Original: Escaping from the Barren Plateau via Gaussian Initializations in Deep Variational Quantum Circuits
Revised: Escaping from the Barren Plateau via Gaussian Initializations in Deep Variational Quantum Circuits

Original: Abstract
Revised: Abstract

Original: Variational quantum circuits have been widely employed in quantum simulationand quantum machine learning in recent years.
Revised: Variational quantum circuits have been widely employed in quantum simulation and quantum machine learning in recent years.

Original: However, quantum circuits withrandom structures have poor trainability due to the exponentially vanishing gradientwith respect to the circuit depth and the qubit number.
Revised: However, quantum circuits with random structures have poor trainability due to the exponentially vanishing gradient with respect to the circuit depth and the qubit number.

Original: This result leads to a generalstandpoint that deep quantum circuits would not be feasible for practical tasks.
Revised: This result leads to a general standpoint that deep quantum circuits would not be feasible for practical tasks.

Original: In this work, we propose an initialization strategy with theoretical guarantees forthe vanishing gradient problem in general deep quantum circuits.
Revised: In this work, we propose an initialization strategy with theoretical guarantees for the vanishing gradient problem in general deep quantum circuits.

Original: Specifically, weprove that under proper Gaussian initialized parameters, the norm of the gradientdecays at most polynomially when the qubit number and the circuit depth increase.
Revised: Speciﬁcally, we prove that under proper Gaussian initialized parameters, the norm of the gradient decays at most polynomially when the qubit number and the circuit depth increase.

Original: Our theoretical results hold for both the local and the global observable cases, wherethe latter was believed to have vanishing gradients even for very shallow circuits.
Revised: Our theoretical results hold for both the local and the global observable cases, where the latter was believed to have vanishing gradients even for very shallow circuits.

Original: Experimental results verify our theoretical findings inthe quantum simulation andquantum chemistry.
Revised: Experimental results verify our theoretical ﬁndings in quantum simulation and quantum chemistry.

Original:  Introduction
Revised:  Introduction

Original: Quantum computing has attracted great attention in recent years, especially since the realization ofquantum supremacy [1, 2] with noisy intermediate-scale quantum (NISQ) devices [3].
Revised: Quantum computing has attracted great attention in recent years, especially since the realization of quantum supremacy [1, 2] with noisy intermediate-scale quantum (NISQ) devices [3].

Original: Due to mildrequirements on the gate noise and the circuit connectivity, variational quantum algorithms (VQAs) [4]become one of the most promising frameworks for achieving practical quantum advantages on NISQdevices.
Revised: Due to mild requirements on the gate noise and the circuit connectivity, variational quantum algorithms (VQAs) [4] become one of the most promising frameworks for achieving practical quantum advantages on NISQ devices.

Original: Specifically, different VQAs have been proposed for many topics, e g., quantum chemistry [5–13], quantum simulations [14–23], machine learning [24–31], numerical analysis [32–36] and linearalgebra problems [37–39].
Revised: Speciﬁcally, different VQAs have been proposed for many topics, e g., quantum chemistry [5– 13], quantum simulations [14–23], machine learning [24–31], numerical analysis [32–36], and linear algebra problems [37–39].

Original: Recently, various small-scale VQAs have been implemented on realquantum computers for tasks such as finding the ground state of molecules [10–12] and exploringapplications in supervised learning [25], generative learning [30] and reinforcement learning [29].
Revised: Recently, various small-scale VQAs have been implemented on real quantum computers for tasks such as ﬁnding the ground state of molecules [10–12] and exploring applications in supervised learning [25], generative learning [30] and reinforcement learning [29].

Original: Typical variational quantum algorithms is a trainable quantum-classical hybrid framework basedon parameterized quantum circuits (PQCs) [40].
Revised: Typical variational quantum algorithms is a trainable quantum-classical hybrid framework based on parameterized quantum circuits (PQCs) [40].

Original: Similar to classical counterparts such as neuralnetworks [41], first-order methods including the gradient descent [42] and its variants [43] are widelyemployed in optimizing the loss function of VQAs.
Revised: Similar to classical counterparts such as neural networks [41], ﬁrst-order methods including the gradient descent [42] and its variants [43] are widely employed in optimizing the loss function of VQAs.

Original: However, VQAs may face the trainability barrierwhen scaling up the size of quantum circuits (i.e., the number of involved qubits or the circuit depth),which is known as the barren plateau problem [44].
Revised: However, VQAs may face the trainability barrier when scaling up the size of quantum circuits (i.e., the number of involved qubits or the circuit depth), which is known as the barren plateau problem [44].

Original: Roughly speaking, the barren plateau describes the phenomenon that the value of the loss functionand its gradients concentrate around their expectation values with exponentially small variances.
Revised: Roughly speaking, the barren plateau describes the phenomenon that the value of the loss function and its gradients concentrate around their expectation values with exponentially small variances.

Original: We remark that gradient-based methods could hardly handle trainings with the barren plateau phenomenon [45].
Revised: We remark that gradient-based methods could hardly handle trainings with the barren plateau phenomenon [45].

Original: Both the machine noise of the quantum channel and the statistical noise induced bymeasurements could severely degrade the estimation of gradients.
Revised: Both the machine noise of the quantum channel and the statistical noise induced by measurements could severely degrade the estimation of gradients.

Original: Moreover, the optimization ofthe loss with a flat surface takes much more time using inaccurate gradients than ideal cases.
Revised: Moreover, the optimization of the loss with a ﬂat surface takes much more time using inaccurate gradients than ideal cases.

Original: Thus,solving the barren plateau problem is imperative for achieving practical quantum advantages with
Revised: Thus, solving the barren plateau problem is imperative for achieving practical quantum advantages with

Original: VQAs.
Revised: VQAs.

Original: In this paper, we propose Gaussian initializations for VQAs which have theoretical guaranteeson the trainability.
Revised: In this paper, we propose Gaussian initializations for VQAs which have theoretical guarantees on the trainability.

Original: We prove that for Gaussian initialized parameters with certain variances, theexpectation of the gradient norm is lower bounded by the inverse of the polynomial term of the qubitnumber and the circuit depth.
Revised: We prove that for Gaussian initialized parameters with certain variances, the expectation of the gradient norm is lower bounded by the inverse of the polynomial term of the qubit number and the circuit depth.

Original: Technically, we consider various cases regarding VQAs in practice,which include local or global observables, independently or jointly employed parameters, and noisyoptimizations induced by finite measurements.
Revised: Technically, we consider various cases regarding VQAs in practice, which include local or global observables, independently or jointly employed parameters, and noisy optimizations induced by ﬁnite measurements.

Original: To summarize, our contributions are fourfold:
Revised: To summarize, our contributions are fourfold:

Original: • We propose a Gaussian initialization strategy for deep variational quantum circuits.By
Revised: • We propose a Gaussian initialization strategy for deep variational quantum circuits.

Original: settingthe variance γ [-
-] = O ( 1 L ) for N -qubit L -depth circuits with independent parameters and localobservables, we lower bound the expectation of the gradient norm by poly( N, L ) − 1 as providedin Theorem 4.1, which outperforms previous 2 −O ( L ) 
Revised: By setting the variance γ [-
-]2 = O ( 1 L ) for N -qubit L -depth circuits with independent parameters and local observables, we lower bound the expectation of the gradient norm by poly( N, L ) − 1 as provided in Theorem 4.1, which outperforms previous 2 −O ( L ) results.

Original: results.• We extend the gradient norm result to the global observable case in Theorem 4.2, which wasbelieved to have the barren plateau problem even for very shallow circuits.
Revised: • We extend the gradient norm result to the global observable case in Theorem 4.2, which was believed to have the barren plateau problem even for very shallow circuits.

Original: Moreover, ourbound holds for correlated parameterized gates, which are widely employed in practical taskslike quantum chemistry and quantum simulations.•
Revised: Moreover, our bound holds for correlated parameterized gates, which are widely employed in practical tasks like quantum chemistry and quantum simulations.

Original:  We provide further analysis on the number of necessary measurements for estimating thegradient, where the noisy case differs from the ideal case with a Gaussian noise.
Revised: • We provide further analysis on the number of necessary measurements for estimating the gradient, where the noisy case differs from the ideal case with a Gaussian noise.

Original: The resultis presented in Corollary 4.3, which proves that O ( Lϵ ) times of measurement is sufficient toguarantee a large gradient.• We conduct various numerical experiments including finding the ground energy and the groundstate of the Heisenberg model and the LiH molecule, which belong to quantum simulationand quantum chemistry, respectively.
Revised: The result is presented in Corollary 4.3, which proves that O ( L(cid:15) ) times of measurement is sufﬁcient to guarantee a large gradient. • We conduct various numerical experiments including ﬁnding the ground energy and the ground state of the Heisenberg model and the LiH molecule, which belong to quantum simulation and quantum chemistry, respectively.

