Quantum simulation

In the paper

“In a recent experiment, the predictions of the two-dimensional Mott-Hubbard model were tested for electrons in an artificial honeycomb lattice (Singha et al. , 2011).”

What the sources say

• Not checked yetcites Singha et al. , 2011

In the paper

“As for scalability, circuits containing 512 qubits have been fabricated (Harris, 2012; Harris et al. , 2010), although quantum coherence was not tested on these circuits in the same way that coherence is commonly tested in other experiments using small numbers of qubits.”

What the sources say

• Not checked yetcites Harris, 2012
• Not checked yetcites Harris et al. , 2010

In the paper

“Recently, it was shown that the propagation of photons in a network of beam splitters is in general a computationally difficult task for classical computers even for a few tens of photons (Aaronson and Arkhipov, 2011), and corresponding experiments with up to four photons were performed (Broome et al. , 2013; Crespi et al. , 2013; Spring et al. , 2013; Tillmann et al. , 2013).”

What the sources say

• Not checked yetcites Broome et al. , 2013
• Not checked yetcites Crespi et al. , 2013
• Not checked yetcites Spring et al. , 2013

In the paper

“For example, the quantum phase transition from a superfluid to a Mott insulator phase, predicted in (Fisher et al. , 1989), was first observed in 2002 (Greiner et al. , 2002) in an ensemble of atoms trapped in an optical lattice.”

What the sources say

• Not checked yetcites Fisher et al. , 1989
• Not checked yetcites Greiner et al. , 2002

In the paper

“As discussed in Section VI.A, an AQS of the BoseHubbard model using atoms in optical lattices was proposed in (Jaksch et al. , 1998) and implemented in (Greiner et al. , 2002).”

What the sources say

• Not checked yetcites Jaksch et al. , 1998
• Not checked yetcites Greiner et al. , 2002

In the paper

“Furthermore, more than 200 superconducting resonators were recently fabricated on a single chip (Houck et al. , 2012).”

What the sources say

• Not checked yetcites Houck et al. , 2012

In the paper

“Photons have been used to calculate the possible fractional statistics of anyons using a six-photon graph state (Lu et al. , 2009), to calculate the energy spectrum of the hydrogen molecule to 20 bits of precision (Lanyon et al. , 2010) and to simulate frustrated spin systems (Ma et al. , 2011).”

What the sources say

• Not checked yetcites Lu et al. , 2009
• Not checked yetcites Lanyon et al. , 2010
• Not checked yetcites Ma et al. , 2011

In the paper

“State-of-the-art superconducting qubits have coherence times exceeding 100 µ s (i.e. decoherence rates below 10 kHz), which is quite high considering that other energy scales in the circuit are typically in the range 10 MHz - 10 GHz.”

In the paper

“As for AQS, optical lattices have been used to experimentally realize a disordered system that may lead to the observation of a Bose glass phase (Fallani et al. , 2007).”

What the sources say

• Not checked yetcites Fallani et al. , 2007

In the paper

“In these experiments, simulations of various spin models with up to six spins have been demonstrated, with one experiment requiring sequences of more than 100 quantum gates.”

What the sources say

• Not checked yetcites Lanyon et al. , 2011

In the paper

“individual control and readout are difficult to realize for atoms in optical lattices”

In the paper

“while for other systems that is not a problem”

In the paper

“With the exception of atoms in optical lattices, quantum simulators cannot yet handle large arrays of qubits.”

In the paper

“The machine would have the capacity to contain an exponentially large amount of information without using an exponentially large amount of physical resources, thus making it a natural tool to perform quantum simulation.”

In the paper

“Quantum simulation could be implemented using quantum computers, but also with simpler, analog devices that would require less control, and therefore, would be easier to construct.”

In the paper

“Other potential application areas include highenergy physics, quantum chemistry, cosmology and nuclear physics.”

In the paper

“With the latest advances in the coherent manipulation of quantum systems (Buluta et al. , 2011; Ladd et al. , 2010), such as atoms in optical lattices, trapped ions, nuclear spins, superconducting circuits, or spins in semiconductors, practical applications of quantum simulation can be expected in the coming years.”

What the sources say

• Not checked yetcites Buluta et al., 2011
• Not checked yetcites Ladd et al., 2010

In the paper

“However, depending on the specifics of the problem under study, good approximations are not always available or they also face some limitations.”

