Storage of Quantum Information in Topological Systems with Majorana Fermions

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1 Storage of Quantum Information in Topological Systems with Majorana Fermions Leonardo Mazza Scuola Normale Superiore, Pisa Mainz September 26th, 2013 Leonardo Mazza (SNS) Storage of Information & Majorana Fermions September 26th, / 21

2 Acknowledgements Max-Planck-Institut für Quantenoptik, München Ignacio Cirac Universität Mainz, Mainz Matteo Rizzi Harvard University, Cambridge MA Misha Lukin Regione Toscana Scuola Normale Superiore, Pisa Europe Leonardo Mazza (SNS) Storage of Information & Majorana Fermions September 26 th, / 21

3 Outline 1 Introduction Storing Quantum Information Zero-Energy Majorana Fermions 2 Theoretical Results The Storage of Information in Presence of Pertubations Optimal Recovery Operations 3 Numerical Results Hamiltonian Quench Perturbations 4 Conclusions Leonardo Mazza (SNS) Storage of Information & Majorana Fermions September 26 th, / 21

4 Outline 1 Introduction Storing Quantum Information Zero-Energy Majorana Fermions 2 Theoretical Results The Storage of Information in Presence of Pertubations Optimal Recovery Operations 3 Numerical Results Hamiltonian Quench Perturbations 4 Conclusions Leonardo Mazza (SNS) Storage of Information & Majorana Fermions September 26 th, / 21

5 The Storage of Quantum Information Idea 1: atomic quantum memory Two Zeeman levels as qubit: 0 = F, m F, 1 = F, m F E i Ψ = α 0 + β 1 Ψ(t) = α 0 + e t β 1 usually E B, external magnetic field Leonardo Mazza (SNS) Storage of Information & Majorana Fermions September 26 th, / 21

The Storage of Quantum Information Idea 1: atomic quantum memory Two Zeeman levels as qubit: 0 = F, m F, 1 = F, m F E i Ψ = α 0 + β 1 Ψ(t) = α 0 + e t β 1 usually E B, external magnetic field Idea 2:

6 The Storage of Quantum Information Idea 1: atomic quantum memory Two Zeeman levels as qubit: 0 = F, m F, 1 = F, m F E i Ψ = α 0 + β 1 Ψ(t) = α 0 + e t β 1 usually E B, external magnetic field Idea 2: topological quantum memory N-particle Hamiltonian Ĥ with: energy gap & degenerate ground space robust degeneracy Ĥ + ˆV Ground space Qubit Robust quasi-degeneracy protects against dephasing Kitaev, arxiv:quant-ph/ v1 (1997) Dennis et al, J. Math. Phys. 43, 4452 (2002) Bravyi et al, J Math Phys (2010) Leonardo Mazza (SNS) Storage of Information & Majorana Fermions September 26 th, / 21

7 The Problem of the Reliability Atomic quantum memory: easy to manipulate, but unreliable Topological quantum memory: difficult to manipulate, but reliable (?) Leonardo Mazza (SNS) Storage of Information & Majorana Fermions September 26 th, / 21

8 The Problem of the Reliability Atomic quantum memory: easy to manipulate, but unreliable Topological quantum memory: difficult to manipulate, but reliable (?) Reliability: the benchmark quantities of the topological quantum memory should improve with the size of the system 1 the intricacies of a N-particles state are paid back by an intrinsic stability of the system 2 N particles collaborate to protect the information 3 no experiments: focus on scalings Leonardo Mazza (SNS) Storage of Information & Majorana Fermions September 26 th, / 21

9 The Problem of the Reliability Atomic quantum memory: easy to manipulate, but unreliable Topological quantum memory: difficult to manipulate, but reliable (?) Reliability: the benchmark quantities of the topological quantum memory should improve with the size of the system 1 the intricacies of a N-particles state are paid back by an intrinsic stability of the system 2 N particles collaborate to protect the information 3 no experiments: focus on scalings This talk: Is a topological quantum memory reliable? Leonardo Mazza (SNS) Storage of Information & Majorana Fermions September 26 th, / 21

The Problem of the Reliability Atomic quantum memory: easy to manipulate, but unreliable Topological quantum memory: difficult to manipulate, but reliable (?

