Joel Yuen-Zhou, Jacob J. Krich, Ivan Kassal, Alan Johnson, and Alán Aspuru-Guzik
Institute of Physics Publishing, Bristol (2014).
This book provides a self-contained introduction to quantum process tomography and nonlinear spectroscopy, which underlie the study of excited state dynamics in molecular aggregates, such as photosynthetic complexes.
Ryan J. MacDonell, Tomas Navickas, Tim F. Wohlers-Reichel, Christophe H. Valahu, Arjun D. Rao, Maverick J. Millican, Michael A. Currington, Michael J. Biercuk, Ting Rei Tan, Cornelius Hempel, and Ivan Kassal
Spectroscopy is one of the most accurate probes of the molecular world. However, predicting molecular spectra accurately is computationally difficult because of the presence of entanglement between electronic and nuclear degrees of freedom. Although quantum computers promise to reduce this computational cost, existing quantum approaches rely on combining signals from individual eigenstates, an approach that is difficult to scale because the number of eigenstates grows exponentially with molecule size. Here, we introduce a method for scalable analog quantum simulation of molecular spectroscopy, by performing simulations in the time domain. Our approach can treat more complicated molecular models than previous ones, requires fewer approximations, and can be extended to open quantum systems with minimal overhead. We present a direct mapping of the underlying problem of time-domain simulation of molecular spectra to the degrees of freedom and control fields available in a trapped-ion quantum simulator. We experimentally demonstrate our algorithm on a trapped-ion device, exploiting both intrinsic electronic and motional degrees of freedom, showing excellent quantitative agreement for a single-mode vibronic photoelectron spectrum of SO2.
In organic photovoltaics, charges can separate efficiently even if their Coulomb attraction is an order of magnitude greater than the available thermal energy. Delocalization has been suggested to explain this fact, because it could increase the initial separation of charges in the charge-transfer (CT) state, reducing their attraction. However, understanding the mechanism requires a kinetic model of delocalized charge separation, which has proven difficult because it involves tracking the correlated quantum-mechanical motion of the electron and the hole in large simulation boxes required for disordered materials. Here, we report the first three-dimensional simulations of charge-separation dynamics in the presence of disorder, delocalization, and polaron formation, finding that even slight delocalization, across less than two molecules, can substantially enhance the charge-separation efficiency, even starting with thermalized CT states. Delocalization does not enhance efficiency by reducing the Coulomb attraction; instead, the enhancement is a kinetic effect produced by the increased overlap of electronic states.
Coherence-enhanced light harvesting has not been directly observed experimentally, despite theoretical evidence that coherence can significantly enhance light-harvesting performance. The main experimental obstacle has been the difficulty in isolating the effect of coherence in the presence of confounding variables. Recent proposals for externally controlling coherence by manipulating the light’s degree of polarization showed that coherent efficiency enhancements would be possible, but they were restricted to light-harvesting systems weakly coupled to their environment. Here, we show that increases in system–bath coupling strength can amplify coherent efficiency enhancements, rather than suppress them. This result dramatically broadens the range of systems that could be used to conclusively demonstrate coherence-enhanced light harvesting or to engineer coherent effects into artificial light-harvesting devices.
Ultrafast chemical reactions are difficult to simulate because they involve entangled, many-body wavefunctions whose computational complexity grows rapidly with molecular size. In photochemistry, the breakdown of the Born-Oppenheimer approximation further complicates the problem by entangling nuclear and electronic degrees of freedom. Here, we show that analog quantum simulators can efficiently simulate molecular dynamics using commonly available bosonic modes to represent molecular vibrations. Our approach can be implemented in any device with a qudit controllably coupled to bosonic oscillators and with quantum hardware resources that scale linearly with molecular size, and offers significant resource savings compared to digital quantum simulation algorithms. Advantages of our approach include a time resolution orders of magnitude better than ultrafast spectroscopy, the ability to simulate large molecules with limited hardware using a Suzuki-Trotter expansion, and the ability to implement realistic system-bath interactions with only one additional interaction per mode. Our approach can be implemented with current technology; e.g., the conical intersection in pyrazine can be simulated using a single trapped ion. Therefore, we expect our method will enable classically intractable chemical dynamics simulations in the near term.
