Pretty good state transfer in qubit chains—The Heisenberg Hamiltonian (original) (raw)
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Pretty good state transfer in discrete-time quantum walks
2021
We establish the theory for pretty good state transfer in discretetime quantum walks. For a class of walks, we show that pretty good state transfer is characterized by the spectrum of certain Hermitian adjacency matrix of the graph; more specifically, the vertices involved in pretty good state transfer must be m-strongly cospectral relative to this matrix, and the arccosines of its eigenvalues must satisfy some number theoretic conditions. Using normalized adjacency matrices, cyclic covers, and the theory on linear relations between geodetic angles, we construct several infinite families of walks that exhibits this phenomenon.
Efficient and perfect state transfer in quantum chains
Journal of Physics A-mathematical and General, 2005
We present a communication protocol for chains of permanently coupled qubits which achieves perfect quantum state transfer and which is efficient with respect to the number chains employed in the scheme. The system consists of MMM uncoupled identical quantum chains. Local control (gates, measurements) is only allowed at the sending/receiving end of the chains. Under a quite general hypothesis on the interaction Hamiltonian of the qubits a theorem is proved which shows that the receiver is able to asymptotically recover the messages by repetitive monitoring of his qubits.
An infinite family of circulant graphs with perfect state transfer in discrete quantum walks
Quantum Information Processing, 2019
We study perfect state transfer in Kendon's model of discrete quantum walks. In particular, we give a characterization of perfect state transfer purely in terms of the graph spectra, and construct an infinite family of 4-regular circulant graphs that admit perfect state transfer. Prior to our work, the only known infinite families of examples were variants of cycles and diamond chains.
Almost perfect state transfer in quantum spin chains
Physical Review A, 2012
The natural notion of almost perfect state transfer (APST) is examined. It is applied to the modelling of efficient quantum wires with the help of XX 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 (PST) and moreover accommodates the unavoidable imperfections. Some examples are discussed.
Factoring Discrete Quantum Walks on Distance Regular Graphs into Continuous Quantum Walks
arXiv: Combinatorics, 2020
We consider a discrete quantum walk, called the Grover walk, on a distance regular graph XXX. Given that XXX has diameter ddd and invertible adjacency matrix, we show that the square of the transition matrix of the Grover walk on XXX is a product of at most ddd commuting transition matrices of continuous quantum walks, each on some distance digraph of the line digraph of XXX.
Perfect State Transfer in Quantum Spin Networks
Physical Review Letters, 2004
We propose a class of qubit networks that admit perfect transfer of any quantum state in a fixed period of time. Unlike many other schemes for quantum computation and communication, these networks do not require qubit couplings to be switched on and off. When restricted to N-qubit spin networks of identical qubit couplings, we show that 2 log 3 N is the maximal perfect communication distance for hypercube geometries. Moreover, if one allows fixed but different couplings between the qubits then perfect state transfer can be achieved over arbitrarily long distances in a linear chain.
Transfer of arbitrary two-qubit states via a spin chain
Physical Review A, 2015
We investigate the fidelity of the quantum state transfer (QST) of two qubits by means of an arbitrary spin-1 2 network, on a lattice of any dimensionality. Under the assumptions that the network Hamiltonian preserves the magnetization and that a fully polarized initial state is taken for the lattice, we obtain a general formula for the average fidelity of the two qubits QST, linking it to the oneand two-particle transfer amplitudes of the spin-excitations among the sites of the lattice. We then apply this formalism to a 1D spin chain with XX-Heisenberg type nearest-neighbour interactions adopting a protocol that is a generalization of the single qubit one proposed in Ref. [Phys. Rev. A 87, 062309 (2013)]. We find that a high-quality two qubit QST can be achieved provided one can control the local fields at sites near the sender and receiver. Under such conditions, we obtain an almost perfect transfer in a time that scales either linearly or, depending on the spin number, quadratically with the length of the chain.
Quantum simulation of perfect state transfer on weighted cubelike graphs
arXiv: Quantum Physics, 2021
A continuous-time quantum walk on a graph evolves according to the unitary operator e−iAte^{-iAt}e−iAt, where AAA is the adjacency matrix of the graph. Perfect state transfer (PST) in a quantum walk is the transfer of a quantum state from one node of a graph to another node with 100100\%100 fidelity. It can be shown that the adjacency matrix of a cubelike graph is a finite sum of tensor products of Pauli XXX operators. We use this fact to construct an efficient quantum circuit for the quantum walk on cubelike graphs. In \cite{Cao2021, rishi2021(2)}, a characterization of integer weighted cubelike graphs is given that exhibit periodicity or PST at time t=pi/2t=\pi/2t=pi/2. We use our circuits to demonstrate PST or periodicity in these graphs on IBM's quantum computing platform~\cite{Qiskit, IBM2021}.
On quantum perfect state transfer in weighted join graphs
2009
We study perfect state transfer on quantum networks represented by weighted graphs. Our focus is on graphs constructed from the join and related graph operators. Some specific results we prove include: • The join of a weighted two-vertex graph with any regular graph has perfect state transfer. This generalizes a result of Casaccino et al. [9] where the regular graph is a complete graph or a complete graph with a missing link. In contrast, the half-join of a weighted two-vertex graph with any weighted regular graph has no perfect state transfer. This implies that adding weights in a complete bipartite graph do not help in achieving perfect state transfer. • A Hamming graph has perfect state transfer between each pair of its vertices. This is obtained using a closure property on weighted Cartesian products of perfect state transfer graphs. Moreover, on the hypercube, we show that perfect state transfer occurs between uniform superpositions on pairs of arbitrary subcubes. This generalizes results of Bernasconi et al. [5] and Moore and Russell [14]. Our techniques rely heavily on the spectral properties of graphs built using the join and Cartesian product operators.
Proceedings of the thirty-third annual ACM symposium on Theory of computing - STOC '01, 2001
We set the ground for a theory of quantum walks on graphsthe generalization of random walks on finite graphs to the quantum world. Such quantum walks do not converge to any stationary distribution, as they are unitary and reversible. However, by suitably relaxing the definition, we can obtain a measure of how fast the quantum walk spreads or how confined the quantum walk stays in a small neighborhood. We give definitions of mixing time, filling time, dispersion time. We show that in all these measures, the quantum walk on the cycle is almost quadratically faster then its classical correspondent. On the other hand, we give a lower bound on the possible speed up by quantum walks for general graphs, showing that quantum walks can be at most polynomially faster than their classical counterparts.