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Toward the first quantum simulation with quantum speedup

Andrew M. Childsa,b,,1, Dmitri Maslovb,c,d, Yunseong Namb,c,e, Neil J. Rossb,c,f, and Yuan Sua,b,c

aDepartment of Computer Science, University of Maryland, College Park, MD 20742; bInstitute for Advanced Computer Studies, University of Maryland, College Park, MD 20742; cJoint Center for and Computer Science, University of Maryland, College Park, MD 20742; dDivision of Computing and Communication Foundations, National Science Foundation, Alexandria, VA 22314; eIonQ, Inc., College Park, MD 20740; and fDepartment of Mathematics and Statistics, Dalhousie University, Halifax, NS B3H 4R2, Canada

Edited by John Preskill, California Institute of Technology, Pasadena, CA, and approved August 10, 2018 (received for review January 30, 2018)

With quantum computers of significant size now on the hori- Arguably, the most natural application of quantum computers zon, we should understand how to best exploit their initially is to the problem of simulating (8). Quan- limited abilities. To this end, we aim to identify a practical prob- tum computers can simulate a wide variety of quantum systems, lem that is beyond the reach of current classical computers, but including fermionic lattice models (9), (10), that requires the fewest resources for a quantum computer. We and quantum field theories (11). However, simulations of consider quantum simulation of spin systems, which could be systems with local interactions likely have less overhead, so we applied to understand condensed matter phenomena. We syn- focus on them as an early candidate for practical quantum sim- thesize explicit circuits for three leading quantum simulation ulation. Note that analog quantum simulation (4, 5) is another algorithms, using diverse techniques to tighten error bounds promising approach to simulating spin systems that may be eas- and optimize circuit implementations. Quantum signal process- ier to realize in the short term. However, analog simulators lack ing appears to be preferred among algorithms with rigorous universal control, and it can be challenging to ensure correctness performance guarantees, whereas higher-order product formu- of their output. Here we focus on digital simulation for its greater las prevail if empirical error estimates suffice. Our circuits are flexibility, for the prospect of invoking fault tolerance, and for its orders of magnitude smaller than those for the simplest classically role as a stepping-stone to other forms of quantum computation. infeasible instances of factoring and quantum chemistry, bringing Efficient quantum algorithms for simulating quantum dynam- practical quantum computation closer to reality. ics have been known for over two decades (12). Recent work has led to algorithms with significantly improved asymptotic | quantum simulation | quantum circuits performance as a function of various parameters such as the evo- lution time and the allowed simulation error (13–17). Our work hile a scalable quantum computer remains a long-term investigates whether these alternative algorithms can be advan- Wgoal, recent experimental progress suggests that devices tageous for simulations of relatively small systems, and aims to capable of outperforming classical computers will soon be avail- lay the groundwork for an early practical application of quantum able (refs. 1–5; www.research..com/ibm-q/). Multiple groups computers. have already developed programmable devices with several 1. Target System and two- gate fidelities around 98% (6), and simi- lar devices with around 50 qubits are under active development. To produce concrete benchmarks, we focus on a specific While the error rates of these early machines severely limit the simulation task. Specifically, we consider a one-dimensional total number of gates that can be reliably performed, future improvements should lead to machines with more qubits and Significance more-reliable gates. This raises the exciting possibility of solv- ing practical problems that are beyond the reach of classical Near-term quantum computers will have limited numbers of computation. Such an outcome would be a landmark in the qubits and will only be able to reliably perform limited num- development of quantum computers and would begin an era in bers of gates. Therefore, it is crucial to identify applications of which they serve not only as test beds for science but as practical quantum processors that use the fewest possible resources. computing machines. We argue that simulating the time evolution of spin sys- Reaching this goal will require not only significant experi- tems is a classically hard problem of practical interest that mental advances but also careful design and is among the easiest to address with early quantum devices. implementation. Here we address the latter issue by develop- We develop optimized implementations and perform detailed ing explicit circuits, and thereby producing concrete resource resource analyses for several leading quantum algorithms for estimates, for practical quantum computations that can outper- this problem. By evaluating the best approaches to small-scale form classical computers. Through this work, we aim to identify quantum simulation, we provide a detailed blueprint for what applications for small quantum computers that help to moti- could be an early practical application of quantum computers. vate the significant investment required to develop scalable, fault-tolerant quantum computers. Author contributions: A.M.C., D.M., Y.N., N.J.R., and Y.S. designed research, performed There has been considerable previous research on compiling research, and wrote the paper.y quantum algorithms into explicit circuits (see SI Appendix, sec- The authors declare no conflict of interest.y tion A for more detail). However, to the best of our knowledge, This article is a PNAS Direct Submission.y none of these studies aimed to identify minimal examples of Published under the PNAS license.y superclassical quantum computation, and typical resource counts Data deposition: The implementations of quantum algorithms for the simulation of were high. Our work is also distinct from recent work on quan- Hamiltonian dynamics in the Quipper language have been tum computational supremacy (7), where the goal is merely deposited on GitHub and are available at https://github.com/njross/simcount.y to accomplish a superclassical task, regardless of its practical- 1 To whom correspondence should be addressed: Email: [email protected] ity. Instead, we aim to pave the way toward practical quantum This article contains supporting information online at www.pnas.org/lookup/suppl/doi:10. computations (which may not be far beyond the threshold for 1073/pnas.1801723115/-/DCSupplemental.y supremacy). Published online September 6, 2018.

