL〉+|LR〉 ➠ entanglement HH 1 0 0 1 1 HV 0 0 0 0 • n qubit states require 4n measurements 2 VH 0 0 0 0 2 qubit states require 16 measurements VV 1 0 0 1
White, James, Eberhard & Kwiat, PRL 83, 3103 (1999) James, Kwiat, Munro & White, PRA 64, 052312 (2001) Quantum'Tomography quantum.info
• State tomography: measure combinations 0 1 0 + 1 0 + i1 of basis states H V D R HH VH DH RH for 2 photon states, bi-photon Stokes parameters HV VV DV RV HD VD DD RD n qubit state requires 22n measurements HR VR DR RR |0+i1〉=|R〉 • Process tomography: measure combinations of basis processes rotations on Poincare sphere I
|0〉= |H〉
|0+1〉=|D〉
White, Gilchrist, Pryde, O’Brien, Bremner, & Langford, JOSA B, 24, 172-183 (2007) Quantum'Tomography quantum.info
• State tomography: measure combinations 0 1 0 + 1 0 + i1 of basis states H V D R HH VH DH RH for 2 photon states, bi-photon Stokes parameters HV VV DV RV HD VD DD RD n qubit state requires 22n measurements HR VR DR RR |0+i1〉=|R〉 • Process tomography: measure combinations of basis processes rotations on Poincare sphere I X
|0〉= |H〉
|0+1〉=|D〉
White, Gilchrist, Pryde, O’Brien, Bremner, & Langford, JOSA B, 24, 172-183 (2007) Quantum'Tomography quantum.info
• State tomography: measure combinations 0 1 0 + 1 0 + i1 of basis states H V D R HH VH DH RH for 2 photon states, bi-photon Stokes parameters HV VV DV RV HD VD DD RD n qubit state requires 22n measurements HR VR DR RR |0+i1〉=|R〉 • Process tomography: measure combinations of basis processes rotations on Poincare sphere I X Z
|0〉= |H〉
|0+1〉=|D〉
White, Gilchrist, Pryde, O’Brien, Bremner, & Langford, JOSA B, 24, 172-183 (2007) Quantum'Tomography quantum.info
• State tomography: measure combinations 0 1 0 + 1 0 + i1 of basis states H V D R HH VH DH RH for 2 photon states, bi-photon Stokes parameters HV VV DV RV HD VD DD RD n qubit state requires 22n measurements HR VR DR RR |0+i1〉=|R〉 • Process tomography: measure combinations of basis processes rotations on Poincare sphere I X Y Z
|0〉= |H〉
|0+1〉=|D〉
White, Gilchrist, Pryde, O’Brien, Bremner, & Langford, JOSA B, 24, 172-183 (2007) Quantum'Tomography quantum.info
• State tomography: measure combinations 0 1 0 + 1 0 + i1 of basis states H V D R HH VH DH RH for 2 photon states, bi-photon Stokes parameters HV VV DV RV HD VD DD RD n qubit state requires 22n measurements HR VR DR RR |0+i1〉=|R〉 • Process tomography: measure combinations of basis processes rotations on Poincare sphere I X Y Z I I X I Y I Z I I X XX YX ZX |0〉= |H〉 I Y XY YY ZY I Z XZ YZ ZR |0+1〉=|D〉
n qubit gate requires 24n measurements
• Reconstructed states and processes are unphysical: effect of uncertainties Maximum likelihood or Bayesian analysis required
White, Gilchrist, Pryde, O’Brien, Bremner, & Langford, JOSA B, 24, 172-183 (2007) quantum.info Quantum process tomography
Measuring a state is not measuring a gate … normally quantum.info Quantum process tomography
Measuring a state is not measuring a gate … normally
Standard n2n input states
^ ^ ρH ρH´ ^ ^ ρV 1 qubit ρV´ ^ ^ ρD gate ρD´ ^ ^ ρR ρR´
General recipe: Nielsen & Chuang, `Quantum Information´ (2003) quantum.info Quantum process tomography
Measuring a state is not measuring a gate … normally
Standard n2n input states Ancilla assisted 2n-qubit input state
^ ^ ρH ρH´ ^ ^ ^ ^ ρ2 ρ´2 ρV 1 qubit ρV´ ^ ^ 1 qubit ρD gate ρD´ ^ ^ gate ρR ρR´
General recipe: Altepeter, Branning, Jeffrey, Wei, Kwiat, Thew, O’Brien, Nielsen & Chuang, `Quantum Information´ (2003) Nielsen, & White, PRL 90, 193601 (2003) quantum.info Quantum process tomography
Measuring a state is not measuring a gate … normally
Standard n2n input states Ancilla assisted 2n-qubit input state
^ ^ ρH ρH´ ^ ^ ^ ^ ρ2 ρ´2 ρV 1 qubit ρV´ ^ ^ 1 qubit ρD gate ρD´ ^ ^ gate ρR ρR´
General recipe: Altepeter, Branning, Jeffrey, Wei, Kwiat, Thew, O’Brien, Nielsen & Chuang, `Quantum Information´ (2003) Nielsen, & White, PRL 90, 193601 (2003)
n qubit gate requires 42n measurements 2 qubit gate (CNOT) requires 256 measurements
2-qubit gate recipes: White, Gilchrist, Pryde, O’Brien, Bremner & Langford, quant-ph/0308115 (2003) quantum.info Single'gate'performance
For a CZ gate: χ matrix: Table of process measurement probabilities, and coherences between them quantum.info Single'gate'performance
For a CZ gate: χ matrix: Table of process measurement probabilities, and coherences between them
Ideal Measured
