Koopman Operator Learning for Accelerating Quantum Optimization and Machine Learning
Authors: Di Luo
Jiayu Shen
Rumen Dangovski
Marin Soljacic
Venue: Submitted to ICLR 2023 (2023)
Keywords: Koopman operators
quantum optimization
machine learning

Abstract: Finding efficient optimization methods plays an important role for quantum optimization and quantum machine learning on near-term quantum computers. While backpropagation on classical computers is computationally efficient, obtaining gradients on quantum computers is not, because the computational complexity scales linearly with the number of parameters and measurements. In this paper, we connect Koopman operator theory, which has been successful in predicting nonlinear dynamics, with natural gradient methods in quantum optimization. We propose a data-driven approach using Koopman operator learning to accelerate quantum optimization and quantum machine learning. We develop two new families of methods: the sliding window dynamic mode decomposition (DMD) and the neural DMD for efficiently updating parameters on quantum computers. We show that our methods can predict gradient dynamics on quantum computers and accelerate the quantum variational eigensolver used in quantum optimization, as well as quantum machine learning. We further implement the learning algorithms on a real quantum computer and demonstrate their practical effectiveness. 

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KOOPMAN OPERATOR LEARNING FOR ACCELERATING QUANTUM OPTIMIZATION AND MACHINE LEARNING
Anonymous authors
Paper under double-blind review
ABSTRACT
Finding efficient optimization methods plays important role for quantum opti- mization and quantum machine learning on near-term quantum computers. While backpropagation classical computers is computationally efficient, obtaining gradients on quantum computers is not, because the computational complexity scales linearly with the number of parameters and measurements. In this paper, we connect Koopman operator theory, which has been successful in predicting nonlinear dynamics, with natural gradient methods in quantum optimization. We propose a data-driven approach using Koopman operator learning to accelerate quantum optimization and quantum machine learning. We develop two new fam- ilies of methods: the sliding window dynamic mode decomposition (DMD) and the neural DMD for efficiently updating parameters on quantum computers. We show that our methods can predict gradient dynamics on quantum computers and accelerate the variational quantum eigensolver used quantum optimization, as well as quantum machine learning. We further implement Koopman opera- tor learning algorithms on a real IBM quantum computer and demonstrate their practical effectiveness.
INTRODUCTION
There has been rapid development of quantum technologies and quantum computation in recent years. number of efforts are put into demonstrating quantum advantages and speedup. Quantum optimization (Moll et al., 2018) and quantum machine learning (QML) (Biamonte et al. 2017). as important applications of quantum technologies, have received increased interest. The Variational
Quantum Eigensolver ( (VQE) (Peruzzo et al., [014bb] (Tilyy et al. (2022). as quantum optimization algorithm, has been developed and applied understanding problems high energy physics (Klco)
et al., 2018 Rinaldi et al. (022), condensed matter physics (Wecker et al. 2015). and quantum chemistry (Peruzzo et al. 2014a). The Variational Quantum Algorithm (VQA) (Cerezo el al. 202T) such the Quantum Approximate Optimization Algorithm (QAOA) (Farhi al.) 2014 Harrigan | (202]) has been applied to the max-cut problem. A recent experiment on 289 qubits has demon-
cct al. strated powerful VQA application in classical optimization by benchmarking against variety of
classical algorithms (Ebadi et al.] (2022). QMI has also been developed for various tasks including
supervised learning (Havlíček et | 2019), unsupervised learning (Kerenidis al. (201)) and rein-
forcement learning (Dong et al. 2008). Theoretical advantages of quantum machine learning have
been investigated (Huang et 2022b Liu al. 202T 2022). and experiments on real quantum
computers have demonstrated encouraging progress (Huang et al. 2022a Rudolph et al. 2022).
In the noisy intermediate-scall quantum (NISQ) era (Preskill, 2018), due the noisy nature of current quantum computer architectures, hybrid classical
schemes have been proposed for quantum optimization and quantum machine learning and become a prominent approach. The key spirit of the hybrid approach relies on performing optimization and machine learning on parame- terized quantum circuits with quantum features while updating the parameters in the circuit is done through classical computers. In classical machine learning, backprogation only requires vector Ja- cobian calculations which share the same complexity as the forward evaluation. Obtaining gradients under the hybrid scheme is much more challenging. Calculating gradients in quantum computers challenging for two reasons: (1) gradient calculation typically scales linearly in the number of parameters O;paaams) and (2) the quantum nature of the gradient itself entails sampling over

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repeated measurements. Despite various research quantum optimization and quantum machine learning in simulation, the implementation of gradient- methods on real quantum computers is computationally inefficient which limits their applications in practice. It is important open prob- lem in the field to develop scalable and efficient optimization methods for quantum optimization applications and quantum machine learning tasks.
In this work, we propose Koopman operator learning for accelerating quantum optimization and QML. The Koopman operator theory is a powerful framework for understanding and predicting non- linear dynamics through linear dynamics embedded into a higher dimensional space (Mezic 1994)
Mezić & Banaszuk] 2004] Mezic 2005 Rowley et al. 2009 Brunton al.) (02I). By viewing parameter optimization on quantum computers as a nonlinear dynamical evolution in the parameter
the Koopman operator theory. In particular, the quantum natural gradient helps to provide a natural embedding of original parame- ters through quantum computtr parameterization into higher-dimennional space, related to linear imaginary-time evolution. We develop new Koopman operator learning algorithms for quantum op- timization and QML, including the sliding window dynamic mode decomposition (SW-DMD) and neural-netwook--ii based DMD that lears the Koopman embedding via neural network parameteri- zation. Our approach is data-driven and based on the information of only a few gradient steps such that the cost of prediction does not scale with paarams-
space, we connect gradient dynamics in quantum optimization
Our methods enable efficient learning of gradient dynamics for accelerating quantum optimization and quantum machine learning. Our experiments are both numerical simulations and real quantum computer. We first demonstrate our Koopman operator learning algorithms for VQE, an important application in quantum optimization. We test the methods for the natural gradient and Adam Kingmaa & Ba (2144) optimizers on quantum Ising model simulations and demonstrate their success on real IBM quantum computer with quasi- graeiiirrdiiet---- based optimizer Simultaneous Perturbation Stochastic Approximation (SPSA) (Spall 1992). Finally, we apply our methods to accelerate QMI on the MNIST dataset.
RELATED WORK
Koopman operators. Koopman operator theory (Koopman 1931; V. Neumann, 1932) was first
proposed by Koopman and von Neumann in early 1930s to understand dynamical systems. Dy- namic mode decomposition (DMD) (Schmid, (2010) was developed to learn the Koopman operator under the linear dynamics assumption of the observed data. Later, more advanced methods such as the extended-DD DMD based on time-delay embedding (Brunton et al.] 2017 Arbabi & Mezic 2017
Kamb et al. 2020 Tu et al. 2014, Brunton al.) 2016), kernel methods (Baddoo al. 2022) and dictionary learning (Li et al. 2017) were introduced to go beyond the linear dynamics assumption, and achieved better performance. Recently, machine learning methods were integrated into Koop- man operator learning where neural networks are used to learn the mapping to a high dimensional
space, in which the dynamics becomes linear (Lusch al., 2018 Li al. 2019 Azencot et al.,
2020 Rice et al. 2020). The machine learning Koopman operator methods were shown to learn nonlinear differential equation dynamics successfully.
In addition to predicting nonlinear dynamics, recently Koopman operator theory was applied to op- timize neural network training (Dogra & Redman 2020) Tano al., 2020) and pruning (Redman et al. 202T). These works take the perspective of viewing the optimization process of neural net- works as a nonlinear dynamical evolution and uses dynamic mode decomposition to predict the parameter updates in the future. In more empirical study, [inhaa et al. (2017) trained convolu- tional neural network (CNN) to predict the future weights neural networks, trained on standard vision tasks.
Besides important applications for classical systems, the Koopman operator theory has natural con- nections to quantum mechanics. Recently, researchers considered Koopman operator theory for quantum control (Goldschmidt et al., (02T) and prediction one particle quantum system evolu- tion (Klus et al.) (022). Since quantum mechanical systems provide natural high dimensional Hilbert space through the wave function, the theory was considered for embedding classical equa- tions for learning and solving differential equations (L et al. 2022 Giannakis al. 2022).
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Optimization methods on quantum computers. Due to the nature of hybrid classical -uaattumm algorithms, taking gradients on quantum computers cannot be done as efficiently as backpropagation on classical computers. In general, the complexity of obtaining gradients scales with the number of parameters on the quantum computer and the number of measurements per parameter. To perform optimization on quantum computers, gradient-free methods such as SPSA and COBYLA (Powell, 1994) are used though they may not scale well to quantum circuits with large number of parameters. For gradient methods, Adam common choice for better scaling though its complexity scales with the number of parameters as discussed above. There are also higher order methods such as the quantum natural gradient method which have been shown have faster convergence while being more challenging to realize experimentally.
