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The Gibbs partition function is an important quantity in describing statistical properties of a system in thermodynamic equilibrium. There are several proposals to calculate the partition functions on near-term quantum computers. However, the existing schemes require many copies of the Gibbs states to perform an extrapolation for the calculation of the partition function, which could be costly performed on the near-term quantum computers. We propose a scheme to calculate the Gibbs function with the imaginary time evolution. After preparing Gibbs states with different temperatures by using the imaginary time evolution, we measure the overlap between them on a quantum circuit, which allows us to calculate the Gibbs partition function. Our scheme requires only 2N qubits to calculate the Gibbs function of N qubits.

Variational quantum algorithms (VQAs) have been considered to be useful applications of noisy intermediate-scale quantum (NISQ) devices. Typically, in the VQAs, a parametrized ansatz circuit is used to generate a trial wave function, and the parameters are optimized to minimize a cost function. On the other hand, blind quantum computing (BQC) has been studied in order to provide the quantum algorithm with security by using cloud networks. A client with a limited ability to perform quantum operations hopes to have access to a quantum computer of a server, and BQC allows the client to use the server's computer without leakage of the client's information (such as input, running quantum algorithms, and output) to the server. However, BQC is designed for fault-tolerant quantum computing, and this requires many ancillary qubits, which may not be suitable for NISQ devices. Here, we propose an efficient way to implement the NISQ computing with guaranteed security for the client. In our architecture, only N+ 1 qubits are required, under an assumption that the form of ansatzes is known to the server, where N denotes the necessary number of the qubits in the original NISQ algorithms. The client only performs single-qubit measurements on an ancillary qubit sent from the server, and the measurement angles can specify the parameters for the ansatzes of the NISQ algorithms. No-signaling principle guarantees that neither parameters chosen by the client nor the outputs of the algorithm are leaked to the server. This work paves the way for new applications of NISQ devices.

Boltzmann machine is a powerful tool for modeling probability distributions that govern the training data. A thermal equilibrium state is typically used for Boltzmann machine learning to obtain a suitable probability distribution. The Boltzmann machine learning consists of calculating the gradient of the loss function given in terms of the thermal average, which is the most time consuming procedure. Here, we propose a method to implement the Boltzmann machine learning by using Noisy Intermediate-Scale Quantum (NISQ) devices. We prepare an initial pure state that contains all possible computational basis states with the same amplitude, and apply a variational imaginary time simulation. Readout of the state after the evolution in the computational basis approximates the probability distribution of the thermal equilibrium state that is used for the Boltzmann machine learning. We actually perform the numerical simulations of our scheme and confirm that the Boltzmann machine learning works well by our scheme., 14 pages, 11 figures

Dipole oscillation is studied in a normal phase of a trapped Bose--Fermi-mixture gas composed of single-species bosons and single-species fermions. Applying the moment method to the linearized Boltzmann equation, we derive a closed set of equations of motion for the center-of-mass position and momentum of both components. By solving the coupled equations, we reveal the behavior of dipole modes in the transition between the collisionless regime and the hydrodynamic regime. We find that two oscillating modes in the collisionless regime have distinct fates in the hydrodynamic regime: one collisionless mode shows a crossover to a hydrodynamic in-phase mode, and the other collisionless mode shows a transition to two purely damped modes. The temperature dependence of these dipole modes are also discussed., 10 pages, 5 figures

We investigate a quantum spatial search problem on fractal lattices, such as Sierpinski carpets and Menger sponges. In earlier numerical studies of the Sierpinski gasket, the Sierpinski tetrahedron, and the Sierpinski carpet, conjectures have been proposed for the scaling of a quantum spatial search problem finding a specific target, which is given in terms of the characteristic quantities of a fractal geometry. We find that our simulation results for extended Sierpinski carpets and Menger sponges support the conjecture for the ${\it optimal}$ number of the oracle calls, where the exponent is given by $1/2$ for $d_{\rm s} > 2$ and the inverse of the spectral dimension $d_{\rm s}$ for $d_{\rm s} < 2$. We also propose a scaling hypothesis for the ${\it effective}$ number of the oracle calls defined by the ratio of the ${\it optimal}$ number of oracle calls to a square root of the maximum finding probability. The form of the scaling hypothesis for extended Sierpinski carpets is very similar but slightly different from the earlier conjecture for the Sierpinski gasket, the Sierpinski tetrahedron, and the conventional Sierpinski carpet., 7 pages, 7 figures

