3 Introduction notesTopics: Quantum information and computation theory Implementation using different technologies Topological Quantum computing Quantum magnetism … 25th Winter School in Theoretical Physics Institute for Advanced Studies Jerusalem 2007
4 Jerusalem notes Unified (east and west) 1967 MoslemChristianJewishArmenianUnified (east and west) 1967Capital city of Israel from 1980Around 800,00 inhabitants : approx 70% Jewish, 29% Muslim, 1% ChristianMostly religious: ‘secular’ population in constant declineOld city around 1km2 :one of the oldest cities in the world
5 Jerusalem notes Dome of the ‘Rock’ 7th century AC Western Wall Retaining wall for the 2nd Temple , 1st century BCHoly SepulchreInitially built by Constantine 4th Century BC
6 Computation –introduction - The general process of computation can be described as an operation performed on initial information and the reading out of the results:InputOutputOpComputation can be performed in classical domain using analogue or digital blocks
9 Classical computation – digital - Digital Sallen Key 5th order low pass filter IIR implementation
10 Quantum limitMoore’s Law (1965): Exponential size shrinking of electronic devices -> atomic limit will be approached around 2015
11 Quantum Computing –introduction - To simulate a probabilistic system (QM system) with a computer requires an exponential increase of resources (i.e. gates, time)The simulation of a probabilistic system (QM system) could be done more efficiently with a probabilistic (QM) machine: Quantum Computer
12 Quantum Computing –introduction - computation in the Quantum domainA composite system of N Quantum bits (Qubits) is the inputoperatorinputoutputAn unitary operator is appliedA non unitary operator is applied to perform measurement
13 Quantum Information –introduction - Quantum information: qubitsA Quantum bit (qubit) describes the states of each individual two-level systems.In the computational bases:Can be depicted as a point on the surface of a Bloch sphere:In theory a Qubit can store infinite amount of information,conserved during evolution. Measurement yields only one of thetwo values.
14 Quantum InformationIn a Quantum computer a unit vector in a Hilbert space H describes the initial state of thesystemAn input state vector of non interacting qubits can be written in the form:A state vector of interacting qubits consists of entangled statesE.g.:
15 Quantum InformationIn a more general case, a state of n qubits can be found in amixed state:The density operator used to describe the state of a subsystem, by tracing out the unwanted system
16 Quantum computing –introduction - Quantum processing: EvolutionTo perform a Quantum computation an unitary operation is performed on the qubits system:The generator Hamiltonian H has to generate this evolution according to Schrodingers’ equation:The Hamiltonian has to be found for a specific operation U:
17 a classical computer does) Quantum computingIf U unitary a solution for H always exists.If classical f not reversible (like universal Boolean classical gates) it can be made reversible by adding extra information. An unitary quantum equivalent can then be builtE.g. Toffoli gate can be used to make any irreversible classical function reversible. Reversible classical gate (in principle) heatless.Quantum computers can simulate any classical deterministic (Toffoli gate) andprobabilistic (Hadamard gate) functions (i.e. they can perform any computations thata classical computer does)
18 Quantum computing Quantum result: Measurement For a given orthonormal bases of HA it is possible to perform Von Neumann measurement:N distinguishable states input to an apparatus that perform a non unitary operation.
