An Application of Quantum Annealing Computing to Seismic Inversion

Alexandre M. Souza,1, ∗ Leonardo J.L. Cirto,1 Eldues O. Martins,2 N. L. Holanda,1 Itzhak

Roditi,1, 3 Maury D. Correia,2 Nahum Sa,1 Roberto S. Sarthour,1 and Ivan S. Oliveira1

1Centro Brasileiro de Pesquisas Fısicas – CBPF,Rua Dr. Xavier Sigaud 150, 22290-180 Rio de Janeiro, Brazil

2Centro de Pesquisa e Desenvolvimento Leopoldo Americo Miguez de Mello – CENPES/PETROBRAS,Av. Horacio Macedo 950, 21941-915 Rio de Janeiro, Brazil

3Institute for Theoretical Physics, ETH Zurich 8093, Switzerland

Quantum computing, along with quantum metrology and quantum communication, are disruptivetechnologies that promise, in the near future, to impact different sectors of academic research andindustry. Among the computational problems with great interest in science and industry are theinversion problems. These kinds of problems can be described as the process of determining the causeof an event from measurements of its effects. In this paper, we apply a recursive quantum algorithmto a D-Wave quantum annealer housed at the University of Southern California to recover a soundwave speed profile, typical in seismic inversion experiments. We compare the obtained results fromthe quantum computer to those derived from a classical computer. The accuracy achieved by thequantum computer is nearly the same as that of the classical computer.

I. INTRODUCTION

Seismic geophysics relies heavily on subsurface mod-eling based on the numerical analysis of data collectedin the field. The computational processing of a largeamount of data generated in a typical seismic experi-ment can take a large amount of time before a consistentsubsurface model is produced. Electromagnetic reservoirdata, like CSEM (Controlled Source Electromagnetic),petrophysical techniques, such as electrical resistivity andmagnetic resonance on multi-wells, and engineering op-timization problems like reservoir flux simulators, wellfield design and oil production maximization also need astrong computational apparatus for analysis.

On the other hand, in the past decade, there has beenmuch progress in the development of quantum computers:machines exploiting the laws of quantum mechanics tosolve hard computational problems faster than conven-tional computers. A concrete example of such progressis the so-called quantum supremacy, that has been re-cently demonstrated by Google using a specific purposequantum computer [1]. The Geoscience field and relatedindustries, such as the hydrocarbon industry, are strongcandidates to benefit from those advances brought byquantum computing.

Currently, different quantum technologies and compu-tational models are being advanced. Giant companies likeIBM, Google, and Intel are developing quantum comput-ers based on superconducting technologies. Other compa-nies are also putting considerable effort into building afully functional quantum computer based on Josephsonjunctions, such as the North American Rigetti, whereas,the also American IonQ and the Austrian AQT are work-ing on computers based on trapped ions. The Canadian

∗Electronic address: amsouza@cbpf.br

company D-Wave, leader in the computational modelknown as quantum annealing, is already trading quan-tum machines. Recent reviews and comparisons amongdifferent quantum computing platforms can be found inreferences [2–6].

In the field of Geoscience, recent works have used quan-tum annealers to hydrology inversion problems [7, 8]. Inthose works, it was shown that, although the size of theproblem that can be solved on a third-generation D-Wavequantum annealer is considered small for modern com-puters, they are larger than the problems solved withsimilar methodology with Intel’s third and fourth genera-tion chips. It is also important to mention that in seis-mic inversions, optimization techniques are widely used,but in general, they get stuck in local minima. Globalminimization techniques, such as simulated annealing,can be employed to find the global minimum. Previousnumerical works [9, 10] have indicated that it can beadvantageous to use quantum annealing to solve seismicproblems. However, the performance of the current avail-able quantum computers has not yet been tested in suchproblems. The potential applications of quantum com-puting in Geoscience has so far been largely unexploitedin the specialized literature.

