PHYSICAL REVIEW RESEARCH 2, 033251 (2020) Hamiltonian assignment for open quantum systems Eugene F. Dumitrescu * and Pavel Lougovski † Quantum Information Science Group, Computational Sciences and Engineering Division, Oak Ridge National Laboratory, Oak Ridge, Tennessee 37831, USA (Received 13 January 2020; accepted 10 July 2020; published 14 August 2020) We investigate the problem of determining the Hamiltonian of a locally interacting open quantum system. To do so, we construct Hamiltonian estimators based on inverting a set of stationary, or dynamical, Heisenberg- Langevin equations of motion which rely on a polynomial number of measurements and model parameters. To validate our Hamiltonian assignment methods we numerically simulate one-dimensional XX-interacting spin chains coupled to thermal reservoirs. We provide general bounds on the scalability and assignment error in the presence of noise. In addition to discussing some details of practical implementations we find that, in a dynamical setting, the Hamiltonian estimator’s accuracy increases when relaxing the environment’s physicality constraints. DOI: 10.1103/PhysRevResearch.2.033251 I. INTRODUCTION learning [5]), for scalable methods (local Hamiltonian tomog- raphy [6]) the estimation task is complicated by interactions Fault tolerant quantum computation provides a framework coupling the principle system of interest to unwanted envi- for digitally decomposing unitary operators using a polyno- ronmental degrees of freedom. To address this outstanding mial number of one- and two-qubit operations drawn from a issue, we study the problem of assigning a Hamiltonian to universal gate set [1]. For noisy intermediate scale quantum an open quantum system, provided the principle system and (NISQ) hardware, characterized by fixed gate fidelities and environmental interactions are both geometrically local. limited coherence times, digitizing a quantum simulation uni- We first formulate the task of Hamiltonian inference and tary is too costly in terms of the polynomial scaling circuit summarize previous results. Afterwards we generalize the depth. Hamiltonian learning protocols to the context of open quan- However, if a programmable quantum device’s many-body tum systems and perform numerical simulations in order to dynamics are described by an underlying Hamiltonian H,itis validate and analyze our techniques in two distinct noisy prudent to consider digital-analog decompositions [2]lever- settings. We conclude by discussing generalizations, practical aging H. It has been proposed that, in such a case, the target constraints, and future directions for Hamiltonian learning. unitary can be decomposed as a sequence of native analog uni- taries U = exp(−iHt) interleaved with programmable single- qubit operations. For certain applications, such as many-body II. BACKGROUND simulations [3], the gate complexity, quantified by the total number of applications of the many-body evolution operator Hamiltonian tomography refers to the task of estimating a U and local rotations, may be significantly smaller than that Hamiltonian H given access to states evolving under H. While of the digitized decomposition. this task is exponentially costly in general, the tomography of The digital-analog quantum simulation’s error must be local Hamiltonians has recently attracted significant attention bounded, e.g., in terms of the distance between the target [6–9] due to its scalability. We are interested in determining and digital-analog unitaries, in order to certify an accurate k-local Hamiltonians of the form simulation. It therefore follows that, in order to upper bound = , the simulation error, one must first precisely characterize the H ciSi (1) Hamiltonian generating the many-body operation. i While some prior results regarding Hamiltonian estimation where each S is an operator supported on k spatially con- exist (e.g., process tomography [4] and Bayesian Hamiltonian i nected sites. We work in the Pauli basis, such that all k- μ = k−1 σ j μ ∈ local operators can be written as Si j=0 j where j {I, X,Y, Z}, where the index j runs over spatially connected *[email protected] sites. |ψ †[email protected] Suppose we have access to either (i) eigenstates n of H, with H|ψn=En|ψn or (ii) thermal states ρ = exp(−βH )/Z Published by the American Physical Society under the terms of the where Z = Tr[exp(−βH )] is the partition function. Aside Creative Commons Attribution 4.0 International license. Further from the trivial phase factors, both states are stationary under distribution of this work must maintain attribution to the author(s) Hamiltonian dynamics. In the Heisenberg picture expectation and the published article’s title, journal citation, and DOI. values taken with respect to these states are likewise stationary 2643-1564/2020/2(3)/033251(6) 033251-1 Published by the American Physical Society EUGENE F. DUMITRESCU AND PAVEL LOUGOVSKI PHYSICAL REVIEW