Outline 1. Optical Quantum Computing 101 2. Where we are 3. - - PowerPoint PPT Presentation

Progress Toward Scalable

Optical Quantum Computing

Paul Kwiat

Linear

Outline

1. Optical Quantum Computing 101 2. Where we are 3. Some theoretical magic 4. Technology Development -- where we need to get to

• Sources
• Detectors
• Integrated optics

Outline

1. Optical Quantum Computing 101 2. Where we are 3. Some theoretical magic 4. Technology Development -- where we need to get to

• Sources
• Detectors
• Integrated optics

Why Optical Quantum Computing?

• Very little/no decoherence -- photon’s don’t interact
• Excellent performance with off-the-shelf optics
• Very fast gates: single-qubit ~10 ps - 5 ns

two-qubit <150 ns

“Photons been very very good to me”

Why not Optical Quantum Computing?

• Photon’s don’t interact -- 2-qubit gates hard
• Linear approach: measurement-induced

nonlinearity

• Nonlinear approach: Zeno and QND gates

Optical Quantum Computing

Linear “KLM” “modified KLM” Nonlinear

• Q. Zeno

(Franson) QND (Munro & Nemoto) graph states Cluster/parity encoding

Optical Quantum Computing

Linear “KLM” “modified KLM” Nonlinear

• Q. Zeno

(Franson) QND (Munro & Nemoto) graph states Cluster/parity encoding

• P. Kok, W. J. Munro, K. Nemoto, T. C. Ralph, J. P. Dowling, and G. J. Milburn, “Linear
• ptical quantum computing with photonic qubits”, Rev. Mod. Phys. 79, 135 (2007)
• J. O’Brien, “Optical Quantum Computing”, Science 318, 1567 (2007)

Grover’s search algorithm with linear optics

• Gates: Linear optical elements
• Nonscalable -- each new qubit

doubles required number of elements

• Seems like you need single-photon

nonlinearities for scalability

PGK et al., J. Mod. Opt. 47, 257 (2000)

Optical realization with single photons: A database of four elements

Grover’s Search algorithm Accuracy: ~97.5% Hosten et al., Nature 439, 949 (2006)

…historical interlude…

Linear optical quantum computing

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

• SINGLE

PHOTONS FAST FEEDFORWARD SINGLE PHOTON DETECTION

Kok, Munro, Nemoto, Ralph, Dowling & Milburn

SINGLE-PHOTON DETECTION

LARGE overhead requirements…(>105/gate)

Introduction

Principles of LOQC

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

NON- DETERMIN GATE

• QUBITS

QUBITS

• SINGLE

PHOTONS SINGLE PHOTON DETECTION & FEEDFORWARD

• Teleportation:

moving information without measuring it

Gottesman & Chuang, Nature 402, 390 (1999)

X Z

|C

X Z

|T

B

X Z

| | |CNOT

NON- DETERMIN GATE

B

Introduction

Principles of LOQC

• Non-deterministic gates
• Don’t always work, but heralded when they do
• Many non-deterministic gates proposed …
• Teleport non-deterministic

gates → deterministic

NON- DETERMIN GATE

• QUBITS

QUBITS

• SINGLE

PHOTONS SINGLE PHOTON DETECTION & FEEDFORWARD

• Teleportation:

moving information without measuring it

Gottesman & Chuang, Nature 402, 390 (1999)

The polarizing beam-splitter -- a parity gate

• The polarizing beam-splitter

and single-photon detection is sufficient to perform universal quantum computation

• Can be used to construct a CNOT gate or cluster states

Franson CNOT gate Browne and Rudolph Fusion gates

Pittman, et. al, PRA 64, 062311 (2001) Gasparoni et al, PRL 93, 020504 (2004) Browne and Rudolph et. al., quant-ph/0405157 Walther et al, Nature 434, 168 (2004)

Introduction

Proposed LOQC gates

+ teleportation + error correction = scalable QC (but still very large

• verhead needs)
• KLM 4-photon

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

• Entangled ancilla 4-photon

Pittman, Jacobs, and Franson, PRL 88, 257902 (2002)

• Simplified 4-photon

Ralph, White, Munro, & Milburn, PRA 65, 012314 (2001)

• Efficient 4-photon

Knill, PRA 66, 052306 (2002)

