Systematic Analysis of Majorization in Quantum Algorithms

Motivated by the need to uncover some underlying mathematical structure of optimal quantum computation, we carry out a systematic analysis of a wide variety of quantum algorithms from the majorization theory point of view. We conclude that step-by-step majorization is found in the known instances of fast and efficient algorithms, namely in the quantum Fourier transform, in Grover's algorithm, in the hidden affine function problem, in searching by quantum adiabatic evolution and in deterministic quantum walks in continuous time solving a classically hard problem. On the other hand, the optimal quantum algorithm for parity determination, which does not provide any computational speed-up, does not show step-by-step majorization. Lack of both speed-up and step-by-step majorization is also a feature of the adiabatic quantum algorithm solving the 2-SAT ``ring of agrees'' problem. Furthermore, the quantum algorithm for the hidden affine function problem does not make use of any entanglement while it does obey majorization. All the above results give support to a step-by-step Majorization Principle necessary for optimal quantum computation.Comment: 15 pages, 14 figures, final versio

Adiabatic quantum computation and quantum phase transitions

We analyze the ground state entanglement in a quantum adiabatic evolution algorithm designed to solve the NP-complete Exact Cover problem. The entropy of entanglement seems to obey linear and universal scaling at the point where the mass gap becomes small, suggesting that the system passes near a quantum phase transition. Such a large scaling of entanglement suggests that the effective connectivity of the system diverges as the number of qubits goes to infinity and that this algorithm cannot be efficiently simulated by classical means. On the other hand, entanglement in Grover's algorithm is bounded by a constant.Comment: 5 pages, 4 figures, accepted for publication in PR

Enhanced energy transfer in a Dicke quantum battery

We theoretically investigate the enhancement of the charging power in a Dicke quantum battery which consists of an array of $N$ two-level systems (TLS) coupled to a single mode of cavity photons. In the limit of small $N$, we analytically solve the time evolution for the full charging process. The eigenvectors of the driving Hamiltonian are found to be pseudo-Hermite polynomials and the evolution is thus interpreted as harmonic oscillator like behaviour. We find that there exists a universal flip duration in this process, regardless to the number of TLSs inside the cavity. Then we demonstrate that the average charging power when using a collective protocol is $\sqrt{N}$ times larger than the parallel charging protocol as for transferring the same amount of energy. Unlike previous studies, we point out that such quantum advantage does not originate from entanglement but dues to the coherent cooperative interactions among the TLSs. Our results provide intuitive quantitative insight into the dynamic charging process of a Dicke battery and can be observed under realistic experimental conditions.Comment: 8 Pages, 3 figure