19 Quantum computing – general process - General Quantum circuit modelUSuperoperator: Quantum operation acts on an inputdensity operator plus ancillary registerIf unchanged dimension of Hinputopoutput
20 Quantum computing Quantum gates An unitary 1-qubit Quantum gate U can be written in terms of rotations around non parallel axes of the Bloch sphere using Pauli gates:A Quantum gate U rotates the Bloch vector on theBloch sphere
21 Quantum computing Universal Quantum gates An universal set of Quantum gate for n-qubits operator is obtained by any 2-qubit entangling gate with an universal set for 1-qubitAn universal set of Quantum gates allows description with arbitrary accuracy of n-qubit unitary operator( i.e. equivalent to classical NAND/NOR)An universal set of Quantum gate for 1-qubit operatorEfficiency of approximation ( number of gates) using G1(includes inverses)
22 U H X Quantum computing Quantum measurement V. Neumann measurement with respect to any orthonormal basisV. Neumann measurementUHXExample: Computational basis to Bell basis
23 Bell X Z Quantum computing Quantum communication - teleportation a Using two classical bits , it is possible to send the state of a qubit: quantum state transmitted using classical channels!aAliceBellbXZBobQuantum teleportation useful to implement 2-Qubit gatesFirst Quantum teleportation experimentally achieved using photons 1998
24 Quantum computing – algorithms - First Quantum algorithm- Deutsch Jozsa OracleUΣ∆Quantum superposition and interference
25 Quantum computing – algorithms - Define the reversible mapping:Deutsch Quantum algorithmProblem:To determine if f is constant or balanced.Classically: 2 queriesDeutsch algorithm: 1 queryInput an eigenstate to the target qubit of an operator and associate the eigenvalue with the control register
27 Quantum computing – algorithms - Deutsch Jozsa Quantum algorithmHHProblem:HHHHTo determine if f is constant or balanced.Classically: 2n-1+1queriesDeutsch algorithm: 1 queryExponential increase in efficiency
28 Quantum computing – algorithms - Quantum algorithm – Quantum Fourier TransformProblem: integer factorizationSplit odd- non prime power NOrders r of integers A co-prime with NSampling estimates to random integer multiple 1/nShor’s algorithm for N factoring could compute 100s digits in secondsOrder of random element in ZNQuantum complexityQFTQFT-1Classical complexityFactorization believed to be NP problem but not demonstrated
29 Quantum computing – algorithms - R(ivest)S(hamir)A(dleman) cryptosystemN,EAliceBobP,Q : N = PQME: GCD(E,(P-1)(Q-1)=1MEmod NE-1mod (P-1)(Q-1)(ME)E-1mod N =MDifficult to factor large numbers: classically ~ weeks for 100 digits N:doubling the digits implies a factorization ~ 106 years!
30 Quantum computing – algorithms - Quantum algorithms remarksIt is not known yet if many NP classical problems can become P using Quantum ComputingQFT (Shor’s algortithm)Deutsch Jozsa algorithmGrover’s search algorithm( N vs sqrt(N))Simon’s algorithmopinputoutputAcyclic quantum gate arrays can compute in polynomial time any function computable in polynomial time by CTM.
31 Quantum computing – further algorithms - Any P problem can be mapped onto a acyclic quantum gateCyclical quantum gate still not investigated:CompactnessPhase delay1-qubit cyclic gate represented by U(2) group2-qubit cyclic gate represented by U(2) group
32 Quantum computing – further algorithms - Perturbation of a 2-qubit cyclic network via C-NOT gateEvolution of a cyclic network after n cycles:In U eigenbases:Simplified QFTSimplified form for Quantum Gates (i.e. FIR vs IIR in DF)Phase delay may lead to instabilityQuantum oscillator!
33 Quantum Computing – implementation techniques Technical issues for building a Quantum Computer (DiVincenzo 2000)Quantum decoherenceInteraction with environment ( I.e. partial measurement operated by the environment)Errors due to decoherence can be recovered, if error rate is around 10-3/ 10-4 (Aharonov 1998)Quantum error correction algorithms are effective if operations are performed 10+3/ faster than decoherence timeReliable representation of Quantum Information (scalable number of Qubits )Setting of initial state of qubitsQuantum gates reliable (decoherence)ReadoutSeveral proposed solution for quantum computationAlternative proposed solution to decoherence and error problem is topological quantum computation
34 Implementation techniques – Quantum Dots InAs/GaAs SA QD grown with MBEQuantum dots using lithography :100nm spacingGood reproducibilityNanotechnology requiredContaminationSelf-assembled Quantum dots using strained epitaxial growth (i.e. Stranski-Krastanov process, growth of material on substrate not lattice matched)10’s nm scaleNo nanothechnology required (etching, implanting)No contaminationNon uniformity in size and positionQuantum well of Si-Ge for bi-dimensional confinement of electrons, top gates for lateral confinementQubits : spins of individual electrons in quantum dotsOrbital coherence time << Spin coherence time (>100ns in T=5K in GaAs)
