(2020) Hamiltonian Assignment for Open Quantum Systems

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 tomography [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 unitaries 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 presence 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 equation 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. The dynamical equa- = = | O (δt ) −O (0) =−iδt[O (0), H] + O(δt 2 ), (4) tions of motion are simply C x W , where C (A B ), W , i j i j i j and A have been defined in the closed systems context. where the trace is taken with respect to an initial state ρ j, which serves as an input degree of freedom. Considering the small time evolution of a set of operators Oi, with respect to IV. MODEL SYSTEM a set of initial states ρ j, we consider the set of heterogeneous Heisenberg equations of motion defined by the measurement We present numerical simulations in order to quantita- tively analyze the behavior of the open system Hamiltonian settings’ vector Wij =Oi(δt ) j −Oi(0) j and matrix ele- = δ , learning methods outlined above. To do so, we consider one- ments Ai, j,l tcl [Sl Oi] j. This can be expressed as dimensional spin chains consisting of N sites with a Hamilto- = + = · σ A c = W , (5) nian H H0 HI . Here, H0 i ci i consists of single- σ = σ x,σy,σz qubit interactions, with i ( i i i ), and the nearest- and, as before, the assigned Hamiltonian will correspond to = σ xσ x neighbor interactions are given by HI i Ji,i+1 i i+1. − 2 the solution vector c optimizing minc A c W 2. We simulate a system with Hamiltonian interactions

033251-2 HAMILTONIAN ASSIGNMENT FOR OPEN QUANTUM SYSTEMS PHYSICAL REVIEW RESEARCH 2, 033251 (2020) c = (0.5, 0, −2.55), Ji,i+1 = 0.25 at each site. The system’s translational symmetry could be used to reduce the size of the model space, but we do not expect this to hold in general and work with the unconstrained model, assigning distinct parameters to each region. In order to mimic environmental effects each spin is then subjected to thermal noise described by thermal excitation √ + and relaxation operators which are written as L+ = g+σ √ − and L− = g−σ . Here, g+ = gn¯/2, g− = g(¯n + 1)/2,n ¯ is a thermal occupation number, and g is the reservoir-spin coupling strength. An operator O supported on a given † g− + D = σ σ− − site evolves dissipatively under n¯ [O] 4 ( O {σ −σ +, }/ + g+ σ − σ + −{σ +σ −, }/ O 2) 4 ( O O 2). Expanding the ladder operators σ ± = (X ± iY )/2 and regrouping the terms we see that this expression may be rewritten in the Pauli D† = D† + D† + D† + D† basis as n¯ X,X [O] Y,Y [O] X,Y [O] Y,X [O], 2 2 where the coefficients are γXX = γYY = (g+ + g−)/4 and γ = γ ∗ = 2 − 2 / XY YX i(g− g+) 4. FIG. 1. Top panel: Steady state estimator error and low-lying singular values as a function of the input {O }’s cardinality [i.e., the V. EQUILIBRIUM LEARNING i number of rows in Eq. (3)]. The gray dashed lines enumerate the In order to the simulate the steady state learning protocol, transitions between input operator localities. The normalized Hamil- we first solve for the fixed-point density operator satisfying tonian (Lindbladian) estimator error H ( H−L) is given by the blue × L[ρ] = 0, where L is defined above [12]. Next, we select (orange triangle) markers while the minimal singular value s0 and the Pauli basis to express the Hamiltonian and Lindbladian gap to the next singular value s are denoted by the green solid and red dashed lines respectively. Note the discontinuity in estimator model spaces and the set of input operators {Oi}. In practice one should begin with a small model space and increase error and singular spectrum upon the correlation matrix becoming its size until adequate convergence, but for simplicity we left invertible (our five-site model contains 141 free parameters). Bottom panel: Measurement complexity |{M }|, which is generated take as our candidate model the space of K = 2, J = 1-local i as the union of unique operators contained in the matrix elements configurations. In order to fully explore the maximal quality of C, in terms of the cardinality |{O }|. The local, XX, and thermal {O } i of learning model we take i to be the overcomplete union interaction parameters are c = (0.5, 0, −2.55), J = 0.25, g = 0.05, of all 1-, 2-, 3-, and 4-local operators. respectively. The top panel in Fig. 1 illustrates the fixed-point model estimation protocol for an N = 5 site chain with open bound- ary conditions. The horizontal axis denotes the cardinality of model with scaling corrections due to finite-size effects. The the set {Oi}. We have swept through operators of each locality, as indicated by the dashed lines, which are enumerated minimal singular value s0 likewise approaches zero, such from the left to the right ends of the chain. Plotted along that Eq. (3) is well approximated by the associated right the vertical axis are the Hamiltonian estimator errors = singular vector. The solution’s uniqueness is indicated by the H = − / || − || /|| || , large singular gap s s1 s0, in the sense that s s0 cT cE 2 cT 2, where T E refer to the true and estimator max(K,2J)+1 models and, recalling that x = (cT , γ T )T , the total estimator 1. A loose upper (lower) bound is given by N(4 ) error H−L =||xT − xE ||2/||xT ||2. The estimators are constructed by decomposing the correlation matrix C = UV T and selecting as a solution the row vector of V T associated with the minimal singular value s0 of = diag(sm−1,...,s0 ). In our numerical examples, given sufficient input data, the reconstruction errors in Fig. 2 are only limited by numerical round-off errors in the singular value decomposition of the correlation matrix. In order to decompose the model space into the image and kernel of the correlation matrix the cardinality of the measurement set Oi should exceed the size of the model space which generally scales as O(V (4K + 42J )), the space of K (J)-local Hamiltonian (Lindbladian) operators can is K(2J) expanded in a 4 -dimensional basis of the Pauli op- FIG. 2. N = 5 equilibrium learning using the parameters of ⊗K(2J) 2 erators (I, X,Y, Z) , where V = N, 3N, N for a one- Fig. 1 in the presence of N (0,σ) distributed measurement noise. The dimensional (1D), ladder, and 2D geometry, respectively. For estimator error grows linearly as a function of σ and is approximately a collection of d-dimensional systems this bound generalizes two orders of magnitude larger than the minimal singular value when O 2K + 4J to (V (d d )). Figure 1 illustrates this convergence, at the gap s is well defined. Results are averaged over 20 disorder the discontinuity, as the input {Oi} space grows for a five-site configurations.

