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Introduction to Quantum Computing Lecture 3: Qubit Technologies Rod Van Meter [email protected] June 27-29, 2005 WIDE University School of Internet with help from K. Itoh, E. Abe, and slides from T. Fujisawa (NTT) Course Outline ● ● ● ● ● Lecture 1: Introduction Lecture 2: Quantum Algorithms Lecture 3: Devices and Technologies Lecture 4: Quantum Computer Architecture Lecture 5: Quantum Networking & Wrapup Lecture Outline ● ● ● ● Brief review DiVincenzo's criteria & tech overview Technologies: All-Si NMR, quantum dots, superconducting, ion trap, etc. Comparison Superposition and ket Notation ● Qubit state is a vector – ● ● two complex numbers |0> means the vector for 0; |1> means the vector for 1; |00> means two bits, both 0; |010> is three bits, middle one is 1; etc. A qubit may be partially both! (just like the cat, but stay tuned for measurement...) complex numbers are wave fn amplitude; square is probability of 0 or 1 1-qubit state and Bloch sphere(Phase) 1-qubit basis states 1 0 0 0 1 1 z Bloch sphere 0 1 superposition 0 2 2 =1 1 , C 0 ei cos 0 ei sin 1 2 global phase has no2 y effect 2 0 i1 x relative phase 2 1 cos 0 2 ei sin 1 2 Reversible Gates NOT Control-NOT x x X or Control-Control-NOT (Toffoli gate) Control-SWAP (Fredkin gate) Bennett, IBM JR&D Nov. 1973 This set can be used for classical reversible computing, as well; brings thermodynamic benefits. Quantum Algorithms ● Deutsch-Jozsa(D-J) – ● Grover's search algorithm – ● Phys. Rev. Lett., 79, 325 (1997) Shor's algorithm for factoring large numbers – D. Deutsch Proc. R. Soc. London A, 439, 553 (1992) SIAM J. Comp., 26, 1484 (1997) R. Jozsa L. K. Grover P. W. Shor Grover's Search N 2n unordered items, we're looking for item “β” Classically, would have to check about N/2 items β Hard task!! N 2 n items To use Grover's algorithm, create superposition of all N inputs N repetitions of the Grover iterator G will produce “β” 1 N 1 N x 0 x Shor's Factoring Algorithm 66554087 ? 6703 9929 An “efficient” classical algorithm for this problem is not known. Using classical number field sieve, factoring an L-bit number is 3 2 k L log L Oe Using quantum Fourier transform (QFT), factoring an L-bit number is O L 3 Superpolynomial speedup! (Careful, technically not exponential speedup, and not yet proven that no better classical algorithm exists.) Summary: Characteristics of QC ● ● ● ● ● Superposition brings massive parallelism Phase and amplitude of wave function used Entanglement Unitary transforms are gates Measurement both necessary and problematic when unwanted So, can we surpass a classical computer with a quantum one? Depends on discovery of quantum algorithms and development of technologies Lecture Outline ● ● ● ● Brief review DiVincenzo's criteria & tech overview Technologies: All-Si NMR, quantum dots, superconducting, ion trap, etc. Comparison DiVincenzo’s Criteria 1. 2. 3. 4. 5. Well defined extensible qubit array Preparable in the “000…” state Long decoherence time Universal set of gate operations Single quantum measurements Qubit initialization Execution of an algorithm Read the result Must be done within decoherence time! Qubit Representations ● ● ● ● ● Electron: number, spin, energy level Nucleus: spin Photon: number, polarization, time, angular momentum, momentum (energy) Flux (current) Anything that can be quantized and follows Schrodinger's equation Problems ● Coherence time – ● Gate time – ● ● NMR-based systems slow (100s of Hz to low kHz) Gate quality – ● nanoseconds for quantum dot, superconducting systems generally, 60-70% accurate Interconnecting qubits Scaling number of qubits – largest to date 7 qubits, most 1 or 2 A Few Physical Experiments ● ● ● ● ● ● ● ● ● IBM, Stanford, Berkeley, MIT (solution NMR) NEC (Josephson junction charge) Delft (JJ flux) NTT (JJ, quantum dot) Tokyo U. (quantum dot, optical lattice, ...) Keio