All-dielectric nanophotonics lies at a forefront of nanoscience and technology as it allows to control light at the nanoscale using its electric and magnetic components. Bulk silicon does not experience any magnetic response, nevertheless, we demonstrate that the metasurface made of silicon parallelepipeds allows to excite the magnetic dipole moment leading to the broadening and enhancement of the absorption. Our investigations are underpinned by the numerical predictions and the experimental verifications. Also surprisingly we found that the resonant electric quadrupole moment leads to the enhancement of reflection. Our results can be applied for a development of absorption based devices from miniature dielectric absorbers, filters to solar cells and energy harvesting devices. Even a few decades ago, silicon films have been used to construct solar cells and energy absorption devices5. Since then, there have been also discovered ways to use plasmonic7 and dielectric11 structures for the energy absorption and accumulation. Thus, the energy can be absorbed in a wide spectral range using low-loss dielectric materials if just would be possible to change their optical properties. The optical properties of dielectric metamaterials have been attracting significant attention in recent years mainly because of their possibility to support the excitation of both electric and magnetic multipole resonances19. Owing to this effect, all-dielectric metasurfaces are widely used for the controllable light manipulation, particularly to control phase21, polarization24 and transmission27. In addition, dielectric materials are in high demand due to their possibility to concentrate the electric field without Joule losses29. The opportunity to control the light scattering can be widely applied to develop optical nanoantennas33, radiation sources37, antireflective coatings39, cloaking techniques40, to improve MRI devices42, sensors29 and many others. The multipole decomposition approach44 is one of the widely-used methods to analyse the optical properties of dielectric metamaterials49. We, therefore, use the multipole decomposition to understand the contribution of multipoles to the absorption effect of the metasurfaces that we study.Absorbing and accumulating energy from light could enable smart sensors to work indefinitely. Therefore, many groups are joining their efforts in the academic and technological level to develop devices which will facilitate the energy absorption51. However, the behavior of particles in an array is not obvious even if their behavior as a single particle is well understood. Therefore, here we are taking a step forward and explore the optical properties of the collective response of the array of particles - the metasurfaces, to obtain the noticeable light absorption. In the current level of technology, the considered silicon parallelepipeds can be fabricated relatively easy using different techniques. We built the numerical model to analyse the transmission and reflection properties of such structures, as well as to calculate the multipole moments excited in every particle of the metasurface. During our analysis we found that the high-order multipole excitations directly affect the optical properties of the metasurface. Thus, for the described structure, the electric quadrupole (EQ) moment can be associated with enhanced reflection and the magnetic dipole (MD) moment provides the extraordinary absorption. Moreover, the interference between the total electric dipole (TED) and MD moments leads to the realization of the Kerker-type effects in the considered metasurface. In addition to the numerical calculations, here we describe the experimental results at the nanoscale. We fabricated the periodic metasurface starting from an amorphous silicon thin film using the focused-ion beam (FIB) technique and analysed its properties. For this, we first deposited the amorphous silicon layer of the height of 214 nm and then created the periodic metasurface of the parallelepipeds with the square base of D = 260 nm and the lattice constant of 400 nm. The transmission, reflection, and absorption spectra of the fabricated metasurface were experimentally measured and explained by the multipole decomposition approach. In addition, the experimental results show the good agreement with theoretical predictions.Here, we study the absorption of light in the silicon metasurfaces on the glass substrate depicted in Fig. 44 however numerically we construct the model with the boundary conditions dictating the periodic nature of the metasurface. Here, we integrate the light induced polarization over a single element of the infinite periodic metasurface to calculate the multipole contributions to the scattering electric field amplitude p, the magnetic dipole (MD) moment m, the electric quadrupole (EQ) moment T and the electric octupole moment 44. The electric and toroidal dipole moments which are the multipole moments of the first and third orders, respectively, can be treated as the total electric dipole (TED) moment εd.To analyse the multipole behavior in the periodic metasurface, we use the same theoretical background as for the single particles we considered in ref.n 52:k0 and kd are the wave numbers in vacuum and in surrounding medium respectively, ε0 is the vacuum dielectric constant.To acquire the multipole decomposition spectrum, we consider the total multipole contributions to the scattering amplitude εd. In the next sections, we present the numerical and experimental results describing the broadband absorption effect obtained with the designed metasurface. For qualitative estimations of the multipole contributions to the transmission and reflection spectra we use Eq. n = .Considering the multipole decomposition spectrum with this approach, we can associate the multipole response of every particle in the array with the total optical response of the entire metasurface. Note that the Eq. To explore the wide-band absorption effect, we consider the metasurface which consists of the parallelepipeds with the square base edge of 260 nm, the height of 214 nm and the lattice constant of 400 nm. First, we calculate the transmission, reflection, and absorption coefficients Fig. . Second,λ = 800 nm corresponds to the dip in the transmission spectrum. At λ = 725 nm, there is the well-pronounced absorption gap which corresponds to the transmission peak around this wavelength. To analyse the contribution of the multipole moments in the absorption broadening effect, we calculated the multipole decomposition of the electric field amplitude D, magnetic dipole m and electric quadrupole Q moments in the considered structure. The area of the broadband metasurface absorption is enclosed in the dashed bluish box.Figure λ = 725 nm. One can note that the TED and MD moments have similar contributions to the electric field amplitude 53. The multipole phases are presented in Fig. λ = 675 nm. The similar phase of these moments is the second condition to achieve the Kerker effect. However, one can see the broad electric quadrupole resonance area around the same wavelength (λ = 675 nm). Due to the EQ excitation, the transmission in this range still does not increase crucially for this structure. The reflection coefficient in Fig. λ ≤ 775 nm can be associated with the dominating EQ resonant contribution to the scattering process. The MQ resonance in this area is not dominant enough to entail the strong interference and to provide the additional suppression of the backscattering in the considered structure54. In the region around λ = 775 nm, the transmission spectrum experiences a dip. This dip appears together with the well-pronounced absorption peak in this area, so the reflection does not crucially increase. It could be noted that the mentioned absorption peak is associated with the excitation of the MD moment resonance at the same wavelengths region as in Fig. λ ≤ 900 nm, the absorption decreases, and despite the magnitudes of the TED and MD moments become different, their phases become similar. Together with the decrease in the EQ moment contribution, the Kerker-type effect is realized and therefore the transmission increases.Let us consider the region around λ ≤ 750 nm dramatically decreases compared to the Fig. λ = 675 nm. This effect can be obtained due to the in-phase interaction of the TED and MD moments as one can see in Fig. λ = 775 nm. It is interesting to note that the small MQ resonance at λ ≈ 630 nm also experiences the blue shift.To study the influence of the glass substrate, we analysed the structure made of the same nanoparticles but in this case, they are embedded in air. In Fig. λ = 775 nm. In Fig. λ = 775 nm. Importantly, the broadening of the absorption effect is realized entirely due to the metasurface parameters and it is not associated with the natural properties of silicon. Worth noting that spectral region of MD resonant excitation corresponds to the enhancement of electric field concentration inside each parallelepipedal particle constructing the metasurface  of the fabricated metasurface shows the fabricated pattern . The substrate influence has been taken into account using the two-step numerical model. The multipole excitations have been analysed using the multipole decomposition approach considering the irreducable Cartesian representations of the multipole moments.The silicon film has been deposited in BGU using the system equipped with the 3 kW 4 pocket e-Gun, the thickness monitoring, the sample heater and the indirect temperature monitoring. The dispersion of n & k parameters measured with this system is presented in Fig. dexp ≈ 0.05 mm through the single mode fiber using the supercontinuum white light source “Fianium WhiteLase”. Transmitted light has been collected with the x10 optical objective and coupled to the multimode fiber which is in turn was connected to the optical spectrum analyser “Yokogawa AQ6370D”. The schematics of this home-made setup is shown in Fig. The ellipsometry measurements have been conducted in BGU with the QDI alpha-SE spectrosopic ellipsometer. The experiment in BGU has been conducted with the home-made setup for the transmission measurements. The sample was illuminated from the close distance


