Learning quantum many-body systems from a few copies Cambyse Rouz´e1, ∗ and Daniel Stilck Fran¸ca2, y 1Department of Mathematics, Technische Universit¨atM¨unchen,85748 Garching, Germany 2QMATH, Department of Mathematical Sciences, University of Copenhagen, Denmark (Dated: August 25, 2021) Estimating physical properties of quantum states from measurements is one of the most funda- mental tasks in quantum science. In this work, we identify conditions on states under which it is possible to infer the expectation values of all quasi-local observables of a given locality up to a relative error from a number of samples that grows polylogarithmically with the system's size and polynomially on the locality of the target observables. This constitutes an exponential improvement over known tomography methods in some regimes. We achieve our results by combining one of the most well-established techniques to learn quantum states, namely the maximum entropy method, with tools from the emerging fields of quantum optimal transport and classical shadows. We conjec- ture that our condition holds for all states exhibiting some form of decay of correlations and establish it for several subsets thereof. These include widely studied classes of states such as one-dimensional thermal and high-temperature Gibbs states of local commuting Hamiltonians on arbitrary hyper- graphs or outputs of shallow circuits. Moreover, we show improvements of the maximum entropy method beyond the sample complexity of independent interest. These include identifying regimes in which it is possible to perform the postprocessing efficiently as well as novel bounds on the condition number of covariance matrices of many-body states. I. INTRODUCTION in learning only physical properties of the state on which tomography is being performed. These are often encoded into the expectation values of The subject of quantum tomography has as quasi-local observables that only depend on re- its goal devising methods for the efficient classi- duced density matrices of subregions of the sys- cal description of a quantum system from access tem. By Helstrom's theorem, obtaining a good to experimental data. However, all tomographic recovery guarantee in trace distance is equivalent methods for general quantum states inevitably to demanding that the expectation value of all require resources that scale exponentially in the bounded observables are close for the two states. size of the system [1, 2], be it in terms of the num- ber of samples required or the post-processing It is in turn desirable to design tomographic needed to perform the task. procedures that can take advantage of the fact that we wish to only approximate quasi-local ob- Fortunately, most of the physically relevant servables, instead of demanding a recovery in quantum systems can be described in terms of trace distance, and some methods in the lit- a (quasi)-local structure. These range from that erature take advantage of that. For instance, of a local interaction Hamiltonian corresponding the overlapping tomography or classical shad- to a finite temperature Gibbs state to that of ows methods of [9{12] allow for approximately a shallow quantum circuit. Hence, locality is a learning all k-local reduced density matrices of physically motivated requirement that brings the an n-qubit state with failure probability δ us- amount of parameters describing the system to a ck −1 −2 arXiv:2107.03333v2 [quant-ph] 24 Aug 2021 ing (e k log(nδ ) ) copies without impos- tractable number. Effective tomographic proce- ingO any assumptions on the underlying state. dures should be able to incorporate this informa- This constitutes an exponential improvement in tion. And, indeed, many protocols in the litera- the system size compared to the previously men- ture achieve a good recovery guarantee in trace tioned many-body setting at the expense of an distance from a number of copies that scales poly- undesirable exponential dependency in the lo- nomially with system size [3{8]. cality of the observables. Unfortunately, with- Furthermore, in many cases one is interested out making any assumptions on the underlying states, this exponential scaling is also unavoid- able [9]. ∗ [email protected] In light of the previous discussion, it is natural y [email protected] 2 to ask if, combining the stronger structural as- chitz observables, transportation cost inequali- sumptions required for many-body tomography ties and the the maximum entropy principle. with techniques like classical shadows, it is pos- sible to obtain the best of the two worlds: a sam- ple complexity that is logarithmic in system size A. Lipschitz observables and polynomial in the locality of the observables we wish to estimate. In the classical setting, given a metric d on In this work, we achieve precisely this demand a sample space , the regularity of a function for a large class of physically motivated states S