Quantum Fourier Networks for solving Parametric PDEs
Authors: Nishant Jain
Jonas Landman
Natansh Mathur
Iordanis Kerenidis
Venue: Submitted to ICLR 2023 (2023)
Keywords: quantum computing
quantum machine learning
quantum deep learnin
fourier transform
fourier neural operator
PDE
partial differential equation

Abstract: Many real-world problems like modelling environment dynamics, physical processes, time series etc., involve solving Partial Differential Equations (PDEs) parameterized by problem-specific conditions. Recently, a deep learning architecture called Fourier Neural Operator (FNO) proved to be capable of learning solutions of given PDE families, for any initial conditions as input. Given the advancements in quantum hardware and the recent results in quantum machine learning methods, we propose three quantum circuits, inspired by the FNO, to learn this functional mapping for PDEs. The proposed algorithms are distinguished based on the trade-off between depth and their similarity to the classical FNO. At their core, we make use of unary encoding paradigm and orthogonal quantum layers, and introduce a new quantum Fourier transform in the unary basis. With respect to the number of samples, our quantum algorithm is proven to be substantially faster than the classical counterpart. We benchmark our proposed algorithms on three PDE families, namely Burger’s equation, Darcy’s flow equation and the Navier-Stokes equation, and the results show that our quantum methods are comparable in performance to the classical FNO. We also show an analysis of the image classification tasks where our proposed algorithms are able to match the accuracy of the CNNs, thereby showing their applicability to other domains.

Under review as a conference paper at ICLR 2023
QUANTUM FOURIER NETWORKS FOR SOLVING PARA- METRIC PDEs
Anonymous authors Paper under double-blind review
ABSTRACT
Many real-world problems like modelling environment dynamics, physical pro- cesses, time series etc., involve solving Partial Differential Equations (PDEs) pa- rameterized by problem-specific conditions. Recently, a deep learning architec- ture called Fourier Neural Operator (FNO) proved to capable of learning solu- tions of given PDE families, for any initial conditions as input. Given the advance- ments in quantum hardware and the recent results in quantum machine learning methods, propose three quantum circuits, inspired by the FNO, to learn this functional mapping for PDEs. The proposed algorithms are distinguished based the trade-off between depth and their similarity to the classical FNO. At their core, we make use of unary encoding paradigm and orthogonal quantum layers, and introduce a new quantum Fourier transform in the unary basis. With respect the number of samples, our quantum algorithm is proven to be substantially faster than the classical counterpart. We benchmark our proposed algorithms on three PDE families, namely Burger's equation, Darcy's 's flow equation and the Navier- Stokes equation, and the results show that our quantum methods are comparable in performance to the classical FNO, We also show analysis of the image clas- sification tasks where our proposed algorithms are able to match the accuracy the CNNs, thereby showing their applicability to other domains.
INTRODUCTION
Solving Partial Differential Equations (PDEs) has been a very crucial step in understanding the dynamics of nature. They have been widely used to understand natural phenomenon such as heat- transfer, modelling the flow of fluids, electromagnetism, etc. and lately have also found their ap- plications in understanding the behavior of dynamical markets. In most of these applications, a closed form solution is difficult to find for the resulting PDEs and thus, classical solvers require lot of evaluations model the solution for a given PDE. To approximate such not-so-easily solvable PDE's, s, there has been an extensive research based on neural networks. group of these methods Yu et al. (2018); RRaissi et al. (2019); Bar & Sochen (2019) aimed at learning the solution function for an instance of PDE and thus requires to be re- re-trained every time the paramoters/conditoon in the PDE change. The other set of methods Zhu & Zabaras 2018); ;; Adler & Öktem (2017); BBhann- gar et al. (019)) targeted at learning over a family of PDES but for a specific resolution dependent, making these methods limited to the discretization the sampling density used in the training data. A recent work Li et al. (2020) addressed both these issues and posed the problem as learning a function-to-functioo mapping for parametric PDEs. Experiments
widely popular PDEs showed that it was effective in learning the mapping from a parametric initial condition function to the so- lution operator for a family of PDEs. The method proposes a Fourier layer, which uses a learnable linear transform sandwiched between Fourier Transform (FT) and an Inverse Fourier Transform (TFT) operation. This is similar to the convolution operation as it also translates to multiplication in the Fourier space.
The major bottleneck which might hinder the scalability of this classical Fourier Neural Operator (FNO) is its time complexity, limited by the classical Fourier Transform (FT) and Inverse Fourier Transform (IFT) operations inside the Fourier Layer. As shown previously by Musk (000)). these operations are much faster when deployed quantum hardware. Similar advantages have led to significant developments in learning approaches based on near term quantum computing. The initial demonstrations of these algorithms involved experiments small-scale hardware Farhi & Neven

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2018 ; Coyle et al. (2000; Cappelletti et al. (2020); Grant et al. (2018) which established their effectiveness in extracting patterns. Following this, many works Abbas et al. (2021): Mari et al.