Our work. We contribute to the development of an efficient data-driven approach that addresses the problems above via Koopman operator learning approach for accelerating optimization on quantum computers. We integrate the state-of-the-art machine learning Koopman operator with the insight of quantum dynamics to achieve our goal. There are several important features of our work that distinguish it from the relevant works above. First, our goal is to accelerate quantum optimiza- tion and quantum machine learning, which compared classical neural networks has much higher complexity of taking gradients, such that the gain of acceleration will be much more prominent. We further connect the Koopman operator theory with the quantum natural gradient, which does not exist in the classical setups and provides well-motivated theoretical foundation for our approach. Second, instead of predicting the gradient dynamics directly, we focus on optimizing the loss func- tion. The previous literature (Dogra & Redman 2020 on Koopman operators for neural networks optimization applies only the standard DMD with no follow-up optimization after the DMD pre- diction, which may not be well-situated for complicated nonlinear training dynamics. Instead, develop and investigate the sliding window DMD (S (SW-DMD) and several variants of neural DMD, including the multi-layer perceptron DMD (MLP-DMD), MLP-SW-DMD and CNN-DMD with an iterative optimization protocol. Our new approach is robust against long-time prediction error and noise, and it is shown to have much better performance compared to the standard DMD. provides acceleration for optimization in general, and enables novel avenues of research.
3 CONNECTING KOOPMAN OPERATOR THEORY AND QUANTUM OPTIMIZATION
3.1 KOOPMAN OPERATOR THEORY
We consider dynamical system with collection of state variables {x(t) RT) with transition function T such that ztt + 1) = T(x(t)). The Koopman operator theory developed by Koopman asserts that there exists linear operator K. and a function g such that
Kg(((tt) ((T(z(t)) g(æ(t + 1))
where K. is the Koopman operator. In general, the Koopman operator can act on an infinite- dimensional space. When constrained to a finite dimensional invariant subspace with g R Rmm the Koopman operator can be presented as a Koopman matrix K E Rmxm.
We now can search for the function g. The standard DMD takes g to be the identity function with the assumption that the underlying dynamics of z is approximately linear, i.e., i' linear operator. The Extended-DMBl method utilizes other feature functions such as polynomial and trigonometric functions the basis functions for g. To improve that, machine learning methods for the Koopman operator adopt neural networks as universal approximators for learning (Lusch et al. 2018).
3.2 VARIATIONAL QUANTUM EIGENSOLVER (VQE)
Consider a Hamiltonian H describing interactions in a physical system. For a quantum mechanical system with N spins or qubits, H is Hermitian operator acting the 2 -dimesooioa Hilbert space for wave functions. A wave function & is an lg-normalized complex-valle valued vector that contains all the information of the state of the system. In particular, the energy of the system is given by a loss function L(v) = (w, Hyp). VQE encodes the wave function as the by a set of parameters E I Sparame via ansatz layer quantum circuit in the quantum computer, is shown in
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the top left part of Figure Rpazams is usually chosen to be polynomial in N, which scales much slower than the 2% -scaling of the dimension of & itself. The goal of VQE is to minimize the loss by minimizing for in the following objective
0 min (()) arg min(we. Hiee)
(2)
0
0
Since the dimension of 0 is usually smaller than the dimension of v itself, the loss (())) may still differ from the minimum of (( However, thanks to the nonlinearity in the encoding the from the nonlinear ansatz layer, in many cases 0 can still be a very good approximation of the true minimum of ( Quantum machine learning has similar setup that targets at minimizing L(0), and parameterized quantum circuits in QML are called quantum neural networks.
We may use a gradient-based classical optimizer such as Adam minimize the loss function, and #. In classical machine learning, the computational cost of backpropagation is only O(1) per iteration. However, in the quantum case, we have to explicitly evaluate the loss with a perturbation in each direction i, for example by using the parameter-shift rule (Mitarai et al. 2018 Schuld et al. 2019) (C(0; + T/2) [(0; m/2))/2, which leads to an O (Hparan))) computational cost per iteration. Hence, quantum optimization is significantly more expensive than classical optimization. As another note, the classical computational components involved in VQE, even including training network typically much cheaper than the quantum cost.
this requires attaining the gradient 70,
algorithms in the following sections,
3.3 QUANTUM FISHER INFORMATION AND QUANTUM NATURAL GRADIENT
The quantum natural gradient method (Stokes et al. 2020) is generalization of the classical natural gradient method in classical machine learning (Amani 1998), where the probability generalized to the complex-valued wave function. The natural gradient for parameter 0 update for Eq. 2 is given by a nonlinear differential equation
_ Ve di
d
(3)
where 77 is the scalar learning rate, and the quantum Fisher Information matrix given by Fij Bhie Bhia thie 000 20, "0, (e))((/ (hée) 20, ) It can be shown that the above nonlinear differential equation for 0 is equivalent to the dynamical equation of (1) as follows
(vo(() = -P.. Hroo (t),
(4)
dt
where Pwo is projector onto the parameterized quantum circuit's manifold (Hackl al. (020).
Notice that H is a linear operator. When the parametrized quantum circuit is sufficiently expressive
such that the projection is within the manifold, Eq. 4 can be approximated by a linear differential equation. By viewing the parameters O(1) as the state variable xr(1) in the Koopman theory, the quantum circuit naturally generates a (wave) function the for 0 that plays the role of g in Eq. [ whose dynamics is close to linear in the Hilbert space. The dimension of the is 2 and we do not construct it explicitly. However, the existence of the and its approximate linear dynamics build the theoretical foundation for Koopman operator learning algorithms for O()) in optimization and the application of accelerating the parameter updates.
KOOPMAN OPERATOR LEARNING ALGORIHTMS
4.1
OUR FRAMEWORK: DATA-DRIVEN APPROACH FOR ACCELERATING OPTIMIZATION
The algorithmic scheme of Koopman operator learning for quantum optimization and quantum machine learning is shown in Figure [ A quantum circuit with parameters 0 is used to perform quantum optimization or QML tasks. The loss function L can be evaluated stochastically through quantum measurements while an optimizer on a classical computer is used to update the param- eters. Due to the probabilistic nature of quantum measurements, the measurement precision out- comes usually scales with the standard quantum limit (sshota) for R'hots quantum measure- ment shots. To compute the gradient, e.g., by using the parameter-shift rule (Mitarai et al. 2018
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(8) Quantum optimization and quanium machine learning
(b) acquisilion
(c) Koopman operator learning
(d) Dala prediction
Queerum
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O ((pensus)) codl per time stop
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( cost per time step
T +0(sa)))
6((
8(6s), P(G)..... B(6.))
B(o (1)a)))) ... (lberec))
((
1
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spamar
(laa
((aa))
panametors optimal loss
((t.) (ftpe)
fape azg min, StSt
(()
Figure I: Koopman operator learning for quantum optimization and quantum machine learning. (a) In quantum optimization and QML, a parameterized quantum circuit processes information, and the loss function is evaluated through measurements on a quantum computer. The parameter updates for quantum circuit are computed by a classical optimizer. (b) The optimization history of the parameters forms a time series, where for each time step the gradient optimization complexity scales as O (Hrparass) (c) The Koopman operator learning takes the time series from (b) as training data
to find an embedding of the original data with approximately linear dynamics. (d) The Koopman
operator predicts the parameter updates where each step has O(1) complexity. The loss from the predicted parameters can be evaluated on quantum computers, and the parameter ((opp) with the optimal loss is used as the starting point for the next iteration in (a).
Schuld al. 2019 usually requires quantum measurements' complexity scaling as (Ipparams) due to the hybrid classical -uaiiicul--uuuummmmmm nature of the algorithm. In addition, the gradient measure- ments per parameter take (Tshots) complexity, so the total cost of quantum measurement scales as (Ipparmss' ]. This is more expensive than the backward propagation in classical machine learning, which has the same complexity as the forward evaluation of the loss function, and the computational cost does not scale as ([Paarams) After m steps of gradient optimization, a time series of parameter updates (to). O(t)), (tmm) is obtained.
The Koopman operator learning algorithm takes the time series as training input to find an em- bedding in which dynamics becomes approximately linear. It further predicts the parameter updates for RDMD future steps for the gradient dynamics and obtains (tm++1). (fm++2)..... 0 (fmm++mmm)) )- For parameter predictions in each time step, the parameters can be set in the quantum circuit di- rectly, and the loss function can be evaluated with quantum measurements. This procedure has O(1) cost in terms of the number of parameters, and hence the total cost is ((nghots), which is the same as the forward evaluation of the loss function. Among the RDMD loss function values (00m), we find the lowest loss and the corresponding time topt = arg min, SlStm+apan L(O(t)). The last VQE iteration 4m is included to avoid degradation, even if the DMD prediction is inaccurate. The optimal scoommnn-predicee
parameter ((lope) is then used as the initial point for the next quan- tum optimization QML step, which completes full cycle of our algorithm. The alternating VQE+DMD is repeated until the loss reaches the target. Our procedure is robust against long-timo prediction error and noise with much fewer gradient measurements, which can efficiently accelerate the gradient-basee methods.