10 The Problem of the Reliability Atomic quantum memory: easy to manipulate, but unreliable Topological quantum memory: difficult to manipulate, but reliable (?) Reliability: the benchmark quantities of the topological quantum memory should improve with the size of the system 1 the intricacies of a N-particles state are paid back by an intrinsic stability of the system 2 N particles collaborate to protect the information 3 no experiments: focus on scalings This talk: Is a topological quantum memory based on Majorana fermions reliable? See also previous work by: Chesi, Loss, Bravyi and Terhal NJP (2010); Cheng, Galitski and Das Sarma PRB (2011); Goldstein and Chamon PRB (2011); Budich, Walter and Trauzettel PRB (R) (2012); König and Bravyi Comm Math Phys (2012); Cheng, Lutchyn and Das Sarma PRB (2012); Rainis and Loss PRB (2012); Schmidt, Rainis and Loss PRB (2012). Leonardo Mazza (SNS) Storage of Information & Majorana Fermions September 26 th, / 21

11 Topological Superconductors 1D Kitaev chain H = µ N N 1 N 1 n j J a j+1 a j + a j+1 a j +H.c. j=1 j=1 Kitaev Physics-Uspekhi (2001) j=1 c j,1 = a j + a j c j,2 = i(a j a j ) GROUND SPACE: g d g Leonardo Mazza (SNS) Storage of Information & Majorana Fermions September 26 th, / 21

12 Topological Superconductors 1D Kitaev chain H = µ N N 1 N 1 n j J a j+1 a j + a j+1 a j +H.c. j=1 j=1 Kitaev Physics-Uspekhi (2001) j=1 c j,1 = a j + a j c j,2 = i(a j a j ) GROUND SPACE: g a g d g a d g Why an ancilla? Fermionic superselection rules on parity Models the low-energy of a Kitaev chain Is taken decoherence-free Leonardo Mazza (SNS) Storage of Information & Majorana Fermions September 26 th, / 21

Topological Superconductors 1D Kitaev chain H = µ N N 1 N 1 n j J a j+1 a j + a j+1 a j +H.c. j=1 j=1 Kitaev Physics-Uspekhi 44 131 (2001) j=1 c j,1 = a j + a j c j,2 = i(a j a j ) QUBIT: 0 = g 1 = a d g Why an ancilla?

13 Topological Superconductors 1D Kitaev chain H = µ N N 1 N 1 n j J a j+1 a j + a j+1 a j +H.c. j=1 j=1 Kitaev Physics-Uspekhi (2001) j=1 c j,1 = a j + a j c j,2 = i(a j a j ) QUBIT: 0 = g 1 = a d g Why an ancilla? Fermionic superselection rules on parity Models the low-energy of a Kitaev chain Is taken decoherence-free Leonardo Mazza (SNS) Storage of Information & Majorana Fermions September 26 th, / 21

14 Outline 1 Introduction Storing Quantum Information Zero-Energy Majorana Fermions 2 Theoretical Results The Storage of Information in Presence of Pertubations Optimal Recovery Operations 3 Numerical Results Hamiltonian Quench Perturbations 4 Conclusions Leonardo Mazza (SNS) Storage of Information & Majorana Fermions September 26 th, / 21

15 Modelling the Effect of Perturbations Leonardo Mazza (SNS) Storage of Information & Majorana Fermions September 26 th, / 21

16 Modelling the Effect of Perturbations Given a decoherence channel, what is the best recovery operation? How much ˆρ q ˆρ q? How does the memory-time associated to the best recovery operation depend on the size N? Leonardo Mazza (SNS) Storage of Information & Majorana Fermions September 26 th, / 21

17 Optimal Recovery Operation Previous work: 1 No recovery operation is applied at all 2 The standard recovery operation is applied once at the end of the storage time Problem: The information could still be in the system but not retrieved with the previous methods Leonardo Mazza (SNS) Storage of Information & Majorana Fermions September 26 th, / 21

18 Optimal Recovery Operation Previous work: 1 No recovery operation is applied at all 2 The standard recovery operation is applied once at the end of the storage time Problem: The information could still be in the system but not retrieved with the previous methods Fidelity of a Recovery Operation F (R( )) = dµ φ φ R D( φ φ ) φ φ is a pure qubit state Optimal Recovery Operation F opt max F (R( )) R Information still in the system Leonardo Mazza (SNS) Storage of Information & Majorana Fermions September 26 th, / 21

