Quantum Annealing for Large MIMO Downlink Vector Perturbation Precoding
Srikar Kasi, Abhishek Kumar Singh, Davide Venturelli, Kyle Jamieson
AAccepted article to appear in the proceedings of IEEE ICC ’21
Quantum Annealing for Large MIMO DownlinkVector Perturbation Precoding
Srikar Kasi (cid:63)
Student Member , IEEE
Princeton University
Abhishek Kumar Singh (cid:63)
Student Member , IEEE
Princeton University
Davide Venturelli
Universities SpaceResearch Association
Kyle Jamieson
Senior Member , IEEE
Princeton University
Abstract —In a multi-user system with multiple antennas atthe base station, precoding techniques in the downlink broadcastchannel allow users to detect their respective data in a non-cooperative manner. Vector Perturbation Precoding (VPP) is anon-linear variant of transmit-side channel inversion that per-turbs user data to achieve full diversity order. While promising,finding an optimal perturbation in VPP is known to be anNP-hard problem, demanding heavy computational support atthe base station and limiting the feasibility of the approach tosmall MIMO systems. This work proposes a radically differentprocessing architecture for the downlink VPP problem, one basedon Quantum Annealing (QA), to enable the applicability of VPPto large MIMO systems. Our design reduces VPP to a quadraticpolynomial form amenable to QA, then refines the problemcoefficients to mitigate the adverse effects of QA hardware noise.We evaluate our proposed QA based VPP (QAVP) technique ona real Quantum Annealing device over a variety of design andmachine parameter settings. With existing hardware, QAVP canachieve a BER of − with 100 µ s compute time, for a 6 × Index Terms —Vector Perturbation, Downlink Precoding,Quantum Computation, Quantum Annealing, Optimization
I. I
NTRODUCTION
Modern wireless networks are experiencing tremendousgrowth in traffic loads at base stations, and hence to meet theresulting computational and latency requirements, designerscontinue to investigate new architectures and hardware fortoday’s 5G and tomorrow’s 6G networks. A large componentof cellular baseband processing comprises of downlink datatraffic due to a significant rise in the popularity and usage ofvideo streaming platforms ( e.g.,
Netflix). To meet the ever-growing user demand, it is critical for the base stations toenhance the quality of downlink data streams in terms ofthroughput, error rate, and latency.In a multi-user multiple-input multiple-output (MIMO)downlink data transmission, precoding techniques can be usedto eliminate the effect of inter-user interference and allow usersto detect their respective data non-cooperatively, minimizingerror-rate and maximizing throughput. In this work, we focuson
Vector Perturbation Precoding (VPP) [1]. VPP is a widely (cid:63)
Co-primary authors, appearing in randomly chosen order.©2021 IEEE. Personal use of this material is permitted. Permission fromIEEE must be obtained for all other uses, in any current or future media,including reprinting/republishing this material for advertising or promotionalpurposes, creating new collective works, for resale or redistribution to serversor lists, or reuse of any copyrighted component of this work in other works. studied non-linear precoding technique that performs transmit-side channel inversion over a perturbed user data vectorto reduce the transmit power scaling. Although VPP hasbeen shown to achieve better error performance comparedto other precoding techniques ( e.g., zero-forcing, Tomlinson-Harashima Precoding [2]), finding an optimal perturbation foruser data in VPP is known to be NP-hard, making its imple-mentation in massive/large MIMO systems to be infeasible.A promising and cost-effective architecture to address theincreased computational burden of wireless networks is a
Centralized Radio Access Network (C-RAN) [3] architecture,which aggregates the computationally-demanding processingat many wireless base stations. Since baseband physical-layerprocessing is highly time-critical and the required streamof baseband signal samples has a high data-rate, this typeof C-RAN deployment imposes both latency and bandwidthrequirements on the interconnect between each base stationand the data center. Related studies to this end are investigatingquantum computation for solving networking problems in theuplink: such as Channel coding [4] and ML detection [5].However, investigating quantum computation for problems inthe downlink remains unexplored to the best of our knowledge.In this paper, we propose a radically different processingto the NP-hard VPP problem in the downlink:
QuantumAnnealing based Vector Perturbation Precoding (QAVP) . Ourproposed technique leverages recent advances in quantumcomputational devices and applies them to the problem ofVPP. QAVP’s approach is to represent the VPP problem as anoptimization problem over a quadratic polynomial with binaryvariables i.e. , Quadratic Unconstrained Binary Optimization (QUBO), which is an optimization form that a
