PoS(LATTICE2018)024 https://pos.sissa.it/ ∗ [email protected] Speaker. Forthcoming exascale digital computers will furtherdynamics, advance our but knowledge formidable of challenges quantum will chromo- are remain. not In particular, well Euclidean suited Montehadronic for Carlo matter methods studying at real-time nonzero temperature evolutionable and in to chemical hadronic achieve potential. accurate collisions, simulations Digital of or computerstheories; such may the phenomena quantum never in properties computers be QCD of will and do otherquantum strongly-coupled so simulation field eventually, of though quantum I’m field notand theory sure will field when. require theorists, the Progress and collaborativestarted toward efforts though of now. the quantumists Today’s physics research payoff canfuels may hasten rapid still progress the in be arrival fundamental of far physics. a away, new it’s era worthwhile in to which get quantum simulation ∗ Copyright owned by the author(s) under the terms of the Creative Commons c Attribution-NonCommercial-NoDerivatives 4.0 International License (CC BY-NC-ND 4.0). The 36th Annual International Symposium on22-28 Lattice July, 2018 Field Theory - LATTICE2018 Michigan State University, East Lansing, Michigan, USA. John Preskill Institute for Quantum Information and Matter Walter Burke Institute for Theoretical Physics California Institute of Technology, Pasadena CA 91125,E-mail: USA Simulating quantum field theory with a quantum computer PoS(LATTICE2018)024 any Quantum John Preskill ]. Instead, I’ll insert some 1 simulated by a classical Turing machine. Church-Turing Thesis, which asserts that any 1 efficiently Strong ]. This is a high-level overview, eschewing technical details. I have tried to , sometimes called the 5 , 4 , 3 , 2 complexity Along the way I’ll tell you about some of the things I’ve done with my collaborators (Stephen Physics underlies computer science, because computation is a physical process. The funda- Though we don’t know how to prove it from first principles, we have good reasons to believe The Church-Turing Thesis asserts that any physically reasonable model of computation can My talk at Lattice 2018 had two main parts. In the first part I commented on the near-term Is the Quantum Church-Turing Thesis true? We don’t know, but either a Yes or No answer Jordan, Hari Krovi, and Keith Lee),these some things of [ the things we still hope to do, and why we are doing general remarks about quantum field theorywho which lack might be familiarity helpful with for this quantum subject. computationalists make it accessible to a broader audience than the attendees of Lattice 2018. 2. Quantum simulation of quantum field theory: Why do2.1 it? The Quantum Church-Turing Thesis mental question about computers is:computations are What intractable? This computations is can athose be question laws. about performed the efficiently, laws and of physics what and the consequences of that there are computational problemson which classical are physics, too yet hard caninterference to be and solve solved using entanglement. by digital computers computers Thisand which based quantum is exploit physics, quantum one and phenomena of it suchclassically deepest poses as hard the distinctions but compelling quantumly ever challenge easy. made to between identify those classical problems which are be simulated by a (classical)been upended Turing by machine. the discovery This ofputational statement fast is quantum still algorithms is widely a believed. stronger statement What regarding has com- physically reasonable of computation can be prospects for useful applications of quantum computing.for In the advancing second part, fundamental I physics assessed thecomputers through prospects and simulations quantum of simulators. quantum Intopic, field this because article theory I based using on have quantum that already talk, written I up will my de-emphasize thoughts the first about that in [ is pleasing. If the answerTuring Thesis is is Yes false), (and then if the quantum asfundamental computer we physics. will suspect, become If an the the indispensable classical tool answerof version for is exploring a of No, quantum the that’s even Strong computer more Church doesn’tof exciting; fully Nature. it capture means the our computational current power concept that’s encoded in the laws Simulating quantum field theory with a quantum computer 1. Introduction This statement is probablyChurch-Turing wrong, Thesis. and should Informally, be thiswe replaced revised plan thesis by to asserts an build that updated andprocess the version, operate that quantum in occurs the computers in the Nature. which coming decades will be capable of efficiently simulating PoS(LATTICE2018)024 , which John Preskill simulate highly entangled sys- highly entangled matter can . In particular, we would like to be sign problem 2 ]. Where should lattice QCD go from there? of high-energy collisions between hadrons, computing from 6 has been a major thrust in computational physics for decades, dynamics Lattice QCD ]. Beyond providing quantitative results, both thinking about and actually perform- 7 , strongly interacting composite particles such as mesons, protons, neutrons, and atomic In fact, the Monte Carlo simulations have serious limitations. Even with further advances in These tasks are difficult because they involve simulations of All the known fundamental interactions in Nature, with the conspicuous exception of gravita- The Hamiltonian of QCD is precisely known, but the Schrödinger equation for this Hamilto- With greater processing power, such static quantities can be computed with improved statistics. Aside from QCD, quantum simulation will also allow us to explore the properties of other hadrons able to simulate the real time digital computing technology, many quantitiesbecause of of a interest fundamental to obstacle physicists known as will the remain out of reach first principles the distribution of outgoingis particles needed produced to in guide searches such for collisions.using new particle This physics accelerators. information beyond We the would also Standard like Modelsity, to in pressure, know experiments the and properties performed temperature. of nuclear This matterstars, at information high and is den- for needed for understanding studies theYet these of behavior tasks supernovae of seem and to hadronic neutron be matter beyond in the reach the of early digital history computation of because of the the universe. sign problem. tion, are encompassed by the Standardthe Model consequences of particle of physics. the Physicists Standard worksults, Model, hard with and to great compute to success. compare The thesetum most predictions chromodynamics daunting with computational (QCD), experimental challenge the re- has quantumof been field the theory study of underlying quan- thenuclei. structure and interactions nian is too hard to solvetions. analytically, Fortunately, some because predictions hadronic of matter QCD isulations. can subject be The to extracted study through large of numerical quantum Monte fluctua- Carlo sim- In a few years, exascalefor Monte many Carlo typical simulations physics of applications lattice [ QCD may provide adequate precision strongly-coupled quantum field theories,physics beyond which the might Standard be Model.stepping important stone to In for simulations addition, of formulating simulations quantumanti-de models gravity, of Sitter which of quantum space) (at field least can in be theory theof formulated provide spacetime case in a [ of terms quantum of gravity a in quantum field theory living on the boundary Simulating quantum field theory with a quantum computer 2.2 The future of computational quantum chromodynamics and many impressive results haveprecisely been computed achieved. and found In to particularphenomenologically match relevant the their matrix masses elements experimentally of of observed local hadrons values; operators furthermore, have have many been been accurately estimated. cannot be succinctly described incoded ordinary in classical conventional language, digital and computers. hencetems cannot Quantum of be many computers particles, accurately