aa r X i v : . [ qu a n t - ph ] J un Hamiltonian simulation in the low energy subspace
Burak S¸ahino˘glu ∗ and Rolando D. Somma † Theoretical Division, Los Alamos National Laboratory, Los Alamos, NM 87545, USA (Dated: June 5, 2020)We study the problem of simulating the dynamics of spin systems when the initial state is sup-ported on a subspace of low energy of a Hamiltonian H . We analyze error bounds induced byproduct formulas that approximate the evolution operator and show that these bounds depend onan effective low-energy norm of H . We find some improvements over the best previous complexitiesof product formulas that apply to the general case, and these improvements are more significant forlong evolution times that scale with the system size and/or small approximation errors. To obtainour main results, we prove exponentially-decaying upper bounds on the leakage or transitions tohigh-energy subspaces due to the terms in the product formula that may be of independent interest. Introduction.
The simulation of quantum systemsis believed to be one of the most important applica-tions of quantum computers [1]. Many quantum al-gorithms for simulating quantum dynamics exist [2–10], with applications in physics [11, 12], quantumchemistry [13–15], and beyond [16]. While these al-gorithms are deemed efficient and run in time poly-nomial in factors such as system size, ongoing workhas significantly improved the performance of suchapproaches. These improvements are important toexplore the power of quantum computers and pushquantum simulation closer to reality.Leading
Hamiltonian simulation methods arebased on a handful of techniques. A main exampleis the product formula, which approximates the evo-lution of a Hamiltonian H by short-time evolutionsunder the terms that compose H [4, 5, 17, 18]. Eachsuch evolution can be decomposed as a sequence oftwo-qubit gates [11] to build up a quantum algo-rithm. Product formulas are attractive for variousreasons: they are simple, intuitive, and their imple-mentations may not require ancillary qubits, whichcontrasts other sophisticated methods as those inRefs. [7, 8]. Product formulas are also the basisof classical simulation algorithms including path-integral Monte Carlo [19].Recent works provide refined error bounds ofproduct formulas [20–23]. These works regard vari-ous settings, such as when H is a sum of spatially-local terms or when these terms satisfy Lie-algebraicproperties. Nevertheless, while these improvementsare important and necessary, a number of shortcom-ings remain. For example, the best-known complex-ities of product formulas scale poorly with the normof H or its terms, which can be very large or un- ∗ [email protected] † [email protected] bounded, even when the evolved quantum systemdoes not explore high-energy states.Motivated by this shortcoming, we investigatethe Hamiltonian simulation problem when the ini-tial state is supported on a low-energy subspace.This is a central problem in physics that has vastapplications, including the simulation of condensedmatter systems for studying quantum phase tran-sitions [24], the simulation of quantum field theo-ries [12], the simulation of adiabatic quantum statepreparation [25, 26], and more. We analyze the com-plexities of product formulas in this setting and showsome improvements with respect to the known com-plexity bounds that apply to the general case.Our main result is that, for a local Hamiltonianon N spins H = P l H l , H l ≥
0, the error induced bya p -th order product formula is O ((∆ ′ s ) p +1 ), where s is a (short) time parameter and ∆ ′ is an effectivelow-energy norm of H . This norm depends on ∆,which is an energy associated with the initial state,but also depends on s and other parameters thatdefine H . The best known error bounds for productformulas that apply to the general case depend onthe k H l k ’s [22]. (Throughout this paper, k . k refersto the spectral norm.) Thus, an improvement inthe complexity of product formulas is possible when∆ ′ ≪ max l k H l k , which can occur for sufficientlysmall values of ∆ and s . Such values of s appearin low-order product formulas (e.g., first order) or,for larger order, when the overall evolution time t issufficiently large and/or the desired approximationerror ε is sufficiently small. We summarize some ofthe complexity improvements in Table I.To obtain our results, we introduce the notion ofeffective Hamiltonians that are basically the H l ’s re-stricted to act on a low-energy subspace. The rel-evant norms of these effective operators is boundedby ∆ ′ . One could then proceed to simulate the evo-lution using a product formula that involves effec- rder Previous result Low-energy simulation p = 1 O ( τ Nε ) ˜ O ( τ ε ) + O ( τ / N / ε / ) p = 2 O ( τ / N / ε / ) ˜ O ( τ / ε / ) + O ( τ / N / ε / ) p = 3 O ( τ / N / ε / ) ˜ O ( τ / ε / ) + O ( τ / N / ε / )TABLE I. Comparison between the best-known complex-ity [22] and the complexity of low-energy simulation for p -th order product formulas. The results are for con-stant ∆ and Hamiltonians on N spins with local interac-tions of constant degree and strength bounded by J , and τ = | t | J . ε is the approximation error. The ˜ O notationhides polylogarithmic factors in τ /ε . tive Hamiltonians and obtain an error bound thatmatches ours. A challenge is that these effectiveHamiltonians are generally non-local and difficult tocompute. Methods such as the local Schrieffer-Wolfftransformation [27] work only at the perturbativeregime and numerical renormalization group meth-ods for spin systems [28, 29] have been studied onlyfor a handful of models, while a general analyticaltreatment does not exist. Thus, efficient methods tosimulate time evolution of effective Hamiltonians arelacking. We address this challenge by showing thatevolutions under the effective Hamiltonians can beapproximated by evolutions under the original H l ’swith a suitable choice of ∆ ′ . This result is key in ourconstruction and may find applications elsewhere.Our main contributions are based on a numberof technical lemmas and corollaries that are provenin detail in Supp. Mat. We state some importantresults and only sketch their proofs in the main text. Product formulas and effective operators.
Fora time-independent Hamiltonian H = P Ll =1 H l ,where each H l is Hermitian, the evolution opera-tor for time t is U ( t ) = e − itH . Product formulasprovide a way of approximating U ( t ) as a productof exponentials, each being a short-time evolutionunder some H l . For p > s ∈ R , a p -thorder product formula is a unitary W p ( s ) = e − is q H lq · · · e − is H l e − is H l , (1)where each s j ∈ R is proportional to s and 1 ≤ l j ≤ L . The number of terms in the product maydepend on p and L , and we assume q = O ( L ). Wedefine | s | = P qj =1 | s j | and also assume | s | = O ( L | s | ).The p -th order product formula satisfies k U ( s ) − W p ( s ) k = O (( Lh | s | ) p +1 ), where h = max l k H l k [4].One way to construct W p ( s ) is to apply a recursionin Refs. [17, 18]. These are known as Trotter-Suzukiapproximations and satisfy the above assumptions. By breaking the time interval t into r steps ofsufficiently small size s , product formulas can ap-proximate U ( t ) as U ( t ) ≈ ( W p ( s )) r . We will referto r as the Trotter number, and this number willdetermine the complexity of product formulas thatsimulate U ( t ) within given accuracy.Known error bounds for product formulas thatapply to the general case grow with h and can belarge. However, error bounds for approximating theevolved state U ( t ) | ψ i may be better under the ad-ditional assumption that | ψ i is supported on a low-energy subspace. We then analyze the case wherethe initial state satisfies Π ≤ ∆ | ψ i = | ψ i , where Π ≤ Λ is the projector into the subspace spanned by eigen-states of H of energies (eigenvalues) at most Λ ≥ H l ≥
0. Our results will be useful when∆ ≪ h , and ∆ will specify the low-energy subspace.The notion of effective operators will be useful inour analysis. Given a Hermitian operator X and∆ ′ ≥ ∆, the corresponding effective operator is¯ X = Π ≤ ∆ ′ X Π ≤ ∆ ′ , which is also Hermitian. Wealso define the unitaries ¯ U ( s ) = e − is ¯ H and ¯ W p ( s )by replacing the H l ’s by ¯ H l ’s in W p ( s ). Note that¯ h = max l k ¯ H l k ≤ ∆ ′ and U ( t ) | ψ i = ¯ U ( t ) | ψ i . Then,using the known error bound for product formulas,we obtain k ( U ( s ) − ¯ W p ( s )) | ψ i k = O (( L ∆ ′ s ) p +1 ).This error bound is a significant improvement overthe general case if ∆ ′ ≪ h , which may occur when∆ ≪ h . However, product approximations of U ( t )require that each term is an exponential of some H l ,which is not the case in ¯ W p ( s ). We will address thisissue and show that the improved error bound is in-deed attained by W p ( s ) for a suitable ∆ ′ . Local Hamiltonians and main result.