033251-3 EUGENE F. DUMITRESCU AND PAVEL LOUGOVSKI PHYSICAL REVIEW RESEARCH 2, 033251 (2020)

[N(4max(K,2J))] measurement settings, as indicated by the dashed line labeled with the numeral “3” (“2”) in Fig. 1.

VI. STABILITY ANALYSIS In practice, state preparation and measurement (SPAM) errors deteriorate the estimator. Assume the correlation matrix including errors is C˜ = C + E, where C is the ideal signal and E is due to SPAM perturbations. Weyl’s theorem, stating that |s˜i − si| ||E||2 [13,14], bounds change in s0 given E.For example, consider measurement noise taking Oi→Oi+ δ Oi , where the Oi ’s are matrix elements of A in Eq. (3) and the open systems generalization. Here, E is the collection μ = σ 2 δ of normally distributed, with 0 and variance, Oi random variables. Figure 2 shows how, for small σ , s0 and the estimator infidelity grow linearly with σ. Intuitively, ||E||2 goes as O(σ ) and Weyl’s bound’s tightness is tested with 100 noise realizations in the gapped regime showing that ||E||2 = 1.669(15)|s˜0 − s0|. The linear scaling continues until s0 > s, i.e., the singular spectrum is effectively degenerate, at which point no unique solution exists. FIG. 3. Top panel: Normalized Hamiltonian and Hamiltonian- To this end, we could also consider steady state pertur- Lindbladian reconstruction errors as a function of chain length using ρ → − ε ρ + ερ ρ bations taking SS (1 ) SS for arbitrary .Due least-squares minimization (×’s and open circles) and minimiza- = − ε ρ + ε ρ to linearity, Oi (1 )Tr[ SSOi] Tr[ Oi], which tion by convex optimization with the Lindbladian model subject to rescales the A → (1 − ε)A with the ρ component generating positive semidefinite constraints (open triangles and squares). We the E noise matrix. We have again numerically verified Weyl’s reuse the earlier model parameters and evolve for a time δt/cz ∼ bound by emulating an anisotropic depolarizing channel, i.e., 1 × 10−3. Bottom panel: Normalized reconstruction errors for first- ρ = ci pîρsspî, by reducing each expectation random value and second-order finite-difference derivative approximations. Here, drawn from a folded normal distribution. Again, the standard the evolution time is varied and the Hamiltonian (open symbols) deviation statistically determines the spectral norm and we and total (lines) estimator errors are given. The least-squares (lst) numerically find ||E||2 = 3.817(18)|s˜0 − s0|. and convex minimization (cvx) Hamiltonians agree at small times as illustrated in the log-linear inset. VII. DYNAMICAL LEARNING Lastly, we simulate the dynamical learning scenario by In practice, it is difficult to evolve for sufficiently small || − |2 solving minx C x W 2. References [7,8] suggested product times. For example, a minimum evolution time for the OPEN- states as a dynamical basis but, as exact product states may not PULSE control framework implementation on IBM’s platform actually be available as a near-term resource, we consider a [15]isδtmin ∼ 14 ns (four pulse steps of dt = 32/9). As generalized scenario. We take the fixed-point density operator noted in Fig. 3, high-fidelity estimators require a sampling as a resource and conjugate it with respect to a set of unitary frequency greater than the interaction frequencies, so the IBM operators {Uj}. In this way, we generate a new state basis timescale is appropriate for interactions below ∼100 MHz. {ρ }={ ρ †} j Uj SSUj . In contrast to stationary inference any set Recent OPENPULSE updates have decreased the pulse interval of states will generate valid correlation matrices, given that by a factor of 16 which would increase the cutoff frequency to the basis is sufficiently large, making dynamical assignment the GHz scale. robust to state perturbations. To ameliorate the small time requirement we employ Choosing {Uj} to consist of all 1-local Pauli operators, and a higher-order finite-difference approximation [−O(2δt ) + 2 beginning a time step δt/cz ∼ 1e − 3, we find that the finite- 4O(δt ) − 3O(0)]/2δt = O˙(0) + O(δt ) such that the error time protocol behaves qualitatively similarly to the steady scales as O(δt 3). The bottom panel of Fig. 3 shows the Hamil- state protocol. That is, the estimator error dramatically van- tonian and composite estimator errors as a function of δt for ishes when the set of equations is complete with respect to the both the linear and quadratic finite-difference approximations. model space and then further converges as more information is Note one major drawback of this modification lies in the collected. The normalized minimum estimator error for both increased variance for the derivative estimator. Denoting V1(2) the Hamiltonian and the total Hamiltonian-Lindbladian esti- as the variance in evaluating the first- (second-) order time mates is plotted in Fig. 3 as a function of system size. Here, we derivative of an operator O, and assuming independence and illustrate the errors for both components of the model system that Var[O(t )] is constant for all t,wehaveV2/V1 = 13/4, using (i) a naive least-squares fit and (ii) imposing positivity which corresponds to an approximately tenfold increase in constraints on the Lindbladian process. Interestingly, Fig. 3 the number of samples to reduce the second-order estimator shows that, given the parameters considered, the Hamiltonian variance to that of the first. estimator error is insensitive to positivity constraints whereas Interestingly, we note that for δt 0.15 both the first- the environmental error is greatly reduced with their inclusion. and second-order least-squares Hamiltonian estimator errors

