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arXiv:1110.6474 · Physical Review A

How to construct spin chains with perfect state transfer

Luc Vinet, Alexei S. Zhedanov

Published Jan 23, 2012• 92 citations
arXivHTMLPDF

Abstract

A method to systematically construct the $XX$ quantum spin chains with nearest-neighbor interactions that allow perfect state transfer (PST) is shown. Sets of orthogonal polynomials (OPs) are in correspondence with such systems. The key observation is that for any admissible one-excitation energy spectrum, the weight function of the associated OPs is uniquely prescribed. This entails the complete characterization of these PST models with the mirror-symmetry property arising as a corollary. A simple and efficient algorithm to obtain the corresponding Hamiltonians is presented. A model connected to a special case of the symmetric $q$-Racah polynomials is offered. An explanation of how additional models with PST can be derived from a parent system by removing energy levels from the one-excitation spectrum of the latter is given. This is achieved through Christoffel transformations and is also completely constructive in regards to the Hamiltonians.

DOI
10.1103/physreva.85.012323

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A method to systematically construct the XXXXXX quantum spin chains with nearest-neighbor interactions that allow perfect state transfer (PST) is shown. Sets of orthogonal polynomials (OPs) are in correspondence with such systems. The key observation is that for any admissible one-excitation energy spectrum, the weight function of the associated OPs is uniquely prescribed. This entails the complete characterization of these PST models with the mirror-symmetry property arising as a corollary. A simple and efficient algorithm to obtain the corresponding Hamiltonians is presented. A model connected to a special case of the symmetric qqq-Racah polynomials is offered. An explanation of how additional models with PST can be derived from a parent system by removing energy levels from the one-excitation spectrum of the latter is given. This is achieved through Christoffel transformations and is also completely constructive in regards to the Hamiltonians.
Abstract New exactly solvable one-dimensional XX spin chain models that exhibit perfect state transfer (PST) are defined. These models have inhomogeneous couplings and magnetic fields determined from the three-term recurrence relations satisfied by the q -Racah and para q -Racah polynomials. Due to this connection with orthogonal polynomials, the one-excitation sector can be solved analytically. This allows us to provide explicit sets of conditions on the polynomial parameters that guarantee the occurrence of PST across these spin chains.
New exactly solvable one-dimensional XX spin chain models that exhibit perfect state transfer are defined. These models have inhomogeneous couplings and magnetic fields determined from the three-term recurrence relations satisfied by the q-Racah and para q-Racah polynomials. Due to this connection with orthogonal polynomials, the one-excitation sector can be solved analytically. This allows us to provide explicit sets of conditions on the polynomial parameters that guarantee the occurrence of perfect state transfer across these spin chains.
We study the time evolution of a single spin excitation state in certain linear spin chains, as a model for quantum communication. Some years ago it was discovered that when the spin chain data (the nearest neighbour interaction strengths and the magnetic field strengths) are related to the Jacobi matrix entries of Krawtchouk polynomials or dual Hahn polynomials, so-called perfect state transfer takes place. The extension of these ideas to other types of discrete orthogonal polynomials did not lead to new models with perfect state transfer, but did allow more insight in the general computation of the correlation function. In the present paper, we extend the study to discrete orthogonal polynomials of q-hypergeometric type. A remarkable result is a new analytic model where perfect state transfer is achieved: this is when the spin chain data are related to the Jacobi matrix of q-Krawtchouk polynomials. The other cases studied here (affine q-Krawtchouk polynomials, quantum q-Krawtchouk polynomials, dual q-Krawtchouk polynomials, q-Hahn polynomials, dual q-Hahn polynomials and q-Racah polynomials) do not give rise to models with perfect state transfer. However, the computation of the correlation function itself is quite interesting, leading to advanced q-series manipulations.
The natural notion of almost perfect state transfer (APST) is examined. It is applied to the modeling of efficient quantum wires with the help of XXXXXX spin chains. It is shown that APST occurs in mirror-symmetric systems, when the 1-excitation energies of the chains are linearly independent over rational numbers. This result is obtained as a corollary of the Kronecker theorem in Diophantine approximation. APST happens under much less restrictive conditions than perfect state transfer and moreover accommodates the unavoidable imperfections. Some examples are discussed.
A simple method for transmitting quantum states within a quantum computer is via a quantum spin chain – that is, a path on n vertices. Unweighted paths are of limited use, and so a natural generalization is to consider weighted paths; this has been further generalized to allow for loops (potentials in the physics literature). We study the particularly important situation of perfect state transfer with respect to the corresponding adjacency matrix or Laplacian through the use of orthogonal polynomials. Low-dimensional examples are given in detail. Our main result is that PST with respect to the Laplacian matrix cannot occur for weighted paths on n≥3 vertices nor can it occur for certain symmetric weighted trees. The methods used lead us to a conjecture directly linking the rationality of the weights of weighted paths on n>3 vertices, with or without loops, with the capacity for PST between the end vertices with respect to the adjacency matrix.
Quantum information transfer is an important part of quantum information processing. Several proposals for quantum information transfer along linear arrays of nearest-neighbor coupled qubits or spins were made recently. Perfect transfer was shown to exist in two models with specifically designed strongly inhomogeneous couplings. We show that perfect transfer occurs in an entire class of chains, including systems whose nearest-neighbor couplings vary only weakly along the chain. The key to these observations is the Jordan-Wigner mapping of spins to noninteracting lattice fermions that display perfectly periodic dynamics if the single-particle energy spectrum is appropriate. After a half-period of that dynamics, any state is transformed into its mirror image with respect to the center of the chain. The absence of fermion interactions preserves these features at an arbitrary temperature and allows for the transfer of nontrivially entangled states of several spins or qubits.