The aim of the research is to draw attention to the accuracy of the pollutant emission research results in driving cycles, and the comparison of pollutant emission results and their repeatability obtained in successive NEDC cycles under cold and hot start conditions. The results of the analysis show that, in the case of a small number of measurements, the results repeatability analysis is necessary for a proper interpretation of the pollutant emission results on the basis of the mean value. According to the authors’ judgment, it is beneficial to determine the coefficient of variation for a more complete assessment of exhaust emission result repeatability obtained from a small number of measurements. This parameter is rarely presented by the authors of papers on exhaust components emission research.Measurement of car engines exhaust pollutants emissions is very important because of their harmful effects on the environment. This article presents the assessment of repeatability of the passenger car engine exhaust pollutants emission research results obtained in the conditions of a chassis dynamometer. The research was conducted in a climate chamber, enabling the temperature conditions to be determined from − 20 to + 30 °C. The emission of CO, CH The harmful effects of transport on the environment are mainly related to the emission of gaseous and particulate pollutants by internal combustion engines. For many years, work has been conducted to minimize energy consumption and reduce exhaust pollutants emissions. This also applies to the introduced emission standards and testing procedures for internal combustion engines. In the case of passenger cars and light commercial vehicles, Euro procedures, which include standards for exhaust pollutants emissions during driving cycles, under controlled conditions, on a chassis dynamometer, and road tests apply in the European Union , which allows for a more accurate assessment of the test results. The authors believe that determination of the coefficient of variation should be helpful in assessing the reproducibility of results obtained by various laboratories. There are currently no standards for specifying a satisfactory value of variation coefficients regarding exhaust emission research in stationary tests of motor vehicles.The research bench is located in the Laboratory of Automotive Ecology at the Faculty of Mechanical Engineering and Aeronautics, of the Rzeszow University of Technology Fig. . The DepThe tests concerning the paper in question were conducted on the chassis dynamometer Zoellner ROADSIM 48 manufactured by AVL built in a climate chamber. The basic specifications of the chassis dynamometer are provided in Table The AVL climate chamber enables temperatures between − 20 and + 30 °C. The measurement bench includes the AVL CVS i60 exhaust dilution system and the AVL AMA i60 exhaust analysis system. The basic data of those systems are summarized in Tables Fo is the road load (N) and V is the vehicle speed (km/h).The object of the test was a passenger car manufactured in 2003 with a mileage of ≈ 60,000 km, whose basic specifications are shown in Table View of the tested car on the research set-up is shown in Fig. The tests were performed according to the NEDC test procedure, which consists of two phases: UDC and EUDC Fig. . The tes4, NMHC, NOX, CO, CO2. The tests were conducted using the AVL PUMA Open system with iGEM Vehicle software and with automatic exhaust emissions measurement.The exhaust pollutants emission measurements were conducted with a constant volume sampling system (CVS i60). Prior to each test, the exhaust analysis system was calibrated. The air diluted exhaust was directed during the bag analysis cycle and the modal analysis, i.e., measurement of diluted exhaust concentrations, was performed in parallel using an AVL AMA i60 exhaust analysis system. After the test completion, the pollutant emissions average values, for the below exhaust components were calculated in individual phases : THC, CH2After a series of three cold start tests (tests 1–3) and three hot start tests (tests 4–6), the measurement method accuracy was analyzed and the repeatability of the test results was assessed.The precision is defined as the degree of compatibility between independent test results obtained under the established conditions .The results for the emission of toxic pollutants in exhaust obtained in subsequent tests according to the NEDC test are presented in Table Figures 2 emission , differ to a greater extent than in the cold start tests (tests 1–3). This may be due to the engine control strategy and the catalytic reactor efficiency. The results of the test included in paper . In the case of the EUDC cycle, the emission factors were similar to those from cold and hot starts. This is due to the heating of the engine during the test . COion Fig.  in the uX, CO, HC) in the catalytic converter, depending on the air/fuel ratio. For lean mixtures (λ > 1), the average conversion efficiency of the NO for exhaust temperature 100 °C constituted ≈ 99%, while for an exhaust temperature of 300 °C it was ≈ 6%. In the case of CO and HC, a low conversion efficiency of the catalytic converter occurs for the rich mixtures. The lower exhaust temperature, regardless of the composition of the blend, caused less variation in conversion efficiency of the catalytic converter , it is apparent that for all toxic components, i.e. CO Fig. , NOX , lower emission for the hot start tests was obtained. For the NEDC tests performed for the hot start , presented by the authors’ paper  and amounted to 0.0031 g/km. A significant value of the standard deviation for CO emissions of ≈ 0.023 g/km was obtained in the case of hot start tests. This value was 4.5 times greater than the value obtained for cold start tests, which may be related to the abovementioned engine control strategy and conversion efficiency of the catalytic reactor. Similar results were obtained by the authors of paper  in relation to those measures obtained from the cold start . For the remaining exhaust components analyzed, the repeatability limit for cold start tests was higher than for hot start tests .Values of ts Table , were hi4 . The lowest CV values were obtained for specific emission of CO2, both for the entire test, as well as for each of its phases . Low variability was also evident in the specific emission of CO . The highest CV values were obtained for specific emission of CH4 (≈ 138% in the EUDC cycle from the cold start and ≈ 95% in the EUDC cycle from the hot start). High variability was also obtained for specific emission results of NOX for the hot start tests in the UDC cycle (≈ 65%). For cold start tests, a high CV value (≈ 21%) was obtained for specific emission of THC in the EUDC cycle. For the entire NEDC cold start, the highest variability (CV = 13%) was observed in the specific emission of NOX. Similarly, for the hot start for the entire NEDC test, the greatest variation was obtained for NOX (CV = 57%). It should be borne in mind that the values of the CVs depend on the absolute values of the pollutants emissions measured, hence their values depend on the accuracy of the measuring equipment used, which is of particular importance for the exhaust components for which the absolute values of the emissions are low.CV values Fig.  for the 4, NMHC, NOX, CO2), the accuracy parameters were indicated for the first phase of the test (UDC), the second phase of the test (EUDC), and for the entire test (NEDC).The paper presents research on the accuracy of the results for exhaust pollutants emission, conducted under repeatability conditions for a small number of tests. For this purpose, a passenger car with Euro 4 homologation was used; moreover, the results of measurements proved that despite the mileage it still met the Euro 4 emission limits. The analysis included tests performed in the climate chamber according to the NEDC test both from cold (coolant temperature of 21 ± 1 °C) and hot (coolant temperature of 90 ± 2 °C) starts. For the measured exhaust components .A higher engine temperature at the test start is associated with greater variability (CV) of the results obtained for the NEDC cycle.The results obtained in phase 1 (UDC) have the greatest impact on the results of the emission for the entire test.4, and NOX emissions.A higher engine temperature at the test start is associated with lower values of repeatability r for the NEDC cycle and CO, CHFor a more complete assessment of the pollutant emission results obtained for a small number of measurements, i.e., when they may be subjected to a large scattering, it is advantageous, according to the authors, to determine the parameter allowing the evaluation of repeatability of obtained values. The use of that coefficient can be helpful in the evaluation of the repeatability of the results obtained by various laboratories.The coefficient of variation (CV) that is used in such cases for assessing variability is unsuitable when very low emission values are recorded. However, such achievements are desirable from environment protection point of view.Since the CV values depend on their average values, the authors suggest to define the threshold coefficient of variation (CVL) determined relatively to the standard emission limit value of a given exhaust compound (L), instead of the average value. The values of the proposed CVL do not depend on the mean values, so they can be used for evaluation of repeatability of the test results.Currently, there are no satisfactory standards specifying the CV values relating to the results of pollutant emission obtained during steady-state driving cycles. In the case of the standard of Euro 4, which was implemented for the tested vehicle, the maximum threshold CVL should be less than 20%.X emissions are respectively equal to 2.6, 0.5, and 3.1%. The obtained values of CVL for harmful exhaust compound emission results  are acceptable for cold start tests.For the conducted cold start tests, the values of the defined here CVL for THC, CO, and NOThe conducted research allows us to formulate the following conclusions:Based on the analysis of obtained test results, authors conclude that the performing procedures for driving cycles have been implemented correctly. The steps taken to establish the satisfactory values of the coefficients of variation separately for each analyzed exhaust component appear to be desired. The research carried out by the authors show that the selection of acceptable values of this coefficient should consider the thermal state of the engine at test start. This objective requires a statistical series of results obtained from various laboratories.