4.2 SLIDING WINDOW DMD (SW-DMD)
DMD (Brunton et al.
2021 uses a linear fit for the dynamics in the original space for the column vector 0 E R as 0 = Kt(()). Concatenating 0 at successive times we get data matrices
(()) = (0(ta) ((t))
(ttm))), ((4)= = (0(t)) O(t2)
O O(tm+1)),
(5)
where (tz)) is the one-step time evolution of (to). In the case of approximate linear dynamics, the matrix K is the same for all times tg. and then ((t++1) = K (tx) extends to ((t2) ~ (t)t)), where K E xm The best fit is at the minimum of the Frobenius loss (* is the pseudo-inverse):
K = arg min (tt ( F = ((t))
(6)
K
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Rasibal
b
K
((1)
charne sracE
Doad CNN
Graral CNV
MLP/-SW-DDD
CNN-UMD
Figure 2: Neural network architectures for our neural DMD approaches. (a) MLP bottleneck archi- tecture with MSE loss for training. (b) CNN bottleneck architecture that operates on simulations as temporal dimension and parameters channel dimension.
When the dynamics of not linear, we can instead consider a time-delay embedding with a sliding window and concatenate the steps to form an extended data matrix (Dylewsky et al. 2022)
(too)
((t))
0 (tm)
((h)
((t2)
0 (t++11)
(O(()) (o(to)
Φ(t1)
Φ(tm)]
:
((aa)
((t++1)
4 is generated by a sliding window of size d + 1 at m - consecutive time steps. Each column of & is time-delay embedding for + and the different columns & in & are embeddings different starting times. The time-delay embedding captures some nonlinearity in the dynamics of 0, with (1a+1) se K ( (to)), where K E [Rnx(d+11) The best fit is given by
= arg min (ta++1) - K ( F ((tz+1)
(8)
The used data from the acquired time series with the largest time in the above equation is (fm+dd+1) During prediction we start with 0(bm+d+2) K (bmm+1). Then update from of 6(mm+1) to (((m++) by removing the oldest data 01m++1) and adding the newly predicted data (lm+d44)) We iteratively repeat prediction via 0(tx+d+1) = (t+t). Our approach different from that of Dylewsky et al. (2022), as we do not use an additional SVD before doing DMD and our matrix KT non-square. We denote DMD performed this way as sliding window DMD (SW-DMD). The standard DMD is special case of SW-DMD when the sliding window size is 1 (i.e,, d = 0).
4.3 NEURAL DMD
General formulation. To provide a better approximation to the nonlinear dynamics, we ask whether the hard-coded sliding window transformation & can be a neural network. Thus, by simply reformulating 4 in Eq. as neural network, we formulate natural neural objective for Koopman operator learning as follows
arg min [10(()))) K&. (0(4)) K.a
where K E N.. Nout is linear Koopman operator and o (tt)) is nonlinear neural embed- ding by a neural network o., with parameters a. 2. := NN o & a composition of the neural network architecture NN, and the sliding window embedding & from the previous section.
MLP-DMD, CNN-DMD and MLP-SW-DMD methods Inspired by the machine learning ad- vancement of DMD (Lusch al. 2018), we introduce MLP-DMD, which uses a simple MI.P ar- chitecture for C. as shown in Figure (a). The architecture consists of two linear layers with an ELU (Clevert et al. 2015) activation and residual connection. We explored an expansion ratio within [1.2) for the hidden layer, but selected as the best one, possibly because higher ratios present a risk of overfitting to the noise the history of the optimization. However, it does not mixing of information between simulation steps because the simulation steps
the batch dimension. We can also consider an ID CNN encoder in Figure 2 (b) to form the CNN-DMD method. In CNN-DMD the simulation steps form the temporal dimension and the parameters form

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(a) Natural gradient 10-qubit results with nai 10, MDMD 40, and nsw 6 for SW-DMD and MLP-SW-MMD. For various DMD methods, the solid piecewise curves are actual gradient steps, and the dashed lines connecting them indicate when the DMD prediction applied.
(b) Adam 12-qubit results with Tism 10, ADDD 40, nsw 6 for SW-DMD and MLP-SW-DMD. For the various DMD methods, curves are actual gradient steps, and the dashed lines connecting them indicate when the DMD prediction is applied.
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(c) Adam 12-qubit results for relative loss versus rela- tive gradient steps. Less relative gradient steps mean less quantum resouree used. Lower relative loss in- dicates better performance of the optimization. See Table. [ for more details.
(d) Experimental results for 5-qubit Ising model us- ing SPSA from the real quantum computer IBM
Lima. Statistical errorbars are plotted but are rela- tively small with 10,000 shots.
Figure 3: Experimental results for the quantum Ising model at h 0.5.
the channel dimension of the CNN. We use causal masking of the CNN kernels on the encoder to utilize parameters information in all previous steps without look-ahead bias. During the inference we look at the history of simulated steps, and look the prediction of the last step. Then we recur- rently feed the last step and resume the predictions. Our architecture consists of two ID-CNN layers with bottleneck middle channel number of (to avoid overfitting) and ELU activation in between. C for more details. We also define an MLP-SW-DMD method which is similar to MLP-DMD but applies the time-delay embedding to the input. MLP-DMD and CNN-DMD both generalize from DMD, and MLP-SW-DMI generalizes from SW-DMD.
See Appendix
5
EXPERIMENTS
5.1
QUANTUM OPTIMIZATION
For quantum optimization, we perform experiments for the 1D quantum Ising model with the trans- verse field h defined in Appendix B The goal to find the wave function that minimizes the cor- responding Hamiltonian H. We use h 0.5 in this section, and show experimental results for = 0.9, 2.0 in Appendix We also implement experiments for the quantum Heisenberg model with results in Appendix H For each Hamiltonian, our alternating VQE+DMD algorithms with n.im VQE iterations and RDMD DMD prediction steps in each piece. Detailed setups of the alternating VQE+DMD are given in Appendix A
perform a full VQE for Noutal iterations, and
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5.1.1 NOISELESS QUANTUM SIMULATIONS
Noiseless quantum natural gradient simulations. In Figure in the case of using natural gra- dient, we show the acceleration effects of the DMD methods on a 10-qubit Ising model with the circular-entanglenae
RealAmplitudes ansatz and reps=1 (2 layers. 20 parameters) explained in Appendix D The learning rate 0.01. We perform Atotal 100 pure VQE iterations. On the DMD curves, the piecewise solid lines are from actual gradient steps, and the dashed lines connecting them are where the DMD algorithms are used to find (fopt) Details of the predicted loss from DMD) and the effect of risw in SW-DMD are discussed in Appendices D
and C respectively All the DMD methods are able to significantly accelerate the quantum optimization, so that the energy at 30 itera- tions is comparable to the final full VQE loss at 100 iterations. DMD and SW-DMD perform better than MLP-DMB the neural networks since the parameters dynamics with natural gradient is theoretically relatively simple according to Sec. 3.3 Results for the quantum Heisenberg model are in Appendix H
HAP-SWWDMD, and CNN-DMD, possibly because of the potential overfitting of
Noiseless Adam simulations. Instead of using the natural gradient method, we show another ap- plication by using the Adam (Kingma & Bal 2014 optimizer based on the gradients obtained from the parameter-shift rule. In real quantum experiments, it is much less challenging to implement the 12-qubit Ising model with the enttulle-nntnnglementt RealAmplitudes ansatz and reps=1 (2 layers, 24 parameters). In Figure 3b| we choose Thtoall 150, Aii 10, ADMD = 40, and for SW-DMD and MLP-SW- DMD, nsw = 6. The Adam learning rate is 0.01. While theoretically the prediction of the Adam update could be more complicated than the natural gradient, the energies with various DMD methods at 60 iterations are comparable to the full VQE final energy at 150 iterations, and the acceleration is very significant. particular, SW-DMD and MLP-SW-DMD converge quickly.
Adam than the quantum natural gradient method. We perform experiments
5.1.2 ABLATIONS
To measure the acceleration effect, we implement series of ablation experiments. We use the full VQE as a reference. For DMD methods, we define Relative gradient steps as the number of gradi- ent steps that DMD methods utilize compared to the full VQE gradient steps, and Relative loss = (Tmin, VQE+ DMD I.min. full / (Tiiiitllll full VQR Timin full vor). where Limin VQE DMD: Lemin, full VQE the minimum loss for VQE+DMD and full VQE respectively, Linitial, full VQE is the initial loss of full VQE. The cost per gradient step has a ((Tpaaamss-scalll --calling, and is the dominant cost for quan- tum optimization. Therefore, we use the relative gradient steps the main metric for the cost of quantum resources. A smaller relative loss means better performance of the optimization.