In this work, we present a formulation of a seismic prob-lem as a binary optimization, and a small scale subsurfaceseismic problem is solved using the D-Wave quantum an-nealer housed at the Information Sciences Institute of theUniversity of Southern California (USC). We have evalu-ated the performance of the quantum computer comparingthe results obtained to those derived from a classical com-puter. Our analysis was focused on the accuracy. It wasfound that the accuracy achieved by the quantum com-puter is nearly the same as that of the classical computerfor the problem we have studied.

This paper is organized as follows. In Sec. II we intro-duce the basic idea of quantum annealing. In Sections IIIand IV, we present the formulation of a seismic inversionas a binary optimization problem and the methods used,

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respectively. In Sec V, the results obtained in the D-Wavequantum annealer are shown. In the last section, we drawthe conclusions.

II. QUANTUM ANNEALING

Currently, there are two main models of quantum com-puting being developed: the logic gate model, or circuitmodel, and the adiabatic model. The gate model is auniversal quantum computation model that is performedprogramming a step-by-step instruction build from basicbuilding blocks, known as quantum gates, similar to theclassical circuit model [11]. This model is exploited bycompanies such as IBM, Intel, Rigetti, IonQ, AQT, andGoogle. The adiabatic computation model [12, 13] is amodel in which the computational problem is mappedinto a Hamiltonian Hfinal, in such a way that the solutionof the computational problem is encoded in the groundstate of a quantum system represented by Hfinal. Thecomputation is performed starting from the ground stateof a known Hamiltonian Hinit. The Hamiltonian is slowlymodified towards to the target Hamiltonian Hfinal:

H(p) = (1− p)Hinit + pHfinal, (1)

for p ∈ [0, 1]. If the evolution is performed adiabati-cally, the quantum state of the system remains in theinstantaneous ground state throughout the entire process.

The adiabatic model is equivalent to the gate model[14], i.e. any problem that can be computed in the gatemodel will also be computable in the adiabatic model.This statement is valid for certain types of k−local Hamil-tonians [14]. For example, considering a system of nqubits, we can perform universal adiabatic computationby choosing [15]:

Hinit =∑i

∆iσxi , (2)

Hfinal =∑i

Hiσzi +

∑j>i

Ji,jσzi σ

zj +

∑j>i

Ki,jσxi σ

xj (3)

where σki , with k = x, y, z, is one of the Pauli matrices

of the ith qubit, ∆i and Hi are local transverse andlongitudinal fields, respectively, and Ji,j and Ki,j arecoupling constants.

The adiabatic model can be viewed as a special case ofthe quantum annealing computing. In quantum anneal-ers, the Hamiltonian change is not adiabatic. Therefore,quantum annealing is a heuristic type of computation.The D-Wave computer is a quantum annealer that usesIsing chains:

Hfinal =∑i

Hiσzi +

∑j>i

Ji,jσzi σ

zj . (4)

The quantum annealing performed with Ising chains is un-likely to implement universal quantum computation [16].Therefore, D-Wave annealers are more restrictive than

the universal adiabatic model. D-Wave computers arespecialized at solving a class of computational problemsknown as NP-hard problems [17]. Here, it is also impor-tant to mention that the advantage of quantum annealersover classical methods is still under debate [18–28].

To solve a problem in the D-Wave quantum annealer [6],it is necessary first to express the problem to be solved asan Ising problem or as a quadratic unconstrained binaryoptimization (QUBO), which is equivalent to Ising but de-fined on the binary values 0, 1, whereas the Ising problemis defined on the binary values ±1/2. The QUBO prob-lem can be written as the minimization of the quadraticfunction

f(q) = qTQq q ∈ {0, 1}n (5)

where Q is a n×n upper (lower) triangular matrix and thevector qT = (q1, q2, · · · , qn) contains n binary variables.QUBO problems are commonly used in machine learn-ing and many important computational problems can betranslated to a QUBO formulation as well [29]. Amongthe examples of problems that have been addressed witha D-Wave quantum annealer we have the classification ofhuman cancer types [30], traffic optimization [31], tran-scription factor DNA binding [24], metamaterial designing[32] and Higgs boson data analysis [33].