QuantumAnnealer (QA) machine takes as input, then refine the QUBOcoefficients to mitigate the adverse effects of QA hardwarenoise and the process of mapping the polynomial onto thephysical QA qubit hardware topology. After these prepro-cessing and embedding steps, QAVP uses a real QuantumAnnealing machine to solve the resultant QUBO problemand then constructs the VPP perturbation from the solutionsreturned by the QA machine. Our results show that, from thestandpoint of computation time, QAVP can outperform popularencoding algorithms for large MIMO systems. a r X i v : . [ c s . N I] F e b ccepted article to appear in the proceedings of IEEE ICC ’21II. V ECTOR P ERTURBATION P RECODING
We consider a general MIMO downlink scenario where abase station equipped with N t transmit antennas communicatedata streams with N r ( ≤ N t ) single-antenna non-cooperativeusers independently and simultaneously.In VPP, the user data symbol vector u ∈ C N r × is perturbedby an integer vector v ∈ G N r × . This maps the data symbolsto a wider constellation space, forming a perturbed transmitvector, d = u + τ v . Here, v is a vector of Gaussian integers and τ = 2( | c max | + ∆ /2) is a constant chosen to providesymmetric decoding regions around the constellation points. | c max | is the magnitude of the largest constellation symboland ∆ is the spacing between the constellation symbols [6].The perturbed vector d is then precoded with a precodermatrix ( P ∈ C N t × N r ), where the choice of P inverts thewireless channel H ∈ C N r × N t and reduces the effect of widerange of eigenvalues of the channel coefficients [6]. Theprecoder matrix P = H H ( HH H ) − is used in Zero Forcing(ZF) precoding. The received symbol vector y correspondingto the transmitted symbol vector x = Pd / √ P t is given by, y = √ P t ( HPd ) + n (1)where the scalar P t = (cid:107) Pd (cid:107) is the transmission powerscaling factor, which is assumed to be known at the receiver,and n is wireless channel noise. The receiver decodes byapplying a modulo τ operation to the received signal y . Thetransmission power scaling ( P t ) causes noise amplificationduring the decoding process. In order to minimize the trans-mission power scaling, an optimal choice of the perturbationvector ( v (cid:63) ) that leads to the smallest P t is computed as in [6]: v (cid:63) = arg min v (cid:13)(cid:13)(cid:13) H H ( HH H ) − ( u + τ v ) (cid:13)(cid:13)(cid:13) . (2)III. R ELATED W ORK
The Sphere encoder [6] builds on the Fincke and Pohst al-gorithm [7] by expressing the precoding matrix P as P = QR by QR decomposition, where Q is unitary and R is uppertriangular. It then performs a tree search, utilizing the uppertriangular structure and limiting the search to the points withina hyper-sphere of a suitably chosen radius, and hence avoidsexhaustive search over all possibilities. This algorithm is verysimilar to the Sphere decoder used for the MIMO receiver.While Sphere encoder is much better than exhaustive search,its expected complexity is still exponential. Park et al. [8]provide an approximation to the VPP problem by minimizingthe real and imaginary parts of the VPP cost function sep-arately. This reduces the complexity of VPP computation atthe cost of error performance. The Thresholded Sphere Searchalgorithm [9] imposes an additional stopping criteria to Sphereencoder algorithm [6] based on an SNR dependent thresholdheuristic. Another scheme for reducing the search space ofSphere encoder is discussed in [10]. It restricts the values of The set of Gaussian integers G = Z + j Z consists of all complex numberswhose real and imaginary parts are integers. each perturbation to four possible values (two for real andtwo for imaginary) and hence reducing the search space of theSphere encoder. Despite improvements in computation costsover Sphere encoder, the search space of all these methods isstill exponential with respect to the MIMO size.Several approximations for VPP (with polynomial com-plexity) exist in literature. The Fixed Complexity SphereEncoder (FSE), adapts the Sphere encoder to have a lowerand fixed polynomial complexity by pruning a large numberof branches during the tree search but leads to degradation inerror performance [11]. Degree-2 Sparse Vector Perturbation(D2VP), presented in [12], is a low complexity algorithm forvector perturbation precoding. It reduces the complexity offinding the VPP solution by assuming that only 2 elements ofthe perturbation vector can be non-zero and then improves thesolution over multiple iterations.IV. Q UANTUM A NNEALING
Quantum Annealing (QA) is a heuristic approach that im-plements in hardware a quantum computing algorithm inspiredby the Adiabatic Theorem of quantum mechanics [13]. Themethod aims to find the lowest energy spin configuration (solution) of the class of quadratic unconstrained binaryoptimization (QUBO) problems in their equivalent
Ising spec-ification. The Ising form is described by: E = ∑ i h i s i + ∑ i < j J ij s i s j (3)where E is the energy , s i is a solution variable, h i and J ij are problem parameters called bias and coupler strength respectively. QUBO form is obtained from Eq.3 by a simplevariable transformation ( s i −→ q i − ), where s i and q i represent Ising and QUBO form variables respectively. Isingform solution variables ( s i ) take on values in {− + } , andQUBO form solution variables ( q i ) take on values in {
0, 1 } .The equivalent QUBO form of Eq. 3 is described by: E = ∑ i f i q i + ∑ i < j g ij q i q j (4) A. Primer: Quantum Annealers.