In the paper

“Quantum simulation promises to have applications in the study of many problems in, e.g., condensed-matter physics, high-energy physics, atomic physics, quantum chemistry and cosmology.”

In the paper

“However, this difficulty may be overcome by using some controllable quantum system to study another less controllable or accessible quantum system, i.e., quantum simulation.”

In the paper

“For instance, in condensedmatter physics, quantum simulation would allow the study of many difficult problems, such as quantum phase transitions, quantum magnetism or highT c superconductivity.”

In the paper

“Simpler quantum devices that mimic the evolution of other quantum systems in an analog manner could be used for this task (note that these are not universal simulators, but rather problem-specific machines).”

In the paper

“A number of quantum systems such as neutral atoms, ions, polar molecules, electrons in semiconductors, superconducting circuits, nuclear spins and photons have been proposed as quantum simulators.”

In the paper

“Moreover, note that ancilla qubits are required, which adds to the resource requirements (see Section V. B).”

In the paper

“This sounds very promising, but the quantum simulation problem is not really solved unless the initial state preparation, the implementation of the time evolution and the measurement are realized using only polynomial resources.”

In the paper

“This question has been thoroughly studied, and several methods have been developed (see, e.g., (Berry et al. , 2007; Bravyi et al. , 2008; Bremner et al. , 2005; Brown et al. , 2011; Dodd et al. , 2002; D¨ ur et al. , 2008; Hastings, 2006; Wang and Zanardi, 2002; Wocjan et al. , 2002a,b,c)), but it still remains a difficult problem.”

What the sources say

• Not checked yetcites Berry et al., 2007
• Not checked yetcites Bravyi et al., 2008
• Not checked yetcites Bremner et al., 2005

In the paper

“In many cases the preparation of the initial state is difficult and an efficient algorithm may not be available.”

In the paper

“As will be discussed later, these are not easy tasks, even for quantum simulators.”

In the paper

“This results in an exponential growth of the statistical error, and hence the required simulation time, with the number of particles, which cancels the advantage of the Monte Carlo methods.”

In the paper

“Unfortunately, in most cases of practical interest [ H l , H l ′ ] = 0 in general.”

In the paper

“Classical stochastic methods, namely Monte Carlo algorithms (Suzuki, 1993), have been developed as a way of tackling the difficult problem of simulating large quantum systems.”

What the sources say

• Not checked yetcites Suzuki, 1993

In the paper

“The Hamiltonian of the system to be simulated, H sys , is directly mapped onto the Hamiltonian of the simulator, H sim , which can be controlled at least to some extent:”

In the paper

“Since any unitary operation can be written in terms of universal quantum gates, it follows that in principle 'anything' can be simulated, i.e. DQS is universal (Lloyd, 1996).”

What the sources say

• Not checked yetcites Lloyd, 1996

In the paper

“In other words, although not all mathematically allowed Hamiltonians can be simulated efficiently, those that appear in most physical theories can be simulated efficiently.”

In the paper

“However, it must be noted that not any unitary operation can be efficiently simulated (that is with polynomial resources) and, therefore, there are Hamiltonians that cannot be efficiently simulated in this way.”

In the paper

“The topic was further discussed in (Poulin et al. , 2011) where it was shown that using randomness, it is possible to efficiently simulate local bounded Hamiltonians with arbitrary time dependence.”

What the sources say

• Not checked yetcites Poulin et al., 2011

In the paper

“In principle, this approximation can be made arbitrarily accurate (by refining the decomposition), but this comes at the cost of an ever-increasing number of gates.”

In the paper

“This procedure provides an arbitrarily accurate estimate of the energy, with the accuracy increasing with increasing number of iterations.”

What the sources say

• Not checked yetcites Aspuru-Guzik et al., 2005

In the paper

“Moreover, according to (Meyer, 2002) it should also be possible to use quantum computers to simulate classical physics more efficiently (see also (Sinha and Russer, 2010; Yung et al. , 2010)).”

What the sources say

• Not checked yetcites Meyer, 2002

In the paper

“As a result, a few imperfections in an ensemble of particles performing AQS might not affect the overall behavior of the ensemble, such that the AQS might still produce useful results even in the presence of these imperfections.”

In the paper

“Note that the simulator may only partly reproduce the dynamics of the system.”

In the paper

“If the quantum simulator suffers from uncertainties in the control parameters, the phase transition under study could still be observed, hence providing the answer to the question of interest.”