Recently, there has been a growing interest in quantumalgorithms that solve systems of linear equations, Ax = b,where A is a n × m matrix and b is a vector. Suchalgorithms may find applications in different researchareas, including Geoscience. In the quantum gate model,the quantum version of such problem is called quantumlinear systems problem (QLSP) [34], and it is defined asthe problem of preparing the state

|ψ〉 =

∑ni |xi〉√∑ni |xi|2

, (6)

where x = (x1, x2, · · · )T, which is the solution of Ax = b.In 2008, Harrow, Hassidim, and Lloyd (HHL) pro-

posed a quantum algorithm for the QLSP problem [35].Given some assumptions [36], the run time of HHL isO(k2s2 log(n)/ε) where k is the condition number of ma-trix A, defined as the absolute value of the ratio betweenthe largest and smallest eigenvalues of A, s is the sparsityof A, defined as the number of nonzero entries per row,and ε is the desired precision.

After the initial HHL proposal, several improvementswere achieved: the condition number dependence was re-duced from k2 to k log3(k) [37], the error dependency wasreduced from 1/ε to a polynomial function in log(1/ε) [38]and a sparsity-independent runtime scaling was achievedin [39]. The QLSP problem can also be solved in runtimesO(k log(k)/ε) and O(k2 log(k)/ε) using the evolution ran-domization method, a simple variant of adiabatic quantumcomputing where the parameter p in 1 varies discretely,rather than continuously [40]. The best general-purposeclassical conjugate gradient algorithm to solve Ax = b

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Figure 1: Schematics of seismic data acquisition. Sound tracescreated in the ith source Si are detected at Di after the timeof travel ti. This figure depicts a simplified scheme with onlym = 3 layers, but we have considered up to m = 24.

has the runtime O(nks log(1/ε)). Here, we must empha-size a fundamental difference between the classical andquantum algorithms. The quantum algorithms do notsolve a system of linear equations, instead, they approx-imately prepare a quantum superposition that containsall the components xi of the solution vector x, while theconjugate gradient returns the solution vector x. Still,the quantum algorithms can be useful as subroutines indifferent applications [36].

In quantum annealers, the problems of solving a systemof linear equations and a system of polynomial equationswere previously studied in [41–43] and [44], respectively.Unlike the previously mentioned quantum algorithms,quantum annealing solves Ax = b completely, i.e. itreturns the vector x. To compare the performances ofa quantum annealer and a classical computer, we musttake into account the procedure to prepare the problem,i.e. to map Ax = b into a QUBO or Ising problem, toperform the annealing and post-process the results. Theperformance of quantum annealers to solve linear systemswas studied in detail in reference [42]. It was shown thatquantum annealers might be competitive if there exists apost-processing method that is polynomial in the size ofthe Matrix A with a degree less then 3.

III. SEISMIC INVERSION WRITTEN AS AQUBO PROBLEM

We considered the propagation of sound waves in amulti-layered medium, as shown in Figure 1. Multiplesources produce sound waves that can be reflected inthe interface of each layer. Assuming that the wavepropagation can be modeled as narrow beams or rays, thesound trace originated in the ith source reaches the ith

detector after the time interval

ti = 2

i∑j=1

dijvj ,

(7)

where dij and vj are the distance traveled by the soundwaves and the sound speed in the jth layer, respectively.If we consider the thickness of each layer as hj and the dis-

tance between two consecutive sources (detectors) as ∆i,we can write dij = hj/ cos θj where θj = arctan(∆j/hj).

The layered model described above is commonly usedin seismic explorations, either offshore or onshore [45]. Inseismic experiments, the goal is to determine the velocities{vj} from time intervals {ti} by solving the system

Ms = t (8)

where tT = (t1, t2, · · · , tm), sT = (1/v1, 1/v2 · · · , 1/vm)is the slowness vector, M is a m × m lower triangularmatrix with nonzero elements given by M.,j = 2hj/ cos θj ,and m is the number of layers.