We describe an application of laser-driven modulation in a dielectric micro-structure for the electron beam in a free-electron laser (FEL). The energy modulation is transferred into longitudinal bunching via compression in a magnetic chicane before entering the undulator section of the FEL. The bunched electron beam comprises a series of enhanced current spikes separated by the wavelength of the modulating laser. For beam parameters of SwissFEL at a total bunch charge of 30 pC, the individual spikes are expected to be as short as 140 as (FWHM) with peak currents exceeding 4 kA. The proposed modulation scheme requires the electron beam to be focused into the micrometer scale aperture of the dielectric structure, which imposes strict emittance and charge limitations, but, due to the small interaction region, the scheme is expected to require ten times less laser power as compared to laser modulation in a wiggler magnet, which is the conventional approach to create a pulse train in FELs. The emission of radiation in a so-called self-amplification of spontaneous emission (SASE) FEL grows exponentially from noise and critically depends on the local properties of the electron beam. A technique proposed to generate an individual or a train of sub-femtosecond X-ray pulses with well-defined separation is the energy modulation of the electron bunch with a laser in the magnetic field of a resonant undulator. This “enhanced SASE” (ESASE) method has been proposed by Zholents8, and is being implemented at LCLS9 and planned at other facilities such as SwissFEL10.Free-electron lasers (FELs) make use of a highly compressed relativistic electron beam to generate electromagnetic radiation in a magnetic undulator. They are the brightest sources of radiation from the VUV11. Athos is planned to deliver FEL radiation for wavelengths ranging from 0.65 nm to 5 nm from 2020 and user operation starting in 202110. Adjusting these chicanes to form overlap between each X-ray pulse with the subsequent slice of the electron bunch, the longitudinal coherence is transferred along the bunch, and the X-ray pulses become phase-locked12. The chamber for the interaction with the laser and dielectric structure is installed in the switchyard to Athos and is currently being commissioned. It is also planned to demonstrate GV/m gradients in these dielectric laser acceleration (DLA) structures with a length of 1 mm at a laser wavelength of 2 μm14.In this paper, we present and compare an alternative method to the conventional undulator modulation scheme. We show, that by modulating the electron bunch in a laser-driven dielectric micro-structure, similar modulation strength can be achieved with significantly lower laser power. The beamline required for the proposed scheme is sketched in Fig. 15. Here, λS denotes the periodicity of the dielectric structure, λL is the wavelength of the laser, β is the normalized electron velocity and n is the order of the spatial harmonic of the evanescent wave. To sustain the interaction over distances greater than the laser pulse length, a pulse-front tilt setup for the laser has to be employed18. Due to the optical phase dependence of the interaction, net-acceleration of electrons can be achieved only for bunches significantly shorter than the wavelength of the driving laser. The momentum distribution of an electron beam longer than the laser wavelength will be modulated sinusoidally. The amplitude of the longitudinal interaction with the evanescent waves inside the channel of a straight grating is proportional to 19. As a result, the modulation amplitude becomes homogeneous over the entire gap and the transverse component vanishes for ultra-relativistic particles . However, compression of the transversely modulated beam leads to a slice emittance increase within the spikes. Since the SASE process strongly depends on the transverse slice emittance21, we focus on longitudinal modulation only to avoid emittance growth after compression.Illuminating a dielectric double grating with a laser creates evanescent waves which travel inside the gap of the structure with a phase velocity defined by the periodicity of the structure and the wavelength of the illuminating laser. For a laser incident normal onto a straight grating a net interaction with a charged particle traveling along the gap is achieved if the resonance condition is fulfilled: 22 to obtain a low emittance and a low energy spread for a bunch charge of 30 pC and an energy of 3 GeV to achieve a final peak current which is sufficient to drive the SASE FEL process. This working point is covered by the parameter range of the SwissFEL accelerator at the location of the ACHIP interaction chamber in the switchyard of the Athos beamline. At this location, the optimized distribution has a peak current of 300 A and a length of around 100 fs. The tracking of the modulated electron beam through the switchyard is done with ELEGANT23 including longitudinal space charge and coherent synchrotron radiation effects. The electromagnetic field distribution created in a laser excited structure is modeled with CST Studio24. We observe that the longitudinal and transverse component of the interaction follow the analytical solution derived in19. The remaining transverse modulation amplitude is 500 times smaller than the transverse momentum spread for the case of the 3 GeV electron beam in Athos at SwissFEL. Detailed information about the field simulations with CST are given in section Methods -. We conclude, that the interaction of an ultra-relativistic electron bunch (19. The symmetry of the field can be even further enhance by illuminating the structure from both sides or by using a distributed Bragg mirror (DBR) behind the structure26. The DBR consists of dielectric layers with well-defined thicknesses to reflect the laser and mimic double-sided illumination. The modulated electron beam is bunched in a subsequent magnetic chicane. Naturally, the achievable peak current after optimal compression depends on the slice energy spread and increases with the modulation strength, up to a certain limit. To illustrate the concept we use a modulation amplitude of 0.5 MeV, which leads to an optimal R56 around 2 mm. This modulation amplitude corresponds to an average acceleration gradient of 0.5 GV/m in a 1 mm long structure. Such gradients have recently been demonstrated in a DLA experiment for relativistic electrons at the Pegasus facility at UCLA in 0.5 mm to 1 mm long structures27. The interaction length was limited to 21.5 μm due to the temporal overlap with the laser pulse since no pulse-front tilt for the laser was used in this experiment. The simulation results for the DLA modulation, transport through the Athos switchyard of SwissFEL and optimal bunching are summarized in Fig. γ is the relativistic mass factor. For SASE FELs the energy spread has to be much smaller than the FEL Pierce parameter ρ, see for instance28. In case of the Athos beamline at SwissFEL, ρ is in the order of 1 × 10−3, this condition is fulfilled. Since the transverse modulation is negligible, we assume the slice emittance  of 26 m−2 and 39 m−2 at a beam energy of 3 GeV13. For the parameters described above, the beam size and charge are plotted against emittance in Fig. σ = 0.3 μm, corresponding to a 4σ-aperture of 1.2 μm, a normalized emittance smaller than 100 nm rad is required, which limits the charge to approximately 50 pC.Typical apertures of wiggler magnets are in the order of 10 mm. Typical electron beam sizes at the end of FEL accelerators are far below 1 mm. Thus, no special focusing elements for the electron beam are needed and the charge which can be transported through the modulating structure is not limited by the geometry of the wiggler. In comparison, the DLA modulation scheme requires the GeV electron beam to be focused into the micrometer scale channel of the dielectric structure. This can be achieved with strong quadrupole focusing in combination with low transverse emittance. The beam size is determined by the emittance and the Twiss parameter γ obtained from a laser with a peak power Ppk and a spot size σL,x, σL,z (distance from the center at which the intensity drops to e−1/2 of the maximum), in a dielectric grating of length LS, is calculated byThe total modulation amplitude Δe1 denotes the structure factor being the ratio of the effective acceleration gradient to the required incident electric field strength, which corresponds to the Fourier coefficient of the first spatial harmonic of the laser field inside the structure19. This formula is derived by integration of the electric field along the structure, and from the relation between the electric field vector I of an electromagnetic wave, Here, z, the dependence is not monotonous and an optimum value can be found. To illustrate this, we plot the effectiveness against x for a Gaussian laser beam is e1 for a relativistic DLA made of fused silica is typically in the order of 1, which is what we use for the calculation, but can be higher (2–3) for a smaller gap size29. Higher e1 however goes along with more resonant structures, prohibiting short pulse operation. Based on these parameters and the formula above we calculate an effectiveness of the DLA modulation technique of around 140. The ESASE scheme proposed for LCLS at Stanford uses a modulation at a wavelength of 2.2 μm with a modulation amplitude of 7 MeV. The required laser peak power is estimated to be 10.7 GW30. The corresponding effectiveness, as previously defined, for this setup is around 12. In comparison, the proposed DLA modulation scheme is around ten times more effective, in terms of required laser power for a fixed modulation amplitude. The requirements for the electron beam and the laser pulse for both schemes are summarized in Table This quantity describes the effectiveness of the modulating process and can be applied to any laser driven modulation scheme. It is a measure for the laser power which is required to achieve a certain modulation amplitude. The effectiveness depends on the laser focal spot size as described by Eqs. mentclass2pt{minimλL is required. In an experiment, this can be realized by a tunable optical parametric amplifier (OPA) in combination with reflective optics for the laser transport to avoid chromatic effects, for example in lenses. Regarding the undulator modulation scheme, the wiggler parameter KW needs to be modified such that λW is the wiggler period8. This is typically achieved by adjusting the gap between the magnetic arrays of the wiggler. In order to change the modulation period in the DLA scheme, the periodicity of the structure needs to be modified according to the resonance condition 31. The length of the individual current spikes can be tuned in both schemes by adjusting the modulation amplitude and the compression factor R56 of the magnetic chicane. For the DLA modulation scheme the modulation amplitude is limited by the damage threshold of the dielectric structure. Irreversible damage of fused silica DLAs has been observed for incident electric fields of around 9 GV/m27, which would limit the modulation amplitude to around 9 MeV for a 1 mm long structure. This boundary is more than one order of magnitude higher than the modulation amplitude we use for the bunching simulation presented above.Another important aspect for both schemes is the tunability of the modulation period. In both scenarios a change of the laser wavelength We investigated a scheme which uses the fields in a dielectric micro-structure excited by a laser to modulate the ultra-relativistic electron beam of an FEL with the purpose of creating a pulse train for the ESASE scheme. Beam dynamics simulations using parameters of the SwissFEL accelerator predict the generation of 140 as (FWHM) electron spikes with peak currents up to 5 kA from a bunch with a total charge of 30 pC and an initial current of 0.3 kA. These beam parameters are suitable to drive the SASE process and create pulsed FEL radiation. Investigation of the properties of the radiation created in the Athos undulators at SwissFEL, through time-dependent FEL simulations, can be subject of a