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arXiv:1205.4294 · Physical Review A

Singlet-state creation and universal quantum computation in NMR using a genetic algorithm

Manu Veliparambil Subrahmanian, Anil Kumar

Published Aug 22, 2012• 24 citations
arXivHTMLPDF

Abstract

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.

DOI
10.1103/physreva.86.022324

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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.
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…
This paper proposes a novel evolutionary computing method called a genetic quantum algorithm (GQA). GQA is based on the concept and principles of quantum computing such as qubits and superposition of states. Instead of binary, numeric, or symbolic representation, by adopting qubit chromosome as a representation GQA can represent a linear superposition of solutions due to its probabilistic representation. As genetic operators, quantum gates are employed for the search of the best solution. Rapid convergence and good global search capability characterize the performance of GQA. The effectiveness and the applicability of GQA are demonstrated by experimental results on the knapsack problem, which is a well-known combinatorial optimization problem. The results show that GQA is superior to other genetic algorithms using penalty functions, repair methods and decoders.
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.
A robust control over quantum dynamics is of paramount importance for quantum technologies. Many of the existing control techniques are based on smooth Hamiltonian modulations involving repeated calculations of basic unitaries resulting in time complexities scaling rapidly with the length of the control sequence. Here we show that bang-bang controls need one-time calculation of basic unitaries and hence scale much more efficiently. By employing a global optimization routine such as the genetic algorithm, it is possible to synthesize not only highly intricate unitaries, but also certain nonunitary operations. We demonstrate the unitary control through the implementation of the optimal fixed-point quantum search algorithm in a three-qubit nuclear magnetic resonance (NMR) system. Moreover, by combining the bang-bang pulses with the crusher gradients, we also demonstrate nonunitary transformations of thermal equilibrium states into effective pure states in three- as well as five-qubit NMR systems.
This paper presents a comparative study regarding the evolutionary design of quantum operators in the form of unitary matrices. Three different techniques allowing the generation of unitary matrices are investigated whose parameters are tuned by means of evolutionary algorithms. Such problem may be considered as a numerical optimisation task because the unitary matrices are represented by complex numbers with real-valued coefficients. Specifically, Genetic Algorithm and Evolution Strategy are applied, each in four different setups, and evaluated on three case studies: the 2-qubit Controlled-NOT gate, 3-qubit entanglement operator and 4-qubit detector of an element with the maximum amplitude. The evolutionary algorithms in combination with the given representation techniques are evaluated in order to determine the abilities of various experimental setups to solve the given tasks. The main contribution is the utilisation of QR decomposition, a technique for generating unitary matrices from theoretically any sequence of complex numbers, that has been applied for the first time in this paper for the evolution of quantum operators. It will be demonstrated that the QR decomposition exhibits a good performance especially in solving more complex quantum operators in comparison with other representations.
We propose a genetic-algorithm-based method to find the unitary transformations for any desired quantum computation. We formulate a simple genetic algorithm by introducing the “genetic parameter vector” of the unitary transformations to be found. In the genetic algorithm process, all components of the genetic parameter vectors are supposed to evolve to the solution parameters of the unitary transformations. We apply our method to find the optimal unitary transformations and to generalize the corresponding quantum algorithms for a realistic problem, the one-bit oracle decision problem, or the often-called Deutsch problem. By numerical simulations, we can faithfully find the appropriate unitary transformations to solve the problem by using our method. We analyze the quantum algorithms identified by the found unitary transformations and generalize the variant models of the original Deutsch’s algorithm.