Exponential speedup of quantum newton optimization algorithm for general polynomials

Quantum optimization algorithms can outperform their classical counterpart and are key in modern technology. The second-order optimization algorithm(the Newton algorithm) is a critical optimization method, speeding up the convergence by employing the second-order derivative of loss functions in addition to their first derivative. Here, we propose a new quantum second-order optimization algorithm for general polynomials with a computational complexity of O(poly(log d)). We use this algorithm to solve the nonlinear equation and learning parameter problems in factorization machines. Numerical simulations show that our new algorithm is faster than its classical counterpart and the first-order quantum gradient descent algorithm. While existing quantum Newton optimization algorithms apply only to homogeneous polynomials, our new algorithm can be used in the case of general polynomials, which are more widely present in real applications.     

Exponential speedup of quantum Newton optimization algorithm for general polynomials

正The gradient-based method plays a central role in optimization problems, and is widely applied in engineering control, financial analysis, weather forecast, and training of machine learning models.     

A simple formula for the average gate fidelity of a quantum dynamical operation

This Letter presents a simple formula for the average fidelity between a unitary quantum gate and a general quantum operation on a qudit, generalizing the formula for qubits found by Bowdrey et al. [Phys. Lett. A 294 (2002) 258]. This formula may be useful for experimental determination of average gate fidelity. We also give a simplified proof of a formula due to Horodecki et al. [Phys. Rev. A 60 (1999) 1888], connecting average gate fidelity to entanglement fidelity.     

Coupling a single atomic quantum bit to a high finesse optical cavity

The quadrupole S(1/2)-D(5/2) optical transition of a single trapped Ca+ ion, well suited for encoding a quantum bit of information, is coherently coupled to the standing wave field of a high finesse cavity. The coupling is verified by observing the ion's response to both spatial and temporal variations of the intracavity field. We also achieve deterministic coupling of the cavity mode to the ion's vibrational state by selectively exciting vibrational state-changing transitions and by controlling the position of the ion in the standing wave field with nanometer precision.     

Security of quantum bit string commitment depends on the information measure

Unconditionally secure nonrelativistic bit commitment is known to be impossible in both the classical and the quantum world. However, when committing to a string of n bits at once, how far can we stretch the quantum limits? In this Letter, we introduce a framework of quantum schemes where Alice commits a string of n bits to Bob, in such a way that she can only cheat on a bits and Bob can learn at most b bits of information before the reveal phase. Our results are twofold: we show by an explicit construction that in the traditional approach, where the reveal and guess probabilities form the security criteria, no good schemes can exist: a + b is at least n. If, however, we use a more liberal criterion of security, the accessible information, we construct schemes where a = 4log2(n) + O(1) and b = 4, which is impossible classically. Our findings significantly extend known no-go results for quantum bit commitment.     

Quantum chaos and random matrix theory for fidelity decay in quantum computations with static imperfections

We determine the universal law for fidelity decay in quantum computations of complex dynamics in presence of internal static imperfections in a quantum computer. Our approach is based on random matrix theory applied to quantum computations in presence of imperfections. The theoretical predictions are tested and confirmed in extensive numerical simulations of a quantum algorithm for quantum chaos in the dynamical tent map with up to 18 qubits. The theory developed determines the time scales for reliable quantum computations in absence of the quantum error correction codes. These time scales are related to the Heisenberg time, the Thouless time, and the decay time given by Fermis golden rule which are well-known in the context of mesoscopic systems. The comparison is presented for static imperfection effects and random errors in quantum gates. A new convenient method for the quantum computation of the coarse-grained Wigner function is also proposed.Received: 13 December 2003, Published online: 18 March 2004PACS: 03.67.Lx Quantum computation - 05.45.Pq Numerical simulations of chaotic systems - 05.45.Mt Quantum chaos; semiclassical methods     

Erasing information and purity of a quantum dot via its spontaneous decay

An analytical solution to the master equation of a system describing a single quantum dot confined in a single-mode microcavity, coupled to its environment, is found. The information loss of the phase space, the purity and the relaxation process are investigated by using the Husimi distribution and its applications. It is found that the spontaneous decay leads to the loss of information of the phase space and the purity. Suggested indicator for the relaxation process is offered by using the Wehrl entropy.     

