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Rutherford Appleton Laboratory Particle Physics Department G. Villani QC seminar RAL Notes on Quantum Computation E.G. Villani STFC Rutherford Appleton Laboratory

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Rutherford Appleton Laboratory Particle Physics Department G. Villani QC seminar RAL Outline Quantum Computation introduction Algorithms examples Quantum Computation Technology Conclusions

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Rutherford Appleton Laboratory Particle Physics Department G. Villani QC seminar RAL Introduction notes Topics: Quantum information and computation theory Implementation using different technologies Topological Quantum computing Quantum magnetism … 25 th Winter School in Theoretical Physics Institute for Advanced Studies Jerusalem 2007

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Rutherford Appleton Laboratory Particle Physics Department G. Villani QC seminar RAL Jerusalem notes Unified (east and west) 1967 Capital city of Israel from 1980 Around 800,00 inhabitants : approx 70% Jewish, 29% Muslim, 1% Christian Mostly religious: ‘secular’ population in constant decline Old city around 1km 2 :one of the oldest cities in the world Christian Armenian Jewish Moslem

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Rutherford Appleton Laboratory Particle Physics Department G. Villani QC seminar RAL Jerusalem notes Dome of the ‘Rock’ 7 th century AC Holy Sepulchre Initially built by Constantine 4 th Century BC Western Wall Retaining wall for the 2 nd Temple, 1 st century BC

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Rutherford Appleton Laboratory Particle Physics Department G. Villani QC seminar RAL The general process of computation can be described as an operation performed on initial information and the reading out of the results: Computation –introduction - InputOutput Op Computation can be performed in classical domain using analogue or digital blocks

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Rutherford Appleton Laboratory Particle Physics Department G. Villani QC seminar RAL Classical Computation – analogue - Analogue computation example: PID controller

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Rutherford Appleton Laboratory Particle Physics Department G. Villani QC seminar RAL Analogue Computation – analogue - Analogue Sallen Key 5 th order low pass filter

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Rutherford Appleton Laboratory Particle Physics Department G. Villani QC seminar RAL Classical computation – digital - Digital Sallen Key 5th order low pass filterIIR implementation

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Rutherford Appleton Laboratory Particle Physics Department G. Villani QC seminar RAL Quantum limit Moore’s Law (1965): Exponential size shrinking of electronic devices -> atomic limit will be approached around 2015

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Rutherford Appleton Laboratory Particle Physics Department G. Villani QC seminar RAL 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 Quantum Computing –introduction -

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Rutherford Appleton Laboratory Particle Physics Department G. Villani QC seminar RAL computation in the Quantum domain operator input output A composite system of N Quantum bits (Qubits) is the input An unitary operator is applied A non unitary operator is applied to perform measurement Quantum Computing –introduction -

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Rutherford Appleton Laboratory Particle Physics Department G. Villani QC seminar RAL 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 the two values. A Quantum bit (qubit) describes the states of each individual two-level systems. In the computational bases: Quantum Information –introduction - Quantum information: qubits

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Rutherford Appleton Laboratory Particle Physics Department G. Villani QC seminar RAL An input state vector of non interacting qubits can be written in the form: E.g.: In a Quantum computer a unit vector in a Hilbert space H describes the initial state of the system Quantum Information A state vector of interacting qubits consists of entangled states

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Rutherford Appleton Laboratory Particle Physics Department G. Villani QC seminar RAL In a more general case, a state of n qubits can be found in a mixed state: The density operator used to describe the state of a subsystem, by tracing out the unwanted system Quantum Information

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Rutherford Appleton Laboratory Particle Physics Department G. Villani QC seminar RAL Quantum computing –introduction - To 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: Quantum processing: Evolution

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Rutherford Appleton Laboratory Particle Physics Department G. Villani QC seminar RAL Quantum computing If U unitary a solution for H always exists. Quantum computers can simulate any classical deterministic (Toffoli gate) and probabilistic (Hadamard gate) functions (i.e. they can perform any computations that a classical computer does) 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 built E.g. Toffoli gate can be used to make any irreversible classical function reversible. Reversible classical gate (in principle) heatless.