Quantum Annealers are specialized quantum computers, es-sentially comprised of two types of resources: qubits (quantumbits) and couplers , whose regular connectivity structure isorganized in unit cells . Fig. 1 shows the hardware structure ofthe QA we adapt in this study, the D-Wave 2000Q (DW2Q)quantum annealing machine. The QA device programs thelinear ( i.e., biases) and quadratic ( i.e., coupler strengths)coefficients of Eq.3 onto qubits and couplers using controllableinductive elements in proximity of superconducting JosephsonJunctions present on the chip [4].
Embedding.
The process of mapping the problem at handonto the physical QA hardware topology follows the graph-theory problem of graph minor embedding . We demonstratehere the embedding process with an example Ising problem: E = J s s + J s s + J s s (5)ccepted article to appear in the proceedings of IEEE ICC ’21 Figure 1:
The figure shows qubits (nodes) and couplers (edges) inDW2Q QA hardware. Each group of eight qubits is called unit cell . The direct connectivity of this example problem is shown inFig.2 (a) . However a three-node complete connectivity graphstructure does not exist in the QA hardware ( cf.
Fig.1).Hence the minor-embedding approach is to map one of theproblem variables in Fig.2 (a) ( e.g., s ) onto two physicalqubits ( e.g., s a and s b ) as Fig.2 (b) shows, such that theresulting connectivity can be realized on the QA hardware.The couplers between the qubits s a and s b are tasked toenforce these physical qubits to be correlated in order to end upwith the same value at the end of the annealing process. Thisis implemented through a strong ferromagnetic interactionenergy called chain strength J F (see in Fig.2 (b) ). Coefficient considerations.
Today’s QA devices providesupport for bias values in [ − + ] and coupler strength valuesin [ − + ] with a bit-precision guaranteed to 4–5 bits only.Further, the QA introduces an analog machine noise distinctfrom communication channel noise called intrinsic controlerrors or ICE . ICE noise, a collection of errors caused by qubitflux noise, susceptibility, among others [14], essentially altersthe problem coefficients ( h i → h i ± δ h i , J ij → J ij ± δ J ij ).Although the errors δ h i and δ J ij are currently on the orderof − , these may degrade the solution quality of someproblems in scenarios where ICE noise erases significantinformation from the ground state of the input problem. B. Annealing process
QA processors simulate systems in the transverse fieldIsing model described by the time-dependent energy functional(Hamiltonian): H = − A ( s ) ∑ i σ xi + B ( s ) (cid:110) ∑ i h i σ zi + ∑ i < j J ij σ zi σ zj (cid:111) (6)where σ x , zi are spin operators (Pauli matrices) acting on the i th qubit, h i and J ij are problem parameters, s (= t / t a ) is called annealing schedule where t is the time and t a is the annealingtime . A ( s ) and B ( s ) are two monotonic scaling signals inthe annealer such that at time t = , A ( ) (cid:29) B ( ) ≈ and at time t = t a , B ( ) (cid:29) A ( ) ≈ . The annealingalgorithm initializes the system in the ground state of ∑ i σ xi where each qubit is in a superposition state √ ( | (cid:105) + | (cid:105) ) ,then adiabatically evolves this Hamiltonian from time t = s s s s s s s Embedding JFJ12J23J13J12 J13J23 (a)
Problem Connectivity (b)
Physical QA Connectivity
Figure 2:
The figure demonstrates the embedding process of Eq. 5.In the figure, (a) shows the direct problem connectivity of Eq. 5 and (b) shows its physical connectivity on QA hardware. until t = t a ( i.e., decreasing A(s) , increasing
B(s) ). The time-dependent evolution described by the Schroedinger Equationdriven by these signals A and B is essentially the annealingalgorithm. While in real processor the dynamics is not idealbut it is dominated by dissipative noise instead [15], theexpectations of the model still hold for the most part. Theprocess of optimizing a problem in the QA is called an anneal ,while the time taken for an anneal is called annealing time .V. D ESIGN