and en- hence solvecomputers the sign will problem. be Eventually essential (I’m for not advancing sure when), our quantum understanding of QCD. 2.3 Beyond QCD PoS(LATTICE2018)024 John Preskill Noisy Intermediate- ]. It stands for 1 [ ]. (3) After many decades of effort by 10 , NISQ 9 ]. The essential idea is that if we want to protect 3 11 ]. (2) Arguments based on complexity theory show (under reasonable 8 . Here “intermediate scale” refers to the size of quantum computers which will Though truly scalable fault-tolerant quantum computing is still a rather distant dream, we But building and operating a large-scale quantum computer is a formidable task. Why is it so Eventually we expect to be able to protect quantum systems and scale up quantum computers Physicists are excited about this NISQ technology, which gives us new tools for exploring the We have at least three good reasons for thinking that quantum computers have capabilities are now entering a pivotaldescribe new this era impending new in era, quantum so technology. I made I up a thought word: we should have a name to a quantum system from damagethe then environment, we interacting should with encode partsinformation it of and in the therefore a system can’t damage very onedoing it. highly at quantum error entangled a Unfortunately, correction, state; time, there so is reliable cannot then not quantum a glimpse likely computers significant the to using overhead be quantum encoded cost error available very correction for soon. are 3.3 The NISQ era unfolds difficult? The core of thecannot problem observe stems a from quantum a system fundamentalThat without feature means producing of that an the if uncontrollable we quantum disturbance want worldwe to in — use need the we a to system. quantum keep system that to system store nearly and perfectly reliably isolated process from information, the then using outside the world. principle of quantum error correction [ physicists to find better ways tosimulate a simulate quantum quantum computer systems, using we a still classical don’t digital know computer. how3.2 to efficiently Why quantum computing is hard Scale Quantum assumptions) that a quantum computer can efficientlycan’t sample sample from probability from distributions efficiently that by we any classical means [ be available in thethese next devices few are years, too with large tosupercomputers. a be number “Noisy” simulated of by emphasizes brute qubits that forcewill we’ll ranging using limit have from the the imperfect most 50 control power powerful to over of existingcircuit a those digital NISQ (say, few qubits. technology, one hundred; because with Noise if morethen we the than signal attempt 1000 in to gates the execute final — readout too that is large is, likely a to 1000 be quantum fundamentalphysics overwhelmed by of two-qubit the many operations), noise. entangled particles. We also hope that NISQ devices will have valuable practical surpassing what classical computers can do.for (1) classical We computers, know of but problems forthese that which are problems quantum believed algorithms easily. to have be been hard Thea discovered large best that composite known could integer solve example [ is the problem of finding the prime factors of Simulating quantum field theory with a quantum computer ing simulations of quantum fieldconceptual theory insights. and quantum gravity may well lead to unanticipated new 3. The potential of quantum computing 3.1 Why we think quantum computing is powerful PoS(LATTICE2018)024 John Preskill — that is, predicting ]. Perhaps we will learn 12 ]. That’s a very large leap from 15 quantum dynamics we mean a system with many qubits whose 4 is a gate-based universal quantum computer which can be ]. 14 analog quantum simulator , 13 digital quantum simulator Throughout the NISQ era it will remain important to strive for lower gate error rates in the We can anticipate, though, that analog quantum simulators will eventually become obsolete. Solving really hard problems (like factoring numbers which are thousands of bits long) using Of the applications for quantum computers that we can currently foresee, the most important is Classical computers are especially bad at simulating When we speak of an Analog quantum simulation has been a very vibrant area of research for the past 15 years, various quantum platforms. With more accurate quantum gates, quantum computers operating where we will be forchasm,” the from next hundreds few of years, physical qubits withtake some to of time, millions order but of a we’ll physical get hundred qubits there qubits. (and eventually. beyond), Crossing is the going “quantum to Because they are hard towhich control, can they be will firmly be controlled surpassed usingoverhead some quantum cost day error by of correction. digital quantum Nevertheless, quantum because errorsist simulators, of for correction, the many hefty the years. reign Whenoverlook of the seeking the potential near-term power analog applications of quantum of analog quantum simulator quantum simulators. technology, may we per- should3.6 not The daunting climb to scalability fault-tolerant quantum computing is not likelyof physical to qubits happen needed. for To a run while,a algorithms because involving number thousands of of of the physical protected large qubits qubits number we which may is need in the millions, or more [ dynamics resembles the dynamics ofcontrast, a a model system we are trying to study and understand. In likely to be studying the propertiesthat of simulating highly entangled such many-particle “strongly quantummany systems. correlated” very good We matter think physicists is have tried a to solve difficult it computational for decades problem, and because have not succeeded. while digital quantum simulationnow with getting started. general Analog purpose quantum circuit-based simulatorsare have quantum already been being getting computers employed notably is to more study sophisticated, just quantumof and dynamics classical in simulators regimes [ which may be beyond the reach Simulating quantum field theory with a quantum computer applications, but we’re not sure aboutmative effects that. on I society eventually, do but think theselong that may it’s going quantum still to be computers take. decades will away. have We transfor- just don’t know3.4 how Quantum simulation interesting things about quantum dynamicsusing noisy using devices NISQ with of technology order in 100 the qubits. relatively3.5 near future, Digital vs. analog quantum simulators used to simulate any physical systemfor of other purposes. interest when suitably programmed, and can also be used how a highly-entangled quantumquantum state computers seem will to change have a with clear advantage time. over classical ones That [ is the natural arena where PoS(LATTICE2018)024 John Preskill 5 ]. The subject took a significant step forward around 1970 16 ]. The Standard Model of particle physics was also formulated ) really works. 