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Rutherford Appleton Laboratory Particle Physics Department G. Villani QC seminar RAL For a given orthonormal bases of H A it is possible to perform Von Neumann measurement: N distinguishable states input to an apparatus that perform a non unitary operation. Quantum result: Measurement Quantum computing

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Rutherford Appleton Laboratory Particle Physics Department G. Villani QC seminar RAL Quantum computing – general process - General Quantum circuit model op input output U Superoperator: Quantum operation acts on an input density operator plus ancillary register If unchanged dimension of H

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Rutherford Appleton Laboratory Particle Physics Department G. Villani QC seminar RAL Quantum gates A Quantum gate U rotates the Bloch vector on the Bloch sphere 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: Quantum computing

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Rutherford Appleton Laboratory Particle Physics Department G. Villani QC seminar RAL Universal Quantum gates An 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 operator An universal set of Quantum gate for n-qubits operator is obtained by any 2-qubit entangling gate with an universal set for 1-qubit Efficiency of approximation ( number of gates) using G1(includes inverses) Quantum computing

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Rutherford Appleton Laboratory Particle Physics Department G. Villani QC seminar RAL Quantum measurement V. Neumann measurementV. Neumann measurement with respect to any orthonormal basis U H X Example: Computational basis to Bell basis Quantum computing

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Rutherford Appleton Laboratory Particle Physics Department G. Villani QC seminar RAL Quantum computing Quantum communication - teleportation Bell Z X Using two classical bits, it is possible to send the state of a qubit: quantum state transmitted using classical channels! Alice Bob a b Quantum teleportation useful to implement 2-Qubit gatesFirst Quantum teleportation experimentally achieved using photons 1998

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Rutherford Appleton Laboratory Particle Physics Department G. Villani QC seminar RAL Quantum computing – algorithms - First Quantum algorithm- Deutsch Jozsa Oracle U Σ∆ Quantum superposition and interference

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Rutherford Appleton Laboratory Particle Physics Department G. Villani QC seminar RAL Deutsch Quantum algorithm Problem: To determine if f is constant or balanced. Classically: 2 queries Deutsch algorithm: 1 query Define the reversible mapping: Input an eigenstate to the target qubit of an operator and associate the eigenvalue with the control register Quantum computing – algorithms -

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Rutherford Appleton Laboratory Particle Physics Department G. Villani QC seminar RAL HH Simultaneous computation result Deutsch Quantum algorithm Quantum computing – algorithms -

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Rutherford Appleton Laboratory Particle Physics Department G. Villani QC seminar RAL To determine if f is constant or balanced. Classically: 2 n-1 +1queries Deutsch algorithm: 1 query Exponential increase in efficiency HH HH HH Problem: Deutsch Jozsa Quantum algorithm Quantum computing – algorithms -

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Rutherford Appleton Laboratory Particle Physics Department G. Villani QC seminar RAL Quantum algorithm – Quantum Fourier Transform Problem: integer factorization Split odd- non prime power N Orders r of integers A co-prime with N Sampling estimates to random integer multiple 1/n Order of random element in ZN Quantum complexity Classical complexity Shor’s algorithm for N factoring could compute 100s digits in seconds Factorization believed to be NP problem but not demonstrated QFTQFT -1 Quantum computing – algorithms -

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Rutherford Appleton Laboratory Particle Physics Department G. Villani QC seminar RAL R(ivest)S(hamir)A(dleman) cryptosystem Alice Bob N,E P,Q : N = PQ E: GCD(E,(P-1)(Q-1)=1 M M E mod N (M E ) E-1 mod N =M E -1 mod (P-1)(Q-1) Difficult to factor large numbers: classically ~ weeks for 100 digits N: doubling the digits implies a factorization ~ 10 6 years! Quantum computing – algorithms -

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Rutherford Appleton Laboratory Particle Physics Department G. Villani QC seminar RAL Quantum algorithms remarks Acyclic quantum gate arrays can compute in polynomial time any function computable in polynomial time by CTM. op input output It is not known yet if many NP classical problems can become P using Quantum Computing QFT (Shor’s algortithm) Deutsch Jozsa algorithm Grover’s search algorithm( N vs sqrt(N)) Simon’s algorithm Quantum computing – algorithms -

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Rutherford Appleton Laboratory Particle Physics Department G. Villani QC seminar RAL qubit cyclic gate represented by U(2) group Any P problem can be mapped onto a acyclic quantum gate Cyclical quantum gate still not investigated: Compactness Phase delay 2-qubit cyclic gate represented by U(2) group Quantum computing – further algorithms -

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Rutherford Appleton Laboratory Particle Physics Department G. Villani QC seminar RAL Perturbation of a 2-qubit cyclic network via C-NOT gate Simplified form for Quantum Gates (i.e. FIR vs IIR in DF) Phase delay may lead to instability Quantum oscillator! Evolution of a cyclic network after n cycles: In U eigenbases: Simplified QFT Quantum computing – further algorithms -