Original: Experiment results show that Gaussian initializationsoutperform uniform initializations, which verify proposed theorems.
Revised: Experiment results show that Gaussian initializations outperform uniform initializations, which verify proposed theorems.

Original: 1.1 Related work
Revised: 1.1 Related work

Original: The barren plateau phenomenon was first noticed in [44], which proves that if the circuit distributionforms unitary 2 -designs [46], the variance of the gradient of the circuit vanishes to zero with therate exponential in the qubit number.
Revised: The barren plateau phenomenon was ﬁrst noticed in [44], which proves that if the circuit distribution forms unitary 2 -designs [46], the variance of the gradient of the circuit vanishes to zero with the rate exponential in the qubit number.

Original: Subsequently, several positive results are proved for shallowquantum circuits such as the alternating-layered circuit [45, 47] and the quantum convolutionalneural network [48] when the observable is constrained in small number of qubits (local observable).
Revised: Subsequently, several positive results are proved for shallow quantum circuits such as the alternating-layered circuit [45, 47] and the quantum convolutional neural network [48] when the observable is constrained in small number of qubits (local observable).

Original: For shallow circuits with N qubits and O (log N ) depth, the variance of the gradient has the orderpoly( N ) − 1 if gate blocks in the circuit are sampled from local 2 -design distributions.
Revised: For shallow circuits with N qubits and O (log N ) depth, the variance of the gradient has the order poly( N ) − 1 if gate blocks in the circuit are sampled from local 2 -design distributions.

Original: Later, severalworks prove an inherent relationship between the barren plateau phenomenon and the complexity ofstates generated from the circuit.
Revised: Later, several works prove an inherent relationship between the barren plateau phenomenon and the complexity of states generated from the circuit.

Original: Specifically, circuit states that satisfy the volume law could lead tothe barren plateau problem [49].
Revised: Speciﬁcally, circuit states that satisfy the volume law could lead to the barren plateau problem [49].

Original: Expressive quantum circuits, which is measured by the distancebetween the Haar distribution and the distribution of circuit states, could have vanishing gradients [50].
Revised: Expressive quantum circuits, which is measured by the distance between the Haar distribution and the distribution of circuit states, could have vanishing gradients [50].

Original: Since random circuits form approximately 2 -designs when they achieve linear depths [46], deepquantum circuits were believed to suffer the barren plateau problem generally.
Revised: Since random circuits form approximately 2 -designs when they achieve linear depths [46], deep quantum circuits were believed to suffer the barren plateau problem generally.

Original: The parameterization of quantum circuits is achieved by tuning the time of Hamiltonian simulations,so the gradient of the circuit satisfies the parameter-shift rule [51].
Revised: The parameterization of quantum circuits is achieved by tuning the time of Hamiltonian simulations, so the gradient of the circuit satisﬁes the parameter-shift rule [51].

Original: Thus, the variance of the lossin VQAs and that of its gradient have similar behaviors for uniform distributions [44, 52].
Revised: Thus, the variance of the loss in VQAs and that of its gradient have similar behaviors for uniform distributions [44, 52].

Original: Onecorollary of the parameter-shift rule is that the gradient of depolarized noisy quantum circuits vanishesexponentially with increasing circuit depth [53], since the loss itself vanishes in the same rate.
Revised: One corollary of the parameter-shift rule is that the gradient of depolarized noisy quantum circuits vanishes exponentially with increasing circuit depth [53], since the loss itself vanishes in the same rate.

Original: Anothercorollary is that both gradient-free [54] and higher-order methods [55] could not solve the barrenplateau problem.
Revised: Another corollary is that both gradient-free [54] and higher-order methods [55] could not solve the barren plateau problem.

Original: Although most existing theoretical and practical results imply the barren plateauphenomenon in deep circuits, VQAs with deep circuits do have impressive advantages from otheraspects.
Revised: Although most existing theoretical and practical results imply the barren plateau phenomenon in deep circuits, VQAs with deep circuits do have impressive advantages from other aspects.

Original: For example, the loss of VQAs is highly non-convex, which is hard to find the globalminima [56] for both shallow and deep circuits.
Revised: For example, the loss of VQAs is highly non-convex, which is hard to ﬁnd the global minima [56] for both shallow and deep circuits.

Original: Meanwhile, for VQAs with shallow circuits, localminima and global minima have considerable gaps [57], which could severely influence the trainingperformance of gradient-based methods.
Revised: Meanwhile, for VQAs with shallow circuits, local minima and global minima have considerable gaps [57], which could severely inﬂuence the training performance of gradient-based methods.

Original: Contrary to shallow cases, deep VQAs have vanishing gapsbetween local minima and global minima [58].
Revised: Contrary to shallow cases, deep VQAs have vanishing gaps between local minima and global minima [58].

Original: In practice, experiments show that overparameterized
Revised: In practice, experiments show that overparameterized

Original: VQAs [59] can be optimized towards the global minima.
Revised: VQAs [59] can be optimized towards the global minima.

Original: Moreover, VQAs with deep circuits havemore expressive power than that of shallow circuits [60–62], which implies the potential to handlemore complex tasks in quantum machine learning and related fields.
Revised: Moreover, VQAs with deep circuits have more expressive power than that of shallow circuits [60–62], which implies the potential to handle more complex tasks in quantum machine learning and related ﬁelds.

Original: Inspired by various advantages of deep VQAs, some approaches have been proposed recently forsolving the related barren plateau problem in practice.
Revised: Inspired by various advantages of deep VQAs, some approaches have been proposed recently for solving the related barren plateau problem in practice [63–65].

Original: For example, the block-identity strategy [63] initializes gate blocks in pairs and sets parameters inversely, such that the initial circuit is equivalentto the identity circuit with zero depth.
Revised: For example, the block-identity strategy [63] initializes gate blocks in pairs and sets parameters inversely, such that the initial circuit is equivalent to the identity circuit with zero depth.

Original: Since shallow circuits have no vanishing gradient problem,the corresponding VQA is trainable with guarantees at the first step.
Revised: Since shallow circuits have no vanishing gradient problem, the corresponding VQA is trainable with guarantees at the ﬁrst step.

Original: However, we remark thatthe block-identity condition would not hold after the first step, and the structure of the circuitneeds to be designed properly.
Revised: However, we remark that the block-identity condition would not hold after the ﬁrst step, and the structure of the circuit needs to be designed properly.

Original: The layerwise training method [64] trains parameters in the circuitlayers by layers, such that the depth of trainable part is limited.
Revised: The layerwise training method [64] trains parameters in the circuit layers by layers, such that the depth of trainable part is limited.

Original: However, this method implementscircuits with larger depth than that of the origin circuit, and parameters in the first few layers arenot optimized.
Revised: However, this method implements circuits with larger depth than that of the origin circuit, and parameters in the ﬁrst few layers are not optimized.

Original: A recent work provides theoretical guarantees on the trainability of deep circuitswith certain structures [65].
Revised: A recent work provides theoretical guarantees on the trainability of deep circuits with certain structures [65].

Original: However, the proposed theory only suits VQAs with local observables,but many practical applications such as finding the ground state of molecules and the quantumcompiling [66, 67] apply global observables.
Revised: However, the proposed theory only suits VQAs with local observables, but many practical applications such as ﬁnding the ground state of molecules and the quantum compiling [66, 67] apply global observables.

Original:  Notations and quantum computing basics
Revised:  Notations and quantum computing basics

Original: We denote by [ N ] the set { 1 , · · · , N } .
Revised: We denote by [ N ] the set { 1 , · · · , N } .

Original: The form ∥ · ∥ 2 represents the ℓ 2 norm for the vector and thespectral norm for the matrix, respectively.
Revised: The form (cid:107) · (cid:107) 2 represents the (cid:96) 2 norm for the vector and the spectral norm for the matrix, respectively.

Original: We denote by a j the j -th component of the vector a .
Revised: We denote by a j the j -th component of the vector a .

Original: Thetensor product operation is denoted as “ ⊗ ".
Revised: The tensor product operation is denoted as “ ⊗ ".

Original: The conjugate transpose of a matrix A is denoted as A † .
Revised: The conjugate transpose of a matrix A is denoted as A † .

Original: The trace of a matrix A is denoted as Tr [ A ] .
Revised: The trace of a matrix A is denoted as Tr [ A ] .

Original: We denote ∇ θ f as the gradient of the function f withrespect to the variable θ .
Revised: We denote ∇ θ f as the gradient of the function f with respect to the variable θ .

Original: We employ notations O to describe complexity notions.
Revised: We employ notations O to describe complexity notions.

Original: Now we introduce quantum computing knowledge and notations.
Revised: Now we introduce quantum computing knowledge and notations.