Internal ancillas

NON- DETERMIN GATE

• 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

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

Cluster- and graph states

Raussendorf & Briegel PRL 86, 5188 (2001)

A cluster state is a collection of qubits that are entangled via nearest-neighbor CZ gates (rectangular lattice).

qubit

Measurement-based computation

• 2004 - Nielsen’s solution: combine KLM non-deterministic gate with

cluster-state model of quantum computation

Nielsen, PRL 93, 040503 (2004)

qubit qubit NS gate qubit NS gate Measurement on qubits qubit

θ=α1 ±α2

conventional circuit Uz(α1) Uz(α2) Uz(α3) Uz(α4) Uz(β1) Uz(β2) Uz(β3) Uz(β4) Uz(γ1) Uz(γ2) Uz(γ3) Uz(γ4)

single qubit gates two-qubit (entangling) gates

flow of quantum information

α1 ±α2 ±α3 ±α4 β1 ±β2 ±β3 ±β4 γ1 ±γ2 ±γ3 ±γ4 cluster/graph circuit

flow of measurement info

entanglement qubit measurement cosθ σx + sinθ σy

±α3

Photons are hard to hold, but with cluster states you can build as you go…

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

Optical quantum computing

OQC Anti-Moore’s Law

Full Fault Tolerance Loss Tolerance

• 3
• 2
• 1

40 50 60 70 80 90 100

resources (dB Bell pairs) threshold for loss

KLM (2001) 10-4 10-3 10-2 10-1 Cluster and Parity (2005) Cluster (2005) Parity (2008) Coherent (2008) Zeno (2007) 3D Cluster non-linear (2008) 1010 109 108 107 106 105 104

Resources (Bell pairs)

Good

Optical Quantum Computing: Improvements in Scaling IARPA- funded

Outline

1. Optical Quantum Computing 101 2. Where we are 3. Some theoretical magic 4. Technology Development -- where we need to get to

• Sources
• Detectors
• Integrated optics

Key Feats

• 1. Cluster-state logic and algorithms with feedforward:

gate time <150 ns (fastest 2-qubit gate; very high decoherence/gate time)

• 2. Several algorithms (Grover, Shor’s, Deutsch)
• 3. Fault-tolerant understanding, techniques for realizing small and

(eventually) large clusters

• 4. Single-photon sources and Detectors under development
• 5. Photonic technologies: micro-, integrated, and adaptive-optics
• 6. Nonlinear approaches: theory & nascent experiments

C T Non-classical interference

✓ No classical interferometers

RH = 1/3 RV = 1

Controlled-Z gate for clusters

Langford, Weinhold, Prevedel, Pryde, O’Brien, Gilchrist and White quant-ph/0506262 (2005) Okamoto, Hofmann, Takeuchi, and Sasaki quant-ph/0506263 (2005) Kiesel, Schmid, Weber, Ursin, and Weinfurter quant-ph/0506269 (2005)

1 2

Non-classical interference between⎟H〉1 and ⎟H〉2

⎟+〉1 ⎟+〉2 3 3 photon cluster ⎟+〉3

Non-classical interference between⎟V〉2 and ⎟H〉3

Repeat unit

N-qubit clusters

Demonstration of 4-photon polarization-entangled Cluster States

Idea: Generation via multi-photon interference and parity gates Requir irements: Multi-photon emission (SPDC), interferometric stability, parity operations Theory Experiment

~ 20,000 two-fold cps ~ 1 four-fold cps Experimental one-way quantum computing Walther, Resch, Rudolph, Schenck, Weinfurter, Vedral, Aspelmeyer, Zeilinger, Nature 434, 169 (2005)

Fast feed-forward Fast feed-forward… …

Pockels Cells: KD*P crystals ~ 6.3 kV

Over 99 % fidelity (500:1)

Feed-Forward Time < 150 ns !!

i.e., 1000x faster than ions 106 faster than NMR Fibers to detector 15ns Detector-Delay 35ns EOM-Delay 65ns Logics-Delay 7.5ns

• Misc. cables 20ns
• R. Prevedel et al., Nature 445, 65 (2007)

Results Results

One-Qubit it Operatio ion: Two-Qubit it Operatio ion:

ideal with FF no FF Prevedel et al. Nature 445, 65 (2007)