Environment-induced decay of teleportation fidelity of the one-qubit state

The one-qubit teleportation protocol is reexamined when it is executed in the presence of various decohering environments. The results revealed that this quantum protocol is more robust under the influence of dephasing environment than those under the influence of dissipative or noisy environment. The environment may deprive the quantum advantage of teleportation over purely classical communication in a finite or infinite lifetime, which is dependent on the type of environment. Also we found that except entanglement, the purity of the entangled state resource is also crucial in determining the quality of the teleported state.     

NMR-like control of a quantum bit superconducting circuit

Coherent superpositions of quantum states have already been demonstrated in different superconducting circuits based on Josephson junctions. These circuits are now considered for implementing quantum bits. We report on experiments in which the state of a qubit circuit, the quantronium, is efficiently manipulated using methods inspired from nuclear magnetic resonance (NMR): multipulse sequences are used to perform arbitrary operations, to improve their accuracy, and to fight decoherence.     

Exponential decay of the power spectrum of turbulence

The analyticity on a strip of the solutions of Navier-Stokes equations in 2D is shown to explain the observed fast decay of the frequency power spectrum of the turbulent velocity field. Some subtleties in the application of the Wiener-Khinchine method to turbulence are resolved by showing that the frequency power spectrum of turbulent velocities is in fact a measure exponentially decaying for frequency ±. Our approach also shows that the conventional procedures used in analyzing data in turbulence experiments are valid even in the absence of the ergodic property in the flow.     

Entanglement fidelity of coherent-state teleportation with asymmetric quantum channel

A strategy for teleporting coherent states with the entanglement fidelity is considered in the general case of an asymmetric teleportation scheme. It is shown that the nonbalanced homodyne detection with the subsequent coherent displacement is required to provide the average teleportation fidelity of entanglement.     

Entanglement fidelity of the standard quantum teleportation channel

We consider the standard quantum teleportation protocol where a general bipartite state is used as entanglement resource. We use the entanglement fidelity to describe how well the standard quantum teleportation channel transmits quantum entanglement and give a simple expression for the entanglement fidelity when it is averaged on all input states.     

Exponential decay of connectivities in the two-dimensional ising model

We prove some results concerning the decay of connectivities in the low-temperature phase of the two-dimensional Ising model. These provide the bounds necessary to establish, nonperturbatively, large-deviation properties for block magnetizations in these systems. We also obtain estimates on the rate at which the finite-volume, plus-boundary-condition expectation of the spin at the origin converges to the spontaneous magnetization.On leave from São Paulo University, Brazil.     

Radioactive decay speedup at T=5 K: electron-capture decay rate of (7)Be encapsulated in C(60)

The electron-capture (EC) decay rate of (7)Be in C(60) at the temperature of liquid helium (T=5 K) was measured and compared with the rate in Be metal at T=293 K. We found that the half-life of (7)Be in endohedral C(60) ((7)Be@C(60)) at a temperature close to T=5 K is 52.47+/-0.04 d, a value that is 0.34% faster than that at T=293 K. In this environment, the half-life of (7)Be is nearly 1.5% faster than that inside Be metal at room temperature (T=293 K). We then interpreted our observations in terms of calculations of the electron density at the (7)Be nucleus position inside the C(60); further, we estimate theoretically the temperature dependence (at T=0 K and 293 K) of the electron density at the Be nucleus position in the stable center inside C(60). The theoretical estimates were almost in agreement with the experimental observations.     

Statistical chain decay of fireballs with quantum numbers

The effects of internal quantum number consevation are considered in the statistical chain decay of fireballs. A simple model containing I = 1, G = ?1, S = 0 mesons (ground state and excited pions) and $\text{I =}\text{1}\text{2}\text{, S = 0}$ baryons (ground state and excited nucleons) is investigated in detail: inclusive distributions, multiplicity distributions and semi-inclusive distributions are determined. The methods are completely general and can be used in more general models with any set of states. As examples, a model with ? and π mesons, and nucleons and another one with π and K mesons, and N and Λ baryons are considered.     

Exponential decay of derivatives of covariance operators on the lattice

We prove exponential decay for derivatives of covariance operators on the lattice. This result is obtained by using random walk methods on the v-dimensional lattice and a certain estimate on the generating function of the one-dimensional random walk. The result is useful in the frame of the cluster and mean field expansion of continuous spin models on the lattice.