Title: Exponential Hardness of Optimization from the Locality in Quantum Neural Networks
Venue: NeurIPS 2023
Accepted: no
Keywords: quantum neural network

training landscape

quantum machine learning

quantum state learning

variational quantum algorithms
Abstract: Quantum neural networks (QNNs) have become a leading paradigm for establishing near-term quantum applications in recent years. The trainability issue of QNNs has garnered extensive attention, spurring demand for a comprehensive analysis of QNNs in order to identify viable solutions. In this work, we propose a perspective that characterizes the trainability of QNNs based on their locality. We prove that the entire variation range of the loss function via adjusting any local quantum gate vanishes exponentially in the number of qubits with a high probability for a broad class of QNNs. This result reveals extra harsh constraints independent of gradients and unifies the restrictions on gradient-based and gradient-free optimizations naturally. We showcase the validity of our results with numerical simulations of representative models and examples. Our findings, as a fundamental property of random quantum circuits, deepen the understanding of the role of locality in QNNs and serve as a guideline for assessing the effectiveness of diverse training strategies for quantum neural networks.

Abstract

Quantum neural networks (QNNs) have become a leading paradigm for establishing near-term quantum applications in recent years. The trainability issue of QNNs has garnered extensive attention, spurring demand for a comprehensive analysis of QNNs in order to identify viable solutions. In this work, we propose a perspective that characterizes the trainability of QNNs based on their locality. We prove that the entire variation range of the loss function via adjusting any local quantum gate vanishes exponentially in the number of qubits with a high probability for a broad class of QNNs. This result reveals extra harsh constraints independent of gradients and unifies the restrictions on gradient-based and gradient-free optimizations naturally. We showcase the validity of our results with numerical simulations of representative models and examples. Our findings, as a fundamental property of random quantum circuits, deepen the understanding of the role of locality in QNNs and serve as a guideline for assessing the effectiveness of diverse training strategies for quantum neural networks.

1 Introduction

Quantum computing is a rapidly growing technology that exploits quantum mechanics to solve intricate problems that classical computers cannot solve. With enormous efforts having been made to develop noisy intermediate scale quantum (NISQ) devices [1], current quantum devices have demonstrated the ability to achieve near-term quantum advantage for practical applications in key areas including many-body physics [2-4], chemistry [5], finance [6-8], and machine learning [9]. Specifically, quantum machine learning (QML) represents an exciting, emerging interdisciplinary field that seeks to enhance machine learning algorithms by harnessing the inherent parallelism of quantum systems [10-20]. Quantum neural networks (QNNs) stand at the forefront of QML, capitalizing on the unprecedented potential of quantum computing to revolutionize data analysis and pattern recognition. Inspired by classical neural networks, QNNs employ quantum gates and quantum states as fundamental building blocks within their computational framework. These networks can be trained using a diverse range of methods, including gradient-based optimization techniques akin to classical neural network training [21-24].

With the aim to show quantum advantage on certain tasks, a critical issue is whether QNNs can be extended to solve large-scale systems, i.e., scalability. Unfortunately, many studies point out that training of QNNs requires exponential resources with the system size under certain conditions [2536]. Besides the practical limitations such as noises [29], even ideal quantum devices will suffer from the so-called barren plateau phenomenon [25], which is the quantum counterpart of vanishing gradient problem in classical machine learning. It was shown that the gradient of the cost function vanishes exponentially in the number of qubits with a high probability for a random initialized QNN with sufficient depth, analogous to the vanishing gradient issue in classical neural networks.

…

…

…

…

Consequently, exponentially vanishing gradients demand exponential precision in the cost function 37 measurement on a quantum device [37] to make progress in the gradient-based optimization, and 38 hence an exponential complexity in the number of qubits. 39

Several attempts have been made to avoid barren plateaus, such as higher order derivatives [38], 40 gradient-free optimizers including gate-by-gate optimization [39, 40], proper initialization [41], 41 pre-training including adaptive methods [42-46], QNN architectures [47, 48] and cost function 42 choices [49, 50]. More efforts are needed to study the general effectiveness of these attempts [26, 27] 43 and develop new strategies to improve the trainability and scalability of QNNs. As a guide for 44 exploring effective training strategies, it is crucial to uncover the essential mechanisms behind the 45 barren plateau phenomenon. 46

However, few rigorous scaling results are known for generic QNNs besides phenomenological calcu47 lations, i.e., gradient analyses and their descendent [26-28]. Instead of just the limited information of 48 vicinity from gradient analyses, it would be quite helpful for designing efficient algorithms if we could 49 gain information on the entire variation range of the cost function when adjusting a single [39, 40] or 50 several parameters. Combined with the fact that parameters usually enter the circuit independently 51 through local quantum gates, all of which motivate our work where we are chiefly concerned with the 52 variation range of the cost function via varying a local unitary within a quantum circuit. 53