We consider the same 12-qubit Ising model the noiseless Adam simulations in Sec. 5.1.1 With ADMD = 90, and nsw = 6 for SW-DMD and MLP-SW-DMD, we show ablations associated with varying Msii- RDMD is chosen large enough so that this hyperparameter does not affect significantly the minimum loss. In Figure [3c] we show the relative loss versus the relative gradient steps (See Appendix F Table. for details). The DMD methods should be compared to the performance of pure VQE (black line). All the DMD methods are able to accelerate quantum optimization, as in Figure 3cl their relative loss curves are below the full VQE curve. SW-DMD works best for relative gradient steps < 10%, and MLP-SW-DMD works best for relative gradient steps 2 10%.
SW-DMD adds the sliding-window window embedding to DMD, and improves the performance signifi- cantly, compared to DMD. MLP-DMD and CNN-DMD both add the neural networks to DMD, and in most cases perform better than DMD. Likewise, MIP-SW-DMD adds the neural networks to SW-DMD, and yields better performance than SW-DMD when the relative gradient steps are rela- tively high. In lower relative gradient steps regimes, since Asim is lower, i.e., less time series data for training, adding the neural networks may be less beneficial.
5.1.3 QUANTUM OPTIMIZATION ON A REAL QUANTUM COMPUTER
To test our approach on real quantum hardware, we further implement the VQE and DMD methods on the 5-qubit Ising model with h 0.5 real IBM quantum computer Lima. The experiments are performed with the 2-layer linear- entannlntttt RealAmplitudes ansatz (10 parameters). SPSA (learning rate 0.04 and perturbation 0.1). 10k quantum shots, and measurement error mitigation.
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CrossEntropy Loss
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(b) Accuracy of binary classification for full QMI. simulation and DMD methods. For the various DMD methods, solid piecewise curves are from actual gradient steps, and the dashed lines necting them indicate the use DMD prediction. Sce Table. 3 for more details.
(a) Quantum machine learning architecture with inter- leaved encoding of 10-qubit quantum circuit for a bi- nary classification task on MNIST.
Figure 4: Quantum machine learning architecture and results.
IBM Lima, we perform full VQE for 60 iterations. SW-DMM and MLP-SW-DMD use nisw = 15. We use aaai = 20, RDMD = 20. We only run the SW-DMD and MLL-SS-DMD methods, because other methods already have unstable performances with the same setups from the noise model simulation on FakeLima which mimics the real Lima, and real quantum computations are costly. With SW-DMD, the optimization is successfully accelerated, but with MLP-SW-DMD the acceleration is less significant. The algorithmically simpler method, SW- DMD, may have more stability and robustness in a realistic setup. Appendix K discusses more details and analysis, and Appendix Cecenddxxcoottiiii contains more simulations with quantum noise effects.
As is shown in Figure 3d
5.2 QUANTUM MACHINE LEARNING
Our approach is also applied to accelerate quantum machine learning. We consider the task of bi- nary classification on filtered MNIST dataset with samples labeled by digits "1" and "9", We use an interleaved block-encoding scheme for QMI, which is shown to have generalization advan- tage (Jerbi et al.. 202T Caro al. 202T Li et al. 2022 Ren al. 2022b) and recently realized in experiment (Ren et al. 2022a). The quantum neural network is shown in Figure 4a and we apply the Koopman operator learning accelerate the learning process. Each image is first downsampled to 16 x 16 pixels, and then used as an input E [0. 1 |206 that is fed into an interleaved encoding quantum gate,. The parameters are also encoded in the interleaved encoding quantum gate. Then the quantum measurements are used as the output for computing the cross-entropy loss. We simulate a 10-qubit quantum computer with Mpparams = 270. The details of the QML data and architecture are in Appendix L To reduce the size of inputs of the neural networks in MLP- MLP-DMD, MLP-SW-DMDD CNN-DMD, we adopt layerwise partitioning in with details in Appendix L Figure 44b shows that all DMD methods can achieve good accuracy at 150 to 200 iterations while saving quantum resources compared to full QMI. at 300 to 400 iterations, which indicates significant acceleration.
Conclusion. Quantum optimization and quantum machine learning have the potential of being powerful tools for finding solutions of certain problems that are difficult for classical algorithms. However, gradient- optimization on quantum computers takes (1 gradient measure- ments for each iteration, and thus can be very expensive, especially in the current era with limited quantum computing resources. Anchored in a theoretical connection between the Koopman oper- ator theory and the quantum natural gradient, we propose a data-driven approach to use Koopman learning for accelerating quantum optimization and quantum machine learning. We further use the power of the time-delay embedding and neural networks to address the nonlinearity in optimization and achieve better prediction, which brings down the expense of quantum optimization and quantum machine learning both simulated and real quantum computer experiments. Our work serves as a bridge between the machine learning and quantum optimization communities and opens up new opportunities explore the efficiency of optimization problems through Koopman operator theory.

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6 REPRODUCIBILITY OF RESULTS
We have stated all of our hyperparameter choices for the experimental settings in the main text and in the appendix. We perform simulations of VQE using Qiskit (ANIS al. 2021), python framework for quantum computation. Our neural network code is based on Qiskit and Pytorch (Paszke et al., 2.19 Our implementation of quantum machine learning is based on Yao (Luo et al.) 2020), framework for quantum algorithms in Julia (Bezanson et al. 2017). We will release our code on Github upon acceptance.
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A SETUP OF VQE+DMD RUNS
In quantum optimization, we start from a random initial point in the parameter space and run VQE for Resi iterations. Then, we apply various DMD methods the trajectory of parameters from VQE, predict the future trajectory for RDDD steps, and evaluate the energies on the predicted trajectory. From the ADMD energy evaluations, the optimal parameters corresponding to the minimum energy are used as the initial point of the next Aaim iterations of VQE. Without particular specification, the word "iteration" " in this work usually refers to a step that requires gradient evaluation. Since per gradient cost scales (fpparrms)) ). it is dominant over the DMD prediction cost, and the number of iterations indicates the main cost of the algorithm. We repeat the alternating runs of VQE and DMD with Asi and RDMD hyperparameters. In SW-DMD and MLP-SW-DMID, the sliding window size nsw is an additional hyperparameter with the requirement nsw < Assim DMD is a special case of SW-DMD with new 1. We benchmark the standard DMD, SW-DMD, MLP-DMD, MLP- SW-DMD and CNN-DMD with Aaim 10. The sliding window size is chosen as nsww 6 for SW-DMD and HLL--SWDDMB (We use Paim 20 and new 15 on the real quantum computer SPSA experiments shown in Sec. (5..3) We also perform a pure VQE run for Rootl iterations, starting from the same initial point as the alternating VQE+DMD runs.
B QUANTUM ISING MODEL AND QUANTUM HEISENBERG MODEL
We first introduce the Pauli matrices X, Y and Z
(10)
The quantum Ising model with transverse field has the following Hamiltonian
N
H = Σ 2.0 Z1+1
X
(11)
The quantum Heisenberg has the following Hamiltonian
N J. H Σ (X, ( + +,, Y,
Σ Zi & Z++1-
(12)
The subscripts the Pauli matrices denote which qubit they are acting on. For both models, use the periodic boundary condition such that the qubits at i = and i = N + 1 are identical.
C FURTHER DETAILS ON NEURAL DMD
Optimization The parameters K and are trained jointly on Eq. 9 from scratch with every new batch of optimization history by using the Adam optimizer for 30k steps with 9k steps of linear warmup from 0 to 0.001 and then cosine decay back to 0 at step 30k. We use the MSE loss, which minimizes the Frobenius norm.
Activation Along with ELU, we also explored cosine, ReLU and tanh activations. We found ELU to be the best, likely because: tanh suffers from vanishing gradients, ReLU biases to positive numbers (while input phases quantum circuit parameters @ are unconstrained) and cosine is periodic.
The importance of learning rate scheduler. We train the neural network 4., from scratch every time we obtain optimization history as training data. It is important that we have stable learning with well converging neural network at every single Koopman operator fitting stage. For that pur- pose, we found it vital to use cosine-decay scheduler with a linear warmup, which is typically useful in the computer vision literature (Loshchilov & Hutter 2016). Namely, for the first 9k steps of the neural network optimization, a cosine-decay decay from 0.001 to until the final step at 30k. In Sec. Sec. for quantum machine learning, use 80k training steps for CNN to achieve better performance.
linearly scale the learning rate from 0 to 0.001. and then use
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The importance of residual connections. In work use residual connection in order to make DMD as a special case of the neural DMD parameterization. The residual connection is indeed very useful, it is driven by the Koopman operator learning formulation. Namely, the residual connection makes it possible for the encoder to learn the identity. If the encoder becomes the identity, then MLP(-SW)-DMI
CNN-DMID become vanilla (SW-)DMI.
