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Optimization Of NMR Experiments Using Genetic Algorithm : Applications In Quantum Infomation Processing, Design Of Composite Operators And Quantitative Experiments

V. S. Manu

Published May 21, 2017• 0 citations
OpenAlex

Abstract

Genetic algorithms (GA) are stochastic global search methods based on the mechanics of natural biological evolution, proposed by John Holland in 1975. Here in this thesis, we have exploited possible utilities of Genetic Algorithm optimization in Nuclear Magnetic Resonance (NMR) experiments. We have performed (i ) Pulse sequence generation and optimization for NMR Quantum Information Processing, (ii ) efficient creation of NOON states, (iii ) Composite operator design and (iv ) delay optimization for refocused quantitative INEPT. We have generated time optimal as well as robust pulse sequences for popular quantum gates. A Matlab package is developed for basic Target unitary operator to pulse sequence optimization and is explained with an example. Chapter 1 contains a brief introduction to NMR, Quantum computation and Genetic algorithm optimization. Experimental unitary operator decomposition using Genetic Algorithm is explained in Chapter 2. Starting from a two spin homonu- clear system (5-Bromofuroic acid), we have generated hard pulse sequences for performing (i ) single qubit rotation, (ii ) controlled NOT gates and (iii ) pseudo pure state creation, which demonstrates universal quantum computation in such systems. The total length of the pulse sequence for the single qubit rotation of an angle π/2 is less than 500µs, whereas the conventional method (using a selective soft pulse) would need a 2ms shaped pulse. This substantial shortening in time can lead to a significant advantage in quantum circuits. We also demonstrate the creation of Long Lived Singlet State and other Bell states, directly from thermal equilibrium state, with the shortest known pulse sequence. All the pulse sequences generated here are generic i.e., independent of the system and the spectrometer. We further generalized this unitary operator decomposition technique for a variable operators termed as Fidelity Profile Optimization (FPO) (Chapter 3) and performed quantum simulations of Hamiltonian such as Heisenberg XY interaction and Dzyaloshinskii-Moriya interaction. Exact phase (φ) dependent experimental unitary decompositions of Controlled-φ and Controlled Controlled-φ are solved using first order FPO. Unitary operator decomposition for experimental quantum simulation of Dzyaloshinskii-Moriya interaction in the presence of Heisenberg XY interaction is solved using second order FPO for any relative strengths of interactions (γ) and evolution time (τ ). Experimental gate time for this decomposition is invariant under γ or τ , which can be used for relaxation independent studies of the system dynamics. Using these decompositions, we have experimentally verified the entanglement preservation mechanism suggested by Hou et al. [Annals of Physics, 327 292 (2012)]. NOON state or Schrodinger cat state is a maximally entangled N qubit state with superposition of all individual qubits being at |0 and being at |1 . NOON states have received much attention recently for their high precession phase measurements, which enables the design of high sensitivity…