future study. Especially, the undulator tapering to increase the pulse energy while keeping the FEL pulses short and phase shifters between the undulator sections to create phase-locked pulses need to be optimized in detail.The micrometer scale aperture of the dielectric structure implies strict limitations for beam focusing and transverse emittance. For the case of the Athos beamline at SwissFEL, the estimated charge limit is around 50 pC. Losses created by the tails of the beam limit the achievable repetition rate. The conventional undulator modulation scheme has no such limitations and is favourable in a high charge, emittance and repetition rate machine. Based on our estimation, dielectric laser modulation for ultra-relativistic electron beams is around a factor of ten more effective than undulator modulation, meaning that only a tenth of laser power is required to achieve the same modulation amplitude of the longitudinal phase-space. Since the laser system is one of the main cost drivers of the ESASE scheme, the proposed DLA modulation scheme presents a significant economic advantage. This aspect is particularly important for future, more compact and less expensive X-ray FEL facilities which will operate at lower charge and emittance than existing facilities. Both schemes allow continuous scanning of the modulation period, by changing the wavelength of the laser and properties of the modulating structure: in the undulator modulation scheme the wiggler parameter needs to be changed; in the DLA modulation scheme the periodicity of the grating needs to be adapted. We propose to use a diverging grating for continuous scanning capabilities.32. In the simulations the assumed intrinsic emittance is 550 nm/mm, accordingly to what was measured at the SwissFEL injector test facility34. Downstream of the injector the beam is compressed in two stages. We optimized the compression parameters to have an optimum balance between the intrinsic energy spread, the peak current and the residual chirp. We simulated 2 × 105 macro-particles for the tracking with ELEGANT23 of the 30 pC-distribution which is used in this study.SwissFEL can run at different bunch charges to accommodate the requests of the users. For each case, the emittance and the optics mismatch along the slices of the beam have been optimized at the injector to maximize the uniformity of the properties along the bunch, and therefore the lasing intensity. The main parameters included in the optimization are the first focusing at the exit of the radio-frequency gun, along the first accelerating structure, and the transverse size of the laser at the cathode. These parameters are determined starting from the layout corresponding to the 200 pC design case of SwissFEL and finely tuned using a simplex optimizer23. Due to the ultra-relativistic energy (3 GeV) transverse space charge effects at the focus can be neglected and the particle tracking code ELEGANT can be applied. The laminarity parameter, as defined for instance in35, must not exceed unity in order to neglect space charge forces. For the parameters of this study the laminarity parameter at the interaction point is indeed around 1 × 10−3. The Twiss parameters  the transverse modulation is expected to be smaller than 0.5 keV. The rms transverse momentum spread at the interaction point is calculated to be around 260 keV, which is a factor of 500 larger than the transverse modulation effect. The large transverse momentum spread is a result of the strong focusing. Hence, the degradation of the slice emittance by transverse forces in the structure can be neglected for an ultra-relativistic electron beam.The evaluation of this integral along the gap using the fields simulated by CST Microwave Studio confirmed the analytical description, i.e. that the transverse effect vanishes, see Fig. 23 show an increase of the spot size by 1% for a typical energy error of 0.1%. For the calculation of the transmitted fraction we consider a maximum position offset of Good stability of beam position and size is required for the DLA modulation scheme, due to its micro-meter sized aperture. The electromagnetic field simulation for the proposed structure shows good homogeneity of the modulation strength (<0.05%) and negligible transverse forces across the channel gap. Therefore, beam position and size jitter will not affect the modulation strength but only the number of particles hitting the boundary of the dielectric grating. In the following, we estimate the fraction of particles being transmitted through the structure geometry defined above. An upper limit for the position jitter of the electron beam in the accelerator of SwissFEL is 10% of the beam size. The beam size measured along the machine typically shows jitter in the order of 3%. In the proposed strong focusing setup, energy jitter adds significantly to the beam size due to chromatic effects of the permanent magnet quadrupoles. Simulations using ELEGANT


Human blood is a complex sample matrix when elemental analysis is considered. In this study, the effects of Na, a natural component of serum samples, was investigated in the quantitative determination of Ca, K, Mg, Cu, Zn, and Fe by microwave plasma atomic emission spectrometry. The robustness of the microwave plasma was tested by evaluating MgII 280.271 nm/MgI 285.213 nm by varying two adjustable operating parameters, the read time, and the nebulizer pressure. The read time has no influence on the robustness while the MgII/MgI ratio decreased when the nebulizer pressure was increased during the analysis. The threshold concentrations of the interfering Na were determined at the analytical lines used for the measurement of other elements. The matrix effect of the commercially available microwave plasma was studied by a series of model experiments with human blood. The increasing concentration of Na in the matrix within the normal ranges reported for blood serum increased the intensities of the measured atomic lines. According to a factorial design—where two applied factors were the concentration of Na matrix and the measured elements as well as their levels were considered as factorial points—it was found that the Na concentration in a serum sample after acid digestion and 10 times dilution affected the intensity values of the measured elements. For Ca, Cu, and Fe, a statistically significant effect was observed, while for Zn, Mg, and K, an interaction effect was also found. However, after calculating the percentage errors caused by the shift, the relative difference was observed to be quite small (< 10%). Atomic spectrometry provides various efficient possibilities for elemental analysis which is mostly designed to measure the analyte from a solution . MicrowaDespite of the disadvantages, the development of MIP discharge has continued to be in the focus of producing a cheaper and easier-to-operate technique and many initial issues have been resolved. In 2008, Hammer reported and described a plasma source excited by the combination of magnetic and electric fields which approached the operation stability of the inductively coupled plasmas . Based oThe elemental analysis of human blood is significant in the routine clinical practice as well as in medical research. The elemental concentration levels of blood serum may indicate certain diseases; therefore, their determination is useful for establishing diagnosis –23. BlooThe matrix effects occurring in the emission source during the measurement is highly important to be studied since it affects the analytical results . Due to For these studies, three macroelements  and three microelements  were chosen which are commonly determined in blood serum due to their essential role in the human body –36. The 3 + H2O2) and from 10 times dilution with ultrapure water [According to our previous paper, blood serum can be conveniently analyzed by MIP for Mg, Ca, K, Zn, Cu, and Fe after acid digestion  , 25, 37.−1 in the digested and 10 times diluted blood serum samples. Such model solutions were prepared from a commercially available mono-element spectroscopic standard of 1000 mg L−1 . In accordance with the typical blood serum composition, the concentration of Na was set to 300–400–500 mg L−1 using reagent-grade solid NaCl . All model solutions were prepared in the presence of the Na matrix applied in the three different concentration values and the absence of it in triplicate, together with blank samples. The measurement of the model samples was carried out using calibration solutions with and without the Na matrix. In order to determine the best emission intensity ratio of the MgII 280.271 nm/MgI 285.213 nm lines, the intensities were measured by varying the read time (1–2–3–5–10–20–30 s) and nebulizer pressure (80–100–120–160–180–200–240 kPa).The expected average concentration of Mg was 3 mg L−1) and measuring the emission intensities of the selected analytical lines of the measured elements  [The threshold concentration for the Na as an interfering element was determined by analyzing the single element solution of Na in the same concentrations as used in the robustness study (300–400–500 mg L and Fe) .The threshold concentration was calculated on the basis of the following expression:blankSDthe standard deviation of the blank sampleCNathe concentration of the Na present as the interfering elementAlSigthe intensity signal obtained at the particular analytical line of the measured elements .To study the effect of Na present on the quantitative determination of Mg, K, Ca, Cu, Zn, and Fe in blood serum samples by MP-AES, a factorial experiment was designed with model solutions. We chose a minimum (MIN), a center (CENT), and a maximum (MAX) concentration of the elements, in accordance with the expected concentrations in the prepared and diluted serum solutions. Tables −1  while the concentration of the Na matrix was set by solutions prepared from reagent-grade solid NaCl . The model solutions were prepared in duplicate and were analyzed for Na, as well as for the microelements and macroelements.The concentration of the measured elements  were set in the prepared solutions by using commercially available ICP standards of 1000 mg LIn the second series of experiments, the effect of increasing Na concentration was studied on the quantitative determination of the measured elements . The model solutions contained the Na matrix in increasing concentration, while the measured elements were present in the center concentration  were determined by the MP-AES method . A five-point calibration was applied , and the solutions were analyzed in a completely randomized sequence. The concentration of each element was determined by measuring the intensities at three or more wavelengths as the average of five independent measurements. The most appropriate analytical lines were selected based on the results of the optimizing method. The emission wavelengths used were those where the best signal/background ratio was obtained and the spectral interferences were the lowest. The purity of applied chemicals and various parts of the equipment was verified by blank samples.In the present study, multivariate statistical methods were used for the optimization of the measuring parameters –43. The n full-factorial design which is an efficient mathematical approach to optimize responses [.). The graphical evaluation of the responses is calculated and created by Statgraphics Centurion XVII software. To study the effects and interactions, ANOVA  test was applied using the SPSS IBM 22 for Windows software package. Limit of detection values for the measured elements were calculated based on the ANOVA results.The first model experiment of the Na matrix effect was a 2esponses –46. The The term robustness refers to the capacity of the plasma source to offset the changes originating from environmental factors, operating conditions, or matrix effects without causing a significant error in the measurement result. The robustness is reached for ICP-OES when the ratio of the signals gained at the ionic and atomic lines of Mg are higher than 6 .Using multivariate optimization design, the analytical performance