The procedure for making predictions using neural DMD. Eq. provides the general objective arg min K.a (ta++) K & a (f) for training the neural-network. DMD including MLP- DMD, CNN-DMD, and MIP-SW-DMDD. First, for MLP-DMD and CNN-DMID, we take d 0 with no sliding window, so the objective gets reduced to arg mink, (11(()) (t1) - K (1o In the phase of training, the optimization history (tt)) ((1)). ................... (fm++1) is first concatenated into e (tg) and (1,) in the same way as in SW-DMD using Eq. 5 In every column, ((1)) is one iteration ahead of (4)) in the future direction. Then, in training, the operator 4. NN,, as neural network architecture, acts only (1)) not on (44). J. K is square Koopman matrix that denotes forward dynamical evolution from 2 (ta) directly to ((z)), rather than from L. (to) to P a (t,). Likewise, in the phase of inference for predicting the future of & beyond +m+ we apply the operator K repeatedly using the evolution (fx++1) = (K (tz), without an explicit inversion of the operator a to bring back the original representation. Next, for MLP-SW-DMD, need to put dl back to the equations and make the operator 4., NN. 4 composition of the neural network architecture NN,, and the sliding window embedding 4. The Koopman operator K of HLP-SW-DMD has the same dimension as K of SW-DMID which is non non-square. The procedure of updating & (t)) by adding the latest data and removing the oldest data is the same as SW- DMD described in Sec. 4.2 The only difference between MLP-SW-DMI
and SW-DMD is the additional neural network architecture NN, in MLP-SW-DMD. The only difference between MLP- SW-DMD and MLP-DMD is the additional sliding window embedding + in MLP-SW-DMD.
D REALAMPLITUDES ANSATZ.
The RealAmplitudes ansatz from Qiskit is an ansatz that always produces a real-valued wave func- tion the without the imaginary part. The minimum loss of the model Hamiltonian is always achievable by a real-valued vh. so this ansatz can reduce the redundancy in the functional form of th and is beneficial for use.
bo as a function of 0 is a nonlinear function, which consists of alternating layers of a rotational-y
gate Ry (0),
cos (0/2)
sin (0/2)
(r) exp (-10Y/2)
(13)
sin (0/2)
cos (0/2)
followed by a 2-qubit controlled-X gates with no parameter
0
0
0
0
0
0
1
CX =
(14)
0
1
0
0
1 0
0
Each layer of the Ry (f) gates has N (the number of qubits) parameters, the total number of parameters is equal to the number of rotational layers times N.
E DMD PREDICTIONS
In this section, we show the details of the DMD predictions that we have used to accelerate VQE on the quantum Ising model at 0.5 with natural gradient and Adam presented in Figures 3a and 3b respectively.
Figure Sashows the natural gradient results with DMD predictions. For the various DMD methods, solid parts VQE runs, and the dashed parts are DMD predictions. The horizontal axis label is the total steps including both the actual gradient steps and prediction steps. For the pure VQE, the total steps are equal to the iterations, which are all actual gradient steps. For the DMD curves, the solid
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(a) Natural gradient 10-qubit results with nsm 10, NDMD = 40, and nsw = 6 for SW-DMD and MLP- SW-DMD. For the various DMD methods, solid parts are VQE runs, and the dashed parts are DMD predic- tions.
(b) Comparison in natural gradient 10-qubit results for Naim = 10 with DMD and Reim 15, nsw = with SW-DMD. Both methods use rDDD 40. For both DMD methods, solid parts are VQE runs, and the dashed parts are DMD predictions.
Full VOU
VIDE SILP-SW-DDD
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(c) Adam 12-qubit results with the standard DMD methad with nsim = 10, ADMD = 40. For the DMD method, solid parts are VQE runs, and the dashed parts are DMD predictions. The red points mark the optimal parameters for each piece of DMD predic- tions. The next piece of VQE starts from optimal parameters rather than the last parameters, as is indi- cated by the red arrows, to guarantee the decrease of energy.
(d) Adam 12-qubit results with the MLP-SW-DNN method with Alm = 10, ADMDD = 40. For the DMD method, solid parts are VQE runs, and the dashed parts are DMD predictions. The red points mark the optimal parameters for each piece of DMD predic- tions, as the initial point for the next piece of VQE.
Figure 5: Experimental results with DMD predictions displayed. The horizontal axis label is the total steps, including the gradient steps and the prediction steps.
parts gradient steps with O paraam))) cost per step, and the dashed parts are prediction steps with ((1) cost per step. Since with the natural gradient, 0 has the quantum wave function as a natural embedding with dynamics close to be linear, Koopman operator learning is hypothesized yield good predictions. All the DMD algorithms are able to achieve the full VQE optimal loss. DMD and SW-DMD both match the full VQE. Our MLP-DMD, MLP-SW-DMD, and CNN-DMD start to deviate from the full VQE when the DMD iterations increase. This could be related to overfitting of the neural network when the parameters dynamics are simple. With asm 10 both for DMD and SW-DMD (niww = 6), DMD has a slight better performance when make predictions, this can be because in SW-DMD, there are only Usim nggw + = 5 columns of data in the time-delay embedding, fewer than 10 independent columns of the standard DMD (equivalent to nsw However, when we use Nsim = 15 and new = 6 for SW-DMD resulting in 10 columns in the time- delay embedding, the performance of SW-DMD is comparable to the standard DMD, as is shown in Figure 5b
1).
In Figure 3c we show the standard DMD predictions in the Adam case. The dashed curves from DMD predictions, which typically first lower the energy but then lead to energy explosion. The energy explosion, however, does not make our algorithm fail, because start the next piece of
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Relotive loss
10-2.
Relstive gradient steps
gradient steps
10-4
6
12SW
Figure 6: Relative loss from SW-DMD versus nisw at 10% and 20% relative gradient steps in the range V nsw S 9. The expriments are all performed on the 12-qubit quantum Ising model at h 0.5 We use Thotal 300 full VQE iterations, Naam 10, and nDMD 90. Smaller relative loss means better performance.
VQE from the optimal point marked by red points rather than the last DMD prediction. In Figure 5d show the ML.P-WWDDMD predictions of Adam as another example. The DMD predictions do not exactly align with the full VQE, but they still successfully lower the energy in general. When the energy close to convergence. the DMD predictions may lead to minor energy increase, but this does not affect the final performance.
F RELATIVE LOSS FOR THE ISING MODEL
Table demonstrates numerically the results from Figure
Relative Gradient Steps
Method
7%
8%
10%
15%
20%
25%
30%
Pure VQE
56.3%
52.2%
44.6%
29.2%
18.4%
11.0%
5.8%
DMD
41.2%
9.2%
32.1%
15.1 %
11.0%
5.7%
1.9%
SW-DMD
4.3
4.1 %
5.7%
1.4%
0.5%
0.2%
0.004%
MLP-DMD
18.8%
12.0%
10.2%
9.9%
4.1%
0.5%
0.09%
MLP-SW-DMD
6.8%
6.6%
3.8%
0.2%
0.04%
0.007% %
0.003 %
CNN-DMD 11.8%
6.9%
5.2%
6.3%
3.0%
2.3%
0.9%
Table Relative loss (in %) as a function of the method used and the relative gradient steps (in %).
Lower relative loss is better. Our methods significantly improve the standard DMD.
The relative loss has trend of decrease when the relative gradient steps increase. However, this trend is not strictly monotonic, because using longer history of optimization (more relative gradient steps) may occasionally lead worse prediction. Because of the complicated space dynamics, may be challenging to fit a longer time series of parameters.
G EFFECTS ON VARYING Nsw
We perform an ablation study for the effects on the performance from varying niww in SW-DMD. We use the quantum Ising model at h 0.5 12 qubits with Adam and choose 7ssm = 10 and use Aottal = 300 steps full VQE and
< nisw (nsw < Rsim a strict requirement).
7DMD
90.
Figure shows the relative loss versus new with 10% and 20% relative gradient steps defined in Sec. 5.1.2 When the relative gradient steps 10% the relative loss is the highest when new which is equivalent to the standard DMD. This shows that SW-DMD in general has a better per- formance than the standard DMD. The relative loss is low in the intermediate regime of new
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PAR SW-DID
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0
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300
400
(a) Natural gradient 10-qubit results with Naim 10, for SW-DMD and MLP-SW-DMD. For various DMD methods, the solid piecewise curves are actual gradient steps, and the dashed lines connecting them indicate when the DMD prediction applied.
MDMD 40, and nsw
(b) Adam 12-qubit results with Neim 10, MDMD 40, nsw = for SW-DMD MLP-SW-DMD. the various DMD methods, solid parts are VQE runs, and the dashed parts are DMD predictions.
