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Spin Chains for Perfect State Transfer
and Quantum Computing
January 17th 2013
Martin Bruderer
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Overview
� Basics of Spin Chains
� Engineering Spin Chains for Qubit Transfer
� Inverse Eigenvalue Problem
� spinGUIn
� Boundary States
� Generating Graph States
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Spin Chains as Quantum Channel
� Alice sends a qubit to Bob via a spin chain
� Spin up = |1⟩ Spin down = |0⟩
� Qubit is tranferred (imperfectly) by ‘natural’ time evolution
______________________________________________________________________________
Quantum Communication through an Unmodulated Spin Chain
Sougato Bose, Phys. Rev. Lett. 91, 207901 (2003)
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Spin Chains
� XX Spin Hamiltonian
� Map to 1d fermionic model using Jordon-Wigner trans.
� Hilbert space seperates into sectors n = 0, 1, 2, …
non-interacting fermions
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Single Fermion States
� Sector of Hilbert space with n = 0 and n = 1
H0 spanned by
H1 spanned by
N × N matrix
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Perfect Transfer of Qubits
� Qubit at t = 0 is prepared at site 1
� After time t = τ want qubit at site N
with time evolution
Have to engineer
Hamiltonian HF
for n = 1 sector
superposition possible
for JW-fermions
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Symmetry Condition
� Mirror symmetry <=> Eigenstates |λk⟩ have deCinite parity
N free parameters ‘fingerprint’ of spin chain
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Eigenvalue Condition
� Condition for eigenvalues λk
anti-symmetric states are flipped
Simplest example:
Double well potential
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Inverse Eigenvalue Problem
� Condition for eigenvalues λk
� Infinitely many solutions e.g. λk = {2, 13, 16, 29, 34, 35}
Structured inverse eigenvalue problem:
Given N eigenvalues λk find the tridiagonal N × N matrix
� Take τ = π and Φ = 0 => eigenvalues λk are integers
very weak!
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Orthogonal Polynomials
� Characteristic polynomial pj of submatrix Hj
� Structure and orthogonality
with weigths
Shohat-Favard theorem
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Orthogonal Polynomials
� Inverse relations
with norm
Gene H. GolubCarl R. de Boor
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Algorithm by de Boor & Golub
Calculate weights wk from λk for scalar product (p0 = 1)
1. Calculate
For j = 1 to ~N/2
2. Find
3. Calculate
End
Computationally
cheap & stable
________________________________________________________________________________________________
The numerically stable reconstruction of a Jacobi matrix from spectral data
C. de Boor and G.H. Golub, Linear Algebr. Appl. 21, 245 (1978)
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Application
� No approximations . . .
Example: If λk symmetrically distributed around zero => aj = 0
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Optimize for Robustness
� Create spin chains with localized boundary states
� Robust against perturbations � Simplified evolution
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Adding Boundary States
� Zero modes ~ Boundary states (cf. Majorana states)
1. Take original spin chain
2. Shift spectrum
3. Calculate new couplings
4. Compare robustness
� Works if eigenvalues λk fulfill
λk = 0
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Optimization Examples
Linear Spectrum
Inverted Quadratic Spectrum
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Test Robustness
� Couplings are uniformly randomized (± few percent)
� Effect on transfer fidelity (numerics)
� Boundary states
=> more high-fidelity chains
=> smooth time evolution
= fidelity averaged over Bloch sphere
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Test Robustness
� Couplings are uniformly randomized (± few percent)
� Effect on transfer fidelity (numerics)
� Boundary states
=> more high-fidelity chains
=> smooth time evolution
= fidelity averaged over Bloch sphere
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Boundary States in Quantum Wires
� Quantum wire with superlattice potential
� Boundary states form double quantum dot
weak link
___________________________________________________________________
Localized End States in Density Modulated Quantum Wires and Rings
S. Gangadharaiah, L. Trifunovic and D. Loss, Phys. Rev. Lett. 108, 136803 (2012)
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spinGUIn
� spin chain Graphical User Interface for Matlab
� Playful approach to spin chains (education)
� Algorithm ‘iepsolve.m’ & GUI
� Some small bugs . . .
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Ex Linear Spectrum
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Ex Boundary States
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Ex Cubic Spectrum
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Ex Three Band Model
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Many Fermion States
_________________________________________________________________________________
Efficient generation of graph states for quantum computation
S.R. Clark, C. Moura Alves and D. Jaksch, New J. Phys. 7, 124 (2005)
� Quantum computation with fermions
� Previous results hold for n ≥ 2 sectors
t = 0
t = τ
� Generate phases between subspaces
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Controlled Phase Gate
t = 0
t = τ
−
=
1000
0100
0010
0001
CZ
Z
Initialize each qubit as
Very robust, but not enough
for quantum computation…
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Generate Graph States
Graph state of n vertices requires at most O(2n) operations
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Summing up
1. For a given spectrum λk we can construct
the tight-binding Hamiltonian
2. Fermionic phases are useful for generating
highly entangled states
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Some People Involved
Stephen R. Clark
Quantum (t-DRMG)
Oxford, Singapore (CQT)
Kurt Franke
g-Factor of Antiprotons
CERN, Geneva
Danail Obreschkow
Astrophysics (SKA)
Perth, Australia
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References
Localized End States in Density Modulated Quantum Wires and Rings
S. Gangadharaiah, L. Trifunovic and D. Loss, Phys. Rev. Lett. 108, 136803 (2012)
A Review of Perfect, Efficient, State Transfer and its Application as a Constructive Tool
A. Kay, Int. J. Quantum Inform. 8, 641 (2010)
Quantum Communication through an Unmodulated Spin Chain
S. Bose, Phys. Rev. Lett. 91, 207901 (2003)
Exploiting boundary states of imperfect spin chains for high-fidelity state transfer
MB, K. Franke, S. Ragg, W. Belzig and D. Obreschkow, Phys. Rev. A 85, 022312 (2012)
The numerically stable reconstruction of a Jacobi matrix from spectral data
C. de Boor and G.H. Golub, Linear Algebr. Appl. 21, 245 (1978)
Fermionic quantum computation
S. B. Bravyi and A. Yu. Kitaev, Annals of Physics 298, 210 (2002)
Efficient generation of graph states for quantum computation
S.R. Clark, C. Moura Alves and D. Jaksch, New J. Phys. 7, 124 (2005)
Graph state generation with noisy mirror-inverting spin chains
S. R Clark, A. Klein, MB and D. Jaksch, New J. Phys. 9, 202 (2007)