In this work, we present a rigorous scaling theorem on the trainability of QNNs beyond gradients 54 from the perspective of QNN locality. As summarized in Fig. 1, we prove that when varying a local 55 unitary within a sufficiently random circuit, the expectation and variance of the variation range of 56 the cost function vanish exponentially in the number of qubits. Then through simple derivations, we 57 show that this theorem implies exponentially vanishing gradients and cost function differences, and 58 hence unifies the restrictions on gradient-based and gradient-free optimizations. Meanwhile, this 59 theorem further delivers extra meaningful information about the training landscapes and optimization 60 possibilities of QNNs. In this sense, we obtain a fundamental limitation on QNN training. Next, we 61 illustrate the applications of our theorem on representative QNN models, where a tighter bound for 62 the fidelity-type cost function is provided specifically even with shallow random circuits. At last, we 63 perform numerical simulations on these representative models, where the scaling exponents coincide 64 with our analytical results almost precisely. 65

Comparison with Previous Works. The advances of our results compared to previous works [25, 66 27, 26, 28] exist in two aspects. Firstly, the exponentially vanishing quantity we claim is the entire 67 variation range of the cost function in the whole parameter subspace corresponding to the local 68 unitary. This provides constraints on multiple parameters at finite intervals simultaneously, instead of 69 an infinitesimal vicinity or two fixed-parameter points. Secondly, our results are irrelevant with the 70 parameterization of the local unitary like e -i Ω θ used previously. Hence, our results are much more 71 general whose only condition is the circuit locality and open a new avenue for analyzing the QNN 72 trainability. 73

74

75

76

77

78

79

80

81

82

83

84

85

86

87

88

89

90

91

92

93

94

95

96

97

98

99

100

101

102

103

104

105

106

107

108

109

110

111

112

113

114

115

116

117

118

119

2 Preliminaries

Quantum State. We first introduce basic concepts and notations in quantum computing. A pure single-qubit quantum state is a linear combination of two computational basis states, represented as | ϕ ⟩ = α | 0 ⟩ + β | 1 ⟩ in Dirac notation, where α, β ∈ C , | α | 2 + | β | 2 = 1 . Here, | 0 ⟩ and | 1 ⟩ denote the basis states [1 , 0] T and [0 , 1] T in the single-qubit space C 2 , respectively. The n -qubit space C 2 n is formed by the tensor product of n single-qubit spaces. Additionally, the quantum state can be represented by a positive semidefinite matrix, also known as a density matrix. The density matrix ρ of a pure state | ϕ ⟩ consisting of n qubits is expressed as ρ = | ϕ ⟩ ⟨ ϕ | , where ⟨ ϕ | = | ϕ ⟩ † . A general mixed quantum state is represented by ρ = ∑ k c k | ϕ k ⟩ ⟨ ϕ k | , where c k ∈ R , ∑ k c k = 1 .

Quantum Gate. Quantum gates are mathematically described as unitary operators. Common single-qubit gates include the Pauli rotations { R P ( θ ) = e -i θ 2 P | P ∈ { X,Y,Z }} , which are in the matrix exponential form of Pauli matrices

Common two-qubit gates include controlled-X gate CNOT = I ⊕ X ( ⊕ is the direct sum) and controlled-Z gate CZ = I ⊕ Z , which can generate quantum entanglement among qubits.

Quantum Measurement. Quantum measurement is a quantum operation to obtain information from the quantum system. For example, for a single-qubit state | ϕ ⟩ = α | 0 ⟩ + β | 1 ⟩ , the outcome of a computational basis measurement is either | 0 ⟩ with probability | α | 2 or | 1 ⟩ with probability | β | 2 . This measurement operation can be mathematically referred to as the average of the observable O = Z under the state | ϕ ⟩ : ⟨ ϕ | O | ϕ ⟩ = tr[ Z | ϕ ⟩ ⟨ ϕ | ] = | α | 2 -| β | 2 . Generally, quantum observables O are Hermitian matrices and O (1 /ε 2 ) times of measurements could give an ε ∥ O ∥ ∞ -error estimation to the value tr[ Oρ ] , where ∥ · ∥ ∞ is the spectral norm of the matrix.

Quantum Neural Network. While classical neural networks operate on classical bits and use classical logic gates, quantum neural networks (QNNs) use quantum bits, or qubits, and quantum gates to process and store information. QNNs are often described as parameterized quantum circuits (PQCs) that are composed of rotation gates with adjustable rotating angles. In general, a QNN takes the mathematical form U ( θ ) = ∏ µ U µ ( θ µ ) W µ , where U µ ( θ µ ) = e -iθ µ Ω µ denotes a parameterized gate, such as a single-qubit rotation gate with Ω µ representing a Hermitian operator, and W µ corresponds to fixed gates like the CNOT gate and SWAP gate. Commonly used templates of QNNs include the hardware efficient ansatz, the alternating-layered ansatz, and the tensor-network-based ansatz [49, 51]. Note that QNNs with intermediate classical controls such as QCNNs [52] can also be included in this general form theoretically.