Grover’s A Alg lgorit ithm:

F > 70%

*

• rder-2
• rder-4
• rder-2
• rder-4

Shorʼs Circuits

Lanyon, Weinhold, Langford, Barbieri, James, Gilchrist, and White, PRL 99, 250505 (2007)

4-photon source

• rder-2
• rder-4

Shor's algorithm

Algorithm performance

algorithm works near perfectly …

Shor’s algorithm is non-deterministic: output is mixed

Algorithm performance cannot distinguish between: mixture from desired entanglement with function-register mixture from undesired entanglement with environment Circuit performance is crucial when used as sub-routine in a larger algorithm measure with quantum state tomography Prime factors of 15 = 3,5

Probability 48 ± 3% Probability 50 ± 2%

circuit does not work perfectly (F = 68±3%)

Lanyon, et al. White, PRL 99, 250505 (2007); Lu, Browne, Yang, and Pan, PRL 99, 250504 (2007)

Quantum controlled-NOT gate realization

Teleportation-based C-NOT gate Goebel et al., in preparation Proposal given by

• D. Gottesman & I.L. Chuang

Nature 402, 390 (1999)

• K. Chan, TODAY, HERE, 4:06pm!

Loss-tolerant quantum coding in the quantum circuit & one-way model

Grassl et al., PRA 56, 33 (1997); Ralph et al., PRL 95, 100501 (2005) Varnava et al., PRL 97, 120501 (2006)

An implementation of the smallest meaningful quantum codes using single photons from SPDC & a linear optics network

F ~ 75±10%

• No. of equations ∝ eparticles

Simulating physics with computers,

• Int. J. Theoretical Physics (1982)

The real problem is simulating quantum mechanics Hopeless task on a classical computer

Richard Feynman

Lets use quantum systems as computational building blocks!

Seth Lloyd

Given initial wavefunction,

• No. qubits required

∝ poly(No. particles) Universal quantum simulators, Science (1996)

Feynman was correct, for very large class of physical quantum systems

Time evolution operator,

• No. gates required ∝ poly(particles)

Approximation

Simulating Quantum Systems

Great ... but how can we learn about physical properties?

H Ψ = E Ψ

Eigenvalue problem

Simulated quantum computation of molecular energies, Science (2005)

U Ψ = eiHt/h Ψ = eiEt/h Ψ = eiφ Ψ

Phase estimation problem

Can calculate energy using the Iterative Phase Estimation Algorithm

Alán Aspuru-Guzik

...can also efficiently simulate chemical reactions

Polynomial-time quantum algorithm for the simulation of chemical dynamics, PNAS 105, 18681 (2008)

quantum.info

H11 H16

H22 H33 H34 H34 H44

H55 H16 H66 U11 U16

U22 U33 U34 U34 U44

U55 U16 U66

Iterative Phase-Estimation Algorithm for Energy

Estimates energy, to fixed precision, using poly scaling resources in molecular size

Eigenstate encoded into many qubits (i.e. using adiabatic method) Single control qubit Read out the binary expansion of phase

• ne bit at a time

powers of the time-evolution operator done with logic gates (using Lloydʼs technique)

• ne

The Simplest Molecule

Hydrogen molecule in a minimal basis:

6 basis states Atom A Atom B |1s〉 atomic orbitals |e〉= |1sA〉+ |1sB〉 |g〉= |1sA〉– |1sB〉 2 molecular orbitals g e Hamiltonian is a 6x6 operator with 2x2 blocks A 2x2 matrix eigenvalue problem

Uij = e

iHijt

Unitary is a 6x6 operator with 2x2 blocks

quantum.info

Every point is correct to the target precision of 20 bits (≈1 part in 105 Eh)

Quantum chemistry of H2

Lanyon, et al., arXiv:0905.0887 (2009)

• Q. How come your algorithm didnʼt give

you the wrong answer occasionally?

It did! but we used a simple classical error correction technique to overcome this.

quantum.info

Experimental data!

• Q. What happens if the input eigenstates arenʼt perfect?