9456–9461 | PNAS | September 18, 2018 | vol. 115 | no. 38 www.pnas.org/cgi/doi/10.1073/pnas.1801723115 Downloaded by guest on October 1, 2021 Downloaded by guest on October 1, 2021 hlse al. et Childs called language description circuit quantum a in in algorithms discussed tion (as overhead greater walk incur D). quantum section to quantum Appendix, on appear digital based 23) to approaches approaches (14, particular, efficient of In most all simulation. the among (16), be the (QSP) to and method in (15), introduced processing (TS) are which series signal (PFs) Taylor quantum the formulas recent of product application high-order direct on (13), imple- based We Hamiltonian 1. algorithms Table for in ment summarized algorithms are which quantum of some distinct simulation, many are There Implementations 2. Hamiltonian sparse algo- on yet literature simulation. applications, the simulation practical in deemphasized for been quantum consideration has it key of a is dependence This rithms. system-size the on accuracy overall time evolution int rpgt costesse,oigt h Lieb–Robinson informa- the to for owing even system, bound). the (or across place propagate to take this tion for to evolve must self-thermalization system for the since long spins, the of number on compa- the times to focus evolution rable consider we We dynamics. measurements, simulating of and cost very preparations only are state involve approaches simple there these of all and simulating Since (19–21). by more), dynamics self-thermalization exploring (and for experimentally proposals several observed be could (18). spins restricted 22 was most, of at aware are difficulty to, we study the indeed, phases numerical computers; extensive is most classical localized the with challenge systems and major quantum thermal simulating A of between understood. poorly transition remain a of details B section (see localization many-body and self-thermalization parameter The (i.e., qubit on matrices where mag- random a the with Hamiltonian in model field netic Heisenberg nearest-neighbor opouecnrt icis eipeetqatmsimula- quantum implement we circuits, concrete produce To pcfial,w rdc aecut o iuain with simulations for counts gate produce we Specifically, that feature any access efficiently can simulation Hamiltonian of studies recent in considered been has Hamiltonian This ~σ n ~σ +1 j (σ = o oedti) ept nesv netgto,the investigation, intensive Despite detail). more for = ~σ 1 j x ,and ), h , t  σ = hrceie h tegho h disorder. the of strength the characterizes 10 = j y z nEq. in QSP simulation Fractional-query algorithms simulation quantum Algorithm of complexities Gate 1. Table hstbesospeiul smttcgt opeiiso unu iuainagrtm safnto fthe of function a as algorithms simulation quantum of time complexities simulation gate asymptotic previously shows table This (23) quantum of combination Linear TS order first PF, unu walk Quantum (2k PF, j , n eips eidcbudr conditions boundary periodic impose We . σ ieto.Ti oe sdsrbdb the by described is model This direction. X aksteps walk h j j tenme fsisi h ytm,and system), the in spins of number (the z =1 n j −3 ) ∈ 1 eepc these expect We C. section Appendix, SI t order )th ( eoe etro Pauli of vector a denotes hs xlctcocshl st focus to us help choices explicit These . ~σ with [−h j · ~σ , t j h = +1 ] t n loe error allowed , scoe nfrl trandom. at uniformly chosen is + n fixed and h j σ j z ), . n h ytmsize system the and , IAppendix, SI x , y e.Gt opeiy(t complexity Gate Ref. (16) (22) (15) (12) (14) (13) and , h 1 = [1] SI z , n o n-iesoa