Progress Toward Scalable Linear Optical Quantum Computing Paul Kwiat
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? “Photons been very very good to me” • 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 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 Nonlinear Q. Zeno QND “KLM” (Franson) (Munro & Nemoto) graph states “modified Cluster/parity KLM” encoding
Optical Quantum Computing Linear Nonlinear Q. Zeno QND “KLM” (Franson) (Munro & Nemoto) graph states “modified Cluster/parity KLM” encoding P. Kok, W. J. Munro, K. Nemoto, T. C. Ralph, J. P. Dowling, and G. J. Milburn, “Linear optical 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 Optical realization with single photons: A database of four elements PGK et al., J. Mod. Opt. 47, 257 (2000) Grover’s Search algorithm Accuracy: ~97.5% - Gates: Linear optical elements Hosten et al., Nature 439, 949 (2006) - Nonscalable -- each new qubit doubles required number of elements - Seems like you need single-photon …historical interlude… nonlinearities for scalability
Linear optical quantum computing Knill, Laflamme and Milburn, Nature 409 , 46 (2001) Kok, Munro, Nemoto, Ralph, Dowling & Milburn FAST SINGLE FEEDFORWARD • • PHOTONS • • • • SINGLE-PHOTON DETECTION SINGLE PHOTON DETECTION LARGE overhead requirements…(>10 5 /gate)
Introduction Principles of LOQC • Non-deterministic gates • • QUBITS QUBITS • • • • NON- • Don’t always work, but heralded when they do DETERMIN GATE SINGLE PHOTON SINGLE DETECTION PHOTONS • • & • • • • FEEDFORWARD • Teleportation: moving information without measuring it Gottesman & Chuang, Nature 402 , 390 (1999)
Introduction Principles of LOQC • Non-deterministic gates • • QUBITS QUBITS • • • • NON- • Don’t always work, but heralded when they do DETERMIN SINGLE PHOTON GATE SINGLE DETECTION PHOTONS • • & • • • • FEEDFORWARD | C � B • Teleportation: moving information | �� without measuring it Z X X NON- | CNOT � DETERMIN X Z GATE Z | �� • Teleport non-deterministic B gates → deterministic | T � • Many non-deterministic gates proposed … 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) Browne and Rudolph et. al ., quant-ph/0405157 Gasparoni et al, PRL 93, 020504 (2004) Walther et al, Nature 434, 168 (2004)
Introduction Proposed LOQC gates External ancillas Internal ancillas NON- DETERMIN GATE • KLM 4-photon • Simplified 2-photon + teleportation Knill, Laflamme, & Milburn, Ralph, Langford, Bell, & White, Nature 409 , 46 (2001) PRA 65 , 062324 (2002) + error correction • Simplified 4-photon • Simplified 2-photon = scalable QC Ralph, White, Munro, & Milburn, Hofmann & Takeuchi, PRA 65 , 012314 (2001) PRA 66 , 024308 (2002) (but still very large • Linear-optical QND • Efficient 4-photon overhead needs) Kok, Lee & Dowling, PRA 66 , Knill, PRA 66 , 052306 (2002) 063814 (2002) • Entangled ancilla 4-photon Gasparoni et al., PRL 93, 020504 (2004) Pittman, Jacobs, and Franson, PRL 88 , 257902 (2002) 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).
Measurement-based computation • 2004 - Nielsen’s solution: combine KLM non-deterministic gate with Nielsen, PRL 93 , 040503 (2004) cluster-state model of quantum computation conventional circuit cluster/graph circuit flow of quantum information flow of measurement info α 1 ± α 2 ± α 3 ± α 4 U z ( α 1 ) U z ( α 2 ) U z ( α 3 ) U z ( α 4 ) U z ( β 1 ) U z ( β 2 ) U z ( β 3 ) U z ( β 4 ) β 1 ± β 2 ± β 3 ± β 4 U z ( γ 1 ) U z ( γ 2 ) U z ( γ 3 ) U z ( γ 4 ) γ 1 ± γ 2 ± γ 3 ± γ 4 two-qubit single (entangling) qubit entanglement qubit measurement gates gates cos θ σ x + sin θ σ y NS gate NS gate ± α 3 ± α 2 θ = α 1 Measurement on qubits qubit qubit qubit qubit qubit
Photons are hard to hold, but with cluster states you can build as you go…
Optical quantum computing OQC Anti-Moore’s Law 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: Improvements in Scaling KLM (2001) 10 10 100 resources Resources (dB Bell pairs) 90 10 9 (Bell pairs) Cluster (2005) 80 10 8 70 10 7 IARPA- Parity (2008) funded 60 10 6 Cluster and Parity (2005) 10 5 50 Coherent (2008) 3D Cluster non-linear (2008) 10 4 40 Zeno (2007) Good -3 -2 -1 10 -1 10 -4 10 -3 10 -2 Full Fault Tolerance threshold for loss Loss Tolerance
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
Controlled-Z gate for clusters C R H = 1 / 3 Non-classical interference R V = 1 T ✓ No classical interferometers 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)
N-qubit clusters ⎟ + 〉 1 ⎟ + 〉 2 ⎟ + 〉 3 3 photon cluster Repeat unit 1 Non-classical interference between ⎟ V 〉 2 and ⎟ H 〉 3 3 2 Non-classical interference between ⎟ H 〉 1 and ⎟ H 〉 2
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 ~ 20,000 two-fold cps ~ 1 four-fold cps Theory Experiment Experimental one-way quantum computing Walther, Resch, Rudolph, Schenck, Weinfurter, Vedral, Aspelmeyer, Zeilinger, Nature 434 , 169 (2005)
Fast feed-forward… … Fast feed-forward R. Prevedel et al., Nature 445, 65 (2007) Pockels Cells: KD*P crystals ~ 6.3 kV Over 99 % fidelity (500:1) Fibers to detector 15ns Feed-Forward i.e., 1000x faster than ions Detector-Delay 35ns Time < 150 ns !! 10 6 faster than NMR EOM-Delay 65ns Logics-Delay 7.5ns Misc. cables 20ns
Results Results One-Qubit it Operatio ion: Two-Qubit it Operatio ion: no FF with FF ideal Grover’s A Alg lgorit ithm: F > 70% Prevedel et al. Nature 445, 65 (2007)
Shor ʼ s Circuits * order-2 order-4 order-2 order-4 4-photon source Lanyon, Weinhold, Langford, Barbieri, James, Gilchrist, and White, PRL 99 , 250505 (2007)
Shor's algorithm Algorithm performance Shor’s algorithm is non-deterministic: output is mixed order-4 Prime factors of 15 = 3,5 order-2 algorithm works near perfectly … Probability 50 ± 2% Probability 48 ± 3% 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 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!
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