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Quantum Computation with Bose-Einstein Condensation and Capable of Solving NP-Complete and #P Problems

Yu Bo Shi

Published Oct 15, 1999• 0 citations
OpenAlex

Abstract

It is proposed that quantum computation can be implemented on the basis of macroscopic quantum coherence of a many-body system, especially the Bose-Einstein condensation. Since a Bose-Einstein condensate is described by a non-linear Schrödinger equation, and the non-linearity is tunable, in principle one may build a quantum computer composed of both linear and non-linear gates. Consequently NP-complete and #P problems can be solved. This idea is illustrated by representing the qubit as the atomic Bose-Einstein condensate trapped in a double-well potential.

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It is proposed that quantum computation can be implemented on the basis of macroscopic quantum coherence of a many-body system, especially the Bose-Einstein condensation. Since a Bose-Einstein condensate is described by a non-linear Schrödinger equation, and the non-linearity is tunable, in principle one may build a quantum computer composed of both linear and non-linear gates. Consequently NP-complete and #P problems can be solved. This idea is illustrated by representing the qubit as the atomic Bose-Einstein condensate trapped in a double-well potential.
For a Bose condensate in a double-well potential or with two Josephson-coupled internal states, the condensate wavefunction is a superposition. Here we consider coupling two such Bose condensates, and suggest the existence of a joint condensate wavefunction, which is in general a superposition of all products of the bases condensate wavefunctions of the two condensates. The corresponding many-body state is a product of such superposed wavefunctions, with appropriate symmetrization. These states may be potentially useful for quantum computation. There may be robustness and stability due to macroscopic occupation of a same single particle state. The nonlinearity of the condensate wavefunction due to particle–particle interaction may be utilized to realize nonlinear quantum computation, which was suggested to be capable of solving NP-complete problems.
An approach to the solution of NP-complete problems based on quantum computing and chaotic dynamics is proposed. We consider the satisfiability problem and argue that the problem, in principle, can be solved in polynomial time if we combine the quantum computer with the chaotic dynamics amplifier based on the logistic map. We discuss a possible implementation of such a chaotic quantum computation by using the atomic quantum computer with quantum gates described by the Hartree-Fock equations. In this case, in principle, one can build not only standard linear quantum gates but also nonlinear gates and moreover they obey to Fermi statistics. This new type of entaglement related with Fermi statistics can be interesting also for quantum communication theory.
We analyze quantum computation using quantum information stored on two component Bose-Einstein condensates (BECs). We construct a general framework for quantum algorithms to be executed using the collective states of the BECs. The use of BECs allows for an increase of energy scales via bosonic enhancement, resulting in two-qubit gate operations that can be performed at a time reduced by a factor of NNN, where NNN is the number of bosons in the BEC. We illustrate the scheme by an application to Grover's algorithm, and discuss possible experimental implementations and decoherence effects.
Quantum computation using qubits made of two component Bose-Einstein condensates (BECs) is analyzed. We construct a general framework for quantum algorithms to be executed using the collective states of the BECs. The use of BECs allows for an increase of energy scales via bosonic enhancement, resulting in two qubit gate operations that can be performed at a time reduced by a factor of N, where N is the number of bosons per qubit. We illustrate the scheme by an application to Deutsch-s and Grover-s algorithms, and discuss possible experimental implementations. Decoherence effects are analyzed under both general conditions and for the experimental implementation proposed.
We present a theoretical formulation of trapped, dilute Bose-Einstein condensates (BEC's) at zero temperature based on ordinary Schr\"odinger quantum mechanics. By a judicious choice of coordinates and of a variational trial wave function we reduce the many-atom problem to a linear Schr\"odinger equation that is easier to handle and interpret than the usual nonlinear Schr\"odinger equation of BEC theory. Ordinary quantum mechanics then reproduces, semiquantitatively, many of the main features of zero-temperature BEC, including the critical number of atoms in a condensate with negative scattering length. The procedure is similar in results, but completely different in spirit, to recent variational approaches to solving the nonlinear Schr\"odinger equation. Moreover, the present procedure represents a step in a systematic alternative method for computing quantitatively accurate wave functions for trapped bosonic atoms.
Photon Bose-Einstein condensates (pBECs) are a platform of interest for both the study of fundamental phenomena in many-body physics and the implementation of next-generation analog simulators for solving optimization problems. In our physical system, the network of coupled pBECs in a microcavity setup is created by individually laser-pumping trap sites that are formed by permanent nanostructuring and connected via waveguides to their nearest neighbours. Due to the roughness and non-flat landscape of the microcavity mirrors, trap sites are excited at different frequencies, making the relative phases undefined. This is crucial for finding the ground state of a network of pBECs - a problem that is NP-complete, which makes this platform a promising candidate for analog optimization problem solver. We propose an approach for in-situ coherence tuning using a permanent nanostructuring technique with active feedback on excited trap site frequencies and report our preliminary results on the creation of an uncoupled coherent network of pBECs using this approach.