Charge transport is well understood in both highly ordered materials (band conduction) or highly disordered ones (hopping conduction). In moderately disordered materials—including many organic semiconductors—the approximations valid in either extreme break down, making it difficult to accurately model the conduction. In particular, describing wavefunction delocalisation requires a quantum treatment, which is difficult in disordered materials that lack periodicity. Here, we present the first three-dimensional model of partially delocalised charge and exciton transport in materials in the intermediate disorder regime. Our approach is based on polaron-transformed Redfield theory, but overcomes several computational roadblocks by mapping the quantum-mechanical techniques onto kinetic Monte Carlo. Our theory, delocalised kinetic Monte Carlo (dKMC), shows that the fundamental physics of transport in moderately disordered materials is that of charges hopping between partially delocalised electronic states. Our results reveal why standard kinetic Monte Carlo can dramatically underestimate mobilities even in disordered organic semiconductors, where even a little delocalisation can substantially enhance mobilities, as well as showing that three-dimensional calculations capture important delocalisation effects neglected in lower-dimensional approximations.
Several kinds of coherence have recently been shown to affect the performance of light-harvesting systems, in some cases significantly improving their efficiency. Here, we classify the possible mechanisms of coherent efficiency enhancements, based on the types of coherence that can characterize a light-harvesting system and the types of processes these coherences can affect. We show that enhancements are possible only when coherences and dissipative effects are best described in different bases of states. Our classification allows us to predict a previously unreported coherent enhancement mechanism, where coherence between delocalized eigenstates can be used to localize excitons away from dissipation, thus reducing the rate of recombination and increasing efficiency.
All photosynthetic organisms convert solar energy into chemical energy through charge separation in dimeric reaction centres. It is unknown why early reaction centres dimerised and completely displaced their monomeric ancestors. Here, we discuss several proposed explanations for reaction-centre dimerism and conclude—with only weak assumptions about the primordial dimerisation event—that the most probable explanation for the dimerism is that it arose because it enhanced light-harvesting efficiency by deepening the excitonic trap, i.e., by enhancing the rate of exciton transfer from an antenna complex and decreasing the rate of back transfer. This effect would have outweighed the negative effect dimerisation would have had on charge transfer within the reaction centre. Our argument implies that dimerisation likely occurred after the evolution of the first antennas, and it explains why the lower-energy state of the special pair is bright.
Quantifying energetic disorder in organic semiconductors continues to attract attention because of its significant impact on the transport physics of these technologically important materials. Here, we show that the energetic disorder of organic semiconductors can be determined from the relationship between the internal quantum efficiency of charge generation and the frequency of the incident light. Our results for a number of materials suggest that energetic disorder in organic semiconductors could be greater than previously reported, and we advance ideas as to why this may be the case.
Spectroscopic experiments have identified long-lived coherences in several light-harvesting systems, suggesting that coherent effects may be relevant to their performance. However, there is limited experimental evidence of coherence enhancing light-harvesting efficiency, largely due to the difficulty of turning coherences on and off to create an experimental control. Here we show that coherence can indeed enhance light harvesting and that this effect can be controlled. We construct a model system in which initial coherence can be controlled using the incident light and which is significantly more efficient under coherent, rather than incoherent, excitation. Our proposal would allow for an unambiguous demonstration of light harvesting enhanced by intermolecular coherence, as well as demonstrate the potential for coherent control of excitonic energy transfer.
