Quantum wires, orthogonal polynomials and Diophantine approximation

Quantum wires, orthogonal polynomials and Diophantine approximation

Introduction Quantum Mechanics (QM) is a linear theory Main idea behind Quantum Information (QI): use the superposition principle of QM to process faster Basic task in QI: transfer quantum states from one location to another communications between Alice and Bob algorithm (operations in a register) connection between quantum computer parts Quantum wires are needed to that end Goal of the talk: discuss one approach to the design of quantum wires Based on joint work with A. Zhedanov, papers can be found on arxiv

Constructing quantum wires is difficult. Need to connect microscopic objects (behaving as per QM) with macroscopic devices doing control Issues of control (micro-macro link) and of noise (decoherence) that control induces One strategy: use one-dimensional chains (more generally graphs) of coupled spin / particles Advantage: no need for control along the chain: the transfer exploits the dynamics of the chain (Noise reduction)

Another challenge: construct quantum wires with high fidelity i.e.: it is required that states are transferred between the two locations with probability equal or very close to When = : perfect state transfer (PST) When = : almost perfect state transfer (APST) Part I: PST and orthogonal polynomials Part II: APST and Diophantine approximation

spin chains = + (σ σ + + σ σ + )+ (σ + ) = = in (C ) + With insight, we are allowing the couplings and magnetic fields to depend on the site Pauli matrices σ = σ = σ =

Action on vectors σ = σ = σ = σ = σ = σ = σ σ = σ σ = σ σ = σ σ = (σ σ + σ σ ) = (σ σ + σ σ ) = Conservation of total -projection [ (σ + )] = = The eigenstates of split in subspaces labeled by the numbers of spins over the chain that are in state

State transfer in chain Most state transfer properties can be derived from the -excitation subspace Initially assume that the register is prepared in the state + = Introduce the unknown state ψ = α + β onto the first site ( = ). We want to recuperate ψ on last site ( = ) after some time Component α must go to final state α +. Automatic because + is an eigenstate of It suffices to consider Allows for restriction to -excitation states

-Excitation dynamics A natural basis for C + ( -excitation state space) =( ) = where the is in the th position Restriction of to the one-excitation subspace is given by the Jacobi matrix......... (Recall the action of Pauli matrices on,, etc.

Its action on the basis vectors Initial conditions = + + + + > = + = Note that the knowledge of prescribes as the couplings and the magnetic fields are then given

Condition for perfect state transfer (PST) and almost perfect state transfer (APST) = : ( spin up at =, all others down) QM time evolution: ( ) = Probability to find at site = the spin up that was at site = after some time is PST: ( )=, i.e. ( )= ( ) ( )= ( ) = = φ( )

APST: ( ) φ( ) < i.e. for APST we ask that ( ) be as close to as desired Problem: Find the matrix and hence the Hamiltonians such that the PST or the APST condition are satisfied Inverse spectral problems

Diagonalization and orthogonal polynomials Eigenbasis : = with = The are real and non-degenerate (i.e. = if = ) Expansion of basis over the other = = + + + + = Hence: = χ ( ) where χ ( ) are polynomials satisfying the recurrence relation and conditions + χ + ( )+ χ ( )+ χ ( )= χ ( ) χ = χ =

Bases and are orthonormal = δ = δ The matrix = χ ( ) is orthogonal: = δ and = δ = = Polynomials χ ( ) are orthonormal on the finite set of spectral points ω χ ( )χ ( )=δ = where the discrete weights ω = are normalized ω = = = =

Dual expansions: = = ω χ ( ) and = ω χ ( = ) Monic polynomials ( )= χ ( )= + ( ) Recurrence relation + ( )+ ( )+ ( )= ( ) where = > Orthogonality relations = ( ) ( )ω = δ =

