Markovian open quantum systems display complicated relaxation dynamics. The spectral gap of the Liouvillian characterizes the asymptotic decay rate toward the steady state, but it does not necessarily give a correct estimate of the relaxation time because the crossover time to the asymptotic regime may be too long. We here give a rigorous upper bound on the transient decay of autocorrelation functions in the steady state by introducing the symmetrized Liouvillian gap. The standard Liouvillian gap and the symmetrized one are identical in an equilibrium situation but differ from each other in the absence of the detailed balance condition. It is numerically shown that the symmetrized Liouvillian gap always gives a correct upper bound on the decay of the autocorrelation function, but the standard Liouvillian gap does not.

In the introductory article of this theme issue, we provide an overview of quantum annealing and computation with a very brief summary of the individual contributions to this issue made by experts as well as a few young researchers. We hope the readers will get the touch of the excitement as well as the perspectives in this unusually active field and important developments there. This article is part of the theme issue 'Quantum annealing and computation: challenges and perspectives'.

We propose a novel type of minor-embedding (ME) in simulated-annealing-based
Ising machines. The Ising machines can solve combinatorial optimization
problems. Many combinatorial optimization problems are mapped to find the
ground (lowest-energy) state of the logical Ising model. When connectivity is
restricted on Ising machines, ME is required for mapping from the logical Ising
model to a physical Ising model, which corresponds to a specific Ising machine.
Herein we discuss the guiding principle of ME design to achieve a high
performance in Ising machines. We derive the proposed ME based on a theoretical
argument of statistical mechanics. The performance of the proposed ME is
compared with two existing types of MEs for different benchmarking problems.
Simulated annealing shows that the proposed ME outperforms existing MEs for all
benchmarking problems, especially when the distribution of the degree in a
logical Ising model has a large standard deviation. This study validates the
guiding principle of using statistical mechanics for ME to realize fast and
high-precision solvers for combinatorial optimization problems.

The gap of the Liouvillian spectrum gives the asymptotic decay rate of a
quantum dissipative system, and therefore its inverse has been identified as
the slowest relaxation time. In contrary to this common belief, we show that
the relaxation time due to diffusive transports in a boundary dissipated
many-body quantum system is determined not by the gap or low-lying eigenvalues
of the Liouvillian but by superexponentially large expansion coefficients for
Liouvillian eigenvectors with non-small eigenvalues at an initial state. This
finding resolves an apparent discrepancy reported in the literature between the
inverse of the Liouvillian gap and the relaxation time in dissipative many-body
quantum systems.

We study a dynamical phase transition in optical bistable systems subject to
a time-periodic driving field. The phase transition occurs in the structure of
limit cycle as a function of the frequency of the driving field. In the
thermodynamic limit, a single limit cycle is divided into two separated limit
cycles at the transition point. In finite-size systems, however, there is
always a single limit cycle due to the quantum tunneling effect. We use a
Floquet dissipative map, which is a time-evolution operator over one period in
a dynamics given by a quantum master equation, and discuss the decay rate of
relaxation dynamics into the limit cycle based on the dominant eigenvalue of
the map. We found that the decay rate exhibits qualitatively different
system-size dependence before and after the phase transition, and it shows a
finite-size scaling of spinodal phenomena around the transition point. The
present work provides a systematic way of studying dynamical phase transition
observed in time-periodically driven open systems in terms of the Floquet
dissipative map.

Persistent homology (PH) was applied to probe the structural changes of glasses under shear. PH associates each local atomistic structure in an atomistic configuration to a geometric object, namely, a hole, and evaluates the robustness of these holes against noise. We found that the microscopic structures were qualitatively different before and after yielding. The structures before yielding contained robust holes, the number of which decreased after yielding. We also observed that the structures after yielding approached those of quickly quenched glass. This work demonstrates the crucial role of robust holes in yielding and provides an interpretation based on geometry.

We investigate steady states of macroscopic quantum systems under dissipation
not obeying the detailed balance condition. We argue that the Gibbs state at an
effective temperature gives a good description of the steady state provided
that the system Hamiltonian obeys the eigenstate thermalization hypothesis
(ETH) and the perturbation theory in the weak system-environment coupling is
valid in the thermodynamic limit. We derive a criterion to guarantee the
validity of the perturbation theory, which is satisfied in the thermodynamic
limit for sufficiently weak dissipation when the Liouvillian is gapped for
bulk-dissipated systems, while the perturbation theory breaks down in
boundary-dissipated chaotic systems due to the presence of diffusive
transports. We numerically confirm these theoretical predictions. This work
suggests a connection between steady states of macroscopic open quantum systems
and the ETH.