Robust for encoded states with greater than 50% overlap with target eigenstate. Psuccess ~ 0.92/bit for 20 bits

Input State Fidelity

Outline

1. Optical Quantum Computing 101 2. Where we are 3. Some theoretical magic 4. Technology Development -- where we need to get to

• Sources
• Detectors
• Integrated Optics

Direct scaling of small LOQC circuits into larger circuits is limited by: * photon production probability * gate probability * circuit characteristics Via concatenated circuit designs and use of additional photonic degrees of freedom major improvements can be obtained. Ralph, Resch, & Gilchrist, Phys.Rev.A 75 022313 (2007)

example: Toffoli Gate

1/32 1/72

1/4096

7 3 15

chained gates new scheme

• no. photons

probability

• f success
• min. photons min. prob.

Improving LOQC Overheads Demonstration

• f Toffoli and

Contolled-U Gates!

UQ Theory

Resource efficient Linear Optical Quantum Computation, D.E. Browne and T. Rudolph, Phys. Rev. Lett. 95, 010501 (2005) Loss tolerance in one-way quantum computation via counterfactual error correction

• M. Varnava, D. E. Browne and T. Rudolph, Phys. Rev. Lett., 97, 120501, 2006

How good must single photon sources and detectors be for efficient LOQC?

• M. Varnava, D.E. Browne and T. Rudolph, (to be published in Phys. Rev. Lett.)
• Dealing with error mechanisms of specific concern to LOQC:

Photon loss, imperfect sources, inefficient detectors, bad memory

• Shown that tree clusters can be grown optically

despite inefficient sources/detectors

• Tradeoff found: If the product of source

efficiency and photodetector efficiency is greater than 2/3 then efficient LOQC is possible! (but with heavy resource requirements…) Detector Efficiency Source Efficiency

Fault-tolerant “limits”

Percolation -- Improving Large-scale Circuits

Fusion success probability =1/2, above percolation threshold. ⇒ get large piece of connected cluster state with high probability Red & Green – not connected Black - connected From the percolated cluster it is easy to compute measurement patterns to produce any desired cluster circuit:

• Every photon undergoes only one Type-I gate and one single-qubit measurement
• Removes requirement for photon rerouting (only requires feedforward to classical

measurement settings)

• Initial resources can be as small as 4-photon cluster states
• K. Kieling, T. Rudolph, and J. Eisert. Phys. Rev. Lett. 99, 130501 (2007)

V H H V V

“Weak” Nonlinearities

• 2004 Nemoto & Munro’s solution: deterministic CNOT gate by combining

measurement-induced nonlinearity with weak optical nonlinearity

Nemoto and Munro, PRL 93, 250502 (2004)

H

Barrett et al., quant-ph/0408117 (2004)

• 2004, also use for

deterministic Bell measurement

Outline

1. Optical Quantum Computing 101 2. Where we are 3. Some theoretical magic 4. Technology Development-- where we need to get to

• Sources
• Detectors
• Integrated Optics

Needed Technologies

1. Sources

a) Single-photon (MURI et al.) b) Entanglement

2. Detectors

a) SPCM/VLPC (MURI) b) Superconducting (NIST) c) Atomic vapor

3. Circuits

a) Storage and switching b) Microoptics c) Mode matching (adaptive opt?)

Photon, but just 1:

• Single emitter
• atom/ion (Kimble) [hard to collect]
• quantum dot (“designer atom”)

[BUT no two exactly alike…]

• Pair sources (SPDC, 4-wave mixing)
• detection of signal photon -->

“heralds” presence of idler photon in well-defined mode

Resources for Photonic Quantum Information Processing

p s i kp ks ki

C

Spontaneous Parametric Downconversion

1 2 0.99999999... P(n) n 0.00000001... Conditional 1-photon per mode Well-behaved spatial modes

Resources for Photonic Quantum Information Processing

NOT “on-demand” (multiplexed sources help)

High-quality OTS components ⇒ >99.9% fidelity ‘gates’

Resources for Photonic Quantum Information Processing

Detectors:

• What we want
• high efficiency (at λ), low noise
• fast (ideally at 100-fs scale)
• photon-number resolving
• What we have now*
• APDs, VLPCs, TES, SSPDs
• η ~ 85-95% (visible to 1550 nm)
• 1 MHz - 1 GHz
• can resolve up to ~10 photons

*But NOT all at once!