In order to use a quantum annealer to solve the aboveseismic problem, it is necessary to translate the probleminto a QUBO formulation. To proceed, first, we rewritethe system 8 as a minimization problem with the objectivefunction

f(s) = ||Ms− t||2 s ∈ Rm. (9)

Next we write the slowness vector as s = s0 + L(x− I),where L defines the bounding limits of s, 0 ≤ xi < 2 ∀ xi,IT = (1, 1, · · · ) and the vector s0 is an initial guess for s.The objective function f(s) is rewritten as

f(x) = ||Mx− b||2 x ∈ Rm, (10)

where b = (t + LMI−Ms0)/L. The matrix M and thevector b are parameters of the objective function while thevector x must be converted into a binary format. Here wediscretized each xi variable with the R-bit approximation

xi =

R−1∑r=0

qi,r2−r. (11)

To formulate our QUBO problem, we construct a newbinary vector q and a new real matrix A in order to formthe binary system of equations Aq = b. It is straightfor-ward to reformulate this system as a binary optimizationproblem [41–43], whose solution vector, q∗, is given by

q∗ = arg minq∈{0,1}R×m

||Aq− b||2 (12)

= arg minq∈{0,1}R×m

qTQq + C. (13)

where

Qii =

m∑k=1

AkiAki − 2Akibk, (14)

Qij =

m∑k=1

2AkiAkj , for i < j, (15)

and C =∑m

k=1 b2k is an additive constant that does not

change the ground state.The precision of the solution depends on how many

binary digits are used to represent the real variables of

4

the problem. In this way a solution with good precisionwould consume a lot of qubits of the quantum hardware.Here we have used a recursive approach similar to whatwas used in [41](see algorithm 1) that can improve thesolution after each iteration using just a few qubits. Next,we will show in our example that using R = 2 and carrying10 iterations is sufficient to reach a good solution.

Algorithm 1 Function to solve the system of equationsMs = t. The vector s0 is the initial guess for s, ε is thetolerance, Nmax is the maximum number of iterations,and L defines the interval where we expect to find the

solutions (see text).

1: function Solve(M, t, s0, L, Nmax,ε)2: I← column vector← [1; 1; 1; · · · ; 1]3: for c← 1 to Nmax do4: b← (t + LMI−Ms0)/L5: Construct the vetor q and the Matrix A6: Convert ||Aq− b||2 to QUBO qTQq7: q∗ ← arg minqTQq8: From q∗ recover x9: s← s0 + L(x− I)

10: if ||s− s0||2 ≤ ε then11: Stop Loop12: else13: s0 ← s14: L← L/215: end if16: end for17: return s18: end function

IV. METHODS

In current versions of the D-Wave quantum annealer,the qubits are connected by a graph called Chimera [46].In this topology, each qubit is coupled to no more than 6other qubits. We can call Chimera as a graph of degree6. A QUBO problem is represented by a graph, whereeach vertex of the graph corresponds to a binary variableqi. When the QUBO problem is represented by a graphwith degree greater than 6, it is necessary to embed theQUBO graph onto the chip topology (see Figure 2). Thepresent seismic problem, for example, is represented bya full graph, where each vertex is connected to all othervertices. In this work, the embeddings were obtainedusing a heuristic algorithm [47] provided by D-Wave.

In our implementation, analog errors are importantissues, i.e. inaccurate implementations of the parametersHi and Ji,j by the quantum hardware. To improve re-sults by reducing the impact of analog errors, we haveused 10 spin-reversal transforms, also known as gaugetransformations. This type of transformation is based onthe fact that the structure of the Ising problem is notaffected when the following transformations are applied:Hi → giHi and Ji,j → gigjJi,j , where gi = ±1. Theoriginal and transformed problems have identical energies.

However, the sample statistics are affected by the spin-reversal transform because the quantum Hardware is aphysical object subject to errors. It is also important tonote that in the real hardware, not all qubits are func-tioning, as shown in figure 2. Therefore, we have useda type of D-Wave’s solver known as Virtual Full-YieldChimera (VFYC) that uses classical post-processing tocompensate for the inoperative qubits.