2220); |Beer et al. (2020); Allcock et al. (2020) proposed small-scale implementations of fully con-
nected quantum neural networks on near term hardware. Other proposals Kerenidis et al. (2019);
Cong al. (2019) for deploying convolution-based learning methods on quantum devices showed effective training in practice. Furthermore, (Chakrabarti et al. (2019) proposed quantum wrantum-hardware implementation for generative adversarial networks. A different approach, where the inputs are en- coded as unary states, using the two-qubits quantum gate RBS (Reconfigurable Beam Splitter) was proposed in a recent work Johri et al.] (202T). This encoding gave rise to the use of orthogonal properties of pure quantum unitaries, as proposed in Kerenidis et al. (2021) for training, for in- stance, orthogonal feed-forward networks to damp the gradient based issues while learning. It used pyramid circuit based on parameterized RBS gates to implement learnable orthogonal matrix as compared to the existing classical approaches which offer approximate orthogonality at the cost of increased training time. This orthogonality in neural networks results in much smoother conver- gence and also lesser parameters as shown by Li et al. (2019) for feed-forward neural networks and Wang et al. (2020) for convolutional nets. The effectiveness of these orthogonal quantum networks was further shown in another work on medical image classification Mathur et al. (2021) problem. We also use a similar idea to do the multi-channel intermediate linear transform using orthogonal matrix but using parameterized butterfly circuits instead of pyramid circuits.
Exploiting these advantages offered by quantum computing, this work, we propose a new kind of Quantum Fourier Transform (QFT), which operates on the unary states and a Icarnable Quantum Linear Transform. We further propose three quantum algorithms inspired by this classical Fourier operation which are faster than the classical operation and require fewer parameters for the same architecture, thereby boosting their scalability. Given the input of dimension N, x No, where N, corresponds to number of samples per PDE and No correspond to feature dimension, the order of time complexity corresponding to Fourier Layer (FL) and proposed algorithms is shown in table T
Table 1: Comparison of order of time/depth complexities (O) of the proposed circuits with the existing clas- sical Fourier Layer (FI.). Here N, denote the sampling dimension, N. denote the feature dimension where N. > No and K (usually in range 4-16) denotes the maximum number of modes allowed [i et al. (2020). This implies that the proposed quantum algorithms would faster than the classical method. Each quantum circuit requires No + N. qubits and K independent parallel circuits are required by the Parallelized QENO.
Method Complexity qubits parallel circuits
Classical FL. Sequential Quantum FL. Parallel Quantum FI. Compound Quantum FI. (log(N.)+ N₂ N,
NogNN))
N. No + N,
log(N,+ K,NNN, No + N,
K
The first algorithm replicates the classical operation on a quantum circuit. The other two algorithms are modifications of the first circuit designed for the noisy learning process offered by the near term quantum hardware. We test all the three proposed algorithms on all the three PDES evaluated in the classical FNO paper Li et al. (2020) namely the Burgers equation, Darcy's Flow equation and Navier Stokes equation on the synthetic datasets used in that paper. We also test our algorithms against the Convolutional neural networks (CNNs) on benchmark datasets for image classification namely MNIST, MASSIO Xiao et al.] (2017), Pneumonia-MNISs all the experiments, three algorithms perform similarly and comparable to state-of-the-an art FNO for PDEs. Also, they perform decently on the image classification tasks.
Yang et al. (2021). In
2 CLASSICAL FOURIER NEURAL OPERATOR
Given a training set comprising the family of a Partial Differential Equation, the classical FNO Li al. (2020) aims to learn a functional mapping from a parameterized initial condition to the solu- tion function for this family. This means given an initial condition function characterizing PDE instance, sampled at different points, it should predict the solution function values at those points at the inference time. To formulate it, given two functional spaces A and U along with a set of observations {az. uy} } (a, ~ 1 is an i.i.d. sequence sampled from some function f E A)), it learns parametric mapping G A x θ U. To achieve this, it proposed learning network based on iteratively applying a new kind of layer which it termed as the Fourier Layer. The layer consists of two parts, the top part involves firstly projecting the input the Fourier domain and then applying
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N A
festres (N))
simppes (N.)
X
Ney
QFT
10F1
(a) Left: Classical Fourier Layer
(b) Right: Sequential Quantum Circuit for FNO.
Figure Left: Intermediate Linear Transform operation performed in the classical FNO, after applying the Fourier Transform Li ct al. (2020). As shown, it performs linear transform on the features corresponding to the first K modes the Fourier Transformed input. Right: The proposed Sequential Quantum Circuit which replicates the classical FNO operation. Further details regarding are given Section 3.2.I] The dashed box incorporates the controlled butterfly incorporating a controlled norm-loader and the parametric butterfly (Pk).
a linear transform (refer fig. D to first K modes and crop the rest and then reproject back to the original domain. This is somewhat similar to convolution but the updates would be taking place in the Fourier space. The lower part is just a simple convolution and doesn't play any significant role in improving performance. In this paper, propose purely quantum algorithms for this top part as a Quantum Fourier Layer and term the network as Quantum Fourier Neural Operator (QFNO). Now, look the mathematical details of the classical Fourier layer for ID PDE Burger's s equation), showing the inputs and outputs of each transforration involved. Given a clas- sical input x N&N, where N₂ corresponds the number of channels/features in the input and N, corresponds to number of samples/observations per function corresponding to a PDE, we denote the corresponding output of this classical operation by RNN.N., As the quantum matri- ces are orthogonal and the 12-norm of any quantum state vector is 1, we first normalize the input (normalization doesn't 't have any significant impact on the optimization process) as follows:
(eg-
Fij
ai
||>||
(1)
Now, following the classical operation, applying the Fourier Transform (FT) along the second di-
mension to this input (N,), FT((;) of where a, (a.)),e(1.N.) and of denote the corresponding fourier-transformed coefficients. Denoting the maximum number of modes with K, intermediate linear transform is a matrix W E x....N and W* E R.N. denotes this matrix W indexed along the last dimension (corresponding to 1hth mode). Here, we assume this matrix W is orthog- onal and thus, can be implemented by quantum layer. This transformation is now applied to the first K modes along the N, dimension resulting in the following output:
(Waa))eenLkK ( (a5)sE[K(L.N.)