RESEARCH 2, 033251 (2020) and we may write III. OPEN SYSTEM GENERALIZATION −i Unfortunately, the inability to evolve by purely unitary O˙= [O, H]=0(2) h¯ dynamics limits the applicability of closed Hamiltonian learn- for any observable O. Inserting Eq. (1), selecting an input op- ing. Realistic quantum systems are open and, in the pres- ence of unknown environmental interactions, evolve dissi- erator basis {O j}, and measuring the commutators [O j, H], , = patively. In order to incorporate environmental couplings we may express the set of linear equations j,i [O j Si] ci 0 concisely in matrix form as in our framework, we consider a Markovian master equa- tion dynamics for a density operator ∂t ρ = L[ρ] generated Ac = 0, (3) by the quantum Liouvillian L. Specifically, we consider a L ρ = −i ,ρ + D ρ Lindblad equation given by [ ] h¯ [H ] [ ], where where we have introduced the matrix A with elements † 1 † D = , γnm(LnρL − {L Ln,ρ}). Motivated by locality, Ai, j =[O j, Si] and the Hamiltonian coefficient vector c = n m m 2 m T we consider the {Ln} operators to form an orthonormal ba- (c1,...,cn ) . Note that in principle A need not be a square matrix as its dimensions are determined by the number of sis spanning the manifold of J-local superoperators. The γ accessible correlation measurements. Since the operators S coefficient matrix is constrained to be positive semidef- i inite in order to represent a physical map between pos- are k-local and we have the freedom to choose O j from a local basis, most correlators will vanish, due to spatially itive semidefinite density operators [10]. An observable’s nonoverlapping (O , S ) pairs, and A will be sparse. dynamics will now be given by the Heisenberg-Langevin j i master equation O˙ = −i [O, H] + D†[O], where D†[O] = In practice, entries of A arise from noisy measurements h¯ D† = γ † − 1 { † , } which may lead to an erroneous evaluation of eigenvalues. n,m n,m[O] n,m nm(LmOLn 2 LmLn O ). As in pre- To improve the numerical stability of the inversion problem, vious sections, a model for the equations of motion is learn- one could reformulate the problem as a convex optimization able with polynomial resources if one imposes a locality 2 constraint on the geometric locality of the Lindbladian dis- problem, minimize Ac 2, which is equivalent to maximiz- ing a Gaussian log likelihood, maximize log e−cT AT Ac.The sipators. In the discussion below we assume that dissipative latter formulation is convenient for incorporating Bayesian correlations are bounded by a length scale J, such that we uncertainty quantification methods, e.g., to treat noise in speak of K-local Hamiltonians and J-local dissipators. L ρ = the matrix A.IfA is a square matrix, then Ac = 0 has a Lindbladian fixed points, satisfying [ ] 0, generalize unique solution only if A has a nondegenerate zero eigenvalue. the notion of Hamiltonian steady states and their stationary Note that Eq. (3) holds for all Hamiltonians aH defined up dynamics can be used to generalize Eq. (3). That is, consider the linear equations Ac + Bγ = 0, where, in addition to A, to the scalar factor a. In order to avoid trivial solutions we , B is defined by its elements Bn m =L†O L − 1 {L†L , O }. may reformulate assignment into the constrained optimiza- i n i m 2 n m i tion task: minimizeAc2, subject toc2 = 1. The solution As before, the set of linear equations can be invoked as 2 2 = = | is then the row vector of V T associated with the minimal sin- Cx 0, where C (A B) acts on a composite Hamiltonian- gular value in A’s singular value decomposition A = UV T . Lindbladian model space spanned by the configuration vector T T T We return to the issue of numerical stability when considering x = (c , γ ) . While Lindbladian learning has been recently noise below. investigated [11], where it was shown that reconstructing While the homogeneous operator equations derived from strongly dephasing jump operators is difficult under certain steady states provide a simple formalism, preparing eigen- conditions, our focus is on precisely estimating the Hamilto- states and thermal states of an unknown Hamiltonian may nian component, possibly at the expense of the environmental be challenging or time consuming. Earlier work has therefore sector. also explored Hamiltonian estimation in a dynamical context Let us also consider finite-time evolution which is applica- [7,8]. In the Heisenberg picture an observable O evolves as ble if Lindbladian fixed points are unavailable. As before, the O(t ) = eitH O(0)e−itH , where O(0) denotes the observable at linearized Heisenberg-Langevin evolution can be constructed 2 time t = 0 where it coincides with its Schrödinger picture at the cost of an approximation error δt . With the input state ρ counterpart. Integrating Eq. (2) over an infinitesimal time δt, degree of freedom j, the matrix elements may be defined n,m = δ † − 1 { † , } we write as Bi, j t LnOiLm 2 LnLm Oi j.