17 137) is weak, it’s sensible to use perturbation theory to work out the con- / 1 . But confusion reigned for 20 years because when one attempts to improve ≈ α α renormalization Ken Wilson is a hero to me and many others because he provided a satisfying answer to the The problem of infinities arises because quantum field theory involves an infinite number The first efforts to turn quantum field theory into a rigorous mathematical subject occurred in Quantum field theory was first invented soon after the arrival of quantum mechanics; Dirac, Because the coupling strength of photons to electrons and other charged particles (the fine- question: What is quantum field theory?final answer, (His but was nevertheless not it the transformed firstmore our answer, deeply understanding nor than of was his it the predecessors the subject.) the complete Wilson meaning and understood of renomalization. of degrees of freedomEven per for unit a volume; finite volume the of fields space, have no modes matter of how arbitrarily small, short simulating wavelength. this infinite system exactly in the 1970s, and still standsdecades as our of best thorough description vetting of in the high-energy-physics strong experiments. and electroweak interactions after 4.2 Universality: Why Ken Wilson is our hero largely through the work of Kenof Wilson, second-order who phase transition taught [ us to think of quantum field theory as a kind sequences of the quantum electrodynamics, andnontrivial this order works pretty in well for computationsthe to accuracy the of first such computationsencountered by which seemed going at to first higher toabruptly order have in no the in late clear perturbation 1940s, physical when theory, interpretation.handle Schwinger, infinities Feynman, That these are Tomonaga, infinities situation and successfully changed Dyson by realized thinking thatcomparing operationally one the about can what result is of being a computed.of computation When other with experimentally experiments, accessible we quantities, shouldinfinities such express conveniently as our cancel. the answers mass in The or terms moralwill charge is: provide of a if the electron; you sensible then askprocedure answer; the (called the however, theory it a took sensible a physical lot question it of combinatoricthe agility 1950s, to when verify Wightman that inby this particular a formulated relativistic quantum a field set theory [ of axioms which define what we mean Heisenberg, Jordan, Pauli, Born and other pioneersformulations of of quantum quantum mechanics field also theory. contributed One toelectromagnetic of early the field first yields things photons; they noticed thatlight is insight quanta, that provided which quantizing the until a free then firmer had foundation been for a the more heuristic concept idea. of structure constant Simulating quantum field theory with a quantum computer without error correction will be ablegates to will go lower further by the executing overheadon larger cost to circuits. fault-tolerant of Furthermore, quantum better doing computing quantum innoise error resilience the into future. correction our when To quantum the we algorithms. extent eventually possible, move we should also build 4. Quantum field theory 4.1 Some history PoS(LATTICE2018)024 10 − John Preskill meters), then we’d really 18 − 6 meters, to compute the properties of the hydrogen atom (10 35 − By attacking problems one energy scale at a time we can make progress, but universality is From today’s post-Wilson viewpoint, the notion that a quantum field theory accommodates an An effective theory has an energy scale associated with it, what we call the theory’s renormal- You might think that the long-distance physics should depend in a very complicated way on Universality makes quantum field theory useful in condensed matter physics, where we typi- Universality is really what makes it possible for us to do physics at all. If we had to understand really a two-edged sword.at Since shorter low-energy distances, phenomena it do is notfeasible hard depend using to very today’s learn particle strongly about accelerators. on very-short-distance This physics physics limitation in frustrates experiments the which high-energy are physicists. infinite number of degrees of freedom perrather unit than volume is a really precise a convenientknown description fiction, physics of an (excluding idealization reality. gravity), but The“effective” we reminds Standard modestly us Model call of accurately the it describes theory’s anit limited almost toward “effective” scope, and field all that higher theory. we and expect The higher it energy. word to fail eventually if weization push scale. As thatto scale provide changes, an accurate the description effectiveory theory of is the should called physics “renormalization-group change, flow.” at too, Givenat that an if shorter scale. effective it theory, distances a This is and theory scale-dependence to flows whichcompletion” of to makes continue of the sense that the the- effective effective theory theory. atcause If longer of an distances universality ultraviolet it is completion need called of notdistances, the the be we “ultraviolet effective unique. need theory to That’s exists, why, do be- to experiments find at out higher how energies. physics behaves at shorter meters) or the physics that’s going onbe at the stuck. Large Hadron Fortunately, that Collider (10 isa not given necessary. energy We scale can in do detail,higher physics energies. while one still scale remaining at largely a ignorant time, about understanding what’s going on at still the short-distance physics that defines the theory,happens. and that’s true All in of a way. the Butparameters” short-distance something needed wonderful details to can specify the be long-distance absorbedconvey phenomena. into that We many a call this models small miracle with number “universality”predictions different of to about properties “renormalized the at physics short at distances long can distances. all give rise tocally the study macroscopic same phenomena which are insensitive todistance the scale. microsopic properties And at it the is atomic alsointeractions. used in For fundamental this physics to latter describe purpose elementarymeans the particles the and quantum same their laws field of theories physics ofanother, govern so interest all there observers are is who “relativistic” no move — preferred at this constant inertial velocity frame relative of to reference. one 4.3 Effective field theory physics at the Planck scale, at 10 Simulating quantum field theory with a quantum computer would be impossible with anyyou finite are machine. interested in But computingwhat the things physicists key call that point “the can about infrared”), be thereunderlying renormalization measured physics is is at limited at that very sensitivity long short if of distances distances that (“the or long-distance ultraviolet”). low physics to energy the (in PoS(LATTICE2018)024 John Preskill superrenormalizable ]. In the early days this 16 shrinks to zero is essentially equiv- a 7 , where the Hamiltonian couples neighboring lattice sites. a ], but that success has mostly been limited to what we call 18 Wilson’s insights largely flowed from his thinking, in the 1960s, about how to simulate a One apologetic note: I have heavily used the word “theory” in this discussion. Physics read- For mathematically inclined readers, I’ll make a comment about rigor. One can rigorously Taking the “continuum limit” in which the lattice spacing Sometimes instead of an “ultraviolet completion” we speak of a “continuum limit.” Following quantum field theory on ainsights digital from computer. thinking It about, and is actually reasonable doing,computers. to simulations Or anticipate of we quantum that might field have we theories an will ondoing, even quantum gain more simulations ambitious further of goal quantum — gravity perhaps on thinkingvexing quantum about, question: computers and can actually What bring is us string closerformulations to of theory? quantum answering field the It theory may (ca. be 1927),(ca. to noteworthy Wilson’s 1970), insights that about is the the comparable nature time to oftheory it the the of took subject span gravity from of (ca. the time first sinceWilson 1974) to string to come theory today. along. was In first the proposed study as of a quantum quantum gravityers we’re are still not waiting for likely our tocomputer be science, perturbed it by can overuse beory,” of disorienting. and this there I word, may have not but notsay be for attempted that, unanimous specialists to thanks agreement in explain to about other what universality, we that fields,which I can even describe mean like identify among similar by broad physicists. long-distance a classes physics, “the- Suffice ofized encoded it microscopic parameters, in model to a and Hamiltonians small we numbersame usually of “theory.” regard In adjustable many two renormal- cases, Hamiltonians just theare in spatial enough the dimensionality data and same for the specifying class symmetry the of as “theory” the in belonging Hamiltonian question. to4.4 the About rigor define what a relativistic quantumand one field can theory prove is nontrivial — theorems that that follow was from point these of axioms the [ Wightman axioms — alent to characterizing whatcompared the to physics the looks lattice like