For = + : characteristic polynomial Important formula + ( )=( )( ) ( ) ω = ( ) + ( )) We assume < < Taking the number of exchanges and ordering into account + ( )=( )( ) ( )( + ) ( ) =( ) + + ( ) + ( )=( ) + ( ) ( (-1) factors because is smallest and for each step one less)

Necessary and sufficient conditions for PST The PST condition: = φ Expand and in terms of the eigenstates and equate coefficients. Using χ ( )= ω = φ = = ω χ ( ) Whence χ ( )= φ = Necessary and sufficient condition for PST

Another formulation Polynomial χ ( ) has real coefficients. Hence χ ( ) =± Interlacing property of zeros of orthogonal polynomials leads to an alternating rule: χ ( ) must have zeros between (the zeros of + ( )) PST condition χ ( )=( ) + = = φ π( + + ) Z = [ φ + π( + + )] Thus the PST condition amounts to the following relation between the neighboring eigenvalues + = π where are odd integers

This implies that the weights satisfy the equivalent condition ω = + ( ) > using ω = ( ) + ( ) + ( )=( ) + + ( ) ( ) χ ( )=( ) +

Summary Given a -excitation spectrum { } such that = + odd integers Knowing that + ( )=( )( ) ( ) The polynomials orthogonal for the weights ω = + ( ) define a Hamiltonian with. We shall provide an algorithm for constructing ( and thus H)

Mirror symmetry Reflected Jacobi matrix: = where is the reflection matrix =.. has the structure:......... with coefficients: =, = +

Corresponding monic orthogonal polynomials + ( )+ ( )+ ( )= ( ) The matrix has the same eigenvalues, = as Property of the weights ω ω = ( + ( )) for = A Jacobi matrix is mirror-symmetric if = = This is equivalent to ω = ω

This in in turn implies that ω = + ( ) > () Conversely, if ω =(), then ω = ω and = Conclusion: () or mirror symmetry = are equivalent necessary conditions for PST

Reconstruction algorithm for the matrix Given a spectrum { } satisfying the PST condition. We then know explicitly the polynomial + ( )=( )( ) ( ) The orthonormal polynomial χ ( ) is prescribed at + points χ ( )=( ) + = Hence χ ( ) can be explicitly found by Lagrange interpolation χ ( )= = ( ) + where are the Lagrange polynomials = =

Starting from monic + ( ) and ( ) it is possible to reconstruct ( ), = by the Euclidean algorithm Recall the -term recurrence relation: + ( )+ ( )+ ( )= ( ) 1 st step: divide the polynomial + ( ) by ( ): + ( )= ( )+ ( ) ( )= ( )= ( ) Next step: repeat this procedure with respect to polynomials ( ) and ( ) and find ( ), and Iteratively, we find all polynomials and all coefficients, = ( ) = This yields

New chains with PST from spectral surgery Can we obtain a spin chain with PST from a parent chain known to possess the property? What kind of spectral surgery is admissible? What are the resulting spin chains? Answer: Christoffel transform Remove : ( ) + ω = ( ) > = ( ) + ω = = ( ) =

This is equivalent to ω = const ( )ω > = which is a Christoffel transform Removing arbitrary ω = const ( )ω However, if = or, then ω is not positive for all How to circumvent: remove a pair of levels, + In general ω = const ( )( + )ω > ω = const ( )( ) ( )ω where must satisfy obvious conditions for positivity of ω

Under the Christoffel transform ω =( )ω, the transformation of the polynomials is: ( )= + ( ) ( ) = + ( ) ( ) Transformation of recurrence coefficients = = + + + These formulas can be repeated step-by-step to obtain polynomials and recurrence coefficients, after Christoffel transforms To sum-up, CT s preserve the PST condition on weights and thus give PST chains directly. The inverse problem gets bypassed

Explicit examples: 1. Krawtchouk polynomials Main idea: start from a spectral grid obeying the PST condition and use the algorithm to determine uniquely Uniform grid: Krawtchouk polynomials = / = PST condition + = π is satisfied for = π =

Recall that: ω = + ( ) = + ( ) = ( ) ( )( ) ( ) =!( )! This yields the (normalized) binomial distribution ω =! ( )!!