100%
VQDE
80%
60%-
Relative
40%
20%
0%
0%
10%
20%
30%
40%
Relative gradient steps
(c) Adam 12-quhit results for relative loss versus rel- ative gradient steps. Less relative gradient steps mean less quantum resource used. Lower relative loss indi- cates better performance of the optimization.
Figure 7: Experimental results for Heisenberg model at J/ = 0.5.
nsw 4 but even lower at large window size nsw = 9. However, the best performance achieved by the maximum new is not universal. As increase the relative gradient steps from 10% to 20%, at each nsw, the relative loss decreases, but nısw = no longer has the lowest relative loss compared to the intermediate window size near niw
Theoretically, the performance versus nsw can be related to the shape of the extended data matrix in Eq. 7 where nsw + 1. The number of rows is nsw "Pparamms- and the number of columns is Dsim 'lw 1. There can be a trade-off in choosing nsw. Increasing nsw may enhance the ability of the time-delay embedding to capture the nonlinear dynamics, but on the other hand, this may also lead to fewer columns which potentially means a less independent information. Heuristically, an intermediate window size new /Pparams 0.5 may have good performance, but this is not strict criterion. In some cases, we find empirically that nsw Ppparams > 0.5 can have better performance, such as risw /ftl///paaamaa 0.9 with 10% relative gradient steps in Figure
H RESULTS OF THE QUANTUM HEISENBERG MODEL
Similar to the experiments we have on the quantum Ising model in Figure 3 C we implement the experiments for the 12-qubit quantum Heisenberg model at J, 0.5 with all the hyperparameters the same as the Ising model experiments.
Figure Fa shows the results from the 10-qubit noiseless natural gradient simulations. All the DMD methods are able to significantly accelerate the quantum optimization. DMD and SW-DMD even lower the energy lower than the final full VQE simulation. This is maybe because the predictions by
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Relative Gradient Steps
Method
7%
8%
10%
15%
20%
25%
30%
Pure VQE
87.2%
84.4%
77.9%
58.8%
40.0%
23.1%
11.9%
DMD
39.2%
40.5%
28.8%
27.1%
45.4%
11.7%
14.3%
SW-DMD
67.0%
10.5%
14.0%
8.0%
5.4%
-0.03%
1.6%
MLP-DMD 19.2%
24.6%
7.8%
13.4%
11.3%
6.0%
2.7%
HLP-SW-DMD
65.4%
12.7%
11.2%
3.1%
4.7%
2.6%
3.4%
CNN-DMD 22.5%
22.0%
1.0%
32.6%
11.5%
13.6%
3.5%
Table 2: Relative loss (in %) from the quantum Heisenberg model at J/ = 0.5 with Adam simula-
tions as function of the method used and the relative gradient steps (in %).. Lower relative loss is better. Our methods significantly improve the standard DMD.
DMD and SW-DMD find a different trajectory from the full VQE trajectory, which takes out of local minimum to a lower energy.
Figure 7b shows the results from the 12-qubit noiseless Adam simulations. All the DMD methods are able to significantly accelerate the quantum optimization. With the fixed number of iterations, the standard DMD method successfully lowers the energy but still does not reach the final loss of full VQE.
Figure 7 shows the results of the ablation study from 12-qubit noiseless Adam simulations, with the same results presented in for better numerical resolution. We use nsw 6 for SW-DMD and MLP-SW-DMD Relative loss and relative gradient steps are defined in Sec. 5.1.2 Almost all points from various DMD methods are below the pure VQE curve, which indicate successful acceleration. The only exception occurs at 20% ralative gradient steps for the standard DMD method, where the relative loss from the standard DMD a little higher than the pure VQE. When we start the follow- up piece of VQE from the DMD optimal prediction (with an energy lower than the last iteration of the previous piece of VQE), the DMD optimal prediction can still be worse position for optimization than the pure VQE in the space of parameters, because the energy landscape can be complicated.
In Table D the best performances among the various DMD methods are achieved by SW-DMD and MIP-SSWDMD, and this also holds in general Table 2 However, here Table 2 at 7% relative gradient steps, neither SW-DMD nor ML.-SWWDMDD have the best performance. This can be because we Rşim = in this case, which leads to the number of columns in the time-delay embedding is only 7aai nsw + 1 2, the minimum number for constructing the dynamics. As is analyzed in Appendix E a small number of columns can sometimes lead to a relatively bad performance. At 8% relative gradient steps, CNN-DMD has the best performance compared to other methods, which indicates the potential power of CNN-DMD if the CNN training can be properly done.
I RESULTS AT MORE VALUES OF h.
We show the 12-qubit noiseless Adam results at h 0.9, 2.0 in Figure with the same setup as we do for = 0.5 in Sec. [51.1] All the DMD methods significantly accelerate the quantum optimization, although some of them in Figure 8b may still need more iterations get closer to the minimum loss full VQE. In Figure Sal the SW-DMD and MLP-SW-DMD find even lower energies than full VQE, similar to what we observe Figure 7a
NOISY QUANTUM SIMULATIONS
To study the problem with more realistic setup, also perform simulations using noise model FakeLima, which mimics a real IBM quantum computer Lima, on a 5-qubit Ising model with h = 0.5 with the periodic boundary condition. The optimizer is Adam with learning rate 0.01. We use
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25.0
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100
150
200
50
100
150
200
Ieerations
Iterations
(a) h 0.9.
(b) h 2.0.
Figure 8: 12-qubit noiseless Adam simulation results at (a) h = 0.9 and (b) = 2.0 with naim = 10, WDDD 90, Nhttal 200. We use nsw 6 for SW-DMD and MLP-SW-DMD. For the various DMD methods, the solid piecewise curves are actual gradient steps, and the dashed lines connecting them indicate when the DMD prediction is applied.
Pal VigE
DMD
SW.2MD
SW.OMD
MLL--TA
MILF nvm
MUP-EW: BAIT
MILF
Energy
Energy
CNN DAII
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150
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(a) 100 shots.
(b) 200 shots.
3..5
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CTON (AVI)
MILP 5WV
Energy
-4.-
4.5-
5.00
50
100
150
200
0
50
100
150
200
Inerations
Iterations
(d) 10,000 shots.
(c) 1,000 shots.
Figure 9: Simulations the noise model FakeLima with different numbers of shots for quantum
measurement: aa 100 shots (b) 200 shots c 1,000 shots (d) 10,000 shots. Optimization histories for energy of full VQE and various DMD methods are shown. Dashed lines indicate that the DMD methods are used to accelerate the optimization.
the RealAmplitudes ansatz with 2 layers (10 parameters) with circular entanglement. We have also applied measurement error mitigation (Barron & Wood 2020) reduce the effect of quantum noise.
The evaluation of loss on the quantum computer based on quantum measurements. With more shots of measurements, the statistical error will be reduced, but the needed quantum resources will increase. In practice, is interesting to see whether our algorithm works with a relatively small number of shots, show below.
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In Fig. we show the simulated optimization histories of the full VQE and various DMD methods. We run the full VQE for Nltotal = 200 iterations. We choose Maim = 10 and RDMD = 20. SW-DMD use window size nsw = 6. Both the loss function accuracy and the gradient precision scale with the number of shots Tishots as (11/ / /llhhtts) Even with a very small number of shots such as nshhos 100, all the DMD methods can accelerate the quantum optimization, as the loss with DMD decreases quicker than the full VQE. The final energy with 100 shots is not as low as the final energy from more shots, but this is not because of any inferiority of the DMD methods, since the full VQE also has difficulty to reach a lower energy. Our noisy simulations demonstrate that our methods are promising in cases closer to a realistic setup.
and MLP-SW-DNS
The spikes in the DMD results occur every aaim 10 iterations at the beginning of every piece of VQE, and are more distinct when the shots are smaller. This may be because it is harder to make predictions of the VQE history when the parameter updates time series from measurement is more noisy. Despite the occasional spikes in energy, the DMD methods still help to get closer to the better parameters for optimization in later steps.