The dynamics of exciton quenching are critical to the operational performance of organic optoelectronic devices, but their measurement and elucidation remain ongoing challenges. Here, we present a method for quantifying small photoluminescence quenching efficiencies of organic semiconductors under steady-state conditions. Exciton quenching efficiencies of three different organic semiconductors, PC70BM, P3HT, and PCDTBT, are measured at different bulk quencher densities under continuous low-irradiance illumination. By implementing a steady-state bulk-quenching model, we determine exciton diffusion lengths for the studied materials. At low quencher densities we find that a secondary quenching mechanism is in effect, which is responsible for approximately 20% of the total quenched excitons. This quenching mechanism is observed in all three studied materials and exhibits quenching volumes on the order of several thousand cubic nanometers. The exact origin of this quenching process is not clear, but it may be indicative of delocalized excitons being quenched prior to thermalization.
Because of the low dielectric constant, charges in organic solar cells must overcome a strong Coulomb attraction in order to separate. It has been widely argued that intermolecular delocalization would assist charge separation by increasing the effective initial electron–hole separation in a charge-transfer state, thus decreasing their barrier to separation. Here we show that this is not the case: including more than a small amount of delocalization in models of organic solar cells leads to an increase in the free-energy barrier to charge separation. Therefore, if delocalization were to improve the charge separation efficiency, it would have to do so through nonequilibrium kinetic effects that are not captured by a thermodynamic treatment of the barrier height.
Although Marcus theory is widely used to describe charge transfer in molecular systems, in its usual form it is restricted to transfer from one molecule to another. If a charge is delocalised across multiple donor molecules, this approach requires us to treat the entire donor aggregate as a unified supermolecule, leading to potentially expensive quantum-chemical calculations and making it more difficult to understand how the aggregate components contribute to the overall transfer. Here, we show that it is possible to describe charge transfer between groups of molecules in terms of the properties of the constituent molecules and couplings between them, obviating the need for expensive supermolecular calculations. We use the resulting theory to show that charge delocalisation between molecules in either the donor or acceptor aggregates can enhance the rate of charge transfer through a process we call supertransfer (or suppress it through subtransfer). The rate can also be enhanced above what is possible with a single donor and a single acceptor by judiciously tuning energy levels and reorganisation energies. We also describe bridge-mediated charge transfer between delocalised molecular aggregates. The equations of generalised Marcus theory are in closed form, providing qualitative insight into the impact of delocalisation on charge dynamics in molecular systems.
In this Letter, we study the role of the donor:acceptor interface nanostructure upon charge separation and recombination in organic photovoltaic devices and blend films, using mixtures of PBTTT and two different fullerene derivatives (PC70BM and ICTA) as models for intercalated and nonintercalated morphologies, respectively. Thermodynamic simulations show that while the completely intercalated system exhibits a large free-energy barrier for charge separation, this barrier is significantly lower in the nonintercalated system and almost vanishes when energetic disorder is included in the model. Despite these differences, both femtosecond-resolved transient absorption spectroscopy (TAS) and time-delayed collection field (TDCF) exhibit extensive first-order losses in both systems, suggesting that geminate pairs are the primary product of photoexcitation. In contrast, the system that comprises a combination of fully intercalated polymer:fullerene areas and fullerene-aggregated domains (1:4 PBTTT:PC70BM) is the only one that shows slow, second-order recombination of free charges, resulting in devices with an overall higher short-circuit current and fill factor. This study therefore provides a novel consideration of the role of the interfacial nanostructure and the nature of bound charges and their impact upon charge generation and recombination.
The origin of photocurrent losses in the power-generating regime of organic solar cells (OSCs) remains a controversial topic, although recent literature suggests that the competition between bimolecular recombination and charge extraction determines the bias dependence of the photocurrent. Here the steady-state recombination dynamics is studied in bulk-heterojunction OSCs with different hole mobilities from short-circuit to maximum power point. It is shown that in this regime, in contrast to previous transient extracted charge and absorption spectroscopy studies, first-order recombination outweighs bimolecular recombination of photogenerated charge carriers. This study demonstrates that the first-order losses increase with decreasing slower carrier mobility, and attributes them to either mobilization of charges trapped at the donor:acceptor interface through the Poole–Frenkel effect, and/or recombination of photogenerated and injected charges. The dependence of both first-order and higher-order losses on the slower carrier mobility explains why the field dependence of OSC efficiencies has historically been attributed to charge-extraction losses.