3 Limitations of Local Unitary Optimization in QNN

We start by introducing a general setting of a QNN model used throughout our analysis. A hybrid quantum-classical framework in QML usually uses a classical optimizer to train a QNN, denoted by U , with an input state ρ by minimizing a task-dependent cost function C , which is typically chosen as the expectation value of some Hermitian operator H :

Note that other cost function forms can be regarded as compositions of observable expectations and some other classical post-processing functions. Here we focus on (2) for simplicity. Divide the whole qubit system into two parts A,B with m qubits and n -m qubits, respectively. Here m is a fixed constant not scaling with n so that we call A a local subsystem. The QNN U is often composed of local unitaries on real devices, such as the single-qubit rotation gates and the CNOT gate. We focus on a local unitary U A within U acting on subsystem A . As shown in Fig. 2, we denote the sub-circuit of U before U A as V 1 and that behind U A as V 2 , such that U = V 2 ( U A ⊗ I B ) V 1 where I B is the identity operator on B . V 1 , V 2 and U A are independent of each other. We also remark that this circuit setting is sufficiently general to cover common representative QNN models, e.g., the variational quantum eigensolver, the quantum autoencoder, and the quantum state learning.

120

121

122

123

124

125

126

127

128

129

130

131

132

133

134

135

136

137

138

139

140

141

142

143

144

145

146

147

148

...

...

...

...

...

...

...

...

To characterize the training landscape beyond the limited information of the vicinity from gradient analyses, we introduce a central quantity throughout this work, i.e., the variation range of the cost function via varying a local unitary.

Definition 1 For a generic cost function C H,ρ ( U ) with a QNN U in Eq. (2) , we define its variation range with given V 1 , V 2 as

where the maximum and minimum with respect to U A are taken over the unitary group U (2 m ) of degree 2 m .

The quantity ∆ H,ρ ( V 1 , V 2 ) intuitively reflects the maximal possible influence that the local unitary U A can have on the cost function. We establish an upper bound on ∆ H,ρ ( V 1 , V 2 ) in the sense of probability by Theorem 1, which thus delivers a limitation on optimizing an arbitrary local unitary. To be specific, we prove that if either V 1 , V 2 , or both match the Haar distribution up to the second moment, i.e., are sampled from unitary 2-designs [53], the expectation of ∆ H,ρ ( V 1 , V 2 ) vanishes exponentially in the number of qubits. See Appendix A for preliminaries on unitary designs.

Theorem 1 Suppose V 1 , V 2 are ensembles from which V 1 , V 2 are sampled, respectively. If either V 1 or V 2 , or both form unitary 2 -designs, then for arbitrary H and ρ , the following inequality holds

where E V 1 ,V 2 denotes the expectation over V 1 , V 2 independently. w ( H ) = λ max ( H ) -λ min ( H ) denotes the spectral width of H , where λ max ( H ) is the maximum eigenvalue of H and λ min ( H ) is the minimum.

Theorem 1 demonstrates that the maximal influence of a local unitary within a random QNN on the cost function diminishes exponentially in the number of qubits, with a high probability. This inherent locality of QNN poses an exponential hardness of optimization in QNN training and we would like to make several remarks to better reveal the underlying implications of the theorem below. The main proof idea of Theorem 1 is to calculate the expectation value over V 1 , V 2 separately. To tackle the maximization over U A , the main technique is to employ Hölder's inequality to extract U A out and bound the remaining part with specific calculations of 2 -design element-wise integrals. For the detailed proof, we defer to Appendix B.

Remark 1 Firstly, due to the non-negativity and boundedness of the variation range, i.e., ∆ H,ρ ∈ [0 , w ( H )] , the variance of ∆ H,ρ can be bounded by its expectation times w ( H ) . Thus from Theorem 1 we know that the variance also vanishes exponentially:

149

150

151

152

153

154

155

156

157

158

159

160

161

162

163

164

165

166

167

168

169

170

171

172

173

174

175

176

177

178

179

180

181

182

183

184

185

186

187

188

189

190

191

192

193

Note that w ( H ) ∈ O (poly( n )) holds for common VQAs. Moreover, Theorem 1 together with Markov's inequality provides an exponentially small upper bound of the probability that ∆ H,ρ ( V 1 , V 2 ) deviates from zero, i.e.,

That is to say, the probability that ∆ H,ρ is non-zero to some fixed precision is exponentially small.

Remark 2 Secondly, we can even establish an exponentially small bound using Theorem 1 for the case where U A is a global unitary satisfying the parameter-shift rule [54-58]. Suppose U A = e -iθ Ω with the Hermitian generator Ω satisfying Ω 2 = I . Since Ω has only two different eigenvalues ± 1 , there exists a unitary W such that We -iθ Ω W † becomes a local unitary acting on a single qubit non-trivially. W and W † could be absorbed into the rest of the circuit with W † V 1 or V 2 W still forming 2 -designs [59]. Therefore, the proof for global unitaries satisfying the parameter-shift rule can be reduced back to the case of local unitaries.