K
ADDITIONAL DISCUSSION ON REAL LIMA EXPERIMENTS
K.1 1 CHOICE OF OPTIMIZER AND HYPERPARAMETERS
We use SPSA instead of Adam, because the required quantum resource for a purely gradient-based optimizer such as Adam is too much at this stage due to our limited access to quantum computers. SPSA is muasi-graident-aaaee based and more realistic to implement on a real quantum computer. With the SPSA optimizer, every iteration of update takes center measurement and 2 gradient measure- ments in only one random direction of the parameter space, so each iteration takes measurements. The smaller number of gradient measurements makes it casier to implement on near-term real quan- tum computers, compared to Adam. However, using SPSA may also require a higher number of iterations. The DMD method only takes I center measurement without gradient measurements. The gradient-based method should have more advantage than SPSA as the number of parameters increases, since SPSA provides much more noisy gradient estimates. There is no fundamental limi- tation to implement pure pradient-based methods, but it is more time and resource consuming given the limited accessibility of quantum computers. Indeed, one of our main motivations develop the Koopman operator learning algorithm is to reduce the cost for gradient- the number of parameters in quantum eircuit usually scales with the nunber of qubits in practice, our approach should be able to demonstrate more gains for large-scale quantum optimization with gradient-based methods.
optimization. Since
To mitigate the randomness in the direction of gradient measurements in SPSA, we choose Assm 20 for Pararms = 10. Within these first 20 VQE simulations, each parameter on average has one forward step gradient measurement and one backward step gradient measurement. We also observe that a relatively large sliding window may help the learning. We choose nsww = 15 for SW-DMD and MLP-SWWDMD and Nss 20 instead of 10 so that new //niu is relatively close to 1. The standard DMD, MLP-DMD, CNN-DMD already have unstable performances with the same setups on Fakel.ima. This probably comes from the fact that SPSA with quantum noise returns very noisy parameter updates compared to the true gradient-base method, since makes random perturbations in the parameter space, and the quantum noise further biases the measurement. This might make the Koopman operator learning much more challenging.
K.2 NOISE ANALYSIS
The noisy decay of the energy in the curve in Figure 3d could be related to multiple sources: (1)
quantum noise, (2) experimental instability in the real lab apparatus, such as day-to-day calibration, (3) statistical error from quantum measurements, (4) randomness from SPSA. (3) is not a dominant factor here, since the statistical errors are relatively small in our experiments. To help probe and di- agnose these various effects, we also perform the same experiments with the same hyperparameters on FakeLima shown in Appendix K. Both SW-DMD and MLP-SW-DMD can successfully accel- erate the quantum optimization on FakeLima. This might indicate that there is in fact a difference between the real Lima and Fakel .im,, since the real Lima is subject daily fluctuation while Fake-
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VIQE
-3.-
SW
MIP-SW-IADD
Enegy
4.0
-4.-
5.0
20 Iterations
40
60
Figure 10: FakeLima results.
Relative Gradient Steps
Method
10%
20%
30%
40%
50%
Pure QMI.
81.8%
92.8%
96.0%
95.8%
95.6%
DMD
95.2%
98.4%
98.8%
98.8%
99.2%
SW-DMD
95.2% %
98.8%
99.0%
99.2% %
99.0%
MLP-DMD 93.6%
96.4%
97.6%
98.4%
98.6%
MLP-SW-DMD
94.0%
98.6%
99.0%
99.2%
99.0%
CNN-DMD 89.2%
95.4%
97.4%
98.6%
98.4%
Table 3: Test accuracy (in %) from quantum machine learning in Sec. 5.2 on the filtered MNIST dataset function of the method used and the relative gradient steps (in %)
Lima is fixed noise model. However, (1) and (4) may still be possible to account for the relatively insignificant acceleration of MLP-SW-DMD on real IBM Lima.
K.3 COMPARISON BETWEEN REAL LIMA AND FAKELIMA
We perform simulation using FikeLima on the same 5-qubit Ising model at h 0.5 with SPSA, as we do on the real Lima in Sec. 5.1.3 and Figure 3d The FakeLima results are shown in Fig- ure IO Both SW-DMD and MLP-SW-DMM FakeLima. The performance in the FakeLima case is better than in the real Lima case, and the diagnosis and analysis are detailed in Sec. 5.1.3
significantly accelerate the quantum optimization on
L QML ARCHITECTURE AND TRAINING DETAILS
In our QMI. example Sec. [.2 the quantum computer has N 10 qubits. The interleaved encoding gate consists of layers. Each layer has rotational layer and linear entanglement layer. On the rotational layer, each qubit has three rotational gates Rx, Rz, Rxx a sequence, where
cos (0/2) Rx (0) exp (-i0X/2) sin (0/2)
sis (0/2)
(15)
cos (0/2)
R2 = exp (-i07/2) = exp (-i0/2) 0
(16)
exp (10/2)
Therefore, each layer has 30 rotational angles, and the whole QML architecture has 270 rotational angles, i.e., Aparaus 270. Since each input example is 256-dimensional, only the first 256 rotational angles to encode the input data. The parameters are also encoded in the rotational angles such that the angles
0.
Quantum measurements, as the last part of the quantum circuit, map the quantum output to the classical probability data. In each quantum measurement, each qubit is in the O-state or the 1-state,
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Tod. accuraayy
CrossEntropy Loss
0.8
qut.
0.6
SELP-5T0- 20D
(0)
quantum measurament
MIPP IVD
FestureMsp(s)
ansatz(
0)
10
20
30
Iterations
(a) Hybrid classical-quantum machine learning archi- (b) Accuracy of binary classification for QML. simulation and DMD methods. For the various DMD methods, solid parts are QML runs, and the dashed parts are DMD predictions.
tecture.
Figure 11: Hybrid classical-quantumg machine learning architecture and results.
and the probability for the qubit to be in the O-state is between 0 and 1. We regard this probability as the probablity for the image to be a digit "I". We only use the probability on the 5th qubit (i 5) as the output and compute the cross-entropy loss with the labels.
In the phase of training, all the training examples share the same but have different æ, and we optimize the final average loss with respect to 0 as
θ arg min
L(x train 0),
(17)
Ptrain
in the case of Teerain training examples. The combination arrain + 0 is entered into the quantum circuit rotational angles, and we need to build Perain separate quantum circuits (which can be built either sequentially layer, and serve dual roles of encoding the classical data æ and containing the machine lcarning parameters O. In the phase of inference, with the optimal parameter 0" feed each test example atest into the quantum circuit as a combination arlas 0" and measure the output probability to compute the test accuracy.
parallel). The interleaved encoding gate whole can be viewed as deep
We use 500 training examples and 500 test examples. During training, we use the stochastic gradient descent optimizer with the batch size 50 and learning rate 0.05. The full QML training has Thotal 400 iterations. We choose Naim DMD.
10, ADMD = 20, and new = for SW-DMD and MLP-SW-
In neural DMD including MLP-DMD, MLP-SW-DMD, CNN-DMD, we use layerwise partitioning that groups 0 by layers in the encoding gate. There are groups with group size 30. We perform neural DMD for each group, so that the number of parameters in the neural networks for DMD is too large. After predicting each group separately, we combine all groups of evaluate the loss and accuracy. In CNN-DMD for QML, we use 80k steps for sufficient CNN training.
The numerical values Figure 4b test accuracy from various methods at relative gradient steps compared to pure QMI. are given in Table 3 All the DMD methods significantly accelerate the QML training.
M HYBRID CLASSICAL-QUANTUM MACHINE LEARNING
We illustrate another example Koopman operator learning for QML using a hybrid quantum- classical neural network with its architecture shown in Figure la We consider supervised learning for binary classification over a filtered MNIST dataset with digits 0 and 1. First the MNIST data is fed into classical neural network for the feature æ. Then, the feature is encoded into the quantum computer via a sequence of quantum gates for feature mapping. Then, the state goes through an ansatz layer with parameter 0, similar the case of quantum optimization.
encoded quantum
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We use the results from quantum measurements the output of the hybrid neural network. We test our Koopman operator learning algorithms with the above architecture. The full QML training has Rrotal 36 iterations. We choose asi 6, RDMD 6, and nsw = 3 for SW-DMD and MI.P-W-DMDD, [Tbjsoss that DMD methods can achieve reasonably good accuracy while using less quantum resource than the full QMI..
The classical neural network the hybrid architecture consists of [Con-2Diin-channel-1, out- channel=2. kernel-size=5), ReLUO, MaxPool2D(siz==2), Con-2D6in-cammell-2, out-channel-2, kernel-size=5), ReLUO, Linear(in-dim=2, out-dim=2), ReLUO, Linear(in-dim=2, out-dim=2)].
The feature mapping is realized through the Ma two qubits with repetition from Qiskit. The ansatz after the feature mapping is the RealAmplitude ansatz with one repetition. The quantum measurements are performed on the Z-basis of two qubits, which return the expectation value of (Z) as an output y. Then, y and 1 are used as logits for the CrossEntropy loss. Adam used as the optimizer with learning rate 5 x 10-4. 50 training examples and 100 test examples are used for each digit.
Summary the functionality of the different components in Figure Il In the beginning classical neural network takes the original input of image data to learn the representation x in classical feature space. Then the rest components in the architecture quantum. The features fed into the FeatureMap(z) quantum gate as layer to encode the classical data into quantum data. Then, the ansatz (0) quantum gate as a follow-up layer serves the same role as the ansatz layer in the of quantum optimization. Finally, the quantum measurement takes the output quantum data into classical outputs.