We areinterested in simulating the time evolution of a local N -spin system on a lattice. Each interaction term in H is of strength bounded by J and involves, at most, k spins. We do not assume that these interactionsare only within neighboring spins but define the de-gree d as the maximum number of interaction termsthat involve any spin. Then, we write H = P Ll =1 H l ,where each H l is a sum of M commuting terms [30]and LM ≤ dN . Each e − isH l in a product formulacan be decomposed as products of M evolutions un-der the local (commuting) terms with no error.These local Hamiltonians appear as importantcondensed matter systems, including gapped andcritical spin chains, topologically ordered systems,and models with long-range interactions [31–34]. Forexample, for a spin chain with nearest neighbour in-teractions, L = 2 and each H l may refer to inter-action terms associated with even and odd bonds,respectively. We will present our results for the case2 = O (1) and d = O (1) in the main text, whichfurther imply L = O (1). Nevertheless, explicit de-pendencies of our results in k , d , L , and other pa-rameters that specify H can be found in Supp. Mat.In general, the assumption H l ≥ H l → H l + a l . This shiftinghas to be implemented carefully to avoid undesiredlarge error bounds from our analysis. It is possiblethat the shifting results in a value of ∆ that dependson some parameters of H such as system size, andour results may not apply to that case. Nevertheless,for many interesting spin Hamiltonians such as theso-called frustration-free Hamiltonians [35, 36], theassumption H l ≥ Theorem 1.
Let H = P Ll =1 H l be a k -local Hamil-tonian as above, H l ≥ , ∆ ≥ , ≤ J | s | ≤ , and W p ( s ) a p -th order product formula as in Eq. (1) .Then, k ( U ( s ) − W p ( s ))Π ≤ ∆ k = O ((∆ ′ s ) p +1 ) , (2) where ∆ ′ = ∆ + β J log( β / ( J | s | )) + β J N | s | andthe β i ’s are positive constants, β ≥ . The proof of Thm. 1 is in Supp. Mat. and we pro-vide more details about it in the next section, butthe basic idea is as follows. There are two contri-butions to Eq. (2) in our analysis. One comes fromapproximating the evolution operator with a prod-uct formula that involves the effective Hamiltoniansand, as long as ∆ ′ ≥ ∆, this error is O ((∆ ′ | s | ) p +1 ),as explained. The other comes from replacing such aproduct formula by the one with the actual Hamilto-nians H l , i.e., W p ( s ). However, unlike ¯ H l , the evolu-tion under each H l allows for leakage or transitionsfrom the low-energy subspace to the subspace of en-ergies higher than ∆ ′ . In Supp. Mat. we use a resulton energy distributions in Ref. [37] to show that thisleakage can be bounded and decays exponentiallywith ∆ ′ . Thus, this effective norm depends on ∆and must also depend on s , as the support on high-energy states can increase as s increases, resultingin the linear contribution to ∆ ′ in Thm. 1.The log( β / ( J | s | )) factor in ∆ ′ only becomes rel-evant when | s | ≪
1. This term appears in our anal-ysis due to the requirement that both contributionsto Eq. (2) discussed above are of the same order.Thus, as s →
0, we require ∆ ′ → ∞ to make theerror due to leakage zero, which is unnecessary andunrealistic. This term plays a mild role when deter-mining the final complexity of a product formula, asthe goal will be to make s as large as possible for atarget approximation error. It may be possible thatthis term disappears in a more refined analysis. Let r = t/s be the Trotter number, i.e., thenumber of steps to approximate U ( t ) as ( W p ( s )) r .Since U ( s )Π ≤ ∆ = Π ≤ ∆ U ( s )Π ≤ ∆ and if k ( U ( s ) − W p ( s ))Π ≤ ∆ k ≤ ǫ , the triangle inequality implies k ( U ( t ) − ( W p ( s )) r )Π ≤ ∆ k ≤ rǫ . Thus, for over-all target error ε >
0, it will suffice to satisfy k ( U ( s ) − W p ( s ))Π ≤ ∆ k = O ( εs/t ). This conditionand Thm. 1 can be used to determine r as follows.Each term of ∆ ′ in Thm. 1 can be dominant de-pending on s and ∆. First, we consider the firsttwo terms, and determine a condition in s to satisfy((∆ + J ) | s | ) p +1 = O ( εs/t ), by omitting the log fac-tor. Then, we consider another term and determinea condition in s to satisfy ( J N | s | ) p +1 = O ( εs/t ).These two conditions alone can be satisfied with aTrotter number r ′ = O ( t (∆ + J )) p ε p + ( tJ √ N ) p +1 ε p +1 ! . (3)Last, we reconsider the second term with log, andwe require ( J log(1 / ( J | s | )) | s | ) p +1 = O ( εs/t ). As thefirst two conditions are satisfied with a value for s that is polynomial in N and tJ/ε , this last condi-tion only sets a correction to the first term in r ′ inEq. (3) that is polylogarithmic in | t | J/ε . Thus, theoverall complexity of the product formula for localHamiltonians is given by Eq. (3), where we need toreplace O by ˜ O to account for the last correction.Note that the number of terms in each W p ( s ) is con-stant under the assumptions and r is proportionalto the total number of exponentials in ( W p ( s )) r .We give a general result on the complexity of prod-uct formulas that provides r as a function of all pa-rameters that specify H in Thm. 2 of Supp. Mat. Comparison with previous results . The bestprevious result for the complexity of product for-mulas (Trotter number) for local Hamiltonians ofconstant degree is O ( τ /p N /p /ε /p ), with τ = | t | J [22]. Our result gives an improvement over thisin various regimes. Note that, a general character-istic of our results is that they depend on ∆, whichis specified by the initial state. Here we assumethat ∆ is a constant independent of other parame-ters that specify H . The comparison for this caseis in Table I. For p = 1, we obtain a strict