In the paper

“The choice of the mapping depends on what needs to be simulated and on the capabilities of the simulator.”

In the paper

“As discussed in the previous sections, fully characterizing a quantum system requires an exponentially large number of parameters.”

In the paper

“so chemical dynamics could be simulated on a quantum computer in O ( N 2 m 2 ) steps, which is exponentially faster than known classical algorithms.”

In the paper

“The analog simulation of the BoseHubbard model using atoms in optical lattices is therefore straightforward.”

In the paper

“The estimation of the requirements for simulating N particles interacting through a pairwise potential has been performed in (Kassal et al. , 2008).”

What the sources say

• Not checked yetcites Kassal et al. , 2008

In the paper

“Although quantum simulators are affected by the interactions with the environment in the same way as quantum computers, it is generally believed that the effects of decoherence are less dramatic.”

In the paper

“Sometimes the mapping is indeed straightforward, but this is not always the case, and quite often clever mappings have to be devised, sometimes involving additional externally applied fields or ancillary systems to mediate various interactions.”

In the paper

“Moreover, directly measuring some physical quantity of the simulator would yield information about its analogue in the simulated system.”

In the paper

“Finding the mapping in an AQS might, at first glance, look simpler than obtaining the most efficient gate decomposition for a given Hamiltonian in DQS.”

In the paper

“With a few tens of qubits, one could perform frustrated-spin simulations or molecular-energy calculations at the limits of present-day supercomputers.”

In the paper

“By analyzing the Ising model as a representative example, and using parameters that are relevant to present-day experiments, the authors concluded that surface codes are superior for quantum simulation.”

What the sources say

• Not checked yetcites You et al. , 2013

In the paper

“The amount of physical resources (i.e., number of qubits, number of operations, number of steps, etc.) needed for the quantum simulation in the case of an N -body problem strongly depends on the type of problem and the particularities of the simulator.”

In the paper

“As a general rule of thumb, it is sometimes said that in order to outperform classical computers quantum simulators require somewhere between 40 and 100 qubits (Buluta and Nori, 2009).”

What the sources say

• Not checked yetcites Buluta and Nori, 2009

In the paper

“This new class of algorithms makes it possible to simulate spin systems for longer times, and to study physical phenomena which would have been inaccessible with previous methods.”

In the paper

“In AQS one is usually emulating an effective many-body model of the simulated system. A controllable 'toy-model' of the system is used to reproduce the property of interest, e.g. the dynamics or ground state.”

In the paper

“Its quantum version allows for the direct sampling from the eigenstates of the Hamiltonian, overcoming in this way the sign problem (Temme et al. , 2011).”

What the sources say

• Not checked yetcites Temme et al. , 2011

In the paper

“Moreover, it has been suggested that the decoherence of the simulator might be useful (Lloyd, 1996) as it could serve as a rough way of modeling the decoherence of the simulated system.”

What the sources say

• Not checked yetcites Lloyd, 1996

In the paper

“Moreover, quantum dots allow flexible control over the confinement potential and they can also be excited optically.”

In the paper

“Electron spins in quantum dots may provide an advantage over atoms in optical lattices due to the very low temperatures (relative to the Fermi temperature) that can be reached and the natural long-range Coulomb interaction.”

In the paper

“The physical implementation of a quantum simulator requires a controllable quantum mechanical system.”

In the paper

“However, it is not very flexible and its main disadvantage is the lack of scalability, one of the main reasons being the spectral crowding that occurs as the number of energy levels increases exponentially with increasing number of spins.”

In the paper

“For many problems in this class, an array of qubits plus their controls (see Figure 8) would make an ideal quantum simulator because it can be thought of as the simplified, magnified lattice structure of a 'solid', that can be manipulated in a number of different ways in order to test various models.”

In the paper

“Ion qubits also have long coherence times of the order of seconds, and sequences of high-fidelity quantum gates have been demonstrated in experiment (Hanneke et al. , 2009; Lanyon et al. , 2011).”

What the sources say

• Not checked yetcites Hanneke et al. , 2009
• Not checked yetcites Lanyon et al. , 2011

In the paper

“Using RF pulses various one-, two- and possibly multi-qubit gates can be implemented.”

In the paper

“Crossvalidation over a number of different physical systems could be used, and in this way the particular imperfections of each implementation could be ruled out as possible sources of error.”

