This article is based on a talk presented at a conference "Quantum Annealing and Other Optimization Methods" held at Kolkata, India on March 2-5, 2005. It will be published in the proceedings "Quantum Annealing and Other Optimization Methods" (Springer, Heidelberg) pp. 39-70.In the present article, we review the progress in the last two decades of the work on the Suzuki-Trotter decomposition, or the exponential product formula. The simplest Suzuki-Trotter decomposition, or the well-known Trotter decomposition [1][2][3][4] is given by and ask what correction terms appear in the exponent of the right-hand side owing to the product in the left-hand side. They hence refer to it as an exponential product formula. (The readers should convince themselves by the Taylor expansion that the second-order correction in Eq. (1) is the same as that in Eq. (2). The higher-order corrections take different forms.) We here ask how we can generalize the Trotter formula (1) to decompositions with higher-order correction terms. We concentrate on the form e x(A+B) = e p1xA e p2xB e p3xA e p4xB · · · e pM xB + O(x m+1 ),or equivalently e p1xA e p2xB e p3xA e p4xB · · · e pM xB = eWe adjust the set of the parameters {p 1 , p 2 , · · · , p M } so that the correction term may be of the order of x m+1 . We refer to the right-hand side of Eq. (3) as an mth-order approximant in the sense that it is correct up to the mth order of x. (See Appendix A for another type of the exponential product formula.)One of the present authors (M.S.) has studied on the higher-order approximant continually . The present article mostly reviews his work on the subject. We first show the importance of the exponential operator in Sect. 1 and the effectiveness of the exponential product formula in Sect. 2. We demonstrate the effectiveness in examples of the time-evolution operator in quantum dynamics and the symplectic integrator in Hamilton dynamics. Section 3 explains a recursive way of constructing higherorder approximants, namely the fractal decomposition. We present in Sect. 4 an application of the fractal decomposition to the time-ordered exponential. We finally review in Sect. 5 the quantum analysis, an efficient way of computing correction terms of general orders algebraically. We can use the quantum analysis for the purpose of finding approximants of an arbitrarily high order by solving a set of simultaneous equations where the higher-order correction terms are put to zero. We demonstrate the prescription in three examples. We mention in Appendix A a type of the exponential product formula different from the form (3); it contains exponentials of commutation relations. We give in Appendix B a short review on the world-line quantum Monte Carlo method with the use of the Trotter approximation (1).1

We provide probabilistic interpretation of resonant states. We do this by showing that the integral of the modulus square of resonance wave functions (i.e., the conventional norm) over a properly expanding spatial domain is independent of time, and therefore leads to probability conservation. This is in contrast with the conventional employment of a bi-orthogonal basis that precludes probabilistic interpretation, since wave functions of resonant states diverge exponentially in space. On the other hand, resonant states decay exponentially in time, because momentum leaks out of the central scattering area. This momentum leakage is also the reason for the spatial exponential divergence of resonant state. It is by combining the opposite temporal and spatial behaviours of resonant states that we arrive at our probabilistic interpretation of these states. The physical need to normalize resonant wave functions over an expanding spatial domain arises because particles leak out of the region which contains the potential range and escape to infinity, and one has to include them in the total count of particles.

We develop a model for a random walker with long-range hops on general graphs. This random multi-hopper jumps from a node to any other node in the graph with a probability that decays as a function of the shortest-path distance between the two nodes. We consider here two decaying functions in the form of the Laplace and Mellin transforms of the shortest-path distances. Remarkably, when the parameters of these transforms approach zero asymptotically, the multihopper's hitting times between any two nodes in the graph converge to their minimum possible value, given by the hitting times of a normal random walker on a complete graph. Stated differently, for small parameter values the multi-hopper explores a general graph as fast as possible when compared to a random walker on a full graph. Using computational experiments we show that compared to the normal random walker, the multi-hopper indeed explores graphs with clusters or skewed degree distributions more efficiently for a large parameter range. We provide further computational evidence of the speed-up attained by the random multi-hopper model with respect to the normal random walker by studying deterministic, random and real-world networks.

The Lindblad equation for a two-level system under an electric field is analyzed by mapping to a linear equation with a non-Hermitian matrix. Exceptional points of the matrix are found to be extensive; the second-order ones are located on lines in a two-dimensional parameter space, while the third-order one is at a point. KEYWORDSNon-Hermitian quantum mechanics, Lindblad operator, exceptional point Introduction: non-Hermitian quantum mechanicsNon-Hermitian quantum mechanics [1] is attracting evermore interest from various points of view. Historically, the research of non-Hermitian quantum mechanics was initiated in the field of nuclear physics; we can go back to Gamow's study in 1928 [2] to find an explanation of resonant scattering in terms of a resonant state with a complex eigenvalue. The resonant state was then defined as an eigenstate of the Schrödinger equation under the Siegert boundary condition [3], which is essentially a non-Hermitian condition, and hence can produce the complex eigenvalue.In 1950s there appear many studies on "the optical model" [4] to explain the nuclear decay phenomenologically. Feshbach [5,6] justified the phenomenology by showing that an effective complex potential comes out when one traces out the environmental space surrounding the central scattering area. We can understand this in terms of the equivalence [7] of the Feshbach formalism to the Siegert boundary condition; the non-Hermiticity implicitly hidden in the Siegert boundary condition for the outer space emerges explicitly when one traces out the environment.This field of research in fact can be called the work on open quantum systems [8] in the modern terminology. Many studies on non-Hermitian random matrices (e.g. Ref.[9]) were motivated by the non-Hermiticity that appears in this way. We should also note that the non-Hermiticity assumed a priori in many recent studies on topological aspects of non-Hermitian systems (e.g. Refs. [10][11][12]) is implicitly based on the work on open quantum systems. Experiments on non-Hermitian quantum systems (e.g. Ref.[13]) also realize the non-Hermiticity by means of the contact to the environment.

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We first show that quantum resonant states observe particle number conservation and hence are consistent with the probabilistic interpretation of quantum mechanics. We then present for a class of quantum open systems, a resonant-state expansion of the sum of the retarded and advanced Green's functions. The expansion is given purely in terms of all discrete eigenstates and does not contain any background integrals. Using the expansion, we argue that the Fano asymmetry of resonance peaks is interpreted as interference between discrete eigenstates. We microscopically derive the Fano parameters for several cases.