Quantum simulation: From basic principles to applications
aa r X i v : . [ c ond - m a t . qu a n t - g a s ] D ec Quantum simulation
Quantum simulation: From basic principles to applications
Laurent Sanchez-Palencia a a CPHT, Ecole Polytechnique, CNRS, Université Paris-Saclay, Route de Saclay, 91128 Palaiseau cedex, France
Envisioned by Richard Feynman in the early 1980s, quantum simulation has received dramatic impetusthanks to the development of a variety of plateforms able to emulate a wide class of quantum Hamiltonians.During the past decade, most of the quantum simulators have implemented rather well-known models,hence permitting a direct comparison with theoretical calculations and a precise benchmarking of theirreliability. The field has now reached a maturity such that one can address difficult problems, which cannot besolved efficiently using classical algorithms. These advances provide unprecedented opportunities to explorepreviously unreachable fields, test theoretical predictions, and inspire novel approaches.This contribution is an elementary introduction to quantum simulation. We discuss the challenges, defineboth digital and analog quantum simulators, and list the demanding conditions they require. We also providea brief account of the contributions gathered in the dossier on
Quantum Simulation of the Comptes-Rendusde Physique of the French Academy of Sciences [1,2,3,4,5,6]. The latter completes excellent reviews thatappeared previously, see for instance Refs. [7,8,9,10,11,12,13,14].
Universal models and the role of simulations in many-body physics
Understanding the behavior of macroscopic quantum systems is a major challenge of modern physics. Thebasic laws of low-energy physics are by now quite well known at the microscopic, say atomic, scale. Con-versely, many fundamental questions remain open, and even debated, about the collective dynamics at themacroscopic scale. By "macroscopic scale", here we mean systems made up of a huge number of constituents,or degrees of freedom, say , , or even more, as relevant in condensed matter physics or in astrophysics,for instance. Such huge systems cannot be treated exactly, be it a the classical level and, even worse, at thequantum level. Yet, it is the main outcome of the thermodynamic approach that the collective behavior of amacroscopic system can drastically differ from that of its elementary constituents. For instance, the elemen-tary interactions between the H O molecules are fundamentally unchanged when a water bucket turns fromthe liquid phase to the solid phase at zero degree Celsius. Similarly, the interactions between the microscopicmagnetic moments do not show any brutal change when a magnetic material gets magnetized underneaththe Curie temperature. Hence, the dramatic effects observed in macroscopic systems are governed by large-scale instabilities, without obvious counterparts at the microscopic level. This observation takes a universalcharacter, summarized in the celebrated motto "More is Different" [15]. Such so-called emerging phenomenaalso appear in quantum systems, where new effects that are impossible in the classical world show up belowsome critical temperature or at zero temperature when some interaction parameter passes through a criticalvalue. Celebrated examples include the superfluid Λ transition in helium or super-conducting transitionsand other metal-insulator transitions in electronic systems. Email address: [email protected] (Laurent Sanchez-Palencia).
Preprint submitted to Elsevier Science January 1, 2019 trikingly enough at first sight, while emerging phenomena only appear in very large scale macroscopicsystems, their germs are contained in the mutual interactions of their elementary constituents. Hence, localtwo-body interactions are sufficient to explain a huge class of phase transitions, such as the liquid-solid andmagnetic transitions mentioned above. The Boltzmann statistical approach proved particularly successfulin describing this connection between the microscopic and macroscopic scales. To understand emergingphenomena on grounds as fundamental as possible, the programme is now well established: One tries toidentify the basic microscopic terms that seem to be relevant and disregards all the other microscopic details.One then elaborates a model, as generic as possible, likely to reproduce the main experimental observations.The most usual examples are the Ising and Heisenberg models for magnetic transitions, or the Hubbardmodel for metal-insulator transitions [16,17,18]. Then all is left to do is to check that the phenomenon ofinterest indeed emerges from the dynamics of the simplified model. The realization of this programme isnothing but a simulation. It consists in building up a simplified system that mimics the main properties ofa real system.