Optimal Recovery Operation Previous work: 1 No recovery operation is applied at all 2 The standard recovery operation is applied once at the end of the storage time

pure qubit state F opt = 2 3 + 1 Dt(ρq,x,+) Dt(ρq,x, ) tr 6 Optimal Recovery Operation F opt max F (R( )) R Information still in the system Decoherence free assumption

19 Optimal Recovery Operation Previous work: 1 No recovery operation is applied at all 2 The standard recovery operation is applied once at the end of the storage time Problem: The information could still be in the system but not retrieved with the previous methods Fidelity of a Recovery Operation F (R( )) = dµ φ φ R D( φ φ ) φ φ is a pure qubit state F opt = Dt(ρq,x,+) Dt(ρq,x, ) tr 6 Optimal Recovery Operation F opt max F (R( )) R Information still in the system Decoherence free assumption look at the equator of the Bloch sphere LM, Rizzi, Lukin, Cirac arxiv: (2012) Leonardo Mazza (SNS) Storage of Information & Majorana Fermions September 26 th, / 21

20 Outline 1 Introduction Storing Quantum Information Zero-Energy Majorana Fermions 2 Theoretical Results The Storage of Information in Presence of Pertubations Optimal Recovery Operations 3 Numerical Results Hamiltonian Quench Perturbations 4 Conclusions Leonardo Mazza (SNS) Storage of Information & Majorana Fermions September 26 th, / 21

Hamiltonian Perturbations Isolated quantum system Hamiltonian time-evolution Cold atoms ρ(t) = e iht ρ(0)e iht The Hamiltonian of the time evolution may be (partially) unknown The Hamiltonian H(λ)

21 Hamiltonian Perturbations Isolated quantum system Hamiltonian time-evolution Cold atoms ρ(t) = e iht ρ(0)e iht The Hamiltonian of the time evolution may be (partially) unknown The Hamiltonian H(λ) depends on a parameter λ out of control, which varies at every experimental measure Statistical mixture D t (ρ q ) = 1 e ih(λ j )t ρ(0)e ih(λ j )t N j NON UNITARY time-evolution Quench problem: not much known on out-of-equilibrium properties of topological states König and Bravyi Comm Math Phys (2012) Leonardo Mazza (SNS) Storage of Information & Majorana Fermions September 26 th, / 21

22 Hamiltonian Perturbations ρ + (t) ρ (t) tr / 2 Initialization with respect to a specific Hamiltonian Time-evolution sampling within the topological region D t (ρ q ) = 1 e ih(λ j )t ρ(0)e ih(λ j )t N j 1.02 N = N = N = N = 8 Efficient computation possible using BCS (fermionic Gaussian) states even for the optimal fidelity time (J 1 ) LM, Rizzi, Lukin, Cirac arxiv: (2012) Leonardo Mazza (SNS) Storage of Information & Majorana Fermions September 26 th, / 21

23 Size Scaling Memory time t N (F 0 ) : first time at which fidelity F 0 < 1.0 in a system of size N 400 t 0 (J 1 ) F 0 = F 0 = F 0 = N Compatible with an exponential scaling with the size N LM, Rizzi, Lukin, Cirac arxiv: (2012) Leonardo Mazza (SNS) Storage of Information & Majorana Fermions September 26 th, / 21

24 Time-Dependent Perturbation Initialization with respect to a specific Hamiltonian /J Time-evolution sampling within the topological region λ(t) depends on time D t (ρ q ) = 1 e ih(λ j )t ρ(0)e ih(λ j )t 2 N j Μ -2 µ/j Μ ρ + (t) ρ (t) tr / 2 t 0 (J 1 ) (a) (b) time (J 1 ) (c) time (J 1 ) N N = 8 N = 12 N = 16 N = 20 N = 24 F 0 = 0.97 F 0 = 0.98 F 0 = 0.99 Μ time 2Π Left: time dependence in µ and Right: time dependence in µ and and randomness in µ i Bottom: fidelity scaling Leonardo Mazza (SNS) Storage of Information & Majorana Fermions September 26 th, / 21