Quantum Annealing based Vector Perturbation (QAVP) en-visions a scenario where a Quantum Annealer (QA) machineis co-located with a centralized data center for computationalprocessing in the C-RAN architecture [3] which allows forlow latency communication between base stations and the QAmachine. QAVP converts a VPP problem to a Quadratic Un-constrained Binary Optimization (QUBO) problem, then fine-tunes the QUBO problem coefficients to mitigate the adverseeffects of QA hardware noise and increase the probability offinding the correct solution with QA. The fine-tuned QUBO isnext embedded onto the physical QA hardware (see Fig.1) forrunning the problem. We conduct multiple anneals for a givenQUBO problem, where each anneal generates a candidatesolution bit-string for VPP. The QUBO solutions are convertedto perturbation vectors by inverting the transform describedin Eq. 7. Among these returned solutions, the solution thatminimizes the VPP objective function is chosen by the basestation. In scenarios where QAVP fails to find a solutionwith lower transmit power scaling than Zero-Forcing (ZF)precoding, the base station discards the QAVP solution anduses ZF precoding instead of VPP.The steps involved in QAVP computation, excluding embed-ding and annealing, can be executed by the physical layer ofbase station or can be offloaded to a server in close proximityto QA machine. Fig. 3 illustrates a typical deployment scenariofor QAVP, where N r users are receiving downlink data streamsfrom a base station with N t antennas. We next demonstrateQAVP’s QUBO formulation for the VPP problem, QAVP’spre-processing, and embedding considerations. A. QAVP’s QUBO Formulation
The downlink VPP problem is to find an optimal perturba-tion vector v (cid:63) that minimizes the transmit power at the baseccepted article to appear in the proceedings of IEEE ICC ’21 Figure 3:
A typical deployment scenario of QAVP. N r users arereceiving downlink data streams from a base station with N t antennas. station, whose NP-hard objective function is represented inEq. 2. As the search objective in Eq.2 is over the vector v ∈ G N r × , we construct each entry in v by a linear combinationof binary variables as follows.Let q i denote the i th QUBO form solution variable, and let a k + j b k be the k th entry in v , where a k and b k are integers.We design each a k and b k ∀ k in an identical fashion using t + distinct solution variables as: a k (cid:46) b k = t ∑ m = m − · q m − t ∗ q t + (7)Since all the solution variables ( q i ) are binary, this for-mulation allows a k and b k to take all integer values in therange [ − t , 2 t − ] , where each integer corresponds to aunique configuration of solution variables. The value of t determines the search range of v . Yuen et al. [16] showthat perturbation values are most likely to be {−
1, 0, 1 } , andhence t = 1 which allows perturbation values in the range [ −
2, 1 ] , is usually sufficient. We substitute the entries of v with their corresponding solution variables into Eq.2 to obtainthe QAVP’s QUBO: arg min ∀ q (cid:110) ∑ ∀ i f i ( H , u ) q i + ∑ ∀ i , j g ij ( H , u ) q i q j (cid:111) (8)where f and g are functions of MIMO channel matrix anduser data vector. The linear and quadratic coefficients obtainedfrom Eq. 8 can be programmed into the QA processor. Thecomputational complexities of the proposed QA techniqueand the optimal case Sphere Encoder are O ( e √ N u ) [17] and O ( e N u ) [18] respectively, where N u is the number of users.We next present our QUBO pre-processing considerations. B. QAVP’s Pre-processing
Our pre-processing scheme aims to mitigate the adverseeffects of QA hardware ICE noise (§IV) by scaling andeliminating minor QUBO coefficients. We note that a genericQUBO form of Eq.4 can be equivalently expressed as: E = ∑ i f i q i + ∑ i < j g ij q i q j = q T Qq (9)where q = [ q , q , ... ] T , and Q is an upper triangular matrixwith Q ij = f i ( i = j ), Q ij = g ij ( i < j ), Q ij = ( i > j ).Let Q max = max i , j | Q ij | be the maximum QUBO coefficient Figure 4:
QAVP’s pre-processing loss for different choices of T high and T low , showing that T high = is near-optimal and T low = − leads to negligible PPL. value, T high and T low be our chosen upper and lower boundsfor the QUBO coefficient values. We define a scaleFactor as: scaleFactor = Q max ≤ T high T high Q max Q max > T high , (10)Our approach is to first scale each entry in Q with the scale-Factor, then eliminate coefficients below a chosen thresholdof T low . This scaling process is summarized below. Step 1:
Set Q ← Q ∗ scaleFactor . Step 2:
For every i ≤ j , if Q ij < T low then set Q ij ← .Let Q pre correspond to the QUBO obtained after this pre-processing step. Let q (cid:63) pre = arg min q q T Q pre q , and q (cid:63) = arg min q q T Qq . We define a pre-processing loss (PPL) as:PPL = | ( q ∗ T pre Qq ∗ pre − q ∗ T Qq ∗ ) || ( q ∗ T Qq ∗ ) | (11)PPL is used to quantify deterioration in the optimal valueof the VPP problem due to the modification of the originalQUBO problem. We see from Fig. 4 that PPL increases withan increase in T low as a higher T low causes more coefficientsto be reduced to zero. We further observe that PPL reducesinitially with T high , and then becomes constant. While having ahigh negative value for T low leads to low PPL, it makes it moresusceptible to QA ICE noise. The optimal choices for T high and T low depend on the distribution of QUBO coefficients,magnitude of QA ICE noise, and problem embedding. Ourempirical studies find that a T high = and a T low = − obtains good solutions with the DW2Q QA. C. QAVP’s Embedding and Parallelism.
We use a D-Wave’s heuristic algorithm described inRef. [19] and implemented in D-Wave MinorMiner softwarelibraries for mapping of QAVP’s QUBO design onto the QAhardware. It is to note that multiple QUBOs or multipleinstances of a QUBO can be parallelly processed by mappingdifferent problems onto distinct physical locations in the QAccepted article to appear in the proceedings of IEEE ICC ’21
Figure 5:
Evaluation of the overall transmission power ( P t ) am-plitudes over various choices of | J F | at T a = µ s and 10 dBchannel SNR. The plot shows that QAVP’s performance variance isnot significant for J F values above J m . chip. For instance, QAVP for 7 × VALUATION
In this section, we first present microbenchmarks for QAmachine parameters. We then investigate QAVP’s end-to-end error performance, comparing head-to-head against ZeroForcing (ZF) and Fixed Complexity Sphere Encoder (FSE)algorithms. The DW2Q QA system performance is majorlyaffected by the choice of annealing time ( T a ), number ofanneals ( N a ), and chain strengths ( J F ). Recall that numberof anneals ( N a ) refers to the number of times annealingprocess is repeated for each QUBO problem. Our evaluationmethodology is as follows: we obtain the VPP solutionsfrom the QA (DW2Q), and use these solutions to simulatean end-to-end downlink data transmission on a trace-drivenMIMO simulator (implemented in MATLAB). We simulatedata transmission over Rayleigh fading wireless channels andempirically measure the BER corresponding to different users. A. Effect of Chain Strength ( J F ) We evaluate QAVP’s performance over an × MIMOsystem with 20 QAVP problem instances. Our channel matrix H is random Rayleigh fading, modulation scheme is BPSK,and wireless channel noise is Gaussian. Let J m be the maxi-mum coupler strength value of our QUBO problem at hand.In Fig. 5 we investigate the sensitivity of chain strengths | J F | over the overall transmission power P t . For this evaluation,we set a high annealing time of µ s and a moderateSNR of dB to ensure minimal disturbance from the timelimit and the channel SNR respectively. We observe that theperformance of QAVP varies with J F and that the sensitivity Figure 6:
Downlink multi-user × MIMO: BER for various QAVPanneal times T a . We compare error performance of QAVP for variousvalues of anneal time ( T a ) while keeping the number of annneals( N a = ) fixed. We set | J F | = J m . Figure 7:
Downlink multi-user × MIMO: Bit Error Rate vs TotalComputation Time ( N a × T a ). We demonstrate the BER performance,at two values of SNR, over different combinations of anneal time ( T a )and number of anneals ( N a ). We set | J F | = J m . of QAVP performance beyond | J F | = J m is not significant.The average P t is much higher for | J F | < J m , and thevariance in performance of QAVP is barely distinguishablefor higher values of | J F | > J m . While the optimal value of J F depends on the parameters of the system ( e.g., modulationand MIMO size), we empirically determine that | J F | = J m is the optimal setting for × and × MIMO systemswith 64 QAM modulation scheme.