Remark 3 Moreover, it is worth noticing that the compact bound in (4) only involves the spectral width w ( H ) and does not depend on any detail of the Hermitian operator H . But if some specific structures about H are known, e.g., the Pauli decomposition of H , a tighter bound could be derived in Appendix B which depends on the coupling complexity of H . In addition, if the cost function reduces to the form of the fidelity between pure states, we could have a tighter bound with scaling O (2 -n ) in Proposition 2. Theorem 1 can be generalized to arbitrary dimensions besides qubit systems of dimension 2 n , e.g., qutrit and qudit systems. The detailed proof is provided in Appendix B.

In fact, Theorem 1 has a natural physical interpretation: the effect of a local operation on a physical observable will vanish exponentially after a chaotic evolution. Remarkably, the concept of local operations yielding minor global influences is a physically intuitive yet mathematically intricate notion. For instance, even a single-qubit unitary is enough to rotate an arbitrary n -qubit pure state to a new state with zero fidelity with the original one, showcasing local operations do make a great global influence. Hence, Theorem 1 may be invaluable as a rigorous formulation of the aforementioned argument within the domain of QNN training, elucidating the locality of QNNs.

4 Unifying the Limitations on Training QNNs

Here we briefly demonstrate how Theorem 1 unifies the restrictions on gradient-based [25, 27] and gradient-free optimizations [26, 28] in a more natural manner, and indicates the extra restrictions besides them on QNN training. In the following, we focus on a PQC applicable for Theorem 1 with M trainable parameters { θ µ } M µ =1 and denote the variation range of the cost function via varying θ µ as ∆ µ .

Consider the gradient-based optimization first. On the one hand, in the case where the parameter-shift rule is valid [54-58], Theorem 1 can strictly deduce vanishing gradients. Suppose { θ µ } M µ =1 are applicable for the parameter-shift rule (e.g., hardware-efficient ansatzes). Namely, θ µ enters the unitary e -iθ µ Ω µ within the circuit where Ω µ is a Hermitian generator satisfying Ω 2 µ = I . From Theorem 1 we know that the expectation of ∆ µ vanishes exponentially. Therefore, the derivative ∂ µ C := ∂C ∂θ µ with respect to θ µ satisfies

where e µ is the unit vector in the parameter space corresponding to θ µ . From Markov's inequality as in (6), we know that the probability that the derivative ∂ µ C deviates from zero by a small constant is exponentially small.

On the other hand, even in the absence of the parameter-shift rule, vanishing gradients could still be obtained approximately by the following arguments. Consider the vicinity of a random initialized parameter point where the linear approximation error is negligible, denoted as an ε -ball B ε of radius ε (here ε plays the same role as the learning rate). As shown in Fig. 3, the linearity in B ε together with Theorem 1 leads to

ΓffΓΛΓΛΓΛΓΛΓΛΓΛΓΛΓfi

{0}(v≠)

[0μC| · 2e

2e

̸

up to the linear approximation error, where 1 /ε is not an essential factor since it reflects the frequencies 194 of the landscape fluctuation rather than magnitudes, similar to the role of the factor tr( V 2 ) in the 195 expression of Var[ ∂ µ C ] [25]. 196

197

198

199

200

201

202

203

204

205

206

207

208

209

210

211

212

213

214

215

216

217

218

219

220

221

222

For the gradient-free optimization based on the cost function difference between any two fixed parameter points θ ′ and θ , Theorem 1 leads to

where θ ( µ ) = θ + ∑ µ ν =1 ( θ ′ ν -θ ν ) e ν for µ = 1 , ..., M and θ ( µ ) = θ for µ = 0 . Thus, as long as the number of parameters satisfies M ∈ O (poly( n )) , the cost function difference between any two points vanishes exponentially with a high probability, demanding an exponential precision to make progress in the gradient-free optimization.

Furthermore, Theorem 1 goes beyond vanishing gradients and vanishing differences between two fixed points. The exponentially vanishing quantity claimed by Theorem 1 is the variation range of the cost function in the whole parameter subspace corresponding to a local unitary, e.g., the subspace of the 3 Euler angles in a single-qubit rotation gate from SU (2) , or the subspace of the 15 parameters in a two-qubit rotation gate from SU (4) , etc. This gives constraints on multiple parameters at finite intervals simultaneously, instead of a vicinity or two fixed parameter points.

5 Application on Representative QNN Models

To better illustrate the meaning of our findings in practice, we investigate the applications of Theorem 1 on three representative QNN models, including the variational quantum eigensolver (VQE), quantum autoencoder, and quantum state learning. The corresponding numerical simulation results are summarized in Fig. 5.

