Rethinking the symmetry-preserving circuits for constrained variational quantum algorithms
Authors: Ge Yan
Hongxu Chen
Kaisen Pan
Junchi Yan
Venue: ICLR 2024 poster (2024)
Keywords: variational quantum algorithm
quantum machine learning
symmetry-preserving circuit
Primary area: applications to physical sciences (physics, chemistry, biology, etc.)

Abstract: With the arrival of the Noisy Intermediate-Scale Quantum (NISQ) era, Variational Quantum Algorithms (VQAs) have emerged as popular approaches to obtain possible quantum advantage in the relatively near future. In particular, how to effectively incorporate the common symmetries in physical systems as hard constraints in VQAs remains a critical and open question. In this paper, we revisit the Hamming Weight (HW) preserving ansatz and establish the links from ansatz to various symmetries and constraints, which both enlarges the usage of HW preserving ansatz and provides a coherent solution for constrained VQAs. Meanwhile, we utilize the quantum optimal control theory and quantum overparameterization theory to analyze the capability and expressivity of HW preserving ansatz and verify these theoretical results on unitary approximation problem. We conduct detailed numerical experiments on two well-studied symmetry-preserving problems, namely ground state energy estimation and feature selection in machine learning. The superior performance demonstrates the efficiency and supremacy of the proposed HW preserving ansatz on constrained VQAs.

Published as a conference paper at ICLR 2024
RETHINKING THE SYMMETRY-PRESERVING CIRCUITS FOR CONSTRAINED VARIATIONAL QUANTUM ALGO-
RITHMS
Ge Yan, Hongxu Chen, Kaiseng Pan, Junchi Yan"
Department of Computer Science and Engineering, Shanghai Jiao Tong University {yange mechta chx, pks0813, yanjunchi jtu. edw..e......m en
ABSTRACT
With the arrival of the Noisy Intermediate-Scall Quantum (NISQ) era, Variational Quantum Algorithms (VQAs) have emerged as a popular paradigm to obtain pos- sible quantum advantage in the relatively near term. In particular, how ef- fectively incorporate the common symmetries in physical systems as hard con- straints in VQAs remains a critical and open question. In this paper, we revisit the Hamming Weight (HW) preserving ansatz and establish the link from ansatz to various symmetries and constraints, which both enlarges the usage of HW pre- serving ansatz and provides a potentially coherent solution for constrained VQAs. Meanwhile, we utilize the quantum optimal control theory and quantum over- parameterization theory analyze the capability and expressivity of HW preserv- ing ansatz and verify these theoretical results on the unitary approximation prob- two well-studied symmetry-
lem. We conduct detailed numerical experiments preserving problems, namely ground state energy estimation and feature selection in machine learning. The superior performance demonstrates the efficiency and supremacy of the proposed HW preserving ansatz on constrained VQAs.
INTRODUCTION
Over the decade, Variational Quantum Algorithms (VQAs) (Cerezo al., 2021; Tilly et al., 2022) have received increasing attention and numerous studies have been conducted to seek potential quan- tum supremacy. With the arrival of the Noisy Intermediate-Seaa 2018; Bharti et al.. 2022), the pace of exploring new VQAs has been further accelerated, as VQAs have shown the potential to obtain quantum advantage the foresecable future on NISQ de- vices (Cerezo et al., 2021). However, typical VQAs such as Quantum Approximate Optimization Algorithm (QAOA) (Farhi al., 2014) for Quadratic Unconstrained Binary Optimization (QUBO), and UCCSD (Romero et al., 2018) for ground state energy estimation are not natively designed to deal with (hard) constraints. It remains an open problem for addressing and incorporating the symmetries and constraints in the VQAs.
Quantum (NISQ) era (Preskill,
We propose to resort to Hamming Weight (HW) preserving ansatz (Kerenidis et al., 2021) to in- troduce hard constraints the VQAs instead of modeling the constraints regularizer in the Hamiltonian as widely done in literature (Chieza et al., 2020; Khumalo et al., 2021). The HW pre- serving ansatz operate in restricted (*) -dimensional subspace with parameterized HW preserving gates. We argue and show that HW preserving ansatz can deal with various symmetries in the phys- ical systems and it could be used sound alternative for the Hardware Efficient Ansatz (HEA) for constrained VQAs. We further utilize the quantum optimal control (Schirmer et al., 2001) and quan- tum overparameterization (Larocca et al., 2023) theories to lay rigorous theoretical foundation for the HW preserving ansatz. Specifically. quantum optimal control theory can help us decide whether a HW preserving gate with certain connectivity is universal. Analysis of the trainability ensures that we can evolve the quantum state without the presence of barren plateaus. While the overparame- terization theory provides us with the guide that how many parameters (layers) one needs to obtain feasible results.
*Correspondence author. This work was in part supported by NSFC (92370201, 62222607).

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Published as a conference paper at ICLR 2024
We provide experimental results two constrained VQA problems, namely ground state energy estimation (Kandala et al., 2017) and feature selection (Chandrashekar Sahin, 2014) in machine learning We use two different HW preserving gates with two different connectivity to illustrate the capability and expressivity of different HW preserving ansatz. Specifically, one the gates is Reconfigurable Beam Splitter (RBS) gate proposed in literature (Cherrat et al., 2023), and the other BS gate is generated through theoretical analysis and then is verified by empirical results. Both HW preserving gates demonstrate great efficiency on constrained VQA problems. BS gate is universal on the (f) -ilmenioioo subspace and is able to solve the unitary approximation problem on the subspace. However, RBS gate shows better convergence rate on easier problem such as the ground state energy estimation but only with full connectivity. Our contributions are:
1) We revisit the Hamming weight (HW) preserving ansatz and link the ansatz to popular symme- tries and constraints in physical systems, thereby expanding the utility of the HW preserving ansatz beyond their currently limited applications (Landman et al., 2022; Cherrat et al., 2023).
2) To our best knowledge, are the few (perhaps the first) to provide theory for the capability, expressivity, and trainability of HW preserving ansatz. We are able to generate a new gate based on our theories and numerical results on the task of unitary approximation verify our hypotheses.
3) We conduct detailed numerical experiments on tion and feature selection in machine learning, which are two popular tasks. The superior perfor- mance demonstrates the efficiency of the HW preserving ansatz on constrained VQAs.
ground state energy estima-
PRELIMINARIES
In this section, we will introduce the definition of the HW preserving quantum circuit and why it is useful in customizing the symmetries and other constraints the VQE problems. We will further introduce several gates that satisfy the constraints of HW preserving.
2.1 HAMMING WEIGHT PRESERVING QUANTUM CIRCUIT
We have discussed several types of symmetries at the physics level and we now consider these symmetries and constraints on the quantum circuits. These symmetries and constraints as discussed in the related work all ensure that the number of Is in the quantum state does not change. Recall the definition Hamming distance and Hamming weight (Def. B. and Def. B.2), we find that these states in the constrained subspace all have the same Hamming weight. Thus we can employ Hamming weight preserving circuit to solve the aforementioned problems
limited subspace.
To make sure the quantum circuit will not generate any quantum state that have different Hamming weight, we need to find proper gates that can preserve the Hamming weight of the initial states. The most common gate can find is the SWAP gate and the RZZ gate with the unitary matrices
0
0
0
U AP
0
0
U Uzz 11
e
0
0
1
0
0
10
0
0 0
1
These two gates ensure that they do not change the amplitudes of state (00) and [11). However, is still not enough to build trainable circuit for HW preserving problems. Cherrat et al. (2023) utilized another gate called the Reconfigurable Beam Splitter (RBS) gate as the basic element, and (Hadfield al., 2019) proposed XY-mixer to construct constrained QAOA. RBS gate and XY-mixer are more trainable compared to RZZ and SWAP gate with a possible decomposition Fig.
