Effects of magnetic tensions on topological quantum memories

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Saeed S. Jahromi, K.N.Toosi University Of Technology, Iran

M. Kargarian, University Of Texas at Austin, USA

K. P. Schmidt, TU Dortmund, Germany

SF. Masoudi, K.N.Toosi University Of Technology, Iran

Effects of magnetic tensions on topological quantum memories

Quantum Information Conference, 4-5 Sep. 2013, Shahrood, Iran.


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Outline

Topological Order

Quantum Computation

Error Correcting Quantum Codes

Color Code

Perturbed Topological Color Code with Magnetic Field

Phase diagram

Discussion and results


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Topological Order

Order

Symmetry

Breaking

Non-Symmetry

Breaking

Landau Theory of

Phase Transition

Quantum

Liquid

String-Net

Condensation

Topological

Order


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Quantum Computation

Quantum

Computation

Quantum

Bits

Error

Correction

Unitary

Operation

Computational

Basis

Quantum

Gates


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Errors in Quantum Computation

DecoherencePhase FlipBit Flip

Search for systemswhich are intrinsically robust against errors.

States carrying non local degrees of freedom.

We can put the quantum information on global degrees of freedom.

Topological Systems


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Error Correcting Quantum Codes

Fault-Tolerant Quantum Computation

Topological Properties protects the system against Errors

Toric Code Color Code

A.Yu. Kitaev, Annals of Physics 303 2-30 (2003). H. Bombin, Phys.Rev.Lett. 97 180501 (2006).


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Topological Color code

H. Bombin, M.A. Martin-Delgado, Phys.Rev.Lett. 97 180501 (2006).

H. Bombin, M.A. Martin-Delgado, Phys. Rev. A 77, 042322 (2008).


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String Operators


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Ground State

X.G. Wen, Quantum Field Theory of Many-body Systems, Oxford Univ. Press, (2004).

String-net Condensation


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Global String Operators

There are another kind of strings which are closed but are not product of

pluquette operators

Global Strings (Fundamental non-contractible loops )


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Topological Degeneracy


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Excitations

Excitations are created at the end points of open strings


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Topological Color Code in magnetic field


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Continuous Unitary Transformations (CUTs)


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Continuous Unitary Transformations (CUTs)

F.J. Wegner, Ann. Phys. (1994)

S.D. Glazek and K.G. Wilson, Phys. Rev. D (1994)

The basic idea is to unitarily transform the

initial problem in a continuous fashion.


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Perturbative Continuous Unitary

Transformations (PCUT)

A. Mielke, EPJB (1998)

C. Knetter and G. S. Uhrig, EPJB (2000)


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Perturbative Continuous Unitary

Transformations (PCUT)

A. Mielke, EPJB (1998)

C. Knetter and G. S. Uhrig, EPJB (2000)

Quasiparticle conserving Generator


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Single parallel field

S. S. Jahromi, M. Kargarian, S. F. Masoudi, K. P. Schmidt,

Phys. Rev. B 87, 094413 (2013).

Mapped to Baxter-Wu model in transverse magnetic field


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Phase Transition

S. S. Jahromi, M. Kargarian, S. F. Masoudi, K. P. Schmidt,

Phys. Rev. B. 87, 094413 (2013)


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Physical Measurement


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Two Parallel field (x,z)


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Transverse field (y)


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Conclusion

Color Code in a single parallel magnetic field is mapped to the

Baxter-Wu model in transverse field.

The problem was solved by mapping the perturbed Hamiltonian to

a quasiparticle conserving effective Hamiltonian, using PCUT

The topological phase of TCC breaks down to a polarized phase

via a first order phase transition in all of the field directions.

TCC in transverse field conserves the parity of the number of

quasiparticles on the plaqquests.

Color Code is More robust than Toric code in parallel fild


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Thanks For Your Attention

Effects of magnetic tensions on topological quantum memories

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