Langford, et al., PRL 95, 210504 (2005) quantum.info Single'gate'performance
For a CZ gate: χ matrix: Table of process measurement probabilities, and coherences between them
Ideal Measured 2004, F = 94 ± 2 % average gate fidelity: 2005, F = 89.3 ± 0.1 % Fp is process fidelity: overlap between ideal & exp processes
1 gate works 90-95% of time At best, 2 gates should work 80-90% of time
Langford, et al., PRL 95, 210504 (2005) O’Brien, Pryde, et al., Nature 426, 264 (2003) Measuring entangling gates quantum.info Quantum process tomography
• Physical interpretation? Change basis
CNOT ⊗ (II, IX, IY, IZ, XI, XY, XZ, YI, YX, YY, YZ, ZI, ZX, ZY, ZZ) ⊗ CNOT Measuring entangling gates quantum.info Quantum process tomography
• Physical interpretation? Change basis
CNOT ⊗ (II, IX, IY, IZ, XI, XY, XZ, YI, YX, YY, YZ, ZI, ZX, ZY, ZZ) ⊗ CNOT
Measured (Re) Measured (Im) Measuring entangling gates quantum.info Quantum process tomography
• Physical interpretation? Change basis
CNOT ⊗ (II, IX, IY, IZ, XI, XY, XZ, YI, YX, YY, YZ, ZI, ZX, ZY, ZZ) ⊗ CNOT
= 87%
Measured (Re) Measured (Im)
O’Brien, Pryde, Gilchrist, James, Langford, Ralph and White, PRL 93, 080502 (2004) quantum.info Single'gate'performance • Gate performance depends on: photon source, circuit quality, detectors dependent photons: not scalable
CZ
F = (89.3 ± 0.1) %
Weinhold, et al., arXiv:0808.0794 (2008) quantum.info Single'gate'performance • Gate performance depends on: photon source, circuit quality, detectors dependent photons: not scalable independent photons: scalable
CZ CZ
circuit & mode- matching is same, F = (89.3 ± 0.1) % source is different F = (82.5 ± 1.5) %
Weinhold, et al., arXiv:0808.0794 (2008) quantum.info Single'gate'performance • Gate performance depends on: photon source, circuit quality, detectors dependent photons: not scalable independent photons: scalable
CZ CZ
circuit & mode- matching is same, F = (89.3 ± 0.1) % source is different F = (82.5 ± 1.5) % • Model source & gate based on measured parameters 2.Beamsplitter 3.Photon loss, reflectivities, Δ=1-2% 90-97%
1. Independent downconversion, 1.2, 3.2% higher order terms
Weinhold, et al., arXiv:0808.0794 (2008) quantum.info Single'gate'performance
1-Fp ideal 0 %
Weinhold, et al., arXiv:0808.0794 (2008) quantum.info Single'gate'performance
1-Fp ideal 0 % ideal srce (2+3) 2.8 % photon source Δ ε=15.8% all (1+2+3) 18.6 %
Weinhold, et al., arXiv:0808.0794 (2008) quantum.info Single'gate'performance
1-Fp ideal 0 % ideal srce (2+3) 2.8 % photon source Δ ε=15.8% all (1+2+3) 18.6 % mode mismatch Δ ε= 3.2% experiment 21.8% ±1.5%
Weinhold, et al., arXiv:0808.0794 (2008) quantum.info Single'gate'performance
1-Fp ideal 0 % ideal srce (2+3) 2.8 % photon source Δ ε=15.8% all (1+2+3) 18.6 % mode mismatch Δ ε= 3.2% experiment 21.8% ±1.5%
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 %
Weinhold, et al., arXiv:0808.0794 (2008) quantum.info Single'gate'performance
1-Fp ideal 0 % ideal srce (2+3) 2.8 % photon source Δ ε=15.8% all (1+2+3) 18.6 % mode mismatch Δ ε= 3.2% experiment 21.8% ±1.5%
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 %
photonics QC: scalability requires sources, detectors are rate limiting factor
Weinhold, et al., arXiv:0808.0794 (2008) COMPRESSED SENSING
5000-pt signal
Candès, Romberg, and Tao, IEEE Transactions on Information Theory, 52 489 - 509 (2006) COMPRESSED SENSING
5000-pt signal
Candès, Romberg, and Tao, IEEE Transactions on Information Theory, 52 489 - 509 (2006) COMPRESSED SENSING
500 random samples from 5000-pt signal
Candès, Romberg, and Tao, IEEE Transactions on Information Theory, 52 489 - 509 (2006) COMPRESSED SENSING
500 random samples from 5000-pt signal
Least-squares reconstruction
Candès, Romberg, and Tao, IEEE Transactions on Information Theory, 52 489 - 509 (2006) COMPRESSED SENSING
500 random samples from 5000-pt signal
Least-squares reconstruction
Candès, Romberg, and Tao, IEEE Transactions on Information Theory, 52 489 - 509 (2006) COMPRESSED SENSING
500 random samples from 5000-pt signal
Compressed sensing (Sparse matrix)
Candès, Romberg, and Tao, IEEE Transactions on Information Theory, 52 489 - 509 (2006) quantum.info COMPRESSED TOMOGRAPHY
d-dimensional quantum system needs for qubits d=2n how many measurement configurations? quantum.info COMPRESSED TOMOGRAPHY
d-dimensional quantum system needs for qubits d=2n how many measurement configurations?