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Acta Physica Sinica•Jan 1, 2015•0 citations
Genetic algorithms (GA) are stochastic global search methods based on the
mechanics of natural biological evolution, proposed by John Holland in 1975. Here
in this thesis, we have exploited possible utilities of Genetic Algorithm optimization
in Nuclear Magnetic Resonance (NMR) experiments. We have performed
(i ) Pulse sequence generation and optimization for NMR Quantum Information
Processing, (ii ) efficient creation of NOON states, (iii ) Composite operator design
and (iv ) delay optimization for refocused quantitative INEPT. We have generated
time optimal as well as robust pulse sequences for popular quantum gates. A
Matlab package is developed for basic Target unitary operator to pulse sequence
optimization and is explained with an example.
Chapter 1 contains a brief introduction to NMR, Quantum computation and Genetic
algorithm optimization. Experimental unitary operator decomposition using
Genetic Algorithm is explained in Chapter 2. Starting from a two spin homonu-
clear system (5-Bromofuroic acid), we have generated hard pulse sequences for
performing (i ) single qubit rotation, (ii ) controlled NOT gates and (iii ) pseudo
pure state creation, which demonstrates universal quantum computation in such
systems. The total length of the pulse sequence for the single qubit rotation of an
angle π/2 is less than 500µs, whereas the conventional method (using a selective
soft pulse) would need a 2ms shaped pulse. This substantial shortening in time
can lead to a significant advantage in quantum circuits. We also demonstrate the
creation of Long Lived Singlet State and other Bell states, directly from thermal
equilibrium state, with the shortest known pulse sequence. All the pulse sequences
generated here are generic i.e., independent of the system and the spectrometer.
We further generalized this unitary operator decomposition technique for a variable
operators termed as Fidelity Profile Optimization (FPO) (Chapter 3) and
performed quantum simulations of Hamiltonian such as Heisenberg XY interaction
and Dzyaloshinskii-Moriya interaction. Exact phase (φ) dependent experimental
unitary decompositions of Controlled-φ and Controlled Controlled-φ are solved
using first order FPO. Unitary operator decomposition for experimental quantum
simulation of Dzyaloshinskii-Moriya interaction in the presence of Heisenberg XY
interaction is solved using second order FPO for any relative strengths of interactions
(γ) and evolution time (τ ). Experimental gate time for this decomposition
is invariant under γ or τ , which can be used for relaxation independent studies of
the system dynamics. Using these decompositions, we have experimentally verified
the entanglement preservation mechanism suggested by Hou et al. [Annals of
Physics, 327 292 (2012)].
NOON state or Schrodinger cat state is a maximally entangled N qubit state
with superposition of all individual qubits being at |0 and being at |1 . NOON
states have received much attention recently for their high precession phase
measurements, which enables the design of high sensitivity…
The experimental implementation of a quantum algorithm requires the decomposition of unitary operators. Here we treat unitary-operator decomposition as an optimization problem, and use a genetic algorithm---a global-optimization method inspired by nature's evolutionary process---for operator decomposition. We apply this method to NMR quantum information processing, and find a probabilistic way of performing universal quantum computation using global hard pulses. We also demonstrate the efficient creation of the singlet state (a special type of Bell state) directly from thermal equilibrium, using an optimum sequence of pulses.
Experimental quantum simulation of a Hamiltonian HHH requires unitary operator decomposition (UOD) of its evolution unitary $U=\text{exp}(\ensuremath{-}iHt)$ in terms of native unitary operators of the experimental system. Here, using a genetic algorithm, we numerically evaluate the most generic UOD (valid over a continuous range of Hamiltonian parameters) of the unitary operator U,U,U, termed fidelity-profile optimization. The optimization is obtained by systematically evaluating the functional dependence of experimental unitary operators (such as single-qubit rotations and time-evolution unitaries of the system interactions) to the Hamiltonian (HHH) parameters. Using this technique, we have solved the experimental unitary decomposition of a controlled-phase gate (for any phase value), the evolution unitary of the Heisenberg XY interaction, and simulation of the Dzyaloshinskii-Moriya (DM) interaction in the presence of the Heisenberg XY interaction. Using these decompositions, we studied the entanglement dynamics of a Bell state in the DM interaction and experimentally verified the entanglement preservation procedure of Hou et al. [Ann. Phys. (N.Y.) 327, 292 (2012)] in a nuclear magnetic resonance quantum information processor.
Quantum simulations, where controllable quantum systems are used to simulate other quantum systems, originally proposed by Richard Feynman, are one of the most remarkable applications of quantum information science. Compared to computation, quantum simulations require much less number of qubits for the m to be practical. In the work described in this thesis, we have performed a few quantum simulations of small quantum systems using Nuclear Magnetic Resonance(NMR) techniques. These simulations have been used to experimentally demonstrate the underlying interesting quantum protocols. All the experiments presented have been carried out using liquid-state or liquid crystal NMR. Numerical pulse optimization techniques have been utilized in some of the experiments, to achieve better control over the spin systems. The first chapter contains “Introduction” to quantum information processing, NMR, and numerical pulse optimization techniques. In chapter 2, we describe quantum simulation of a 3-spin Heisenberg-XY spin chain having only nearest neighbour interactions. Recently, spin chains having pre-engineered short-range interactions have been proposed to efficiently transfer quantum information between different parts of a quantum information processor. Other important proposals involving these spin chains include generating entangled states and universal quantum computation. However, such engineered interactions do not occur naturally in any system. In such a scenario, the experimental viability of these proposals can be tested by simulating the spin chains in other controllable quantum systems. In this work, we first theoretically study the time evolution of bipartite and tripartite entanglement measures for a 3-spin open ended XY spin chain. Then, by simulating the XY interactions in a 3-spin nuclear spin system, we experimentally generate, (i)a bipartite maximally(pseudo-)entangled state(Bell state) between end qubits, and(ii) multipartite(pseudo-)entangled states(Wand GHZ states),starting from separable pseudo-pure states. Bell state has been generated by using only the natural unitary evolution of the XY spin chain. W-state and GHZ-state have been generated by applying a single-qubit rotation to the second qubit, and a global rotation of all the three qubits respectively after the unitary evolution of the spin chain. In chapter 3, we simulate a 3-spin quantum transverse Ising spin system in a triangular configuration, and show that multipartite quantum correlations can be used to distinguish between the frustrated and non-frustrated regimes in the ground state of this spin system. The ground state of the spin system has been prepared by using adiabatic state preparation method. Gradient ascent pulse engineering technique has been utilized to efficiently realize the adiabatic evolution of the spin system. To analyse the experimental ground state of the system, we employ two different multipartite quantum correlation measures, generated from monogamy studies of bipartite quantum correlations. Chapter 4 contains a digital…
