Cheeger inequalities bound the spectral gap $\gamma$ of a space by isoperimetric properties of that space and vice versa. In this paper, I derive Cheeger-type inequalities for nonpositive matrices (aka stoquastic Hamiltonians), real matrices, and Hermitian matrices. For matrices written $H = L+W$, where $L$ is either a combinatorial or normalized graph Laplacian, each bound holds independently of any information about $W$ other than its class and the weighted Cheeger constant induced by its ground-state. I show that independently of $\lVert W \rVert$: (1) when $W$ is diagonal and $L$ has maximum degree $d_{\max}$, $2h \geq \gamma \geq \sqrt{h^2 + d_{\max}^2}-d_\max$; (2) when $W$ is real, we can route negative weighted edges along positive weighted edges such that the Cheeger constant of the resulting graph obeys an inequality similar to that above; and (3) when $W$ is Hermitian, the weighted Cheeger constant obeys $2h \geq \gamma$. The weighted Cheeger constant reduces bounds on $\gamma$ to information contained in the underlying graph and the Hamiltonian's ground-state.

If efficiently computable, the constant opens up a very clear path towards adaptive quantum adiabatic algorithms, those that adjust the adiabatic path based on spectral structure. I sketch a bashful adiabatic algorithm that aborts the adiabatic process early, uses the resulting state to approximate the weighted Cheeger constant, and restarts the process using the updated information. Should this approach work, it would provide more rigorous foundations for adiabatic quantum computing without a priori knowledge of the spectral gap.

The classical Second Law of Thermodynamics demands that an isolated system evolves with a non-diminishing entropy. This holds as well in quantum mechanics if the evolution of the energy-isolated system can be described by a unital quantum channel. At the same time, the entropy of a system evolving via a non-unital channel can, in principle, decrease. Here, we analyze the behavior of the entropy in the context of the H-theorem. As exemplary phenomena, we discuss the action of a Maxwell demon (MD) operating a qubit and the processes of heating and cooling in a two-qubit system. We further discuss how small initial correlations between a quantum system and a reservoir affect the increase in the entropy under the evolution of the quantum system.

The Second Law of Thermodynamics states that temporal evolution of an isolated system occurs with non-diminishing entropy. In quantum realm, this holds for energy-isolated systems the evolution of which is described by the so-called unital quantum channel. The entropy of a system evolving in a non-unital quantum channel can, in principle, decrease. We formulate a general criterion of unitality for the evolution of a quantum system, enabling a simple and rigorous approach for finding and identifying the processes accompanied by decreasing entropy in energy-isolated systems. We discuss two examples illustrating our findings, the quantum Maxwell demon and heating-cooling process within a two-qubit system.

Microresonator-based nonlinear processes are fundamental to applications including microcomb generation, parametric frequency conversion, and harmonics generation. While nonlinear processes involving either second- ($\chi^{(2)}$) or third- $\chi^{(3)}$) order nonlinearity have been extensively studied, the interaction between these two basic nonlinear processes has seldom been reported. In this letter, we demonstrate a coherent interplay between second- and third- order nonlinear processes. The parametric ($\chi^{(2)})$ coupling to a lossy ancillary mode shortens the lifetime of the target photonic mode and suppresses its density of states, preventing the photon emissions into the target photonic mode via Zeno effect. Such effect is then used to control the stimulated four-wave mixing process and realize a suppression ratio of $34.5$.

Optics naturally provides us with some powerful mathematical operations. Here we reveal that a single planar interface can compute spatial differentiation to paraxial coherent beams under oblique incidence. We show that intrinsically the spatial differentiation results from the spin Hall effect of light with preparing and postselecting polarization states, in both quantum and classical levels. Since the spin Hall effect of light is a geometrically protected effect, the spatial differentiation generally accompanies light reflection and refraction and occurs at any optical interface, regardless of composition materials and incident angles. We experimentally demonstrate the generality of spatial differentiation and use such an energy-efficient method to perform ultra-fast edge detection. Compared with recent developments in miniaturizing spatial differentiation computing devices from metamaterials and layered structures to surface plasmonic structures and photonic crystal slabs, the proposed spin-optical method with a single optical interface is as compact as possible and moreover offers a simple but powerful mechanism to vectorial-field based computation.