In the paper

“Indeed, optical lattices provide the highly desirable properties of being easily tunable and essentially defect-free.”

In the paper

“Alternative systems that can be used for quantum simulation include Rydberg atoms (Weimer et al. , 2010) and polar molecules (Lewenstein, 2006; Micheli et al. , 2006).”

What the sources say

• Not checked yetcites Weimer et al. , 2010
• Not checked yetcites Lewenstein, 2006
• Not checked yetcites Micheli et al. , 2006

In the paper

“Nuclear spins manipulated by means of NMR have been among the first experimental systems to implement small quantum algorithms and quantum simulation (Li et al. , 2011; Peng and Suter, 2010; Peng et al. , 2009; Zhang et al. , 2012).”

What the sources say

• Not checked yetcites Li et al. , 2011
• Not checked yetcites Peng and Suter, 2010
• Not checked yetcites Peng et al. , 2009

In the paper

“Nevertheless, nuclear spins provide a very good testbed for various small simulation problems and allow the implementation of both DQS and AQS.”

In the paper

“Any physical system that can be used as a quantum computer would also be a universal machine for DQS.”

In the paper

“If the region is roughly the same size as the wavelength of the charge carrier, the energy levels are quantized and the quantum dot becomes very similar to a real atom (and can therefore be referred to as an 'artificial atom').”

In the paper

“Ions can be trapped by electric (or magnetic) fields, laser-cooled and manipulated with high precision for realizing quantum simulation (Blatt and Wineland, 2008; Johanning et al. , 2009; Sch¨ atz et al. , 2007, 2004).”

What the sources say

• Not checked yetcites Blatt and Wineland, 2008
• Not checked yetcites Johanning et al. , 2009
• Not checked yetcites Sch¨ atz et al. , 2007, 2004

In the paper

“several groups have observed signatures of the crossover between a BCS and a BEC superfluid as the strength of attractive interactions between fermionic particles is varied”

What the sources say

• Not checked yetcites Regal et al. , 2004
• Not checked yetcites Zwierlein et al. , 2005
• Not checked yetcites Lewenstein et al. , 2007

In the paper

“The frequencies can also be tuned in situ by adjusting an external parameter (typically an external magnetic field), and the coupling energy between two qubits can be turned on and off at will.”

In the paper

“Photons can carry quantum information over long distances, hardly being affected by noise or decoherence.”

In the paper

“The Hubbard model is the simplest model of interacting particles on a lattice. However, for large numbers of particles in more than one dimensions, the model is difficult to treat using classical computers.”

In the paper

“In general, with a quantum simulator one could address problems that are either intractable on classical computers or experimentally challenging .”

In the paper

“Currently only with neutral atoms in an optical lattice is it possible to perform quantum simulations with more than a few particles, and these systems can at present be considered the most advanced platform for AQS.”

In the paper

“They naturally possess the ability to encode qubit states, e.g. in the polarization of the photon, and one-qubit gates can be easily realized with linear optical components.”

In the paper

“it has been simulated with a trapped ion (Gerritsma et al. , 2010).”

What the sources say

• Not checked yetcites Gerritsma et al. , 2010

In the paper

“Another system that could be used for the quantum simulation of condensed matter physics is electrons trapped on the surface of liquid helium (Mostame and Sch¨ utzhold, 2008). This setup could be used to simulate the Ising model.”

What the sources say

• Not checked yetcites Mostame and Sch¨ utzhold, 2008

In the paper

“A recent experiment (L¨ ahteenm¨ aki et al. , 2013) used a superconducting circuit to simulate a massless Klein-Gordon field with tunable speed of propagation, which was used to simulate the generation of photons in the dynamical casimir effect.”

What the sources say

• Not checked yetcites L¨ ahteenm¨ aki et al. , 2013

In the paper

“A quantum simulation of this phenomenon was recently implemented with trapped ions (Gerritsma et al. , 2011).”

What the sources say

• Not checked yetcites Gerritsma et al. , 2011

In the paper

“Furthermore, superconducting circuits can be coupled to 'cavities', which are actually electrical resonators (and the 'photons' are, for the most part, electron-density oscillations). This setup is very useful for the study of electric-circuit analogues of cavity quantum electrodynamics (circuit QED) (Schoelkopf and Girvin, 2008; You and Nori, 2005, 2011).”

