On the diagonalization of quadratic Hamiltonians

A new procedure to diagonalize quadratic Hamiltonians is introduced. We show that one can establish the diagonalization of a quadratic Hamiltonian by changing the frame of reference by a unitary transformation. We give a general method to diagonalize an arbitrary quadratic Hamiltonian and derive a few of the simplest special cases in detail.

A real expectation value of the time-dependent non-Hermitian Hamiltonians

With the aim to solve the time-dependent Schr ̈odinger equation associated to a time-dependent non-Hermitian Hamiltonian, we introduce a unitary transformation that maps the Hamiltonian to a time-independent PT-symmetric one. Consequently, the solution of time-dependent Schrödinger equation becomes easily deduced and the evolution preserves the C(t)PT -inner product, where C(t) is a obtained from the charge conjugation operator C through a time dependent unitary transformation. Moreover, the expectation value of the non-Hermitian Hamiltonian in the C(t)PT normed states is guaranteed to be real. As an illustration, we present a specific quantum system given by a quantum oscillator with time-dependent mass subjected to a driving linear complex time-dependent potential.

An Efficient Improved OGWSBI Algorithm for Accurate Off-Grid DOA Estimation of Coherent Signals

The performance of the weighted sparse Bayesian inference (OGWSBI) algorithm for off-grid coherent DOA estimation is not satisfactory due to the inaccurate weighting information. To increase the estimation accuracy and efficiency, an improved OGWSBI algorithm based on a higher-order off-grid model and unitary transformation for off-grid coherent DOA estimation is proposed in this paper. Firstly, to reduce the approximate error of the first-order off-grid model, the steering vector is reformulated by the second-order Taylor expansion. Then, the received data is transformed from complex value to real value and the coherent signals are decorrelated via utilizing unitary transformation, which can increase the computational efficiency and restore the rank of the covariance matrix. Finally, in the real field, the steering vector higher-order approximation model and weighted sparse Bayesian inference are combined together to realize the estimation of DOA. Extensive simulation results indicate that under the condition of coherent signals and low SNR, the estimation accuracy of the proposed algorithm is about 50% higher than that of the OGWSBI algorithm, and the calculation time is reduced by about 60%.

Four-level double V-type atomic system solved by virtue of supersymmetric unitary transformation

We consider a four-level double V-type atom with two closely-separated top levels and two closely-separated lower levels interacting with a single mode field via multi-photon processes and in the presence of Kerr medium. We show that this atomic system possesses supersymmetric structure and construct its supersymmetric generators. We diagonalize the Hamiltonian of this system using supersymmetric unitary transformation and obtain the corresponding eigenstates and eigenvalues. The atom–field wave functions are obtained when the atom and the field mode are initially in two different cases. The evolution of both the quasi-probability distribution Q-function and the Mandel Q-parameter of the field are studied when the input field is in a coherent state. The influence of the Kerr medium and the detuning parameters on the behavior of these quantum effects is analyzed. The results show that they play a prominent role on the Poissonian statistics of the field. Also, the Kerr medium changes the behavior of the quasi-probability distribution Q-function. We end with discussion and conclusions.

Time-Dependent Unitary Transformation Method in the Strong-Field-Ionization Regime with the Kramers-Henneberger Picture

Time evolution operators of a strongly ionizing medium are calculated by a time-dependent unitary transformation (TDUT) method. The TDUT method has been employed in a quantum mechanical system composed of discrete states. This method is especially helpful for solving molecular rotational dynamics in quasi-adiabatic regimes because the strict unitary nature of the propagation operator allows us to set the temporal step size to large; a tight limitation on the temporal step size (δt<<1) can be circumvented by the strict unitary nature. On the other hand, in a strongly ionizing system where the Hamiltonian is not Hermitian, the same approach cannot be directly applied because it is demanding to define a set of field-dressed eigenstates. In this study, the TDUT method was applied to the ionizing regime using the Kramers-Henneberger frame, in which the strong-field-dressed discrete eigenstates are given by the field-free discrete eigenstates in a moving frame. Although the present work verifies the method for a one-dimensional atom as a prototype, the method can be applied to three-dimensional atoms, and molecules exposed to strong laser fields.