This paper discusses exact relations in Bose--Einstein condensates (BECs), starting from basic properties of an ideal Bose gas. In particular, focused on are the Hugenholtz--Pines relation, Nepomnyashchii--Nepomnyashchii identity, and identities for the density response function. After introducing these exact relations, a few approaches of many-body approximations are discussed, which satisfy the exact relations in BECs. This paper will serve as a bridge between theories on exact relations and those on approximations in BECs., 10 pages

We investigate collective excitations of density fluctuations and a dynamic density structure factor in a mixture of Bose and Fermi gases in a normal phase. With decreasing temperature, we find that the frequency of the collective excitation deviates from that of the hydrodynamic sound mode. Even at temperature much lower than the Fermi temperature, the collective mode frequency does not reach the collisionless limit analogous to zero sound in a Fermi gas, because of collisions between bosons and fermions.

We investigate a quantum spatial search problem on a fractal lattice. A recent study for the Sierpinski gasket and tetrahedron made a conjecture that the dynamics of the search on a fractal lattice is determined by spectral dimension. We tackle this problem for the Sierpinski carpet, and our simulation result supports the conjecture. We also propose a scaling hypothesis of oracle calls for the quantum amplitude amplification., 6 pages, 5 figures

We investigate multiparticle excitation effect on a collective density excitation as well as a single-particle excitation in a weakly interacting Bose--Einstein condensate (BEC). We find that although the weakly interacting BEC offers weak multiparticle excitation spectrum at low temperatures, this multiparticle excitation effect may not remain hidden, but emerges as bimodality in the density response function through the single-particle excitation. Identification of spectra in the BEC between the single-particle excitation and the density excitation is also assessed at nonzero temperatures, which has been known to be unique nature in the BEC at absolute zero temperature., 6 pages, 4 figures

We investigate tunneling properties of collective modes in the polar phase of a spin-1 spinor Bose-Einstein condensate. This spinor BEC state has two kinds of gapless modes, i.e., Bogoliubov mode and spin-wave. Within the framework of the mean-field theory at T=0, we show that these Goldstone modes exhibit the perfect transmission in the low-energy limit. Their anomalous tunneling behaviors still hold in the presence of superflow, except in the critical current state. In the critical current state, while the tunneling of Bogoliubov mode is accompanied by finite reflection, the spin-wave still exhibit the perfect transmission, unless the strengths of a spin-dependent and spin-independent interactions take the same value., 8 pages, 3 figures

We present the simplest theory of perfect tunneling of an excitation in a Bose-Einstein condensate (BEC) through an impurity potential with an arbitrary shape in the low-momentum limit. That is for the transverse spin wave in the ferromagnetic phase of a spin-1 BEC. This mode obeys a Schr\"odinger-type equation; yet, effects of the potential on its transmission coefficient $T$ and on its scattering cross section $sigma$ vanish in that limit. The order parameter determines $T$, and the momentum $p$-dependence of $sigma$ exhibits a Rayleigh scattering type ($sigma propto p^{4}$). These properties are common between two types of Nambu-Goldstone modes: this spin wave and the Bogoliubov mode., Comment: 5 pages

On the basis of a moment method, general solutions of a linearized Boltzmann equation for a normal Fermi system are investigated. In particular, we study the sound velocities and damping rates as functions of the temperature and the coupling constant. In the extreme limits of collisionless and hydrodynamic regimes, eigenfrequency of sound mode obtained from the moment equations reproduces the well-known results of zero sound and first sound. In addition, the moment method can describe crossover between those extreme limits at finite temperatures. Solutions of the moment equations also involve a thermal diffusion mode. From solutions of these equations, we discuss excitation spectra corresponding to the particle-hole continuum as well as collective excitations. We also discuss a collective mode in a weak coupling case., 38 pages, 5 figures

We determine the adiabatic phase diagram of a resonantly-coupled system of Fermi atoms and Bose molecules confined in the harmonic trap by using the local density approximation. The adiabatic phase diagram shows the fermionic condensate fraction composed of condensed molecules and Cooper pair atoms. The key idea of our work is conservation of entropy through the adiabatic process, extending the study of Williams et al. [Williams et al., New J. Phys. 6, 123 (2004)] for an ideal gas mixture to include the resonant interaction in a mean-field theory. We also calculate the molecular conversion efficiency as a function of initial temperature. Our work helps to understand recent experiments on the BCS-BEC crossover, in terms of the initial temperature measured before a sweep of the magnetic field., 13 pages, 8 figures. In press, "Journal of the Physical Society of Japan", Vol.76, No.6