25 Topological Protection Initialization with respect to a specific Hamiltonian Time-evolution sampling outside the topological region D t (ρ q ) = 1 e ih(λ j )t ρ(0)e ih(λ j )t N 1 j 00 ρ + (t) ρ (t) tr / time (J 1 ) N = 8 N = 16 N = 24 N = 32 N = 40 N = 48 LM, Rizzi, Lukin, Cirac arxiv: (2012) Leonardo Mazza (SNS) Storage of Information & Majorana Fermions September 26 th, / 21

26 Is Disorder Necessary? General widespread idea: disorder and localization improve the stability of a memory. localization of errors dephasing happens on longer time-scales Leonardo Mazza (SNS) Storage of Information & Majorana Fermions September 26 th, / 21

27 Is Disorder Necessary? General widespread idea: disorder and localization improve the stability of a memory. localization of errors dephasing happens on longer time-scales 1 Initialize in g + a d g 2 w.r.t. H 0 2 Time-evolution with H 0 + V 3 Apply one specific physically-motivate error-correction procedure V can be disordered or not disordered from: Bravyi and König, Comm. Math. Phys (2012) Leonardo Mazza (SNS) Storage of Information & Majorana Fermions September 26 th, / 21

28 Is Disorder Necessary? General widespread idea: disorder and localization improve the stability of a memory. localization of errors dephasing happens on longer time-scales 1 Initialize in g + a d g 2 w.r.t. H 0 2 Time-evolution with H 0 + V 3 Apply one specific physically-motivate error-correction procedure V can be disordered or not disordered from: Bravyi and König, Comm. Math. Phys (2012) Information is anyway preserved, but the recovery operation does not find it See other works on toric code: Wotton, Pachos, PRL 107, (2011); Stark, Pollet, Imamoglu, Renner, PRL 107, (2011). Leonardo Mazza (SNS) Storage of Information & Majorana Fermions September 26 th, / 21

29 Towards an Explanation 1 Two ground states: 0 g and 1 d g 2 Hamiltonians for the time evolution: {H(λ j )} N dis j=1 Overlap matrices for out-of-equilibrium states G(0) j,k 0 e ih(λ j )t e ih(λ k )t 0 G(1) j,k 1 e ih(λ j )t e ih(λ k )t 1 Leonardo Mazza (SNS) Storage of Information & Majorana Fermions September 26 th, / 21

30 Towards an Explanation 1 Two ground states: 0 g and 1 d g 2 Hamiltonians for the time evolution: {H(λ j )} N dis j=1 Overlap matrices for out-of-equilibrium states G(0) j,k 0 e ih(λ j )t e ih(λ k )t 0 G(1) j,k 1 e ih(λ j )t e ih(λ k )t 1 [ ] 1 2 ˆρ x,+(t) ˆρ x, (t) 1 = Tr G(0) G(1) N dis N dis = G(0) N dis, G(1) N dis HS Recoverable information: 1 excitations propagate in the two parity sectors of the theory in the same way N 2 [G(0) G(1)] j,k 0 Leonardo Mazza (SNS) Storage of Information & Majorana Fermions September 26 th, / 21

31 Towards an Explanation 1 Two ground states: 0 g and 1 d g 2 Hamiltonians for the time evolution: {H(λ j )} N dis j=1 Overlap matrices for out-of-equilibrium states G(0) j,k 0 e ih(λ j )t e ih(λ k )t 0 G(1) j,k 1 e ih(λ j )t e ih(λ k )t 1 [ ] 1 2 ˆρ x,+(t) ˆρ x, (t) 1 = Tr G(0) G(1) N dis N dis = G(0) N dis, G(1) N dis HS Recoverable information: 1 excitations propagate in the two parity sectors of the theory in the same way N 2 [G(0) G(1)] j,k 0 [G(0) G(1)] j,k = det X (j, k) Topological H(λ j ) Non-Topological H(λ j ) Leonardo Mazza (SNS) Storage of Information & Majorana Fermions September 26 th, / 21

