Introduction to Topological Error Correction and Computation. James R. Wootton Universität Basel

Transcription

1 Introduction to Topological Error Correction and Computation James R. Wootton Universität Basel

2 Overview Part 1: Topological Quantum Computation Abelian and non-abelian anyons Quantum gates with Abelian anyons Part 2: Anyons in the real world Quantum error correcting codes The Planar code Part 3: Errors and Thresholds Error models Active error correction Self-correction

3 Part 1: The abstract theory of TQC

4 Bosons and Fermions Bosons: R R 2 Fermions: R R 2 A full braiding of bosons or fermions has a trivial effect This is a general property of point particles in a universe with >2 spacial dimensions

5 Anyons In a 2D universe non-trivial braiding is allowed Abelian anyons: R 2 e i e i Non-Abelian anyons: R 2 U U

6 Why should we care? If anyons only exist in 2D universes, and ours is 3D, why should we care? We are not just limited to real particles (electrons, protons, billiard balls) Excitations in many-body systems can also behave like particles These are quasiparticles In a 2D many-body system, or 2D surface of a 3D system, these quasiparticles can behave like anyons! Spin lattice models Fractional quantum hall effect We can build 2D 'universes' in which anyons exist, and get them to work for us!

7 Why should we care? Anyons are well suited to fault-tolerant computation Gates can be implemented by braiding anyons The effects only depend on the topology of the path: stable against local noise U

8 QC with Abelian Anyons It's not only non-abelian anyons that are useful for QC! Imagine a universe with 2 spacial dimensions and 2 particle types: e and m These are their own antiparticles They have the following braiding behaviour R 2 These are anyons There are many other possible 'universes' of anyons we can construct, but this is the one present in the surface codes

9 QC with Abelian Anyons Anyons typically localized in the universe we build However, we can also build large delocalized 'holes' for them to live in 1 'Rough' holes for e anyons and 'smooth' for m anyons + - Since these cannot be accessed by local noise, they are suitable for storing qubits

10 QC with Abelian Anyons Holes can be moved by deforming the physical substrate What is the effect of braiding holes? R 2 Nothing with nothing R Nothing with e anyon R m anyon with nothing R m anyon with e anyon R This is a CNOT: braiding implements useful gates

11 QC with Abelian Anyons Abelian anyons are not universal by braiding alone Universality is achieved by introducing preparing 'magic states' This is noisy, but magic states can be distilled by the fault-tolerant gates R. Raussendorf, J. Harrington Phys. Rev. Lett. 98, (2007) R. Raussendorf, J Harrington, K. Goyal New Journal of Physics 9, 199 (2007) They can then be used to implement fault-tolerant QC More kinds of Abelian anyons exist, and have been considered for QC H. Bombin, M. A. Martin-Delgado Phys. Rev. Lett. 97, A. Kitaev, Annals Phys. 303 (2003) 2-30

12 Part 2: Anyons in the real world A.G. Fowler et al., Phys. Rev. A 80, (2009)

13 Error Correcting codes Anyons can be realized on quantum error correcting codes Many-body system with multi-spin observables (all commute) These detect the effects of Pauli operators Degenerate subspaces that are indistinguishable to these observables For surface codes we have B p operators that detect endpoints of strings A v x z operators that detect endpoints of strings The names for the codes derived from their boundary conditions Toric code has has periodic boundary conditions Planar code has open boundary conditions Surface codes have obc's as well as 'holes'

14 Planar Code Defined on a spin lattice Observables defined for each plaquette and vertex A v iv i x B p i p i z The A and B operators have eigenvalues +1 and -1 and mutually commute Their common eigenstates can be interpreted in terms of quasiparticles If A v B p we say that there are no anyons on p or v If If B p A v we say that there is an m anyon on p we say that there is an e anyon on v

15 Anyons e anyons can exist on top and bottom edges, m anyons on left and right edges

16 Anyons These anyons have the braiding behaviour discussed earlier R 2 z x z z z z B p i x j z i x j z B p i x j z

17 Stabilizer Measurements To determine correct errors we need to measure stabilizers How to measure four body operators? Additional ancilla spin in center of each plaquette and vertex A v iv i x B p i p i z Act with nearest neighbour CNOT's to map stabilizer value onto ancilla Measure the ancilla A.G. Fowler et al., Phys. Rev. A 80, (2009) Many-body stabilizer measurements require only spin-spin gates and single spin measurements

18 Holes Planar codes can be defined on any lattice, not just regular square For example, we could have a lattice with big plaquettes: these can serve as smooth holes Small plaquettes can be combined into large ones by single spin measurements Reverse can be done by measuring B p and clearing stray anyons Holes can be moved by expansion and contraction

19 Alternative method: Twists B. Brown et al., Phys. Rev. Lett. 111, Altering the stabilizers can lead to deformations of the lattice Pairs of these deformations can store quantum information H. Bombin, Phys. Rev. Lett. 105, Braiding and fusing them can be used to perform gates

20 Alternative method: Lattice Surgery C. Horsman, et al., New J. Phys (2012) By altering stabilizers we can merge two codes into one We can also split one into two These operations can be used to perform useful entangling gates

21 Alternative Codes Another code based on a 2-level spin lattice is the Color code More anyons: effectively two copies of the surface code anyons More transversal gates possible Similar methods (holes, twists, surgery) can be applied H. Bombin, M. A. Martin-Delgado Phys. Rev. Lett. 97, Generalizations of surface codes to qudits (higher dimensional spins) also exist These are more robust against noise A. Kitaev, Annals Phys. 303 (2003) 2-30 The braiding and fusion behaviour is more complex There are also the more complex non-abelian models