of a technique can be improved by selecting the most relevant variables affecting the measurement process .The adjustable operating parameters of the MP-AES were considered as factors (read time and nebulizing pressure), and Fig. Since it was already described that in the wavelength region of 280 and 285 nm the response is increasing relatively slow, a factor of 1.15 must be used to multiply with the intensity ratio data of ionic to atomic lines for Mg , 49. TheAs expected based on previously published results, the ratio of the Mg ion to atomic line intensities varies between 0.5 and 1.0, while for ICP-OES the values are in the range of 8–14. The reason for this difference is explained by the lower temperature and electron densities present in the microwave plasma , 49.It is clearly shown that the value of read time has no influence on the gained intensity data and the calculated ratios. However, a higher ratio was achieved with decreasing nebulizer pressure which shows agreement with Chalyavi et al. . This teThe threshold value determined for a selected analytical line is the concentration of the interfering element above which it statistically can cause a significant effect on the measurement results. The threshold values gained are indicated in Table The results confirm that only the 210.857 nm line of zinc and 285.213 nm line of magnesium are free of the interference by the Na matrix in the concentrations expected in the pre-treated blood serum samples. In the other cases, a statistically distinguishable signal was generated by the matrix.The elemental analysis of human blood is a difficult task due to its relatively complex sample matrix and the low concentration of the analyte to be measured. The microwave plasma technique may prove to be a suitable alternative of ICP-OES for routine blood analysis if by lower temperature MIP can handle the residual matrix components in the acid-digested human serum.In the first series of model experiments, a full-factorial design was used to test the effect of the Na matrix by MP-AES in a concentration range that is expected in a human blood serum sample after digestion and dilution. Analytical techniques such as ICP-OES can generate a number of responses  which isFactorial designs are used to investigate how the variables and/or interactions between them influence the gained responses, and even the magnitude of the possible effects can be estimated. In our case, the number of the studied variables is quite small and hence the full-factorial design could easily be applied. It is one of the most used screener methods to study the significance of experimental variables in ICP-OES , 50.Figure .It was found that the Na concentration present in a blood serum sample after acid digestion and 10 times dilution affected the intensity values of the measured elements. For Ca, Cu, and Fe, a statistically significant effect was observed, while for Zn, Mg, and K, an interaction effect was also found. The graphic evaluation of the surface responses for the other elements is indicated in Fig. 42-MIP systems but reported to be rather similarly to the AAS, and the observation height of 12 mm [It is long known that the presence of EIEs in the ICP excitation source under certain conditions—despite the higher temperature and electron number density in the Ar plasma—may result in a shift of the ionization equilibrium state affecting the gained intensity values –53. UsuaInterferences are even higher when the plasma is viewed axially . The MP-The EIEs can shift the ionization equilibrium and enhance the atomic line emission in AAS measurements. According to the literature data, EIEs causes an enhancement effect at atomic lines and a suppression effect at ionic lines in ICP-OES. In the lower-temperature nitrogen plasma of the MP-AES method, much more atomic lines are available than ionic ones compared to the ICP-OES. For example, out of the 20 available lines of Ca, only three are ionic and exactly the same is true for Mg. For Cu, there are 20 lines present to choose from but only two are ionic, while from the 20 lines of Fe only one. Thus, our measurements were carried out at the atomic lines of Ca (430.253 nm), K (766.491 nm), Mg (285.219 nm), Cu (324.754 nm), Zn , and Fe (371.993 nm), respectively. It was proved that an enhancement of the measured intensities must be considered when digested blood serums are analyzed by MP-AES, similarly to ICP measurements.−1), the percentage differences of the shift in the intensity values were calculated, as indicated in Table The observed effects were further evaluated by ANOVA. Based on the average expected Na concentration in the digested and diluted serum samples (400 mg L−1 in the samples occurred in the case of Mg—the intensity results gained for Mg were shifted with − 7.8% when the matrix concentration of Na was in the minimum (300 mg L−1) and + 7.8% when it was in the maximum (500 mg L−1) of the range naturally present in the human serum after the 10 times dilution compared to the central value (400 mg L−1). The statistical analysis proved the intensity shift and therefore the matrix effect of Na on the intensity results of the measured elements from human blood. However, after the error is quantified numerically, one can conclude that the relative difference is quite small (< 10%).The highest observed difference from the central concentration expected when the Na matrix content is 400 mg L2-MIP system [It is important to note that the effects of nitric acid, another matrix component left in the serum samples after dilution, were not considered in this paper. A high concentration of nitric acid is reported to affect the microwave plasma in a more simple way compared to the EIEs and very similarly to the ICP measurements . The MP-P system .During the measurement of the second matrix effect study from every sample solution, the intensity of all six elements were registered —even those elements were determined which were not present in the model solutions. Thus, a huge amount of data were available from which the background of the measurements was monitored and it proved to be constant during the analysis. The registered background is indicated in Fig. The amount of registered background intensity values provided the opportunity to calculate the limit of detection results from the average and standard deviation of background intensity data which are indicated for each element in Table The aim of this paper is to describe the application of the MP-AES instrument for the cost-effective elemental analysis of digested human blood serum samples. The potential matrix effect is investigated caused by the relatively high concentration of the easily ionized element, Na, naturally present in the serum sample. According to a series of model experiments, the suppressing effect of Na in the expected concentration range in blood serum on the intensity values of the measured elements is statistically proved. The matrix effect was expressed in percentage difference showing the magnitude of the potential errors caused by the lower or higher level of Na in the naturally occurring concentration range, and it was found that the error of measurement for the Cu, Zn, and Fe intensities is under 3%, while for the Ca, Mg, and K intensities it is under 8%.Due to the slight differences observed in the measurement data caused by the shift and the low enough limit of detection (LOD) values, MP-AES can be considered as a coeffective alternative to ICP-OES in routine blood analysis, similarly as proven in the case of other applications . Matrix


We also provide a demonstration of improved performance on an unencoded hydrogen molecule.With rapid developments in quantum hardware comes a push towards the first practical applications. While fully fault-tolerant quantum computers are not yet realized, there may exist intermediate forms of error correction that enable practical applications. In this work, we consider the idea of post-processing error decoders using existing quantum codes, which mitigate errors on logical qubits using post-processing without explicit syndrome measurements or additional qubits beyond the encoding overhead. This greatly simplifies the experimental exploration of quantum codes on real, near-term devices, removing the need for locality of syndromes or fast feed-forward. We develop the theory of the method and demonstrate it on an example with the perfect ] code, which exhibits a pseudo-threshold of  Fault-tolerant quantum computation is still far, but there could be ways in which quantum error correction could improve currently available devices. Here, the authors show how to exploit existing quantum codes through only post-processing and random measurements in order to mitigate errors in NISQ devices. Experimental and theoretical proposals have explored the potential for performing a well-defined computational task faster than a classical computer on as few as 50 qubits, a task often referred to as “quantum supremacy”4. It remains an open question, however, if these results can be extended to applications of interest outside the domain of pure computation.Rapid developments in both the theory and hardware for quantum computation push us closer than ever to the dream of practically useful quantum computing. However, while a key development in the road map of quantum computing was the concept of quantum error correction, the hardware requirements to implement fully fault-tolerant schemes for non-trivial algorithms may still be some years away. A natural question that arises from this realization is whether it will be possible to perform meaningful computations on non-fault tolerant or noisy intermediate scale quantum computers (NISQ)18, which are known to experience a natural form of robustness against certain types of noise. In conjunction with this, much progress has been made in reducing the gate overhead required for practical applications, especially in the domain of quantum chemistry22. However, the impact of incoherent noise remains daunting for the accuracy thresholds specified23. Moreover, while there has been some success in implementing early quantum error correcting code experiments in various architectures25, a full implementation remains daunting. As a result, in order to reach practical applications, it may be necessary to implement some form of partial error correction for NISQ computations. The exact form of this error correction could take to achieve success is yet unknown; however, it has been suggested that one of the best applications for early quantum computers is using them to study and optimize error correcting codes in real conditions26. Yet despite great theoretical progress, most quantum codes are difficult to study experimentally on NISQ devices due to the need for complicated syndrome measurements, fast feedback, and decoding capabilities.Many early proposals for practical applications have advocated the use of variational algorithms30. If one specializes to the quantum structure of fermionic problems, notably the N-representability conditions, enforcing these as constraints alone can reduce the impact of noise in simulations31. More generally within quantum simulation32, an error mitigation technique known as the quantum subspace expansion (QSE)27 was predicted and experimentally confirmed to both approximate excited states and reduce errors through additional measurements and the solution of a small offline eigenvalue problem14. Since then there have been variations leveraging QSE that use both additional techniques from quantum chemistry for excited states33 and imaginary time evolution34.An alternative approach that strays from traditional ideas of error correction and targets NISQ devices is “error mitigation”. This term largely refers to techniques that reduce the influence of noise on a result using only batch measurements and offline classical processing as opposed to active measurement and fast feedback type corrections. While they are not believed to lead to scalable, fault-tolerant computation, it is hoped that sufficient mitigation may open the possibility of practical applications or