Single-Qubit Error Mitigation by Simulating Non-Markovian Dynamics.

Quantum simulation is a powerful tool to study the properties of quantum systems. The dynamics of open quantum systems are often described by completely positive (CP) maps, for which several quantum simulation schemes exist. Such maps, however, represent only a subset of a larger class of maps: the general dynamical maps which are linear, Hermitian preserving, and trace preserving but not necessarily positivity preserving. Here we present a simulation scheme for these general dynamical maps, which occur when the underlying system-reservoir model undergoes entangling (and thus non-Markovian) dynamics. Such maps also arise as the inverse of CP maps, which are commonly used in error mitigation. We illustrate our simulation scheme on an IBM quantum processor, demonstrating its ability to recover the initial state of a Lindblad evolution. This paves the way for a novel form of quantum error mitigation. Our scheme only requires one ancilla qubit as an overhead and a small number of one and two qubit gates. Consequently, we expect it to be of practical use in near-term quantum devices.

Dynamical Coulomb Blockade as a Local Probe for Quantum Transport.

Quantum fluctuations are imprinted with valuable information about transport processes. Experimental access to this information is possible, but challenging. We introduce the dynamical Coulomb blockade (DCB) as a local probe for fluctuations in a scanning tunneling microscope (STM) and show that it provides information about the conduction channels. In agreement with theoretical predictions, we find that the DCB disappears in a single-channel junction with increasing transmission following the Fano factor, analogous to what happens with shot noise. Furthermore we demonstrate local differences in the DCB expected from changes in the conduction channel configuration. Our experimental results are complemented by ab initio transport calculations that elucidate the microscopic nature of the conduction channels in our atomic-scale contacts. We conclude that probing the DCB by STM provides a technique complementary to shot noise measurements for locally resolving quantum transport characteristics.

Quantum Brownian Motion at Strong Dissipation Probed by Superconducting Tunnel Junctions.

We have investigated the phase dynamics of a superconducting tunnel junction at ultralow temperatures in the presence of high damping, where the interaction with environmental degrees of freedom represents the leading energy scale. In this regime, theory predicts the dynamics to follow a generalization of the classical Smoluchowski description, the quantum Smoluchowski equation, thus, exhibiting overdamped quantum Brownian motion characteristics. For this purpose, we have performed current-biased measurements on the small-capacitance Josephson junction of a scanning tunneling microscope placed in a low impedance environment at milli-Kelvin temperatures. We can describe our experimental findings with high accuracy by using a quantum phase diffusion model based on the quantum Smoluchowski equation. In this way we experimentally demonstrate that overdamped quantum systems follow quasiclassical dynamics with significant quantum effects as the leading corrections.

Lamb-shift enhancement and detection in strongly driven superconducting circuits.

It is shown that strong driving of a quantum system substantially enhances the Lamb shift induced by broadband reservoirs, which are typical for solid-state devices. By varying drive parameters the impact of environmental vacuum fluctuations with continuous spectral distribution onto system observables can be tuned in a distinctive way. This provides experimentally feasible measurement schemes for the Lamb shift in superconducting circuits based on Cooper pair boxes, where it can be detected either in shifted dressed transition frequencies or in pumped charge currents.

Tracking a spin-polarized superconducting bound state across a quantum phase transition.

The magnetic exchange coupling between magnetic impurities and a superconductor induce so-called Yu-Shiba-Rusinov (YSR) states which undergo a quantum phase transition (QPT) upon increasing the exchange interaction beyond a critical value. While the evolution through the QPT is readily observable, in particular if the YSR state features an electron-hole asymmetry, the concomitant change in the ground state is more difficult to identify. We use ultralow temperature scanning tunneling microscopy to demonstrate how the change in the YSR ground state across the QPT can be directly observed for a spin-1/2 impurity in a magnetic field. The excitation spectrum changes from featuring two peaks in the doublet (free spin) state to four peaks in the singlet (screened spin) ground state. We also identify a transition regime, where the YSR excitation energy is smaller than the Zeeman energy. We thus demonstrate a straightforward way for unambiguously identifying the ground state of a spin-1/2 YSR state.