What the sources say

• Not checked yetcites Schoelkopf and Girvin, 2008
• Not checked yetcites You and Nori, 2005, 2011

In the paper

“Although macroscopic in size, these circuits can display quantum behavior and can be seen as 'artificial atoms'.”

In the paper

“Unlike other systems studied in the context of quantum computation, however, NV centers in diamond have not received much attention as potential quantum simulators, which might be due to difficulties in coupling the NV centers to each other and future scalability.”

In the paper

“it is also possible to simulate classically the Zitterbewegung of a free Dirac electron in an optical superlattice (Dreisow et al. , 2010).”

What the sources say

• Not checked yetcites Dreisow et al. , 2010

In the paper

“In principle it could be scaled up, but the control would be very difficult.”

In the paper

“Although the difficulty in implementing two-qubit gates is a serious drawback for photonic systems in the context of quantum computation, there have been some notable achievements for quantum simulation using photons.”

In the paper

“Quantum information can be encoded in different ways: in the number of superconducting electrons on a small island, in the direction of a current around a loop, or in oscillatory states of the circuit.”

In the paper

“Both have discrete energy levels and exhibit coherent quantum oscillations between those levels.”

What the sources say

• Not checked yetcites You and Nori, 2005, 2011

In the paper

“the quantum baker's map, a prototypical example in quantum chaos, has an efficient realization in terms of quantum gates (Schack, 1998).”

What the sources say

• Not checked yetcites Schack, 1998

In the paper

“In (Aspuru-Guzik et al. , 2005) it was shown how to compute molecular energies using DQS and the required number of qubits was estimated (see also Figure 4).”

What the sources say

• Not checked yetcites Aspuru-Guzik et al. , 2005

In the paper

“Simulating the dynamics of open quantum systems is even more costly than that of closed quantum systems because solving the Lindblad equation requires quadratically more resources than solving the Schr¨ odinger equation for the same physical system.”

In the paper

“In (Piilo and Maniscalco, 2006) it was theoretically shown that a driven harmonic oscillator can act as a quantum simulator for the non-Markovian damped harmonic oscillator.”

What the sources say

• Not checked yetcites Piilo and Maniscalco, 2006

In the paper

“Another recent experiment simulated non-markovian Dynamics using a linear-optics setup (Chiuri et al. , 2012).”

What the sources say

• Not checked yetcites Chiuri et al. , 2012

In the paper

“Simulating open quantum systems with quantum simulators can be done in two ways.”

In the paper

“More recently, chemical reaction dynamics have been investigated using NMR DQS (Lu et al. , 2011).”

What the sources say

• Not checked yetcites Lu et al. , 2011

In the paper

“A simulation of decoherence caused by classical noise that was artificially added to the control signal of a superconducting qubit was reported in (Li et al. , 2013).”

What the sources say

• Not checked yetcites Li et al. , 2013

In the paper

“A recent experiment with trapped ions demonstrated the possibility of engineering open-system dynamics using the dissipative preparation of entangled states (Barreiro et al. , 2011).”

What the sources say

• Not checked yetcites Barreiro et al. , 2011

In the paper

“The algorithm would run in polynomial time while any exact classical simulation is bound to exhibit exponential scaling.”

What the sources say

• Not checked yetcites Lidar and Wang, 1999

In the paper

“The authors showed how a quantum dot with one electron can be considered an artificial hydrogen atom and a quantum dot containing four electrons can be viewed as an artificial oxygen atom.”

What the sources say

• Not checked yetcites Smirnov et al. , 2007

In the paper

“But whereas natural atoms are driven using visible or microwave photons that excite electrons from one state to another, the artificial atoms in the circuits are driven by currents, voltages, and microwave photons.”

In the paper

“Differences between quantum circuits and natural atoms include how strongly each system couples to its environment (the coupling is weak for atoms and strong for circuits) and the energy scales of the two systems.”

In the paper

“an efficient algorithm for calculating the thermal rate constant was given in (Lidar and Wang, 1999).”

What the sources say

• Not checked yetcites Lidar and Wang, 1999

In the paper

“The molecular energies of the hydrogen molecule have been calculated using a small photonic DQS (Lanyon et al. , 2010).”

What the sources say

• Not checked yetcites Lanyon et al. , 2010

In the paper

“The number of qubits scales linearly with the number of basis functions, and the number of gates grows polynomially with the number of qubits, so simulations with a few tens or a hundred qubits would already outperform classical computers.”

