SlowGuess/NeurIPS_2023_OCR huggingface dataset:
Connection to Overparameterization Theory. Recent advancement on overparameterization theory [37, 44, 87] finds that when the number of parameters in the quantum circuit exceeds a certain limit, linear convergence of the variational quantum optimization under gradient descent is guaranteed. You et al. [87] shows that the normalized  $\psi_{\theta}$  from the parameterized quantum circuit in the overparatermization regime follows a dynamical equation which has a dominant part linear in  $\psi_{\theta}$  with additional perturbation terms. Similar to the quantum natural gradient, the overparameterization regime could provide an effective structure to simplify the Koopman operator learning with approximate linear dynamics.
5.1 Stability Analysis
We first note that the direct application of DMD without alternating the controlled scheme will lead to unstable or trivial performance in the asymptotic limit.
Theorem 5.1. Asymptotic DMD prediction for quantum optimization is trivial or unstable.
Proof. The asymptotic dynamics from DMD prediction is given by  $\pmb{\theta}(T) = K^T\pmb{\theta}(t_0)$  for  $T \to \infty$ . It follows that  $\pmb{\theta}(T) \to w_m^T v_m$ , where  $w_m$  and  $v_m$  are the largest magnitude eigenvalue and the corresponding eigenvector of the Koopman operator  $K$ . If  $|w_m| < 1$ , then  $\pmb{\theta}(T)$  will converge to zero, and the quantum circuit will reach a trivial fixed state. If  $|w_m| \geq 1$ ,  $\pmb{\theta}(T)$  will keep oscillating or growing unbounded. For a unitary gate  $U(\theta) = e^{-i\theta P}$  where  $P$  is a Pauli string,  $U(\theta)$  is periodic with respect to  $\theta$ . The oscillation or unbounded growing behavior of  $\pmb{\theta}(T)$  will lead to oscillation of the unitary gate in the asymptotic limit resulting in unstable performance.
The above issue is also found to exist in numerical results plotted in Appendix, indicating that the eigenvalues of the Koopman operators from quantum optimization dynamics can lead to trivial or unstable DMD prediction, which motivates us to develop QuACK. Indeed, our QuACK has controllable performances given by the following
Theorem 5.2. In each iteration of QuACK, the optimal parameters  $\theta(t_{\mathrm{opt}})$  yield an equivalent or lower loss than the  $m$ -step gradient-based optimizer.
Proof. From  $t_{\mathrm{opt}} = \arg \min_{t_m \leq t \leq t_{m+n_{\mathrm{DMD}}}} \mathcal{L}(\pmb{\theta}(t))$ , we have  $\mathcal{L}(\pmb{\theta}(t_{\mathrm{opt}})) \leq \mathcal{L}(\pmb{\theta}(t_m))$  where  $\mathcal{L}(\pmb{\theta}(t_m))$  is the final loss from the  $m$ -step gradient-based optimizer.
As long as QuACK is able to capture certain dynamical modes that decrease the loss, then it will produce a lower loss even when the predicted dynamics does not exactly follow the same trend of the  $m$ -step gradient descent updates. We also note that it is possible for QuACK to converge to a different
local minimum than the baseline gradient-based optimizer, but our experiments generally demonstrate that our QuACK achieves accuracy the same as or even better than the baseline with much faster convergence. Our procedure is robust against long-time prediction error and noise with much fewer gradient calculations, which can efficiently accelerate the gradient-based methods. Furthermore, our scheme has the following important implication, the proof of which we provide in Appendix.
Corollary 5.3. QuACK achieves superior performance over the asymptotic DMD prediction.
5.2 Complexity and Speedup Analysis
To achieve a certain target loss  $\mathcal{L}_{\mathrm{target}}$  near convergence, the traditional gradient-based optimizer as a baseline takes  $T_{b}$  gradient steps. To achieve the same  $\mathcal{L}_{\mathrm{target}}$ , QuACK takes  $T_{Q,t} = T_{Q,1} + T_{Q,2}$  steps where  $T_{Q,1}(T_{Q,2})$  are the total numbers of QuACK training (prediction) steps. We denote the ratio between the computational costs of the baseline and QuACK as  $s$ , which serves as the speedup ratio from QuACK. Our definition of speedup has some difference with the speedup defined in Ref. [16] although shares a similar spirit. In gradient-based optimization, the computational costs of each step of QuACK training and prediction are different, with their ratio defined as  $f(p)$  where  $p \coloneqq n_{\mathrm{params}}$ . In general,  $f(p)$  is  $\Omega(p)$  since only QuACK training, not prediction, involves gradient steps.