The difficulty of simulating generic quantum systems using classical computers
Unfortunately, severe problems usually appear at the last stage because limiting ourselves to a simplemodel does not guarantee that the solution is straightforward. In a few cases, exact solutions are known, asignificant class of which is Bethe-ansatz integrable models. In some other cases, it is possible to make relevantapproximations and build up tractable theories that yield accurate predictions. This is the case of mean fieldapproximations, which work well to explain some superfluid transitions for instance. In most cases, however,exact or quasi-exact solutions are not known. One then traditionally resorts to numerical simulations. Inthis respect, the development of advanced approaches, such as Monte Carlo techniques, density functionaltheory, molecular dynamics, tensor-network approaches, and dynamical mean field theory, to name a few,have dramatically contributed to enlarging the class of models whose solution is known. However, even themost advanced numerical approaches have their own limitations and use approximations or representationsthat do not hold in all cases. Then, it is necessary to turn to exact computations. While it may be possiblefor some reasonably large classical systems, it is practically impossible for a quantum system.Consider the most simple example put forward by Richard Feynman in Ref. [19] of a system made of N spins / . Since each spin can be either in the spin-up or spin-down state, there are N possible configurations.A pure classical state is parametrized by N binary numbers, the values of which, or , represent thestate of each spin. A computer with a memory of Go ∼ bits can thus efficiently simulate about spins / . Conversely, a generic pure quantum state is the coherent superposition of all the possibleconfigurations. This requires N C -coefficients and the same memory of Go can only store the spin stateof only log(10 ) / log(2) ∼ particles. It is thus impossible to simulate the exact quantum state of morethan a few tens of spins. More generally, it is practically impossible to store and manipulate the state of amacroscopic quantum system, owing to the exponential growth of its Hilbert space in the system’s physicalsize.A careful reader might note that a similar issue occurs if one wants to simulate the full statistical dis-tribution of the classical counterpart of the same spin systems. In this case, one would need to store theprobabilities of each of the N configurations, the number of which also grows exponentially with the numberof constituents, N . However, this issue can be easily solved by using a stochastic algorithm, which amountsto introduce random jumps between the configurations. This is what Monte Carlo algorithms do for in-stance. Then, relevant quantities may be found by running the simulation a large, but not exponentiallylarge, number of times, and averaging the results. As noticed by Feynman, this just simulates what Natureindeed does when we acquire experimental data [19].The issue is much more serious in the quantum world, because even pure states cannot be simulatedefficiently on truly large scales. Then, Feynman pointed out that the only reasonable thing to do is tomake the computer quantum itself. Then, the register would naturally be exponentially large in the numberof bits and one would be able to store an exponentially large number of coefficients [19]. Feynman alsopostulated that it should be possible to simulate the time evolution of any quantum system in a reasonablecomputer time. This statement was proven for local quantum Hamiltonians a few years later [20]. Seith Lloydshowed that the simulation can be performed with arbitrary precision in a computer time that grows at2ost polynomially with the physical time, using Trotter-Suzuki’s decomposition of the many-body evolutionoperator into sequences of local Hamiltonian evolutions. A machine, exploiting quantum resources to simulatea given quantum model, is called a quantum simulator .Up to now, we have focused on the dynamics of many-body systems at thermodynamic equilibrium.However, similar questions and challenges appear when one considers the unitary time evolution of largesystems. Realizing quantum simulations in this case is actually even more important for several reasons. Onthe one hand, we do not know (yet) a universal approach, which would be the out-of-equilibrium counter-part of what is statistical physics for equilibrium thermodynamics. It follows that many questions are stillcompletely open, regarding for instance the spreading of information or the thermalization of these systems.On the other hand, out-of-equilibrium quantum systems are likely to develop an entanglement entropy thatgrows unbounded during the time evolution. It results that simulations using classical machines rapidlyreach their limits. What is a quantum simulator?