Resources for Photonic Quantum Information Processing

Superconducting bolometric detectors

• system efficiency (at 1550 nm) ~95%

(pushing toward 99%)

• near-perfect photon-number resolution
• slow-ish (0.1 - 1 µs)

A.E. Lita, A. J. Miller, and

• S. W. Nam, Opt. Exp. 16,

3032 (2008)

#2 V-polarized (from #2)

Maximally entangled state

(Polarization-) Entangled Source:

Tune pump polarization:  Nonmax. entangled, mixed states Stable, simple  Used to test QM in various undergrad labs New ultra-bright versions, narrow bandwidth, …

Not on-demand, unwanted entanglement in other DOFs

PRL 75, 4337 (1995)

Fiber-Based Sources (4-wave mixing)

@ NIST, Northwestern,…

LOQC Gates

• Pairs created in fiber

i.e., naturally single-mode

• Low-loss
• Exploits existing telecom

infrastructure

• 1550 nm or 1310 nm
• Require cryogenic cooling

Chen, et al. Phys. Rev. Lett. 100, 133603 (2008). Medic, et al. CLEO Conference 2009, paper ITuE7.

Degenerate Entanglement F = 96%

Moore’s law for entanglement

Polarization-entangled pairs @ 2,000,000 s-1, with F ~98%, T > 96% R e I m Φ(−) ∼ |HH〉 − |VV〉

F >99.5%

Next main limitation: detector saturation

|Sexpt| = 2.7260 ± 0.0008 (216σ in 0.8 s) SLHV ≤ 2 |SQM, max| = 2√ 2 = 2.828 |Sexpt| = 2.826 ± 0.005 165σ Optimized Bell test: Bell-Ineq. Tests

• Opt. Exp. 13, 8951 (2005)

Now: Various tests with 2-5 photons (GHZ), with different DOFs, “qudits”, etc. More to come…

~25mm

Integrated Quantum Photonics

Politi, Cryan, Yu, Rarity and O’Brien Science 320, 646 (2008) News & Views Review: Kwiat, Nature 453 294 (2008)

V = 0.948 ± 0.005

High-fidelity Quantum Integrated Optic Operation

Laing, Peruzzo, Politi, Rodas Verde, Halder, Ralph, Thompson, O’Brien arXiv…

F = 0.994 ± 0.002 Quantum interference: V = 0.995 ± 0.007 CNOT chip:

2 cm

Laing, Peruzzo, Politi, Rodas, Thompson, O’Brien

Politi, Matthews, O’Brien Science to appear (2009)

Compiled Shorʼs Algorithm on Chip

F = 0.99 ± 0.01

On-chip Phase Control: Reconfigurable Quantum Circuits

Output A Output B

A B

State preparation Favg = 0.99984 ± 0.00004

Interferometer Phase- control ≡ variable BS ⇒tunable two-photon (HOM) interference Also used in 4-photon entangled state metrology

Matthews, Politi, Stefanov, O’Brien Nature Photonics 3, 346 (2009)

LiNbO3 Integrated Quantum Photonics

• Linear (in applied V) electro-optic effect < 1ns → much faster gates
• Can support a single polarization
• Operation at visible and telecom wavelengths possible
• Integration of sources (PPLN)
• Enables complex circuits, including multi-stage and parallel paths,

composed of MZIs and directional couplers; control and synchronization among multi sections in the circuit, and external sources, detectors, etc.

LiNbO3 device fabrication at Bristol

• Established process for single-polarization single-mode waveguides
• 800-nm light coupled into devices
• Currently under test:

Laser Direct-Write Quantum Photonic Circuits with CUDOS – Macquarie University

Marshall, Politi, Matthews, Dekker, Ams, Withford, O’Brien, Opt. Exp. 17, 12546 (2009)

• Starting to explore

3D photonic circuits

• Couple to different inputs

(circuits, memories, detectors, etc.) using fibers, MEMs, adaptive optics…

V = 95.8 ± 0.5%

Single- & entangled- photon sources (synchronized)

Notional Photonic Quantum Processor

• Q. memory/ delay*

Photon detectors Reconfigurable ‘circuitry’

• phase shifts apply 1-qubit gates
• swap in source/memory/detector upgrades

*short delays (~1-100ns) on chip; sychronization memory off-chip Not shown: classical feedback from detectors to all other components