inspire more near-term error correction ideas. A number of these techniques have been developed both within an application specific and general context35 and post-processing in this framework, which we then generalize using quantum subspace expansions. Although a connection to symmetries was explored in the original work27 and this connection was extended in subsequent work37 that has also been verified by experimental implementation38, these papers have focused on application specific contexts. Here we generalize this to any circuit performed within a quantum code, and show how subspace expansions may be used to then correct some logical errors within the code space, as well as be applied to approximate symmetries of unencoded Hamiltonians. We provide a concrete example using the perfect ] code to demonstrate post-processed quantum state recovery. When applied at the highest level, this recovery exhibits a p ≈ 0.50 pseudo-threshold for an uncorrelated depolarizing channel applied to all qubits. An example of an unencoded hydrogen molecule is also demonstrated across the entire range of depolarizing errors. We close with an outlook and potential applications of this methodology.In this work, we show that it is possible both to use existing quantum error correcting codes to mitigate errors on NISQ devices and to study the performance of these codes under experimental conditions using classical post-processing and additional measurements. We briefly review the theory of stabilizer codesn physical qubits. Quantum error correcting codes utilize entanglement to encode a set of k < n logical qubits, with the hope of improving robustness to probable errors. A code that requires at least a weight d Pauli operator to induce a logical error is said to have distance d. These three numbers are often used to define a quantum error correcting code, with the notation ].We begin by briefly reviewing and establishing notation for the relevant topics of quantum error correcting codes in the stabilizer formalism, and using this formalism to develop a set of projection operators. Consider a set of k logical operators formally written S is the full stabilizer group implying 40.The set of 2Pi = (I + Mi) ∕ 2 that removes components of the state outside the  +1 eigenspace of the stabilizers or code space. One may use this to construct a projector that is a linear combination of projectors to remove desired errors outside the code space. We note that this clearly cannot remove logical errors made within the code space. This is related to the idea of error detection and post selection which has made recent progress both in theory and experiment49. Quantum error detection typically discards results based on syndrome measurements without using correction, however we will avoid the need for direct syndrome measurements, which can be cumbersome on geometrically local qubit layouts and are challenging to do in a fault tolerant fashion for complex codes. For a stabilizer group with generators Si, the complete projector can be formed from iPi is the sum of all elements of the stabilizer group with a constant coefficient which we fix to 1∕2m,m is the number of stabilizer generators used. For the case of full projection, this will be the full stabilizer group which contains 2m terms. While this is generally an exponential number of terms, it will be shown that the number of terms is not an explicit factor in the cost when a stochastic sampling scheme is used to apply the corrections. Rather the correction cost will depend on the volume of the state outside the code space. The group structure allows projective correction of the density matrix When not performing active error correction, one must rely on projection operations. Namely, members of the stabilizer group, X1X2 and Z1Z2 commute, measurement of Z1 and Z2 transversally to determine Z1Z2 destroys the ability to recover X1X2. Our scheme allows us to bypass this difficulty through stochastic operator sampling and works with codes of arbitrary structure. This transvesal measurement scheme that avoids the need for ancilla is also at the core of the method’s relatively high pseudothreshold for a given code.It is important to emphasize here the reason this expansion of projectors is used here, as opposed to traditional measurement of stabilizer generators, which are not exponential in number. In particular, we are prescribing the use of transversal, destructive measurement to avoid the need for measurement syndrome qubits and associated fault-tolerant gadgets. As such, while stabilizer generators commute, transversal measurement of their components may not. For example, while Γ is expressed as a sum of Pauli operators Γi as Γ = ∑iγiΓi. Then the corrected value for the expectation of Γ may be computed fromΓ commute with stabilizer group elements Mi, and if Suppose that some logical Hermitian operator X1Z2Z3X4 as a stabilizer, a true stabilizer measurement would require extracting only the  ±1 measurement using an ancilla. However in this scheme, we are free to use repeated preparations of the state and construct any unbiased estimator of 〈X1Z2Z3X4〉 we desire, including those which might destroy the encoded state. This dramatically simplifies the use of codes with non-local stabilizer measurements.We emphasize a distinction between this measurement scheme and traditional error correction/detection is that we do not need to measure the stabilizers in earnest. As this is a post-processing procedure, we are free to destroy the information in the state by measuring qubit-wise across Pauli operators. To be explicit, if one had the Pauli operator 27. In this section, we show how these constructions can be relaxed for greater flexibility and power with simple relations to approximations of these projectors within a subspace. A schematic comparison of the deterministic subspace expansion and the stochastic variant we will use is shown in Fig. L is the number of terms in the linear ansatz and the check operators, Mi, still come from the stabilizer group, however it no longer needs to be true that ci = 1∕2m and let the sum run over the entire stabilizer group, we recover the expression from the projection technique. To find coefficients ci, we formulate this problem as minimizing the distance to the code space subject to a normalization constraint. Using the Hamiltonian formulation of the code space, this is equivalent to approximating the ground state of the code space byIn the previous section, we showed how explicit projectors from quantum error correcting codes can be used to correct observables in post processing. We generally define the expansion of the problem around a reference quantum state into a small surrounding subspace as a quantum subspace expansion. Expanding the code projectors to the full group (as opposed to their stabilizer generators) is one restricted form of this, but the construction is much more generalρ and choice of Hc in general. From the linear ansatz and normalization constraint, this problem is equivalent to minimization of a quadratic form on the surface of a sphere with a non-orthogonal metric. This general problem has a well-known solution in mathematics50 and is commonly used within linear variational methods in chemistry27 and physics derived from the Ritz method51 or closely related Galerkin discretizations in applied mathematics. The solution is given by the solution of the generalized eigenvalue problemH here forms a representation of the action of the code Hamiltonian in this stabilizer projector basis, the matrix S is the overlap or metric matrix defining the subspace geometry, C is the matrix of eigenvectors, and E is the diagonal matrix of eigenvalues. We note that for cases where the check operators Mi are built from projectors from generators, the solutions coincide with the previous formalism. When it is not the case that one builds the check operators from the product of stabilizer projectors, it provides an optimal solution that interpolates between different numbers of projectors in the subspace given. This type of expansion about a state is referred to as a quantum subspace expansion (QSE). The ground state eigenvector of this NM × NM eigenvalue problem, forms the optimal solution of the above problem within this subspace. In general, the optimal solution of this eigenvalue problem may not be a strict projector. However, this is not necessarily undesirable. In some cases, a lower energy state may be found for a specific problem Hamiltonian that corresponds to a rotation in the logical space.This optimization is dependent both on the state ci may be used to obtain a corrected estimate of any desired logical observable. Alternatively, one may build a representation of an operator in the same basis defined by the expansion operator, and use it to perform further symmetry projections or improve the estimate of a logical observable as in ref. 27.When the solution of this eigenvalue problem is obtained, it may be used in a number of ways. In connection to the stochastic technique prescribed above, the values As in typical quantum error correction, the degeneracy of the ground state of the full code Hamiltonian prevents referencing a single state within the code space. This makes removing logical errors with the above procedure impossible. However, when considered in conjunction with a problem Hamiltonian such as that from a quantum physical system like an electronic system, it becomes possible to correct logical errors as well if the goal is to prepare an eigenstate of this Hamiltonian or minimize its energy. As a simple example, if one has a single encoded spin with a problem Hamiltonian Hp. Here we show how the machinery developed so far can be applied to this case.So far we have considered the case of decoding within an error correcting code that redundantly encodes quantum information via engineered symmetries. However, this strategy inevitably involves some overhead due to the encoding in the execution of gates, and in some near-term experiments, it will still be most practical to work directly in the space of a physical problem Hamiltonian Hp may have symmetries that are often known about the desired state ahead of time. For example in the case of an interacting fermion system, the total number of fermions, the total spin and Sz component, and symmetries related to spatial degrees of freedom in a system are often good candidates. The application of these symmetries has been explored previously, symmetries52, and as these are expected to be exact symmetries, it is always safe to apply them when the symmetry is known.In this case, the physical problem Hamiltonian Np, or While these symmetries are exact and effective to apply, they are often more expensive to implement than one might desire. For example, the problem of number symmetry in a fermion Hamiltonian can take eigenvalues that range from 0 to the number of spin orbitals in the system. Thus to select just the correct particle number, one may have to construct a projector which removes all the components except the desired particle number iZi. An extension of this, is to use both the up-spin (α) and down-spin (β) number parities, which generate the full number parity, and offer additional power in their projection. These are simply given by ∏i∈αZi and ∏i∈βZi respectively. These simple parity symmetries have been utilized before to reduce the number of qubits52, however just as in subsystem error correcting codes, retaining these redundancies can sometimes be beneficial for gate depth or efficiency of representation. That work also contained a general algorithm for searching for unknown As a result, it is much simpler and effective to start with symmetries that have only two distinct eigenvalues, also referred to as p = 3∕4. Although a single qubit depolarizing channel is not a perfect error model for any system, experimental data suggests that correlated errors are much weaker in many architectures than independent, single-qubit errors. In addition, it is expected that in the gate model, use of randomized compiling is advisable to decohere errors, which tends to map other error