im-provement of order N / over the best-known result.For higher values of p , the improvement appears forlarger values of τ /ε that may scale with N , e.g., τ /ε = ˜Ω( N p − / ( p +1) ). In Supp. Mat. we providea more detailed comparison between our results andthe best previous results for product formulas as afunction of ∆ and other parameters that specify H .3 more recent method for Hamiltonian simula-tion uses a truncated Taylor series expansion of e − iHt/r ≈ U r = P Kk =0 ( − iHt/r ) k /k ! [7]. Here, r isthe number of “segments”, and U ( t ) is approximatedas ( U r ) r . A main advantage of this method is that,unlike product formulas, its complexity in terms of ε is logarithmic, a major advantage if precise compu-tations are needed. The complexity of this methodfor the low-energy subspace of H can only be mildlyimproved. A small ∆ allows for a truncation value K that is smaller than that for the general case [7].Nevertheless, the complexity of this method is dom-inated by r , which depends on a certain 1-norm of H that is independent of ∆. Disregarding logarith-mic factors in τ N/ε , the overall complexity of thismethod is ˜ O ( τ N ) when d = O (1). Our results onproduct formulas provide an improvement over thismethod in various regimes, if ε is constant. Leakage to high-energy states . A key ingredientfor Thm. 1 is a property of local spin systems, wherethe leakage to high-energy states due to the evolu-tion under any H l can be bounded. Let Π > Λ ′ be theprojector into the subspace spanned by eigenstatesof energies greater than Λ ′ . Then, for a state | φ i that satisfies Π ≤ Λ | φ i = | φ i , we consider a questionon the support of e − isH l | φ i on states with energiesgreater than Λ ′ . This question arises naturally inHamiltonian complexity and beyond, and Lemma 1below may be of independent interest. A general-ization of this lemma will allow one to address theHamiltonian simulation problem in the low-energysubspace beyond spin systems. Lemma 1 (Leakage to high energies) . Let H = P Ll =1 H l be a k -local Hamiltonian as above, H l ≥ ,and Λ ′ ≥ Λ ≥ . Then, ∀ s ∈ R and ∀ l , k Π > Λ ′ e − isH l Π ≤ Λ k ≤ e − λ (Λ ′ − Λ) (cid:16) e α | s | M − (cid:17) , (4) where λ = 1 / (2 Jdk ) and α = eJ . The proof is in Supp. Mat. It follows from a resultin Ref. [37] on the action of a local interaction termon a quantum state of low energy, in combinationwith a series expansion of e − isH l . While the localinteraction term could generate support on arbitrar-ily high-energy states, that support is suppressed bya factor that decays exponentially in Λ ′ − Λ.Another key ingredient for proving Thm. 1 is theability to replace evolutions under the H l ’s in a prod-uct formula by those under their effective low-energyversions (and vice versa) with bounded error. Thisis addressed by Lemma 2 below, which is a conse-quence of Lemma 1. The proof is in Supp. Mat. Lemma 2.
Let H = P Ll =1 H l be a k -local Hamilto-nian as above, H l ≥ , and ∆ ′ ≥ Λ ′ ≥ Λ ≥ . Then, ∀ s ∈ R and ∀ l , k Π ≤ Λ ′ ( e − is ¯ H l − e − isH l )Π ≤ Λ k≤ e − λ (Λ ′ − Λ) ( e α | s | M −
1) (5) and k Π > Λ ′ e − is ¯ H l Π ≤ Λ k ≤ e − λ (Λ ′ − Λ) ( e α | s | M − , (6) where λ = 1 / (2 Jdk ) and α = eJ . The consequences of these lemmas for Hamilto-nian simulation are many-fold and we only sketchthose that are relevant for Thm. 1. Consider anyproduct formula of the form W = Q qj =1 e − is j H lj .Then, there exists a sequence of energies Λ q ≥ . . . ≥ Λ = ∆ such that the action of W on the initiallow-energy state | ψ i can be well approximated bythat of W Λ = Q qj =1 Π ≤ Λ j e − is j H lj on the same state.Furthermore, each Π ≤ Λ j e − is j H lj in W Λ can be re-placed by Π ≤ Λ j e − is j ¯ H lj and later by e − is j ¯ H lj withinthe same error order, as long as Λ q ≤ ∆ ′ .In particular, for sufficiently small evolution times s j and ∆ ≪ h , the resulting effective norm satis-fies ∆ ′ ≪ h for local Hamiltonians. This is for-malized by several corollaries in Supp. Mat. Start-ing from W , we can construct the product formula¯ W = Q qj =1 e − is j ¯ H lj . Lemmas 1 and 2 imply thatboth product formulas produce approximately thesame state when acting on | ψ i , for a suitable choiceof ∆ ′ as in Thm. 1. If ¯ W is a product formulaapproximation of ¯ U ( s ) = e − is ¯ H , it follows that U ( s ) | ψ i = ¯ U ( s ) | ψ i ≈ ¯ W | ψ i ≈ W | ψ i . Conclusions and open problems.
We providedimproved error bounds and complexities for productformulas that approximate the evolution operatorwhen the initial state belongs to a subspace of lowenergy. These formulas are at the root of variousquantum and classical methods that simulate quan-tum systems and our results can be translated intocomplexity improvements of such methods as well.The obtained complexities are an improvement aslong as the energy ∆ of the initial state is sufficientlysmall. As we discussed, the assumption H l ≥ H l (cid:3) − H instead. Indeed, certain spin models4ossess a symmetry that connects the high-energyand low-energy subspaces via a simple transforma-tion. Whether such (high-energy) improvement ispossible or not remains open. Additionally, knowncomplexities of product formulas are polynomial in1 /ε . This is an issue if precise computations are re-quired as in the case of quantum field theories orQED. Whether this complexity can be improved asin Refs. [6–8] is also open.Our work is an initial attempt to this problem.We expect to motivate further studies on improvedHamiltonian simulation methods in this setting by refining our analyses, assuming other structures suchas interactions that are geometrically local, or im-proving other simulation approaches. Acknowledgements.