Original: The pure state of a qubit could bewritten as | ϕ ⟩ = a | 0 ⟩ + b | 1 ⟩ , where a, b ∈ C satisfy | a | 2 + | b | 2 = 1 , and | 0 ⟩ = (1 , 0) T , | 1 ⟩
Revised: The pure state of a qubit could be written as | φ (cid:105) = a | 0 (cid:105) + b | 1 (cid:105) , where a, b ∈ C satisfy | a | 2 + | b | 2 = 1 , and | 0 (cid:105) = (1 , 0) T , | 1 (cid:105) =

Original: = (0 , 1) T .
Revised: (0 , 1) T .

Original: The N -qubit space is formed by the tensor product of N single-qubit spaces.
Revised: The N -qubit space is formed by the tensor product of N single-qubit spaces.

Original: For pure states, thecorresponding density matrix is defined as ρ = | ϕ ⟩⟨ ϕ | , in which ⟨ ϕ | 
Revised: For pure states, the corresponding density matrix is deﬁned as ρ = | φ (cid:105)(cid:104) φ | , in which (cid:104) φ | =

Original: = ( | ϕ ⟩ ) † .
Revised: ( | φ (cid:105) ) † .

Original: We use the densitymatrix to represent general mixed quantum states, i.e., ρ = (cid:80)
Revised: We use the density matrix to represent general mixed quantum states, i.e., ρ = (cid:80)

Original: k c k | ϕ k ⟩⟨ ϕ k | , where c k ∈ R and112 (cid:80) k
Revised: k c k | φ k (cid:105)(cid:104) φ k | , where c k ∈ R and (cid:80) k

Original: c k = 1 .
Revised: c k = 1 .

Original: A single-qubit operation to the state behaves like the matrix-vector multiplicationand can be referred to as the gate in the quantum circuit language.
Revised: A single-qubit operation to the state behaves like the matrix-vector multiplication and can be referred to as the gate in the quantum circuit language.

Original: Specifically, single-qubitoperations are often used as R X ( θ ) =
Revised: Speciﬁcally, single-qubit operations are often used as R X ( θ ) =

Original: e − iθX , R Y ( θ ) =
Revised: e − iθX , R Y ( θ ) =

Original: e − iθY , and R Z ( θ ) =
Revised: e − iθY , and R Z ( θ ) =

Original: e − iθZ , where
Revised: e − iθZ , where

Original: X = (cid:18)

Original: [Equation]

Original: (cid:19) , Y = (cid:18)

Original: [Equation]
Revised: [Equation]

Original: (cid:19)

Original: [Equation]

Original: (cid:18)

Original: [Equation] [Equation] (cid:19) . Pauli matrices will be referred to as { I, X, Y, Z } = { σ 0 , σ 1 , σ 2 , σ 3 } for the convenience.
Revised: Pauli matrices will be referred to as { I, X, Y, Z } = { σ 0 , σ 1 , σ 2 , σ 3 } for the convenience.

Original: Moreover,two-qubit operations, such as the CZ gate and the √ i SWAP gate, are employed for generatingquantum entanglement:
Revised: Moreover, two-qubit operations, such as the CZ gate and the √ i SWAP gate, are employed for generating quantum entanglement:

Original: [Equation]
Revised: [Equation]

Original: We could obtain information from the quantum system by performing measurements, for example,measuring the state | ϕ ⟩ = a | 0 ⟩ + b | 1 ⟩ generates 0 and 1 with probability p (0)=
Revised: We could obtain information from the quantum system by performing measurements, for example, measuring the state | φ (cid:105) = a | 0 (cid:105) + b | 1 (cid:105) generates 0 and 1 with probability p (0)

Original:  | a | 2 and p (1) = | b | 2 ,respectively.
Revised: = | a | 2 and p (1) = | b | 2 , respectively.

Original: Such a measurement operation could be mathematically referred to as calculating theaverage of the observable O =
Revised: Such a measurement operation could be mathematically referred to as calculating the average of the observable O =

Original: σ 3 under the state | ϕ ⟩ :
Revised: σ 3 under the state | φ (cid:105) :

Original: [Equation]
Revised: [Equation]

Original: Mathematically, quantum observables are Hermitian matrices.
Revised: Mathematically, quantum observables are Hermitian matrices.

Original: Specifically, the average of a unitaryobservable under arbitrary states is bounded by [ − 1 , 1] .
Revised: Speciﬁcally, the average of a unitary observable under arbitrary states is bounded by [ − 1 , 1] .

Original: We remark that O ( 1 ϵ 2 ) times of measurementscould provide an ϵ ∥ O ∥ 2 -error estimation to the value
Revised: We remark that O ( 1 (cid:15) 2 ) times of measurements could provide an (cid:15) (cid:107) O (cid:107) 2 -error estimation to the value

Original: Tr [ Oρ ] .
Revised: Tr [ Oρ ] .

Original:  Framework of general VQAs
Revised:  Framework of general VQAs

Original: In this section, we introduce the framework of general VQAs and corresponding notations.
Revised: In this section, we introduce the framework of general VQAs and corresponding notations.

Original: A typicalvariational quantum algorithm can be viewed as the optimization of the function f , which is definedas the expectation of observables.
Revised: A typical variational quantum algorithm can be viewed as the optimization of the function f , which is deﬁned as the expectation of observables.

Original: The expectation varies for different initial states and differentparameters θ used in quantum circuits.
Revised: The expectation varies for different initial states and different parameters θ used in quantum circuits.

Original: Throughout this paper, we define
Revised: Throughout this paper, we deﬁne

Original: [Equation]
Revised: [Equation]

Original: [Equation] as the loss function of VQAs, where V ( θ ) denotes the parameterized quantum circuit, the hermitianmatrix O denotes the observable, and ρ in denotes the density matrix of the input state.
Revised: as the loss function of VQAs, where V ( θ ) denotes the parameterized quantum circuit, the hermitian matrix O denotes the observable, and ρ in denotes the density matrix of the input state.

Original: Next, weexplain observables, input states, and parameterized quantum circuits in detail.
Revised: Next, we explain observables, input states, and parameterized quantum circuits in detail.

Original: Both the observable and the density matrix could be decomposed under the Pauli basis.
Revised: Both the observable and the density matrix could be decomposed under the Pauli basis.

Original: We define thelocality of a quantum observable as the maximum number of non-identity Pauli matrices in the tensorproduct, such that the corresponding coefficient is not zero.
Revised: We deﬁne the locality of a quantum observable as the maximum number of non-identity Pauli matrices in the tensor product, such that the corresponding coefﬁcient is not zero.

Original: Thus, the observable with the constantlocality is said to be local , and the observable that acts on all qubits is said to be global .
Revised: Thus, the observable with the constant locality is said to be local , and the observable that acts on all qubits is said to be global .

Original: The observable and the input state in VQAs could have various formulations for specific tasks.
Revised: The observable and the input state in VQAs could have various formulations for speciﬁc tasks.

Original: For thequantum simulation or the quantum chemistry scenario, observables are constrained to be the system
Revised: For the quantum simulation or the quantum chemistry scenario, observables are constrained to be the system

Original: Hamiltonians, while input states are usually prepared as computational basis states.
Revised: Hamiltonians, while input states are usually prepared as computational basis states.

Original: For example,( | 0 ⟩⟨ 0 | ) ⊗ N is used frequently in quantum simulations [17, 18].
Revised: For example, ( | 0 (cid:105)(cid:104) 0 | ) ⊗ N is used frequently in quantum simulations [17, 18].

Original: Hartree–Fock (HF) states [9, 10],which are prepared by the tensor product of {| 0 ⟩ , | 1 ⟩} , serve as good initial states in quantumchemistry tasks [9, 11–13].
Revised: Hartree–Fock (HF) states [9, 10], which are prepared by the tensor product of {| 0 (cid:105) , | 1 (cid:105)} , serve as good initial states in quantum chemistry tasks [9, 11–13].

Original: For quantum machine learning (QML) tasks, initial states encode theinformation of the training data, which could have a complex form.
Revised: For quantum machine learning (QML) tasks, initial states encode the information of the training data, which could have a complex form.

Original: Many encoding strategies havebeen introduced in the literature [24, 68, 69].
Revised: Many encoding strategies have been introduced in the literature [24, 68, 69].

Original: In contrary with the complex initial states, observablesemployed in QML are quite simple.
Revised: In contrary with the complex initial states, observables employed in QML are quite simple.

Original: For example, π 0 = | 0 ⟩⟨ 0{+
 (cid:105)(cid:104) 
+} | serves as the observable in most QMLtasks related with the classification [24–26] or the dimensional reduction [70].
Revised: For example, π 0 = | {+
 (cid:105)(cid:104) 
+} | serves as the observable in most QML tasks related with the classiﬁcation [24–26] or the dimensional reduction [70].