channels closer to this model on average and allows one to treat phenomena like measurement errors as bit-flip errors. Application of these techniques and analysis to true devices is, of course, the eventual goal in order to both understand their applicability and optimize existing quantum codes.Here we both exhibit some of the performance of the presented techniques and clarify their construction through the use of simple examples. Both a general error correcting code and specific problem Hamiltonian systems with symmetries are studied. We note that in our numerical studies we use a single qubit depolarizing channel defined byp. The logical state is selected at random within the space to not exhibit any special properties with regards to errors. In connection with the formalism above, we evaluate the expectation value of the logical operator To see how the general recovery process using stabilizer codes can work in practice, let us consider the concrete example of the perfect ] code, which is a distance 3 code that encodes 1 logical qubit in 5 physical qubits. To evaluate the performance in practice, we perform the following numerical experiment. A logical state This code has logical operators and stabilizer generatorsWe denote the two states of the logical qubit as p. In this situation, we have up to 5 qubit errors, which we do not expect the code can recover from without introduction of a problem Hamiltonian, however they occur with probability p5, which can be quite small for modest p.We denote the hierarchy of check operators as the elements in the sum generated by FL where the physical line denotes the trivial encoding into one qubit and compare the two for a range of values of p. The infidelity is calculated with respect to the correct logical state, which is a single state in the physical Hilbert space. This is done as we are seeking to discard all non-codeword states, but cannot do so when encoded yet uncorrected. Hence for the encoded but uncorrected state, fully depolarizing each qubit individually will lead to a fidelity of 1∕2n, where n is the number of physical qubits. We define the pseudo-threshold to be the value of p for which the logical infidelity in the encoded space is lower than the physical infidelity for the unencoded system, and we see that at both levels there is a pseudo-threshold in this model. For S(4), the pseudo-threshold is numerically found to be p = 0.50 for this code and symmetric depolarizing channel. We see here that the pseudo-threshold generally depends on the truncation level, however it converges to the pseudo-threshold of the full code for the given operation and error model. If we are sufficiently below the corresponding pseudo-threshold, in the case where no recovery operations are used, we may obtain up to a pd−1 suppression of the error rate. A topic of interest is to investigate how this pseudothreshold may relate to the distance in the more general case using our technique which avoids the need for ancilla measurement qubits.The cross lines in Fig. α and β number parity operators are an efficient choice of projectors. In the Jordan-Wigner representation, these are given by ∏i∈evenZi and ∏i∈oddZi when an even-odd ordering of orbitals are used.We now look at an example of an unencoded Hamiltonian, which has become a canonical test case for quantum computing in quantum chemistry. This is the second quantized hydrogen molecule in a minimal, 4 qubit basis. In this case, the S = {Z0Z2, Z1Z3, X0X1X2X3} which up- and down-spin number parity operators as well a non-local operator which need not be an exact symmetry of the Hamiltonian. The ordering of l matches that given here. The logical infidelity as a function of the depolarizing probability is plotted in Fig. The effectiveness of the technique on this system is evaluated numerically by preparing the exact ground state of the hydrogen molecule and subjecting it to an independent depolarizing channel on all 4 qubits. We choose as the generating operators 26. The modeling of true noise within the device is incredibly difficult as the system size grows, and studying which codes excel under natural conditions and how to optimize them may lead to progress towards fully fault tolerant computation. Indeed, knowledge of biased noise sources can vastly increase the threshold of a given code53. The tool we have provided here gives a method to experimentally study the encoding through post-processing while removing the complication of fault-tolerant syndrome measurement or fast feedback. This allows one to explore a wider variety of codes experimentally before worrying about these final details. We propose that one can run simple gate sequences in the logical space with known results, and use the post processing decoder here to study the decay of errors as stabilizers are added. This limit will inform the propagation of logical errors in the system and allow one to make code optimizations before full fault tolerant protocols are available.As has been conjectured before, one of the best uses of early quantum devices may be to tune quantum error correcting codes under actual device conditions54 or variations of Bravyi-Kitaev superfast55 thought to be good candidates for near-term simulations, without the need for complicated decoding circuits or ancilla for syndrome measurements.We note that this type of decoder benefits greatly from the fact that check measurements need not be geometrically local to be implementable in a realistic setting. This allows one to explore and utilize codes that are not geometrically local on the architecture in use, which may have nicer properties with respect to distance and rate than geometrically local codes. Moreover, they naturally allow implementation of recent fermionic based codes, such as Majorana loop stabilizer codes30. In these techniques, one artificially introduces additional noise to extrapolate to a lower noise limit. To apply this technique to quantum subspace expansions, one simply needs to perform the extrapolation on each of the desired matrix elements, then proceed as normal.Moreover, as this method is a post-processing method, it is entirely compatible with the extrapolation techniques introduced for error mitigationp ≈ 0.50 under a depolarizing channel acting on single qubits in the ] code and made connections to the traditional theory of stabilizer codes. We believe this method has the potential to play a role in the development and optimization of quantum codes under realistic noise conditions as well as the ability to remove errors from early application initiatives.In this work, we have introduced a method for mitigating errors and studying error correcting codes using a post processing technique based on quantum subspace expansions. We showed that in implementations of this method, one can achieve a pseudo-threshold of As suggested in the exposition on the correction formalism, it is key for efficiency to sample the terms for the correction in a way that reflects the state rather than number of terms. We outline and analyze a simple stochastic scheme for performing this sampling here.Γ, which can be decomposed into Pauli operators Γj as jγj = 1 and γj ≥ 0 from absorbing required signs into Γj. Projecting this into the code space of some selected code as before, we haveχ =  is a bit string that we use to conveniently enumerate the stabilizer group operators and we definec as well for Γ = I. We will also assume here that as in the case of stabilizer projectors and the Mi is a symmetry of the encoding, commuting with the desired Hamiltonian and physical observables.Suppose that we want to measure the corrected expectation value of some logical operator χ as in an importance sampling scheme below, however taking the uniform distribution is perhaps the most straightforward and μΓ, x ∈ {+1, −1} from measuring the Pauli operator Γχ,j, and we have lumped this into the probability distribution as Ns terms with the probabilities from this distribution which will yield either  +1 or  −1 from the Pauli measurements, add them together and divide by Ns. The variance in the estimate will be given by the variance of this estimator divided by Ns.To sample the trace stochastically, we may use the coefficients of the terms as a normalized probability distribution. One may choose the distribution to depend on x ∈ {−1, +1} derived from marginalizing over the joint distribution for projector terms χ and Pauli decomposition terms j to findFrom our construction, we see that we can view the estimator as a binomial distribution with a probability for the two results As a result, one may write down a particularly simple form of the variance for the estimator given byΓ, with w, we haveTo understand how the state influences the variance, we consider a simple example case using the total depolarizing channel and a single Pauli operator w = 0 we have that all states are in the code space, and hence the sum over χ trivially collapses, and we have that px = qx, which gives the same statistics as the original measurement of Γ. Thus in such a case one has no dependence on the number of stabilizer terms used in the expansion. Hence, adding this procedure to a perfect state is expected to incur no additional cost on average.In the limit of w > 0, marginalizing over χ is equivalent to applying the code space projector and hence geometrically analogous to determining the volume of the state in the code space. This must correspond equate to a portion of the average being 0, however as we are only capable of measuring 1 and  −1, it then must constitute an equal probability of being in  +1 and  −1 that is determined by the volume of non-code space the state occupies. More explicitlyΓ, so that q+1 = 1, thenw = 0) the variance is minimal and independent of the number of terms, and that the variance increases as the state quality degrades.Considering the imperfect case χ as reflecting the volume of space attached to the projector lets one easily reason about the generalization of this scheme to sampling with recovery. In that case, we simply attach one more probability which allows us to sample over the different selected errors Ei, with probability χ is now better conditioned, as it is determined by the ratio of the volume of recovered space to the volume of Hilbert space rather than the volume of the code space to the volume of Hilbert space. As a result, the sample variance for recovery may be lower than the sample variance for strict projection. However as discussed, the ultimate quality of recovery is expected to be superior for strict projection due to removing errors up to weight d − 1 instead of (d − 1)∕2.The general picture of viewing the sum over Γχ,j,x, and samples the high weight terms preferentially while applying a correction to the measured values to remain unbiased. One approach is to sample the bit string χ according to its Pauli-weight Wχ, i.e, the number of qubits that the stabilizer operator acts non-trivially on. For the single-qubit depolarizing channel with error probability p, one may sample the bit string χ with probability proportional to One potential way to suppress the numbers of samples required is to use variance reduction techniques such as importance sampling. This approach requires a priori knowledge of the values of Eα issα,j = 0, 1 and sα =  are the syndromes of the α-th error. The recovered state takes the formTo be more explicit in the construction for the random sampling method discussed here applied to the recovery procedure discussed in Section “Corrections with recovery operations”. Suppose that the projector on the subspace corresponding to the error operator αbα = 1 and bα > 0 by choice of the recovery operators. In this case, the stochastic sampling algorithm is given by choosing a Pauli operator defined by pχ,j,α, recording the series of  +1,  −1 results, and finding their average as before. The calculation of the variance follows as in the previous case. To reduce variance, the frequency of sampling α maybe chosen to be proportional to the error probability of Eα for the expected errors on the physical system of interest.The expectation value