f : R can be quantified by its Lipschitz by combining recent techniques from quantum constantS![29, Chapter 3] optimal transport [13{18], the well-established maximum entropy method [19] to learn quan- f Lip = sup f(x) f(y) =d(x; y) : (1) tum states and the method of classical shadows. k k x;y2S j − j This results in an exponential improvement over For instance, if we consider functions on the n- known methods of many-body tomography [3{ dimensional hypercube 1; 1 n endowed with 8] and recent shadow tomography or overlapping the Hamming distance,{− the Lipschitzg constant tomography techniques [9{12], as summarized in quantifies by how much a function can change Table I. per flipped spin. It should then be clear that Our revisit of the maximum entropy princi- physical quantities like average magnetization ple is further motivated by recent breakthroughs have a small Lipschitz constant. Some recent in Hamiltonian learning [5, 20], shadow to- works [14, 16] extended this notion to the non- mography [9], the understanding of correlations commutative setting. Although the definition and computational complexity of quantum Gibbs put forth in [14] and the approach in [16] are not states [21{25] and quantum functional inequal- equivalent, they both recover the intuition dis- ities [18, 26] that shed new light on this sea- cussed previously. E.g., for the approach followed soned technique. Examples where we obtain ex- in [16], the Lipschitz constant of an observable on ponential improvements include thermal states of n qudits is defined as [30] 1D systems and high-temperature thermal states of commuting Hamiltonians on arbitrary hyper- O Lip; k k := max max tr [O(ρ σ)] ; graphs and outputs of shallow circuits. Further- pn 1≤i≤n ρ,σ2Ddn − more, based on results by [24], we conjecture that tri[ρ]=tri[σ] our results should hold for any high-temperature (2) Gibbs state, even with long-range interactions, where n denotes the set of n-qudit states. and give evidence in this direction. Dd That is, O Lip; quantifies the amount by which The main ingredient to obtain our improve- the expectationk k value of O changes for states ments are so-called transportation cost (TC) in- that are equal when tracing out one site. It equalities [27]. They allow us to relate by how is clear that O Lip 2pn O 1 always holds much the expectation values of two states on by H¨older'sinequality,k k ≤ butk itk can be the case Lipschitz observables, a concept we will review that O Lip; pn O 1. For instance, con- shortly, differ from their relative entropy. Such siderk fork somek > k0k the n-qubit observable inequalities constitute a powerful tool from the n i+k O = i=1 j=i Zj, where for each site j, Zj de- theory of optimal transport [28] and are tradi- notes the Pauli⊗ observable Z acting on site j and tionally used to prove sharp concentration in- we takeP addition modulo n. It is not difficult equalities [29, Chapter 3]. Moreover, they have to see that O Lip; = 2kpn, while O 1 = n. been recently extended to quantum states [14, We refer tok thek discussion in Fig. 1k fork another 16, 18]. By combining such inequalities with the example. maximum entropy principle, we are able to eas- Moreover, one can show that shallow local ily control the relative entropy between the states circuits or short-time local dynamics satisfying and, thus, the difference of expectation values of a Lieb-Robinson bound cannot substantially in- Lipschitz observables. crease the Lipschitz constant of an observable Before we summarize our contributions in when evolved in the Heisenberg picture. That is, more detail, we first define and revise the main ∗ if we have that Φt is the quantum channel that concepts required for our results, namely Lips- describes some local dynamics at time/depth t 3 Structure Assumptions on State Assumptions on Observable Samples Many-body tomography Many-body none poly(n; −1) [5] Classical Shadows none k-local poly(eck; log(n); −1) [9] This work Many-body+transportation k-local poly(k; log(n); −1) TABLE I. Summary of underlying assumptions and sample complexity of other approaches to perform tomog- raphy on quantum many-body states. in the Heisenberg picture and it satisfies a Lieb- can define a Wasserstein-1 distance on states by Robinson bound, then we have: duality [14, 16]. The latter quantifies how well we can distinguish two states by their action on cer- Φ∗(O) = (evt O ) ; t Lip; Lip; tain regular or local observables and is given k k O k k LΛ where v denotes the Lieb-Robinson velocity. This by result is discussed in more detail in Section B 1 of the supplemental material. Thus, averages W1;Λ(ρ, σ) := sup tr [O(ρ σ)] : (4) O2LΛ − over local observables and short-time evolutions thereof all belong to the class of observables that y Here we will consider Λ = O = O ; O Lip;Λ have a small Lipschitz constant when compared L f k k ≤ 1 for : Lip;Λ : Lip; ; : Lip;r and to generic observables.