Building up a quantum simulator is still a tour de force . Of course, the aim here is not to reproduce exactlythe initial system, for instance a complete material with all its microscopic details. Should we do this,we would loop back to the initial problem and we would gain no information about the relevance of thetheoretical model. In the best case, we would address the specific initial problem and miss the basic physicalingredients at the origin of the considered phenomenon. Instead, the aim here is to build up a clean system,exactly governed by a basic model Hamiltonian. The latter is proposed in a theoretical context to reproducesome physical phenomenon observed in a class of real systems. This is the basic idea of quantum simulation.More precisely, a quantum simulator is a controlled device that allows us to (i) engineer a class of quantumHamiltonians exactly, (ii) control its dynamics, and (iii) make sufficiently precise measurements to considerthat the problem is solved. In practice, a quantum simulator is useful provided it is able to solve a hardproblem. In this context, one considers that a problem is "hard" when it cannot be solved, either analyticallyor numerically with a classical computer in a time that grows at most as a power of the system’s size. Thisfeature is often retained as the main criterion for a quantum simulator. However, it cannot be considereda perennial definition owing to continuous progress in many-body analytical and numerical approaches. Infact, it is very hard to prove that a given quantum problem cannot be solved with a classical algorithm inpolynomial time. However, it is a common belief that a large amount of entanglement is a good workingcriterion. Such a situation appears, for instance, in the vicinity of quantum phase transitions, in gaplesssystems at equilibrium or in far-from-equilibrium systems where the entanglement may grow unboundedduring the time evolution. The applications of quantum simulation are potentially unlimited and range fromcondensed-matter and high-energy physics to chemistry, biology, and cosmology, for instance [7,8,21].To be more specific, one distinguishes two classes of quantum simulators:
Digital quantum simulators –
A digital quantum simulator is a universal machine, fully reprogrammable tosimulate the thermodynamics or the real-time evolution of any quantum model. This is the kind of simulatorsenvisioned by Feynman and Lloyd. It was proven that, at least for local Hamiltonians, i.e., Hamiltonians con-taining only finite-range interactions, such a universal quantum simulator can indeed be implemented [19,20].Such a machine would exploit a fully reconfigurable register of qubits and a programmable sequence of log-ical gates to realize the desired simulation. This is actually nothing but a quantum computer [22,23,24,25].Hence, the distinction between a digital quantum simulator and a quantum computer mainly lies on the usewe make of it. While a "quantum computer" would be used to implement a variety of quantum algorithms,the most celebrated example of which are the Shor factorization algorithm or the Grover search algorithm,a "universal quantum simulator" would be dedicated to optimization problems, such as the determinationof the ground state of some Hamiltonian, or to its unitary, real-time, dynamics. In principle, building auniversal quantum simulator would thus be as difficult as building a quantum computer. In particular, itwould be sensitive to the same decoherence issues and require the implementation of error-correction codes.However, real or imaginary time evolution by a local Hamiltonian usually requires much less resources thangeneric quantum algorithms, and, in practice, are significantly more robust.3 nalog quantum simulators –
An alternative approach consists in building up a physical system from scratchto simulate each specific model. Assume one wants to determine the ground state or the time evolution ofsome Hamiltonian ˆ H , say a spin- / model. The idea behind analog quantum simulation is to create anensemble of elements with two well-identified states, e.g., the two polarizations of a photon, the two internalstates of an atom, or a quantum dot. These two states are used to represent the two spin states. Then isolatethe system from its environment and engineer interactions between these two-state elements according tothe Hamiltonian ˆ H . This may be realized by coupling the system with a cavity in the case of photons orvia laser fields in the case of atoms, for instance. In order to find the ground state, cool down the system;in order to determine its time evolution, prepare it in a well-defined initial state and let it run. Then, onerealizes from scratch a system exactly governed by the desired model Hamiltonian and Nature simply worksfor us, with all its quantum properties. The last thing to be done is to measure the outcome.Analog quantum simulation has pros and cons compared to digital quantum simulation. On the one hand,one needs to build up a new analog simulator for each studied model, while a unique digital simulator wouldbe sufficient. On the other hand, the architecture of the analog simulator can be optimized for the consideredproblem, while that of a digital simulator is the result of a compromise between all possible cases. In practice,an analog quantum simulator is much easier to build than a digital quantum simulator and there are nowmany examples of successful implementations of the former, see for instance Ref. [21] and the contributionsin this special issue [1,2,3,4,5,6]. Requirements and challenges
By analogy with standard numerical simulation, one may say that digital quantum simulation assumes thatthe hardware is available and focuses on the software part. Conversely, analog quantum simulation worksdirectly on the former and the software part is built on the latter. In both cases, realizing an efficientquantum simulator requires to take up several major challenges that we summarize here (see also Ref. [9]):(i)
Build up.