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Reviews:
[{"review_id": "Hv2O_3DyvY", "reviewer": "Reviewer_XcoP", "summary": "", "questions": "", "limitations": "", "rating": 6, "confidence": 3, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "", "claims_and_evidence": "", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "main_review": "", "summary_of_the_paper": "The author has discussed the issue of backpropagation and obtaining the gradient of a quantum computer. The gradient computation during backpropagation in a quantum computer is not similar to a classical computer because the complexity increase with the number of parameters and measurements. The author discusses the Koopman operator theory for predicting nonlinear dynamics with the natural gradient method in quantum optimization to address this issue. The author proposed a data-driven approach using the Koopman operator to accelerate quantum optimization and Quantum machine learning. The author proposed Sliding window dynamic mode decomposition (DMD) and Neural DMD for efficiently updating parameters in quantum computers. The author shows that the method can predict gradient dynamics and accelerate the quantum variational eigensolver in quantum optimization and quantum ML.  \n \n", "summary_of_the_review": "The overall writing quality is ok, and the proposed method is simple and beneficial. However, the experiment comparison lacks justification, and the technical contribution is plain.\n\n", "strength_and_weaknesses": "Strength:\n\nThe paper discusses an essential problem in the literature.\n  \nAlso, the author provides an excellent solution to tackle the problem, but the analysis and intuition part is very poorly represented in this paper.\n  \nThe paper is easy to follow and understand. The authors introduce several quantum machine learning strategies, advantages, and weaknesses. Based on this, the motivation of this paper is reasonable.\n\n\nWeaknesses:\n\nThe author has discussed accelerating quantum mechanics and quantum optimization algorithms compared to classical operators. Did the author quantify or qualify this using some experiment? \n\nHow did the Koopman operator help to accelerate the gradient calculation in quantum computers compared to classical computers? Can the author provide a brief explanation for this?\n\nIn section5.2, the author has discussed the sequence of the quantum gate. Is this similar to layers? Can the author discuss the objective of each block in Figure 4a (like the objective of block \"ansatz\" and its input and output)? Similarly need to discuss other blocks. Can the author describe all the blocks and their functionality in the figure-1 for MNIST dataset example (Like, for an image sample)? The author should describe the complete algorithm and discuss which part is classic machine learning and which is quantum machine learning.\n\nAlong with MNIST, the author also discuss Figure 1 with the time series task. Did the author report any comparative results on time series?\n\nThe author should report the performance and analysis of results on the MNIST dataset.\n\nThe author can add a small preliminary section that describes the steps for the quantum ML algorithm (Training and Inference)  \n\nThe experiment section is not designed correctly. The author should compare the results with SOTA and do an ablation analysis on the Koopman operation. The analysis section of the paper needs to improve with more analysis results.\n\nTake home: Koopman operator  for stability and robustness and \nMNIST on QML\n", "limitations_and_societal_impact": "", "review_text": "", "extra_scores": {"correctness": "3: Some of the paper’s claims have minor issues. A few statements are not well-supported, or require small changes to be made correct.", "technical_novelty_and_significance": "3: The contributions are significant and somewhat new. Aspects of the contributions exist in prior work.", "empirical_novelty_and_significance": "3: The contributions are significant and somewhat new. Aspects of the contributions exist in prior work."}, "extra_text": {}}, {"review_id": "CNbYxjDPB5c", "reviewer": "Reviewer_5VYL", "summary": "", "questions": "", "limitations": "", "rating": 3, "confidence": 4, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "", "claims_and_evidence": "", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "main_review": "", "summary_of_the_paper": "The authors present two algorithms for quantum optimization and machine learning based on the Koopman operator. They test their algorithm with simulations and a small hardware experiment.", "summary_of_the_review": "The paper has an interesting idea of using Koopman operator theory as a way to avoid gradient calculations in training quantum neural networks. More work and evidence is needed for the impact of this methods", "strength_and_weaknesses": "+ the Koopam operator optimization method may turn out to be interesting for learning\n- there is hardly any strong evidence in this work of the performance of the methods\n- the experiments are trivial (3 qubits) and no conclusions can be drawn from them\n- no theoretical reasons why this methods should outperform other gradient-free optimization methods are given", "limitations_and_societal_impact": "", "review_text": "", "extra_scores": {"correctness": "3: Some of the paper’s claims have minor issues. A few statements are not well-supported, or require small changes to be made correct.", "technical_novelty_and_significance": "2: The contributions are only marginally significant or novel.", "empirical_novelty_and_significance": "2: The contributions are only marginally significant or novel."}, "extra_text": {}}, {"review_id": "APfUgZxg7d", "reviewer": "Reviewer_TgmW", "summary": "", "questions": "", "limitations": "", "rating": 3, "confidence": 4, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "", "claims_and_evidence": "", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "main_review": "", "summary_of_the_paper": "This paper proposes to model the variational quantum eigensolver optimization as a dynamical system, and approximate its iterative updates via Koopman-based approaches. In particular, the authors propose a sliding window DMD method, and a neural DMD approach. The authors evaluate their approach on quantum machine learning tasks, and compare it to standard DMD. ", "summary_of_the_review": "In summary, while the application domain of this paper is interesting and refreshing, it is essentially a different application of an existing paper. Moreover, the proposed methods are similar or equivalent to existing work in the field. I suggest the authors to conduct a proper literature search in Koopman and DMD communities (see e.g., Hankel DMD and Koopman Autoencoder), and to position their contribution accordingly. Due to these reasons, I think the paper should not be published in its current form.", "strength_and_weaknesses": "In my opinion, the strongest point of the paper is the application domain of quantum machine learning (QML) and the incorporation of Koopman-based methodologies in it. Having said that, it should be noted that the technical problem solved in this paper is essentially the ability to approximate complex dynamics with Koopman methods, and the relation to QML is only by application.\n\nUnfortunately, there are several weak points. First, the same observation and application was already considered in Dogra et al. (and other papers, some of which cited in this paper). In this context, the main difference between this paper and the paper of Dogra et al. is in the evaluation on QML tasks vs. vision tasks. While this difference may be significant in certain applications and algorithms, I do not believe this is the case here, as they essentially optimize a standard neural network, similar to those considered in Dogra et al.\n\nSecond, the proposed approaches to approximate the Koopman operator generally ignore many similar if not identical ideas in the literature. Both sliding windows with time delay coordinates and neural DMD have been heavily studied before. Properly discussing these approaches and positioning the proposed method in this context is essential.", "limitations_and_societal_impact": "", "review_text": "", "extra_scores": {"correctness": "3: Some of the paper’s claims have minor issues. A few statements are not well-supported, or require small changes to be made correct.", "technical_novelty_and_significance": "1: The contributions are neither significant nor novel.", "empirical_novelty_and_significance": "1: The contributions are neither significant nor novel."}, "extra_text": {}}, {"review_id": "76WPm7TxW0", "reviewer": "Reviewer_zjTr", "summary": "", "questions": "", "limitations": "", "rating": 6, "confidence": 2, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "", "claims_and_evidence": "", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "main_review": "", "summary_of_the_paper": "The paper propose using Koopman operator to speed up the optimization of hybrid quantum computing algorithm. The motivation is gradient step in quantum computer takes many forward evaluations (linear in number of parameters). Using Koopman operator I the classical computing sense, to predict future optimal quantum machine learning parameters. then pick the best, optimal time, and then do more predictions from there.\n", "summary_of_the_review": "Good motivation for this work. Not absolutely sure if the equations are theoretically sound due to my ignorant.\nMy sense is this paper is in between (6) Marginally above acceptance threshold and (8) accept, good paper.\n\n", "strength_and_weaknesses": "The motivation of the paper is good. It is one plausible solution to an important problem. The method of Koopman operator learning using sliding window and neural network to learn dictionary, can potentially be use for optimising other classical computing algorithm. its advantage is most in quantum computing.\n\nI have not thought through thoroughly if using the sliding window approach is theoretically sound. Certainly did not sit down, write the required equations and convince myself this is a theoretically sound approach. Loss function of Eq. (8) (or similar) can be constructed in many ways, sliding window way etc, seems arbitrary to me. I understand several other works in the area of Koopman operator also uses similar approach of constructing some measurable / observable and then do optimization. In the limit of infinite observables, one can agree that the trained Koopman operator can represent the underlying dynamics accurately. There is also an issue of how to invert the \"g\" function to get back the underlying \"x\" (Eq. (1)). In this paper, it seems g is the identity function.\n\n\n", "limitations_and_societal_impact": "", "review_text": "", "extra_scores": {"correctness": "3: Some of the paper’s claims have minor issues. A few statements are not well-supported, or require small changes to be made correct.", "technical_novelty_and_significance": "3: The contributions are significant and somewhat new. Aspects of the contributions exist in prior work.", "empirical_novelty_and_significance": "2: The contributions are only marginally significant or novel."}, "extra_text": {}}]

Author rebuttal:

Decision: Reject