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Rutherford Appleton Laboratory Particle Physics Department G. Villani QC seminar RAL Quantum Computing – implementation techniques Technical issues for building a Quantum Computer (DiVincenzo 2000) Reliable representation of Quantum Information (scalable number of Qubits ) Setting of initial state of qubits Quantum gates reliable (decoherence) Readout Quantum decoherence Interaction with environment ( I.e. partial measurement operated by the environment) Errors due to decoherence can be recovered, if error rate is around / (Aharonov 1998) Quantum error correction algorithms are effective if operations are performed / faster than decoherence time Alternative proposed solution to decoherence and error problem is topological quantum computation Several proposed solution for quantum computation

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Rutherford Appleton Laboratory Particle Physics Department G. Villani QC seminar RAL Implementation techniques – Quantum Dots Quantum well of Si-Ge for bi-dimensional confinement of electrons, top gates for lateral confinement Qubits : spins of individual electrons in quantum dots Orbital coherence time 100ns in T=5K in GaAs) Self-assembled Quantum dots using strained epitaxial growth (i.e. Stranski-Krastanov process, growth of material on substrate not lattice matched) 10’s nm scale No nanothechnology required (etching, implanting) No contamination Non uniformity in size and position Quantum dots using lithography : 100nm spacing Good reproducibility Nanotechnology required Contamination InAs/GaAs SA QD grown with MBE

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Rutherford Appleton Laboratory Particle Physics Department G. Villani QC seminar RAL Qubits : spins of individual electrons in quantum dots For universal set G, coupling J of spins (qubits) needed Quantum operation performed by acting on gate voltages (2-qubit) to control J ESR (electron spin resonance) ( 1-qubit phase rotation) Dv/v<10^-6 only at low frequency… RO using Spin to Q technique Implementation techniques – Quantum Dots e-e- e-e- AC SEM of double Q-dot device

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Rutherford Appleton Laboratory Particle Physics Department G. Villani QC seminar RAL Implementation techniques – Ion traps - Ions are confined in free UHV space using electromagnetic field (Paul trap) Qubits : ground and excited state level or hyperfine levels Very long decoherence time Initial state by optical pumping Measurement using laser pulses coupled to one of the qubit states: emitted photons read using CCD camera Not easily scalable Chip size planar Ions Trap: 6+6 traps of Mg on a flat alumina surface Field applied through gold electrodes Tens of trapped ions feasible Limitations in minimum trap size (~ 5µm) Low temperature ( ~ -150C) CNOT operation demonstrated Trap region

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Rutherford Appleton Laboratory Particle Physics Department G. Villani QC seminar RAL 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 qubit Adiabatic 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

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Rutherford Appleton Laboratory Particle Physics Department G. Villani QC seminar RAL Topological properties are deformation invariant (i.e. physically unaffected by perturbations) : this would render quantum computation almost error-free Topological Quantum computation using non-abelian anyons (K. Shtengel, UC Riverside) ≡ ≠ CBA Idea is to perform Quantum Computation using topological properties of quasi-particles

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Rutherford Appleton Laboratory Particle Physics Department G. Villani QC seminar RAL 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, fermions anyonsIn 2D the phase can take any value:

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Rutherford Appleton Laboratory Particle Physics Department G. Villani QC seminar RAL Classes of trajectories taking N anyons from A to B are isomorphic to B N Non abelian anyons A B Multiplication of elements of Bn is the successive execution of the trajectories Topological Quantum computing –

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Rutherford Appleton Laboratory Particle Physics Department G. Villani QC seminar RAL Topological Quantum Computing Anyons might arise in some low dimensions confined many particles systems Quasiparticles = Localized disturbances of the quanto- mechanical ground state of the two-dimensional system To check if quasiparticles are anyons: 1.Take quasiparticles around each other adiabatically (i.e. intial positions = final positions with interchange) 2.The adiabatic interchange applies a unitary transformation on the ground state (phase): 3.If anyons Experimental evidence that quasiparticles occurring in fractional Quantum Hall effect are (non abelian ) anyons is still debated (J. Goldman et al. 2006)

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Rutherford Appleton Laboratory Particle Physics Department G. Villani QC seminar RAL Quantum Information – Topological techniques – Pairs of Anyons are brought together: degeneracy is lifted -> 2 states = qubit Qubits in U U3U3 U1U1 U2U2 Qubits out t Mapping of unitary operations to braids not trivial Readout of anyons not trivial

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Rutherford Appleton Laboratory Particle Physics Department G. Villani QC seminar RAL 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 to the 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

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Rutherford Appleton Laboratory Particle Physics Department G. Villani QC seminar RAL 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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