Original: Apart from the input states and the observable choices, parameterized quantum circuits employedin different variational quantum algorithms have various structures, which are also known asansatzes [71–73].
Revised: Apart from the input states and the observable choices, parameterized quantum circuits employed in different variational quantum algorithms have various structures, which are also known as ansatzes [71–73].

Original: Specifically, the ansatz in the VQA denotes the initial guess on the circuit structure.
Revised: Speciﬁcally, the ansatz in the VQA denotes the initial guess on the circuit structure.

Original: For example, alternating-layered ansatzes [71, 74] are proposed for approximating the Hamiltonianevolution.
Revised: For example, alternating-layered ansatzes [71, 74] are proposed for approximating the Hamiltonian evolution.

Original: Recently, hardware efficient ansatzes [7, 75] and tensor-network based ansatzes [76, 77],which could utilize parameters efficiently on noisy quantum computers, have been developed forvarious tasks, including quantum simulations and quantum machine learning.
Revised: Recently, hardware efﬁcient ansatzes [7, 75] and tensor-network based ansatzes [76, 77], which could utilize parameters efﬁciently on noisy quantum computers, have been developed for various tasks, including quantum simulations and quantum machine learning.

Original: For quantum chemistrytasks, unitary coupled cluster ansatzes [78, 79] are preferred since they preserve the number ofelectrons corresponding to circuit states.
Revised: For quantum chemistry tasks, unitary coupled cluster ansatzes [78, 79] are preferred since they preserve the number of electrons corresponding to circuit states.

Original: In practice, ansatz is deployed as the sequence of single-qubit rotations { e − iθσ k , k ∈ { 1 , 2 , 3 }} andtwo-qubit gates.
Revised: In practice, ansatz is deployed as the sequence of single-qubit rotations { e − iθσ k , k ∈ { 1 , 2 , 3 }} and two-qubit gates.

Original: We remark that the gradient of the VQA satisfies the parameter-shift rule [51, 80, 81];namely, for independently deployed parameters θ j , the corresponding partial derivative is
Revised: We remark that the gradient of the VQA satisﬁes the parameter-shift rule [51, 80, 81]; namely, for independently deployed parameters θ j , the corresponding partial derivative is

Original: [Equation]
Revised: [Equation]

Original: where θ + and θ − are different from θ only at the j -th parameter: θ j → θ j ± π 4 .
Revised: where θ + and θ − are different from θ only at the j -th parameter: θ j → θ j ± π 4 .

Original: Thus, the gradient of fcould be estimated efficiently, which allows optimizing VQAs with gradient-based methods [82–84].
Revised: Thus, the gradient of f could be estimated efﬁciently, which allows optimizing VQAs with gradient-based methods [82–84].

Original:  Theoretical results about Gaussian initialized VQAs
Revised:  Theoretical results about Gaussian initialized VQAs

Original: In this section, we provide our theoretical guarantees on the trainability of deep quantum circuitsthrough proper designs for the initial parameter distribution.
Revised: In this section, we provide our theoretical guarantees on the trainability of deep quantum circuits through proper designs for the initial parameter distribution.

Original: In short, we prove that the gradient of the
Revised: In short, we prove that the gradient of the

Original: L -layer N -qubit circuit is upper bounded by 1 / poly( L, N ) , if initial parameters are sampled from a
Revised: L -layer N -qubit circuit is upper bounded by 1 / poly( L, N ) , if initial parameters are sampled from a

Original: Gaussian distribution with O (1 /L ) variance.
Revised: Gaussian distribution with O (1 /L ) variance.

Original: Our bounds significantly improve existing results ofthe gradients of VQAs, which have the order 2 −O ( L ) for shallow circuits and the order 2 −O ( N ) fordeep circuits.
Revised: Our bounds signiﬁcantly improve existing results of the gradients of VQAs, which have the order 2 −O ( L ) for shallow circuits and the order 2 −O ( N ) for deep circuits.

Original: We prove different results for the local and global observable cases in Section 4.1 and
Revised: We prove different results for the local and global observable cases in Section 4.1 and

Original: Section 4.2, respectively.
Revised: Section 4.2, respectively.

Original: 4.1 Independent parameters with local observables
Revised: 4.1 Independent parameters with local observables

Original: First, we introduce the Gaussian initialization of parameters for the local observable case.
Revised: First, we introduce the Gaussian initialization of parameters for the local observable case.

Original: We usethe quantum circuit illustrated in Figure 1 as the ansatz in this section.
Revised: We use the quantum circuit illustrated in Figure 1 as the ansatz in this section.

Original: The circuit in Figure 1performs L layers of single qubit rotations and CZ gates on the input state ρ in , followed by a R X layerand a R Y layer.
Revised: The circuit in Figureperforms L layers of single qubit rotations and CZ gates on the input state ρ in , followed by a R X layer and a R Y layer.

Original: We denote the single-qubit gate on the n -th qubit of the ℓ -th layer as e − iθ ℓ,n G ℓ,n ,∀ ℓ ∈ { 1 , · · · , L + 2 } and n ∈ { 1 , · · · , N } , where θ ℓ,n is the corresponding parameter and G ℓ,n is a
Revised: We denote the single-qubit gate on the n -th qubit of the (cid:96) -th layer as e − iθ (cid:96),n G (cid:96),n , ∀ (cid:96) ∈ { 1 , · · · , L + 2 } and n ∈ { 1 , · · · , N } , where θ (cid:96),n is the corresponding parameter and G (cid:96),n is a

Original: Hermitian unitary.
Revised: Hermitian unitary.

Original: To eliminate degenerate parameters, we require that single-qubit gates in the first
Revised: To eliminate degenerate parameters, we require that single-qubit gates in the ﬁrst

Original: L layers do not commute with the CZ gate.
Revised: L layers do not commute with the CZ gate.

Original: After gates operations, we measure the observable
Revised: After gates operations, we measure the observable

Original: [Equation]
Revised: [Equation]

Original: L times
Revised: L times

Original: R ( θ ℓ, 1 )
Revised: R ( θ (cid:96), 1 )

Original: CZ ℓ
Revised: CZ (cid:96)

Original: R X ( θ L +1 , 1 )
Revised: R X ( θ L +1 , 1 )

Original: R Y ( θ L +2 , 1 )
Revised: R Y ( θ L +2 , 1 )

Original: R ( θ ℓ, 2 )
Revised: R ( θ (cid:96), 2 )

Original: R X ( θ L +1 , 2 )
Revised: R X ( θ L +1 , 2 )

Original: R Y ( θ L +2 , 2 ) ...
Revised: R Y ( θ L +2 , 2 ) ...

Original: R ( θ ℓ,N )
Revised: R ( θ (cid:96),N )

Original: R X ( θ L +1 ,N )
Revised: R X ( θ L +1 ,N )

Original: R Y ( θ L +2 ,N )   ρ in   where i j ∈ { 0 , 1 , 2 , 3 } , ∀ j ∈ { 1 , · · · , N } , and i contains S non-zero elements.
Revised: R Y ( θ L +2 ,N )   [Figure] in   where i j ∈ { 0 , 1 , 2 , 3 } , ∀ j ∈ { 1 , · · · , N } , and i contains S non-zero elements.

Original: Figure 1 provides ageneral framework of VQAs with local observables, which covers various ansatzes proposed in theliterature [65, 85, 61, 64].
Revised: Figure 1 provides a general framework of VQAs with local observables, which covers various ansatzes proposed in the literature [65, 85, 61, 64].

Original: The bound of the gradient norm of the Gaussian initialized variationalquantum circuit is provided in Theorem 4.1 with the proof in Appendix.
Revised: The bound of the gradient norm of the Gaussian initialized variational quantum circuit is provided in Theorem 4.1 with the proof in Appendix.

Original: Theorem 4.1.
Revised: Theorem 4.1.

Original: Consider the L -layer N -qubit variational quantum circuit V ( θ ) defined in Figure 1 and the cost function f ( θ ) =
Revised: Consider the L -layer N -qubit variational quantum circuit V ( θ ) deﬁned in Figureand the cost function f ( θ ) =

Original: Tr (cid:2) σ i V ( θ ) ρ in V ( θ ) † (cid:3) , where the observable σ i follows thedefinition (3).
Revised: Tr (cid:2) σ i V ( θ ) ρ in V ( θ ) † (cid:3) , where the observable σ i follows the deﬁnition (3).

Original: Then,
Revised: Then,

Original: [Equation]
Revised: [Equation]

Original: (4) where S is the number of non-zero elements in i , and the index j
Revised: where S is the number of non-zero elements in i , and the index j

Original: = ( j 1 , j 2 , · · · , j N ) such thatj m = 0 , ∀ i m = 

Revised: = ( j 1 , j 2 , · · · , j N ) such that j m = 0 , ∀ i m =

Original: and j m = 3 , ∀ i m ̸ = 0 .
Revised: and j m = 3 , ∀ i m (cid:54) = 0 .