Theorem 5.4. With a baseline gradient-method, the speedup ratio  $s$  is in the following range
$$
a \leq s \leq a \frac {f (p) \left(n _ {\text {s i m}} + n _ {\mathrm {D M D}}\right)}{f (p) n _ {\text {s i m}} + n _ {\mathrm {D M D}}}. \tag {3}
$$
where  $a \coloneqq T_b / T_{Q,t}$ . In the limit of  $n_{\mathrm{iter}} \to \infty$  the upper bound can be achieved.
We present the proof in the Appendix.  $a$  is a metric of the accuracy of Koopman prediction. Higher  $a$  means better prediction, and a perfect prediction has  $a = 1$ . The exact form of  $f(p)$  depends on the details of the gradient method. For example, we have  $f(p) = 2p + 1$  for parameter-shift rules and  $f(p) \sim p^2 + p$  for quantum natural gradients [49]. If  $a$  is fixed, then the upper bound of  $s$  in Eq. 13 increases when  $p$  increases with an asymptote  $a(n_{\mathrm{sim}} + n_{\mathrm{DMD}}) / n_{\mathrm{sim}}$  at  $p \to \infty$ . The upper bound in Eq. 13 implies  $s \leq af(p)$  where the equal sign is achieved in the limit  $n_{\mathrm{DMD}} / n_{\mathrm{sim}} \to \infty$ . If  $a = 1$  one could in theory achieve  $f(p)$ -speedup, at least linear in  $n_{\mathrm{params}}$ .
In practice, the variable  $a$  is influenced by  $n_{\mathrm{sim}}$  and  $n_{\mathrm{DMD}}$ . A decrease in  $a$  can occur when  $n_{\mathrm{sim}}$  diminishes or  $n_{\mathrm{DMD}}$  enlarges, as prediction accuracy may drop if the optimization dynamics length for QuACK training is insufficient or the prediction length is too long. Moreover, estimating the gradient for each parameter requires a specific number of shots  $n_{\mathrm{shots}}$  on a fixed quantum circuit. Quantum measurement precision usually follows the standard quantum limit  $\sim 1 / \sqrt{n_{\mathrm{shots}}}$ , hence a finite  $n_{\mathrm{shots}}$  may result in noisy gradients for Koopman operator learning, influencing  $a$ . Intriguingly, when the prediction aligns with the pure VQA, QuACK's loss might decrease faster than pure baseline VQA, leading to  $a > 1$  and a higher speedup. This could be due to dominant DMD spectrum modes with large eigenvalues aligning with the direction of fast convergence in the  $\theta$ -space.
We adopt the Relative loss metric for benchmark, which is  $(\mathcal{L} - \mathcal{L}_{\mathrm{min,fullVQA}}) / (\mathcal{L}_{\mathrm{initial,fullVQA}} - \mathcal{L}_{\mathrm{min,fullVQA}})$ , where  $\mathcal{L}$  is the current loss, and  $\mathcal{L}_{\mathrm{initial,fullVQA}}$  and  $\mathcal{L}_{\mathrm{min,fullVQA}}$  are the initial and minimum loss of full VQA. We use pure VQE [59] and pure QML [63] as the baseline. We use  $\mathcal{L}_{\mathrm{target}}$  at  $1\%$  relative loss for VQE and the target test accuracy (0.5% below maximum) for QML to compute the computational costs and the speedup  $s$  as a metric of the performance of QuACK. Our experiments are run with Qiskit [2], Pytorch [57], Yao [47] (in Julia [7]), and Pennylane [6]. Details of the architectures and hyperparameters of our experiments are in Appendix. More ablation studies for hyperparameters are also in Appendix.
Quantum Ising Model. Quantum Ising model with a transverse field  $h$  has the Hamiltonian  $\mathcal{H} = -\sum_{i=1}^{N} Z_i \otimes Z_{i+1} - h \sum_{i=1}^{N} X_i$  where  $\{I_i, X_i, Y_i, Z_i\}$  are Pauli matrices for the  $i$ -th qubit. On the quantum circuit, we then use the RealAmplitudes ansatz and hardware-efficient ansatz [31]
(a) Quantum natural gradients.
(b) Overparameterization.
(c) Smooth optimization regimes.
Figure 2: Performance of our QuACK with the standard DMD in the following cases. (a) For quantum natural gradient, with short training (4 steps per piece) and long prediction (40 steps per piece), DMD accurately predicts the intrinsic dynamics of quantum optimization, and QuACK has  $20.18\mathrm{x}$  speedup. (b) In the overparameterization regime, QuACK has  $>200\mathrm{x}$  speedup with 2-5 qubits. (c) In smooth optimization regimes, QuACK has  $>10\mathrm{x}$  speedup with 2-12 qubits.