ers-egbrsi oe as model spin nearest-neighbor one-dimensional a for bei ulcrpstr ( repository avail- public is a in implementation large-scale able Our for (25). developed optimization we circuit tool quantum automated an process using also circuits We implementation. all the parame- optimize algorithm manually of and analysis ters the tighten we possible, Wherever (see (24) Quipper ncoe om eepiil vlaeti on o h first-, the for bound this formulas. product evaluate fourth-order explicitly and second-, We bound form. the commutator closed the exploit compute in that to Hamiltonian techniques the develop of structure we practical, approach this O constant. to the of complexity one-dimensional time a For Hamiltonian. of target system the relations in commutation terms exploits we among that Thus, bound (28–30). tight improved from (13). an far develop analysis are type previous this of strengthen bounds However, slightly that bounds, imized” details. implementation to these is concrete precise. error. algorithms arbitrarily the these is explicit making an simulation in choose the challenge that main ensure The split- can By we into (27). large, evolution formulas can the Suzuki approach individ- higher-order ting the this with of of exponentials improved complexity of be asymptotic product The a operators. by ual operators of sum a Algorithm. PF measured that spe- assume later. for reused we refined overview be qubits, can be broad ancillas counting can When a that systems. give costs cific to implementation potential aiming of instead constraints, tectural fault-tolerant for count gates can expensive we most a that the computation. on so typically circuits are (26) Clifford+T physical-level which synthesis produce of circuit also cost optimal We the than device. assessing error universal for more the incur merit and of Thus, perform two- (2) gates. to technologies, ions longer both single-qubit trapped take In both gates 2). (1, qubit with circuits level superconducting physical and the at implemented single-qubit and gates, R Clifford single-qubit gates, O(5 z (n av optto ftecmuao on ae time takes bound commutator the of computation Naive “min- and “analytic” the call we which bounds, two present We archi- as such details practical many ignores analysis Our two-qubit of set the over circuits our express We O (θ O(t O O O 2k 2k (n exp(−i = : )    t t O(t t t t + O(t +1 1+/2k = 3+2/(2k o log(t log log(t log log(t log log log log log(1/ / hc a epoiiieee o small for even prohibitive be can which ), n 2 3. √ n 2 2 / eto F section Appendix, SI 5 hscmuao on mrvsteasymptotic the improves bound commutator this , (t (t (t ) /) /) ) / /) pn ihnaetniho opig,eovn for evolving couplings, nearest-neighbor with spins /) /) /) h Fapoc prxmtsteepnnilof exponential the approximates approach PF The 1/2k +1) )) PNAS    , σ ) ) aecmlxt (n) complexity Gate ) z r (2k hl losgicnl mrvn h leading the improving significantly also while θ/ eto E section Appendix, SI htesrssm eie pe on on bound upper desired some ensures that | 2) hodrP loih from algorithm PF )th-order etme 8 2018 18, September r for emnsadmaking and segments https://github.com/njross/simcount θ O O O O(5 ∈ O(n O(n CNOT    uhgtscnb directly be can gates Such R. n n n 2k ie ealddsrpinof description detailed a gives O(n 4 3 4 3 4 n o log log log log log log log log log 3+1/k log log 5 ) 2 on saueu figure useful a is count n n n) n) n | n n n )    o.115 vol. o oedetails). more for | icisusing circuits r o 38 no. n z O sufficiently omake To . rotations (n T 3+1/k CNOT | gates, 9457 ). )