Original: The expectation is taken with the Gaussian distribution
Revised: The expectation is taken with the Gaussian distribution

Original: N (cid:16) 0 , 1 4 S ( L +2) (cid:17) for the parameters θ .
Revised: N (cid:16) 0 , 1 4 S ( L +2) (cid:17) for the parameters θ .

Original: Compared to existing works [44, 45, 47, 48, 65], Theorem 4.1 provides a larger lower bound of thegradient norm, which improves the complexity exponentially with the depth of trainable circuits.
Revised: Compared to existing works [44, 45, 47, 48, 65], Theorem 4.1 provides a larger lower bound of the gradient norm, which improves the complexity exponentially with the depth of trainable circuits.

Original: Different from unitary 2 -design distributions [44, 45, 47, 48] or the uniform distribution in theparameter space [52, 86, 65] that were employed in existing works, we analyze the expectation of thegradient norm under a depth-induced Gaussian distribution.
Revised: Different from unitary 2 -design distributions [44, 45, 47, 48] or the uniform distribution in the parameter space [52, 86, 65] that were employed in existing works, we analyze the expectation of the gradient norm under a depth-induced Gaussian distribution.

Original: This change follows a natural idea thatthe trainability is not required in the whole parameter space or the entire circuit space, but only on theparameter trajectory during the training.
Revised: This change follows a natural idea that the trainability is not required in the whole parameter space or the entire circuit space, but only on the parameter trajectory during the training.

Original: Moreover, large norm of gradients could only guarantee thetrainability in the beginning stage, instead of the whole optimization, since a large gradient for trainedparameters corresponds to non-convergence.
Revised: Moreover, large norm of gradients could only guarantee the trainability in the beginning stage, instead of the whole optimization, since a large gradient for trained parameters corresponds to non-convergence.

Original: Thus, the barren plateau problem could be crucial ifinitial parameters have vanishing gradients, which has been proved for deep VQAs with uniforminitializations.
Revised: Thus, the barren plateau problem could be crucial if initial parameters have vanishing gradients, which has been proved for deep VQAs with uniform initializations.

Original: In contrary, we could solve the barren plateau problem if parameters are initializedproperly with large gradients, as provided in Theorem 4.1.
Revised: In contrary, we could solve the barren plateau problem if parameters are initialized properly with large gradients, as provided in Theorem 4.1.

Original: Finally, Gaussian initialized circuitsconverge to benign values if optima appear around θ = 0 , which holds in many cases.
Revised: Finally, Gaussian initialized circuits converge to benign values if optima appear around θ = 0 , which holds in many cases.

Original: For example,over-parameterized quantum circuits have benign local minima [58] if the number of parametersexceeds the over-parameterization threshold.
Revised: For example, over-parameterized quantum circuits have benign local minima [58] if the number of parameters exceeds the over-parameterization threshold.

Original: Moreover, over-parameterized circuits have exponentialconvergence rates [87, 88] on tasks like quantum machine learning and the quantum eigensolver.
Revised: Moreover, over-parameterized circuits have exponential convergence rates [87, 88] on tasks like quantum machine learning and the quantum eigensolver.

Original: These works indicate that quantum circuits with sufficient depths could find good optimums near theinitial points, which is similar to the classical wide neural network case [89].
Revised: These works indicate that quantum circuits with sufﬁcient depths could ﬁnd good optimums near the initial points, which is similar to the classical wide neural network case [89].

Original: 4.2 Correlated parameters with global observables
Revised: 4.2 Correlated parameters with global observables

Original: Next, we extend the Gaussian initialization framework to general quantum circuits with correlated parameters and global observables.
Revised: Next, we extend the Gaussian initialization framework to general quantum circuits with correlated parameters and global observables.

Original: Quantum circuits with correlated parameters have wide applications in quantum simulations and quantum chemistry [9, 11–13].
Revised: Quantum circuits with correlated parameters have wide applications in quantum simulations and quantum chemistry [9, 11–13].

Original: One example is the Givens rotation
Revised: One example is the Givens rotation

Revised: R Givens

Original: [Equation]
Revised: [Equation]

Original: P
Revised: θ

Original: [Equation]
Revised: [Equation]

Original:  which preserves the number of electrons in parameterized quantum states [11].
Revised: = [Equation] [Equation] which preserves the number of electrons in parameterized quantum states [11].

Original: To analyze VQAs with correlated parameterized gates, we consider the ansatz V ( θ ) =
Revised: To analyze VQAs with correlated parameterized gates, we consider the ansatz V ( θ ) =

Original: (cid:81) 

Revised: (cid:81)

Original: j = L V j ( θ j ) ,which consists of parameterized gates { V j ( θ j ) }
Revised: j = L V j ( θ j ) , which consists of parameterized gates { V j ( θ j ) }

Original: Lj =1 .
Revised: Lj =1 .

Original: Denote by h j the number of unitary gates thatshare the same parameter θ j .
Revised: Denote by h j the number of unitary gates that share the same parameter θ j .

Original: Thus, the parameterized gate V j ( θ j ) consists of a list of fixed andparameterized unitary operations
Revised: Thus, the parameterized gate V j ( θ j ) consists of a list of ﬁxed and parameterized unitary operations

Original: [Equation]
Revised: [Equation]

Original: with the term a j ∈ R / { 0 } , where the Hamiltonian G jk and the fixed gate W jk are unitary ∀ k ∈
Revised: with the term a j ∈ R / { 0 } , where the Hamiltonian G jk and the ﬁxed gate W jk are unitary ∀ k ∈

Original: [ h j ] .
Revised: [ h j ] .

Original: Moreover, we consider the objective function
Revised: Moreover, we consider the objective function

Original: [Equation]
Revised: [Equation]

Original: where ρ in and O denote the input state and the observable, respectively.
Revised: where ρ in and O denote the input state and the observable, respectively.

Original: In practical tasks of quantumchemistry, the molecule Hamiltonian H serves as the observable O .
Revised: In practical tasks of quantum chemistry, the molecule Hamiltonian H serves as the observable O .

Original: Minimizing the function (7)provides the ground energy and the corresponding ground state of the molecule.
Revised: Minimizing the function (7) provides the ground energy and the corresponding ground state of the molecule.

Original: We provide thebound of the gradient norm of the Gaussian initialized variational quantum circuit in Theorem 4.2with the proof in Appendix.
Revised: We provide the bound of the gradient norm of the Gaussian initialized variational quantum circuit in Theorem 4. with the proof in Appendix.

Original: Similar to the local observable case, we could bound the norm of thegradient of Eq.
Revised: Similar to the local observable case, we could bound the norm of the gradient of Eq.

Original: (7) if parameters are initialized with O ( 1 L ) variance.
Revised: (7) if parameters are initialized with O ( 1 L ) variance.

Original: Theorem 4.2 provides nontrivialbounds when the gradient at the zero point is large.
Revised: Theorem 4.2 provides nontrivial bounds when the gradient at the zero point is large.

Original: This condition holds when the mean-field theoryprovides a good initial guess to the corresponding problems, e.g. the ground energy task in quantumchemistry and quantum many-body problems [90].
Revised: This condition holds when the mean-ﬁeld theory provides a good initial guess to the corresponding problems, e.g. the ground energy task in quantum chemistry and quantum many-body problems [90].

Original: Theorem 4.2.
Revised: Theorem 4.2.

Original: Consider the N -qubit variational quantum algorithms with the objective function (7).
Revised: Consider the N -qubit variational quantum algorithms with the objective function (7).

Original: Then the following formula holds for any ℓ ∈ { 1 , · · · , L } ,
Revised: Then the following formula holds for any (cid:96) ∈ { 1 , · · · , L } ,

Original: [Equation]
Revised: [Equation]

Original: where 0 ∈ R L is the zero vector.
Revised: where 0 ∈ R L is the zero vector.

Original: The expectation is taken with Gaussian distributions N (0 , γ 2 j ) forparameters in θ = { θ j } Lj =1 , where the variance γ 2 j ≤
Revised: The expectation is taken with Gaussian distributions N (0 , γ 2 j ) for parameters in θ = { θ j } Lj =1 , where the variance γ 2 j ≤

Original: [Equation]
Revised: [Equation]

Original: We remark that Theorem 4.2 not only provides an initialization strategy, but also guarantees theupdate direction during the training.
Revised: We remark that Theorem 4.2 not only provides an initialization strategy, but also guarantees the update direction during the training.

Original: Different from the classical neural network, where the gradientcould be calculated accurately, the gradient of VQAs, obtained by the parameter-shift rule (2), isperturbed by the measurement noise.
Revised: Different from the classical neural network, where the gradient could be calculated accurately, the gradient of VQAs, obtained by the parameter-shift rule (2), is perturbed by the measurement noise.