(a) LiH molecule.
Figure 3: Experimental results for (a) LiH molecule with 10 qubits using Adam (b) Quantum Ising model with 12 qubits using Adam (c) test accuracy of binary classification in QML. The solid piecewise curves are true gradient steps, and the dashed lines connecting them indicate when the DMD prediction is applied to find  $\theta(t_{\mathrm{opt}})$  in our controlled scheme. Our QuACK with all the DMD methods bring acceleration, with maximum speedups (a)  $4.63\mathrm{x}$  (b)  $3.24\mathrm{x}$  (c)  $4.38\mathrm{x}$ .
(b) Quantum Ising model.
(c) Quantum machine learning.
which then define  $\psi_{\theta}$ , and then  $\mathcal{L}(\pmb{\theta}) = \langle \psi_{\pmb{\theta}}|\mathcal{H}\psi_{\pmb{\theta}}\rangle$ . We implement the VQE algorithm and QuACK for  $h = 0.5$  using gradient descent, quantum natural gradient, and Adam optimizers.
Quantum Chemistry. Quantum chemistry is of great interest as an application of quantum algorithms, and has a Hamiltonian  $\mathcal{H} = \sum_{j=0}^{n_P} h_j P_j$  with  $n_P$  polynomial in  $N$ , where  $P_j$  are tensor products of Pauli matrices, and  $h_j$  are the associated real coefficients. The loss function of quantum chemistry is typically more complicated than the quantum Ising model. We explore the performance of QuACK by applying it to VQE for the 10-qubit Hamiltonian of a LiH molecule with a interatomic distance 2.0 Å provided in Pennylane, with Adam and the RealAmplitudes ansatz.
Quantum Machine Learning. In addition to VQE, we consider the task of binary classification on a filtered MNIST dataset with samples labeled by digits "1" and "9". We use an interleaved block-encoding scheme for QML, which is shown to have generalization advantage [29, 13, 40, 63] and recently realized in experiment [62]. We use a 10-qubit quantum circuit with stochastic gradient descent for QML and QuACK, and a layerwise partitioning [16] in  $\theta$  for neural DMD. Similar to the target loss, we set the target test accuracy as  $0.5\%$  below the maximum test accuracy to measure the closeness to the best test accuracy during the QML training, and use the ratios of cost between pure QML and QuACK achieving the target test accuracy as the speedup ratio.
6.2 Accurate Prediction for Quantum Natural Gradients by QuACK
In Figure 2a, we present the 5-qubit quantum Ising model results (learning rate 0.001) with the standard DMD method in the QuACK framework. We observe that the DMD prediction is almost perfect, i.e.  $a \approx 1$ . The speedup in Figure 2a is 20.18x close to 21.19x, the theoretical speedup from the upper bound in Eq. 13 under  $a = 1$ . This experiment shows (1) the success of DMD in predicting the dynamics (2) the power of QuACK of accelerating VQA (3) the precision of our complexity and speedup analysis in Sec. 5.2. We show 10-qubit results for all DMD methods in the Appendix.
Noise System	n_shots	Speedup
DMD (ours)	SW- DMD (ours)	MLP- DMD (ours)	MLP- SW-DMD (ours)	CNN- DMD (ours)
10-qubit shot noise	100	5.51x	5.54x	3.25x	3.23x	1.99x
1,000	3.20x	4.36x	2.38x	4.36x	2.54x
10,000	1.59x	3.49x	1.59x	2.41x	1.96x
5-qubit shot noise	100	1.50x	2.64x	1.83x	4.57x	1.47x
1,000	1.39x	3.84x	1.45x	2.25x	1.95x
10,000	1.49x	3.43x	1.80x	2.00x	1.78x
5-qubit FakeLima	100	2.44x	2.24x	2.43x	2.15x	2.34x
1,000	1.50x	2.61x	2.07x	2.41x	1.84x
10,000	1.48x	2.32x	1.93x	2.56x	1.92x
5-qubit FakeManila	100	2.14x	2.51x	2.54x	2.91x	1.25x
1,000	1.49x	2.02x	1.90x	2.09x	1.89x
10,000	1.95x	2.13x	2.13x	2.27x	1.82x
Table 1: Speedup ratios of our QuACK with all DMD methods for various noise systems.
Figure 4: Noisy quantum optimization with  $n_{\mathrm{shots}} = 100$ .