We discuss a surprisingly simple scheme for accounting (and removal) of error in observables determined from quantum algorithms. A correction to the value of the observable is calculated by first measuring the observable with all error sources active and subsequently measuring the observable with each error source removed separately. We apply this scheme to the variational quantum eigensolver, simulating the calculation of the ground state energy of equilibrium H$_2$ and LiH in the presence of several noise sources, including amplitude damping, dephasing, thermal noise, and correlated noise. We show that this scheme provides a decrease in the needed quality of the qubits by up to two orders of magnitude. In near-term quantum computers, where full fault-tolerant error correction is too expensive, this scheme provides a route to significantly more accurate calculation

Photodetection is a process in which an incident field induces a polarization current in the detector. The interaction of the field with this induced current excites an electron in the detector from a localized bound state to a state in which the electron freely propagates and can be classically amplified and detected. The induced current can interact not only with the applied field, but also with all of the initially unpopulated vacuum modes. This interaction with the vacuum modes is assumed to be small and is neglected in conventional photodetection theory. We show that this interaction contributes to the quantum efficiency of the detector. We also show that in the Purcell enhancement regime, shot noise in the photocurrent depends on the bandwidth of the the vacuum modes interacting with the detector. Our theory allows design of sensitive detectors to probe the properties of the vacuum modes.

Topological phase transition plays a significant role in modern condensed matter physics, since it is beyond Landau symmetry-breaking theory. Quantum walks have been demonstrated to be a powerful method in exploring topological quantum matter. Here, we investigate a special quantum walk in a line in terms of coherent state representation. By detecting average number of photons of the system, quantum phase transition can be observed, and the properties of coherent space are utilized. In addition, we propose an experimental protocol in a circuit quantum electrodynamics architecture, where a superconducting qubit is a coin while the cavity mode is used for quantum walk.

Steane's seven-qubit quantum code is a natural choice for fault-tolerance experiments because it is small and just two extra qubits are enough to correct errors. However, the two-qubit error-correction technique, known as "flagged" syndrome extraction, works slowly, measuring only one syndrome at a time. This is a disadvantage in experiments with high qubit rest error rates. We extend the technique to extract multiple syndromes at once, without needing more qubits. Qubits for different syndromes can flag errors in each other. This gives equally fast and more qubit-efficient alternatives to Steane's error-correction method, and also conforms to planar geometry constraints.

We further show that Steane's code and some others can be error-corrected with no extra qubits, provided there are at least two code blocks. The rough idea is that two seven-qubit codewords can be temporarily joined into a twelve-qubit code, freeing two qubits for flagged syndrome measurement.

In the description of quantum key distribution systems, much attention is paid to the operation of quantum cryptography protocols. The main problem is the insufficient study of the synchronization process of quantum key distribution systems. This paper contains a general description of quantum cryptography principles. A two-line fiber-optic quantum key distribution system with phase coding of photon states in transceiver and coding station synchronization mode was examined. A quantum key distribution system was built on the basis of the scheme with automatic compensation of polarization mode distortions. Single-photon avalanche diodes were used as optical radiation detecting devices. It was estimated how the parameters used in quantum key distribution systems of optical detectors affect the detection of the time frame with attenuated optical pulse in synchronization mode with respect to its probabilistic and time-domain characteristics. A design method was given for the process that detects the time frame that includes an optical pulse during synchronization. This paper describes the main quantum communication channel attack methods by removing a portion of optical emission. This paper describes the developed synchronization algorithm that takes into account the time required to restore the photodetectors operation state after the photon has been registered during synchronization. The computer simulation results of the developed synchronization algorithm were analyzed...

Group-IV -- Vacancy color centers in diamond are fast emerging qubits that can be harnessed in quantum communication and sensor applications. There is an immediate quest for understanding their magneto-optical properties, in order to select the appropriate qubits for varying needs of particular quantum applications. Here we present results from cutting edge \emph{ab initio} calculations about the charge state stability, zero-phonon-line energies, spin-orbit and electron-phonon couplings for Group-IV -- Vacancy color centers. Based on the analysis of our results, we develop a novel spin Hamiltonian for these qubits which incorporates the interaction of the electron spin and orbit coupled with phonons beyond perturbation theory. Our results are in good agreement with previous data and predict a new defect for qubit applications with thermally initialized ground state spin and long spin coherence time.

A new laboratory bound on the axial-vector mediated interaction between electron spins at micrometer scale is established with single nitrogen-vacancy centers in diamond. A single crystal of p-terphenyl doped pentancene-d$_{14}$ under laser pumping provides the source of polarized electron spins. Based on the measurement of polarization signal via nitrogen-vacancy centers, we set a constraint for the exotic electron-electron coupling, $g_A^eg_A^e$, within the force range from 10 to 900 $\mu$m. The obtained upper bound of the coupling at 500 $\mu$m is $|g_A^eg_A^e / 4\pi\hbar c |\leq 5.7\times 10^{-19} $, which is one order of magnitude more stringent than previous experiment. Our result shows that the NV center can be a promising platform for searching for new particles predicted by theories beyond the standard model.

We investigate the routing of a single-photon in a modulated cavity optomechanical system, in which the cavity is driven by a strong coupling field, the mechanical resonator (MR) is modulated by a weak coherent field, and the signal photon is made up of a sequence of pulses with exactly one photon per pulse. We demonstrate that, when there is no a weak coherent field modulating the MR, the system cannot act as a single-photon router, since the signal will be completely covered by the quantum and thermal noises. With the existence of the weak coherent field, we can achieve the routing of the single-photon by changing the frequency of the weak coherent field, and the system can be immune to the quantum noises and thermal noises when the MR is cooled to its quantum ground state.

An optical probe of cesium Rydberg atoms generated in a thermal vapor cell is used to retrieve a baseband signal modulated onto a 16.98-GHz carrier wave in real-time, demonstrating an atom-based quantum receiver suitable for microwave communication. The 60$S_{1/2}$ Rydberg level of cesium atoms in the cell is tracked via electromagnetically induced transparency (EIT), an established laser-spectroscopic method. The microwave carrier is resonant with the 60$S_{1/2}$ $\rightarrow$ 60$P_{1/2}$ Rydberg transition, resulting in an Autler-Townes (AT) splitting of the EIT signal. Amplitude modulation of the carrier wave results in a corresponding modulation in the optically retrieved AT splitting. Frequency modulation causes a change in relative height of the two AT peaks, which can be optically detected and processed to retrieve the modulation signal. The optical retrieval of the baseband signal does not require electronic demodulation. The method is suitable for carrier frequencies within a range from $\sim 1$~GHz to hundreds of GHz. The baseband bandwidth, which is $\sim$~20~Hz in the present demonstration, can be increased by faster spectroscopic sampling.

We present a full operator approach to treatment of the cross-Kerr interaction combined with parametric amplification. It is shown that this problem can be exactly integrated using the method of higher-order operators. While the initial basis is infinite-dimensional, an orthogonal transformation can reduce the problem exactly into a six-dimensional basis which can be integrated conveniently.

Lov{\'a}sz Local Lemma (LLL) is a very powerful tool in combinatorics and probability theory to show the possibility of avoiding all "bad" events under some "weakly dependent" condition. Over the last decades, the algorithmic aspect of LLL has also attracted lots of attention in theoretical computer science. A tight criterion under which the abstract version LLL holds was given by Shearer [shearer1985problem]. It turns out that Shearer's bound is generally not tight for variable version LLL (VLLL) [he2017variable]. Recently, Ambainis et al. introduced a quantum version LLL (QLLL), which was then shown to be powerful for quantum satisfiability problem.