Persistence of coherent quantum dynamics at strong dissipation.

The quantum dynamics of a two-state system coupled to a bosonic reservoir with sub-Ohmic spectral density is investigated for strong friction. Numerically exact path integral Monte Carlo methods reveal that a changeover from coherent to incoherent relaxation does not occur for a broad class of spectral distributions. In nonequilibrium coherences associated with substantial system-reservoir entanglement exist even when strong dissipation forces the thermodynamic state of the system to behave almost classically. This may be of relevance for current experiments with nanoscale devices.

From Coulomb-blockade to nonlinear quantum dynamics in a superconducting circuit with a resonator.

Motivated by recent experiments on superconducting circuits consisting of a dc-voltage-biased Josephson junction in series with a resonator, quantum properties of these devices far from equilibrium are studied. This includes a crossover from a domain of incoherent to a domain of coherent Cooper pair tunneling, where the circuit realizes a driven nonlinear oscillator. Equivalently, weak photon-charge coupling turns into strong correlations captured by a single degree of freedom. Radiated photons offer a new tool to monitor charge flow and current noise gives access to nonlinear dynamics, which allows us to analyze quantum-classical boundaries.

Improvements to Kramers turnover theory.

The Kramers turnover problem, that is, obtaining a uniform expression for the rate of escape of a particle over a barrier for any value of the external friction was solved in the 1980s. Two formulations were given, one by Mel'nikov and Meshkov (MM) [V. I. Mel'nikov and S. V. Meshkov, J. Chem. Phys. 85, 1018 (1986)], which was based on a perturbation expansion for the motion of the particle in the presence of friction. The other, by Pollak, Grabert, and Hänggi (PGH) [E. Pollak, H. Grabert, and P. Hänggi, J. Chem. Phys. 91, 4073 (1989)], valid also for memory friction, was based on a perturbation expansion for the motion along the collective unstable normal mode of the particle. Both theories did not take into account the temperature dependence of the average energy loss to the bath. Increasing the bath temperature will reduce the average energy loss. In this paper, we analyse this effect, using a novel perturbation theory. We find that within the MM approach, the thermal energy gained from the bath diverges, the average energy gain becomes infinite implying an essential failure of the theory. Within the PGH approach increasing the bath temperature reduces the average energy loss but only by a finite small amount of the order of the inverse of the reduced barrier height. Then, this does not seriously affect the theory. Analysis and application for a cubic potential and Ohmic friction are presented.

A protocol for the use of cloud-based quantum computers for logical network analysis of biological systems.

Boolean networks are commonly used in systems biology to dynamically model gene regulatory interactions. Here, we present a protocol for implementing Boolean network dynamics as quantum circuits. We describe steps for accessing cloud-based quantum processing units offered by IBM and IonQ and downloading and parsing logic for gene regulatory networks. We then detail procedures for performing simulations of quantum circuits on local devices and visualizing measurement results. For complete details on the use and execution of this protocol, please refer to Weidner et al.1.

A quantum physics layer of epigenetics: a hypothesis deduced from charge transfer and chirality-induced spin selectivity of DNA.

BACKGROUND: Epigenetic mechanisms are informational cellular processes instructing normal and diseased phenotypes. They are associated with DNA but without altering the DNA sequence. Whereas chemical processes like DNA methylation or histone modifications are well-accepted epigenetic mechanisms, we herein propose the existence of an additional quantum physics layer of epigenetics. RESULTS: We base our hypothesis on theoretical and experimental studies showing quantum phenomena to be active in double-stranded DNA, even under ambient conditions. These phenomena include coherent charge transfer along overlapping pi-orbitals of DNA bases and chirality-induced spin selectivity. Charge transfer via quantum tunneling mediated by overlapping orbitals results in charge delocalization along several neighboring bases, which can even be extended by classical (non-quantum) electron hopping. Such charge transfer is interrupted by flipping base(s) out of the double-strand e.g., by DNA modifying enzymes. Charge delocalization can directly alter DNA recognition by proteins or indirectly by DNA structural changes e.g., kinking. Regarding sequence dependency, charge localization, shown to favor guanines, could influence or even direct epigenetic changes, e.g., modification of cytosines in CpG dinucleotides. Chirality-induced spin selectivity filters electrons for their spin along DNA and, thus, is not only an indicator for quantum coherence but can potentially affect DNA binding properties. CONCLUSIONS: Quantum effects in DNA are prone to triggering and manipulation by external means. By the hypothesis put forward here, we would like to foster research on "Quantum Epigenetics" at the interface of medicine, biology, biochemistry, and physics to investigate the potential epigenetic impact of quantum physical principles on (human) life.