We acknowledge supportfrom the LDRD program at LANL and the U.S. De-partment of Energy, Office of Science, High EnergyPhysics and Office of Advanced Scientific ComputingResearch, under the Accelerated Research in Quan-tum Computing (ARQC) program. Los Alamos Na-tional Laboratory is managed by Triad National Se-curity, LLC, for the National Nuclear Security Ad-ministration of the U.S. Department of Energy un-der Contract No. 89233218CNA000001. [1] R. P. Feynman, International Journal of TheoreticalPhysics , 467 (1982).[2] S. Lloyd, Science , 1073 (1996).[3] D. Aharonov and A. Ta-Shma, Proc. of the 35thACM Symp. on Theory of Comp. , 20 (2003).[4] D. Berry, G. Ahokas, R. Cleve, and B. Sanders,Comm. Math. Phys. , 359 (2007).[5] N. Wiebe, D. Berry, P. Hoyer, and B. Sanders, J.Phys. A: Math. Theor. , 065203 (2010).[6] D. W. Berry, A. M. Childs, R. Cleve, R. Kothari,and R. D. Somma, in Proc. of the 46th ACM Sympo-sium on Theory of Computing (2014) pp. 283–292.[7] D. W. Berry, A. M. Childs, R. Cleve, R. Kothari,and R. D. Somma, Phys. Rev. Lett. , 090502(2015).[8] G. Low and I. Chuang, Phys. Rev. Lett. , 010501(2017).[9] G. H. Low and I. L. Chuang, Quantum , 163(2019).[10] E. Campbell, Phys. Rev. Lett. , 070503 (2019).[11] R. D. Somma, G. Ortiz, J. E. Gubernatis, E. Knill,and R. Laflamme, Phys. Rev. A , 042323 (2002).[12] S. P. Jordan, K. S. Lee, and J. Preskill, Science , 1130 (2012).[13] D. Wecker, B. Bauer, B. K. Clark, M. B. Hastings,and M. Troyer, Phys. Rev. A , 022305 (2014).[14] R. Babbush, D. W. Berry, I. D. Kivlichan, A. Y.Wei, P. J. Love, and A. Aspuru-Guzik, quant-ph:1506.01020 (2015).[15] R. Babbush, D. W. Berry, I. D. Kivlichan, A. Y.Wei, P. J. Love, and A. Aspuru-Guzik, quant-ph:1506.01029 (2015).[16] A. M. Childs, R. Kothari, and R. D. Somma, SIAMJ. Comp. , 1920 (2017).[17] M. Suzuki, Phys. Lett. A , 319 (1990).[18] M. Suzuki, J. Math. Phys. , 400 (1991).[19] M. Newman and G. Barkema, Monte Carlo Meth-ods in Statistical Physics (Oxford University Press,1998). [20] R. D. Somma, Journal of Mathematical Physics ,062202 (2016).[21] A. M. Childs and Y. Su, Phys. Rev. Lett. ,050503 (2019).[22] A. M. Childs, Y. Su, M. C. Tran, N. Wiebe, andS. Zhu, arXiv: 1912.08854 (2019).[23] L. Clinton, J. Bausch, and T. Cubitt,arXiv:2003.06886 (2020).[24] S. Sachdev, Quantum Phase Transitions, 2nd Edi-tion (Cambridge University Press, Cambridge,2011).[25] E. Farhi, J. Goldstone, S. Gutmann, and M. Sipser,arXiv:quant-ph/0001106 (2000).[26] S. Boixo, E. Knill, and R. Somma, arXiv:1005.3034(2010).[27] S. Bravyi, D. P. DiVincenzo, and D. Loss, Annalsof physics , 2793 (2011).[28] Z.-C. Gu, M. Levin, and X.-G. Wen, Physical Re-view B , 205116 (2008).[29] G. Evenbly and G. Vidal, Physical Review B ,144108 (2009).[30] R. L. Brooks, in Mathematical Proceedings of theCambridge Philosophical Society , Vol. 37 (Cam-bridge University Press, 1941) pp. 194–197.[31] E. Lieb, T. Schultz, and D. Mattis, Annals ofPhysics , 407 (1961).[32] I. Affleck, T. Kennedy, E. H. Lieb, and H. Tasaki,in Condensed Matter Physics and Exactly SolubleModels (Springer, 2004) pp. 249–252.[33] A. Y. Kitaev, Annals of Physics , 2 (2003).[34] H. J. Lipkin, N. Meshkov, and A. Glick, NuclearPhysics , 188 (1965).[35] S. Bravyi and B. Terhal, SIAM J. Comp. , 1462(2009).[36] N. de Beaudrap, M. Ohliger, T. Osborne, andJ. Eisert, Phys. Rev. Lett. , 060504 (2010).[37] I. Arad, T. Kuwahara, and Z. Landau, Journalof Statistical Mechanics: Theory and Experiment , 033301 (2016). UPPLEMENTARY MATERIAL
In the following, we let H be a k -local Hamiltonian of N spins, where each interaction term involves atmost k > H = P Ll =1 H l , where each H l is a sum of at most M k -local and commutinginteraction terms. The H l ’s may be obtained via graph coloring [30], where a graph can be constructed withvertices that are labeled according to the subset of spins in each interaction term and the edges connectvertices associated with the same spins, but more efficient constructions may be possible. Indeed, in manyinteresting examples such as spins on the square lattice, H is already given in the desired form. The degreeof H , i.e. the highest number of interaction terms that act non-trivially on any spin, is d >
0. The strengthof each local interaction term is bounded by
J >
0, hence k H l k ≤ JM and k H k ≤ JM L . Throughout thispaper, k . k refers to the spectral norm (largest eigenvalue for positive semidefinite and Hermitian operators).The total number of local terms in H is then upper bounded by M L and dN . We will assume that H containsexactly M L terms and thus N ≤ M L ≤ dN with no loss of generality (e.g., we can add or subtract trivialterms to H l and each spin appears, at least, in one term). Furthermore, following the coloring proceduredescribed above, we may assume L ≤ kd + 1 [30].For Λ ′ ≥ Λ ≥
0, the operators Π ≤ Λ and Π > Λ ′ are the projectors into the subspaces spanned by theeigenstates of H with energies (eigenvalues) at most Λ and larger than Λ ′ , respectively. For a given ∆ ′ ≥ ′ -effective (or simply effective) Hamiltonians are then defined as ¯ H = Π ≤ ∆ ′ H Π ≤ ∆ ′ , ¯ H l = Π ≤ ∆ ′ H l Π ≤ ∆ ′ ,and ¯ H = P Ll =1 ¯ H l . We assume H l ≥
0, and thus ¯ H l ≥ k ¯ H l k ≤ k ¯ H k = ∆ ′ .
