Optimal State Encoding for Quantum Walks and Quantum

Optimal State Encoding for Quantum Walks and Quantum Communication over Spin Systems Henry Haselgrove, School of Physical Sciences, University of Queensland, Australia The Problem: - How can a fixed network of interacting spins be used as a conduit for high-fidelity quantum communication, when only limited external control is available? 1 — MESSAGE ENCODING Alice wishes to send ®|0 i + ¯|1 i to Bob. They each control some subset, A and B, of the spins in a network. Example: A B (graph edges represent fixed interactions between spins) The communication procedure: 2 — DYNAMICAL CONTROL In this scheme Alice and Bob control the interactions on just two spins each. The interaction strengths on the control spins are modulated throughout the communication procedure. We consider two types of “limited external control”: 1) The sender encodes the message onto several spins, or, 2) The sender and receiver modulate the interactions on a just two spins each. (Bulk of the spin network— fixed known interactions, arbitrary graph) BA 1(t) A 1 We assume: -- The Hamiltonian conserves total Z-spin -- The system can be initialised in all-|0 i 1. The state of the system is pre-prepared: Quant-ph/0404152 – submitted to PRA 2. Alice encodes the message onto her spins: BA 2(t) A 2 JA(t) C The message state is placed onto this spin by Alice (encoding of the |1 i message – to be optimised) (encoding of the |0 i message – fixed at |0… 0 i, for convenience) … A 1 A 2 3. The system evolves for time T. If |1 ENCi was chosen well, the state is: 4. Bob decodes: “extended Bob” “extended Alice” C B 2 “phantom” spins B 1 … 1) Imagine that Alice and Bob control many more spins (see figure, left) 2) Find the optimal encoding for |1 i on these spins (see LHS of poster) 3) Simulate the evolution. Use the results to find control functions for actual system above. NB: Steps 2 and 3 can be done efficiently, in a restricted subspace. Alice Bob JA(t) A ´ space of states of form B ´ space of states of form EXAMPLE: The evolution of best encoding |1 ENCi, when Alice and Bob are joined by a Heisenberg chain Total # spins = 300 # control spins = 20 + 20 Avg. fidelity = 0. 99999 NB: j is the coefficient of The message is received by Bob here Method for deriving good control functions: Result: The best choice for |1 ENCi is given by the first rightsingular vector of where PB and PA are the projectors onto: BB 2(t) BB 1(t) B 2 B 1 JB(t) (All fixed bonds have strength 1) Fidelity versus time, no control (JA and JB fixed at 1): EXAMPLE: Irregular XY chain Note: the B(·)(t) functions may be set to zero in this simple example Derived control functions: --- JA(t) --- JB(t) Fidelity versus time, with control: