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Topological quantum computation JKP Pavia, January 2013 Why? Topology promises to solve the problem of errors that inhibit the experimental realisation of quantum computers… …and it is lots of fun :-) Geometry – Topology • Geometry – Local properties of object • Topology – Global properties of object geom. ⇔ ⇔ topo. Quantum Physics Quantum mechanics Particle state -> wave function Ψ(x, t) d Ψ(t) i = H Ψ(t) dt Ψ(t) = U(t) Ψ(0) Ψ(x) x P(x) = Ψ(x) Uncertainty: € • Intrinsic ΔxΔp ≥ 2 • Control errors H → H ' ⇒ U → U€ '€ ⇒ Ψ(t) → Ψ '(t) give measurement errors P(x) ≠ P'(x) 2 (Classical) computers • Encoding: binary numbers 0 and 1: 10101110 • Binary gates: 10101110 -> 01101000 • Error correction error 1011 ! !! →1001 – Error: – Avoid errors by redundant encoding (copy) 1011 !enciding !!→(111)(000)(111)(111) !error !! →(011)(010)(110)(011) !correction !!! →(111)(000)(111)(111) !decoding !!→1011 Quantum computers • Quantum binary states 0 , 1 (spin,...) Ψ = a 0 +b1 , a 0 ⊗ 1 ⊗ 0 ⊗ ...+ ... • Quantum binary€gates-> Unitary matrices € U 0 ⊗ 0 = a 0 ⊗ 0 + b1 ⊗ 1 • Use them for computation (can find faster paths to get to the answer) € |input> |answer> Quantum computers: Why? • Factoring quantum hackers exponentially better than classical hackers! • Searching objects: where is ❥? ¢®¶♬ΙΠÃ≥⅙⏎✜Ψ✪❥⅖ű Quantum error correction • Error correction 1011 !error !! → 1001 – Error – Avoid errors by redundant encoding Copying or “cloning” quantum states is not allowed! UΨ ⊗ 0 ≠ Ψ ⊗ Ψ Quantum error correction (Shor ‘96, Steane ’96) € Topological quantum computers... Quantum error correction • Quantum redundancy: 1 = 0 L 1 O1 €2 € € 6 O2 € 3 € €4 € € 4 =1L 5 Too costly… Topological quantum effects Aharonov-Bohm effect Magnetic flux Φ and charge Ψ(x) → e € ineΦ Ψ(x) Φ e e− The phase is a function of winding number €n € Topological effect: n is€the integer number of rotations € Particle statistics Exchange two identical particles: x1 x2 Statistical symmetry: Physics stays the same, but Ψ could change! € 2× € € Ψ(x1, x 2 ) = ???Ψ(x 2 , x1 ) € = Anyons and statistics 3D 2D Bosons Ψ → Ψ Fermions Ψ → ei 2π Ψ Ψ → ei 2ϕ Ψ Ψ →U Ψ Anyons Anyons ~ particles with mag. flux and el. charge Statistical phase ~ Aharonov-Bohm effect Anyons and physical systems 2D Ψ → ei 2ϕ Ψ Ψ →U Ψ Anyons Anyonic properties can be found in 2-dimensional topological physical systems: (c)! (a)! (b)! • Fractional quantum Hall effect • Topological insulators • Bose-Einstein condensates [Giandomenico Palumbo & JKP, “CSM theory from lattice fermions”, arXiv:1301.2625] Anyonic evolutions are topological Anyonic evolutions have topological properties like the Aharonov-Bohm effect. Φ (b)! e− € € Many different states of physical system give the Redundant encoding same anyonic states Topological anyonic states Quantum states of anyons similar to spin, but cannot be locally determined. σ σ σ σ € time These states change when we exchange anyons. 1L 0L Measure: σ × σ = 1+ ψ fuse #1 1 & % €⊗ = 0 +1( € $2 2 ' 1 vacuum 1 ψ ψ Topological quantum computers 67 - Quantumtinformation encoded in state of anyons. - Their statistics is used for quantum gates. € ! 4.2 Anyonic quantum computation ! € € σa σa a σ a σ σa ! ! aσ ! ! € € € 1 3 - avoid errors: -- Redundancy equivalence between states -- Hamiltonian e e e e e e 1 tFig. 4.10 A possible configurationMeasure protects information of topological quantum computation. Initially, pairs of information 2 4 5 6 created from the vacuum. Braiding operations between them unitarily evolve Finally, fusing the gives set of outcomes ! anyons ! a! ! ! ei , i = 1, ..., which ! together ! of the computation. by fusing. Conclusions The problem of error correction in QC can be resolved if we find a way to engineer topological systems that can support anyons. http://quantum.leeds.ac.uk/~jiannis