1. Proof of Lemma 1
We employ Theorem 2.1 in Ref. [37] that, for an operator A , states k Π > Λ ′ A Π ≤ Λ k ≤ k A k · e − λ (Λ ′ − Λ − R ) . (7)Here, λ = 1 / (2 gk ), where g is an upper bound on the sum of the strengths of the interactions associatedwith any spin. In our case, we take g = dJ and λ = 1 / (2 Jdk ). If E A is the subset of local interaction termsin H that do not commute with A , R is the sum of the strengths of the terms in E A . For any H l , we notethat ( H l ) n is a sum of, at most, M n terms, each of strength bounded by J n and containing, at most, kn spins. For each such term, R ≤ Jdkn , and we obtain k Π > Λ ′ ( H l ) n Π ≤ Λ k ≤ ( M J ) n e − Jdk (Λ ′ − Λ − Jdkn ) (8) ≤ ( eM J ) n e − Jdk (Λ ′ − Λ) . (9)We now consider the Taylor series expansion of the exponential, e − isH l = ∞ X n =0 ( − isH l ) n n ! . (10)The triangle inequality and Eq. (9) imply k Π > Λ ′ e − isH l Π ≤ Λ k ≤ ∞ X n =1 | s | n k Π > Λ ′ ( H l ) n Π ≤ Λ k n ! (11) ≤ e − Jdk (Λ ′ − Λ) ∞ X n =1 ( | s | eM J ) n n ! (12)= e − Jdk (Λ ′ − Λ) (cid:16) e | s | eMJ − (cid:17) (13)= e − λ (Λ ′ − Λ) (cid:16) e α | s | M − (cid:17) , (14)where α = eJ . 6 . Proof of Lemma 2: To prove the first result, we will use the identity e − is ¯ H l − e − isH l = − i Z s ds ′ e − i ( s − s ′ ) ¯ H l ( ¯ H l − H l ) e − is ′ H l . (15)We note that, from the assumptions, Π ≤ Λ ′ = Π ≤ Λ ′ Π ≤ ∆ ′ = Π ≤ ∆ ′ Π ≤ Λ ′ and [Π ≤ ∆ ′ , e − is ¯ H l ] = 0 for all s .Then, we can express Π ≤ Λ ′ ( e − is ¯ H l − e − isH l )Π ≤ Λ as − i Π ≤ Λ ′ (cid:18)Z s ds ′ e − i ( s − s ′ ) ¯ H l Π ≤ ∆ ′ ( ¯ H l − H l )(Π ≤ ∆ ′ + Π > ∆ ′ ) e − is ′ H l (cid:19) Π ≤ Λ . We can simplify this expression since Π ≤ ∆ ′ ( ¯ H l − H l )Π ≤ ∆ ′ = 0. Observing that k Π ≤ Λ ′ k = 1, k e − i ( s − s ′ ) ¯ H l k =1, and using standard properties of the spectral norm, we obtain k Π ≤ Λ ′ ( e − is ¯ H l − e − isH l )Π ≤ Λ k ≤ Z | s | ds ′ k Π ≤ ∆ ′ ( ¯ H l − H l )Π > ∆ ′ kk Π > ∆ ′ e − is ′ H l Π ≤ Λ k (16) ≤ k Π ≤ ∆ ′ ( ¯ H l − H l )Π > ∆ ′ k e − λ (∆ ′ − Λ) Z | s | ds ′ ( e αs ′ M −
1) (17)= k Π ≤ ∆ ′ ( ¯ H l − H l )Π > ∆ ′ k e − λ (∆ ′ − Λ) ( e α | s | M − − αM | s | ) αM , (18)where λ = 1 / (2 Jdk ), α = eJ , and Eq. (17) follows from Lemma 1. Note that k Π ≤ ∆ ′ ( ¯ H l − H l )Π > ∆ ′ k = k Π ≤ ∆ ′ H l Π > ∆ ′ k (19) ≤ k H l k (20) ≤ JM (21) ≤ αM . (22)We obtain k Π ≤ Λ ′ ( e − is ¯ H l − e − isH l )Π ≤ Λ k ≤ e − λ (∆ ′ − Λ) ( e α | s | M − − αM | s | ) . (23)To simplify our analysis, we will use a looser upper bound in the statement of the Lemma 2, which followsdirectly from Eq. (23) and ∆ ′ ≥ Λ ′ : k Π ≤ Λ ′ ( e − is ¯ H l − e − isH l )Π ≤ Λ k ≤ e − λ (Λ ′ − Λ) ( e α | s | M − . (24)To prove the second result, we use Lemma 1 together with Eq. (24) and standard properties of the spectralnorm, and obtain k Π > Λ ′ e − is ¯ H l Π ≤ Λ k = k Π ≤ ∆ ′ Π > Λ ′ e − is ¯ H l Π ≤ Λ k (25)= k Π ≤ ∆ ′ Π > Λ ′ ( e − is ¯ H l − e − isH l + e − isH l )Π ≤ Λ k (26) ≤ k Π ≤ ∆ ′ Π > Λ ′ ( e − is ¯ H l − e − isH l )Π ≤ Λ k + e − λ (Λ ′ − Λ) ( e α | s | M −
1) (27)= k (Π ≤ ∆ ′ − Π ≤ Λ ′ )( e − is ¯ H l − e − isH l )Π ≤ Λ k + e − λ (Λ ′ − Λ) ( e α | s | M −
1) (28) ≤ k Π ≤ ∆ ′ ( e − is ¯ H l − e − isH l )Π ≤ Λ k + k Π ≤ Λ ′ ( e − is ¯ H l − e − isH l )Π ≤ Λ k + e − λ (Λ ′ − Λ) ( e α | s | M −
1) (29) ≤ ( e − λ (∆ ′ − Λ) + 2 e − λ (Λ ′ − Λ) )( e α | s | M −
1) (30) ≤ e − λ (Λ ′ − Λ) )( e α | s | M − . (31)7 . Approximation errors for product formulas We consider generic product formulas of q > W ( s ) = e − is q H lq · · · e − is H l e − is H l , (32)¯ W ( s ) = e − is q ¯ H lq · · · e − is ¯ H l e − is ¯ H l , (33)where s = s , . . . , s q , s j ∈ R , and 1 ≤ l j ≤ L . We also define W Λ ( s ) = Π ≤ Λ q e − is q H lq · · · Π ≤ Λ e − is H l Π ≤ Λ e − is H l , (34)¯ W Λ ( s ) = Π ≤ Λ q e − is q ¯ H lq · · · Π ≤ Λ e − is ¯ H l Π ≤ Λ e − is ¯ H l , (35)where Λ = (Λ , . . . , Λ q ), and Λ j ≥ j . Using Lemmas 1 and 2, we now prove a number of results(corollaries) on the approximation errors for these product formulas that will be required for the proof ofThm. 1. First, we will prove that W ( s ) produces approximately the same state as W Λ ( s ) when the initialstate is supported on the low-energy subspace and for a suitable choice of Λ . Next, we will show that theapproximation error from replacing W Λ ( s ) by ¯ W Λ ( s ) is of the same order as that of the first approximationfor a suitable choice of ∆ ′ and effective Hamiltonians. A similar result is obtained if we further replace¯ W Λ ( s ) by ¯ W ( s ). Combining these results we show that the state produced by W ( s ) approximates thatproduced by ¯ W ( s ) for a suitable choice of ∆ ′ . Corollary 1.
Let δ > , ∆ ≥ , λ = 1 / (2 Jdk ) , and α = eJ . Then, if Λ satisfies Λ j − Λ j − ≥ λ ( α | s j | M +log( q/δ )) and Λ = ∆ , k ( W Λ ( s ) − W ( s ))Π ≤ ∆ k ≤ δ . (36) Proof.
We use the identity W ( s ) − W Λ ( s ) = Π > Λ q e − is q H lq Π ≤ Λ q − · · · Π ≤ Λ e − is H l + · · · + e − is q H lq · · · Π > Λ e − is H l Π ≤ Λ e − is H l + e − is q H lq · · · e − is H l Π > Λ e − is H l . (37)Since k e − is j H lj k = k Π ≤ Λ j k = 1, we can use the triangle inequality and Lemma 1 to obtain k ( W Λ ( s ) − W ( s ))Π ≤ ∆ k ≤ q X j =1 k Π > Λ j e − is j H lj Π ≤ Λ j − k (38) ≤ q X j =1 e − λ (Λ j − Λ j − ) ( e α | s j | M −
1) (39) ≤ q X j =1 δ/q (40) ≤ δ . (41) Corollary 2.