Although organic heterojunctions can separate charges with near-unity efficiency and on a sub-picosecond timescale, the full details of the charge-separation process remain unclear. In typical models, the Coulomb binding between the electron and the hole can exceed the thermal energy kT by an order of magnitude, making it impossible for the charges to separate before recombining. Here, we consider the entropic contribution to charge separation in the presence of disorder and find that even modest amounts of disorder have a decisive effect, reducing the charge-separation barrier to about kT or eliminating it altogether. Therefore, the charges are usually not thermodynamically bound at all and could separate spontaneously if the kinetics otherwise allowed it. Our conclusion holds despite the worst-case assumption of localised, thermalised carriers, and is only strengthened if mechanisms like delocalisation or 'hot' states are also present.
The remarkable rotational symmetry of the photosynthetic antenna complexes of purple bacteria has long been thought to enhance their light harvesting and excitation energy transport. We study the role of symmetry by modeling hypothetical antennas whose symmetry is broken by altering the orientations of the bacteriochlorophyll pigments. We find that in both LH2 and LH1 complexes, symmetry increases energy transfer rates by enabling the cooperative, coherent process of supertransfer. The enhancement is particularly pronounced in the LH1 complex, whose natural geometry outperforms the average randomized geometry by 5.5 standard deviations, the most significant coherence-related enhancement found in a photosynthetic complex.
Blends of electron-donating and -accepting organic semiconductors are widely used as photoactive materials in next-generation solar cells and photodetectors. The yield of free charges in these systems is often determined by the separation of interfacial electron–hole pairs, which is expected to depend on the ability of the faster carrier to escape the Coulomb potential. Here we show, by measuring geminate and non-geminate losses and key transport parameters in a series of bulk-heterojunction solar cells, that the charge-generation yield increases with increasing slower carrier mobility. This is in direct contrast with the well-established Braun model where the dissociation rate is proportional to the mobility sum, and recent models that underscore the importance of fullerene aggregation for coherent electron propagation. The behaviour is attributed to the restriction of opposite charges to different phases, and to an entropic contribution that favours the joint separation of both charge carriers.
Transport phenomena on a quantum scale appear in a variety of systems, ranging from photosynthetic complexes to engineered quantum devices. It has been predicted that the efficiency of coherent transport can be enhanced through dynamic interaction between the system and a noisy environment. We report an experimental simulation of environment-assisted coherent transport, using an engineered network of laser-written waveguides, with relative energies and inter-waveguide couplings tailored to yield the desired Hamiltonian. Controllable-strength decoherence is simulated by broadening the bandwidth of the input illumination, yielding a significant increase in transport efficiency relative to the narrowband case. We show integrated optics to be suitable for simulating specific target Hamiltonians as well as open quantum systems with controllable loss and decoherence.
Photosynthetic complexes improve the transfer of excitation energy from peripheral antennas to reaction centers in several ways. In particular, a downward energy funnel can direct excitons in the right direction, while coherent excitonic delocalization can enhance transfer rates through the cooperative phenomenon of supertransfer. However, isolating the role of purely coherent effects is difficult because any change to the delocalization also changes the energy landscape. Here, we show that the relative importance of the two processes can be determined by comparing the natural light-harvesting apparatus with counterfactual models in which the delocalization and the energy landscape are altered. Applied to the example of purple bacteria, our approach shows that although supertransfer does enhance the rates somewhat, the energetic funnelling plays the decisive role. Because delocalization has a minor role (and is sometimes detrimental), it is most likely not adaptive, being a side-effect of the dense chlorophyll packing that evolved to increase light absorption per reaction center.
