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. non-interacting fermions Hilbert space seperates into sectors n = 0, 1, 2,
Single Fermion States Sector of Hilbert space with n = 0and n = 1 H0 spanned by H1 spanned by N N matrix
Perfect Transfer of Qubits Qubit at t = 0 is prepared at site 1 superposition possible for JW-fermions After time t = τ want qubit at site N Have to engineer Hamiltonian HF for n = 1 sector with time evolution
Symmetry Condition Mirror symmetry <=> Eigenstates λk have decinite parity N free parameters fingerprint of spin chain
Eigenvalue Condition Condition for eigenvalues λk Simplest example: Double well potential anti-symmetric states are flipped
Inverse Eigenvalue Problem Condition for eigenvalues λk very weak! Take τ = π and Φ = 0 => eigenvalues λk are integers 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
Orthogonal Polynomials Characteristic polynomial pj of submatrix Hj Structure and orthogonality Shohat-Favard theorem with weigths
Orthogonal Polynomials Inverse relations with norm Carl R. de Boor Gene H. Golub
Algorithm by de Boor & Golub Calculate weights wkfrom λkfor scalar product (p0= 1) For j = 1 to ~N/2 1. Calculate Computationally cheap & stable 2. Find 3. Calculate End 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 λk= 0 Works if eigenvalues λk fulfill
Optimization Examples Linear Spectrum Inverted Quadratic Spectrum
Test Robustness Couplings are uniformly randomized (± few percent) Effect on transfer fidelity (numerics) = fidelity averaged over Bloch sphere Boundary states => more high-fidelity chains => smooth time evolution
Test Robustness Couplings are uniformly randomized (± few percent) Effect on transfer fidelity (numerics) = fidelity averaged over Bloch sphere Boundary states => more high-fidelity chains => smooth time evolution
Boundary States in Quantum Wires Quantum wire with superlattice potential weak link Boundary states form double quantum dot 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 Quantum computation with fermions Previous results hold for n 2 sectors t = 0 t = τ Generate phases between subspaces Efficient generation of graph states for quantum computation S.R. Clark, C. Moura Alves and D. Jaksch, New J. Phys. 7, 124 (2005)
Controlled Phase Gate t = 0 t = τ = 1 0 0 0 0 1 0 0 0 0 1 0 0 0 0 1 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 λkwe 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 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) Localized End States in Density Modulated Quantum Wires and Rings S. Gangadharaiah, L. Trifunovic and D. Loss, Phys. Rev. Lett. 108, 136803 (2012)