0
0
0
0
HRRS
0
-1
0
U ( = 10HHnn
0 cos 0
sin (0) cos(
0
(2)
0
0
0
sin
0
0
0
0
0
0
0
Hxy
0
1
0
Uxr = i0Hxy
cos
- -isin())
(0)
0
(3)
0
(in
cos
0
0
0
1
2

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Consider n-qubit quantum circuit with HW k. have dg =(3) basis states in the HW subspace. The SWAP, RZZ and RBS gates all preserve the subspace with HW as k. The main character of these 2-qubit gates is that they only act two basis states (01) and 110). Following this condition, we can theoretically construct our own gate that is HW preserving. Note that the dimension of the HW preserving circuit is (*), which is smaller than the dimension of the whole Hilbert space 2 (especially when & is relative small compared to n)). This suggests some interesting properties.
2.2 DYNAMICAL LIE ALGEBRA
To understand the expressivity and capability of the HW preserving ansatz, we need utilize mathematics tool, which is Dynamical Lie Algebra (DLA) (Zeier & Schulte-Herbrüggeen, 2011; Alesssandro, 2021). Since quantum circuits unitary transformations, we can use Lie algebra and Lie group to analyze the properties of them. However, quantum machine learning introduces parameters into the unitary transformations, so we need to further extend the Lie algebra dynam- ical Lie algebra. The unitary matrix U()) of QML circuit with L layers is of the form:
U II U.(O.)
IIII H,, 1=1p=1
(4)
where indicates the layer, Hp are the Hermitian matrices that generate the unitary matrix U,( 0: {θn1 -1 , 0tp) is the parameters in the 1-th layer and 0 {01, 02..... 0L) is the parame- ters in the whole circuit. The generator of each layer in the ansatz can defined as follows:
Definition 2.1 The set of generators: Consider a single layer of parameterized quantum circuit (,) in Eq. 4. We define group of Hermitian matrices that generate the unitary matrices U(0) as a set of generators G = -Hp)p=m, where [
With the set of generators by Definition 2.1, introduce well-established concept:
Definition 2.2 Dynamical Lie Algebra (DLA): Consider the set of generators defined in Defini- tion 2.1, the DLA g is defined as:
spon Cran(iHi (HH, iH,
(5)
where (-) denotes the Lie closure. The set of generators is obtained by repeated take the commu- tators of the elements in the set.
We denote the corresponding Dynamical Lie group of DLA g as G. The reachable unitary matrices from arbitrary parameters 0 have the following properties (Larocca et al., 2023):
(U())) G SU(N),
(6)
where SU(N) denotes the special unitary group and d denotes the dimension.
INTERPRETABILITY OF THE HW PRESERVING ANSATZ
In this section, we will discuss the capability and expressivity of the proposed HW preserving ansatz. We also introduce definitions and theories for dynamical lie algebra, quantum control system and quantum overparameterrizttoo. With these knowledge, we are able to determine if gate is universal the dimdinnnso subspace, and estimate the number of parameters (circuit depth) needed the circuits. We conduct experiments on the unitary approximation problem to verify these theories.
3.1 QUANTUM OPTIMAL CONTROL ON HW PRESERVING SPACE
Now we introduce the quantum control system and how to get complete controllability over a given quantum system. Consider N N-dimensional quantum system, the Hamiltonian of which can be denoted as H. If we denote the initial state (vo). the time-independent Schrödinger equation is
HU (b) (U) (u5o)
where E is scalar and stands for the energy. Notice that not all the states in the space can be reached by the dynamical unitary matrix, unless the dynamical Lie group generated by & is equal to the special unitary group (UN.
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Lemma 3.1 A quantum system H is completely controllable if (0 GG SU(N).
Complete controllability indicates that any target state in the space can be dynamically reached from any initial state with a proper unitary matrix. We can further conclude that complete controllability implies that any unitary matrices in the space can be approximated to an arbitrary precision by U()). Ramakrishna al. (1995) has shown that if the dimension of DLA g generated by the operators is 222 then the dynamical Lie group of the system is SUNN.. Therefore, we have
Theorem 3.2
(Ramakrishna et al., 1995) A necessary and sufficient condition for complete con- trollability of a quanium system H is that the dimension of the DLA g is N2.
This theorem gives simple way to verify the controllability of quantum system, namely by com- puting the dimension of the Lie algebra generated by the set of generators G. The detailed computa- tional method to calculate the dimension of DLA is described in Alg. 1 (Schirmer et al., 2001).
Notice that different connectivity of the physical qubits can lead to different dimensions in the DLA example, if all the qubits
since they will have different set of generators. Take RBS gate fully connected (we can apply 2-qubit gates to any pair of qubits), we will have (n(n-13 - 1) generators in G and the corresponding dimension of DLA is 1d (dz 1). However, if all the physical qubits linked as a circle (each qubit can only be connected with its two neighbors), we will have n generators in 9 and the dimension of DLA is nn(n 1) (illustrated in Fig. 1(b)).
3.2 OVERRARRMETTRRZATION FOR HW PRESERVING ANSATZ
Once we know which ansatz can have full controllability of the system, critical to know the number of parameters or layers we need to build such ansatz. Larocca et al. (2023) provides a theory to link the overparameterization phenomenon to the dimension of DLA. Overparameterization has certain implications for the trainability of an ansatz. Underparameterrzzed ansatz will lead to local minima and overparameterized ansatz can have better properties and lead to a better landscape for optimization. We first define crucial concept in the overparameterization theory.
Definition 3.3 Quantum Fisher Information Matrix (QFIM): Consider quantum state Wp) and set of parameters {01, 0- The QFIM is defined M x M matrix:
(0)le 4R (00, (0,)), (0) (0r,, (0),,
(8)
where M is the number of parameters, 1 (0) 0 10 (0)) 800 a (0)) 0, e
The overparameterization phenomenon can then be linked to the rank of QFIM.
Definition 3.4 Overparameterized QNN: A QNN defined overparameterized if increasing the
number of parameters M past minimal bound M. does not further increase the rank of any QFIM:
max rank (F,,)(((()))))))))) = Ry
(9)
M..
Larocca et al. (2023) further shows that for the general case of QNN, we have:
M dim(g).
(10)
where Md is the bound of the number of parameters reach overparameterrizatio,, and g is the DLA correspond to the ansatz. When the number of parameters is larger than Mc, the ansatz will reach its maximum capacity and further adding parameters will not improve the performance. The theory of overparameterizatioo is important for us to decide the number of gates layers we need to solve certain problems using a given HW preserving ansatz.
3.3 GENERATING UNIVERSAL HW PRESERVING GATES
Now we have rigorous theories for HW preserving gates as well as HW preserving ansatze. We can use the quantum control system theory to generate gate and decide the connectivity of physical qubits we need to have full controllability and then use the overparameterizztion theory to decide the number of gates (layers) need in the ansatz (see Fig. for details).