Excitonic couplings between (bacterio)chlorophyll molecules are necessary for simulating energy transport in photosynthetic complexes. Many techniques for calculating the couplings are in use, from the simple (but inaccurate) point-dipole approximation to fully quantum-chemical methods. We compared several approximations to determine their range of applicability, noting that the propagation of experimental uncertainties poses a fundamental limit on the achievable accuracy. In particular, the uncertainty in crystallographic coordinates yields an uncertainty of about 20% in the calculated couplings. Because quantum- chemical corrections are smaller than 20% in most biologically relevant cases, their considerable computational cost is rarely justified. We therefore recommend the electrostatic TrEsp method across the entire range of molecular separations and orientations because its cost is minimal and it generally agrees with quantum-chemical calculations to better than the geometric uncertainty. Understanding these uncertainties can guard against striving for unrealistic precision; at the same time, detailed benchmarks can allow important qualitative questions—which do not depend on the precise values of the simulation parameters—to be addressed with greater confidence about the conclusions..
It has been argued that excitonic energy transport in photosynthetic complexes is efficient because of a balance between coherent evolution and decoherence, a phenomenon called environment-assisted quantum transport (ENAQT). Studies of ENAQT have usually assumed that the excitation is initially localized on a particular chromophore, and that it is transferred to a reaction center through a similarly localized trap. However, these assumptions are not physically accurate. We show that more realistic models of excitation and trapping can lead to very different predictions about the importance of ENAQT. In particular, although ENAQT is a robust effect if one assumes a localized trap, its effect can be negligible if the trapping is more accurately modeled as Forster transfer to a reaction center. Our results call into question the suggested role of ENAQT in the photosynthetic process of green sulfur bacteria and highlight the subtleties associated with drawing lessons for designing biomimetic light-harvesting complexes.
The conventional picture of photocurrent generation in organic solar cells involves photoexcitation of the electron donor, followed by electron transfer to the acceptor via an interfacial charge-transfer state (Channel I). It has been shown that the mirror-image process of acceptor photoexcitation leading to hole transfer to the donor is also an efficient means to generate photocurrent (Channel II). The donor and acceptor components may have overlapping or distinct absorption characteristics. Hence, different excitation wavelengths may preferentially activate one channel or the other, or indeed both. As such, the internal quantum efficiency (IQE) of the solar cell may likewise depend on the excitation wavelength. We show that several model high-efficiency organic solar cell blends, notably PCDTBT:PC70BM and PCPDTBT:PC60/70BM, exhibit flat IQEs across the visible spectrum, suggesting that charge generation is occurring either via a dominant single channel or via both channels but with comparable efficiencies. In contrast, blends of the narrow optical gap copolymer DPP-DTT with PC70BM show two distinct spectrally flat regions in their IQEs, consistent with the two channels operating at different efficiencies. The observed energy dependence of the IQE can be successfully modeled as two parallel photodiodes, each with its own energetics and exciton dynamics but both having the same extraction efficiency. Hence, an excitation-energy dependence of the IQE in this case can be explained as the interplay between two photocurrent-generating channels, without recourse to hot excitons or other exotic processes.
Large-scale quantum computers will require the ability to apply long sequences of entangling gates to many qubits. In a photonic architecture, where single-qubit gates can be performed easily and precisely, the application of consecutive two-qubit entangling gates has been a significant obstacle. Here, we demonstrate a two-qubit photonic quantum processor that implements two consecutive CNOT gates on the same pair of polarisation-encoded qubits. To demonstrate the flexibility of our system, we implement various instances of the quantum algorithm for solving of systems of linear equations.
Recent observations of coherence in photosynthetic complexes have led to the question of whether quantum effects can occur in vivo, not under femtosecond laser pulses but in incoherent sunlight and at steady state, and, if so, whether the coherence explains the high exciton transfer efficiency. We distinguish several types of coherence and show that although some photosynthetic pathways are partially coherent processes, photosynthesis in nature proceeds through stationary states. This distinction allows us to rule out several mechanisms of transport enhancement in sunlight. In particular, although they are crucial for understanding exciton transport, neither wavelike motion nor microscopic coherence, on their own, enhance the efficiency. By contrast, two partially coherent mechanisms—ENAQT and supertransfer—can enhance transport even in sunlight and thus constitute motifs for the optimisation of artificial sunlight harvesting. Finally, we clarify the importance of ultrafast spectroscopy in understanding incoherent processes.
