  The eigenspectrum of a non-normal matrix, which does not commute with its
Hermitian conjugate, is a central issue of non-Hermitian physics that has been
extensively studied in the past few years. There is, however, another
characteristic of a non-normal matrix that has often been overlooked: the
pseudospectrum, or the set of spectra under small perturbations. In this paper,
we study the dynamics driven by the non-normal matrix (Hamiltonian) realized as
a continuous quantum trajectory of the Lindblad master equation in open quantum
systems and point out that the dynamics can reveal the nature of unconventional
pseudospectrum of the non-normal Hamiltonian. In particular, we focus on the
transient dynamics of the norm of an unnormalized quantum state evolved with
the non-normal Hamiltonian, which is related to the probability for observing
the trajectory with no quantum jump. We formulate the transient suppression of
the decay rate of the norm due to the pseudospectral behavior and derive a
non-Hermitian/non-normal analog of the time-energy uncertainty relation. We
also consider two methods to experimentally realize the non-normal dynamics and
observe our theoretical findings on quantum computers: one uses a technique to
realize non-unitary operations on quantum circuits and the other leverages a
quantum-classical hybrid algorithm called variational quantum simulation. Our
demonstrations using cloud-based quantum computers provided by IBM Quantum
exhibit the frozen dynamics of the norm in transient time, which can be
regarded as a non-normal analog of the quantum Zeno effect.