Microwave excitation of atomic scale superconducting bound states.

Magnetic impurities on superconductors lead to bound states within the superconducting gap, so called Yu-Shiba-Rusinov (YSR) states. They are parity protected, which enhances their lifetime, but makes it more difficult to excite them. Here, we realize the excitation of YSR states by microwaves facilitated by the tunnel coupling to another superconducting electrode in a scanning tunneling microscope (STM). We identify the excitation process through a family of anomalous microwave-assisted tunneling peaks originating from a second-order resonant Andreev process, in which the microwave excites the YSR state triggering a tunneling event transferring a total of two charges. We vary the amplitude and the frequency of the microwave to identify the energy threshold and the evolution of this excitation process. Our work sets an experimental basis and proof-of-principle for the manipulation of YSR states using microwaves with an outlook towards YSR qubits.

Dissipation can enhance quantum effects.

Usually one finds that dissipation tends to make a quantum system more classical in nature. We study the effect of momentum dissipation on a quantum system. The momentum of the particle is coupled bi-linearly to the momenta of a harmonic oscillator heat bath. For a harmonic oscillator system we find that the position and momentum variances for momentum coupling are, respectively, identical to momentum and position variances for spatial friction. This implies that momentum coupling leads to an increase in the fluctuations in position as the temperature is lowered, exactly the opposite of the classical-like localization of the oscillator, found with spatial friction. For a parabolic barrier, momentum coupling causes an increase in the unstable normal mode barrier frequency, as compared to the lowering of the barrier frequency in the presence of purely spatial coupling. This increase in the frequency leads to an enhancement of the thermal tunneling flux, which below the crossover temperature becomes exponentially large. The crossover temperature between tunneling and thermal activation increases with momentum friction so that quantum effects in the escape are relevant at higher temperatures.

Sensing the quantum limit in scanning tunnelling spectroscopy.

The tunnelling current in scanning tunnelling spectroscopy (STS) is typically and often implicitly modelled by a continuous and homogeneous charge flow. If the charging energy of a single-charge quantum sufficiently exceeds the thermal energy, however, the granularity of the current becomes non-negligible. In this quantum limit, the capacitance of the tunnel junction mediates an interaction of the tunnelling electrons with the surrounding electromagnetic environment and becomes a source of noise itself, which cannot be neglected in STS. Using a scanning tunnelling microscope operating at 15 mK, we show that we operate in this quantum limit, which determines the ultimate energy resolution in STS. The P(E)-theory describes the probability for a tunnelling electron to exchange energy with the environment and can be regarded as the energy resolution function. We experimentally demonstrate this effect with a superconducting aluminium tip and a superconducting aluminium sample, where it is most pronounced.

Semiclassical theory of energy diffusive escape in a Duffing oscillator.

Motivated by recent experimental progress to readout quantum bits implemented in superconducting circuits via the phenomenon of dynamical bifurcation, transitions between steady orbits in a driven anharmonic oscillator, the Duffing oscillator, are analyzed. In the regime of weak dissipation a consistent diffusion equation in the semiclassical limit is derived to capture the intimate relation between finite tunneling and reflection and bath induced quantum fluctuations. From the corresponding steady-state distribution an analytical expression for the switching probability is obtained. It is shown that a reduction of the transition rate due to finite reflection at the phase-space barrier is overcompensated by an increase due to environmental quantum fluctuations that are specific for diffusion processes over dynamical barriers. The scaling behavior of the rate is discussed and it is revealed that close to the bifurcation threshold the escape dynamics enters an overdamped domain such that the quantum-mechanical energy scale associated with friction even exceeds the thermal energy scale.