Create a quantum system that can be manipulated almost at will using external fields. Itshould be isolated from its environment to avoid decoherence issues. Depending on the problem athand, the system can be made of bosons, fermions, spins, or mixtures of the latter.(ii)
Quantum engineering.
Design the desired Hamiltonian with at least one adjustable relevant parameter.One should be able to tune the latter from a regime where the problem can be solved by other means toa regime where it cannot be solved easily. It allows benchmarking of the quantum simulator, similarlyas it is done in traditional numerical work.(iii)
Initialization.
Prepare the system in a well-defined state. It allows one to target the ground stateusing cooling techniques or to prepare some initial state to explore a specific trajectory in the system’sHilbert space, for instance. The initial state is often a pure state, but one can also prepare a mixedstate by interaction with a controlled bath. Furthermore, the bath may be used for simulating openquantum systems.(iv)
Detection.
Measure relevant, local or non-local, observables that yield sufficient information to "solve"the problem with sufficient fidelity. Depending on the problem at hand, one may use destructive ornon-destructive measurement techniques.Since the goal is to use quantum simulators to solve problems than cannot be solved by other means, a majorconcern is of course the reliability of the used simulator. There are several ways to address this issue [26].First, as mentioned above, one can make sufficient benchmarking of the simulator by addressing regimeswhere other solutions exists. Second, in the case of isolated systems, one can check that quantum coherenceis maintained using an adiabaticity property: one back-propagates the system and checks the fidelity of thefinal state to the initial state. Third, it is crucial to solve a given problem on several simulators based onsignificantly different platforms. Should the results be consistent, one would consider that the solution isreliable. In this respect, dramatic progress has been realized starting by the early 2000s in a variety of fields,including:(*) Ultracold quantum gases,(*) Artificial ion crystals,(*) Photonic systems, including polaritons in cavities,(*) Superconducting circuits, 4*) Magnetic insultors,(*) Electronic spins in quantum dots,for instance. During the past decade, a number of quantum simulators have been demonstrated. So far,the results of most of them could be directly compared to theoretical calculations, which allowed extensivebenchmarking. Now, some of the most advanced implementations are likely to address really hard problems,which cannot be solved efficiently using classical algorithms. To make this exciting perspective a reality,continued development of a variety of platforms, including those mentioned above and hopefully new ones,will be pivotal. It will allow us to address complementary questions as well as to compare results obtainedby different approaches.
Contributions to this dossier
The dossier on
Quantum simulation makes a point on recent advances of the field and discusses perspectivesvia a selection of contributions in various areas from ultracold atoms and quantum optics to statistical physicsand condensed matter. Tarruell and myself review progress on the quantum simulation of the celebratedHubbard model using ultracold Fermi gases. Aidelsburger et al. provide an introduction to novel approachesto engineer artificial gauge fields within a wide class of systems ranging from quantum optics to solid-state systems. Lebreuilly and Carusotto discuss realizations of strongly correlated quantum fluids of lightin driven-dissipative photonic devices with applications to the generation of Mott insulator and fractionalquantum Hall states of light. Le Hur et al. review advances in the study of real-time dynamics of impuritymodels and their realizations in quantum devices, including superconducting circuits, quantum electricalcircuits, and ultracold-atom architectures. Bell et al. propose a novel platform based on superconductingquantum interference devices (SQUIDs) to emulate quantum phase transitions in one dimension, as well asperspectives to address non-integrable and disordered systems. Finally, Alet and Laflorencie discuss recentadvances on many-body localization in isolated quantum systems and current experimental efforts to probingthis physics.