32 Outline 1 Introduction Storing Quantum Information Zero-Energy Majorana Fermions 2 Theoretical Results The Storage of Information in Presence of Pertubations Optimal Recovery Operations 3 Numerical Results Hamiltonian Quench Perturbations 4 Conclusions Leonardo Mazza (SNS) Storage of Information & Majorana Fermions September 26 th, / 21

33 Hints from Quantum Information to Many-Body Physics 1 Quantum quench with a set of Hamiltonians real-time evolution + average over realizations 1 testing the robustness of information & other quantities to unknown changes in the parameters of the Hamiltonian 2 it does not necessarily means: average over disorder Leonardo Mazza (SNS) Storage of Information & Majorana Fermions September 26 th, / 21

34 Hints from Quantum Information to Many-Body Physics 1 Quantum quench with a set of Hamiltonians real-time evolution + average over realizations 1 testing the robustness of information & other quantities to unknown changes in the parameters of the Hamiltonian 2 it does not necessarily means: average over disorder 2 Equilibration picture in topological vs non-topological systems 1 equilibration of zero-energy modes in non-disordered systems Leonardo Mazza (SNS) Storage of Information & Majorana Fermions September 26 th, / 21

35 Hints from Quantum Information to Many-Body Physics 1 Quantum quench with a set of Hamiltonians real-time evolution + average over realizations 1 testing the robustness of information & other quantities to unknown changes in the parameters of the Hamiltonian 2 it does not necessarily means: average over disorder 2 Equilibration picture in topological vs non-topological systems 1 equilibration of zero-energy modes in non-disordered systems 3 Many-body systems and information 1 a huge Hilbert space may possibly be useful for conserving important properties over long times 2 complexity of dealing with a many-body system 3 is disorder the only solution? Leonardo Mazza (SNS) Storage of Information & Majorana Fermions September 26 th, / 21

36 Thank you for your attention! Results: Is a quantum memory based on Majorana fermions reliable? Extreme sensibility to an environment No topological protection against particle losses 1 Information still in the system Optimal recovery operation Protection against Hamiltonian perturbation Hamiltonian must be topological Spectral protection, not only ground state Perspectives: Relaxation of several hypotheses 1 Ancilla is not decoherence free 2 Two-dimensional system p x + ip y as qubit 3 Other topological models: Quantum Hall effect LM, Rizzi, Lukin, Cirac arxiv: (2012) Leonardo Mazza (SNS) Storage of Information & Majorana Fermions September 26 th, / 21

37 Optimal Gaussian Recovery Operation Problem: How complicated is the optimal recovery operation? Problem: Can it be realized in practice? Optimal Gaussian Recovery Operation F opt max F (R( )) R is a Gaussian channel What is a Gaussian channel? Addition and discard of fermionic modes Manipulation of two fermionic modes at time Example: time-evolution under quadratic Hamiltonian Covariance matrix of a state Γ ρ two points correlators Tr[ρ a k a l] or Tr[ρ a k a l ] Botero and Reznik Phys Lett A (2004) Bravyi Quantum Info Comp (2005) Information retrievable from the system F opt G = Γ D t(ρ q,x,+) Γ Dt(ρ q,x, ) op LM, Rizzi, Lukin, Cirac arxiv: (2012) Leonardo Mazza (SNS) Storage of Information & Majorana Fermions September 26 th, / 21

Optimal Gaussian Recovery Operation Initialization with respect to a specific Hamiltonian Time-evolution sampling within the topological region D t (ρ q ) = 1 e ih(λ j )t ρ(0)e ih(λ j )t N j Γ + (t)

38 Optimal Gaussian Recovery Operation Initialization with respect to a specific Hamiltonian Time-evolution sampling within the topological region D t (ρ q ) = 1 e ih(λ j )t ρ(0)e ih(λ j )t N j Γ + (t) Γ (t) op / N = 24, 32, 40, N = 8 N = time (J 1 ) Leonardo Mazza (SNS) Storage of Information & Majorana Fermions September 26 th, / 21

Detecting and using Majorana fermions in superconductors

Detecting and using Majorana fermions in superconductors Anton Akhmerov with Carlo Beenakker, Jan Dahlhaus, Fabian Hassler, and Michael Wimmer New J. Phys. 13, 053016 (2011) and arxiv:1105.0315 Superconductor