COMPUTER SCIENCES Fig. 1. Gate counts for optimized implementations of the PF algorithm (using the fourth-order formula with commutator bound and the better of the fourth- or sixth-order formulas with empirical error bound), the TS algorithm, and the QSP algorithm (using the segmented version with analytic error bound and the nonsegmented version with empirical Jacobi–Anger error bound) for system sizes between 10 and 100. (Left) CNOT gates for Clifford+Rz circuits. (Right) T gates for Clifford+T circuits.

Unfortunately, even the commutator bound can be very loose. By carefully choosing a sequence of rotation angles for that qubit, To address this, we report empirical error estimates by extrap- we induce the desired evolution. olating the error seen in direct classical simulations of small The circuit for each phased iterate is built from subroutines instances [as also explored in previous work on simulating many- similar to the TS algorithm. However, computing the M rota- body dynamics (28) and quantum chemistry (29, 30)]. While tion angles for the phased iterates requires finding the roots of a these empirical bounds do not provide rigorous guarantees on polynomial of degree 2M , and these roots must be determined the simulation error, they may nevertheless be useful in practice, to high precision. Because of these challenges, we were unable to and they improve the cost of PF algorithms by several orders of compute the parameters of the algorithm explicitly except in very magnitude. small instances. Instead, we produced estimates of the gate count (but not a complete implementation) by synthesizing a version of TS Algorithm. The TS algorithm directly implements the (trun- the algorithm with placeholder values of the parameters. cated) Taylor series of the evolution operator for a carefully One way to alleviate this problem is to consider a segmented chosen constant time using a procedure for implementing linear implementation of the QSP algorithm. In this approach, we combinations of unitary operations (15). This segment is then divide the evolution time into r segments, each of which is simply repeated until the entire evolution time has been simu- lated. The circuit for a segment is built using three subroutines: a state preparation procedure, a reflection about the |0 state, and an operation denoted select(V ) (discussed further below). Our implementation of the TS algorithm (described in detail in SI Appendix, section G) also includes a concrete error analysis that establishes rigorous, non-asymptotic bounds on the simulation parameters. The aforementioned select(V ) operation applies a unitary

Vj conditioned on a control register being in the state |j , for j ∈ {1, ... ,Γ}. We develop an improved implementation of this operation by designing an optimized walk on a binary tree, saving a factor of about log2 Γ in the gate count. For our sim- ulations of systems with 10 to 100 spins, this reduces CNOT and T gate counts over a naive implementation by a factor of between 5 and 9, significantly improving the overall com- plexity. Furthermore, the cost of our select(V ) implementation meets a previously established asymptotic lower bound (31). This improvement may be more broadly applied to any algorithm using the select(V ) procedure, such as others based on linear combinations of unitaries.

QSP Algorithm. The QSP algorithm of Low and Chuang (16, 17) effectively implements a linear combination of unitary opera- Fig. 2. Numbers of qubits used by the PF, TS, and QSP algorithms. See SI Appendix, Eqs. 184 and 195 for the precise qubit counts of the TS and QSP tors by a different mechanism. This algorithm applies a sequence algorithms, respectively. Jumps in the qubit count of the TS algorithm cor- of operations called “phased iterates” that manifest each eigen- respond to increases in the size of each of K registers used to index one of L value of the Hamiltonian as a rotation acting on an ancilla qubit. terms in the Hamiltonian, as detailed in SI Appendix, section G.