Quantum compiling by deep reinforcement learning

The general problem of quantum compiling is to approximate any unitary transformation that describes the quantum computation as a sequence of elements selected from a finite base of universal quantum gates. The Solovay-Kitaev theorem guarantees the existence of such an approximating sequence. Though, the solutions to the quantum compiling problem suffer from a tradeoff between the length of the sequences, the precompilation time, and the execution time. Traditional approaches are time-consuming, unsuitable to be employed during computation. Here, we propose a deep reinforcement learning method as an alternative strategy, which requires a single precompilation procedure to learn a general strategy to approximate single-qubit unitaries. We show that this approach reduces the overall execution time, improving the tradeoff between the length of the sequence and execution time, potentially allowing real-time operations.

Causal and compositional structure of unitary transformations

The causal structure of a unitary transformation is the set of relations of possible influence between any input subsystem and any output subsystem. We study whether such causal structure can be understood in terms of compositional structure of the unitary. Given a quantum circuit with no path from input system A to output system B, system A cannot influence system B. Conversely, given a unitary U with a no-influence relation from input A to output B, it follows from [B. Schumacher and M. D. Westmoreland, Quantum Information Processing 4 no. 1, (Feb, 2005)] that there exists a circuit decomposition of U with no path from A to B. However, as we argue, there are unitaries for which there does not exist a circuit decomposition that makes all causal constraints evident simultaneously. To address this, we introduce a new formalism of `extended circuit diagrams', which goes beyond what is expressible with quantum circuits, with the core new feature being the ability to represent direct sum structures in addition to sequential and tensor product composition. A causally faithful extended circuit decomposition, representing a unitary U, is then one for which there is a path from an input A to an output B if and only if there actually is influence from A to B in U. We derive causally faithful extended circuit decompositions for a large class of unitaries, where in each case, the decomposition is implied by the unitary's respective causal structure. We hypothesize that every finite-dimensional unitary transformation has a causally faithful extended circuit decomposition.

Time-Dependent Unitary Transformation Method in the Strong-Field-Ionization Regime With the Kramers-Henneberger Picture

Time evolution operators of a strongly ionizing medium are calculated by a time-dependent unitary transformation (TDUT) method. The TDUT method has been employed in quantum mechanical system composed of discrete states. This method is especially helpful for solving molecular rotational dynamics in quasi-adiabatic regimes because the strict unitary nature of the propagation operator allows us to set the temporal step size large; a tight limitation on the temporal step size ($\delta t &lt;&lt;1$) can be circumvented by the strict unitary nature. On the other hand, in a strongly ionizing system where the Hamiltonian is not Hermitian, the same approach cannot be directly applied because it is demanding to define a set of field-dressed eigenstates. In this study, the TDUT method was applied to the ionizing regime using the Kramers-Henneberger frame, in which the strong-field-dressed discrete eigenstates are given by the field-free discrete eigenstates in a moving frame. Although the present work verifies the method for a one-dimensional atom as a prototype, the method can be applied to three-dimensional atoms, and molecules exposed to strong laser fields.

Threefold spin helicity as possible origin of SU(3) gauge symmetry

In this paper, the notion of spin helicity is generalized into threefold spin helicity. It appears to be a useful means to extend the standard Dirac equation to describe coloured fermions. The threefold generalization of helicity is derived from the mass shell condition, spin, and kinetic helicity in a natural way. It is found that the three different types of helicities are associated with the rotation degrees of freedom in 3-D coordinate space. Moreover, threefold helicity can by unitary transformation be connected with the empirical SU(3) colour symmetry of the quarks, and thus be brought into the mathematical form of the SU(3) Yang–Mills gauge theory of the standard model. This offers an alternative picture of the physical origin of quark symmetry in compliance with the Coleman–Mandula theorem.