We have acquired 1000 samples for each run, the an-nealing time was the default value of 20µs. Here theoutputs were post-processed with the Multi-qubit Cor-rection Algorithm (MQC) [48] implemented in MATLAB.MQC is a post-processing method proposed to find alower energy solution to a QUBO problem, from a set ofD-Wave output samples. The method transforms a set ofsamples into a single sample that is at least as good asthe best output D-Wave sample. The method scaling isO(n2 logm), where n is the number of qubits used andm is that of annealing samples [8, 48].

V. RESULTS

We have applied the above formulation to an under-water seismic inversion problem. Artificial data weregenerated by simulating sound traces in the ocean. Wehave used a typical sound speed profile found in oceans,as shown in Figure 3. From the simulation, we obtainedthe travel times between the sources and detectors. Theseismic inversion model was constructed using 24 layerswhile the real variables were digitalized with R = 2 bits.This model yields 48 binary variables, for the present typeof problem, it is the maximum number of variables thatwe can minimize in the D-Wave quantum annealer hostedat USC.

To perform the seismic inversion we have considered twodifferent situations. In the first case, we have consideredall layers in the model with the same thickness, hj = 83m(Figure 4), and in the second case, the thickness of thelayers varies with the depth of the ocean (Figure 5). Whilein both cases the shape of the speed profile was wellreproduced by the quantum computer, better results werefound in the second case. The solutions are obtained withless than 10 iterations, as shown in the Figure 6.

We also performed the inversions with a classical com-puter running the forward substitution algorithm to invertthe lower triangular matrix in system 8. When classicaland quantum inversions are compared, we found thatthe relative error between them was ≈ 10−3. This resultshows that the quantum computer using the recursiveapproach to solve a system of linear equations has enoughcontrol to find solutions with good precision. The discrep-ancy between the seismic inversions and the real speedprofile is due to the choice of the model’s parametersrather than the quantum computation itself.

The results should also be compared to the benchmarkprovided by the condition number of the numerical prob-lem at hand. When performing a numerical inversion

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S1 D1S2 D2S3 D32∆1 ∆2 ∆3

Seismic Model

Graph Representation

Figure 2: Minor embedding. In the left panel, we show a full graph representing the QUBO problem associated with theseismic inversion. For clarity, it is shown a problem with 12 binary variables. In the middle panel, we show the chimera topologyof the D-Wave quantum annealer housed at USC. The missing vertices correspond to inoperative qubits. In the right panel, weshow the seismic inversion problem with 48 binary variables embedded onto the quantum hardware. We use a full chimeratopology and post-processing to compensate the inoperative qubits (see text).

Position (km)Speed (km/s)

Dep

th (k

m)

S1S6 S5 S4 S3 S2 D1 D6D5D4D3D2

Figure 3: Acoustic sound traces. In the left panel, we show a typical sound speed profile found in oceans. In the right panel weshow the results of numerical simulations of sound traces with incident angle θ0 = 88o and all 24 sources (receptors) equallyspaced (∆i = 800m ∀i). For clarity, we only show the simulations involving 6 sources (receptors). The color bar indicates thesound’s velocity.

procedure on the lower triangular matrix M to recoverthe solution s in Equation 8, a straightforward estimationof the lower bound of the relative error εs in s arisingfrom the relative error εt in the righthand side vector t

due to the numerical conditioning of M is given by

εs = κ∞εt (16)

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Speed (Km/s) Speed (Km/s) Error (x 10-3)

Dep

th (

Km)

Figure 4: Seismic inversion considering a model containing 24layers, each one with thickness hj = 83m. In the left panel, wecompare the ocean sound speed profile (black solid line) andthe profile recovered by the quantum computer (blue dashedline). In the middle panel, we show the results obtained ina classical computer (red dashed line). In the right panel,we show the relative error between quantum and classicalinversions.

where κ∞ is the condition number with norm ||.||∞,

κ∞ ≥maxj(hj/ cos θj)

minj(hj/ cos θj). (17)