Original: A guide on the size of acceptable measurement noise couldbe useful for the complexity analysis of VQAs.
Revised: A guide on the size of acceptable measurement noise could be useful for the complexity analysis of VQAs.

Original: Specifically, define θ ( t − 1) as the parameter at thet − 1 -th iteration.
Revised: Speciﬁcally, deﬁne θ ( t − 1) as the parameter at the t − 1 -th iteration.

Original: Denote by θ ( t ) and ˜ θ ( t ) the parameter updated from θ ( t − 1) for noiseless and noisycases, respectively.
Revised: Denote by θ ( t ) and ˜ θ ( t ) the parameter updated from θ ( t − 1) for noiseless and noisy cases, respectively.

Original: Then ˜ θ ( t ) differs from θ ( t ) by a Gaussian error term due to the measurementnoise.
Revised: Then ˜ θ ( t ) differs from θ ( t ) by a Gaussian error term due to the measurement noise.

Original: We expect to derive the gradient norm bound for ˜ θ ( t ) , as provided in Corollary 4.3.
Revised: We expect to derive the gradient norm bound for ˜ θ ( t ) , as provided in Corollary 4.3.

Original: Thus,1 γ 2 = O ( Lϵ ) number of measurements is sufficient to guarantee a large gradient.
Revised: Thus,γ 2 = O ( L(cid:15) ) number of measurements is sufﬁcient to guarantee a large gradient.

Original: Corollary 4.3.
Revised: Corollary 4.3.

Original: Consider the N -qubit variational quantum algorithms with the objective function (7).
Revised: Consider the N -qubit variational quantum algorithms with the objective function (7).

Original: Then the following formula holds for any ℓ ∈ { 1 , · · · , L } ,
Revised: Then the following formula holds for any (cid:96) ∈ { 1 , · · · , L } ,

Original: [Equation]
Revised: [Equation]

Original: The expectation is taken with Gaussian distributions N (0 , γ 2 j ) for parameters
Revised: The expectation is taken with Gaussian distributions N (0 , γ 2 j ) for parameters

Original: δ = { δ j } Lj =1 , wherethe variance γ 2 j ≤
Revised: δ = { δ j } Lj =1 , where the variance γ 2 j ≤

Original: [Equation]
Revised: [Equation]

Original: [Figure]
Revised: [Figure]

Revised: 5 rounds of optimizations.

Original: Corollary 4.3 is derived by analyzing the gradient of the function g ( δ ) =
Revised: Corollary 4.3 is derived by analyzing the gradient of the function g ( δ ) =

Original: f ( δ + θ ( t ) ) via Theorem 4.2.
Revised: f ( δ + θ ( t ) ) via Theorem 4.2.

Original: For any number of measurements such that the corresponding Gaussian noise δ satisfies the conditionin Corollary 4.3, the trainability at the updated point is guaranteed.
Revised: For any number of measurements such that the corresponding Gaussian noise δ satisﬁes the condition in Corollary 4.3, the trainability at the updated point is guaranteed.

Original:  Experiments
Revised:  Experiments

Original: In this section, we analyze the training behavior of two variational quantum algorithms, i.e., findingthe ground energy and state of the Heisenberg model and the LiH molecule, respectively.
Revised: In this section, we analyze the training behavior of two variational quantum algorithms, i.e., ﬁnding the ground energy and state of the Heisenberg model and the LiH molecule, respectively.

Original: Allnumerical experiments are provided using the Pennylane package [91].
Revised: All numerical experiments are provided using the Pennylane package [91].

Original: 5.1 Heisenberg model
Revised: 5.1 Heisenberg model

Original: In the first task, we aim to find the ground state and the ground energy of the Heisenberg model [92].
Revised: In the ﬁrst task, we aim to ﬁnd the ground state and the ground energy of the Heisenberg model [92].

Original: The corresponding Hamiltonian matrix is
Revised: The corresponding Hamiltonian matrix is

Original: [Equation]
Revised: [Equation]

Original: where N is the number of qubit, X i
Revised: where N is the number of qubit, X i

Original: = I ⊗ ( i − 1) ⊗ X ⊗
Revised: = I ⊗ ( i − 1) ⊗ X ⊗

Original: I ⊗ ( N − i ) , Y i
Revised: I ⊗ ( N − i ) , Y i

Original: = I ⊗ ( i − 1) ⊗ Y ⊗
Revised: = I ⊗ ( i − 1) ⊗ Y ⊗

Original: I ⊗ ( N − i ) , and
Revised: I ⊗ ( N − i ) , and

Original: Z i =
Revised: Z i =

Original: I ⊗ ( i − 1) ⊗ Z ⊗ I ⊗ ( N − i ) .
Revised: I ⊗ ( i − 1) ⊗ Z ⊗ I ⊗ ( N − i ) .

Original: We employ the loss function defined by Eq.
Revised: We employ the loss function deﬁned by Eq.

Original: (1) with the input state( | 0 ⟩⟨ 0 | ) ⊗ N and the observable (10).
Revised: (1) with the input state ( | 0 (cid:105)(cid:104) 0 | ) ⊗ N and the observable (10).

Original: Thus, by minimizing the function (1), we can obtain the leasteigenvalue of the observable (10), which is the ground energy.
Revised: Thus, by minimizing the function (1), we can obtain the least eigenvalue of the observable (10), which is the ground energy.

Original: We adopt the ansatz with N = 15qubits, which consists of L 1 = 10 layers of R Y R X CZ blocks.
Revised: We adopt the ansatz with N =qubits, which consists of L 1 = 10 layers of R Y R X CZ blocks.

Original: In each block, we first employ the
Revised: In each block, we ﬁrst employ the

Original: CZ gate to neighboring qubits pairs { (1 , 2) · · · , ( N, 1) } , followed by R X and R Y rotations for allqubits.
Revised: CZ gate to neighboring qubits pairs { (1 , 2) · · · , ( N, 1) } , followed by R X and R Y rotations for all qubits.

Original: Overall, the quantum circuit has 300 parameters.
Revised: Overall, the quantum circuit has 300 parameters.

Original: We consider three initialization methods forcomparison, i.e., initializations with the Gaussian distribution N (0 , γ 2 ) and the uniform distributionin [0 , 2 π ] , respectively, and the zero initialization (all parameters equal to 0 at the initial point).
Revised: We consider three initialization methods for comparison, i.e., initializations with the Gaussian distribution N (0 , γ 2 ) and the uniform distribution in [0 , 2 π ] , respectively, and the zero initialization (all parameters equal to 0 at the initial point).

Original: Weremark that each term in the observable (10) contains at most S = 2 non-identity Pauli matrices,which is consistent with the ( S, L ) =
Revised: We remark that each term in the observable (10) contains at most S = 2 non-identity Pauli matrices, which is consistent with the ( S, L ) =

Original: (2 , 18) case of Theorem 4.1.
Revised: (2 , 18) case of Theorem 4.1.

Original: Thus, we expect that the Gaussianinitialization with the variance γ 2 = 1 4 S ( L +2)
Revised: Thus, we expect that the Gaussian initialization with the variance γ 2 = 1 4 S ( L +2)

Original: = 1160 could provide trainable initial parameters.
Revised: = 1160 could provide trainable initial parameters.

Original: In the experiment, we train VQAs with gradient descent (GD) [93] and Adam optimizers [94],respectively.
Revised: In the experiment, we train VQAs with gradient descent (GD) [93] and Adam optimizers [94], respectively.

Original: The learning rate is 0 .
Revised: The learning rate is 0 .

Original: 01 and 0 .
Revised: 01 and 0 .

Original: 01 for both GD and Adam cases.
Revised: 01 for both GD and Adam cases.

Original: Since the estimation of gradients on real quantum computers could be perturbed by statistical measurement noise, wecompare optimizations using accurate and noisy gradients.
Revised: Since the estimation of gradients on real quantum computers could be perturbed by statistical measurement noise, we compare optimizations using accurate and noisy gradients.

Original: For the latter case, we set the varianceof measurement noises to be 0 .
Revised: For the latter case, we set the variance of measurement noises to be 0 .

Original: The numerical results of the Heisenberg model are shown in the
Revised: The numerical results of the Heisenberg model are shown in the

Original: Figure 2.
Revised: Figure 2.

Original: The loss during the training with gradient descents is shown in Figures 2(a) and 2(b) for theaccurate and the noisy gradient cases, respectively.
Revised: The loss during the training with gradient descents is shown in Figures 2(a) and 2(b) for the accurate and the noisy gradient cases, respectively.

Original: The Gaussian initialization outperforms the othertwo initializations with faster convergence rates.
Revised: The Gaussian initialization outperforms the other two initializations with faster convergence rates.