We give a microscopic description of the optical bistability, where the
transmission coefficient has two different values as a function of input light
intensity, and the system exhibits a discontinuous jump with a hysteresis loop.
We developed an efficient numerical algorithm to treat the quantum master
equation for hybridized systems of many photons and a large number of two-level
atoms. By using this method, we characterize the bistability from the viewpoint
of eigenmodes and eigenvalues of the time evolution operator of the quantum
master equation. We investigate the optical bistability within the low
photon-density regime, where the hybridization of photon and atom degrees of
freedom occurs and the resonance spectrum has a double peak structure. We
compared it with the standard optical bistability between the low
photon-density regime and the high photon-density regime, where the photons can
be treated as a classical electromagnetic field and the resonance spectrum has
a single peak structure. We discuss the steady-state properties of the optical
bistability: dependencies of the photon number density on the intensity and the
double peak structure of the photon number distribution inside the bistable
region. As for the dynamical properties, we find that the relaxation timescale
shows an exponential growth with the system size, and reveal how the hysteresis
loop of the optical bistability depends on the size of the system and the
sweeping rate of the driving amplitude. Finally, by investigating the effects
of detuning frequency of the input field, we clarify the characteristic
properties of the present optical bistability within the low photon-density
regime, which are qualitatively different from the standard optical bistable
phenomena.

A periodically driven quantum system, when coupled to a heat bath, relaxes to
a non-equilibrium asymptotic state. In the general situation, the retrieval of
this asymptotic state presents a rather non-trivial task. It was recently shown
that in the limit of an infinitesimal coupling, using so-called rotating wave
approximation (RWA), and under strict conditions imposed on the time-dependent
system Hamiltonian, the asymptotic state can attain the Gibbs form. A
Floquet-Gibbs state is characterized by a density matrix which is diagonal in
the Floquet basis of the system Hamiltonian with the diagonal elements obeying
a Gibbs distribution, being parametrized by the corresponding Floquet
quasi-energies. Addressing the non-adiabatic driving regime, upon using the
Magnus expansion, we employ the concept of a corresponding effective Floquet
Hamiltonian. In doing so we go beyond the conventionally used RWA and
demonstrate that the idea of Floquet-Gibbs states can be extended to the
realistic case of a weak, although finite system-bath coupling, herein termed
effective Floquet-Gibbs states.

We study the quantum dynamics of a spin ensemble coupled to cavity photons. Recently, related experimental results have been reported, showing the existence of the strong coupling regime in such systems. We study the eigenenergy distribution of the multi-spin system (following the Tavis-Cummings model) which shows a peculiar structure as a function of the number of cavity photons and of spins. We study how this structure causes changes in the spectrum of the admittance in the linear response theory, and also in the frequency dependence of the excited quantities in the stationary state under a probing field. In particular, we investigate how the structure of the higher excited energy levels changes the spectrum from a double-peak structure (the so-called vacuum-field Rabi splitting) to a single-peak structure. We also point out that the spin dynamics in the region of the double-peak structure corresponds to recent experiments using cavity ringing, while in the region of the single-peak structure, it corresponds to the coherent Rabi oscillation in a driving electromagnetic field. Using a standard Lindblad-type mechanism, we study the effect of dissipation on the line width and separation in the computed spectra. In particular, we study the relaxation of the total spin in the general case of a spin ensemble in which the total spin of the system is not specified. The theoretical results are correlated with experimental evidence of the strong coupling regime, achieved with a spin-1/2 ensemble.

We study non-equilibrium stationary states of a cavity system consisting of
many atoms interacting with a quantized cavity field mode, under a driving
field in a dissipative environment. We derive a quantum master equation which
is suitable for treating systems with a strong driving field and a strong
atom-photon interaction. We do this by making use of the fact that the
mean-field dynamics are exact in the thermodynamic limit thanks to a uniform
coupling between atoms and photons. We find ordered states with symmetry-broken
components of the photon field and atomic excitation driven by the external
field. The mechanism by which these ordered states arise is discussed from the
viewpoint of the quantum interference effect.