In this particular application of seismic inversion, wheredistances are generally in the scale of 103m, it is reason-able to assume that the figures are accurate up to theorder of ≈ 10m, with the next order of magnitude (1m)giving the scale of the error εt. Thus in this problem wehave εt ≈ 10−3 and, given that the condition number ofM is κ ≈ 1, equation 16 becomes

εs ≈ εt ≈ 10−3. (18)

Therefore the lower bound of the estimated relative er-ror in the solution originating from numerical inversion ina realistic scenario is of the same order of the discrepancybetween the recovered solutions from the quantum andclassical algorithms obtained here. That this lower boundcan be achieved using only a small number of qubits is atestimonial to the adequancy of the quantum approach.

VI. CONCLUSIONS

The quantum computer represents a fundamentallydifferent paradigm, an entirely different way to performcalculations. The Geoscience field and related industriesare strong candidates to benefit from quantum comput-ing. However, the performance of Geoscience inversionproblems on current available quantum computers has sofar been largely unexploited.

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Figure 5: Seismic inversion considering a model containing24 layers, with h1 = 20m, h2 = 100m, h3 = 145m, h4 = 145mand hj = 145m − 7(j − 5)m from j = 5 to j = 24. In theleft panel, we compare the ocean sound speed profile (blacksolid line) and the profile recovered by the quantum computer(blue dashed line). In the middle panel, we show the resultsobtained in a classical computer (red dashed line). In theright we panel show the relative error between quantum andclassical inversions.

In this paper, we have implemented a seismic inversionproblem in a D-Wave quantum annealer. The seismicproblem was written as a system of linear equations andthen translated into a QUBO formulation. The resultspresented here indicate that the current available quantumannealers can solve a seismic inversion at a relativelysmall size with nearly the same accuracy as a classicalcomputer. The proof-of-principle computations performedhere show some promise for the use of quantum annealingin Geosciences.

We believe that the development of hybrid quantum-classical methods will be essential to implement real seis-mic problems on quantum annealers in the near future.Promising methods are those that can embed large-scaleproblems on the D-Wave [49]. Another interesting ap-proach is the reverse annealing. Within this method, onestarts from known local solutions which can be obtainedin a classical computer. The annealing is performed back-ward from the known classical state to a state of quantumsuperposition, then proceeding forward it is possible toreach a new classical state that is a better solution thanthe initial one. With recursive applications of reverseannealing, we can refine local solutions [50–53].

Finally, we should mention that the problem solved hereis well-conditioned, however, often Geoscience problemsare ill-conditioned. An interesting question is whetherquantum computers can solve ill-conditioned problemsefficiently. In reference [54], it was theoretically proposedthat preconditioning methods can expand the numberof linear systems problems that can achieve exponentialspeedup over classical computers. In future works, weplan to study the performance of the quantum anneal-

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th (K

m)

Initial Guess 1st Iteration 4th Iteration 8th Iteration 10th Iteration

Figure 6: Seismic inversion obtained in the quantum annealer for different iterations. The seismic model used is the same as inthe Figure 5.

ers to solve ill-conditioned problems. Another attractiveprospect for future work is the implementation of Geo-science problems in gate model computers, based eitheron superconducting or trapped ions technologies.

VII. ACKNOWLEDGMENTS

This work was supported by the Brazilian NationalInstitute of Science and Technology for Quantum Infor-mation (INCT-IQ) Grant No. 465469/2014-0, the Co-

ordenao de Aperfeioamento de Pessoal de Nvel Supe-rior - Brasil (CAPES) - Finance Code 001, Conselho Na-cional de Desenvolvimento Cientfico e Tecnolgico (CNPq)and PETROBRAS: Projects 2017/00486-1, 2018/00233-9and 2019/00062-2. A. M. S. acknowledges support fromFAPERJ (Grant No. 203.166/2017). I.S.O acknowledgesFAPERJ (Grant No. 202.518/2019). We are in debt withProf. Daniel Lidar and the Information Sciences Instituteof the University of Southern California, for giving usaccess to the D-Wave Quantum Annealer.

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