Full QT state O(d2)
process O(d4) quantum.info COMPRESSED TOMOGRAPHY
d-dimensional quantum system needs for qubits d=2n how many measurement configurations?
Full QT Compressed QT quadratically faster state O(d2) O(d r log2 d) blind, r is matrix rank
process O(d4)
Gross, Liu, Flammia, Becker, & Eisert, PRL 105, 150401 (2010). quantum.info COMPRESSED TOMOGRAPHY
d-dimensional quantum system needs for qubits d=2n how many measurement configurations?
Full QT Compressed QT quadratically faster state O(d2) O(d r log2 d) blind, r is matrix rank exponentially faster process O(d4) O(s log d) needs prior, s is matrix sparsity
Gross, Liu, Flammia, Becker, & Eisert, Shabani, Kosut, Mohseni, Rabitz, Broome, Almeida, Fedrizzi, & White, PRL 105, 150401 (2010). PRL 106, 100401 (2011) quantum.info COMPRESSED TOMOGRAPHY
d-dimensional quantum system needs for qubits d=2n how many measurement configurations?
Full QT Compressed QT quadratically faster state O(d2) O(d r log2 d) blind, r is matrix rank exponentially faster process O(d4) O(s log d) needs prior, s is matrix sparsity
Gross, Liu, Flammia, Becker, & Eisert, Shabani, Kosut, Mohseni, Rabitz, Broome, Almeida, Fedrizzi, & White, PRL 105, 150401 (2010). PRL 106, 100401 (2011) quantum.info COMPRESSED TOMOGRAPHY
d-dimensional quantum system needs for qubits d=2n how many measurement configurations?
Full QT Compressed QT quadratically faster state O(d2) O(d r log2 d) blind, r is matrix rank exponentially faster process O(d4) O(s log d) needs prior, s is matrix sparsity engineered quantum systems aim to implement a unitary process which is maximally-sparse in its eigenbasis
Gross, Liu, Flammia, Becker, & Eisert, Shabani, Kosut, Mohseni, Rabitz, Broome, Almeida, Fedrizzi, & White, PRL 105, 150401 (2010). PRL 106, 100401 (2011) quantum.info COMPRESSED TOMOGRAPHY
d-dimensional quantum system needs for qubits d=2n how many measurement configurations?
Full QT Compressed QT quadratically faster state O(d2) O(d r log2 d) blind, r is matrix rank exponentially faster process O(d4) O(s log d) needs prior, s is matrix sparsity engineered quantum systems aim to implement a unitary process which is maximally-sparse in its eigenbasis in practice—as observed in QPT experiments in liquid-state NMR, photonics, ion traps, and superconducting circuits—nearly-sparse, still compressible!
Gross, Liu, Flammia, Becker, & Eisert, Shabani, Kosut, Mohseni, Rabitz, Broome, Almeida, Fedrizzi, & White, PRL 105, 150401 (2010). PRL 106, 100401 (2011) COMPRESSED TOMOGRAPHY two-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 TOMOGRAPHY two-photon CZ gate low-noise, purity 91% Compressed tomo Full tomo
Shabani, Kosut, Mohseni, Rabitz, Broome, Almeida, Fedrizzi, and White, Physical Review Letters 106, 100401 (2011) COMPRESSED TOMOGRAPHY two-photon CZ gate low-noise, purity 91% Compressed tomo Full tomo
32 combinations
16 input combinations of H,V,D,R
2 measurement F=95% combinations of RI and IR
Shabani, Kosut, Mohseni, Rabitz, Broome, Almeida, Fedrizzi, and White, Physical Review Letters 106, 100401 (2011) COMPRESSED TOMOGRAPHY four-photon CZ gate high-noise, purity 62% Compressed tomo Full tomo
32 combinations
16 input combinations of H,V,D,R
2 measurement F=85% combinations of RI and IR
Shabani, Kosut, Mohseni, Rabitz, Broome, Almeida, Fedrizzi, and White, Physical Review Letters 106, 100401 (2011)