6.3 More than 200x Speedup near the Overparametrization Regime by QuACK
As it is used by the overparameterization theory in You et al. [87], we use the 250-layer hardware-efficient ansatz [31] on the quantum Ising model with gradient descent, with numbers of qubits  $N = 2,3,4,5$  so  $n_{\mathrm{params}} = 1000,1500,2000,2500$ , all in the regime of  $n_{\mathrm{params}} \geq (2^N)^2$ , near the overparameterization regime [37]. In addition, we use a small learning rate 5e-6 so that the dynamics in the  $\theta$ -space is approximately linear dynamics. We expect QuACK with the standard DMD to have good performance in this regime. In Figure 2b, for each  $n_{\mathrm{params}}$ , we randomly sample 10 initializations of  $\theta$  to calculate mean values and errorbars and obtain  $>200x$  speedup in all these cases. The great acceleration from our QuACK is an empirical validation and an application of the overparameterization theory for quantum optimization. On the other hand, the overparameterization regime is difficult for near-term quantum computers to realize due to the large number of parameters and optimization steps, and our QuACK makes its realization more feasible.
6.4 More than 10x Speedup in Smooth Optimization Regimes by QuACK
For the quantum Ising model with gradient descent and 2-layer RealAmplitudes, rather than the overparameterization regime, we consider a different regime with lower  $n_{\mathrm{params}}$  with learning rate 2e-3 so that the optimization trajectory is smooth and the standard DMD is still expected to predict the dynamics accurately for a relatively long length of time. With the number of qubits  $N \in [2,12]$  ( $n_{\mathrm{params}} \in [4,24]$ ) and 100 random samples for initialization of  $\theta$  for each  $n_{\mathrm{params}}$ , in Figure 2c, our QuACK achieves  $>10x$  speedup in all these cases. This shows the potential of our methods for this regime with fewer parameters than overparameterization, which is more realistic on near-term quantum hardware.
6.5 More than 3x Speedup in Non-smooth Optimization by QuACK
We further demonstrate speedup from QuACK in the non-smooth optimization regimes with learning rate 0.01 for examples of quantum Ising model, quantum chemistry, and quantum machine learning with performance shown in Figure 3 with numerical speedups in Appendix. Only the gradient steps are plotted, but the computational cost of QuACK prediction steps are also counted when computing speedup. All the 5 DMD methods are applied to all the examples. Our QuACK with all the DMD methods accelerates the VQA, with maximum speedups (a) LiH molecule:  $4.63\mathrm{x}$  (b) quantum Ising model:  $3.24\mathrm{x}$  (c) QML:  $4.38\mathrm{x}$ . These applications are of broad interest across different fields and communities and show that our QuACK works for a range of loss functions and tasks.
6.6 2x to 5.5x Speedup from Ablation the Robustness of QuACK to Noise
Near-term quantum computers are imperfect and have two types of noise: (1) shot noise from the probabilistic nature of quantum mechanics, which decreases as  $1 / \sqrt{n_{\mathrm{shots}}}$ , (2) quantum noise due to the qubit and gate errors which cannot be removed by increasing  $n_{\mathrm{shots}}$ . Therefore, we consider two categories of ablation studies (1) with shot noise only (2) with both shot noise and quantum noise. We apply our QuACK with all 5 DMD methods to 4 types of noise systems: 10-qubit and 5-qubit systems with only shot noise, FakeLima, and FakeManila. The latter two are noise models provided by Qiskit to mimic 5-qubit IBM real quantum computers, Lima and Manila, which contain not only shots noise but also the machine-specific quantum noise. We use the quantum Ising model with Adam and show generic dynamics in Figure 4 with statistical error from shots as error bands. The fluctuation and error from baseline VQE in Figure 4(a) 100-shot 10-qubit system are less than Figure 4(b) 100-shot FakeManila. Our QuACK works well in (a) up to  $5.54\mathrm{x}$  speedup and in (b) up to  $2.44\mathrm{x}$  speedup. In all examples, we obtain speedup and show them in Table 1, which demonstrate the robustness of our QuACK in realistic setups with generic noisy conditions. We have also implemented an experiment to accelerate VQE on the real IBM quantum computer Lima with results in Appendix.
We developed QuACK, a novel algorithm that accelerates quantum optimization. We derived QuACK from connecting quantum natural gradient theory, overparameterization theory, and Koopman operator learning theory. We rigorously tested QuACK's performance, robustness, spectral stability, and complexity on both simulated and real quantum computers across a variety of scenarios. Notably, we observed orders of magnitude speedups with QuACK, achieving over 200 times faster in overparameterized regimes, 10 times in smooth regimes, and 3 times in non-smooth regimes. This research highlights the significant potential of Koopman operator theory for accelerating quantum optimization and lays the foundation for stronger connections between machine learning and quantum optimization. We elaborate more on the broader impact of our work in the Appendix.