spacing.becomes at This a distance limit boring may scales noninteracting fail whichinteracting (“free”) to are effective exist, theory theory arbitrarily or cannot in be large rather, this completed it limit. in might the be ultraviolet. Correspondingly, that in the that theory case the program was nonconstructive; the only“free” known theories examples of of noninteracting theories particles,there satisfying were which the notable are successes axions not at where the very rigorously interesting. axioms constructing [ nontrivial Starting interacting in theories the that 1970s satisfy This theory defined on the lattice, withsaid a to finite be number a of lattice “regulated” sites versionas per of unit the the of continuum “short-distance physical theory, volume, cutoff,” where is the thethe shortest lattice theory spacing distance is can scale be regulated, to regarded it whichan has the infinite a theory number, finite is number which applicable.regulated, of makes Once degrees the it of theory easier freedom can to perscientists. unit be define, volume, simulated understand, rather using and than methods study. which Furthermore, are once well known to computational Wilson, we make senselattice of with the a quantum nonzero field lattice theory spacing by replacing continuous space by a spatial Simulating quantum field theory with a quantum computer PoS(LATTICE2018)024 like John Preskill correlation functions transport properties 4 spacetime dimensions isn’t a vacuum-to-vacuum persistence = D . Given some description of an initial state 8 ]. Though there is much evidence in favor, this conjecture 7 scattering processes ]. For now that’s the best we can do. Our goal is for the analysis to be 4 , 3 , 2 . If classical sources are coupled to local observables, where the sources turn on and off 4 is marginal; we don’t think it exists as an interacting theory in the continuum satisfying = When we analyze classical or quantum algorithms for simulating quantum field theory, we By the way, while there is a rigorous version of relativistic quantum field theory, and a set of When we simulate quantum field theory, what kinds ofWe might, problems for might example, study we want to solve? With rare exceptions, quantum field theory in more than All these simulations of dynamics are hard to do with classical computers. But no “sign D sometimes need to settle forsimulations heuristic scales arguments. with For the example, lattice in spacing,we estimating we can’t how might fully the justify make error [ use in of our perturbation theory in ways that very interesting subject because onlyin free theories exist . Theall case the of Wightman a axioms, self-coupled but scalarexceeding it theory the is short-distance still cutoff potentially by interesting a to reasonably large simulate factor. its behavior at distances axioms specifying properties which any suchtheory. theory A should famous obey, conjecture we asserts don’t thatspacetime yet quantum have is gravity that in precisely for negatively equivalent curved string (or anti-defined Sitter “dual”) on (AdS) the to boundary a of conformally the spacetime invariant [ quantum field theory de- 5. Quantum simulation of quantum field theory: What is it? 5.1 What problem does the algorithm solve? There are many options. precise when possible, and nonrigorous only when necessary. is a bit dissatisfying at presentAdS because aside we from have saying no it alternative is way the of thing defining which is quantum dual gravity to in the field theory! such as particles thatfinal are states about produced to by collisions. collide We with might one want to another, compute we the can sample accurately from the Simulating quantum field theory with a quantum computer theories, which have especially weak sensitivitywhether to asymptotically the free short-distance quantum physics. chromodynamics ItWightman (QCD) remains axioms, unsettled exists though as we a physicists have theory no that particular satisfies reason the to doubt it. amplitude in some way that dependsground on state of position the in theory spaceof to and be the time, preserved. local we This observables. can amplitudethe calculate can matrix We the element be can between amplitude related also initial to for calculate andat the final such specified ground correlation points states in for functions a spacetime. directly, string determining Such of local correlators observables can inserted be related to bulk problem” prevents us from doing the simulation if we have a quantum computer. thermal conductivity or viscosity. Orthe we bulk can dynamics. compute transport Toinvestigate properties study these by properties phenomena by directly of doing simulating an interest, experiment?algorithm it modeled Your answer on is may that guide often potential the experiment. fruitful formulation of to an ask: How would you PoS(LATTICE2018)024 H (in a John Preskill geometric locality . When the quantum algorithm ex- can be expressed as a sum of terms H local 9 is local, they usually mean that H Experts on lattice gauge theory should be reminded that when we speak of algorithms for That’s too bad, because imaginary time evolution has advantages for some purposes like ef- In quantum simulations we work with Hamiltonians, not with a manifestly covariant formula- The Hamiltonians that arise in physics are typically Quantum field theories are geometrically local in this sense; if the theory is defined on a spatial Here’s a typical task that we might perform in a quantum simulation. (1) Prepare the initial The goal of this procedure is to sample accurately from the probability distribution of possible simulating quantum systems on a quantumevolution computer in we real usually have time. in mind Today’sEuclidean lattice simulating rather experts quantum than prefer Lorentzian (classical) spacetime. simulationsdon’t know in We how quantumists imaginary to wish time, simulate we in imaginary could time do evolution with the a same, quantum but computer. ficiently we preparing ground states ofevolves in quantum real systems. time, too, and But oursimulation it’s ultimate goal not of is really real-time to simulate evolution so how for bad, Naturequantum highly behaves! because field Furthermore, entangled Nature the theories, many-particle is quantum believed systems,clidean to computations including be are a useful for hard computing problemelements static classically, of properties, yet local like operators, quantumly the but mass tractable. are spectrumquantum ill Eu- and computer suited matrix should for have studying a dynamical huge phenomena. advantage. That’s where the tion of the theory likethe an action Hamiltonian formulation. evolution Typically in wedepend that pick on some frame. that inertial choice frame, of If reference andas frame. we simulate observed In are in any a careful case, particular our we frame, goal so might can it be extract to makes simulate sense results an to which experiment simulate the do dynamics not as seen in that frame. Simulating quantum field theory with a quantum computer 5.2 Real time vs. imaginary time lattice, only fields at neighboring lattice sitesthan are coupled. arbitrary Hamiltonians, Local Hamiltonians but are the easier simulation to task simulate can still be quite5.3 challenging! Prototypical task state whose evolution we want tostate of study two in or our more particles, quantum ifward computer, we in such are time as trying to using an study the incoming a Hamiltonian,specified scattering scattering in process. time the (2) interval. particular Evolve the reference state (3) frameperformed for- Measure that by we’ve a an chosen, particle observable, for detector for some in example an idealized by laboratory. simulating theoutcomes measurement for that finalin measurement. the simulation If as the the simulationsample distance from is (in in imperfect, practice, some and then suitable the norm) we ideal between distribution quantify the for the the probability error noiseless distribution system. we perts say a Hamiltonian such that each term couplesnumber together of only degrees a of constant freedom number whichis increases of local, with degrees the they of size usually freedom, of have rather thespecified in than system. number mind a of When a spatial physicists say dimensions). stronger For property,dimension example, sometimes if a called the Hamiltonian degrees is of geometricallycouples local freedom only in can degrees one be of arranged freedom on located inside a an line, interval and of each constant term length. in the Hamiltonian PoS(LATTICE2018)024 ]. 