Noise-assisted transport in quantum systems occurs when quantum time evolution and decoherence conspire to produce a transport efficiency that is higher than what would be seen in either the purely quantum or purely classical cases. In disordered systems, it has been understood as the suppression of coherent quantum localization through noise, which brings detuned quantum levels into resonance and thus facilitates transport. We report several new mechanisms of environment-assisted transport in ordered systems, in which there is no localization to overcome and where one would naively expect that coherent transport is the fastest possible. Although we are particularly motivated by the need to understand excitonic energy transfer in photosynthetic light-harvesting complexes, our model is general—transport in a tight-binding system with dephasing, a source and a trap—and can be expected to have wider application.
Topological phases exhibit some of the most striking phenomena in modern physics. Much of the rich behaviour of quantum Hall systems, topological insulators, and topological superconductors can be traced to the existence of robust bound states at interfaces between different topological phases. This robustness has applications in metrology and holds promise for future uses in quantum computing. Engineered quantum systems—notably in photonics, where wavefunctions can be observed directly—provide versatile platforms for creating and probing a variety of topological phases. Here we use photonic quantum walks to observe bound states between systems with different bulk topological properties and demonstrate their robustness to perturbations—a signature of topological protection. Although such bound states are usually discussed for static (time-independent) systems, here we demonstrate their existence in an explicitly time-dependent situation. Moreover, we discover a new phenomenon: a topologically protected pair of bound states unique to periodically driven systems.
The difficulty of simulating quantum systems, well known to quantum chemists, prompted the idea of quantum computation. One can avoid the steep scaling associated with the exact simulation of increasingly large quantum systems on conventional computers, by mapping the quantum system to another, more controllable one. In this review, we discuss to what extent the ideas in quantum computation, now a well-established field, have been applied to chemical problems. We describe algorithms that achieve significant advantages for the electronic-structure problem, the simulation of chemical dynamics, protein folding, and other tasks. Although theory is still ahead of experiment, we outline recent advances that have led to the first chemical calculations on small quantum information processors.
Quantum walks have a host of applications, ranging from quantum computing to the simulation of biological systems. We present an intrinsically stable, deterministic implementation of discrete quantum walks with single photons in space. The number of optical elements required scales linearly with the number of steps. We measure walks with up to 6 steps and explore the quantum-to-classical transition by introducing tunable decoherence. Finally, we also investigate the effect of absorbing boundaries and show that decoherence significantly affects the probability of absorption.
Benjamin P. Lanyon, James D. Whitfield, Geoff G. Gillet, Michael E. Goggin, Marcelo P. Almeida, Ivan Kassal, Jacob D. Biamonte, Masoud Mohseni, Ben J. Powell, Marco Barbieri, Alán Aspuru-Guzik, and Andrew G. White
Nature Chem.2, 106 (2010).
One of 17 “First anniversary highlights” of Nature Chemistry and subject of a “News and Views” (Nature Chemistry2, 76 (2010)).
Exact first-principles calculations of molecular properties are currently intractable because their computational cost grows exponentially with both the number of atoms and basis set size. A solution is to move to a radically different model of computing by building a quantum computer, which is a device that uses quantum systems themselves to store and process data. Here we report the application of the latest photonic quantum computer technology to calculate properties of the smallest molecular system: the hydrogen molecule in a minimal basis. We calculate the complete energy spectrum to 20 bits of precision and discuss how the technique can be expanded to solve large-scale chemical problems that lie beyond the reach of modern supercomputers. These results represent an early practical step toward a powerful tool with a broad range of quantum-chemical applications.
Quantum computers, if available, could substantially accelerate quantum simulations. We extend this result to show that the computation of molecular properties (energy derivatives) could also be sped up using quantum computers. We provide a quantum algorithm for the numerical evaluation of molecular properties, whose time cost is a constant multiple of the time needed to compute the molecular energy, regardless of the size of the system. Molecular properties computed with the proposed approach could also be used for the optimization of molecular geometries or other properties. For that purpose, we discuss the benefits of quantum techniques for Newton’s method and Householder methods. Finally, global minima for the proposed optimizations can be found using the quantum basin hopper algorithm, which offers an additional quadratic reduction in cost over classical multi-start techniques.