where the vector af (a())EE(I,N)) In the classical paper, they discard the rest of modes (make the vectors zero) and we leave them unchanged here. We verified that this doesn't impacts the optimization significantly. For these transformed K vectors of e R-N,, we define bf = W or more explicitly b/8 (M"(WWa. Given W', a, and by. we can now write the input after Fourier Transform and the intermediate linear transform as:
(0/) (b))SEL.K)) (c)))e[K+
(3)
Finally, applying Inverse Fourier Transform (IFT) operation on this transformed input, along the N, dimension, results the following output as the output the complete operation:
yi IFT ([06))EEAAR)) (at (aL))e(K+L..)))
where y: (B5)) The Time Complexity of this complete Fourier Layer (FT+linear Iransform+/FT) is O(Ne Nylog(N.)).
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3 QUANTUM FOURIER NETWORKS
In this section we propose three quantum circuits to replace the Fourier layer of the FNO, namely the Sequential (Section 3.2.I), Parallel (Section 3.2.2p. and the Compound (Section 3.2.3 We compare their computational complexities (see Table I and their efficiency in practice in the following sec- tions, Before this, we also define a learnable orthogonal linear transform, using quantum circuit, called parameterized butterffy. Using this as the building block, we propose the Quantum Circuit to carry out the intermediate linear transform proposed in the classical FNO (Section 2
3.1 QUANTUM BUTTERFLY CIRCUITS
An efficient implementation of FFT, proposed by Cooley & Tukey (1965), involves connections in shape of radix-2 butterflies resulting in a logarithmic depth complexity. Täking inspiration from this, propose such butterfly shaped Quantum Circuits carry out: a) A learnable linear transform and b) A Quantum Fourier Transform (QFT), both operating only on the unary states. Here each radix- 2 operation is implemented using RBS-gates Foxen et al. (2020). Refer supplementary for details regarding the unitary corresponding to this gate.
Parameterized Quantum Butterfly. Due the logarithmic depth offered by the butterfly circuits for FT, which is important given the noisy near-term quantum hardware, we propose such a butterfly shaped Quantum Circuit to carry out a learnable linear transform. Here, each radix-2 -2 butterfly shaped connection is replaced by parameterized RBS gates similar to concurrent work [Chrraa] jet al. (2022). This results in a orthogonal linear transform only on the unary quantum states similar to the pyramidal circuit proposed by Kerenidis et al. (2021). We further propose controlled version of these linear quantum layers to perform the operation similar to the intermediate linear transform operation in the fourier layers.
Unary Quantum Fourier Transform. Here, we propose a butterfly-shaped Quantum Circuit to earry out the QFT unary states. It involves replacing each radix-2 operation in the Butterfly FT by single qubit gate and an RBS-gate with -T/4 /4 as the angle. Refer supplementary for more details and circuit diagrams for the Parameterized Butterfly layer and Unary QFT.
3.2 QUANTUM CIRCUITS FOR FOURIER LAYER
To perform the classical FNO operation using quantum circuit, the final output state of the circuit should correspond to the quantum-state encoding of the output resulting from this classical oper- ation. Therefore, we load this classical output into a quantum state and for this we use the unary datalouding similar to the one used in Johri et al. (202I) due to its logarithmic depth. loads an N -ddimensional classical vector into normalized N quantum states corresponding to the unary rep- resentation of numbers from 1 to N. The circuit consists of only the Reconfigurable Beam Splitter (RBS) gates. Using controlled version of this circuit, matrix loader can be defined which dis- cuss in the supplementary and it loads the normalized vectors comprising the matrix into quantum states. Thus, for the classical output y (y1, ,N.,, the encoded quantum state (y) is:
No. N.
(v) = Σ lee) (yx) = Σ Σ IFT ((bb)eeenn
((t))
les) lez)
(5)
where le,) denotes the state in unary notation with 1 being the 7th position. Note there is no normalization factor, as the inputs (a1,... aN.) were assumed normalized in the previous subsec- tion. Now, we discuss the three proposed circuit namely sequential, parallelized and the compound circuit. The sequential circuit replicates the classical operation discussed above and the other two are modifications to it to make the quantum algorithm more efficiently deployable on hardware.
3.2. SEQUENTIAL QUANTUM CIRCUIT FOR FOURIER LAYER
Figure shows the diagram for this circuit. Lower register of N, qubits correspond to the second dimension and upper register of N₂ qubits correspond to the N. dimension used in the above math- ematical description of the classical operation. We begin by discussing the quantum dataloading followed by quantum transforms corresponding to the classical ones formulated above. Dataloading. Similar the quantum-state encoding of the classical output defined above, here also
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the unary matrix loader to load the given input into quantum state. As shown in Figure I the circuit initially contains Ne controlled unary dataloaders, each encoding a row of N₂ inputs into the quantum state the lower register. The loaded state, after controlled operation Xv, in the circuit, can be written as:
Nc ΣΣ ai (e,) le,)
(6)
where the coefficients aij correspond to normalized matrix elements. Denoting the given classical input as x E N.N.N we have (refer supplementary for details):
Rij
|
Now, to apply the QFT operation on lower register, the state can be re-arranged as follows:
N.
N.
Σ
lea)
Σ
ai (ej)))))))))
(8)
Quantum Fourier Transform (QFT). Given a normalized real vector a = (z:)i=-1...Mr M. and its Fourier Transform x²= )i=)i,,.. Mr the QFT operation in unary basis and its inverse, the IQFT operation, can be defined as follows:
QFT 24 (e))) = Σ lee) and IQFT Σ x (e)) = Σ xs les)
(9)
Applying this QFT operation to the lower register, on the state in eq. we have:
8
No Σ
N.