Original: Figures 2(c) and 2(d) verify that Gaussian initialized
Revised: Figures 2(c) and 2(d) verify that Gaussian initialized

Original: VQAs have larger gradients in the early stage, compared to that of uniformly initialized VQAs.
Revised: VQAs have larger gradients in the early stage, compared to that of uniformly initialized VQAs.

Original: Wenotice that zero initialized VQAs cannot be trained with accurate gradient descent, since the initialgradient equals to zero.
Revised: We notice that zero initialized VQAs cannot be trained with accurate gradient descent, since the initial gradient equals to zero.

Original: This problem is alleviated in the noisy case, as shown in Figures 2(b) and2(d).
Revised: This problem is alleviated in the noisy case, as shown in Figures 2(b) and 2(d).

Original: Since the gradient is close to zero at the initial stage, the update direction mainly depends onthe measurement noise, which forms the Gaussian distribution.
Revised: Since the gradient is close to zero at the initial stage, the update direction mainly depends on the measurement noise, which forms the Gaussian distribution.

Original: Thus, the parameter in the noisy zeroinitialized VQAs is expected to accumulate enough variances, which takes around 10 iterations basedon Figure 2(h).
Revised: Thus, the parameter in the noisy zero initialized VQAs is expected to accumulate enough variances, which takes around 10 iterations based on Figure 2(h).

Original: As illustrated in Figure 2(b), the loss function corresponding to the zero initializationdecreases quickly after the variance accumulation stage.
Revised: As illustrated in Figure 2(b), the loss function corresponding to the zero initialization decreases quickly after the variance accumulation stage.

Original: Results in Figures 2(e) and 2(h) show similartraining behaviors using the Adam optimizer.
Revised: Results in Figures 2(e) and 2(h) show similar training behaviors using the Adam optimizer.

Original: 5.2 Quantum chemistry
Revised: 5.2 Quantum chemistry

Original: In the second task, we aim to find the ground state and the ground energy of the LiH molecule.
Revised: In the second task, we aim to ﬁnd the ground state and the ground energy of the LiH molecule.

Original: Wefollow settings on the ansatz in Refs.
Revised: We follow settings on the ansatz in Refs.

Original: For the molecule with n e active electrons and n o freespin orbitals, the corresponding VQA contains N = n o qubits, which employs the HF state [9, 10]| ϕ HF ⟩ = | 1 ⟩ ⊗ · · · | 1 ⟩ (cid:124) (cid:123)(cid:122) (cid:125) n e ⊗ | 0 ⟩ ⊗ · · · | 0 ⟩ (cid:124) (cid:123)(cid:122)
Revised: For the molecule with n e active electrons and n o free spin orbitals, the corresponding VQA contains N = n o qubits, which employs the HF state [9, 10] | φ HF (cid:105) = | 1 (cid:105) ⊗ · · · | 1 (cid:105) (cid:124) (cid:123)(cid:122) (cid:125) n e ⊗ | 0 (cid:105) ⊗ · · · | 0 (cid:105) (cid:124) (cid:123)(cid:122)

Original: [Equation]
Revised: [Equation]

Original: as the input state.
Revised: as the input state.

Original: We construct the parameterized quantum circuit with Givens rotation gates [12],where each gate is implemented on 2 or 4 qubits with one parameter.
Revised: We construct the parameterized quantum circuit with Givens rotation gates [12], where each gate is implemented on 2 or 4 qubits with one parameter.

Original: Specifically, for the LiHmolecule, the number of electrons n e = 2 , the number of free spin orbitals n o = 10 , and the numberof different Givens rotations is L = 24 [13].
Revised: Speciﬁcally, for the LiH molecule, the number of electrons n e = 2 , the number of free spin orbitals n o = 10 , and the number of different Givens rotations is L = 24 [13].

Original: We follow the molecule Hamiltonian H LiH defined in
Revised: We follow the molecule Hamiltonian H LiH deﬁned in

Original: Ref.
Revised: Ref.

Original: Thus, the loss function for finding the ground energy of LiH is defined as
Revised: Thus, the loss function for ﬁnding the ground energy of LiH is deﬁned as

Original: [Equation]
Revised: [Equation]

Original: where V Givens ( θ ) = (cid:81) 24 i =1 R Givens i ( θ i ) denotes the product of all parameterized Givens rotationsof the LiH molecule.
Revised: where V Givens ( θ ) = (cid:81) 24 i =1 R Givens i ( θ i ) denotes the product of all parameterized Givens rotations of the LiH molecule.

Original: By minimizing the function (11), we can obtain the least eigenvalue of the
Revised: By minimizing the function (11), we can obtain the least eigenvalue of the

Original: Hamiltonian H LiH , which is the ground energy of the LiH molecule.
Revised: Hamiltonian H LiH , which is the ground energy of the LiH molecule.

Original: In practice, we initialize parameters in the VQA (11) with three distributions for comparison, i.e.,the Gaussian distribution N (0 , γ 2 ) , the zero distribution (all parameters equal to 0 ), and the uniformdistribution in [0 , 2 π ] .
Revised: In practice, we initialize parameters in the VQA (11) with three distributions for comparison, i.e., the Gaussian distribution N (0 , γ 2 ) , the zero distribution (all parameters equal to 0 ), and the uniform distribution in [0 , 2 π ] .

Original: For 2-qubit Givens rotations, the term ( h, a ) =
Revised: For 2-qubit Givens rotations, the term ( h, a ) =

Original: (2 , 2) as shown in Eq.
Revised: (2 , 2) as shown in Eq.

Original: For4-qubit Givens rotations, the term ( h, a ) = (8 , 8) [95].
Revised: For 4-qubit Givens rotations, the term ( h, a ) = (8 , 8) [95].

Original: Thus, we set the variance in the Gaussiandistribution γ 2 = 8 2 × 12 48 × 8 4 × 24 , which matches the ( L, h, a, ϵ )=
Revised: Thus, we set the variance in the Gaussian distribution γ 2 = 8 2 × 12 48 × 8 4 × 24 , which matches the ( L, h, a, (cid:15) )

Original:  (24 , 8 , 8 , 12 ) case of Theorem 4.2.
Revised: = (24 , 8 , 8 , 12 ) case of Theorem 4.2.

Original: Similar to the task of the Heisenberg model, we consider both the accurate and the noisy gradientcases, where the variance of noises in the latter case is the constant 0 .
Revised: Similar to the task of the Heisenberg model, we consider both the accurate and the noisy gradient cases, where the variance of noises in the latter case is the constant 0 .

Original: Moreover, we consider thenoisy case with adaptive noises, where the variance of the noise on each partial derivative ∂f∂θ ℓ (cid:12)(cid:12) θ
Revised: Moreover, we consider the noisy case with adaptive noises, where the variance of the noise on each partial derivative ∂f∂θ [Equation]

Original: = θ ( t )in the t -th iteration is
Revised: θ [Equation] in the t -th iteration is

Original: [Equation]
Revised: [Equation]

Original: The variance in Eq.
Revised: The variance in Eq.

Original: (12) matches the ( L, h, a, ϵ )
Revised: (12) matches the ( L, h, a, (cid:15) )

Original: = (24 , 8 , 8 , 12 ) case of Corollary 4.3 when the VQAis nearly converged:
Revised: = (24 , 8 , 8 , 12 ) case of Corollary 4.3 when the VQA is nearly converged:

Original: [Equation]
Revised: [Equation]

Original: In the experiment, we train VQAs with gradient descent and Adam optimizers.
Revised: In the experiment, we train VQAs with gradient descent and Adam optimizers.

Original: Learning rates areset to be 0 . 1 and 0 .
Revised: Learning rates are set to be 0 . 1 and 0 .

Original: 01 for GD and Adam cases, respectively.
Revised: 01 for GD and Adam cases, respectively.

Original: The loss (11) during training iterationsis shown in Figure 3.
Revised: The loss (11) during training iterations is shown in Figure 3.

Original: Optimization results with gradient descents are shown in Figures 3(a)-3(c)for the accurate gradient case, the adaptive noisy gradient case, and the noisy gradient case with theconstant noise variance 0 .
Revised: Optimization results with gradient descents are shown in Figures 3(a)-3(c) for the accurate gradient case, the adaptive noisy gradient case, and the noisy gradient case with the constant noise variance 0 .

Original: 001 , respectively.
Revised: 001 , respectively.

Original: The variance of the noise in the adaptive noisy gradientcase follows Eq.
Revised: The variance of the noise in the adaptive noisy gradient case follows Eq.

Original: Figures 3(a) and 3(b) show similar performance, where the loss f with the
Revised: Figures 3(a) and 3(b) show similar performance, where the loss f with the

Original: [Figure]
Revised: [Figure]

Original: Gaussian initialization and the zero initialization converge to 10 −  the global minimum f 
Revised: Gaussian initialization and the zero initialization converge to 10 − 4 over the global minimum f ∗ .