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(c)
(0)
X
Inilial State Isb))
Hamming
Hermitia Matrix
connectivity
Set Generetors
Weight
0
= Hr.Hooy My.m
(0]
Preserving
(0)
Ansatz
(b)
dim(f) < d,
Algorihhm
S
B
B
B
S
B
B
Decide the Number Parameters and Layers
Dimension
M, dx
dim(s) d
= span(G)
S
NN connectvityy
Full connectivity
Overparameterzatiin
Quantum Optimal Control
Figure 1: Pipeline generate universal HW preserving gates. (a) The overall circuit of the HW preserving ansatz. We require k pauli-x gates to generate an initial state and then apply the HW preserving ansatz. (b) Examples of two different connectivity, (c) Pipeline generate universal HW preserving gates and ansatz. We utilize the quantum optimal control and quantum overparame- terization theory help design the gate well the ansatz.
Following a similar pattern as SWAP, RZZ and RBS gate, we can generate a two-qubit HW preserv- ing gate by only operating states (01) and [10).The The Hermitian matrix can be of the form:
0
0
0
0
Haw
(1I)
0
0
0
0
where a, c IR E C, and b denotes the complex conjugate of since H H We only have three variables in the matrix which is simple for us to create new gates. We can then convert the Hermitian matrix to a Unitary matrix that can be further decomposed and applied on a quantum circuit. As an embodiment without loss of generality, we give the following example of Hamiltonian for HW preserving gate as BS. We start from HRBS and then repeatedly alter a, and c to make it universal:
0
0
141
+1)
BS
2/2
U (0)
2/2
(12)
2
0
2
0
0
(I-I)(
2
+1)
0
1
where e00 = cos(() + (O). We put the detailed derivation of this gate in Apx. D. We then decide the connectivity of the circuit to calculate the size of the set generators G. For an n-qubit circuit, have P for Nearest Neighbor connectivity and P = nn 1) for full connectivity. With the set of generators, we can compute the dimension of the corresponding DLA Note that The dimension of DI.A is de for both full connectivity (BS-full) and NN connectivity (BS-NN) for BS gate (how to connect the gate is shown in Fig. 1(b),, which indicates that BS gate is universal in the diddimensional subspace (a possible decomposition for BS gate is in Fig. 7).
We can further utilize the overparameterization theory to determine the number of parameters or layers we need using BS gate. From Eq. 10, we can see that M, for BS gate with both connectivity is de which can provide us with an approximation amount of parameters we need to build an ansatz. We are also able to estimate the number of layers we need, e.g. we need around [dk /n" layers for NN connectivity and [2d]/ (n 1))1 layers for full connectivity, It is vital for to tell whether the ansatz have enough parameters and have reached its maximum capability.
3.4 UNITARY APPROXIMATION
To better illustrate the connection between the dimension of DLA and the above mentioned theories, we introduce the task of unitary approximation. As described Sec. 3.1, we have linked the com- plete control of quantum system to the unitary approximation. The unitary approximation aims to solve the problem of whether (e))) is equal to SU (see lemma. 3.1). For a target unitary matrix Ü in da dimensional HW subspace, the loss function for unitary approximation is
Lu Tr (()
(13)
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1.0
Success Probability
0.8
0.6
d
10 BS-MI
0.4
10 BS NN
0.2
15 BS-NN
di 20 BS-ful
0.0
d, 20
100
200
300
400
500
600
Number Parameters
Figure 2: Results for unitary approximation. For each dg. we randomly generate 50 unitary matrices based on Haar measure'. For each matrix, if the loss function L < 10 10-10, we say that we successfully approximate this matrix. The success probability indicates the portion of matrices we can approximate to 10- from all the 50 randomly generated matrices. The vertical line indicate the dimension of the DLA which is &. The results show that Nearest Neighbor (NN) connectivity and full connectivity for BS gate have similar performance with only slight difference.
We conduct the experiments the challenging and widely accepted task of unitary approximation in the HW preserving subspace with dimension de- The purpose is to compare the performance of different gates and different connectivity. We first consider RBS gate. The DLA dimensions of RBS are less than di for full connectivity (RBS-full) and Nearest Neighbor connectivity (RBS-NN). Theoretically the RBS gate can form ansatz that can approximate an arbitrary unitary matrix.
On the contrary, we can use BS gate to get complete control over the de-dimensional subspace. To better illustrate the thcoretical results, we test RBS and BS gate on randomly generated unitary matrices. We set d₂ (i)) (2) ( ( (5, 10, 15, 20) and the results are shown in Fig. 2.
The results verify the theoretical analysis for both complete controllability and overparameterization phenomenon. RBS gate has failed all the tests thus we omit the RBS gates from Fig. However, both connectivity for BS gate are able to approximate all the given matrices. Moreover, the exper- iments have shown a clear evidence that we need around df parameters to approximate the unitary matrices. This verifies Eq. 10 that the dimension of the DLA for both connectivity of BS gate is di
Note that the BS gate surpasses RBS gate on the unitary approximation task because of the dimen- sion of DLA (according to Theorem 3.2, and only the dimension of BS gate reached df ), but it not necessarily better choice on any problem. Unitary approximation is the hardest problem in dy- di-dimensional subspace and if the problem is easy enough, RBS could also be better choice. We will further illustrate this phenomenon in the numerical experiments.
3.5 THE TRAINABILITY OF THE HW PRESERVING ANSATZ
Now that we have the optimal control theory to determine whether the target state is within the reachable subspace of the ansatz, and the overparameteriizaion theory to determine how many layers we need to construct a proper ansatz, we need to further discuss the trainability of the HW preserving ansatz make sure that presence of barren plateaus problem. A variational quantum circuit is said to have barren platcaus problem the gradients decay exponentially with the number of qubits (McClean al., 2018). Thus, we have the following theorem:
find path evolve the initial state the target state without the
Theorem 3.5 Consider n--qubit quantum circuit operating in the subspace with Hamming Weight k. The variance of the cost function partial derivative Varolü;C] x
16A (n-k)²
n,
Detailed proof of this theorem is provided in Apx. H. From the above results, we can further
clude that if k is equal to 1, then Vare [] x
16 n3 If k = on the other hand, V aar s
Utilizing quanntum info.random unitary in IBM Qiskit

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Published as a conference paper at ICLR 2024
0 which is approximate exponentially small. This result is consistent with the conjecture n/22 that the trainability of the circuit is closely related dg and smaller dg will lead to better trainability.
4 EXPERIMENTS
We present results on two well-studied constrained VQAs: symmetry- preeeerrigrgs ground state en- ergy estimation and feature selection. Experiments are performed on a machine with 190GB mem- ory, one physical CPU with 32 cores AMD Ryzen 3970X CPU, 5 GPUs (NV GeForce RTX 3090).
4.1 LYMMETRY-PRESERVE STATE PREPARATION
Background. State preparation is well-studied problem in quantum chemistry with ground state energy estimation as a popular application. The ground state of a molecule its stationary state with the lowest allowed energy, i.e. ground state energy En. which can be estimated given the types and relative coordinates of its atoms as well as the number of orbitals and electrons. The energy is closely related to the molecular Hamiltonian H.. which is an Hermitian matrix embodying the energy of electrons and nuclei in the molecule. Given the final state (قا) of the circuit, we have:
(1bl Hmm (16)
Eo S
(14)
(1b1))
where the equality holds if and only if (1b) is the ground state. Notice that there are several well- defined symmetries from the physical level (Gard et al., 2020) such as the particle number, total spin and time-reversal, etc. Consider molecule with n orbitals and & electrons, the ground state is bounded in the di-dimensional subspace.