N.
No
les QFTQ
au (ej)) Σ
o le,) = Σ
Σ
o lei) lez)
(10)
Quantum Linear Transform. To realize the intermediate transform used in the classical operation, we implement an orthogonal matrix W*, corresponding to the one in the classical operation, and realize it using a parameterized butterfly circuit. Furthermore, we propose "k-butterfly" which is a parameterized butterfly circuit (P&) on the top register controlled by the 1tth qubit of the lower register. As shown in Figure [ it involves using an ancilla qubit (top-most) initialized to state [1). When this qubit is in state [1), then combining it with upper register results in hamming weight basis (/2)). Any transformation on this using the RBS gates would result in an another state of the hamming weight 2 basis. We can ignore all these hamming weight states using a post select operation. To apply the k-butterfly, we flip this qubit back to if the 4tth qubit of the lower register is (1), using CNOT gate. Thus, all the states of upper qubits are in (b2) basis, except for the ones corresponding to thth unary state of the lower register:
No de))))))) >E
a 10) les) (eg) +
ΣΣ 57t
20 le)
(11)
j=k
Now, applying K such k- k-butterfly circuits, perform the operation P, both the unary and the (h2) states. We consider the unary states in our analysis since we discard the states in other basis:
of
o 0) lei))
Σ j=K+1
a (e)))
(12)
Considering only the top register, the k-butterfly operation P' corresponds to the sub-matrix W N.N.N The overall matrix (a/)) can be decomposed into N, vectors af = (af))ee),Na) Then, for the first vectors of E Ne we will have bf = Wia where by will be the same as in the classical case. Finally, the output state of this circuit after IQFT the lower register becomes:
No
N.
Σ
ee) IQFT
Σ
M (
Σ
a los))))
(13)
-=K+1
Since IQFT(E (e.)) > (e)), where IFT(x!) = x, this implies that jth component of IFT would be same as the coefficient of jth state in IQFT. From this we can conclude that the state in eq. is equivalent to the state in eq. 13 and thus, this circuit replicates the classical operation. The depth complexity of this circuit is O(KK + 1))og((N)) + (Ne + 2))))
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[x
P
Konmiiie Dunatt
Perrzetrind
Jemple (h)
94
OFF
Munple (N)
I0F
Ng Utory
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Datalaaton
Datikascen
(a) Left: Parallelized QENO Circuit
(b) Right: Compound QENO Circuit
Figure 2: Left: Parallelized version of the Sequential Quantum Circuit to minimize the depth of the learning part, thus making it more efficient when deployed the noisy hardware. For each mode (out of the top K) in the transformed input, there is different circuit to perform the parameterized matrix transform. Right: Another variant of the sequential circuit where instead of controlled but- terfly circuits, there is a Compound Butterfly Circuit spanning upper register and top K qubits of lower register.
3.2.2 PARALLELIZED QUANTUM CIRCUIT FOR FOURIER LAYER
Given the multiplicative noise model for the NISQ devices, the depth of the learnable part, which is proportional K for the sequential circuit, might be a hindrance to learning and also increases the computation time. A useful modification then can be parallelizing the learnable butterfly circuits. Figure Suureessooe shows this modified version of the sequential circuit, which consists of quantum circuits operating parallely and each implementing only one learnable circuit controlled by one of the top K qubits in the lower register. As all the circuits upto the learnable part are similar to the sequential circuit, we can directly write the state after the QFT, using eq. I0 as:
N N
Σ
Σ
(off)e lev) les)
(14)
k=1
where the index i denotes the 4 parallel circuit. Also, in the 4th parallel circuit, the learnable butterfly part is controlled by the ws qubit of the lower register. We recall that the pyramid applied on the top register is effectively mapping the vector of to by (see Eq updated state of the circuits as:
13 and thus we can write the
N. Eoffdeeed lex)
ΣΣ
(a led les) +
(15)
Jyk
k=1
Now, applying IQFT on the lower register in each of the circuits independently:
K
Σ ledh IQFT
(b/wi ekd + E(afyl les)
(16)
jtk
In supplementary, derive that this joint output of these circuits cannot lead the output of the classical operation even after measurement and classical post-processing. We also show that how measuring without applying the TQFT operation and instead applying the classical IFT, after some post-processing, can lead to the same output as the classical operation. The depth of this parallel version of sequential circuit is (2)log(N, (N)) + (No + 2)log(NN, and a total of K quantum circuits required to execute this parallely.
3.2.3 COMPOUND QUANTUM CIRCUIT FOR FOURIER LAYER
As highlighted in previous subsection, depth of the parameterized part of the sequential circuit might make the learning process difficult currently available noisy quantum hardware. To deal with

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this, we propose another variant of the sequential circuit, where, instead of having the parameterized circuit controlled by top K qubits of lower register, we span à a single bigger pyramid over the upper register (N₂) and top K qubits in the lower register. Note the upper and lower registers are unary independently. Therefore, if we jointly consider the upper register and top K from the lower register, then there would be states with hamming weight two as well, corresponding the scenario when is in one of the top K qubits of the lower register. Thus, instead of unary basis (which corresponds to having hamming weight only one), we have hamming weight one and hamming weight two basis. Lets imagine butterfly circuit on N: + K qubits. The complete unitary will be N x N block diagonal matrix with each block corresponding to a subspace with fixed hamming weight Kerenidis & Prakash| (2022), B = B1 @ B2 @ @ Br, where B; correspond to the block diagonal unitary for subspace with hamming weight i. Since our input has hamming weight or 2, we only care about unitaries B1 and B2. B1 will be of size (Nc+K) x K)) and Bg of (Nv**)) x (N=+*)) Given these unitaries, we take the input state and split into (Ne + K) hamming weight states and (No**) hamming weight 2 states. Then we apply the B1 and Bg to these states. Also, B₁ is a butterfly matrix and B2 is the corresponding compound order two matrix Hoonn & Johnson (2012).