B. Effect of Anneal Time ( T a ) Another critical QA parameter that affects the QAVP per-formance is the anneal time ( T a ), along with its associatednumber of anneals ( N a ). We heretofore quantify the QAperformance with the total compute time: T a × N a . Fig. 6compares the BER performance of FSE, ZF and QAVP, andFig. 7 reports the BER performance of QAVP at different QAcompute times, for a N t = , N r = MIMO system with 64QAM modulation, for various choices of T a and N a .We first observe in Fig. 6 that a higher anneal time leads tobetter BER performance from QA. Although a higher annealtime allows QA to return better quality solutions, it increasesthe communication latency (due to a higher computation time)as Fig. 7 depicts. The tradeoff between BER performanceand anneal time represents a fundamental design problem inQAVP. We observe from Figs. 6 and 7 that QAVP achieves anccepted article to appear in the proceedings of IEEE ICC ’21 Figure 8:
Downlink multi-user × MIMO: BER and throughputperformance. We compare the BER performance of QAVP againstFSE and ZF. We set | J F | = J m , T a = µ s and N a = . acceptable BER with a low anneal time of T a = µ s , whilekeeping the computation time lower by an order of magnitude. C. End-to-end BER performance
In this section, we consider a large × MIMO systemwith Rayleigh fading channel, White Gaussian noise, and 64QAM modulation. Fig. 8 reports results (with | J F | = J m , T a = µ s, N a = ) for mean BER and normalizedthroughput (ratio of achieved to maximum possible through-put), with packets size of 1500 bytes, for uncoded datatransmission. We observe from Fig. 8 that QAVP outperformsZF and FSE by achieving a 1–2 orders of magnitude lowerBER at E b / N of 27dB. We see from the throughput curvesthat QAVP achieves a 3 dB gain over FSE and around 6 dBgain over ZF. It is to note that a 12 ×
12 MIMO system withVPP, using theoretically optimal Sphere Encoder, is practicallyinfeasible due to its high computational complexity.VII. C
ONCLUSION
In this paper we propose QAVP, a novel technique thatperforms Vector Perturbation Precoding using Quantum An-nealing. We evaluate QAVP on a real DW2Q QA device over avariety of machine parameters. Our studies show that for largeMIMO systems, if the analysis is restricted to computationtime, QAVP can outperform practically feasible state-of-the-arttechniques like FSE and ZF algorithms for VPP. While priorwork investigates QA technology for problems in the uplink[4], [5], our studies investigate its potential in the downlink.Our analysis disregards the engineering and system integrationoverheads of currently available commercial QA systems (e.g.latency, programming time and thermalization times betweenthe runs), since they can be optimized heavily if the system isbuilt for a specific application deployment. Nevertheless, thetechniques we propose in this work, in the more distant future,may enable the applicability of VPP to large MIMO systemsby exploiting stronger QA devices and parallelism.
A. Future Work
Our work has several possible improvements along thelines of QUBO pre-processing, problem embedding, and QAmachine parameter selection. It is of interest to further refineour pre-processing step to optimize the QUBO representation based on the embedding of the problem. Further, a moresophisticated tuning and selection of QA parameters such aschain strengths and annealing times can potentially improvethe QAVP performance. Porting our model to other alternativeemerging Ising-model related technologies such as quantum-inspired algorithms, reverse quantum annealing, and quantum-classical hybrid approaches is also a potential work direction.A
CKNOWLEDGEMENTS
This research is supported by National Science Founda-tion (NSF) Awards CNS-1824357 and CNS-1824470. Supportfrom USRA Cycle 3 Research Oppurtunity Program allowedmachine time on a D-Wave Quantum Annealer machine hostedat NASA Ames Research Center.R
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