The corresponding OPs are the symmetric Krawtchouk polynomials with recurrence coefficients = = ( + ) First example found by Albanese et al. Krawtchouk polynomials are associated to Ehrenfest urn model Equivalence to random walk on hypercube

Explicit examples: 2. Para-Krawtchouk polynomials Bi-lattice: Para-Krawtchouk polynomials Take odd, consider for the -excitation spectrum the bi-lattice = + (γ )( ( ) ) = Bi-lattice because it is the union of two uniform grids separated by γ: = and + = + γ When γ =, uniform grid and Krawtchouk polynomials

We have + = γ + + = γetc. PST condition + = π odd integer is satisfied if γ = N odd and = π

The weights ( + γ/ ) ( ) ( γ/ ) ω = ( / )!( γ/ ) ( + γ/ ) ( ) (γ/ ) ω + = ( / )!( + γ/ ) with =( )/ and ( ) = ( + ) ( + ). One has ω = = = ω + = / Recurrence coefficients = + γ = ( + )(( + ) γ ) ( )( + )

A commercial break: the Bannai-Ito scheme It is nice when physically motivated problem lead to pieces of new mathematics Askey-Scheme of hypergeometric orthogonal polynomials

A commercial break: the Bannai-Ito scheme It is nice when physically motivated problem lead to pieces of new mathematics It is possible to systematically explore the orthogonality grids of the finite OPs to find new PST chains The OPs of the Askey scheme are bispectral. In addition to -term recurrence relation, they obey Differential Difference q-difference eigenvalue equation

Recently with A. Zhedanov, S. Tsujimoto and V.X. Genest, I explored the bispectral OPs that satisfy eigenvalue equations of Dunkl type Dunkl operators: differential (difference) operators + reflections e.g: µ = + µ ( ) ( )= ( ) At the top of this scheme: Bannai-Ito polynomials (BI) The BI polynomials have four parameters Introduced in 1984 in algebraic combinatorics We found them to be eigenfunctions of the most general operator of st order in Dunkl shifts (discrete)

The analog of the Askey scheme for Dunkl or is as follows; we call it the Bannai-Ito scheme polynomials

The analog of the Askey scheme for Dunkl or is as follows; we call it the Bannai-Ito scheme polynomials We found the para-krawtchouk polynomials in our search for PST chains We now understand better where they fit There are also PST chains associated with other Ops of the BI scheme

Part II: Almost perfect state transfer Question of practical importance: what is the robustness of the properties of theoretically designed chains with PST against errors? Many sources of error: imperfect input/output operations, fabrications defects, additional interactions, systematic biases, etc. Since measurements always have some imprecisions, it practically suffices to require that the transfer probability be as close to as we desire and hence to consider APST Question: Is APST more broadly realized in spin chains and stabler against perturbations?

APST condition: >, such that ( ) φ ( ) < Using the same occupation basis similarly found that the condition and eigenbasis, it is χ ( )=( ) + is also necessary for APST As we saw, this implies that = (mirror symmetry) ω = + ( ) As a result, (and ) is reconstructed from the spectrum with same algorithm as in PST case

Difference between PST and APST resides in the condition on the spectrum Recall: amplitude in : ( )= = ω ω χ ( )χ ( ) = ω χ ( ) = ω ( ) + ( ) i.e. ( ) is almost periodic function amounts to ( ) + φ ( ) since ω =

APST condition can be restated: δ >, φ and such that or π + φ < δ (mod π) δ < π + φ + π < δ integers that may depend on Condition on spectrum of for APST Question: what properties must the eigenvalues possess to ensure that and integers can be found so that this condition is verified?