Figure 2 shows the diagram for this circuit. Given the circuit is similar to the sequential circuit till the QFT operation, the state of this circuit would be same the one in eq. [0] We now separate this complete state into two sets of states corresponding to hamming weight and 2:
K
N
Σ
o ea) les)
Σ
Σ j=K+1
dg les) leg)
(17)
where the first term corresponds to hamming weight 2 states (22) and similarly the second term corresponds to hamming weight 1 states l/t)). Lets first focus on the term corresponding to /4)) It does not contain the states where the qubits in upper register are all and the lies in the top K qubits of the lower register. It implies that the coefficients of all these states should be taken as zero Therefore, the state corresponding to this (h1) can also be written as:
Σ
[ea))ess (eg)
Σ
Σ j=K+1
eo) leiy)
(18)
j=K+1
where leo) denotes the state corresponding to no ones in the upper No register and legy) denotes the hamming weight 2 state for the lower register, where i and j denote the positions of 1. Similarly, if we consider the first term in eq. 17 corresponding to (h2), we further have to include states where both ones are in upper register or both ones in top K of the lower register. These new states again would have zero coefficients. As a result, we can write the term corresponding to (h2) in eq. 17 as:
No 0 leys lco) + + de lea) (e)))e))))))
K (eo) lei)
ΣΣ
Σ> j>
(19)
This results in a total of (No+t)) states. Now, we apply the butterfly circuit, corresponding to unitary B1, to the (11) state in eq.
18 For notational consistency we denote this operation as a multiplication
with matrix W E (V.
K (N++K) results in the transformed coefficients byy:
N
N
US = Σ ( (Waa(() +
Σ (W3 x i E KJj I=N++1
(20)
Furthermore, also apply post select operation preserve the basis, selecting only the states with non-zero coefficient before applying the B, Now, for the hamming weight state, the second order compound unitary Bz, denoted by the matrix 2 E Rox4 where q = (Motk). has each of its elements corresponding to the determinant of a 2 x 2 submatrix of W1
W W:2, 2
W W2+ Wa,b+k
(21)
for some a, b, < No + K. After applying this unitary B, on (22) states, their coefficients Cij are:
(N
N.K
4j =
(
0) + Σ
(M+) x k//)) +
(W2
Σ
(W3 x 0)
(22)
t= N No)
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Similar to the case of hamming weight I, here also post select operation to discard the states which initially had coefficients zero thereby preserving the basis. Combining the transformed Ihi), (22) states and applying IQFT on the lower register, the final output state of this circuit is:
K Σ le,)
N, Σ j= k=
Σ (e) IQFT
bes les)
(23)
and the order of the depth complexity is log(NVe + K) + log NN) + (Ne + 2)log(Naa)
4 EXPERIMENTS
In this section, we analyze our proposed Quantum algorithms for solving PDEs and Image clas- sification tasks. We compare them against the otate-of-the-arts in both the domains, i.e., classical Fourier Networks (PDEs) and CNNs (image classification), for both the tasks. All the details related to architecturehhypeparaaaate
are provided at the end of the paper. All the experiments shown in this section are simulated, i.e., the quantum operations have been simulated using classical matrices corresponding to quantum unitaries. The currently available quantum hardware is limited to 8-10 qubits and is also too noisy for deeper circuits. Deploying these algorithms on a quantum hardware will involve noise which can be either due to noisy quantum gates, measurement, environment, etc.
4.1 PARTIAL DIFFERENTIAL EQUATIONS
We show results on all the three PDEs used in the classical Fourier Layer paper Li et al. (2020); Burger's equation, Darcy's 's Flow equation and Navier-Stokes equation, using the datasets proposed in that paper. All of these equations were designed for modelling the flow of fluids and have found their applications in other domains
well. We abstractly describe the three tasks. For equations
and other details please refer Li et al. (2020). We analyze the performance of the trained networks across different resolutions (N₂) for the first and for different viscosity values for the third.
Burger's ' Equation. It is a ID-Partial Differential Equation for modeling fluid motion. The initial condition of the fluid represented by a function. We need to learn the mapping from this function to the solution function at time one, for a given viscosity. Figure (left) shows the comparison
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Figure 3: Left: Performance comparison (relative eπror as used in Li et al. (0200) of the classical fourier networks, CNNs and the three circuit proposals for a quantum fourier layer on the Burger's ID PDE equation across different resolutions. The quantum fourier circuits are quite close to the performance of the classical fourier baseline and much better than classical CNNs in minimizing the error. Middle: Same comparison on the Darcy's s 2D PDE for different resolutions. A similar relative performance is observed where error in CNNs is much larger and is increasing w.r.t. resolution, whereas the error in the other four (3 quantum circuits and classical layer) is quite similar. Right: Convergence comparison for the Navier-Stole equation with = le 3. trained for 500 epochs.
of relative error in estimating this mapping among the classical Fourier Layer, classical CNNs and proposed quantum circuits for the Fourier Layers, across different resolutions. The quantum circuits perform comparably to the classical Fourier Layer and much better than classical CNNs.