Let δ > , ∆ ≥ , λ = 1 / (2 Jdk ) , and α = eJ . Then, if Λ satisfies Λ j − Λ j − ≥ λ ( α | s j | M +log( q/δ )) , Λ = ∆ , and ∆ ′ ≥ Λ q , k ( ¯ W Λ ( s ) − W Λ ( s ))Π ≤ ∆ k ≤ δ . (42) Proof.
We use the identity¯ W Λ ( s ) − W Λ ( s ) =Π ≤ Λ q ( e − is q ¯ H lq − e − is q H lq )Π ≤ Λ q − · · · Π ≤ Λ e − is ¯ H l + . . . + Π ≤ Λ q e − is q H lq Π ≤ Λ q − · · · Π ≤ Λ ( e − is ¯ H l − e − is H l ) . (43)8ince k e − is j H lj k = k e − is j ¯ H lj k = k Π ≤ Λ j k = 1, we can use the triangle inequality and Eq. (24) in Lemma 2to obtain k ( ¯ W Λ ( s ) − W Λ ( s ))Π ≤ ∆ k ≤ q X j =1 k Π ≤ Λ j ( e − is j ¯ H lj − e − is j H lj )Π ≤ Λ j − k (44) ≤ q X j =1 e − λ (Λ j − Λ j − ) ( e α | s j | M −
1) (45) ≤ q X j =1 δ/q (46) ≤ δ . (47) Corollary 3.
Let δ > , ∆ ≥ , λ = 1 / (2 Jdk ) , and α = eJ . Then, if Λ satisfies Λ j − Λ j − ≥ λ ( α | s j | M +log( q/δ )) , Λ = ∆ , and ∆ ′ ≥ Λ q , k ( ¯ W ( s ) − ¯ W Λ ( s ))Π ≤ ∆ k ≤ δ . (48) Proof.
We use the identity¯ W ( s ) − ¯ W Λ ( s ) = Π > Λ q e − is q ¯ H lq Π ≤ Λ q − · · · Π ≤ Λ e − is ¯ H l + · · · + e − is q ¯ H lq · · · Π > Λ e − is ¯ H l Π ≤ Λ e − is ¯ H l + e − is q ¯ H lq · · · e − is ¯ H l Π > Λ e − is ¯ H l . (49)Since k e − is j ¯ H lj k = k Π ≤ Λ j k = 1, we can use the triangle inequality and Eq. (31) in Lemma 2 to obtain k ( ¯ W ( s ) − ¯ W Λ ( s ))Π ≤ ∆ k ≤ q X j =1 k Π > Λ j e − is j ¯ H lj Π ≤ Λ j − k (50) ≤ q X j =1 e − λ (Λ j − Λ j − ) ( e α | s j | M −
1) (51) ≤ q X j =1 δ/q (52) ≤ δ . (53) Corollary 4.
Let δ > , ∆ ≥ , λ = 1 / (2 Jdk ) , α = eJ , and | s | = P qj =1 | s j | . Then, if ∆ ′ ≥ ∆ + λ ( α | s | M + q log( q/δ )) , k ( ¯ W ( s ) − W ( s ))Π ≤ ∆ k ≤ δ . (54) Proof.
We define the energies Λ q ≥ . . . ≥ Λ = ∆ viaΛ j − Λ j − = 1 λ ( α | s j | M + log( q/δ )) . (55)In particular, ∆ ′ ≥ Λ q = ∆ + λ ( α | s | M + q log( q/δ )). We use the identity¯ W ( s ) − W ( s ) = ( ¯ W ( s ) − ¯ W Λ ( s )) + ( ¯ W Λ ( s ) − W Λ ( s )) + ( W Λ ( s )) − W ( s )) . (56)9he triangle inequality and Corollaries 3, 2, and 1 imply k ( ¯ W ( s ) − W ( s ))Π ≤ ∆ k≤ k ( ¯ W ( s ) − ¯ W Λ ( s ))Π ≤ ∆ k + k ( ¯ W Λ ( s ) − W Λ ( s ))Π ≤ ∆ k + k ( W Λ ( s )) − W ( s ))Π ≤ ∆ k (57) ≤ δ + δ + δ (58)= 5 δ . (59)
4. Proof of Thm. 1
For some ∆ ′ ≥ ∆ ≥
0, let ¯ U ( s ) = e − is ¯ H be the evolution operator with the effective Hamiltonian and¯ W p ( s ), p ≥
1, be the corresponding p -th order product formula obtained by replacing H l → ¯ H l in W p ( s ) ofEq. (1). Since k ¯ H l k ≤ ∆ ′ , we obtain k ( ¯ U ( s ) − ¯ W p ( s ))Π ≤ ∆ k ≤ k ¯ U ( s ) − ¯ W p ( s ) k (60) ≤ ǫ (∆ ′ ) , (61)where ǫ (∆ ′ ) = γ ( L ∆ ′ | s | ) p +1 is an upper bound of the error induced by product formulas using effectiveoperators [22] and γ = O (1) is a constant. This error bound grows with ∆ ′ . It does not exploit any structureof the effective Hamiltonians so it may be possible to improve it under further constraints. Additionally, k ( U ( s ) − ¯ U ( s ))Π ≤ ∆ k = 0 , (62)and then k ( U ( s ) − ¯ W p ( s ))Π ≤ ∆ k ≤ ǫ (∆ ′ ) . (63)The other contribution to the error is due to Cor. 4, which can be turned around to obtain a bound on theerror that depends on ∆ ′ . Let λ = 1 / (2 Jdk ), α = eJ , and q > W p ( s ). Then, Cor. 4 implies k ( ¯ W p ( s ) − W p ( s ))Π ≤ ∆ k ≤ δ (∆ ′ ) , (64)and δ (∆ ′ ) = e − q ( λ (∆ ′ − ∆) − α | s | M − q log q ) . (65)This error bound decreases with ∆ ′ . It is now valid for all ∆ ′ ≥ ′ ≤ ∆. We assume that our product formula is such that | s | ≤ κL | s | for a constant κ ≥ α ′ = κα . Then δ (∆ ′ ) = e − q ( λ (∆ ′ − ∆) − α ′ | s | ML − q log q ) . (66)Thus, for any ∆ ′ ≥ ∆, the triangle inequality implies k ( U ( s ) − W p ( s ))Π ≤ ∆ k ≤ ǫ (∆ ′ ) + 5 δ (∆ ′ ).For given | s | , we can search for ∆ ′ ≥ ∆ that minimizes the overall error bound. Let that ∆ ′ be ∆ ′ min ,which satisfies ǫ (∆ ′ min ) + 5 δ (∆ ′ min ) ≤ ǫ (∆ ′ ) + 5 δ (∆ ′ ) , (67)for all ∆ ′ ≥ ∆. Then, we can fix any value of ∆ ′ ≥ ∆ and obtain a bound for the overall error fromcomputing ǫ (∆ ′ ) + 5 δ (∆ ′ ). In particular, we choose∆ ′ = ∆ + 1 λ α ′ | s | M L + qλ log q + qλ ( p + 1) log (cid:18) J | s | (cid:19) , (68)10nd, assuming J | s | ≤ q >
1, we obtain δ (∆ ′ ) = ( J | s | ) p +1 (69) ≤ (∆ ′ | s | ) p +1 (70) ≤ ( L ∆ ′ | s | ) p +1 . (71)The constraint in J | s | is to avoid errors larger than 1: it is sufficient for δ (∆ ′ ) ≤ ′ ≥ ∆. Therefore,if J | s | ≤ k ( U ( s ) − W p ( s ))Π ≤ ∆ k ≤ ǫ (∆ ′ min ) + 5 δ (∆ ′ min ) (72) ≤ ǫ (∆ ′ ) + 5 δ (∆ ′ ) (73) ≤ ( γ + 5)( L ∆ ′ | s | ) p +1 (74) ≤ ˜ γ ( L ∆ ′ | s | ) p +1 (75)= ˜ ǫ (∆ ′ ) , (76)where ∆ ′ was determined in Eq. (68) and ˜ γ = γ + 5 is a constant.An upper bound for ∆ ′ can be given in terms of three factors β , β , and β , which can be easily computedfrom the parameters that define H as ∆ ′ = ∆+ β J log( β / ( J | s | ))+ β J N | s | . This is the expression providedin Thm. 1. Using the properties q > p ≥
1, and N ≤ M L ≤ dN , the factors satisfy β = 2 qdk ( p + 1) (77) β = q / ( p +1) , (78) β ≤ ekd κ . (79)When d , k , L , and q are O (1) constants, we obtain β = O (1), β = O (1) and β = O (1).Note that the error ˜ ǫ (∆ ′ ) approaches zero as s → | s | p +1 , as in the case of p -th order productformulas. The appearance of log(1 / ( J | s | )) in Eq. (68) is due to our error bounds, where we are requiringthat the “leakage” δ (∆ ′ ) vanishes as | s | →