In this paper, we prove that Shearer's bound is tight for QLLL, affirming a conjecture proposed by Sattath et. al. Our result shows the tightness of Gily{\'e}n and Sattath's algorithm, and implies that the lattice gas partition function fully characterizes quantum satisfiability for almost all Hamiltonians with large enough qudits.

Commuting LLL (CLLL), LLL for commuting local Hamiltonians which are widely studied in literature, is also investigated here. We prove that the tight regions of CLLL and QLLL are generally different. Thus, the efficient region of algorithms for CLLL can go beyond shearer's bound. Our proof is by first bridging CLLL and VLLL on a family of interaction bipartite graphs and then applying the tools of VLLL, e.g., the gapless/gapful results, to CLLL. We also provide a sufficient and necessary condition for deciding whether the tight regions of QLLL and CLLL are the same for a given interaction bipartite graph.

We demonstrate the role of measurement back-action of a coherent spin environment on the dynamics of a spin (qubit) coupled to it, by inducing non-classical (Quantum Random Walk like) statistics on its measurement trajectory. We show how the long-life time of the spin-bath allows it to correlate measurements of the qubit over many repetitions. We have used Nitrogen Vacancy centers in diamond as a model system, and the projective single-shot readout of the electron spin at low temperatures to simulate these effects. We show that the proposed theoretical model, explains the experimentally observed statistics and their application for quantum state engineering of spin ensembles towards desired states.

Pairing between fermionic quasi-particles in solids through the exchange of virtual bosonic excitations is one of the most studied phenomena in the solid state because it can lead to remarkable emergent phases of matter like superconductivity. This phenomenon has been extensively investigated for the case of attractive interactions induced by phonons and magnetic excitations. Vacuum fluctuations of the electromagnetic field can also provide attractive interactions, albeit with far lower efficiency. Here, we show that photon-mediated superconductivity, which is not observed in free space, is possible in unpumped optical cavities in which spontaneous vacuum fluctuations may lead to pairing at equilibrium. We study the case of a generic material with the parameters of GaAs, embedded in a nanoplasmonic terahertz cavity. We show the possibility of electron-photon superconductivity with critical temperatures on the order of 0.5 Kelvin. Because the electron-photon pairing interaction is effectively long-range, both singlet and triplet pairing are possible. Consequently, cavity-mediated electron-photon superconductors with nontrivial topology may be induced in 2D materials.

A recent experiment [J. L. Garrett, D. A. T. Somers, and J. N. Munday, Phys. Rev. Lett {\bf 120}, 040401 (2018)] measured for the first time the gradient of the Casimir force between two gold spheres in vacuum at room temperature, and placed a bound on the magnitude of the deviation of the measured force from the proximity force approximation (PFA). The present work extends a previous theoretical analysis of this experiment [G. Bimonte, Phys. Rev. D {\bf 97}, 085011 (2018)], by analyzing in detail how the magnitude of the deviation from PFA is affected by the inclusion or neglect of ohmic dissipation at zero frequency, a much debated issue in the current Casimir literature, which goes by the name of the Drude vs plasma controversy. We analyze as well the effect of connecting the plates to charge reservoirs, which is the standard configuration used in Casimir experiments. We describe a simple and effective decimation procedure, allowing for a faster computation of the Casimir force for large aspect ratios of the system.

We unravel the dynamical stability of a fully polarized one-dimensional ultracold few-fermion spin-1/2 gas subjected to inhomogeneous driving of the itinerant spins. Despite the unstable character of the total spin-polarization the existence of an interaction regime is demonstrated where the spin-correlations lead to almost maximally aligned spins throughout the dynamics. The resulting ferromagnetic order emerges from the build up of superpositions of states of maximal total spin. They comprise a decaying spin-polarization and a dynamical evolution towards an almost completely unpolarized NOON-like state.