Experimental Setting. We select three well- studied molecules, i.e. Hydrogen (H2), Lithium Hydride (LiH), Water (H₂O) and Ammonia (NHz). We obtain the Hamiltonian of these molecules from the Python package Open- Fermion (McClean et al., 2020). The compu- tational basis for all the molecules is STO-3G. Detailed statistics of the molecules are listed in Tab. 1. LiH-8 is simplified molecule from the original LiH with only 8 orbitals and 2 electrons. To better illustrate the efficiency of the HW pre- serving ansatz, we select HEA ansatz (Kandala et al., 2017) and UCCSD (Romero et al., 2018) as the baselines. Among all the methods, only HEA acts in the whole Hilbert space and use [Pu) 0 as the initial state, all the HW preserving ansatz and UCCSD use Hartree-Fock state (bнP) as the initial state. Thus, the loss for all the ansatz except HEA is:
Table 1: Statistics of molecules. n and & are the number of orbitals and electrons respectively.
Molecules
H2
LiH-8
LiH
H₂O
NH₃
4
8
12
14
16
2
4
10
10
dR
6
28
495
1001
8008
Cm (0) = (WHFl (O) U(0) ({HF)
(15)
Results and Discussion. We conduct two groups of experiments on 4 of the above mentioned molecules (NH3 apart). We first use a fixed bond length and vary the number of layers. For all the molecules as shown in Fig. 3, our method can achieve an error below 1 x 10 10 which is much smaller than the chemical accuracy at 1.6 x HH (we suspect the eиror this level might be caused by the accuracy error of python). HEA is able to achieve comparable results with HW preserving ansatz on H2, but with the problem size increases, the 2n space for HEA to explore grows faster than the dy-dimensional subspace. Thus, HEA will need much more parameters than HW preserving ansatz, which leads to the poor performance on LiH and H2O.
Recall that we have three different dimensions of DLA for the HW preserving ansatz. The dimension of RBS with NN connectivity is n - 1), RBS with full connectivity is dd - 1), and BS with both connectivity dZ. From the results, we can see that RBS with NN connectivity to solve the ground state energy estimation problem, and the other three can achieve consistent good estimation. We can thus conclude that the state preparation problem is much easier than the unitary approximation problem with the dimension of DLA be somewhere between [/ - 1) and ddy 1). We can also observe from Fig. 3 that RBS gate with full connectivity performs better than BS gate with less parameters required to reach the exact energy. This reflects the importance of selecting proper gate considering the physical qubit connectivity and problem difficulty.
unable
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(a)a
(b)
(c)o
(d)
AE (Hartree)
H2
LiH-8
LiH
H2O
-10
-14
5 10 15 20
25
100
200
300
400
1030
3000
5000
2006
4006
8000 8000
Number of Parameters
Number of Parameters
Number of Parameters
Number of Parameters
BS-ful fullull
BS-NN
RBS-full
RBS
HEA
Figure 3: The energy errors w.r.t. the number of parameters. Note that HEA operates in the whole space, so it is possible for HEA to converge with more parameters. The number of parameters of UCCSD does not align with the circuit depth thus we omit the comparison in the figure.
(a)
(b)
(d)
Energy (Hartree)
-74
00000
-12
-7.0
-1.3
-7.6
(e)
(f)
(g)
(h)
10a
20004
10a
AMAA
101
(Hartree)
10-1
10-1
16-4
10-1
H2
LiH-8
LiH
H2O
10-33
10-12
10-11
10-34
10:30
1/0
1.5
20
25
30
1.0
13
Interatomic Distance (A] Interatomic Distance [A]
Interatomic Distance [A]
Interatomic Distance [A]
Ground
Figure 4: Energy curve for different molecules. (a-d) Potential energy curves of the bond distance.
HF stands for the energy with Hartree-Fock state. Ground stands for the exact energy. (e-h) Absolute errors compared to the exact energy. Blue region indicates energy is within chemical accuracy.
As for NHg. it is too large and takes unaffordable time to finish all the tests. We have the following data that using BS gate with full connectivity, we reach an energy error 10 with 21000 parameters. This verify the capability of BS gate to more complex molecules but the number of parameters required are indeed very large.
We then vary the bond distance of these molecules to see if the results are consistent on these methods. We fix the number of parameters for each method the minimum value that reaches overparameterrizaioon based on the results in Fig. 3. From the results in Fig. 4, we can see that all methods are consistent with different bond distances. better) results to UCCSD, which indicates that HW preserving ansatz is efficient and steady substitution for HEA ansatz. The HW preserving gates can be decomposed into basic gates and the NN connectivity is the same the connectivity of HEA. We on the actual structure the quantum hardware and provide ansatz with better capability, which is lower bounded by the NN connectivity ansatz.
are able to provide comparable (or
adjust the connectivity based
4.2 FEATURE SELECTION
Background. We then study feature selection (Kira & Rendell, 1992; Kumar & Minz, 2014; Li et al., 2017), which has been widely used in machine learning, serving to mitigate the deleterious effects caused by cast dimension, simplify models for enhanced predictability, decrease redundant and irrelevant and speed up training. The feature selection problem can be mapped to a QUBO problem. Since Variational Quantum Eigensolvers solvers on QUBO problems (Tilly et al., 2022), numerous papers have utilized quantum algorithms to solve feature selection (Nguyen al., 2014; Milne et al., 2017; Ferrari Dacrema et al., 2022).
have potential advantages against classical
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Table 2: Statistics of the datasets for the task of feature selection.
Dataset
Instances
Features
Feature Type
Wine Quality (Cortez et al., 2009)
4898
11
Real
Hcart Disease (Janosi et al., 1988)
303
13
Categorical, Integer, Real
Titanic (Vanschoren el al., 2013)
1045
14
Boolean, Integer Integer, Real
Dry Bean (Koklu & Ozkan, 2020)
13611
16
Table Results for the feature selection problem with best in bold. The numbers with striking lines denote over-selected feature dimensions due to the soft constraints problem-solving.
PyQUBO a = 10
RBS
a 0.5
a=
0 = 5
a 50
a 100
o 1000
Wine Quality 1.136342
1,135580
1.295212
1.080423
1,080423
1,099222
1.001571
0,983495
0.997417
Heart Disease
3.4183388
3.418183
8:736747
5-635489
3.223445
2.963659
2.792978
2.697520
2.717723
Titunic
1.197799
1.197159
4-230309
0.962090
1,009051
0.936198
0.929956
0,916599
0.923464
Dry 8.616844
8.616916
67.350723
20.765887
8.823824
7.790612
7.928807
7.864919
7,870827
Experimental Setting. One of the main methods evaluate feature selection algorithm is to utilize the mutual information (Guyon & Elisseeff, 2003) (see Apx. G for more details), We then use the following feature selection score evaluate different algorithms:
SFs = x TQ..
(16)
Thus, the target is maximize the score function SEs. However, selecting more than & features can lead to larger scores than possible. To address the constraint, we add a penalty term to the information matrix Q the QUBO problem as a soft constraint Hadfield al. (2017):
x x=xx= (xx==xQx-aaFIinn
- (HHW(x)-
(17)
where a is the penalty coefficient and i is the number of features to select. This constrained infor- mation matrix Q' can be converted into Hamiltonian Hrs for the feature selection problem. The initial state should be an arbitrary state in the di-dimensional subspace. Notice that HW preserving ansatze do not need the penalty term so we can set a for all the HW preserving ansatze.
We select four open-source datasets to test the HW preserving ansatz as well as classical QUBO solver py DUBBO (Tanahashi et al., 2019; Zaman et al., 2021). Statistics for the datasets are listed in Tab. 2. We set & for all the datasets 3 indicating that select the top 3 features from all the datasets, which can also lead to various dy.