0, implying ∆ ′ → ∞ . Tighter error bounds for this leakage thatavoid this complication may be obtained. Nevertheless, as our purpose is to make | s | as large as possiblein the product formula, while still satisfying the desired error bound (see Thm. 2), the above analyses willsuffice.For the special case of Trotter-Suzuki product formulas, the constant γ that appears in ǫ (∆ ′ ) has beenpreviously studied in Ref. [4]. In this case, the number of terms satisfies q ≤ p L and | s | ≤ c p L | s | , for someconstant c ≈ .
32. Thus, we can take κ = c p and κ = O (1) for p = O (1).
5. Complexity of product formulas for local Hamiltonians
We now determine the complexity of product formulas, which is the total number of exponentials of the H l ’s to approximate the evolution operator U ( t ) within given precision ε >
0. We give an explicit dependenceof this complexity in terms of the relevant parameters that specify H . If a quantum algorithm is constructedto implement such product formula, then our result will determine the complexity of the quantum algorithm(number of two-qubit gates) from multiplying it by the complexity of implementing the exponential of H l .The latter is linear in kM following the results in Ref. [11]. Theorem 2.
Let ε > , ∆ ≥ , t ∈ R , H = P Ll =1 H l a k -local Hamiltonian as above, H l ≥ , and W p ( s ) a p -th order product formula as in Eq. (1) . Then, there exists r = ˜ O t p ε p ( L ∆ + Ldkq (log q ) J ) p ! + O t p +1 ε p +1 ( L dM J ) + p +2 ! , (80)11 uch that k ( U ( t ) − ( W p ( t/r )) r )Π ≤ ∆ k ≤ ε . (81) The ˜ O notation hides a polylogarithmic factor in ( | t | JLqdk/ε ) .Proof. Let r = t/s be the Trotter number, i.e., the number of “segments” in the product formula, eachapproximating the evolution U ( s ) for short time s . We assume s ≥ t ≥ t ≤ t → | t | . Note that U ( s )Π ≤ ∆ = Π ≤ ∆ U ( s )Π ≤ ∆ and U ( t ) = ( U ( s )) r .We use the identity( U ( t ) − ( W p ( s )) r )Π ≤ ∆ = r − X r ′ =0 ( W p ( s )) r ′ ( U ( s ) − W p ( s ))( U ( s )) r − r ′ − Π ≤ ∆ (82)= r − X r ′ =0 ( W p ( s )) r ′ (Π ≤ ∆ + Π > ∆ )( U ( s ) − W p ( s ))Π ≤ ∆ ( U ( s )) r − r ′ − Π ≤ ∆ (83)= r − X r ′ =0 ( W p ( s )) r ′ (Π ≤ ∆ ( U ( s ) − W p ( s ))Π ≤ ∆ + Π > ∆ ( U ( s ) − W p ( s ))Π ≤ ∆ ) ( U ( s )) r − r ′ − (84)= r − X r ′ =0 ( W p ( s )) r ′ (Π ≤ ∆ ( U ( s ) − W p ( s ))Π ≤ ∆ − Π > ∆ W p ( s )Π ≤ ∆ ) ( U ( s )) r − r ′ − . (85)If ∆ ′ is given by Eq. (68) and Js ≤
1, Thm. 1 implies k Π ≤ ∆ ( U ( s ) − W p ( s ))Π ≤ ∆ k ≤ k ( U ( s ) − W p ( s ))Π ≤ ∆ k (86) ≤ ˜ ǫ (∆ ′ ) , (87)and k Π > ∆ W p ( s )Π ≤ ∆ k = k Π > ∆ ( U ( s ) − W p ( s ))Π ≤ ∆ k (88) ≤ k ( U ( s ) − W p ( s ))Π ≤ ∆ k (89) ≤ ˜ ǫ (∆ ′ ) , (90)where ˜ ǫ (∆ ′ ) = ˜ γ ( L ∆ ′ s ) p +1 and ˜ γ is a constant that can be determined from the error bounds of productformulas – see Thm. 1, Eq. (76). Using the triangle inequality with Eq. (85) and k W p ( s ) k = k U ( s ) k = 1 forunitary operators, we obtain k ( U ( t ) − ( W p ( s )) r )Π ≤ ∆ k ≤ r ( k Π ≤ ∆ ( U ( s ) − W p ( s ))Π ≤ ∆ k + k Π > ∆ W p ( s )Π ≤ ∆ k ) (91) ≤ r ˜ ǫ (∆ ′ ) . (92)Thus, for overall error bounded by ε , it suffices to choose ˜ ǫ (∆ ′ ) ≤ ε/ (2 r ) or, equivalently,˜ γ ( L ∆ ′ s ) p +1 ≤ ε t s . (93)To set some conditions in s , in addition to Js ≤
1, we note that each of the terms in Eq. (68) that definethe effective norm can be dominant depending on s , ∆, and other parameters. Then, to obtain the overallcomplexity of product formulas, we will analyze four different cases as follows. In the first case, we assumethat ∆ is the dominant term in ∆ ′ , and we require˜ γ (4 L ∆ s ) p +1 ≤ ε t s , (94)12hich is satisfied as long as s ≤ s = (cid:18) ε γt (cid:19) /p L ∆) /p . (95)In the second case, we assume that α ′ sM L/λ is the dominant term in Eq. (68) and impose˜ γ (cid:18) L α ′ M Ls λ (cid:19) p +1 ≤ ε t s, (96)which implies s ≤ s = (cid:18) ε γt (cid:19) p +1 (cid:18) λ α ′ M L (cid:19) + p +2 . (97)In the third case, we assume that ( q log q ) /λ is the dominant term in Eq. (68) and impose˜ γ (cid:18) L q log q sλ (cid:19) p +1 ≤ ε t s, (98)which implies s ≤ s = (cid:18) ε γt (cid:19) p (cid:18) λ Lq log q (cid:19) /p . (99)In the fourth case, we assume that qλ ( p + 1) log (cid:0) Js (cid:1) is the dominant term in Eq. (68) and impose˜ γ (cid:18) L qλ ( p + 1) log (cid:18) Js (cid:19) s (cid:19) p +1 ≤ ε t s , (100)under the assumption Js ≤
1. Equivalently, if z = Js ≤ f ( z ) = (log(1 /z )) p +1 z p , p ≥
1, we impose f ( z ) ≤ X , (101)where X = ε γt (cid:18) λ Lq ( p + 1) (cid:19) p +1 J p . (102)To set a fourth condition in s we could then compute X from the inputs of the problem and find a rangeof values of z for which Eq. (101) is satisfied. We can also obtain such a range analytically as follows. Thefunction f ( z ) increases from f (0) = 0, attains its maximum at z M = e − p +1 p (hence e − ≤ z M ≤ e − for all p ≥ f (1) = 0. Additionally, f ( z ) ≤ ((1+1 /p ) /e ) p +1 ≤ /e ≈ .