Application on VQE. The variational quantum eigensolver is the most famous implementation of a hybrid quantum-classical algorithm with the goal to prepare the ground state of a given Hamiltonian ˆ H of a physical system [60]. The cost function is the energy expectation with respect to an ansatz state U | 0 ⟩ , i.e.,

For most physical models with local interactions, the spectral width is proportional to the system size, i.e., w ( ˆ H ) ∈ O ( n ) . For common repeated-layer-type ansatzes, e.g., the hardware-efficient ansatzes [61], linear depth O ( n ) is enough to make a randomly initialized circuit to be a sample from an approximate 2 -design ensemble [25, 62, 63]. Hence from Theorem 1 we know that ∆ VQE ( V 1 , V 2 ) vanishes exponentially with a high probability for random circuits forming 2 -designs. We conduct

223

224

225

226

227

228

229

230

231

232

233

234

235

236

237

238

239

240

241

242

243

244

245

...

...

...

...

numerical simulations for the variation range of the VQE cost function ∆ VQE using the 1 -dimensional spin1 / 2 antiferromagnetic Heisenberg model:

with periodic boundary condition, as shown in Fig. 5(a).

Application on Quantum Autoencoder. The quantum autoencoder (QAE) is an approach for quantum data compression [64, 65]. As shown in Fig. 4, a QNN U is trained as an encoder to compress a given state ρ QR on a bipartite system QR into a reduced state σ Q = tr R ( U ρ QR U † ) on subsystem Q , such that ρ QR can be reproduced from σ Q by the decoder isometry ⟨ 0 | R U † with a high fidelity. According to the monotonicity of the fidelity under partial trace, an easy-to-measure cost function could be reduced from the fidelity between ρ QR and the reconstructed state as

where the second term is exactly the fidelity between the state of the discarded part σ R = tr Q ( U ρ QR U † ) and the zero state | 0 ⟩ R on subsystem R . The spectral width for the QAE cost function (12) is w ( H QAE ) = 1 with H QAE = I QR -| 0 ⟩ ⟨ 0 | R ⊗ I Q . Thus again from Theorem 1 we know that ∆ QAE ( V 1 , V 2 ) vanishes exponentially in the number of qubits, specifically with the scaling O (2 -n/ 2 ) as shown in Fig. 5(b).

Application on Quantum State Learning. The fidelity between pure states is a special case of the cost function in (2) with a low-rank observable. Many QML applications make use of fidelity as their cost functions [66-68]. Here we uniformly call them quantum state learning (QSL) tasks. Denote the input state as | ψ ⟩ and the target state as | ϕ ⟩ . The QSL cost function can be written as

Theorem 1 can be applied here with H QSL = I -| ϕ ⟩ ⟨ ϕ | and w ( H QSL ) = 1 . Here a tighter bound for ∆ QSL is provided in Proposition 2, which generally holds for the Bures fidelity. The proof of Proposition 2 is detailed in Appendix C.

Proposition 2 If either V 1 or V 2 , or both form unitary 1 -designs, then for the variation range of the fidelity-type cost function ∆ QSL , the following inequality holds

Compared with Theorem 1, the bound O (2 -n ) becomes tighter and the demanded randomness 246 becomes weaker in this special case. Notably, even a random circuit of constant depth is enough 247 to form a 1 -design, which is much shallower than 2 -designs. Like in (5) and (6), the variance and 248 the probability that ∆ QSL deviates from zero also vanish exponentially, but only require random 249 circuits forming unitary 1 -designs. Moreover, still with 1 -designs, Proposition 2 implies exponentially 250 vanishing cost gradients and cost differences in the same way as Theorem 1, which may be considered 251 as the underlying mechanism behind the severe barren plateaus for global cost functions even with 252 shallow quantum circuits [49]. 253

254

255

256

257

258

259

260

261

262

263

264

265

266

267

268

269

270

271

272

273

274

275

276

277

2

3

4

5

6

7

8

9

the number of qubits

n

10 -2

10 -1

10 0

10 1

V

1

, V

2

[∆VQE(

V

1

, V

2

)]

/w

( ˆ

H

)

(a)

1

≈

2-design

2

≈

2-design

1

,

2

≈

2-designs

Upper bound

2

3

4

5

6

7

8

9

the number of qubits

n

10 -1

10 0

V

1

, V

2

[∆QAE(

V

1

, V

2

)]

(b)

1

≈

2-design

2

≈

2-design

1

,

2

≈

2-designs

Upper bound

2

3

4

5

6

7

8

9

the number of qubits

n

10 -3

10 -2

10 -1

10 0

V

1

, V

2

[∆QSL(

V

1

, V

2

)]

(c)

1

=

1-design

2

=

1-design

1

,

2

=

1-designs

Upper bound

6 Numerical Simulations of Experiments

Previously, we have theoretically shown that with a high probability, the maximal influence of a local unitary within a random QNN on the cost function will vanish exponentially in the number of qubits. We further demonstrate the validity of our results with numerical simulations of experiments on the three representative QNN models. All of these experimental results show the exponentially vanishing variation range in the number of qubits, which is consistent with Theorem 1 and Proposition 2.