Darcy's 's Flow Equation. For this case, it is a 2D PDE where the aim is to learn the mapping from the diffusion coefficient function to the solution function in presence of forcing function. All of them are functions of positional coordinates only. Figure 3 shows the relative error for the 2D-version of all the methods in solving this PDE, across different resolutions. Here also the three quantum
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Figure 4: Left: Performance comparison of the CNNS, classical fourier layer and the proposed quantum circuits on the MNIST dataset. It can be observed that all of them perform quite similarly, classical CNNs being the best. Middle: A similar comparison on the Pneumonia- Miiiia-MIIITTTaaa (2021) dataset. The performance of CNNs is somewhat noisy here whereas it is smoother in case of the sequential circuit, both converging to a similar value. The compound quantum circuit and the classical Fourier baseline are also quite close to the CNNs in convergence. Right: Same comparison on the FashionMNIST Xiao et al. (2017) data. Here, a significant difference in the performance is observed with CNNS being the best followed by the Sequential circuit.
al.
eircuits and the classical fourier layer show similar performance, consistent across resolutions, and the CNNs show much worse results, having error increased with resolution.
Navier Stokes Equation. Similar to Li et al. (2020), we consider the 2D-navier Stokes Equation where the aim is to model the fluid vorticity upto instant T(> 10) given the vorticity upto time 10. Figure (right) (right) shows the performance comparison for this equation between our proposed circuits and classical methods. It shows the convergence comparison for this family with viscosity fixed to le - 3 for all the methods. Here again, it can be observed that all the proposed circuits and the classical Fourier method perform significantly better than CNNs. Also, from Table [ sequential circuit performs similarly to the classical method and the others converge at slightly higher error.
Table 2:
Comparison of parameters required by one layer of the proposed circuits and the existing classical Fourier Layer along with error analysis for different and values for the 2D case of Navier Stokes equation.
Method
Classical FNO
Sequential QFNO
Parallelized QFNO
Compound QFNO
Parameters
294,912
23,040
23,040
6,144
lc 3: T 50
0.0139
0.0148
0.0167
0.0186
V le 4; T 30
0.1603
0.1618
0.1633
0.1660
V == le 5; T 20
0.1601
0.1615
0.1626
0.1638
4.2 IMAGE CLASSIFICATION
We further compare our proposed Quantum algorithms on the downstream image classification tasks, as side experiment, on benchmark datasets including the MNIST FashionMNIST Xiao et al. (2017) and PneumoniaMNIST Yang et al. (2021) datasets. Figure shows the results for this evaluation. can observed that our proposed algorithms perform decently in the image- classification task well.
5 CONCLUSION
We proposed quantum algorithm to carry out the recently proposed classical Fourier Neural Oper- ator on the quantum hardware. We further proposed two more quantum algorithms, which perform different operation than the classical algorithm and can be much more efficiently deployed quantum hardware. The aim is to make the learning process more efficient when using the noisy quantum hardware. Experimental results further verify that proposed quantum circuits perform ef- ficiently when solving PDEs. The sequential circuit is quite similar to the best performing classical algorithm for PDEs and performs decently on image classification as well. An interesting future di- rection can be further modifying the learning process of the compound circuit so that it outperforms the sequential circuit at the same time being more efficient to deploy.

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Under review as a conference paper at ICLR 2023
REPRODUCIBILITY STATEMENT
We have used the JAX framework to develop the methods discussed the paper. All the experi- ments shown in the paper are simulated by defining the matrices using the unitaries corresponding to the quantum gates used in the paper. The architectural and hyperparameter details for both classical and quantum Fourier Layers are same as the ones used in Li al. (2220) All the experiments volve a 4-layered architecture corresponding to both classical Fourier baseline and quantum Fourier circuits. The architecture for CNNs used in the Burger and Darey Flow PDE experiement a Fully Convolution Networks based architecture proposed in Zhu & Zabaras 2018). For the Navier stokes, we use the ResNet architecture for CNNs, comprising For the CNNs in the image classification tasks, we use four layered architecture with 64 channels and also use a max- max-pooling layer after the second and fourth layer. For both the tasks, each classical quantum layer is immediately followed by ReLU activation and Batch Normalization opera- tion. For the classification tasks, finally the output each of the architectures is fed to a linear layer with projection heads equal number of classes for that task. For the classical FNOs and proposed quantum FNOs except the compound quantum FNO, K is fixed to 16 for 1D PDE and image classification and 12 for 2D PDBs, and N. 64 for the 1D PDE and 32 for the 2D PDEs. For image classification, .o is set to 16. For the compound quantum FNO, due to less parameteric requirements K to 16 and N2 to 48 for all the tasks. The input for the PDEs having Nv initially 1 is first projected to the the required N₂ using a linear layer for all the classical and quantum fourier methods similar to to et al. (2020). The final output of all the layers of N₂ features is back projected to N₁ using another linear layer. For all the PDEs, unless specified, 1000 train instances and 200 test instances are used and training is carried out for 500 epochs using Adam optimizer with learning rate initialized to 0.001 and halved after every 100 epochs. It is fixed to 0.001 for the classification task. Number of epochs for the classification tasks can be seen in the plots.
Residual Blocks as used in|Li et (2020).