for the recovered state readsThe power of error correction extends beyond the simple identification of errors and includes recovery operations that restore some states to the original code space. The formalism here built on projectors and post-processing would seem at first glance unable to take advantage of such unitary projection operations; however, we will show how one can use these recovery operations to some advantage in sampling complexity over the unrecovered projections.Ei} which is known to be correctable within the chosen code. These errors will either commute or anti-commute with the stabilizers of the code to produce a syndrome of the error that has happened, which we denote j’th syndrome measurement of the i’th error. We will assume the recovery operation for this error within the code is known, and is denoted as Ri.Consider a set of Pauli errors on the system of physical qubits {Ei acting on the code space. This is given byEi and stabilizer generator Sj. Once one has projected into this space, we can now use the recovery operation, Ri to map the state back into the code space before using it. If we take the set of all correctable errors, including no error as the identity, then we get an updated correction formula for projection with recovery asRi, however in many cases this again reduces to a simple sum over Pauli operators that may be stochastically sampled, where many of the same simplifications resulting from commutation of logical operators with stabilizers are possible.The formalism presented here avoids direct stabilizer measurement by design to favor implementation on NISQ devices, hence we need to specify how one uses recovery operations within the projection formalism. Similar to a projector on the code space, we may formulate a projector onto the error subspace that corresponds to error c, and leads in practice to lower errors with small, finite samples. However a tradeoff is being made in including these values in that it may reduce the overall projection quality. Consider for example a distance d code. If one has a Pauli error with weight greater than (d − 1)∕2, a recovery operation may become a logical error which is not then removed by this procedure. In contrast, strict projection is capable of removing errors of all the way up to weight d − 1, which is a significant boost in maximum potential. However, as mentioned, the tradeoff of finite sampling complexity with potential for correction must be carefully balanced in real implementations.The consequences of including recovery on top of projection are interesting. The immediate practical benefit in increasing the size of Hilbert space over which one attains signal. This is reflected in the estimation of the value


Several devastating forest pathogens are suspected or known to have entered the UK through imported planting material. The nursery industry is a key business of the tree trade network. Variability in demand for trees makes it difficult for nursery owners to predict how many trees to produce in their nursery. When in any given year, the demand for trees is larger than the production, nursery owners buy trees from foreign sources to match market demand. These imports may introduce exotic diseases.We have developed a model of the dynamics of plant production linked to an economic model. We have used this to quantify the effect of demand variability on the risk of introducing an exotic disease.We find that: (a) When the cost of producing a tree in a UK nursery is considerably smaller than the cost of importing a tree , the risk of introducing an exotic disease is hardly affected by an increase in demand variability. (b) When the cost of producing a tree in the nursery is smaller than, but not very different from the cost of importing a tree, the risk of importing exotic diseases increases with increasing demand variability.Synthesis and applications. Our model and results demonstrate how a balanced management of demand variability and costs can reduce the risk of importing an exotic forest disease according to the management strategy adopted. For example, a management strategy that can reduce the demand variability, the ratio of production to import cost or both, optimizes the nursery gross margin when mainly own‐produced trees are commercialized. This can also translate into a reduction of the risk of introducing exotic forest diseases due to the small number of imported trees for sale. Our model and results demonstrate how a balanced management of demand variability and costs can reduce the risk of importing an exotic forest disease according to the management strategy adopted. For example, a management strategy that can reduce the demand variability, the ratio of production to import cost or both, optimizes the nursery gross margin when mainly own‐produced trees are commercialized. This can also translate into a reduction of the risk of introducing exotic forest diseases due to the small number of imported trees for sale. This disease killed over 45 million trees in Britain, leaving very few elm trees alive . This pest was likely introduced to England through the movement of live oak plants in trade from continental Europe  the market demand, (2) production cost and import costs, and (3) nursery owner decisions on tree production volume.The details and mathematical formulation of the described model are found in Supporting Information Appendix 2.1B seedlings or cuttings are planted. Seedlings develop into saplings, then grow to become small trees and subsequently develop into medium size trees /√12. Due to the absence of detailed data on the demand distribution, we use a uniform distribution. Moreover, the example analysed in the manuscript is drawn from a homogeneous distribution. However, in earlier simulations, we investigated the effect of a normal distribution on the results and found that there is no qualitative difference. This can be seen in Supporting Information Appendix The demand for trees is a random variable. Each year, the demanded number of trees is drawn from a homogeneous distribution with mean demand 2.4The probability of importing an invasive pathogen is linked to the number of imports acquired over time. Each imported tree has a very small probability of being infected with an exotic pathogen that goes unnoticed during the import process. We calculate the mean probability per year that a tree infected with the pathogen is imported. The rationale is that when infected trees are imported more frequently the probability that this disease escapes and causes an epidemic also increases. Our models also generalize to the case of imported pests in which there is a low probability of detection on imported trees.2.5R, as the main decision point for growers to maximize their profit.Nursery growers aim to maximize their profit. Every year, the nursery owners must decide what their production volume will be. This production volume depends on production costs and on the expected demand. We take the number of yearly planted seeds or cuttings, The costs that a nursery owner incur are the cost of seedling planting material, the costs of tree maintenance (which can vary with size class), and the cost of importing a tree when demand is larger than stock.Nursery owners obtain some profit from selling both the home‐produced and the imported trees. We refer to the captions of Figures As we are not including overheads and operating expenses in our calculations, we calculate the gross margin instead of the profit of the nursery. The gross margin or gross profit of a nursery is calculated as the difference between income from the sales and the costs of producing and importing trees. We express the gross margin as the yearly mean gross margin of a nursery and we will assume that the nursery owner aims to maximize the expected (mean) gross margin.3We quantify how the probability of importing an exotic disease is affected by the planting rate and the demand variability. To this end, we first simulate the dynamics of the growth of a tree population in a nursery over time. We also simulate how demand variability and imports affect the economic dynamics of the nursery. Combining these two, we obtain contour plots of the mean gross margin as a function of the demand variability and the planting rate chosen by the nursery owner. Linking these results with a contour plot of the mean probability to import an exotic pathogen, we see how demand variability and the aim of the grower to maximize the nursery gross margin affect the probability to introduce an exotic disease.3.1Figure Figure 3.2α) and planting rate (R). The dotted white line represents the maximum gross margin obtainable.Figure κP < (κI/2)), optimum gross margin contours bend towards planting rates larger than the mean demand, (b) when production costs are larger than half the importing costs (κP > (κI/2)), optimum gross margin contours bend towards planting rates smaller than the mean demand, and (c) when production costs are equal to half the importing costs (κP = (κI/2)), the gross margin contours do not bend from the mean demand planning rate. In Supporting Information Appendix Considering first the gross margin contours, we see that for each level of demand variability there is a planting rate that maximizes the gross margin. For the parameter values with small demand variability, this planting rate is the same as the demand rate because in these simulations trees are assumed not to die. We have run simulations with other parameters values and concluded that our qualitative conclusions do not change. For any fixed planting rate, the gross margin always decreases as the demand variability increases. Interestingly, the planting rate that maximizes the gross margin can either decrease with increasing demand variability Figure a or incrIf the cost of producing a tree in a nursery is less than half the cost of importing a tree Figure a, the deIf the cost of producing a tree in a nursery is equal to half the cost of importing a tree Figure b, we seeWhen the cost of producing a tree in a nursery is larger than half the cost of importing a tree Figure c, the siNow let us consider the contours of yearly mean probability to import an infected tree  as a study case due to its’ ecological and economic importance for the sustainability of English broadleaved forests. Broadleaves account for about half (49%) of the UK woodland area   in the UK. The nursery sector chosen is nursery growers selling and trading tree sizes between 20–40 cm and 40–60 cm which is one of the most commonly sold oak planting material in the UK. Table The study case is built upon limited information about the growth and commercialization of Common Oak  at year t and μi is the mean demand of trees from stage i.We obtained a time series of the number of broadleaf trees planted from 1976 to 2017 from the Forestry Commission statistics , if we follow the line of optimum gross margin.There is a range of measures that can be taken by policymakers and stakeholders to manage the fluctuations in demand and the difference in cost of production and cost of import. For example, longer term grant schemes that allow nursery grower to plan their tree planting can reduce this variability. Policies that incentivize growers to sell home‐grown plants can also help to manage these fluctuations.Our results suggest that a balanced management of demand variability and costs can significantly reduce the risk of importing an exotic forest disease.V.A.C., C.A.G., and F.B. conceived the project; F.B. and V.A.C. developed the model and conducted the analysis; V.A.C. led the writing of the manuscript. All authors contributed critically to the drafts and gave final approval for publication.https://doi.org/10.5061/dryad.g446vj1 (Alonso Chavez, Gilligan, & van den Bosch, Data are available via the Dryad Digital Repository  Click here for additional data file. Click here for additional data file. Click here for additional data file. Click here for additional data file. Click here for additional data file. Click here for additional data file. Click here for additional data file. Click here for additional data file.