54 for all 0 ≤ z ≤ X ≥ ((1 + 1 /p ) /e ) p +1 then Eq. (101) is readily satisfied for all z ≤ s will be required in this case (this happens, for example, for sufficiently small values of t ). Moregenerally, for a given X , we can solve for f ( z ) = X . If there are two solutions z , ≤ z , we consider thesmaller one ( z < z ) and the relevant range for z to satisfy Eq. (101) is [0 , z ]. It will then suffice to imposethat z belongs to a range [0 , z ′ ], where z ′ ≤ z . To this end, we define z ′ = X /p / ( e log (cid:0) e /X (cid:1) ) ( p +1) /p and,13nder the assumption X . .
54, we have z ′ < e − . Additionally, f ( z ) = (log(1 /z )) p +1 z p (103) ≤ log e log ( p +1) /p ( e /X ) X /p !! p +1 X ( e log( e /X )) p +1 (104) ≤ (cid:18) log (cid:18) e log ( e /X ) X (cid:19)(cid:19) p +1 X ( e log( e /X )) p +1 (105)(106) ≤ (cid:0) (cid:0) e /X (cid:1)(cid:1) p +1 X ( e log( e /X )) p +1 (107) ≤ X , (108)where we used X /p ≥ X and log (cid:0) e /X (cid:1) ≤ e /X for X ≤
1. This is the condition of Eq. (101). Then, forthe fourth condition in s , we impose z ≤ z ′ or, equivalently, s ≤ s = (cid:18) ε γt (cid:19) /p (cid:18) λ Lq ( p + 1) (cid:19) /p (cid:18) e log (cid:18) e γtεJ p (cid:16) Lq ( p +1) λ (cid:17) p +1 (cid:19)(cid:19) /p (109)= (cid:18) ε γt (cid:19) /p (cid:18) λ Lq ( p + 1) (cid:19) /p (cid:16) e log (cid:16) e γtJε (8 Lq ( p + 1) dk ) p +1 (cid:17)(cid:17) /p . (110)Except for a mild polylogarithmic correction in tJLqdk/ε – the third factor – this condition is similar tothe first and third ones.Then, if Js ≤ s additionally satisfies Eqs. (95), (97), (99), and (110), we obtain the desired conditionof Eq. (93). The product formulas under consideration [Eq. 1] are such that α ′ = κα = O ( J ) and ˜ γ = O (1),and we consider the case where p is a O (1) constant. The conditions in s allow us to obtain a sufficientcondition for the Trotter number as follows: r = t/s (111)= ˜ O t p ε p ( L ∆ + Ldkq (log q ) J ) p ! + O t p +1 ε p +1 ( L dM J ) + p +2 ! , (112)where the ˜ O notation hides a polylogarithmic factor in tJLqdk/ε coming from Eq. (110). For the case when q is O ( L ), k = O (1), d = O (1), and hence L = O (1) and M = O ( N ), and considering the asymptotic limit,we obtain r = ˜ O ( t (∆ + J )) p ε p ! + O ( tJ √ N ) p +1 ε p +1 ! . (113)
6. Comparison with known results on product formulas
We compare our result on the complexity of product formulas with those in Ref. [22] that are the state ofthe art. When no assumption is made for the initial state, the Trotter number stated in Ref. [22] for k -localHamiltonians is ˜ r = O (cid:18) t /p ε /p k H k ind − k H k /p (cid:19) . (114)14ere, k H k is the 1-norm of H , given by P Ll =1 k H l k in our case, and k H k ind − is the so-called induced 1-norm of H . The latter is defined as follows. We write H = P Nj ,...,j k =1 h j ,...,j k , where each h j ,...,j k includesthe k -local interaction terms of qubits labeled as j , . . . , j k in H . Then, k H k ind − := max l max j l N X j ,...,j l − ,j l +1 ,...,j k =1 k h j ,...,j k k . (115)That is, we fix certain qubit j l and consider all the interaction terms that contain that qubit. For a degree d Hamiltonian with k -local interaction terms (not necessarily geometrically local), each of strength at most J , k H k ind − ≤ dJ . Furthermore, k H k can be upper bounded as k H k ≤ JM L ≤ JdN . As a result, for a k -local Hamiltonian as above, the best known upper bound for the Trotter number is˜ r = O (cid:18) t /p ε /p ( dJ ) /p N /p (cid:19) . (116)To compare our main result with Eq. (116), we express Eq. (112) in terms of d and N . We recall that thetotal number of local terms in H is M L ≤ dN , L ≤ dk + 1, and we assume that k = O (1), M = O ( N ), and q = O ( L ) = O ( d ) for the p -th order product formula. Then, in the asymptotic limit, r = ˜ O t p ε p (cid:0) d ∆ + d J (cid:1) p ! + O t p +1 ε p +1 ( d N J ) + p +2 ! . (117)The results for various values of p and k = O (1) are in Table II. Comparison of asymptotic complexities (Trotter number) as a function of ∆ , J, d, N, ε, t