10

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Under review as a conference paper at ICLR 2023
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Appendix
A QUANTUM BUTTERFLY CIRCUITS
As discussed in the paper, replacing each radix-2 butterfly shaped operation the butterfly diagram for Fast Fourier Transform (FFT) by parameterized 2-qubit gate, Reconfigurable Beam Splitter (RBS) gate, results in a parameterized butterfly circuit (See Figure 5 The gate is parameterized
by single parameter 0 and the corresponding unitary is:
0 C08
0 sin
0
RBS
(24)
sin 0
COs
0
0
1
It can be observed that it applies transformations on the (01} and [10) basis and is an identity op- eration on the remaining. These both comprise the unary basis or hamming weight I basis for two qubits. It can then be understood that it applies transformation between the states with same hamming weight basis and thus, preserves the hamming weight. Therefore, if we load our input to basis with certain hamming weight, then after any transformation made of these RBS gates, basis states corresponding to non-zero coefficients weight. Corresponding to N qubits would be only N unary states and thus, if an N-dimensional input is encoded into -uuaary states using the unary loading similar to Johri et al. (202I), then the transformation resulting from this butterfly circuit can be seen multiplication of the coefficients of these N unary states with an orthogonal matrix W E TR N A recent work by Kerenidis et al. (222)) further explains this orthogonal matrix multiplication for pyramid-shaper circuit.
the output will have that same hamming
Unary QFT. As shown in Figure 5 (Right), replacing the RBS gate in parameterized butterfly cir- cuit with fixed single qubit gate and a fixed angle RBS gate results in circuit which can perform QFT in the unary basis. The depth complexity of this circuit is still O(log (M) where M is the number of qubits and also the total number of unary states. The single qubit gate just a Phase gate which rotates about the Z- axis with angle set to N
2 and matrix corresponding to this gate
is: ()
2gg where N and corresponding to each are same as in the corresponding radix-2
operation in the classical butterfly Fourier Transform (.
Replaced by
Transform
Figure 5: Left: Parameterized Quantum Circuit generated by replacing each connection in the But- terfly Diagram for FFT by 2-qubit RBS(0) gate, where is a learnable parameter unique to each gate. As discussed in the text, it transforms the coefficients of unary basis states. Right: Replacing the 2-qubit parameterized RBS gate by single qubit rotation gate and RBS gate, with angle fixed - results in the circuit for doing QFT on the unary states.
orthogonal feed-forward neural network layer which
B DATALOADERS
Since input is No N, matrix, load it into quantum state, the matrix can be interpreted as N. vectors each of size N.. The resultant should be a superposition these vectors. In the paper discussed recent work Johri et al. (202T) which used dataloaders to load N dimensional input
13

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02
04
Figure 6: Quantum circuit for loading a classical vector into unary basis Johri et al. (2021). ).
comprises of (N 1) RBS gates where the angles for each of these gates (ax,) pre-calculated using the input. See Johri et al. (2021) for further details.
to N unary quantum states. One of the proposed dataloaders is shown in Figure 6 for qubits. It involves firstly flipping the first qubit to state [1). Then it involves applying (N-1) RBS gates with
precomputed angles (ax)) for loading this vector. It can be observed that it log-depth eircuit. For a classical input vector a (.),... it results in the following quantum state:
(0)
a3 le))
(25)
a
where les) denotes the unary basis states for N qubits. There will be total N such state and coeffi- cient of each of these states would correspond the normalized classical vector element. To load superpose its vectors b,, where each of these vectors would be first loaded using some controlled vector-loader. To do so, we first load the norms corresponding to each of these rows ||b,l| using the norm-loader shown in the circuit diagrams in the paper. It results the following state on the upper register:
input matrix b E NoN,,
N [b,llle))
(26)
66
Now, for each of the unary states les) in the upper register, we load the corresponding row vectors (bi))je(L,N.) in the lower register using a controlled version of the unary vector loader (controlled operations labelled ||X,]| the circuit diagrams) which involves applying the unary vector loader
and its adjoint along with CNOT gates. For more details this loader, refer Cherrat et al. 2022 The final loaded state after N₂ of these controlled unary vector loaders
N₂
N. di les) lcz)
Σ
Σ
(27)
|bb|
C PARALLEL QUANTUM CIRCUIT
Given the final state, after the IQFT, for the parallelized circuit in Eq 16 the paper:
K
N
(1h)e led) + E'agyee les)a
Σ
led IQFT
(28)
J#k
k=1
We denote coefficients of this state corresponding to 4th circuit by (ciy)a and thus re-writing it as:
K
No. N.
Σ
(cy)2 (es) les)
(29)
k=1
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Under review as a conference paper at ICLR 2023
where (ctj) explicitly given by using the equation for the fourier transform as follows:
(cig)a Similarly writing Cij for the sequential circuit discussed in the paper (using eq. 13
1 N. JZR
Eiafiee (b( 30
(30)
b Zni NA
Cj N
(31)
j=k+1
Comparing the above two equations leads to the observation that coefficients in Eg. 31 wouldn's it be subset of coefficients in Eq. and there no closed form classical processingttrnnsoormatii achieve this. Thus, this parallel circuit results in somewhat different operation which is intuitively similar to the sequential circuit. Given, the experimental results, this operation also effective in dealing with PDEs/Images, at the same time being more effective than sequential circuit under noisy scenarios. However, if remove the TQPT operation from this eircuit and instead apply the classical IFT, measuring after eq. 15 get the following K matrices after applying the square root operation:
(b))es (a.) |
(32)
km
where tf and af have been defined previously. There are K such N. x N, matrices. In case combine of from all of the K matrices with (4))ee(tINN.) from any of the K matrices, suppose the first one, it leads the following No N, matrix:
(00) (E))) (a (aDDEE[+1.NN)
(33)
which is exactly same eq. Thus, this circuit (without the IQFT) followed by some classical post-processing and IFT can replicate the classical fourier layer operation.