Results. The results are listed in Tab. 3, where some numbers stroked for denoting "invalidity" as they are obtained with more than l features. This also indicates that the penalty term is not large enough to ensure legal results. The results show that the selection of a is important in the large 2" Hilbert space. A small a will lead to illegal states and an overlarge a can lead to hard optimization landscape. Both HW preserving ansatz can achieve better results than the soft constrained QUBO solver, which again verifies the supremacy of searching in hard constrained subspace.
CONCLUSION AND LIMITATION
This paper mainly focus on the symmetries and constraints in the VQAs, which lead to a smaller subspace, namely de dd-dimensional subspace, coherent fit for the HW preserving ansatz. Thus, we revisit the HW preserving ansatz and link the ansatz with these constrained VQAS. We also pro- vide theoretical analysis the capability and expressivity of the HW preserving ansatz and verify these theoretical results on unitary approximation problem. With these theories, we can generate new gate that is universal in the subspace. We conduct numerical experiments ground state energy estimation and feature selection problems, both with superior performance.
The main limitation of this paper is that the BS gate proposed in this paper is quite deep if we decompose the gate with basic gates. It is possible that there is perfect HW preserving gate which universal in the subspace and also easy decompose. We have provided all the theories needed to verify the gate, but finding and decomposing the gate is still remains an open question.

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A RELATED WORKS
Symmetries and Constraints in Quantum Systems. We first focus on the quantum chemistry problem where we have several well-defined symmetries to illustrate the necessity of considering hard constraints. The VQE algorithm on ground state energy estimation problems can be divided into two main categories. One is based on chemical properties such the unitary coupled cluster method (Bartlett et al., 1989; Ryabinkin et al., 2018; Shkolnikov et al., 2023) and utilizes Trot- ter decomposition to form the ansatz. This approach will lead to deep circuits which is unrealiz- able on current hardware. The other approach does not follow these chemical properties and only uses parameterized gates that are available on the processor such as the Hardware Efficient Ansatz (HEA) (Kandala et al., 2017; Bian et al., 2019) and the Quantum Architecture Search (QAS) (Wu al., 2023) methods. These methods can produce relatively shallow and feasible ansatz, but they can also generate unphysical states. The most important symmetries (or invariants) in the quantum chemistry problem are the particle number and total spin (Gard et al., 2020). This kind of symmetry is widely used in various problems such VQE (Gard et al., 2020), Feature selection (Nembrini et al., 2021; Mücke et al., 2023), constrained QUBO (Hadfield et al., 2019), etc.
Previous HW preserving related works. For the above-mentioned constraints, we can easily ad- dress them by adding regularizers in the Hamiltonian to give a penalty if we break the constraints. However, this method can only provide soft constraints and final states. Thus, finding an ansatz that only operates in the constrained subspace can guarantee the results are admissible. There several works that are related to the idea of HW preserving to enable hard constraints on the circuits.
still have illegal results in the
The HW preserving gate is previously used in several literature where (Kerenidis et al., 2021) first proposed the idea of quantum orthogonal neural networks. The authors proposed RBS gate with a pyramid structure on the quantum circuit and called it the Pyramidal Circuit. This eircuit can act as a substitution for the classical orthogonal neural networks. The Pyramidal Circuit is used as data loader with experiments on image classification for MNIST and medical images (Landman et al., 2022). The quantum orthogonal layer can be taken as a special case for the HW preserving circuit with k = 1.
Cherrat et al. (2023) further utilizes the RBS gate to solve the hedging problem. This paper in- vestigates the quantum orthogonal layer as well as the quantum compound layer. The quantum orthogonal layer is used as an encoding layer with different architectures of RBS gates arrangement. Results on financial problems especially the hedging problem on trapped-ion quantum processor provided in the paper. The authors also provide theoretical analysis that the variance of the gradients decays only polynomially with the number of qubits, which indicates the HW preserving circuit can have better trainability than HEA.
Notice that there is a concurrent work Monbroussou et al. (2023), which shared a similar thought with by utilizing DLA analyze the trainability and expressivity of the HW preserving ansatz. They mainly focused on building quantum data loader with HW preserving ansatz and the two proposed HW preserving gates RBS and FBS are able to fulfill the task. To compare Monbroussou et al. (2023) and this work, we have the following differences. (I) We analyze the expressivity from a more general perspective instead of only as dataloader; (2) we introduce the overparameteriization theory to figure out the number of parameters needed in the ansatz; (3) we focus more on constrained VQE problems as Monbroussou et al. (2023) focus on building dataloader for supervised quantum deep learning models; (4) we provide a more thorough background check for HW preserving ansatz by introducing other gates such as XY-mixer which is widely used in constrained QAOA.
These above papers share an opinion to use HW preserving gate as data loader for quantum deep learning. It obvious that HW preserving ansatz can only load data in the dy-dimensional subspace. For deep learning problem. it's rare to find all the data in the dataset falling within such limited subspace. Furthermore, when additional quantum layers are added to the circuit, the circuit no longer maintains its HW preserving property, which negates the advantages of using the HW preserving ansatz.
Though not using the name of HW preserving ansatz, there are also works utilizing specific gates to replace the mixing operator in QAOA so that they can tackle a certain kind of constraint (Hadfield et al., 2019). The proposed mixiie as well some other mixers in (Hadfield al., 2019) can
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be classified as special kinds of HW preserving ansatz. However, those gates in (Hadfield al., 2019) are found to have HW preserving properties, and we would like to know we can generate HW preserving gates with thorough understanding the properties of those generated gates. Thus, we rethink the HW preserving ansatz and link this ansatz to constrained VQEs to enlarge the usage of HW preserving ansatz.
Existing gates that are HW preserving. Apart from the aforementioned RBS gate that we have discussed in detail in the paper, also have several HW preserving gates from previous literature. Hadfield et al. (2019) proposed a quantum alternating operator ansatz which utilize the X-miixer and RZZ gate form constrained quantum ansatz. The XY-mixer is a HW preserving gate with its Hamiltonian constructed by XX and YY rotation HXY = X for XX, YY, and 7Z. gates are
+ Y X Y. The Hamiltonians X X
0
1
0
0
0
0
0
1
0
0
0
1
0
0
-
0
0
Hxx- =
Hyy
H
1
0
0
:
0
1
0
1 0
0
0
-1
0
0
0
0
(18)
Thus, see that XX, YY. and ZZ gates all HW preserving gates, and might use XX. YY, and ZZ to construct other HW preserving gates. However, if we consider the general form of HW preserving gate in Eq. 11, That is exactly why we need more general form to construct and analyze the HW preserving gates so that we can fully understand the HW preserving circuit with the ability to generate any specific gates we want.
only construct HW preserving gates with
and b b.
In line with the RBS gate, here provide brief analysis the X-miixer utilizing all the theo- retical tools provided in the paper. First of all, the hamiltonian and unitary matrix of the XY-mixer
0
0
(
0
0
0
1
0
cos cos(O)
si)
0
H
UxY =
(19)
0
1
0
-isi
cos cos(
(0)
0
0
0
0
0
0
0
0
1
For the NN connectivity, the dimension of DLA of the XY-mixer is
(n + 1)(n 10n-11 1)
n is odd
dim(gxx)
(20)
2 n(n - -1) n even
For the FC connectivity, the dimension of DLA of the XY-mixer is
(de 1)(2 - 1) n * 2k
dim(gxv) =
(21)
2 (dx + 2) 2) n = 2k
Thus, we can conclude that XY-mixer with full connectivity is quite capable of solving most of the problems except for some rare cases and maybe that exactly why it is famous for solving constrained QAOA. However, the XY-mixer on NN-connectivity is not very good even with the RZZ gate as the phase operator (the dimension DI.A is still not full). Therefore, we can conclude that the XY-mixer might not be so capable considering the physical qubit topology. One possible solution to this is to combine other gates to increase the initial generators and gain full controllability under both connectivities.