9458 | www.pnas.org/cgi/doi/10.1073/pnas.1801723115 Childs et al. Downloaded by guest on October 1, 2021 Downloaded by guest on October 1, 2021 on ie h etaalbeP loih iharigorous a with algorithm commutator sixth- the PF with bounds, error available formula empirical as Using best guarantee. fourth-order qubits, performance the of The hundreds gives 3. to bound Fig. tens performance in with best that system shown the find benchmark had we our formulas sizes, for sixth-order system and small fourth- for the advantageous be not would especially of dominant, is order count. algorithm qubit an PF lower the its about considering that the by indicates still in algorithm but advantage QSP The full magnitude. the the over it the improves not making are magnitude, For guarantees of performance required. rigorous orders if three approach to preferred two by algorithm mrvn vrteP loih yaota re fmagni- of order an about T by algorithm for PF tude performance, the best the over algo- has improving analyzed algorithm rigorously QSP segmented the the among rithms, particular, small In for sizes. even system algorithm PF bounded rigorously the outperforms PF the than requirements space more greater algo- requiring QSP algorithm. slightly the also only contrast, In while has preferred. 2) clearly rithm is Fig. latter the in the so shown than gates, qubits (as more and ver- algorithm significantly nonsegmented uses algorithm, QSP and algorithm com- TS TS segmented (with The both the algorithm sions). (in bounds), PF algorithm QSP error the the for empirical counts and gate mutator compares 1 Fig. Results 3. detail. in algorithms in so improvement bound much give error empirical not case). an analytic did that because bound an (and of error rigorous algo- version used Jacobi–Anger remains segmented overall algorithm instead the the the we that For for significant. algorithm, bound be the error not would empirical additional rithm an the from that suggests savings modest a evidence giving Numerical expansion, Jacobi–Anger improvement. the on bound error cal complete of values a moderate develop high. for too to asymp- not overhead us higher the allows has and it as implementation, approach However, evolution segmented complexity. entire the totic the segment, sim- simulating Hamiltonian single to on tractable. a approach essentially is QSP relies the preprocessing ulation of classical optimality the the Since that short sufficiently 500. and 13 between sizes system for bounds, 3. Fig. hlse al. et Childs lhuhw xetdta ihrodrpoutformulas product higher-order that expected we Although PF the of performance the improve bounds error Empirical upiigy ept en oeivle,teQPalgorithm QSP the involved, more being despite Surprisingly, H section Appendix, SI empiri- an consider we algorithm, the of version full the For count. oa aecut nteClifford+R the in counts gate Total CNOT on n yams w reso antd for magnitude of orders two almost by and count CNOT icse u mlmnaino QSP of implementation our discusses on,teeprclP algorithm PF empirical the count, T z ai o rdc oml loihsuigte(Left the using algorithms formula product for basis on sls significant, less is count n is 3)]A hssz,tesgetdQPagrtmi h etrigor- best about the using is approach, algorithm analyzed QSP ously work segmented our the has in size, this considered work At those (34).] than [Recent for depth smaller 50. only uses much but of only algorithm computations, circuits uses 56-qubit QSP of algorithm the simulation PF and demonstrated the qubits 171 whereas uses 67, (33)— algorithm ours as TS such circuits the for simulation classical direct of limits been usually not those have 32). they as (30, though considered such even simulations, may chemistry, classical quantum quantum formulas for other of higher-order tol- for that be reach method advantageous can suggest be the the bound results error former beyond These heuristic the erated). just a making provided simulations (again, qubits, computers for more choice or of 30 systems about for formula of fourth-order the outperforms formula order r infiatylre,a lutae nFg .Frt consider factorization the First, beyond 4. is Fig. which in number, 1,024-bit illustrated a as factoring simulations larger, chemistry significantly quantum are and logarithms, discrete toring, Clifford+R about uses algorithm PF Clifford+T of set Clifford+R of set the ain ecie nti ae sgetdQPi re;sxhodrP with PF red). sixth-order in green; bound in error QSP (segmented empirical simu- paper 50-spin this and in (orange), described (30) lations FeMoco of simulation (purple), (35) number 4. Fig. o ytmo 0qbt—hc speual ls othe to close presumably is qubits—which 50 of system a For o oprsn rvosetmtso aecut o fac- for counts gate of estimates previous comparison, For oprsno eorerqieet o atrn 1,024-bit a factoring for requirements resource of Comparison z n Clifford+T and PNAS iiie,(Center minimized, ) ae) sn h miia ro on,the bound, error empirical the Using gates). | z etme 8 2018 18, September ae)and gates) 3 × 10 respectively). , 6 omttr n (Right and commutator, ) CNOT 2.4 1.8 × and s | × 10 o.115 vol. 10 9 8 1.8 T CNOT ae oe the (over gates × | 10 o 38 no. ae (over gates 8 empirical ) T (over s | 9459