4 , 3 , 2 ]. John Preskill 21 , the mass of the lightest mass gap 10 . For example, suppose we would like to prepare the ground state ], as hard as any problem whose solution can be checked efficiently 19 , as hard as any problem whose solution can be checked efficiently using a ], and QMA-hard instances can occur even in one dimension [ 20 adiabatic method QMA-hard To get the simulation started, we need to load the initial state into our quantum computer. That That sounds scary, but we don’t need to be too discouraged if our goal is to simulate Na- With admitted hubris, we declare that if Nature has managed to prepare the initialOne state is effi- the In analyzing the algorithm, we determine what computational resources would suffice to If we use a classical computer to perform this task, a brute force simulation would require using a classical computer.harder: For it quantum is local Hamiltonians,quantum preparing computer the [ ground state is even might not be so easy. For somenian local is Hamiltonians, computationally even hard, preparing too the hard groundwe’ve state for known of a for the quantum decades, Hamilto- computer just to finding solvetwo the the dimensions lowest problem is energy efficiently. NP-hard state As [ of a classical Ising-like spin glass in ture. States existing in Naturewere, are that not would ones be which aremark are deep applies and NP-hard to very or ground surprising QMA-hard states, discovery tochemical about and prepare. potential, the also (If or history to they of to equilibrium the stateslow-temperature Gibbs universe.) Gibbs states far state This at from might nonzero be equilibrium. hard, temperatureof but and For that the a won’t spin prevent spin us glass glass, from observedit simulating for in takes the example, behavior an an preparing actual exponentially a long experimentbehavior time performed will for in not be the a observed system reasonable in to amount the reach experiment. of thermal time; equilibrium, if thenciently, then equilibrium we should be able tobut do there the are same a in our few simulation. tricks at But how? our disposal. No method is foolproof, If the resources scale polylogarithmically ratherwe than are polynomially even with the happier. input Constant parameters,the factors then algorithm are might also be important, practical using especially relatively if near-term we devices. wish5.4 to Preparing assess the whether initial state of a massive interacting scalar field theory.free; If then we its turn off ground the state coupling isof constant, the Gaussian, the theory and state becomes hence as easy the toquantum coupling prepare. adiabatic constant Then theorem, slowly we the increases simulate system tothe the is the coupling evolution desired likely changes nonzero to slowly, on value. stay a inparticle. Thanks time its This to scale method ground the determined may state by fail during if the the a quantum evolution phase if transition is encountered during the excursion of achieve a specified error.the number The of gates, resources perhaps we alsoresource the typically cost circuit care depth depends if on about the many computation arephysical features is volume, the of highly the parallelizable. the number evolution input This time, of to thecess, qubits the number and problem. and of the particles These desired involved, include error. theon the total particle The simulated energy masses cost of and also the interaction depends strengths. pro- on properties of the Hamiltonian,resources scaling in exponential particular with the system size.at the If very we least use that a the quantum resource computer costhave instead, scales verified polynomially we this with hope all scaling of for these scattering input parameters, problems and involving we scalar particles and fermions [ Simulating quantum field theory with a quantum computer PoS(LATTICE2018)024 is ) x ( φ , which , which x John Preskill -dimensional N of mutually orthog- ]. Alternatively, we 5 , are sources of error. N a of the Hilbert space at N tensor-network methods ], which can be then be compiled as 23 in space. For the theory to be simulable, ]. ]. More informally, a scalar field x , the quantum state of the field at a site can 24 16 n increases we need to make the lattice spacing 2 E , the field momentum at the lattice site = ) , which are relatively insensitive to the precise x a ( N 11 π . , then as . For E x E , as well as the nonzero lattice spacing ) x ( π . a and ) x ( , but rather the quantum Fourier transform applied to the φ x by Fourier transforming. Here I mean, not the Fourier transform with respect ) qubits. at each lattice site by a discrete variable with a finite number x ) ( n ]. x φ ( 3 , φ 2 . The lattice is an artifice introduced for convenience, and our goal is to extract physical ], or of the incoming state for a scattering process [ a 22 How should we regulate a quantum field theory and turn it intoThere an object are that potential we know advantages how to to working in momentum space, since transforming to mo- In our algorithms for simulating scalar field theories, we introduce a spatial lattice, with lattice Formally, fields are operator-valued distributions [ An alternative approach is to leverage the success of classical Of course, the vacuum does not evolve, but we can adapt the adiabatic method to prepare This truncation of smaller in physical units, and we also need to increase the dimension simulate? There are a lot of options, and it is notmentum immediately space obvious diagonalizes which the way Hamiltonian is of best. there a are free powerful translation-invariant theory, tools and for furthermore for studying simulating renormalization-group interacting theories flow the in position-space momentumcause formulation the space. of interactions the among theory However, fields is are more spatiallyto natural, local; make be- therefore the working simulation in more the efficient. position basis tends spacing properties at distance scales much larger than can prepare wave packetspackets in to the prevent free them theorycoupling from [ (which propagating is or fairly spreading easy), during and the then adiabatic “tether” ramping the up wave of the a quantum circuit to initialize a quantum computation [ 5.5 How to regulate structure of the theory at scale more interesting states. Forcan example, simulate after carefully the shaped vacuum pulsesparticle of wave coupled the packets to with interacting the specified theory (approximate) fields is momentum which prepared, and excite position we the [ creation of localized Simulating quantum field theory with a quantum computer the coupling, where the mass gap goesa to finite zero. volume, Even in since that the case finite-size wethe system might very has succeed small when a gap simulating nonzero due energy towhile gap the even crossing finite at the volume the critical means critical point, that point, which the but makes coupling the must method change much especially less slowly efficient. work especially well in one spatialtems. dimension, for For preparing example, low-energy classical statesstate methods of [ many-body can sys- be used to find an efficient description of the ground an unbounded quantum continuous variable at each point we replace onal eigenstates. There is a conjugate variable to the spatial variable is related to Hilbert space residing at each lattice site be encoded in each site, to keep theof error a fixed. fixed This physical volume means increases that with the number of qubits needed for the simulation a If we want to study a process with energy PoS(LATTICE2018)024 0 ]. D m 28 (5.3) (5.1) (5.2) , 3 may be , making a 2 D ) − John Preskill 4 0 φ m H 0 . ε i ) λ y − , ( x ( is the lattice spacing. δ exp a ) 1 . ) − π  D may have a much different 4 H ( ) ε 0 − i x , . However, the parameters ( m 2 ia − D basis. We can switch back and φ (  − 0 ) ) as a sum of two terms — one is ) 4 0 4! j λ x ( )] = m e 5.1 y / φ + direction, and ( 0 − 2 j as exp π λ ) x , ) ( x ) ( φ . x H φ 1 ( a ε i − , and the gradient term is defined by 2 0 − 2 φ in the 2 [ ) a Λ m − j x ( e ], but I don’t know of persuasive evidence that + , , + 2 0 12  26 ) x 2 x , ( ) ( x φ φ 25 )] = ( y  π ∇ where ( 1 1 1 2 2 1 π − = obey the canonical commutation relations ∑ j , D   , ) ) 1 1 x φ x = ( − − ( H 2 π D D term vanishes, then the Hamiltonian density is a quadratic func- π [ ) a a + 4 x Λ Λ ( π spacetime dimensions φ , we can express exp ∈ ∈ , ∑ ∑ φ and H x x ε 0 ]). It is easy to simulate time evolution governed by the diagonal D ∇ ) 1 spatial dimensions): x = = = 27 ( − of the π φ φ )] = H . Thus we can evolve the system using the (field) Fourier transform to D y 0 H H ( φ λ is summed over a cubic lattice basis, while the other is diagonal in the variables. In that case the theory is Gaussian; it is easy to diagonalize the φ H x , parameters appearing in the lattice Hamiltonian; ) ) π theory in x x or ( 4 ( denotes the nearest neighbor of site π φ j π φ bare [ and e H error (and furthermore the error can be systematically suppressed using higher-order φ + ) x 2 ε ( are the O 0 If the coefficient Note that we have written the Hamiltonian density in eq.( To be a bit more concrete, consider the Hamiltonian for a self-interacting scalar field in There may be more clever ways of regulating that would improve the efficiency of the sim- λ is nonzero these particles interact and the physics is more interesting. The dimensionless number 0 which characterizes the strength of the these interactions is tion of the Hamiltonian and solve the theoryλ exactly — it describes massive noninteracting particles. When and Here the lattice site The conjugate variables value than the actual physical mass of the particle described by this theory, and (I emphasize again thattransform.) this Decomposing Fourier the Hamiltonian transform this shouldwe way choose is not a handy be when small confused we timesmall simulate step with time the evolution. If spatial Fourier Hamiltonian Suzuki-Trotter formulae [ spacetime dimensions ( where the much different than the dimensionless couplingstrength constant for which particles characterizes with the energy actual small interaction compared to diagonal in the ulation. Alternatives have been proposed [ Simulating quantum field theory with a quantum computer an alternative method wouldphysics significantly questions improve of the interest. resource cost of simulations that5.6 address Example: forth between these two bases by Fourier transforming the field variables at each site [ PoS(LATTICE2018)024 ) x ( φ interaction John Preskill 4 φ to measure occu- basis and the ) x ( π quantum phase estimation 13 bases, and applying a diagonal evolution operator in φ and of the lightest particle in the theory to be small compared π m ]. This theory is superrenormalizable, so we know it has a 3 , 2 is the lattice spacing, and fine tuning of the bare Hamiltonian is needed for this a , where a In the Hamiltonian formulation, introducing a lattice breaks Lorentz invariance badly. There- Suppose we simulate the high-energy scattering of two single-particle wave packets in the What potential sources of error should we worry about when we follow this protocol? Some To be a bit more concrete, consider the case of the scalar field theory with a / continuum limit and a relatively well behaved perturbation expansion. A perturbative analysis Simulating quantum field theory with a quantum computer alternate back and forth between the to 1 fore, we need to carefullymate tune the the Lorentz-invariant theory bare we parameters wishphysics in to the we study. lattice need In Hamiltonian the addition, to to physical closely accurately mass approxi- describe continuum purpose as well. (That we needto to be tune close light to compared a second-order tophysics phase at the transition the ultraviolet for Large a cutoff Hadron scalar Collider was particle at the energy reason scales not for5.7 far expecting above Sources the the of Higgs error discovery boson of mass.) new self-coupled scalar field theory.packets As of described the in free §5.4, theory,coupling we and constant, might while then keeping do “dressing” the this the waveOnce by packets the wave tethered wave first packets packets to by preparing are prevent fully adiabatically wave propagation dressed,in and turning we §5.6, spreading. untether on we them, the can and simulate allowtum them time to Fourier evolution collide; using transform as Trotter described to product switch formulas, back while applying and the forth quan- repeatedly between the each small time step. basis. Finally we sampleadiabatically the turning final off state the by coupling measuring and suitable then observables. using We might do this by in 2+1 spacetime dimensions [ of these are already familiar fromhas experience a with Euclidean nonzero lattice lattice computations.capture spacing, The the simulation and continuum we physics need accurately.to to We extrapolate simulate extrapolate to only to the a theeach limit finite limit lattice spatial of of site volume, infinite zero and are volume. lattice so unboundedfinite need spacing number continuous In of variables, to addition, bits which of the wewith precision. fields approximate a Aside and by from finite conjugate retaining all quantum only momenta that,Finally, circuit, when we a at we approximate introducing use continuous the an time adiabatic method error evolution a for serious which state contribution preparation, depends to diabatic particle on the production errorall the can if these be size sources the of Hamiltonian of error changes determines the toothe how time quickly. the properties computational The of step. resources combined the scale effect incoming with of scatteringwhen the state. error the and Finally, quantum with all circuit of the usednoise errors in in described the the so execution of far simulation quantum arise is gates even executed will be perfectly. a further In troublesome a source realistic of error. simulation, pation numbers of the modes ofare the produced.) free theory. Here (In we a takeadiabatic typical method; for high-energy potentially collision, granted these many that could particles there ariseduring are because the no of excursion a avoided of quantum level the phase crossingsbound coupling, transition obstructing states or encountered the rather because than the dressed lightest versions particles of in the the single-particle interacting states of theory the are free theory. PoS(LATTICE2018)024 , E
6 E . The , but it ) O 2 E a = ( ], based on of the sim- John Preskill 3 O G with , E is large. (It is 2 = G ε m / as E a can cancel, improving gates. qubits, while the ground splits into three particles a
; therefore the number of ) n when 2 m a 373 E / . 