U. (silicon NMR, quantum dot) Caltech, Berkeley, Stanford (quantum dot) Australia (ion trap, linear optics) Many others (cavity QED, Kane NMR, ...) Physical Realization Cavity QED Ion trap Magnetic resonance Superconductor Examples of Qubits Neutral atoms Atomic Flux states in in optical lattices cavity QED superconductors Optically driven electronic states Entanglement (4) in quantum dots Trapped ions Solution NMR Optically driven Solid-state Shor (7) systems All-silicon quantum computer Crystal lattice spin states in Rabi (1) Charge states in superconductors quantum dots Electronically driven electronic states in quantum dots Electrons floating on liquid helium Single-spin Impurity spins in Electronically driven Others: Nonlinear MRFM semiconductors spin states in quantum dots optics, STM etc Lecture Outline ● ● ● ● Brief review DiVincenzo's criteria & tech overview Technologies: All-Si NMR, quantum dots, superconducting, ion trap, etc. Comparison Technologies Reviewed ● ● ● ● ● ● ● Liquid NMR Solid-state NMR Quantum dots Superconducting Josephson junctions Ion trap Optical lattice All-optical Liquid Solution NMR Billions of molecules are used, each one a separate quantum computer. Most advanced experimental demonstrations to date, but poor scalability as molecule design gets difficult and SNR falls. Qubits are stored in nuclear spin of flourine atoms and controlled by different frequencies of magnetic pulses. Used to factor 15 experimentally. Vandersypen, 2000 All-Silicon Quantum Computer 105 29Si atomic chains in 28Si matrix work like molecules in solution NMR QC. Many techniques used for solution NMR QC are available. 2 1 4 3 6 5 A large field gradient separates Larmor frequencies of the nuclei within each chain. Copies 8 No impurity dopants or 7 9 electrical contacts are needed. T.D.Ladd et al., Phys. Rev. Lett. 89, 017901 (2002) Overview L 400Î ¼ m W 4Î ¼ m H 10Î m ¼ s 2.1Î ¼ m l 300Î ¼ m t 0.25Î m ¼ w 4Î ¼ m dBz/dz = 1.4 T/mm B0 = 7 T Active region 100m x 0.2m Keio Choice of System and Material Two incompatible conditions to realize quantum computers: 1. Isolation of qubits from the environment 2. Control of qubits through interactions with the environment System: NMR 1. 2. Weak ensemble measurement Established rf pulse techniques for manipulation Material: silicon 1. 2. Longest possible decoherence time Established crystal growth, processing and isotope engineering technologies Fabrication: 29Si Atomic Chain Regular step arrays on slightly miscut 28Si(111)7x7 surface (~1º from normal) Steps are straight, with about 1 kink in 2000 sites. 28Si 29Si STM image J.-L.Lin et al., J. Appl. Phys 84, 255 (1998) Kane Solid-State NMR Qubits are stored in the spin of the nucleus of phosphorus atoms embedded in a zero-spin silicon substrate. Standard VLSI gates on top control electric field, allowing electrons to read nuclear state and transfer that state to another P atom. Kane, Nature, 393(133), 1998 Semiconductor nanostructure High electron mobility transistor (HEMT) Quantum dot structure # of electrons N < 30 http://www.fqd.fujitsu.com/ Quantum dot in the Coulomb blockade regime Single-electron transistor (SET) Coulomb blockade region, (the number of electrons is an integer, N) orbital degree of freedom & spin degree of freedom (spin qubit) double quantum dot bonding & anti-bonding orbitals (charge qubit) Quantum dot artificial atom 500 nm |(r)|2 1s 2p 3d S. Tarucha et al., Phys. Rev. Lett. 77, 3613 (1996). Environment surrounding a QD dissipation (T1), decoherence (T2) & manipulation coherency of the system Double quantum dot device A double quantum dot fabricated in AlGaAs/GaAs 2DEG (schematically shown by circles) cf. CPB charge qubit (Y. Nakamura ’99) Approximate number of electrons: 10 ~ 30 Charging energy of each dot: ~ 1 meV Lattice temp. ~ 20 mK (2 ueV) Typical energy spacing in each dot: ~100 ueV Electron temp. ~100 mK (9 ueV) Electrostatic