B FURTHER INTRODUCTION ON PRELIMINARIES
B. QUANTUM COMPUTING AND QUANTUM MACHINE LEARNING
In quantum computing. 'qubit' (abbreviation of "quantum bit') is key concept which is similar to a classical bit with a binary state. The two possible states for a qubit are the state (0) and [1), which correspond to the state 0 and for classical bit respectively. We refer the readers to the
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textbook (Nielsen & Chuang, 2002) for comprehension of quantum information and quantum com- puting. Here we give a brief introduction to the background.
A quantum state is commonly denoted in bracket notation. It is also common to form linear combination of states, which we call a superposition: (e) = a + 311). Formally, a quantum system qubits is n-fold tensor product Hilbert space 7/ (C²)od with dimension 2 For any (0)) E H. the conjugate transpose (æb = 1t)). The inner product (1b1e) = || denotes the square of the 2-norm of vb The outer product ( is rank 2 tensor. Computational basis states are given by [0] (1,0), and [1] = (0, 1). The composite basis states are defined by e.g.
(01) = 0) @ 11) 0..
Analog to classical computer, quantum computer is built from a quantum circuit containing wires and elementary quantum gates to carry around and manipulate the quantum information. These gates can be parameterized quantum gates such as Rx(0), Ry(O), R2()) or basic gates as oz: By. U. CNOT, CZ. For an initial state vo) and L layers of quantum circuit, the final state ((l") can be denoted as
( = U (vo)
(22)
B.2 HAMMING DISTANCE AND HAMMING WEIGHIT
In this section, we recall the definition of Hamming distance and Hamming weight, which con- stantly used in this paper. Considering two binary vectors a and b with a, b E {0, 13. where N the dimension of the vectors. We can define the corresponding Hamming distance of these two vectors as follows:
Definition B.1 Hamming distance: The Hamming distance 'D of two binary vectors and b is:
c=a@b b,
(23)
D(a, Σ C₁.
where @ stands for exclusive OR operator.
With the definition of Hamming distance, we can further define the Hamming weight of given binary vector
Definition B.2 Hamming weight: Let 0 = OV The Hamming weight HW of binary vector is:
HW (a) ((a,0).
(24)
C DETAILED ALGORITHM FOR COMPUTING THE DIMENSION OF DLA
We define a transformation which satisfies h = (H) (H E CNNN, h x1), where h is column vector obtained by concatenating the columns of vertically, o stands for the inversed transformation which maps column vector E CN*x1 back to matrix e CN D., denotes the j-th column of matrix D, and rank(D, h) the rank of matrix appending column vector the right. [s denotes the commutator between two matrices with [A, BJ = AB BA. The pseudo code for the algorithm is shown in Alg. 1.
D CONVERTING HAMILTONIAN TO UNITARY MATRIX
0
0
0
0
0
1ti
0
H ps
2/2
(25)
0
2v2
2
0
0
0
0
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Algorithm 1 Computing the dimension of DLA
Require: Generator set G (H1, H2, HHp, h, o(H.).
let D h1;
let rank(D);
for i = to P do
if' rank h:) then
Dha;pend(h:);
r -=
end if
end for
let Tout 0: while Tout # r and # N2
for I = Tout 1 to r do
for j j= 1= to do
let Hamp lo- DD. (D.,), (D.,)),
hemp (Hemp);
rank (D, hkmp)
> then
D.append(htmp)
r +=
end if end for end for let Tout let r rank (D);
end while return Tout
We can get the unitary matrix of the BS gate by the Taylor expansion:
U Un$(0) = e BB Σ 5! 1
(26)
When we design the Hermitian matrix HBs for the BS gate, we set
H Bs HBs.
(27)
Thus, we have special character for the matrix Hns that
H BS = H BS = =.. H2s = H
(28)
The unitary matrix can be simplified follows:
Σ (i0)' =I Σ 11-111-1 (0)25-1
(0)
(0)
U BB) 1 H
1)
(29)
(2j 1)!
(2j)
We Use the Taylor expansion of (in cos (0) =0
UBS H His sin 9 cos
(30)
Therefore, the unitary matrix for BS is
0 (coa P+i sin 9+11
0
(1os) 0+i 8 8-1)
0
U (0) =
2v2
(31)
[I -i) (oos 94isin 8-1)
(cos e - 0+1)
0
2
1
E REMARKS ON THE BIT-FLIP ERROR
has occurred to us that the bit flip error can easily destroy the Hamming Weight of the quantum states and thus affect the subspace that operating in. However, this problem can be tackled
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H
(8/2)
H
(-122)
Figure 5: A decomposition of RBS(O) gale.
R (20)
Figure 6: A decomposition of XY-mixer(0) gate. Notice that controlled-R Rx gate requires at least one two-qubit gate to implement.
R20 + x2222
Rg (TT2)
R2(-33/4)
(10 x)/2)
Ry((F 0)/2)
R2(4/4)
Figure 7: A possible decomposition of BS(() gate.
10)
qubits
Figure 8: Circuit for the parity check.
by utilizing parity check to make the whole HW preserving circuit bit-flip fault tolerant. The Hamming weight l indicates that we know whether there are odd number even number of Is in the state. We can utilize one ancilla qubit as shown Fig. to enable the parity check. The ancilla qubit is set to 10) in the beginning and all the working qubits are linked to the ancilla qubit with CNOT gate, with the ancilla qubit as the target bit. If is odd, then the measurement outcome of the ancilla qubit should be [1). Otherwise. there is bit flip occurring in the working qubits. Thus, it much easier to detect bit-flip error on the HW preserving circuit compared to Shor's code with two ancilla qubits required to protect one qubit from bit-flip error. We can either rerun the circuit or locate the qubit to correct the bit-flip error.
F FURTHER RESULTS ON UNITARY APPROXIMATION
In this section, we provide further results on the unitary approximation problem. To better demon- strate the convergence of the algorithms with different number of layers, we list the following results in Fig. 9. We also include RBS gate to better illustrate the best precision RBS can get. It is clear that matter how many parameters the comparison between the first column and the second column, we can see that NN connectivity requires more iterations get similar results compared ull connectivity. The results BS also
add, RBS gate is unable to go further than 10 From
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(a)
4000
(b)
6000
(c)
400
10
BS-Full
50
10'
BS-NN
30
1094
RBS-full
50
mmtters Number Ferameters Number of Parameters sumber of Parameters
dy 5
40
d2
de 5
-40
10-1
10-5
c(0)
18
10-92
10
1000
2000
3000
2500
5000
7500
250
500
750
150
150
BS-full
BS-NN
RBS-full
dy
120
dk 10
125
dx 10
120
10-5
10-5
10:
((7)
90
100
-90
10
10
10-42
60
75
10-11
30
10-11
(g)
(h)
100
BS-full
107
BS-NN
RBS-full de = 15
de 15
d
155
10-103
C(8)
20-115
10-120
120
10-121
10-12
60
10-12
60
0
2000
4000
(k)
1000
2000
3000
4000
50
100
150
200
10
BS-full
720
10a
BS-NN
750
10
RBS-full
720
dy 20
600
c/ 20
600
dy = 201
600
10-1
10 -5
10-313
2(0)
480
450
10-11s
480
10
300
Number
10
10-11
Iteration Figure 9: The energy errors w.r.t. iterations with the different number of parameters.