35 Implementation techniques – Quantum Dots ACSEM of double Q-dot deviceQubits : spins of individual electrons in quantum dotsFor universal set G, coupling J of spins (qubits) neededQuantum operation performedby acting on gate voltages (2-qubit) to control JESR (electron spin resonance) ( 1-qubit phase rotation)Dv/v<10^-6 only at low frequency…RO using Spin to Q technique
36 Implementation techniques – Ion traps - Trap regionChip size planar Ions Trap: 6+6 traps of Mg on a flat alumina surfaceField applied through gold electrodesTens of trapped ions feasibleLimitations in minimum trap size (~ 5µm)Low temperature ( ~ -150C)CNOT operation demonstratedIons are confined in free UHV space using electromagnetic field (Paul trap)Qubits : ground and excited state level or hyperfine levelsVery long decoherence timeInitial state by optical pumpingMeasurement using laser pulses coupled to one of the qubit states: emitted photons read using CCD cameraNot easily scalable
37 Alternative implementation techniques Nuclear Magnetic Resonance: Qubits are the spin states of the nuclei of the molecules of the liquid used (demonstrated up to 8 qubits)Superconductors QC based on Josephson junctions: (~ 1K required), Charge qubit/Flux qubitAdiabatic Quantum Computer (D.Aharonov, W. Van Dam et al) based on Adiabatic Theorem (simulated)A Quantum System in its ground state remains in it along an adiabatic transformation in which the Hamiltonian is varied slowly enough from an initial to a final one.Idea: to vary the Hamiltonian slowly from initial to final state as if an U was performed on the initial state. The final ground state encodes the solution
38 Topological Quantum computation using non-abelian anyons (K Topological Quantum computation using non-abelian anyons (K. Shtengel, UC Riverside)Topological properties are deformation invariant (i.e. physically unaffected by perturbations) : this would render quantum computation almost error-free≡≠CBAIdea is to perform Quantum Computation using topological properties of quasi-particles
39 Topological Quantum Computing Topological differences between 2 and 3 dimensions Quantum systems (Leinaas & Myrheim, 1977)(i.e. if two particles are confined in 2D, their trajectories involve non-trivial winding if their positions are interchanged twice)Two identical particles exchange their position anticlockwise:Boson, fermionsIn 2D the phase can take any value:anyons
40 Topological Quantum computing – Classes of trajectories taking N anyons from A to B are isomorphic to BNBAMultiplication of elements of Bn is the successive execution of the trajectoriesNon abelian anyons
41 Topological Quantum Computing Anyons might arise in some low dimensions confined many particles systemsTo check if quasiparticles are anyons:Take quasiparticles around each other adiabatically (i.e. intial positions = final positions with interchange)The adiabatic interchange applies a unitary transformation on the ground state (phase):If anyonsQuasiparticles = Localized disturbances of the quanto-mechanical ground state of the two-dimensional systemExperimental evidence that quasiparticles occurring in fractional Quantum Hall effect are (non abelian ) anyons is still debated (J. Goldman et al. 2006)
42 Quantum Information – Topological techniques – Pairs of Anyons are brought together: degeneracy is lifted -> 2 states = qubitMapping of unitary operationsto braids not trivialtReadout of anyons not trivialU3U2U1QubitsoutQubitsinU
43 Quantum Information – Conclusions and relevance to Particle physics Quantum Computing promises breakthroughs in solving complex mathematical problems,some hard or insolvable classically (but still investigated)Computational power is an obvious benefit for all scientific fields, including Particle Physics(e.g. searching through immense databases)Simulation of Quantum Mechanical systems may be another area of research: essentially,to simulate a quantum mechanical system means really to simulate nature with its laws. This applies tothe world of nanotechnology as Particle Physics too.Theoretically one could think of modified laws of Quantum Mechanics(e.g. ‘ad’ hoc’ terms,non linearities etc)Use of Quantum technology for next generation of detectors
44 Quantum Information – Backup slides - Turing machine:Unbounded tape;Head that can read from the tape and can write on it, with infinite number of states;Instruction table.Given the initial head’s state and initial input the head reads, the table computes:The symbol the head writes on the tape;Where the head moves next on the tape.Church-Turing thesis: any effectively calculable function can be computed by a Turing machine
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