22 Requirements for surface code QC Whichever method we use (holes, twists or surgery) quantum computation with surface codes requires: 2D lattice of 2-level spins (code and ancilla qubits) Nearest neighbour interactions (i.e. controlled-not gate) Single spin measurements L. Trifunovic, et al., Phys. Rev. X 2, (2012) With these we can measure and deform the stabilizers as required The stabilizer deformations can be used for topologically protected quantum gates (by braiding holes or twists, or lattice surgery) With this we can do fault-tolerant quantum computation

23 Experimental Progress Anyonic braiding demonstrated on small surface codes 4 photons 6 cold atoms 6 photons J. K. Pachos, H. Weinfurter, et al., New J. Phys. 11, (2009) Y.-J. Han, R. Raussendorf, and L.-M. Duan, Phys. Rev. Lett. 98, (2007) Chao-Yang Lu, Jian-Wei Pan, et al., Phys. Rev. Lett. 102, (2009) Correction of an error on 8 photon surface code X-C Yao, J-W Pan, et al., Nature 482, (2012) Error detection and Clifford gates on 7 trapped ion Color code D. Nigg, R. Blatt, et al., Science, 345 (6194): (2014) IQOQI/Harald Ritsch

24 Part 3: Errors and Thresholds Fowler et al., Phys. Rev. A 80, (2009)

25 Active Error Correction Logical qubits are stored in the subspace of states with no anyons on the bulk 0 The effect of errors is to then create and move around anyons The job of error correction is to remove these, and their effects For this we must first measure the anyon positions Given this info, we work out a clever way to remove them: decoding algorithm 1 For sufficiently weak noise, probability of getting a logical error decays exponentially with system size Constantly repeating this error correction leads to exponentially long coherence times

26 Errors and correction There is a threshold noise level under which good error correction is possible Value of this depends on code, noise model and decoding algorithm Typically we consider i.i.d. noise on each spin with probability p: For perfect measurement of the stabilizers 5% p c 20%, p c 10% For measurement noise also with probability p p c 3%, p c 1 % Correlated errors have only just started to be studied A. Hutter, D. Loss, Phys. Rev. A 89, (2014), Poster Here A. Fowler, J. Martinis, Phys. Rev. A 89, (2014) As has error correction of non-abelian anyons JRW, et al., Phys. Rev. X 4, (2014), C. Brell, et al., arxiv: (2013)

27 Decoding Algorithms Many types of decoding algorithm exist, optimized for different codes and noise model MWPM decoders Most suitable for surface codes Also applicable to color codes Parallelizable to O(1) time E. Dennis, A. Landahl, et al., J. Math. Phys. 43, (2002). A. Hutter, JRW, D. Loss, Phys. Rev. A 89, (2014) A. Fowler, et al., Phys. Rev. A 86, (2012) SDRG decoders Applicable to all (Abelian) codes with p.b.c.'s Can give arbitrarily good approximation of optimal decoding Computational cost for better approximation can be high Can choke on highly correlated errors G.Duclos-Cianci, D. Poulin, Phys. Rev. Lett (2010) HDRG decoders Proven threshold for all Abelian codes Applicable to all codes Fast but low thresholds S. Bravyi, J. Haah, arxiv: (2011) JRW, arxiv: (2013) H. Anwar, B. Brown, et al., New J. Phys (2014) JRW, D. Loss, Phys. Rev. Lett. 109, (2012) MCMC decoders Applicable to all codes A. Hutter, JRW, D. Loss, Phys. Rev. A 89, (2014) Can give arbitrarily good approximation of optimal decoding Computational cost for better approximation can be high (not as bad as SDRG) Optimal decoders Bravyi, et al., arxiv: (2014) Known for surface codes with i.i.d. noise and no measurement errors Version for measurement errors thought to exist

28 Hamiltonian Protection Hamiltonian can be defined using stabilizer operators Ground state space is the anyonic vacuum, where QI is stored Creation of anyons has an energy penalty H J v A v J p B p For T=0, the effects of all (weak) local perturbations are suppressed when initially in GS Disordered couplings extend this to noisy GS preparation For T>0, diffusion of anyons leads to lifetime e Bravyi, et al., J. Math. Phys (2010) JRW & Pachos, Phys. Rev. Lett. 107, (2011) Stark, et al., Phys. Rev. Lett. 107, (2011) : does not increase with system size

29 Self Correction Can a Hamiltonian be engineered such that active error correction is not needed? For the 4D version of the surface code: yes! But this is not realistic Multiple no-go theorems for 2D In 3D there, some proof of principle models exist: Cubic code: Lifetime scales as 'partial' self correction e 2 for large L S. Bravyi, J. Haah, Phys. Rev. Lett. 111, (2013) Toric code coupled to bosonic bath: F. Pedrocchi, A. Hutter, JRW, D. Loss., Phys. Rev. A 88, (2013) Induces long-range interactions between anyons Lifetime scales as e L 2

30 Thanks for your attention Review articles for surface codes A.G. Fowler et al., Phys. Rev. A 80, (2009) D.P. DiVincenzo, Phys. Scr (2009) J. R. Wootton, J. Mod. Opt. 59, 20 (2012) A. Fowler, et al., Phys. Rev. A 86, (2012) H. Bombin, arxiv: (2013) Review articles for TQC J. Preskill, Lecture notes on TQC, G. Brennen and J.K. Pachos, Why should anyone care about computing with anyons Proc. R. Soc. A /rspa (2007) Jiannis K. Pachos, Steven H. Simon, Focus on topological quantum computation New J. Phys (2014)