Circuit Setting. Consider subsystem A only containing a single qubit, namely m = 1 , and parameterize the local unitary U A ∈ U (2) with 3 Euler angles up to a global phase, i.e., U A ( ϕ, θ, α ) = R z ( ϕ ) R y ( θ ) R z ( α ) , where R y and R z are single-qubit rotation gates with generators being Y and Z Pauli matrices. To construct random circuits forming 2 -designs as V 1 or V 2 used in the VQE and QAE examples, we employ the following hardware-efficient ansatz as in [25] for comparison.

R

y

(

π

4

)

R

P

1

,

1

(

θ

1

,

1

)

•

· · ·

R

y

(

π

4

)

R

P

1

,

2

(

θ

1

,

2

)

•

•

· · ·

R

y

(

π

4

)

R

P

1

,

3

(

θ

1

,

3

)

•

•

· · ·

· · ·

· · ·

· · ·

· · ·

R

y

(

π

4

)

R

P

1

,n

(

θ

1

,n

)

•

· · ·

×

10

n

(15)

A single layer of R y ( π/ 4) = exp( -iY π/ 8) gates are laid at the very beginning of the circuit to make the three rotation axes have equal status, then followed by 10 × n repeated layers. Each layer consists of n single-qubit rotation gates R P ( θ ) on each qubit together with n -1 controlled phase gates between nearest neighboring qubits aligned as a 1-dimensional array, where the rotation axes P ∈ { x, y, z } is chosen with uniform probability and θ ∈ [0 , 2 π ) is also chosen uniformly. A such random circuit with O ( n ) repeated layers could be considered as an approximate 2 -design (here we employ 10 × n ) [25, 62, 63]. Experimental results with different numbers of layers are also presented in Appendix D to show how the expectation of the cost variation range ∆ H,ρ vanishes with the circuit depth. To construct random circuits forming 1 -designs used in the QSL example, we just replace the repeated layers above with a single layer of SU (2) elements R z ( ϕ ) R y ( θ ) R z ( α ) on each qubit with ϕ, θ, α ∈ [0 , 2 π ) are chosen with uniform probability.

Implementation Details. To compute max U A C and min U A C in the definition of ∆ H,ρ ( V 1 , V 2 ) with respect to U A , we employ the Adam optimizer to update U A iteratively until convergence for

278

279

280

281

282

283

284

285

286

287

288

289

290

291

292

293

294

295

296

297

298

299

300

301

302

303

304

305

306

307

308

309

310

311

312

313

314

315

316

317

318

319

320

321

322

323

324

325

326

327

each of the 100 samples of V 1 , V 2 . We consider the converged value as a good estimation with a tolerable error at least for circuits with a small number of qubits ( ≤ 10 ) and a modest depth ( ≤ 10 × n ). We repeat this procedure for different numbers of qubits and different statistics of V 1 and V 2 , i.e., V 1 or V 2 being a 2 -design ( 1 -design) while the other being identity.

Numerical Results. We summarize the simulation results of the three examples in Fig. 5. The slopes of the lines imply the rates of exponential decay. The data points represent the sample averages of the cost variation range ∆ H,ρ via varying U A over w ( H ) , and the error bars represent the standard deviations over samples. We specially rescale the error bar in the QSL example as a quarter of the standard deviation for better presentation on semi-log plots. One can see that in all the cases, the expectations of ∆ H,ρ ( V 1 , V 2 ) vanish exponentially in the number of qubits. The data lines are almost parallel to the dashed lines depicting the theoretical upper bounds. That is to say, the scaling behaviors almost coincide with the predictions from Theorem 1 and Proposition 2. These results suggest that while optimizing a local unitary within a random QNN, the cost function exhibits fluctuations within an exponentially small range relative to the number of qubits. It is this phenomenon that elucidates the vanishing gradient issue and contributes to the exponential difficulty of training as the QNN scales up. A detailed derivation can be found in Appendix B for the tighter task-dependent upper bounds used in Fig. 5(a) and (b).

7 Conclusion and Discussion

We have shown that the maximal possible influence of a local unitary within a QNN on the cost function vanishes exponentially in the number of qubits with a high probability. This finding unveils the exponential hardness associated with training QNNs as they scale up. The randomness required is just a 2 -design for the generic cost function and a 1 -design for the fidelity-type cost function, in spite that the integrand ∆ H,ρ ( V 1 , V 2 ) is not necessarily a polynomial of degree at most 2 or 1 in the entries of V 1 and V 2 . We remark that a 2 -design circuit can be achieved approximately by only O ( n ) depth [25, 62, 63] for common repeated-layer-type ansatzes, e.g., the hardware-efficient ansatzes [61], and a 1 -design circuit can be achieved more easily by only O (1) depth.