Iteration
Iteration
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implies the overparameterizztion phenomenon. We need more than df parameters reach 10-15 and further adding parameters will not lead to better performance but less iterations.
G FILTER METHODS AND MUTUAL INFORMATION
Out of three primary categories of feature selection techniques, which are wrapper methods, filter methods and embedded methods, we choose filter methods to address the task. Filter methods are particularly advantageous for their ability to execute swiftly and independently Our chosen filter method is based on information theory, specifically utilizing mutual information to quantify the relativity between individual features and the target variable. A higher mutual infor- mation indicates that the selected feature has greater predictive power for the target. The mutual information between feature X and target Y is in the form:
quantum circuits.
I(X;)) Dxx Px 8 Py)
= =Σ Σ (x, v) log( Pxy(2z1))
Px(x)
(32)
DEY TEX
= (X) + H(Y) -(X,Y)
And we apply conditional mutual information to quantify the relationship between a specific fea- ture, denoted as X, and the target variable condition of another feature Z. A higher value of conditional mutual information signifies that the feature X exhibits a greater degree of relevance to target and is independent of Z. Conversely, if the conditional mutual information is lower, it shows that the relativity between Z and the target primarily mediated by the presence of Z, and thus, Z may be considered redundant variable if already picked Z. The conditional mutual information is in the form:
1(x; = =E II Px|z & Py 2)2
= -ΣΣ Px,Y.2 2) log( Pz Pz(z) Px(2))xxxz(2.ubb))
2x, Pr.((v)))))
(33)
=EZ WEY
H(X + H(YIZ) H(X,YIZ)
With the acquisition of mutual information and conditional mutual information, we can encode all the n features into binary vector Z1:...,, In {0, 13" where signifies the i-th feature with means feature chosen. Subsequently, this information can be amalgamated into a n x n matrix Q where the items the diagonal Qu - [[x;; Y) correspond the mutual information between the i-th feature X; and target Y and the off-diagonal elements Qiz = -I(X; X|)))- Then reformulate the feature selection problem, seeking maximize both mutual information and conditional mutual information as well as considering the penalty, into the form of a new QUBO problem Q', and add the soft constrain Hadfield et (2017):
x QQxx x'x==xx = Qx - a (HW(())-
(34)
x² arg max (0,1)
TQQ al (H) (HW-o(HW(x))) (x)))
arg max (x Texx)
(35)
#
H DETAILED PROOF FOR THE TRAINABILITY
Theorem H.1 Consider n-qubit quantum circuit operating in the subspace with Hamming Weight
k. The variance of the cost function partial derivative is Varold:C) ~
16k (n-A)²
Proof. Consider the partial derivative the cost function with respect to the parameters For some parameter 0 in the 1-th RBS gate, we have:
19

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Published as a conference paper at ICLR 2024
0,C(0) -(T0)100)))0000000) = =((TT(U-NU000)) =UTr\U-pU_(A, 0+]
where P is the input state, O is the observable measure. U_ denotes the unitary matrix of the circuit before the 1-th gate and U+ denotes the unitary matrix after gate I. O+ U-OU.. The variance of the partial derivative is
{ dU+ /
Vara 10:000:01=
Ju dU-(0,C(0))
Jut
{ dU+ ( dU dU-(IIN(U-U' 0+1)),
Jut
where [: denotes the commutator of two matrices.
Vara (orc( S.. dU+ Trip² TT 0+1
Tr Trrip]Tr[HH, 0+1
dz
d4 (d 1)
ds *Tr(p²] - Tro1p] [, dU+(Tr[[H, 0+1²1
Ju,
dx(d] 1)
di - O.Tr([H
da (d% 1)
The initial state (vo) is set to be the uniform superposition over all computational basis in the da subspace. Thus, we have Trip] = 1, 1,Tr|p"]= 1
1
(
Varald/C] de(de 1)
du dU,Tr[[H,,41P 0+12]
2
1 Tr(HA)TH(O9) dR
dx(dx +
2
Tr(H7) Tr|
dd + +1)
dz
dx(dg 11)
21T[1111
Tv2 - d,Tr(0
de(de
We take Zo as the observable, then Tr(O] = du(n (u(n-2k) Tr( = dR. (other observables will also hold with the same magnitude) Thus, we have
where Tr[H?] = 2 =
2k(n 2k(n-k) dr. n(n-1) 1)
2k(n - k)dE
di
VaroloiC] = (dl + 1)
X
n
n(n 1)
4k(n - k)
dg(ni² (n 2k
(dz + 1)m(n 1)
(d. 1)n2
4k(n - k)
RR(Ank: 41:2)
(d: + 1)m(n -1)
(dl 1)n2
16k - df
P 16 (n k)²
(dz + 1)n3(n 1)( 1)
nd
We can further analyze that if the & is only 1, then Varo[a x na V Vara [a,C] z (,72)== which is approximate to exponentially small. This result is consistent with the conjecture that the trainability of the eircuit is closely related to de and smaller d, will lead to better trainability.
16 the kt = on the other hand,
20


Reviews:
[{"review_id": "dkzvw999iH", "reviewer": "Reviewer_ADjN", "summary": "This work discusses incorporating symmetries as hard constraints in ansatz design for Variational\nQuantum Algorithms (VQAs). In particular, the authors revisit the Hamming Weight (HW) preserving ansatz. By adopting Dynamical Lie Algebra (DLA) as a quantum optimal control theory and over-parameterization theory based on the maximum rank of quantum Fisher information matrix, one can quantify (and further ensure) the expressivity and capability of the proposed symmetry-preserving ansatz. These theoretical guarantees are verified on the unitary approximation problem. Moreover, the authors can demonstrate better performance on ground-state energy estimation ( need to preserve electron number) and feature selection as a constrained QUBO problem (preserve number of features), both using VQE.", "questions": "Can you explain why you use gate infidelity as the performance metric for the unitary approximation problem but plot Success Probability in Fig. 2? I guess the authors are converting infidelity of 1e-5 as log10(1e-5)/log10(1e-10)=0.5. Please make it clear in the paper.\nIt would be nice if the author could discuss connections of their HW preserving VQE with XY-mixer QAOA or other constrained QAOA. And maybe apply the theoretical tools used here to improve those work?\nWhat is the basis set used in ground-state energy estimation (such as STO-3G) to discretize the computation space?\nI believe the second test case, feature selection, is not a typical benchmark for VQAs. Is this a classically hard problem? Can you say more about why choose this test case?\n\nSome Typos:\n--Right below Lemma 4.1. \"Ramakrishna et al. (1995) has shown that if the dimension of DLA\" instead of \"Ramakrishna et al. (1995) has shown that if the dimension of the dimension of DLA\"\nThe y-axis in Fig. 3 and Fig. 7 should be in logarithmic scale\nRight below Appendix B.1. 'qubit'", "limitations": "", "rating": 8, "confidence": 5, "soundness": 3, "presentation": 2, "contribution": 3, "strengths": "A novel idea to use Dynamical Lie Algebra and over-parameterization theory for Parametrized Quantum Circuit (PQC) to design symmetry-preserving VQA ansatz.\nVery clear explanation of the theoretical tools used.\nThe authors conduct extensive numerical studies showing positive results.\nA bottom-up approach to guide quantum circuit design compared to the previous top-down approach [1]. The latter uses a different sampling-based definition of expressivity and entanglement capability to guide general-purpose circuit design, but actual task performance could vary a lot.\nHard constraints are more favorable than soft constraints in some industrial use cases, where solution validity and robustness are priorities.\n[1] Sim, Sukin, Peter D. Johnson, and Alán Aspuru‐Guzik. \"Expressibility and entangling capability of parameterized quantum circuits for hybrid quantum‐classical algorithms.\" *Advanced Quantum Technologies* 2.12 (2019): 1900070.", "weaknesses": "Regarding the literature review, it is not clear to me why the authors chose to categorize some previous work based on XY-mixer QAOA [1-2] as soft constraints. Their circuit is composed of a problem Hamiltonian layer (ZZ gates) and a mixer layer, both preserving the Hamming weight instead of adding a penalty to the cost function.\nOne of my biggest concerns is when we run this framework on NISQ hardware or noisy simulation. A bit-flip (X) error would suddenly break the Hamming distance. Will this be a big problem or not? It will be better to see more results along this consideration.