Perfect State Transfer in Quantum Walks on Graphs

2011 ◽  
Vol 8 (3) ◽  
pp. 422-433 ◽  
Author(s):  
Vivien M. Kendon ◽  
Christino Tamon
Keyword(s):  
Quantum Walks ◽  
Perfect State ◽  
State Transfer ◽  
Perfect State Transfer

scholarly journals Quantum walks on graphs of the ordered Hamming scheme and spin networks

2019 ◽  
Vol 7 (1) ◽  
Author(s):  
Hiroshi Miki ◽  
Satoshi Tsujimoto ◽  
Luc Vinet
Keyword(s):  
Magnetic Fields ◽  
Quantum Walks ◽  
Perfect State ◽  
Spin Networks ◽  
Krawtchouk Polynomials ◽  
Energy Eigenstates ◽  
State Transfer ◽  
Perfect State Transfer ◽  
Local Magnetic Fields ◽  
Basis Vectors

It is shown that the hopping of a single excitation on certain triangular spin lattices with non-uniform couplings and local magnetic fields can be described as the projections of quantum walks on graphs of the ordered Hamming scheme of depth 2. For some values of the parameters the models exhibit perfect state transfer between two summits of the lattice. Fractional revival is also observed in some instances. The bivariate Krawtchouk polynomials of the Tratnik type that form the eigenvalue matrices of the ordered Hamming scheme of depth 2 give the overlaps between the energy eigenstates and the occupational basis vectors.

scholarly journals PERFECT STATE TRANSFER, GRAPH PRODUCTS AND EQUITABLE PARTITIONS

2011 ◽  
Vol 09 (03) ◽  
pp. 823-842 ◽  
Author(s):  
YANG GE ◽  
BENJAMIN GREENBERG ◽  
OSCAR PEREZ ◽  
CHRISTINO TAMON
Keyword(s):  
Continuous Time ◽  
Quantum Walks ◽  
Perfect State ◽  
Graph Products ◽  
State Transfer ◽  
Time Quantum ◽  
Perfect State Transfer

We describe new constructions of graphs which exhibit perfect state transfer on continuous-time quantum walks. Our constructions are based on generalizations of the double cones and variants of the Cartesian graph products (which include the hypercube). We also describe a generalization of the path collapsing argument (which reduces questions about perfect state transfer to simpler weighted multigraphs) for graphs with equitable distance partitions.

scholarly journals Periodicity and perfect state transfer in quantum walks on variants of cycles

2014 ◽  
Vol 14 (5&6) ◽  
pp. 417-438
Author(s):  
Katharine E. Barr ◽  
Tim J. Proctor ◽  
Daniel Allen ◽  
Viv M. Kendon
Keyword(s):  
Discrete Time ◽  
Initial Conditions ◽  
Quantum Walks ◽  
High Fidelity ◽  
Perfect State ◽  
Initial State ◽  
State Transfer ◽  
Time Quantum ◽  
Perfect State Transfer ◽  
Very High

We systematically investigated perfect state transfer between antipodal nodes of discrete time quantum walks on variants of the cycles $C_4$, $C_6$ and $C_8$ for three choices of coin operator. Perfect state transfer was found, in general, to be very rare, only being preserved for a very small number of ways of modifying the cycles. We observed that some of our useful modifications of $C_4$ could be generalised to an arbitrary number of nodes, and present three families of graphs which admit quantum walks with interesting dynamics either in the continuous time walk, or in the discrete time walk for appropriate selections of coin and initial conditions. These dynamics are either periodicity, perfect state transfer, or very high fidelity state transfer. These families are modifications of families known not to exhibit periodicity or perfect state transfer in general. The robustness of the dynamics is tested by varying the initial state, interpolating between structures and by adding decoherence.

scholarly journals Perfect state transfer on signed graphs

2013 ◽  
Vol 13 (5&6) ◽  
pp. 511-530
Author(s):  
John Brown ◽  
Chris Godsil ◽  
Devlin Mallory ◽  
Abigail Raz ◽  
Christino Tamon
Keyword(s):  
Regular Graph ◽  
Complete Graph ◽  
Quantum Walk ◽  
Double Cover ◽  
Quantum Walks ◽  
Signed Graph ◽  
Perfect State ◽  
Signed Graphs ◽  
State Transfer ◽  
Perfect State Transfer

We study perfect state transfer of quantum walks on signed graphs. Our aim is to show that negative edges are useful for perfect state transfer. First, we show that the signed join of a negative $2$-clique with any positive $(n,3)$-regular graph has perfect state transfer even if the unsigned join does not. Curiously, the perfect state transfer time improves as $n$ increases. Next, we prove that a signed complete graph has perfect state transfer if its positive subgraph is a regular graph with perfect state transfer and its negative subgraph is periodic. This shows that signing is useful for creating perfect state transfer since no complete graph (except for the $2$-clique) has perfect state transfer. Also, we show that the double-cover of a signed graph has perfect state transfer if the positive subgraph has perfect state transfer and the negative subgraph is periodic.Here, signing is useful for constructing unsigned graphs with perfect state transfer. Finally, we study perfect state transfer on a family of signed graphs called the exterior powers which is derived from a many-fermion quantum walk on graphs.

scholarly journals Perfect state transfer on quotient graphs

2012 ◽  
Vol 12 (3&4) ◽  
pp. 293-313
Author(s):  
Rachel Bachman ◽  
Eric Fredette ◽  
Jessica Fuller ◽  
Michael Landry ◽  
Michael Opperman ◽  
...  
Keyword(s):  
Cartesian Product ◽  
Quantum Walk ◽  
Physical Review ◽  
Quantum Walks ◽  
Discrete Mathematics ◽  
Perfect State ◽  
Equitable Partition ◽  
Quotient Graphs ◽  
State Transfer ◽  
Perfect State Transfer

We prove new results on perfect state transfer of quantum walks on quotient graphs. Since a graph G has perfect state transfer if and only if its quotient G/\pi, under any equitable partition \pi, has perfect state transfer, we exhibit graphs with perfect state transfer between two vertices but which lack automorphism swapping them. This answers a question of Godsil (Discrete Mathematics 312(1):129-147, 2011). We also show that the Cartesian product of quotient graphs \Box_{k} G_{k}/\pi_{k} is isomorphic to the quotient graph \Box_{k} G_{k}/\pi, for some equitable partition \pi. This provides an algebraic description of a construction due to Feder (Physical Review Letters 97, 180502, 2006) which is based on many-boson quantum walk.

Laplacian Perfect State Transfer in Extended Neighborhood Coronas

2021 ◽  
Vol 37 (12) ◽  
pp. 1921-1932
Author(s):  
Yi Peng Li ◽  
Xiao Gang Liu ◽  
Sheng Gui Zhang
Keyword(s):  
Perfect State ◽  
State Transfer ◽  
Perfect State Transfer

scholarly journals Perfect state transfer on abelian Cayley graphs

2019 ◽  
Vol 563 ◽  
pp. 331-352 ◽  
Author(s):  
Ying-Ying Tan ◽  
Keqin Feng ◽  
Xiwang Cao
Keyword(s):  
Cayley Graphs ◽  
Perfect State ◽  
State Transfer ◽  
Perfect State Transfer

scholarly journals Perfect state transfer in long-range interacting spin chains

2008 ◽  
Vol 78 (2) ◽  
Author(s):  
Giulia Gualdi ◽  
Vojtech Kostak ◽  
Irene Marzoli ◽  
Paolo Tombesi
Keyword(s):  
Long Range ◽  
Spin Chains ◽  
Perfect State ◽  
State Transfer ◽  
Perfect State Transfer
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