Original: Theloss with  uniform initialization is higher than 10 −  global minimum.
Revised: The loss with the uniform initialization is higher than 10 − 1 over the global minimum.

Original: Figure 3( ) showsthe training with constantly perturbed gradients.
Revised: Figure 3(c) shows the training with constantly perturbed gradients.

Original: The Gaussian initialization and the zero initializationinduce the 10 − 3 convergence, while the loss function with the uniform initialization is still higherthan 10 − 1 over the global minimum.
Revised: The Gaussian initialization and the zero initialization induce the 10 − 3 convergence, while the loss function with the uniform initialization is still higher than 10 − 1 over the global minimum.

Original: Figures 3(d)-3(f) show similar training behaviors using the
Revised: Figures 3(d)-3(f) show similar training behaviors using the

Original: Adam optimizer.
Revised: Adam optimizer.

Original: Based on Figures 3(a)-3(f), the Gaussian initialization and the zero initializationoutperform the uniform initialization in all cases.
Revised: Based on Figures 3(a)-3(f), the Gaussian initialization and the zero initialization outperform the uniform initialization in all cases.

Original: We notice that optimization with accurate gradientsand optimization with adaptive noisy gradients have the same convergence rate and the final valueof the loss function, which is better than that using constantly perturbed gradients.
Revised: We notice that optimization with accurate gradients and optimization with adaptive noisy gradients have the same convergence rate and the ﬁnal value of the loss function, which is better than that using constantly perturbed gradients.

Original: We remark thatthe number of measurements T 
Revised: We remark that the number of measurements T =

Original: = O ( 1 Var(noise) ) .
Revised: O ( 1 Var(noise) ) .

Original: Thus, finite number of measurements with thenoise (12) for gradient estimation is enough to achieve the performance of accurate gradients, whichverifies Theorem 4.2 and Corollary 4.3.
Revised: Thus, ﬁnite number of measurements with the noise (12) for gradient estimation is enough to achieve the performance of accurate gradients, which veriﬁes Theorem 4.2 and Corollary 4.3.

Original:  Conclusions
Revised:  Conclusions

Original: In this work, we provide a Gaussian initialization strategy for solving the vanishing gradient problemof deep variational quantum algorithms.
Revised: In this work, we provide a Gaussian initialization strategy for solving the vanishing gradient problem of deep variational quantum algorithms.

Original: We prove that the gradient norm of N -qubit quantum circuitswith L layers could be lower bounded by poly( N, L ) − 1 , if the parameter is sampled independentlyfrom the Gaussian distribution with the variance O ( 1 L ) .
Revised: We prove that the gradient norm of N -qubit quantum circuits with L layers could be lower bounded by poly( N, L ) − 1 , if the parameter is sampled independently from the Gaussian distribution with the variance O ( 1 L ) .

Original: Our results hold for both the local and theglobal observable cases, and could be generalized to VQAs employing correlated parameterizedgates.
Revised: Our results hold for both the local and the global observable cases, and could be generalized to VQAs employing correlated parameterized gates.

Original: Compared to the local case, the bound for the global case depends on the gradient performanceat the zero point.
Revised: Compared to the local case, the bound for the global case depends on the gradient performance at the zero point.

Original: Further analysis towards the zero-case-free bound could be investigated as futuredirections.
Revised: Further analysis towards the zero-case-free bound could be investigated as future directions.

Original: Moreover, we show that the necessary number of measurements, which scales O ( L ϵ ) ,suffices for estimating the gradient during the training.
Revised: Moreover, we show that the necessary number of measurements, which scales O ( L (cid:15) ) , sufﬁces for estimating the gradient during the training.

Original: We provide numerical experiments on findingthe ground energy and state of the Heisenberg model and the LiH molecule, respectively.
Revised: We provide numerical experiments on ﬁnding the ground energy and state of the Heisenberg model and the LiH molecule, respectively.

Original: Experimentsshow that the proposed Gaussian initialization method outperforms the uniform initialization methodwith a faster convergence rate, and the training using gradients with adaptive noises shows the sameconvergence compared to the training using noiseless gradients. 
Revised: Experiments show that the proposed Gaussian initialization method outperforms the uniform initialization method with a faster convergence rate, and the training using gradients with adaptive noises shows the same convergence compared to the training using noiseless gradients. Acknowledgement

Revised: This work is supported in part by ARC FL-170100117, IC-190100031, and LE-200100049.

Original: Checklist
Revised: Checklist

Original: For all authors... (a) Do the main claims made in the abstract and introduction accurately reflect the paper’s contributions and scope?
Revised: For all authors... (a) Do the main claims made in the abstract and introduction accurately reﬂect the paper’s contributions and scope?

Original: [Yes] (b) Did you describe the limitations of your work?
Revised: [Yes] (b) Did you describe the limitations of your work?

Original: [Yes] (c) Did you discuss any potential negative societal impacts of your work?
Revised: [Yes] (c) Did you discuss any potential negative societal impacts of your work?

Original: [N/A] (d) Have you read the ethics review guidelines and ensured that your paper conforms to them?
Revised: [N/A] (d) Have you read the ethics review guidelines and ensured that your paper conforms to them?

Original: [Yes] 
Revised: [Yes] 2.

Original: If you are including theoretical results... (a) Did you state the full set of assumptions of all theoretical results?
Revised: If you are including theoretical results... (a) Did you state the full set of assumptions of all theoretical results?

Original: [Yes] (b) Did you include complete proofs of all theoretical results?
Revised: [Yes] (b) Did you include complete proofs of all theoretical results?

Original: [Yes] 
Revised: [Yes] 3.

Original: If you ran experiments... (a) Did you include the code, data, and instructions needed to reproduce the main experi- mental results (either in the supplemental material or as a URL)?
Revised: If you ran experiments... (a) Did you include the code, data, and instructions needed to reproduce the main experimental results (either in the supplemental material or as a URL)?

Original: [Yes] (b) Did you specify all the training details (e.g., data splits, hyperparameters, how they were chosen)?
Revised: [Yes] (b) Did you specify all the training details (e.g., data splits, hyperparameters, how they were chosen)?

Original: [Yes] (c) Did you report error bars (e.g., with respect to the random seed after running experi- ments multiple times)?
Revised: [Yes] (c) Did you report error bars (e.g., with respect to the random seed after running experiments multiple times)?

Original: [Yes] We run each experiment independently for 5 times.
Revised: [Yes] We run each experiment independently for 5 times.

Original: (d) Did you include the total amount of compute and the type of resources used (e.g., type of GPUs, internal cluster, or cloud provider)?
Revised: (d) Did you include the total amount of compute and the type of resources used (e.g., type of GPUs, internal cluster, or cloud provider)?

Original: [No] Each experiment finishes quickly on CPUs with a few hours.
Revised: [No] Each experiment ﬁnishes quickly on CPUs with a few hours.

Original: If you are using existing assets (e g., code, data, models) or curating/releasing new assets... (a) If your work uses existing assets, did you cite the creators?
Revised: If you are using existing assets (e.g., code, data, models) or curating/releasing new assets... (a) If your work uses existing assets, did you cite the creators?

Original: [N/A] (b) Did you mention the license of the assets?
Revised: [N/A] (b) Did you mention the license of the assets?

Original: [N/A] (c) Did you include any new assets either in the supplemental material or as a URL?
Revised: [N/A] (c) Did you include any new assets either in the supplemental material or as a URL?

Original: [N/A] (d) Did you discuss whether and how consent was obtained from people whose data you’re using/curating?
Revised: [N/A] (d) Did you discuss whether and how consent was obtained from people whose data you’re using/curating?

Original: [N/A] (e) Did you discuss whether the data you are using/curating contains personally identifiable information or offensive content?
Revised: [N/A] (e) Did you discuss whether the data you are using/curating contains personally identiﬁable information or offensive content?

Original: [N/A] 
Revised: [N/A] 5.

Original: If you used crowdsourcing or conducted research with human subjects... (a) Did you include the full text of instructions given to participants and screenshots, ifapplicable?
Revised: If you used crowdsourcing or conducted research with human subjects... (a) Did you include the full text of instructions given to participants and screenshots, if applicable?

Original: [N/A]
Revised: [N/A]

Original: (b) Did you describe any potential participant risks, with links to Institutional Review Board (IRB) approvals, if applicable?
Revised: (b) Did you describe any potential participant risks, with links to Institutional Review Board (IRB) approvals, if applicable?

Original: [N/A] (c) Did you include the estimated hourly wage paid to participants and the total amountspent on participant compensation?
Revised: [N/A] (c) Did you include the estimated hourly wage paid to participants and the total amount spent on participant compensation?

Original: [N/A]
Revised: [N/A]