In other words: given the set of real numbers = π φ, when is it possible to find values of for which is approximated with any accuracy in terms of integers by π? This is where Diophantine approximation comes in Definition A set α, =, of reals are linearly independent over the field of rationals if for any, the only rational values of such that are = = = α + α + + α =

The theorem of Kronecker is perfectly suited to answer our question which is: given δ and φ, when are there a and such that π + φ + π < δ Version 1: Assume that the reals, = are linearly independent over rational numbers. Let be fixed arbitrary reals. Then δ, and integers such that π < δ = Conclusion: provided that the are linearly independent, for every φ it is possible to find a time such that ( ) is as close to as we wish

In many cases, the relations are NOT linearly independent; i.e. exist ( ) + ( ) + + ( ) = = () where ( ) are integers ( integers equivalent to rationals) In such cases we can use the (generalized) Kronecker theorem Theorem Assume that the reals, = are all distinct and that there are non-trivial relations of type (), then π < δ holds δ if and only if the reals satisfy ( ) + ( ) + + ( ) (mod π)

Necessary and sufficient conditions for APST Let be the (assumed) distinct eigenvalues of -excitation restriction of If there are 6 relations ( ) + + ( ) = = 6 with ( ) = for some for every APST occurs if and only if 1 is mirror symmetric = 2 The linear relations = are compatible = = + ( ) (π φ) = (mod π) =

Spectral surgery and APST Almost obvious that APST survives spectral surgery Weights have same form as for PST Appropriate Christoffel transforms preserve their class and positivity What happens with the spectrum? If the were linearly independent over rationals, a reduced set will have the same property If there are linear relations, all that can happen under the removal of levels is that some of these relations will disappear In fact, spectral surgery could generate chains with APST from chains without the property

Special case: the uniform spin chain = σ σ + + σ σ + = = and = and =...... This matrix is easily diagonalized (with Chebyshev polynomials) and the spectrum is ( + )π = + It is checked that PST occurs only for 6, i.e. for at most four spins. ( Hence the interest in the non-uniform chains)

APST, if it is to occur, will put constraints on + the number of spins which is the only parameter Godsil et al. have proved using the Kronecker theorem that APST occurs if and only if + = or + = where is a prime or if + = Those are the values of for which the are linearly independent over the rationals Occurrence of APST in testing However, waiting time chain can be related to primality grows with Puts in question the practicality of APST

Special case: the para-krawtchouk chain An example where state transfer can be achieved with arbitrary high fidelity in finite time = + = + γ = ( )/ PST for γ = /, odd Clearly we have the relations = + = + So, to have APST, the conditions = + = + must be satisfied for φ = (mod π) = π φ Compatible for

γ = ζ + ζ + ζ + If γ is irrational no further conditions on, and APST will occur Possible to estimate waiting time for given accuracy with Diophantine approximation methods Expand irrational γ in continued fractions with ζ integers The convergents / are rational numbers which provide approximations of γ We can choose an infinite set of convergents (intermediate if necessary) { / / } such that is odd

Let = γ It is known that < Choose = π Substitute in the amplitude ( ): ( )= ω ( ) + = and + = + γ, odd

We get ( )= ( )/ = ( )/ = ( )/ ω π ( ) + + = ( )/ ω + πγ = = ω + ω + ( +γ)π ( ) + + πγ = π( + / ) = π (since is odd) Thus with ( )/ = ω = ( )/ = ω + = / ( )= ( + π ) ( ) = (π / ) π

Consider γ = = + + + + = First appropriate convergents = with odd It is checked that the rd convergent / already give an accuracy of % for a waiting time = π

Conclusions Good idea of the design of quantum wires with Study the conditions for PST and APST. Formulation as inverse spectral problem Role of OP theory and Diophantine approximation Fertilization of OP theory Still many questions: other chains, higher dimensions (lattices), inverse spectral problem for block 3-diagonal matrices, stability of APST under perturbations, etc. spin chains