With the development of quantum information and quantum computation science, quantum information processor has been widely used in different areas such as quantum simulation, quantum computation and quantum metrology and so on. To make quantum computer come true, we need to increase the number of controllable qubits of the system and improve the controllability to perform complex quantum manipulation. As a good experimental testbed for quantum information processing, nuclear magnetic resonance (NMR) spin system provides rich and sophisticated quantum control methods. In recent years a lot of multi-qubit experiments have been performed on the platform and a series of experimental technologies have been developed. In this paper, we firstly explain the difficulties of multi-qubit NMR experiments. Then by focusing on the experiment of 7-qubit labelled pseudo-pure state preparation and other relevant experiments, we review the technologies in multi-qubit experiments. Using the radio frequency selective method, the inhomogeneities of the radio frequency pulses are reduced and the spectral resolution is improved. After performing 1/2 spin selective sequence, we can regard the three methyl protons in the sample of crotonic acid as a single 1/2 spin nucleus and treat the whole molecule as a 7-qubit quantum information processor. We utilize Gauss pulses, Hermite pulses, composite pulses and gradient ascent pulse engineering (GRAPE) pulses to implement basic /2 and rotation operations. The GRAPE pulses are calculated by subspace GRAPE program to speed up the computation greatly. The errors of the basic pulses caused by chemical shift and J-coupling evolution can be estimated by the program of pulse compilation. It divides the errors of the pulses into a series of post-errors and pre-errors. A program of sequence compilation is used to eliminate the accumulated error of the whole pulse sequence, reduce the number of pulses and optimize the experimental duration. A variety of methods of quantum state tomography have been proposed to improve the efficiency of reading out information about quantum state. As an experimental example, we combine the above experimental technologies and perform the experiment of 7-qubit labelled pseudo-pure state preparation by using the method of cat state preparation. The sequence of cat state preparation consists of three steps:encoding procedure, phase cycling and decoding procedure. We use 14 experiments to realize the phase cycling and acquire the final 7-qubit labelled pseudo-pure state. The total duration of experimental sequence is about 132 ms. All the readout spectra have the similar shapes to the theoretical expectations. Finally we give an outlook for further research in this direction.
We develop a new class of genetic algorithm that computationally determines efficient pulse sequences to implement a quantum gate U in a three-qubit system. The method is shown to be quite general, and the same algorithm can be used to derive efficient sequences for a variety of target matrices. We demonstrate this by implementing the inversion-on-equality gate efficiently when the spin-spin coupling constants J12=J23=JJ_{12}=J_{23}=JJ12​=J23​=J and J13=0J_{13}=0J13​=0. We also propose new pulse sequences to implement the Parity gate and Fanout gate, which are about 50% more efficient than the previous best efforts. Moreover, these sequences are shown to require significantly less RF power for their implementation. The proposed algorithm introduces several new features in the conventional genetic algorithm framework. We use matrices instead of linear chains, and the columns of these matrices have a well defined hierarchy. The algorithm is a genetic algorithm coupled to a fast local optimizer, and is hence a hybrid GA. It shows fast convergence, and running on a MATLAB platform takes about 20 minutes on a standard personal computer to derive efficient pulse sequences for any target 8X8 matrix UUU.
Quantum Fourier transform (QFT) is a quantum analogue of the classical discrete Fourier transform. It is a fundamental quantum gate in quantum algorithms which has an exponential advantage over the classical computation and has been excessively studied. Normally, an n-qubit quantum Fourier transform could be resolved into the tensor product of n single-qubit operations, and each operation could be implemented by a Hadamard gate and a controlled phase gate. Then the complexity of an n-qubit QFT is of order O(n2). To reduce the complexity of quantum operations, optimal control (OC) method has recently been used successfully to find the minimum time for implementing a quantum operation. Up to now, two types of quantum optimal control methods have been presented, i.e. analytical and numerical methods. The analytical approach is to change the problem of efficient synthesis of unitary transformations into the geometrical one of finding the shortest paths. Numerical optimal control procedures are based on the gradient methods (GRAPE, Gradient Ascent Pulse Engineering) and Krotov methods. Notable application mainly focus on nuclear magnetic resonance fields, including imaging, liquid-state NMR, solid-state NMR, and NMR quantum computation. One obvious advantage of optimal control NMR quantum computation is that the OC unitary evolution transformation pulse sequences are normally shorter than the conventional corresponding ones. Here we use the optimal control method to find the minimum duration for implementing QFT quantum gate. A linear spin chain with nearest-neighbor Ising interaction is used to find the optimization. And the optimized pulse sequence is experimentally demonstrated on an NMR quantum information processor. By using optimal control method with numerical calculation, a three-qubit QFT in an indirect-linear-coupling chain system is optimized. The duration of the OC QFT is obviously shorter than that of conventional approaches. The OC pulse sequence has been experimentally implemented on a liquid-state NMR spectrometer. To verify the optimally controlled pulse sequence for the three-qubit QFT, different initial states are assumed. The accuracy of the OC pulse sequence could be demonstrated by the consistency of theoretical simulation spectra with the experimental results. The good consistency between the simulation and the experimental spectra demonstrates that the OC QFT is of high fidelity.