\nNeed more discussion on the trainability of the proposed ansatz and Barren Plateau (BP) phenomenon. It makes intuitive sense to think of better trainability compared to HEA since it is only exploring a constrained subspace. The optimization problem should be easier. Previous work [3] showed that a compound layer consisting of FBS gates could lead to a gradient decay only polynomially. Will the same conclusion hold for this work?\nSome numerical experiment details don't make sense to me. For example, why did the authors choose the penalty weight in Eq. 16 as \\alpha = [0.5, 1, 5, 10]. To my best knowledge, people usually set this empirical parameter to a much larger value like 100.\n\n1] Hadfield, Stuart, et al. \"From the quantum approximate optimization algorithm to a quantum alternating operator ansatz.\" *Algorithms* 12.2 (2019): 34.\n[2] He, Zichang, et al. \"Alignment between initial state and mixer improves qaoa performance for constrained portfolio optimization.\" *arXiv preprint arXiv:2305.03857* (2023).\n[3] Cherrat, El Amine, et al. \"Quantum Deep Hedging.\" *arXiv preprint arXiv:2303.16585* (2023).", "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": "", "extra_scores": {}}, {"review_id": "cKvRQIZmFc", "reviewer": "Reviewer_Eu68", "summary": "Different families of Hamming weight (HW) preserving quantum circuits are examined in the context of general variational algorithms. A theoretical framework is developed for understanding when a given circuit family is sufficiently expressive to generate arbitrary symmetric transformations, and these various circuit families are experimentally compared against each other and against different baseline models in a variety of problems.", "questions": "I don't have any particular questions for the authors.", "limitations": "", "rating": 8, "confidence": 3, "soundness": 4, "presentation": 4, "contribution": 3, "strengths": "* The paper includes a large number of results of both a theoretical and experimental nature, that should be useful for someone wanting to use HW-preserving quantum circuits. Overall, this will likely encourage the use of these type of symmetric circuits in the context of more general classes of variational problems than they have been used in the past.\n\n* The experimental tasks are well chosen, with the unitary approximation, ground state energy estimation, and feature selection problems being representative of problems from quantum computing, quantum chemistry, and machine learning. The baseline methods used for comparison seem to be well chosen, and to the best of my knowledge are representative of the methods actually used in these various subject areas.\n\n* The paper is well-written, both in terms of its overall structure and its writing. Given the amount of material presented, this is critical for being able to understand the paper's results. The appendices are quite helpful for providing background and additional detail about the results.", "weaknesses": "* The premise of the paper is a bit atypical for conference submissions, in that it isn't \"selling\" a new model or method developed by the authors. Rather, the goal of the paper seems to be improving the community's understanding regarding the expressivity and performance of HW-preserving circuits in general. I think this is a valuable contribution, but I could see other reviewers pointing to a lack of novelty due to this unorthodox aim.\n\n* There is a lot of material that is packed into a limited space, and while I found the presentation to help a lot with this, the results still take some time to digest and understand in detail.\n\n* Figure 4 compares a large number of different circuit ansatzes, but many of these overlap with each other and are barely visible (e.g. the RBS-full points). It would helpful to revise this figure to permit easier viewing of all of the baseline results, for example by varying the colors, size, and/or ordering of the plot markers.", "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": "", "extra_scores": {}}, {"review_id": "QLOxZQc9gD", "reviewer": "Reviewer_Yeih", "summary": "This paper studies ansatz for variational quantum algorithms with the requirement that they preserve Hamming weight. They test the expressibility for the task of unitary approximation, where they show that the expressibility of these ansatz is the same as predicted (in terms of the number of parameters needed). In addition, experiments demonstrate the effectiveness of their approach on two different tasks: ground state preparation and feature selection.", "questions": "- In Equation 8, should the maximization be over $M$ such that $M\\geq M_c$?\n- Where does the construction for the BS gate come from?\n- Throughout this paper, it is stated that this is a \"revisiting\" of the Hamming weight preserving ansatz. What is the status of the previous work using these ansatz, and what are the specific improvements made in this paper compared to previous proposals?", "limitations": "", "rating": 6, "confidence": 4, "soundness": 4, "presentation": 3, "contribution": 2, "strengths": "- This paper nicely summarizes and makes clear the idea of using Hamming weight (or in physics terminology, excitation) preserving ansatz for variational quantum algorithms. The numerical results verify the accuracy of the theory, which characterizes the expressibility of these ansatz.\n- The experiments are nice, showing a modest improvement over existing methods.", "weaknesses": "- Typos: \"relative\" should be \"relatively\" throughout\n- Typo: In Figure 2, \"Haar measurement\" should be \"Haar measure\"\n- Typo: \"Hartree-Fork\"\n- Based on common gate sets of real machines, I feel the BS gate is quite artificial. It might make more sense to use parameterized $XX$ rotations with single-qubit $Z$ rotations. If the intention is to perform VQAs on near-term machines, then one might want to use more native instructions than the decompositions shown in Fig 5 and Fig 6.\n- Related to the above point, the decomposition of the BS gate is addressed as a possible limitation. I am pretty sure a better decomposition is possible using $XX$, $YY$ and $ZZ$ rotations.\n- I think this idea of preserving the Hamming weight is not very new, and the theoretical results are not very surprising to me. For this reason, I feel the contribution of this work is not so large, although the proposed ansatz appear to perform better than existing ones.", "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": "", "extra_scores": {}}, {"review_id": "YiBf0VYvxU", "reviewer": "Reviewer_obpr", "summary": "The authors review the known Hamming Weight (HW) preserving ansatz and discuss how it fits into the dynamical Lie algebra framework of (Larocca et al., 2023). The authors give an elementary gate set that they show generates all HW preserving gates (i.e., that gives \"full controllability\"). The authors then perform numerics demonstrating the practical utility of the HW preserving ansatz.", "questions": "What are some advantages to directly linking trainability to subspace controllability given the trainability of the HW preserving ansatz was already known?", "limitations": "", "rating": 5, "confidence": 4, "soundness": 4, "presentation": 3, "contribution": 2, "strengths": "The authors give a nice review on the HW preserving ansatz and link it to the well-known line of results discussing when quantum machine learning (QML) models are trainable by examining their controllability. The authors also describe a set of gates to parameterize the HW preserving ansatz in a controllable way.", "weaknesses": "Most of the paper is review. For instance, arXiv:2303.16585 (which the authors cite) already demonstrates that the HW preserving ansatz is efficiently trainable. Though never explicitly linked to (Larocca et al., 2023) in arXiv:2303.16585, the intuition that the trainability comes from the large symmetry of the ansatz is stated and described: see, e.g., their Figure 3.", "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": "", "extra_scores": {}}]

Author rebuttal:

Decision: Accept (poster)