033251-4 HAMILTONIAN ASSIGNMENT FOR OPEN QUANTUM SYSTEMS PHYSICAL REVIEW RESEARCH 2, 033251 (2020) are smaller than their positive semidefinite counterparts. We measurement infidelity is ∼1%, meaning that either these attribute this counterintuitive result to the finite-difference ap- rates must fall by around two orders of magnitude or the as- proximation error. The least-squares minimization is afforded signment must be improved with other sources of information. greater freedom in generating a nonphysical environment, It is also of great interest to determine how such methods which partially absorbs the finite-time error, thus estimating can be applied to locally interacting non-Markovian envi- Hamiltonians more accurately. ronments and to better understand the decoupling between Hamiltonian and environmental estimation. Lastly, from an VIII. CONCLUSION information theoretic sense, it is of interest to determine, given state resources, which sets of measurement configura- In this work we have studied the task of assigning a local tions saturate Cramer-Rao bounds for Hamiltonian parameter Hamiltonian to open quantum systems in a variety of settings. estimation. By restricting ourselves to a Lindbladian formulation, with a polynomial number of model parameters describing the evolution, we are able to generalize and implement previous ACKNOWLEDGMENTS local model estimation techniques in both a steady state and dynamical setting. We thank A. Seif, P. T. Bhattacharjee, and R. S. Ben- To validate our constructions, we have performed numer- nink for helpful conversations and acknowledge DOE ASCR ical simulations of open system Hamiltonian assignments funding under the Quantum Computing Application Teams in the context of an XX-interacting spin chain subject to program, FWP No. ERKJ347. This research used quantum thermal relaxation. Our results verify how in the clean limit computing system resources supported by the US Department and, contingent upon an appropriately chosen model space, of Energy, Office of Science, Office of Advanced Scientific Hamiltonians may be inferred both in and out of equilib- Computing Research program office. rium. Furthermore, we have bounded the effects of noise This manuscript has been authored by UT-Battelle, LLC, and discussed simple modifications to increase the estimator under Contract No. DE-AC0500OR22725 with the US De- accuracy in the dynamical context. partment of Energy. The United States Government retains Our work paves the way for the determination of many- and the publisher, by accepting the article for publication, ac- body Hamiltonians in open quantum systems. The extension knowledges that the United States Government retains a non- of our results to a greater diversity of open quantum sys- exclusive, paid-up, irrevocable, world-wide license to publish tems remains an open research direction. For example, it or reproduce the published form of this manuscript, or allow will be of great interest to understand how our current work others to do so, for the United States Government purposes. can be enhanced by incorporating existing techniques such The Department of Energy will provide public access to these as Bayesian learning [5], and quantum process identifica- results of federally sponsored research in accordance with the tion [16]. Note that for current superconducting devices, the DOE Public Access Plan.

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