Low-temperature quantum fluctuations in overdamped ratchets.

At low temperatures and strong friction the time evolution of the density distribution in position follows a quantum Smoluchowski equation. Recently, also higher-order contributions of quantum fluctuations to drift and diffusion coefficients have been systematically derived. As a nontrivial situation to reveal the impact of subleading quantum corrections and to demonstrate convergence properties of the perturbation series, directed transport in ratchets is studied. It is shown that the perturbation series typically has a nonmonotonous behavior. Depending on symmetry properties higher-order contributions may even compensate current reversals induced by leading quantum fluctuations. This analysis demonstrates how to consistently treat the dynamics of overdamped quantum systems at low temperatures also in numerical applications.

Quantum Smoluchowski equation: a systematic study.

The strong-friction regime at low temperatures is analyzed systematically starting from the formally exact path-integral expression for the reduced dynamics. This quantum Smoluchowski regime allows for a type of semiclassical treatment in the inverse friction strength so that higher-order quantum corrections to the original quantum Smoluchowski equation [J. Ankerhold, P. Pechukas, and H. Grabert, Phys. Rev. Lett. 87, 086802 (2001); J. Ankerhold and H. Grabert, Phys. Rev. Lett. 101, 119903 (2008)] can be derived. Drift and diffusion coefficients are determined by the equilibrium distribution in position and are directly related to the corresponding action of extremal paths and fluctuations around them. It is shown that the inclusion of higher-order corrections reproduces the quantum enhancement above crossover for the decay rate out of a metastable well exactly.

Non-Markovian dissipative semiclassical dynamics.

The exact stochastic decomposition of non-Markovian dissipative quantum dynamics is combined with the time-dependent semiclassical initial value formalism. It is shown that even in the challenging regime of moderate friction and low temperatures, where non-Markovian effects are substantial, this approach allows for the accurate description of dissipative dynamics in anharmonic potentials over many oscillation periods until thermalization is reached. The problem of convergence of the stochastic average at long times, which plagues full quantum mechanical implementations, is avoided through a joint sampling of the stochastic noise and the semiclassical phase-space distribution.

Detecting charge noise with a Josephson junction: a problem of thermal escape in presence of non-Gaussian fluctuations.

Motivated by several experimental activities to detect charge noise produced by a mesoscopic conductor with a Josephson junction as on-chip detector, the switching rate out of its zero-voltage state is studied. This process is related to the problem of thermal escape in presence of non-Gaussian fluctuations. In the relevant case of weak higher than second order cumulants, an effective Fokker-Planck equation is derived, which is then used to obtain an explicit expression for the escape rate. Specific results for the rate asymmetry due to the third moment of current noise allow to analyze experimental data and to optimize detection circuits.

How to detect the fourth-order cumulant of electrical noise.

It is proposed to measure the current noise generated in a mesoscopic conductor by macroscopic quantum tunneling (MQT) in a current biased Josephson junction placed in parallel to the conductor. The theoretical description of this setup takes into account the complete dynamics of detector and noise source. Explicit results are given for the specific case of current fluctuations in an oxide layer tunnel junction, and it is shown how the device allows to extract the fourth-order cumulant of the noise from the MQT data for realistic experimental parameters.

Path-integral Monte Carlo simulations for electronic dynamics on molecular chains. II. Transport across impurities.

Electron transfer (ET) across molecular chains including an impurity is studied based on a recently improved real-time path-integral Monte Carlo (PIMC) approach [L. Mühlbacher, J. Ankerhold, and C. Escher, J. Chem. Phys. 121 12696 (2004)]. The reduced electronic dynamics is studied for various bridge lengths and defect site energies. By determining intersite hopping rates from PIMC simulations up to moderate times, the relaxation process in the extreme long-time limit is captured within a sequential transfer model. The total transfer rate is extracted and shown to be enhanced for certain defect site energies. Super-exchange turns out to be relevant for extreme gap energies only and then gives rise to different dynamical signatures for high- and low-lying defects. Further, it is revealed that the entire bridge compound approaches a steady state on a much shorter time scale than that related to the total transfer. This allows for a simplified description of ET along donor-bridge-acceptor systems in the long-time range.