2 / V ) 2 ε m is / ], we concluded that 3 V V ( , and mass 2 E O . scales with the total energy E ) = G is larger.) Such unwanted particle production 373 . m 2 n ( 14 O = increases more computational steps are needed during for a fixed spatial volume, assuming we keep fixed the G ) E ε / V ( O = n in the simulation scales with the lattice spacing ε . Furthermore, our naive estimate for the Gaussian preparation step assumes ε . E needed scales like scales with 32 lattice would already require thousands of logical qubits and millions of logical gates. n G × Studies of quantum field theory in one spatial dimension will require more modest resources Most likely a less conservative analysis would find more favorable scaling of This simulation algorithm has not been carefully costed — in our initial work we only studied By being more clever we might do better. For example, with a careful choice for the lattice One may also wonder how the number of gates But the dominant effect is that as . Second, we choose a smaller Trotter step size to maintain a fixed Trotter error; that’s another E than simulations in higher dimensions,well-motivated so physics it questions is in natural simulationsquantum to of simulations explore one-dimensional of the quantum opportunities one-dimensional field for dynamicsParticularly theory. answering powerful might What classical surpass tools what exist can for be simulating done one-dimensional classically? physics, especially for also possible for diabatic effects tothat produce process a is pair less of likely particles because anywhere the in energy the gap sample 2 volume, but may be difficult to do accurate andsimulation convincing algorithms estimates until of the we computational are cost able of to such run quantum them on5.8 actual Simulations quantum of devices. one-dimensional physics number of qubits per latticeof site. the algorithm For is processes the at initialalgebra preparation task, relatively of which low the can energy, be Gaussian the achieved state most with for expensive the part free theory. This is a matrix how we use a general procedure whichstate would of apply the to free any theory Gaussian has statealso a of hope special and property expect – that translation there invariance are – more which efficient reduces protocols the than cost. those We described in [ moving almost collinearly has an energy gap which scales like is adequately suppressed if theincreases, adiabatic accounting dressing for the takes three longer additional (hence powers requiring of more gates) as the asymptotic scaling of the cost.a 32 But rough estimates indicateThat’s that discouraging. simulating dynamics on, say, Hamiltonian the leading dependence of the error on the lattice spacing factor of Simulating quantum field theory with a quantum computer indicates that the error number of sites needed toqubits simulate a physical spatial volume different ideas. ulated scattering process. In ourthough analysis this of estimate the is algorithmsmaller [ probably lattice spacing too in pessimistic. order toof resolve First, shorter-wavelength physics; as this accounts we for increase two factors the energy we choose a the adiabatic dressing of thecoupling single-particle constant, wave the packet Hamiltonian states. remains During translationbut the diabatic invariant, slow jumps so across turning that an on momentum energy of isexample, gap conserved, the the can occur process if in the which Hamiltonian a changes single quickly particle enough. with For energy PoS(LATTICE2018)024 N 10. In ≈ N John Preskill and the total number of particles E 15 . This may already become classically intractable for N e , then the number of particles produced, and the thermodynamic entropy ) 1 ]. In both cases, one uses a matrix-product state (MPS) approximation to the ( 23 O would scale like N ]. The bond dimension needed in an MPS description grows, and classical simulation ≈ optimization algorithm. Once found, that MPS description can be compiled as a quantum 29 S For either the scattering process or the quench, the initial state in the simulation need not be More generally, because of the limitations of NISQ technology, we may anticipate that the It is also noteworthy that, even for non-relativistic particles at weak coupling in one spatial Alternatively, we might simulate a “quench” in which the Hamiltonian suddenly changes. If we consider a high-energy collision between two particles in one dimension, how entangled Time evolution in a one-dimensional systems can be simulated classically using, for example, produced. The bond dimension neededentropy to accurately describe a state with bipartite entanglement highly entangled, so that it admitsclassical a succinct classical MPS description which can be found using a early advances in computationalclassical-quantum physics algorithms, enabled in by which quantumhanced the by devices power a will of quantum co-processor. a makeinitial Using classical use a state supercomputer of classical in is computer hybrid a to somehowcapabilities find quantum en- of the simulation the MPS classical is description and of one quantum the computers. example of how wedimension, can multiple scattering exploit events the can build complementary up enough entanglement for classical simulation to may eventually become infeasible if the energy density is nonzero. circuit for preparing the initial state, atfor a the lower cost state in preparation. quantum gatesparticles, than the Even a initial fully state in quantum of the protocol theaccurate incoming case particles MPS of is description only with a modestly a highly entangled, reasonable and bond relativistic therefore dimension. collision has an that produces many of the outgoing particles,state scales is linearly pure, with then theand so left-moving total is particles. the energy outgoing of Theis state, the thermodynamic really with process. entropy entanglement the of momentum entropy, the Ifopposite balanced arising direction. particles the between According from moving incoming to right-moving in their this naive one correlations picture,from then, direction the with we collision would the expect to the particles increase entanglement linearly moving arising with in the the total energy After the quench, a state whichevolves [ is initially slightly entangled becomes more highly entangled as it contrast, the quantum simulation remains efficient even when the state becomes highly entangled. quantum state, which is continually updatedmal as feasible the bond state dimension evolves. of Accuracy theof is tensors, the limited and state by grows the the beyond approximation maxi- what fails the badly maximal if bond the dimension entanglement can accommodate. are the outgoing particles? Aapproximately crude thermal picture particles are is emitted that asof the the collision the fireball creates fireball cools a and is “fireball” dissipates. and If that the many temperature the time-evolving block decimation (TEBD)(TDVP) method method [ or the time-dependent variational principle Simulating quantum field theory with a quantum computer systems with an energy gap.we should Therefore, strike a in balance definingsimulator, between yet the choosing sufficiently task a hard achieved for task the by whichmight best is a existing have as classical quantum a simulators easy simulation, that clear as the possible advantage.become quantum for so simulator highly our The entangled quantum key that is classical methods to fail. make sure that the quantum states to be simulated PoS(LATTICE2018)024 John Preskill ]; if we could 5 ]. This trick has been 30 lattice, with a standard encoding we might 3 ], with the goal of formulating more efficient 33 16 ]. So far, though, there have not been attempts to do 32 , 31 ]. These proposed experimental protocols are certainly challenging, but still may 38 , 37 , 36 ]. The resources scale polynomially with total energy, particle number, and accuracy, but , The first crude (and probably overly pessimistic) estimates of asymptotic resource scaling 4 , Meanwhile, there are also ingenious suggestions for using analog simulation platforms like The study of quantum algorithms for simulating quantum field theory• is still in its very early The focus of this meeting is the exploration of quantum chromodynamics using computational A more efficient encoding of quantum states might reduce this cost substantially. Perhaps 35 3 , , 2 34 quantum simulations of dynamicalto processes simulate that classically. produce highly PerhapsUnfortunately, entangled this this method states will of that be integrating are possibleway, out in gauge soon, hard higher fields for dimensional does one-dimensional theories notother gauge with work, ways theories. dynamical at of least gluons not reducing or in photons. the the gauge It’s same important redundancy to [ explore are nonetheless sobering. ultracold atoms in optical[ lattices to simulate gaugebe theories more in powerful one than digital oryet quantum more clear, however, simulations spatial to of what dimensions extent gauge these theoriesinformation methods surpassing for can what achieve the can sufficient time accuracy be to being. attained extract using valuable It’s classical not simulation tools. 