From the perspective of quantum information theory, our results can be regarded as a basic property of random quantum circuits. That is, a local unitary within a random circuit of polynomial depth has an exponentially small impact on the expectation of physical observables, which is expected to have potential applications in other areas involving random quantum circuits. This property may also provide insight into QNN design to address the critical trainability issue.

For the training of QNN, our results unify the restrictions on gradient-based and gradient-free optimizations in a natural way and hence can be regarded as the underlying mechanism behind the barren plateau phenomenon. Therefore, a fundamental limitation is unraveled in training QNNs, which can serve as a guide for designing better training strategies to improve the scalability of QNNs. A direct consequence is that the gate-by-gate optimization strategy [39, 40] is ineffective no matter what optimizers are utilized. Reparameterization within local unitaries is also unhelpful. For future research, it will be of great interest to explore potential solutions via proper initialization [41], pre-training including adaptive methods [42-46], circuit architectures [47, 48] and cost function choices [49, 50].

References

John Preskill. Quantum Computing in the NISQ era and beyond. Quantum , 2:79, aug 2018. ISSN 2521-327X. doi: 10.22331/q-2018-08-06-79. URL https://quantum-journal.org/papers/ q-2018-08-06-79/ .

Dave Wecker, Matthew B. Hastings, and Matthias Troyer. Progress towards practical quantum variational algorithms. Physical Review A , 92(4):042303, oct 2015. ISSN 1050-2947. doi: 10.1103/PhysRevA.92. 042303. URL https://link.aps.org/doi/10.1103/PhysRevA.92.042303 .

Wen Wei Ho and Timothy H. Hsieh. Efficient variational simulation of non-trivial quantum states. feb 2018. doi: 10.21468/SciPostPhys.6.3.029. URL http://arxiv.org/abs/1803.00026http://dx. doi.org/10.21468/SciPostPhys.6.3.029 .

Alexey Uvarov, Jacob Biamonte, and Dmitry Yudin. Variational Quantum Eigensolver for Frustrated 328 Quantum Systems. Physical Review B , 102(7):075104, may 2020. ISSN 2469-9950. doi: 10.1103/ 329 PhysRevB.102.075104. URL https://link.aps.org/doi/10.1103/PhysRevB.102.075104http: 330 //arxiv.org/abs/2005.00544 . 331

Sam McArdle, Suguru Endo, Alán Aspuru-Guzik, Simon C. Benjamin, and Xiao Yuan. Quantum 332 computational chemistry. Reviews of Modern Physics , 92(1):015003, mar 2020. ISSN 0034-6861. 333 doi: 10.1103/RevModPhys.92.015003. URL https://link.aps.org/doi/10.1103/RevModPhys. 334 92.015003 . 335

Daniel J. Egger, Claudio Gambella, Jakub Marecek, Scott McFaddin, Martin Mevissen, Rudy Raymond, 336 Andrea Simonetto, Stefan Woerner, and Elena Yndurain. Quantum Computing for Finance: State-of-the337 Art and Future Prospects. IEEE Transactions on Quantum Engineering , 1:1-24, 2020. ISSN 2689-1808. 338 doi: 10.1109/TQE.2020.3030314. URL https://ieeexplore.ieee.org/document/9222275/ . 339

Dylan Herman, Cody Googin, Xiaoyuan Liu, Alexey Galda, Ilya Safro, Yue Sun, Marco Pistoia, and Yuri 340 Alexeev. A Survey of Quantum Computing for Finance. arXiv preprint arXiv: 2201.02773 , jan 2022. URL 341 http://arxiv.org/abs/2201.02773 . 342

Adam Bouland, Wim van Dam, Hamed Joorati, Iordanis Kerenidis, and Anupam Prakash. Prospects 343 and challenges of quantum finance. arXiv:2011.06492 , nov 2020. URL http://arxiv.org/abs/2011. 344 06492 . 345

Jacob Biamonte, Peter Wittek, Nicola Pancotti, Patrick Rebentrost, Nathan Wiebe, and Seth Lloyd. 346 Quantum machine learning. Nature , 549(7671):195-202, sep 2017. ISSN 0028-0836. doi: 10. 347 1038/nature23474. URL http://dx.doi.org/10.1038/nature23474http://www.nature.com/ 348 articles/nature23474 . 349

Maria Schuld, Alex Bocharov, Krysta M. Svore, and Nathan Wiebe. Circuit-centric quantum classi350 fiers. Physical Review A , 101(3):032308, mar 2020. ISSN 2469-9926. doi: 10.1103/PhysRevA.101. 351 032308. URL http://arxiv.org/abs/1804.00633http://dx.doi.org/10.1103/PhysRevA.101. 352 032308https://link.aps.org/doi/10.1103/PhysRevA.101.032308 . 353