D WHY COMPOUND MATRIX ?
The intuition behind why a different operation than classical, form of this compound matrix, might work lies in the expressive power of this operation. The matrix for the sequential eircuit can be interpreted as a special case of this compound matrix where instead of trainable connections, upper and lower registers interact using fixed control gate. Due this special connection, this transform is making the convergence efficient. This compound matrix, thus, is more expressive version of the linear transform and also with a lower depth. Using it was further backed by the ussumption that if optimized carefully it might lead to either a more parallelized version of the sequential circuit or a completely different and more efficient learned matrix. Also, this is more operation which would be quite complex to carry out classically and thus can provide a better demonstration of quantum-ad advantage.
15


Reviews:
[{"review_id": "u6YxTnO6lc", "reviewer": "Reviewer_URvi", "summary": "", "questions": "", "limitations": "", "rating": 5, "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": "In this paper, the authors extend the classical Fourier neural operator with Quantum circuits. It applies butterfly circuits as a model compression. The paper studies Burgers, Darcy, and Navier-Stokes equations. The Quantum FNO achieves similar accuracy with less amount of parameters. It prepares scientific computing for near term quantum device.", "summary_of_the_review": "Overall, it's good to see the connection. However, it seems the real application is still quite distant. The review feels the paper is marginally below the threshold of acceptance.", "strength_and_weaknesses": "While it is interesting to see connections between quantum algorithms and classical PDEs, I am not sure if this extension is well-motivated. Empirically, in classical FNO, the FFT is already quite fast and the matrix multiplication takes the most computation since the channel dimension of FNO can be very large. It's not very sure if quantum FNO can be more efficient in practice.\n\nStrength:\n- nice observation to speed up FFT with quantum algorithms.\n- empirically, the Quantum FNO has comparable performance, with a small number of parameters.\n\nWeakness:\n- The Quantum FFT seems not to be justified. Unnecessary to use quantum language when everything is classical.\n- empirically, the performance is not as good as the original FNO.\n- the Mnist example looks unnecessary. There exists a previous work on Imagenet (https://arxiv.org/pdf/2111.13587.pdf).\n- there exists a previous work that uses matrix butterfly decomposition for neural operators (https://proceedings.mlr.press/v162/dao22a.html). \n", "limitations_and_societal_impact": "", "review_text": "", "extra_scores": {"correctness": "4: All of the claims and statements are well-supported and 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": {}}, {"review_id": "gbij_Paf-ho", "reviewer": "Reviewer_EV9o", "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": "This paper proposes three tailored quantum circuits, inspired by the FNO (Fourier Neural Operator), to learn the functional mapping for PDEs (Partial Differential Equations). The authors evaluate the proposed methods on three PDE families, with results showing that the quantum methods are comparable in performance to the classical FNO. This method is further verified on image classification task with comparble performance to CNN, which shows the potential value of their method to other domains.", "summary_of_the_review": "The paper addresses an interesting and promising area for adapting quantum computing to solve the machine learing-based PDE solvers. Such AI solvers themselves have attracted intensive attention in both computer science and applied math communities and derive two main lines of research: ONet and FNO. This paper presents a quantum version of FNO based the natural quantum-friendly computing of Fourier transformation and its inverse. The experiments over three typical PDEs and it can also be applied in other domains as preliminarily verified on the image classification task. The experiments are all done on simulation based classic computers for the unavailability to high-end quatum computers (with challenges in terms of noise and qubit limites etc.), which I think is understandable.", "strength_and_weaknesses": "Strength:\n1) good topic and quamtum AI solver for PDEs is promising and worh study\n2) complete evaluation on three typical kinds of PDEs with additional test on image classification\n3) the technical approach seems convincing with good justification\n\nWeakness:\n1) the AI PDE solvers e.g. ONet and FNO are all open-sourced, this work may also release the code for advancing the research in this intersection area?\n2) some details might be missing in the paper, e.g. the image classification implementation part.\n3) it still gives me an impression as a combiation of existing techniques i.e. FNO and quantum Foruier transformation\n\nOverall, I think this is a good paper, and I vote for acceptance.", "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": "RLcuuZA6kBW", "reviewer": "Reviewer_ExUz", "summary": "", "questions": "", "limitations": "", "rating": 3, "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": "Fourier Neural Operator (Li 2020) has become a popular tool in solving PDEs numerically. The main contribution of the paper is to replace the classic Fourier Layer in FNO with the quantum Fourier layer. The authors argue that the proposed algorithm is provably substantially faster (using quantum hardware) than the classic counterpart. The authors also provide several experimental results (solving 3 PDEs and one simple image classification) to show that the proposed method is comparable to the classical FNO. \n\nI am not familiar with quantum computing and won't be able to judge the significance of the theoretical contribution. Nevertheless, the overall quality of the paper seems not to be very high, which will be detailed below. Moreover, the experimental results from the paper do not support the strong claims made by the authors.", "summary_of_the_review": "Overall, the quality of the paper seems not to be high and I have raised several concerns. ", "strength_and_weaknesses": "- Presentation in Sec 2 (the classical FNO) is confusing, both notational-wisely and mathematically. \n- The experimental results are not convincing enough to support many claims in the paper. \n- The complexity bound, at least for the classical setting, seems not correct \n\nPlease see the details below. ", "limitations_and_societal_impact": "", "review_text": "", "extra_scores": {"correctness": "2: Several of the paper’s claims are incorrect or not well-supported.", "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": {}}]

Author rebuttal:

Decision: Reject