6. Challenges and opportunities in quantum simulation of6.1 quantum field Where theory are we now? stages. Some achievements so far include: (number of qubits and gates) for simulations of[ high energy scattering in scalar and Yukawa theories simulation protocols for dynamical phenomena in QCD. tools. For reasons already discussed,but quantum it’s computers not will clear eventually when. be used Even for for this a purpose, relatively small 32 that is possible, in view ofone-dimensional the gauge huge theory, gauge where redundancy “photons” that or arises “gluons”of using are charges, nondynamical, brute-force the together encodings. spatial with In position boundary a can conditions, solve suffice to the determine Gauss the lawnonlocal gauge constraint Hamiltonian fields; exactly, involving eliminating therefore only we the charged gauge matter fields degrees to of freedom obtain [ a geometrically solve that problem efficiently with a classicalthe computer number (using resources of scaling particles polynomially with efficiently and solve the any number problem that of apossible, scattering quantum and events), computer can then therefore solve the have efficiently.dimension classical good We (even don’t at computer reason think weak could to that’s coupling) is believe acomputers. that hard problem simulating for classical a computers, scalar yet easy field for quantum theory5.9 in The one challenge of gauge theories need millions of logical qubits, taking into account the multiple colors and flavors. exploited successfully to study bothclassical static and and dynamical quantum properties simulators in one [ dimension, using both Simulating quantum field theory with a quantum computer be hard. In fact, simulating many particles which scatter many times is BQP-hard [ PoS(LATTICE2018)024 - ) 1 + D ( ]. 38 John Preskill , 37 , ]. It seems likely ]. This argument 5 36 46 , 35 , 34 -dimensional edges of a D ]. Both static properties and dynamical pro- 42 , 17 41 , 40 , -dimensional chiral theory on the lattice, we can introduce 39 D ]. (2) To realize a 45 , if we try to introduce left-handed fermions with a specified charge we also get e.g. ]. 33 ]. , We need better methods and analysis for dealing with systems that have massless particles Few-site quantum simulations of 1D QED using trapped ions and superconducting circuits Chiral fermions pose another important challenge. The standard model is chiral; that is, for It would be useful to do more serious costing of the algorithms proposed so far, and to seek Proposals for analog quantum simulations using ultracold atoms [ Proposals for digital quantum simulation of nonabelian gauge theories in higher dimensions We need to think through what physics insights could be drawn from simulations with NISQ For simulations of gauge theories beyond one spatial dimension, reducing gauge redundancy A demonstration that simulating a scalar quantum field theory in one dimension is BQP hard, Successful applications of tensor network methods in classical simulations of one-dimensional • • • What should we be thinking about• next? • • • • This long-standing problem may be nearing a resolution, guided in part by recent insights • • 44 32 , , -dimensional chiral theory on the lattice, we can introduce an extra spatial dimension, so that 43 31 dimensional bulk [ the left-handed and right-handed fermions live on two different strong interactions for the expressgiving them purpose large of masses) removing while preserving the the unwanted massless right-handed left-handed fermions fermions [ (by gauge theories with massive fermions [ [ [ improvements. More clever methodslattice of Hamiltonians regularizing might and help a renormalization lot. group improvement of in encoding of quantum states could improve resource requirements substantially. (like the photon). Long-wavelength massless particleschanges will in be the copiously Hamiltonian; produced we during adiabatic shouldinsensitive design to our these algorithms spurious to soft address quanta. physics questions that are both quarks and leptons, the masslesscharges. left-handed Yet existing and methods right-handed for fermions formulating carry fermion theoriestheories different on gauge — the lattice always yield nonchiral unwanted right-handed particles with the sameyears, charge. but We’ve been there facing is this still problem no for accepted over method 40 for regulating a chiralregarding theory. symmetry-protected That’s topological embarrassing. phases ofD matter. Two old ideas are: (1) To realize a bolsters the case thatclassical quantum methods for simulations simulating quantum of field dynamics theory. using quantum computers can surpass cesses (like the breaking of an electricfermion flux pairs) tube have due been to studied. spontaneous creation of electrically charged 6.2 Where should we go? devices, perhaps by exploiting hybrid quantum-classicaltially methods. promising, but Studies it of may be dynamics crucialprotocols, are to poten- if improve we the noise hope resilience to ofusing run digital quantum quantum useful error-correcting simulation simulations codes. using gates with relatively high noise and without even at weak coupling, in scenarios where many particles scatter many times [ Simulating quantum field theory with a quantum computer PoS(LATTICE2018)024 ], 48 John Preskill ]. That’s be- 47 ]. 18 49 ˘ glu, Martin Savage, Frank Verstraete, and Erez Zohar. My work is supported by Realizing supersymmetry on the lattice will also create new opportunities for informative In any case, we should continue the quest for alternative paradigms which will enrich our Also of great interest is the dynamical behavior of conformal field theories, with or without • Quantum simulation of quantum field theory will be a long term project, and it may be a while My remarks here have been influenced by many colleagues, most of all my collaborators The efficacy of this method still needs to be demonstrated convincingly, but if it works that • • [1] John Preskill, Quantum computing in the NISQ era and beyond, Quantum 2, 79 (2018), arXiv:1801.00862. supersymmetry. Part of my own not-so-secretI motivation hope for that pursuing some quantum day simulations we istions will that of enrich holographic our CFTs. understanding Possibly ofsimulation this holographic considered will duality in be through this achieved the article, not simula- butin just also particular, through using that quite the the different conformal kind methods. bootstrap ofCFTs program It brute-force by allows may approximately us be solving to noteworthy, semidefinte extract programs, valuablewith a information quantum task about computers that than might classical be ones performed [ much faster but so far realizations of supersymmetry in the Hamiltonian setting have received less attention. before transformative physics results can bewill achieved. be Ultimately, seen though, as I an think especially such importantmake simulations consequence progress, of the the quantum development algorithm of quantum experts technology.and and To the pool lattice their field theorists expertise. willahead. need There to So join let’s is forces get much going — to this be is going learned to as be the fun! subjectAcknowledgments advances over the decades Stephen Jordan, Hari Kvoi,(among and others) Alex Keith Buser, Lee. David Kaplan,Burak Alexei I’m Kitaev, ¸Sahino Natalie also Klco, grateful Junyu Liu, for Benni illuminating Reznik, discussions with conceptual grasp of quantumfirmly field believe theory that the as circle welltanglement, of and as ideas emergent geometry suggesting relating offers quantum new great opportunities chaos, computationalphysics; for quantum major quantum methods. steps information, simulation forward of quantum in strongly-coupled I theoretical en- dynamics may help to fuel further progress. ARO, DOE, IARPA, NSF, and the SimonsMatter Foundation. (IQIM) The is Institute an for NSF Quantum Physics Information Frontiers and Center. References will settle the longstandingreally open exist, and question will whether also quantumstrongly-coupled open chiral the field gauge door theories theories, for with with classicalmodel. potential and chiral applications quantum fermions to studies physics of beyond the the rich standard dynamics of studies of dynamics. 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