Acknowledgment
I am grateful to Jean Dalibard and Daniel Estève for suggesting the dossier on
Quantum Simulation and toChristophe Salomon for tacking over the editorial process of the contribution I am an author of [1]. I alsothank Jean Dalibard for useful comments on this manuscript.
References [1] Tarruell, L. & Sanchez-Palencia, L. Quantum simulation of the Hubbard model with ultracold fermions in optical lattices.
C. R. Phys. , 365–393 (2018).[2] Aidelsburger, M., Nascimbene, S. & Goldman, N. Artificial gauge fields in materials and engineered systems. C. R. Phys. , 394–432 (2018).[3] Lebreuilly, J. & Carusotto, I. Quantum simulation of zero temperature quantum phases and incompressible states of lightvia non-Markovian reservoir engineering techniques. C. R. Phys. , 433–450 (2018).[4] Le Hur, K. et al. Driven dissipative dynamics and topology of quantum impurity systems.
C. R. Phys. , 451–483 (2018).[5] Bell, M., Douçot, B., Gershenson, M., Ioffe, L. & Petkovic, A. Josephson ladders as a model system for 1d quantum phasetransitions. C. R. Phys. , 484–497 (2018).[6] Alet, F. & Laflorencie, N. Many-body localization: An introduction and selected topics. C. R. Phys. , 498–525 (2018).[7] Buluta, I. & Nori, F. Quantum simulators. Science , 108–111 (2009).[8] Georgescu, I. M., Ashhab, S. & Nori, F. Quantum simulation.
Rev. Mod. Phys. , 153–185 (2014).[9] Cirac, J. I. & Zoller, P. Goals and opportunities in quantum simulation. Nat. Phys. , 264–266 (2012).[10] Bloch, I., Dalibard, J. & Nascimbène, S. Quantum simulations with ultracold quantum gases. Nat. Phys. , 267–276(2012).[11] Blatt, R. & Roos, C. F. Quantum simulations with trapped ions. Nat. Phys. , 277–284 (2012).
12] Aspuru-Guzik, A. & Walther, P. Photonic quantum simulators.
Nat. Phys. , 285–291 (2012).[13] Houck, A. A., Tureci, H. E. & Koch, J. On-chip quantum simulation with superconducting circuits. Nat. Phys. , 292–299(2012).[14] Ward, S. et al. Spin ladders and quantum simulators for Tomonaga-Luttinger liquids.
J. Phys.: Cond. Matt. , 014004(2013).[15] Anderson, P. W. More is different. Science , 393–396 (1972).[16] Mahan, G.
Many Particle Physics (Springer, New York, 2000).[17] Bruus, H. & Flensberg, K.
Many-body quantum theory in condensed matter physics: An introduction (Oxford universitypress, Oxford, 2004).[18] Tsvelik, A. M.
Quantum field theory in condensed matter physics (Cambridge university press, Cambridge, 2007).[19] Feynman, R. P. Simulating physics with computers.
Int. J. Theor. Phys. , 467–488 (1982).[20] Lloyd, S. Universal quantum simulators. Science , 1073–1078 (1996).[21] Nature Physics Insight on Quantum Simulation.
Nat. Phys. , 263–299 (2012).[22] Benioff, P. The computer as a physical system: A microscopic quantum mechanical Hamiltonian model of computers asrepresented by Turing machines. J. Stat. Phys. , 563–591 (1980).[23] Deutsch, D. Quantum theory, the Church-Turing principle and the universal quantum computer. Proc. Roy. Soc. A: Math.Phys. Eng. Sci. , 97–117 (1985).[24] DiVincenzo, D. P. Quantum computation.
Science , 255–261 (1995).[25] DiVincenzo, D. P. The physical implementation of quantum computation.
Fortschr. Phys. , 771–783 (2000).[26] Hauke, P., Cucchietti, F. M., Tagliacozzo, L., Deutsch, I. & Lewenstein, M. Can one trust quantum simulators? Rep. Prog.Phys. , 082401 (2012)., 082401 (2012).