COMPUTER SCIENCES of RSA-768 that was achieved classically in 2009 (36). The best We have also identified some avenues for future improve- implementation we are aware of uses 3,132 qubits and about ment of quantum simulation algorithms. We saw that rigorous 5.7 × 109 T gates (when realized over the set of Clifford+T error bounds for product formulas are very loose, even with our gates) (35). (Ref. 35 does not give explicit resource counts; newly developed commutator bound. This motivates attempting we estimate them as described in SI Appendix, section A.) to prove stronger rigorous error bounds for product formulas. Quantum algorithms for classically hard instances of the ellip- Also, the difficulty of computing the angles needed to perform tic curve discrete logarithm problem have roughly compara- the QSP algorithm prevents us from taking full advantage of the ble cost (37). For quantum chemistry, a natural target for algorithm in practice, so it would be useful to develop a more a problem just beyond the reach of classical computing is a efficient classical procedure for specifying these angles. simulation of FeMoco, the primary cofactor of nitrogenase, After the initial version of this paper was posted, Haah et al. an enzyme that catalyzes the nitrogen fixation process. Even (42) developed a new quantum algorithm for simulating the for a fairly low-precision simulation, and using nonrigorous dynamics of spatially local Hamiltonians. Their approach uses estimates of the product formula error, the best implemen- bounds on the propagation of information in a local system to tation we are aware of uses 111 qubits and 1.0 × 1014 T give a simulation with complexity only linear in the product of gates (30). Thus, it appears that simulation of spin systems is the system size and the evolution time (up to log factors), asymp- indeed a significantly easier task for near-term practical quantum totically improving over all previous methods. While the gate computation. count estimates presented in figure 3 of ref. 42 suggest that, for For a more detailed discussion of the results, see SI Appendix, the example considered in this paper, the PF and QSP methods section I. remain favorable for systems with up to about 100 spins, the tech- niques developed in ref. 42 are likely to be useful in relatively 4. Discussion near-term simulations. The work described in this paper represents progress toward Further reduction of the gate count could be especially sig- a possible early application of quantum computers, solving a nificant if it led to a simulation with sufficiently few gates to be practical problem that is beyond the reach of classical compu- performed without invoking fault tolerance. With our current tation. Our optimized implementations can perform a 50-spin estimate of millions of CNOT gates for a superclassical simu- quantum simulation using fewer qubits, and dramatically fewer lation, this is likely out of reach at present. However, further gates (by over five orders of magnitude in the case of quantum improvement could obviate the need for error correction in a sys- chemistry), than comparable instances of other classically hard tem with highly accurate gates, making an early demonstration of problems. Of course, our results only represent upper bounds. superclassical simulation more accessible. While we attempted to optimize the implementation wherever Finally, our work has considered an idealized system. Prac- possible, it is likely that further improvements can be found, and tical devices will come with architectural constraints, may use it is conceivable that another algorithm (or computational task) different basic operations than those considered here, may allow may offer better performance. Our work establishes a concrete parallelization of gates, and will likely require fault tolerance. By set of benchmarks that we hope can be improved through future incorporating such features, we hope the work begun here will studies. lead to a blueprint for an early practical quantum computation. Demonstrations of digital quantum simulation performed to date (38–41) have been limited in scope, primarily using the ACKNOWLEDGMENTS. We thank Zhexuan Gong, Alexey Gorshkov, Guang Hao Low, Chris Monroe, and Nathan Wiebe for helpful discussions. This first-order formula [except for some limited applications of the work was supported, in part, by the Army Research Office (Multidisciplinary second-order formula (38, 39)]. Our results show that higher- University Research Initiative Award W911NF-16-1-0349), the Canadian Insti- order formulas are useful even for simulations of small systems. tute for Advanced Research, and the National Science Foundation (Grant In the near term, it could be fruitful to demonstrate the utility 1526380). This material was partially based on work supported by the National Science Foundation during D.M.’s assignment at the foundation. of these formulas experimentally. Even relatively small experi- Any opinion, finding, and conclusions or recommendations expressed in this ments might be able to probe the validity of our empirical error material are those of the authors and do not necessarily reflect the views of bounds. the National Science Foundation.

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