While quantum computers are capable of simulating many quantum systems efficiently, the simulation algorithms must begin with the preparation of an appropriate initial state. We present a method for generating physically relevant quantum states on a lattice in real space. In particular, the present algorithm is able to prepare general pure and mixed many-particle states of any number of particles. It relies on a procedure for converting from a second-quantized state to its first-quantized counterpart. The algorithm is efficient in that it operates in time that is polynomial in all the essential descriptors of the system, the number of particles, the resolution of the lattice, and the inverse of the maximum final error. This scaling holds under the assumption that the wave function to be prepared is bounded or its indefinite integral is known and that the Fock operator of the system is efficiently simulatable.
Transport phenomena at the nanoscale are of interest due to the presence of both quantum and classical behavior. In this work, we demonstrate that quantum transport efficiency can be enhanced by a dynamical interplay of the system Hamiltonian with pure dephasing induced by a fluctuating environment. This is in contrast to fully coherent hopping that leads to localization in disordered systems, and to highly incoherent transfer that is eventually suppressed by the quantum Zeno effect. We study these phenomena in the Fenna–Matthews–Olson protein complex as a prototype for larger photosynthetic energy transfer systems. We also show that the disordered binary tree structures exhibit enhanced transport in the presence of dephasing.
The computational cost of exact methods for quantum simulation using classical computers grows exponentially with system size. As a consequence, these techniques can be applied only to small systems. By contrast, we demonstrate that quantum computers could exactly simulate chemical reactions in polynomial time. Our algorithm uses the split-operator approach and explicitly simulates all electron-nuclear and interelectronic interactions in quadratic time. Surprisingly, this treatment is not only more accurate than the Born–Oppenheimer approximation but faster and more efficient as well, for all reactions with more than about four atoms. This is the case even though the entire electronic wave function is propagated on a grid with appropriately short time steps. Although the preparation and measurement of arbitrary states on a quantum computer is inefficient, here we demonstrate how to prepare states of chemical interest efficiently. We also show how to efficiently obtain chemically relevant observables, such as state-to-state transition probabilities and thermal reaction rates. Quantum computers using these techniques could outperform current classical computers with 100 qubits.
Samples from Thai manuscripts dated to the 18th to 20th century were analyzed by Raman spectroscopy and Fourier-transform infrared spectroscopy (FTIR) to determine the pigments used. This suggested a change in palette from the 18th to 20th century, with use of imported pigments in the later manuscripts. In the 18th century, the main green used was an organic copper salt, which was replaced by emerald green and mixtures of Prussian blue with gamboge, chrome yellow and zinc yellow (zinc potassium chromate). Chrome yellow was used in addition to gamboge in one later 19th century manuscript. Similarly, indigo in the 18th century manuscripts was replaced by Prussian blue and then synthetic ultramarine in the 19th century manuscripts. Lead white was the main white pigment in all but one manuscript, which contained huntite, a magnesium calcium carbonate. Huntite also occurred in mixtures with other pigments in two other manuscripts. In all the works studied, red lead, vermilion and red earth were used for red, orange and pink shades and red earth in brown areas. The organic copper salt used in the 18th century gave good FTIR spectra but could not initially be matched with any published compound. X-ray diffraction (XRD) suggested this was a copper citrate phase, and examination of the literature showed that the FTIR spectra matched those published for a hydrated copper citrate. Raman spectra were obtained from this organic copper salt, which showed close agreement with those obtained from synthetic copper citrate. Copper citrate has not been identified previously as an artist's material, although its use has been postulated on the basis of historical texts. Minor copper formate and/or copper chloride were also identified by XRD and scanning electron microscopy (SEM) in some green samples containing copper citrate.