coupling energy:~200 ueV Measurement system dilution refrigerator Tlat ~ 20 mK Telec ~ 100 mK B = 0.5 T Double quantum dot AlGaAs/GaAs 2DEG EB litho., fine gates, ECR etching 1 mm “thin coax” filtering Josephson Junction Charge (NEC) One-qubit device can control the number of Cooper pairs of electrons in the box, create superposition of states. Superconducting device, operates at low temperatures (30 mK). Nakamura et al., Nature, 398(786), 1999 Two-qubit device Pashkin et al., Nature, 421(823), 2003 JJ Phase (NIST, USA) Qubit representation is phase of current oscillation. Device is physically large enough to see! J. Martinis, NIST JJ Flux (Delft) The qubit representation is a quantum of current (flux) moving either clockwise or counter-clockwise around the loop. Ion Trap Ions (charged atoms) are suspended in space in an oscillating electric field. Each atom is controlled by a laser. NIST, Oxford, Australia, MIT, others Ion Trap Number of atoms that can be controlled is limited, and each requires its own laser. Probably <100 ions max. NIST, Oxford, Australia, MIT, others Optical Lattice (Atoms) Neutral atoms are held in place by standing waves from several lasers. Atoms can be brought together to execute gates by changing the waves slightly. Also used to make high-precision atomic clocks. Deutsch, UNM All-Optical (Photons) All-optical CNOT gate composed from beam splitters and wave plates. O'Brien et al., Nature 426(264), 2003 Lecture Outline ● ● ● ● Brief review DiVincenzo's criteria & tech overview Technologies: All-Si NMR, quantum dots, superconducting, ion trap, etc. Comparison Comparison ● ● ● NMR (Keio, Kane): excellent coherence times, slow gates – nuclear spin well isolated from environment – Kane complicated by matching VLSI pitch to necessary P atom spacing, and alignment Superconducting: fast gates, but fast decoherence Quantum dots: ditto – electrons in solid state easily influenced by environment Comparison (2) ● ● ● Ion trap: medium-fast gates, good coherence time (one of the best candidates if scalability can be addressed) Optical lattice (atoms): medium-fast gates, good coherence time; gates and addressability of individual atoms need work All-Optical (photons): well-understood technology for individual photons, but hard to get photons to interact, hard to store By the Numbers Technology All-Si NMR Kane NMR Ion Trap Optical Lattice Quantum Dot Josephson Junction All-Optical Decoherence Time 25 seconds seconds seconds seconds low microsecs microseconds N/A Gate Time high millisecs high millisecs low millisecs low millisecs nanosecs nanosecs N/A Apples-to-apples comparison is difficult, and coherence times are rising experimentally. Gates 10000? 1000? 1000? ? 100? 10-10000? N/A Quantum Error Correction and the Threshold Theorem Our entire discussion so far has been on “perfect” quantum gates, but of course they are not perfect. Various “threshold theorems” have suggested that we need 10^4 to 10^6 gates in less than the decoherence time in order to apply quantum error correction (QEC). QEC is a big enough topic to warrant several lectures on its own. Wrap-Up ● ● Qubits can be physically stored on electrons (spin, count), nuclear spin, photons (polarization, position, time), or phenomena such as current (flux); anything that is quantized and subject to Schrodinger's wave equation. More than fifty technologies have been proposed; all of these described are relatively advanced experimentally. Wrap-Up (2) ● ● ● ● ● Many technologies depend on VLSI Most are one or two qubits Have not yet started our own Moore's Law doubling schedule Several years yet to true, controllable, multiqubit demonstrations 10-20 years to a useful system? Upcoming Lectures ● Quantum computer architecture – ● Quantum networking – – ● how do you scale up? how do you build a computer out of this? what matters? Quantum key distribution Teleportation Wrap-Up and Review