What the sources say

• Not checked yetcites Aspuru-Guzik et al. , 2005

In the paper

“These latter phenomena can be realized using quasi-one-dimensional open systems where photons are injected from one side and move towards the opposite side of the device.”

In the paper

“for some problems where bulk properties are of interest, individual control and measurement may not be required”

In the paper

“it was shown that six to ten qubits would be sufficient for a simulation of the single-particle Schr¨ odinger equation.”

What the sources say

• Not checked yetcites Benenti and Strini, 2008

In the paper

“The demonstration of quantum simulations using more than a few tens of qubits would mark the point where quantum computers (whether digital or analog) surpass their classical counterparts, at least for certain applications.”

In the paper

“Several phenomenological models have been developed, one of which is the superfluid model of the atomic nucleus.”

In the paper

“Progress on the experimental implementation of quantum simulation would also be relevant for measurement-based quantum computation.”

In the paper

“adiabatic quantum computation can be viewed as a special case of quantum simulation where the ground state of the simulated Hamiltonian encodes the solution to a computational problem”

What the sources say

• Not checked yetcites Farhi et al. , 2001

In the paper

“From the experimental point of view, in all proposed quantum simulators improved controllability and scalability are required.”

In the paper

“Even though N is not as large as in condensed matter physics (in this case N , the atomic mass number, is smaller than 300), the calculation of the nuclear force is difficult and therefore nuclear physics simulations require significant computing power.”

In the paper

“Several recent experiments have demonstrated boson sampling using photons (Broome et al. , 2013; Crespi et al. , 2013; Spring et al. , 2013; Tillmann et al. , 2013)”

What the sources say

• Not checked yetcites Broome et al. , 2013
• Not checked yetcites Crespi et al. , 2013
• Not checked yetcites Spring et al. , 2013

In the paper

“These qubits, operating as tunable mirrors, can change the reflection and transmission coefficients of photons confined in waveguides.”

In the paper

“Recent theoretical and experimental results on quantum simulation lead us to believe that practical quantum simulators will be built in the near future.”

In the paper

“Superconducting qubits can be used, for example, to perform Landau-Zener-St¨ uckelberg interferometry (Shevchenko et al. , 2010) and Fano and Fabry-Perot interferometry (Ian et al. , 2009; Liao et al. , 2010; Zhou et al. , 2008a).”

What the sources say

• Not checked yetcites Shevchenko et al. , 2010
• Not checked yetcites Ian et al. , 2009
• Not checked yetcites Liao et al. , 2010

In the paper

“Along the way, the photons interact with either one or two qubits acting as tunable mirrors, controlled by changing the applied electric and/or magnetic fields on the qubits.”

In the paper

“Boson sampling (Aaronson and Arkhipov, 2011) is another interferometry problem that is closely related to quantum simulation.”

What the sources say

• Not checked yetcites Aaronson and Arkhipov, 2011

In the paper

“Some recent topics in physics research, such as Majorana fermions (Casanova et al. , 2011; You et al. , 2011), graphene (Gibertini et al. , 2009) or neutrino oscillations (Noh et al. , 2012) are now discussed in the context of quantum simulation.”

What the sources say

• Not checked yetcites Casanova et al. , 2011
• Not checked yetcites You et al. , 2011
• Not checked yetcites Gibertini et al. , 2009

In the paper

“This was among the first experimental realizations of quantum simulation.”

What the sources say

• Not checked yetcites Leibfried et al. , 2002

In the paper

“Nonlinear interferometers were first investigated in trapped-ion experiments (Leibfried et al. , 2002).”

What the sources say

• Not checked yetcites Leibfried et al. , 2002

In the paper

“It should exhibit observable interesting behavior even with only eight qubits.”

In the paper

“The quantum baker's map has been experimentally realized in NMR (Weinstein et al. , 2002) and with linear optics (Howell and Yeaze, 2000).”

What the sources say

• Not checked yetcites Weinstein et al. , 2002
• Not checked yetcites Howell and Yeaze, 2000

In the paper

“The possibility of simulating the Schr¨ odinger equation on a DQS was first discussed in 1996 (Lloyd, 1996), with concrete algorithms given in 1998 (Boghosian and Taylor, 1998a), where a quantum lattice-gas model for the many-particle Schr¨ odinger equation in d dimensions was proposed.”

What the sources say

• Not checked yetcites Lloyd, 1996
• Not checked yetcites Boghosian and Taylor, 1998a