The authors acknowledge helpful discussions with Hao He, Charles Roques-Carmes, Eleanor Crane, Nathan Wiebe, Zhuo Chen, Ryan Levy, Lucas Slattery, Bryan Clark, Weikang Li, Xiuzhe Luo, Patrick Draper, Aida El-Khadra, Andrew Lytle, Yu Ding, Ruslan Shaydulin, Yue Sun. DL, RD and MS acknowledge support from the NSF AI Institute for Artificial Intelligence and Fundamental Interactions (IAIFI). DL is supported in part by the Co-Design Center for Quantum Advantage (C2QA). JS acknowledges support from the U.S. Department of Energy, Office of Science, Office of High Energy Physics QuantISED program under an award for the Fermilab Theory Consortium "Intersections of QIS and Theoretical Particle Physics". This material is also in part based upon work supported by the Air Force Office of Scientific Research under the award number FA9550-21-1-0317. We acknowledge the use of IBM Quantum services for this work. The views expressed are those of the authors, and do not reflect the official policy or position of IBM or the IBM Quantum team. In this paper we used ibmq_lima, which is one of the IBM Quantum Falcon Processors.
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The structure of our Appendix is as follows. Appendix A gives an introduction to quantum mechanics and the overparameterization theory as preliminaries of our paper. Appendix B provides more details of our QuACK framework introduced in Sec. 4 of main text. Appendix D provides proofs of our theoretical results in Sec. 5 of main text. Appendix E discusses detailed information on Sec. 6 Experiments of main text. Appendix F contains additional information.
A.1 Introduction to Quantum Mechanics for Quantum Computation
We introduce quantum mechanics for quantum computation in this section. In quantum mechanics, the basic element to describe the status of a system is a quantum state (or a wave function)  $\psi$ . A pure quantum state is a vector in a Hilbert space. We can also use the Dirac notation for quantum states  $\langle \psi |$  and  $|\psi \rangle$  to denote the row and column vectors respectively. By convention,  $\langle \psi |$  is the Hermitian conjugate (composition of complex conjugate and transpose) of  $|\psi \rangle$ .
In a quantum system with a single qubit, there are two orthonormal states  $|0\rangle$  and  $|1\rangle$ , and a generic quantum state is spanned under this basis as  $|\psi \rangle = c_0|0\rangle + c_1|1\rangle$  where  $c_0, c_1 \in \mathbb{C}$ . Therefore,  $\psi$  itself is in a 2-dimensional Hilbert space as  $\psi \in \mathbb{C}^2$ . For a quantum mechanical system with  $N$  qubits, the basis of quantum state contains tensor products (Kronecker products) of the single-qubit bases, i.e.,  $|b_1\rangle \otimes |b_2\rangle \otimes \dots \otimes |b_N\rangle$ , where  $b_i \in \{0,1\}$  ( $i = 0,1,\dots,N$ ). There are in total  $2^N$  basis vectors, and  $\psi$  is a linear combination of them with complex coefficients. Therefore,  $\psi$  for an  $N$ -qubit system is in a  $2^N$ -dimensional complex-valued Hilbert space  $\mathbb{C}^{2^N}$ . For a normalized quantum state, we further require the constraint  $\| \psi \|_2^2 = 1$  where  $\| \cdot \|_2$  is the  $l_2$ -norm.
For the  $N$ -qubit system, the Hamiltonian  $\mathcal{H}$  is a Hermitian  $2^{N} \times 2^{N}$ -matrix acting on the quantum state (wave function)  $\psi$ . The energy of  $\psi$  is given by  $\mathcal{L}(\psi) = \langle \psi | \mathcal{H} \psi \rangle$ , which is the inner product between  $\langle \psi|$  (a row vector, the Hermitian conjugate of  $|\psi\rangle$ ) and  $|\mathcal{H} \psi\rangle$  (matrix multiplication between the matrix  $\mathcal{H}$  and the column vector  $|\psi\rangle$ ).
Pauli matrices (including the identity matrix  $I$  by our convention) are the  $2 \times 2$  matrices
$$
I = \left[ \begin{array}{l l} 1 & 0 \\ 0 & 1 \end{array} \right], \quad X = \left[ \begin{array}{l l} 0 & 1 \\ 1 & 0 \end{array} \right], \quad Y = \left[ \begin{array}{l l} 0 & - i \\ i & 0 \end{array} \right], \quad Z = \left[ \begin{array}{l l} 1 & 0 \\ 0 & - 1 \end{array} \right]. \tag {4}
$$