Spin Chains for Perfect State Transfer

and Quantum Computing

January 17th 2013

Martin Bruderer

Overview

� Basics of Spin Chains

� Engineering Spin Chains for Qubit Transfer

� Inverse Eigenvalue Problem

� spinGUIn

� Boundary States

� Generating Graph States

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)

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

Single Fermion States

� Sector of Hilbert space with n = 0 and n = 1

H0 spanned by

H1 spanned by

N × N matrix

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

Symmetry Condition

� Mirror symmetry <=> Eigenstates |λk⟩ have deCinite parity

N free parameters ‘fingerprint’ of spin chain

Eigenvalue Condition

� Condition for eigenvalues λk

anti-symmetric states are flipped

Simplest example:

Double well potential

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!

Orthogonal Polynomials

� Characteristic polynomial pj of submatrix Hj

� Structure and orthogonality

with weigths

Shohat-Favard theorem

Orthogonal Polynomials

� Inverse relations

with norm

Gene H. GolubCarl R. de Boor

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)

Application

� No approximations . . .

Example: If λk symmetrically distributed around zero => aj = 0

Optimize for Robustness

� Create spin chains with localized boundary states

� Robust against perturbations � Simplified evolution

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

Optimization Examples

Linear Spectrum

Inverted Quadratic Spectrum

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

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

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)

spinGUIn

� spin chain Graphical User Interface for Matlab

� Playful approach to spin chains (education)

� Algorithm ‘iepsolve.m’ & GUI

� Some small bugs . . .

Ex Linear Spectrum

Ex Boundary States

Ex Cubic Spectrum

Ex Three Band Model

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

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…

Generate Graph States

Graph state of n vertices requires at most O(2n) operations

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

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

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)