Ryan LaRose and Brian Coyle. Robust data encodings for quantum classifiers. Physical Review A , 102(3): 354 032420, sep 2020. ISSN 2469-9926. doi: 10.1103/PhysRevA.102.032420. URL https://link.aps. 355 org/doi/10.1103/PhysRevA.102.032420 . 356

Junyu Liu, Francesco Tacchino, Jennifer R Glick, Liang Jiang, and Antonio Mezzacapo. Representation 357 learning via quantum neural tangent kernels. PRX Quantum , 3(3):030323, 2022. 358

Matthias C Caro, Hsin-Yuan Huang, Marco Cerezo, Kunal Sharma, Andrew Sornborger, Lukasz Cincio, 359 and Patrick J Coles. Generalization in quantum machine learning from few training data. Nature 360 communications , 13(1):4919, 2022. ISSN 2041-1723. 361

MCerezo, Guillaume Verdon, Hsin-Yuan Huang, Lukasz Cincio, and Patrick J Coles. Challenges and 362 opportunities in quantum machine learning. Nature Computational Science , 2(9):567-576, 2022. ISSN 363 2662-8457. 364

Valentin Gebhart, Raffaele Santagati, Antonio Andrea Gentile, Erik M Gauger, David Craig, Natalia Ares, 365 Leonardo Banchi, Florian Marquardt, Luca Pezzè, and Cristian Bonato. Learning quantum systems. Nature 366 Reviews Physics , 5(3):141-156, feb 2023. ISSN 2522-5820. doi: 10.1038/s42254-022-00552-1. URL 367 https://www.nature.com/articles/s42254-022-00552-1 . 368

Ge Yan, Yehui Tang, and Junchi Yan. Towards a Native Quantum Paradigm for Graph Representation 369 Learning: A Sampling-based Recurrent Embedding Approach. In Proceedings of the 28th ACM SIGKDD 370 Conference on Knowledge Discovery and Data Mining , pages 2160-2168, 2022. 371

Xuchen You, Shouvanik Chakrabarti, Boyang Chen, and Xiaodi Wu. Analyzing Convergence in Quantum 372 Neural Networks: Deviations from Neural Tangent Kernels. arXiv preprint arXiv:2303.14844 , 2023. 373

Tongyang Li, Chunhao Wang, Shouvanik Chakrabarti, and Xiaodi Wu. Sublinear classical and quantum 374 algorithms for general matrix games. In Proceedings of the AAAI Conference on Artificial Intelligence , 375 volume 35, pages 8465-8473, 2021. ISBN 2374-3468. 376

Hsin-Yuan Huang, Richard Kueng, Giacomo Torlai, Victor V Albert, and John Preskill. Provably efficient 377 machine learning for quantum many-body problems. Science , 377(6613):eabk3333, 2022. ISSN 0036378 8075. 379

380

381

382

383

384

385

386

387

388

389

390

391

392

393

394

395

396

397

398

399

400

401

402

403

404

405

406

407

408

409

410

411

412

413

414

415

416

417

418

419

420

421

422

423

424

425

426

427

428

429

430

431

432

433

Zhan Yu, Hongshun Yao, Mujin Li, and Xin Wang. Power and limitations of single-qubit native quantum neural networks. In 36th Conference on Neural Information Processing Systems (NeurIPS 2022) , 2022. URL https://arxiv.org/abs/2205.07848 .

Jarrod R. McClean, Jonathan Romero, Ryan Babbush, and Alán Aspuru-Guzik. The theory of variational hybrid quantum-classical algorithms. New Journal of Physics , 18(2):023023, feb 2016. ISSN 13672630. doi: 10.1088/1367-2630/18/2/023023. URL https://iopscience.iop.org/article/10. 1088/1367-2630/18/2/023023 .

M. Cerezo, Andrew Arrasmith, Ryan Babbush, Simon C. Benjamin, Suguru Endo, Keisuke Fujii, Jarrod R. McClean, Kosuke Mitarai, Xiao Yuan, Lukasz Cincio, and Patrick J. Coles. Variational quantum algorithms. Nature Reviews Physics , 3(9):625-644, sep 2021. ISSN 2522-5820. doi: 10.1038/s42254-021-00348-9. URL https://www.nature.com/articles/s42254-021-00348-9 .

Kishor Bharti, Alba Cervera-Lierta, Thi Ha Kyaw, Tobias Haug, Sumner Alperin-Lea, Abhinav Anand, Matthias Degroote, Hermanni Heimonen, Jakob S Kottmann, Tim Menke, Wai-Keong Mok, Sukin Sim, Leong-Chuan Kwek, and Alán Aspuru-Guzik. Noisy intermediate-scale quantum (NISQ) algorithms. arXiv:2101.08448 , pages 1-82, jan 2021. URL http://arxiv.org/abs/2101.08448 .

Suguru Endo, Zhenyu Cai, Simon C Benjamin, and Xiao Yuan. Hybrid Quantum-Classical Algorithms and Quantum Error Mitigation. Journal of the Physical Society of Japan , 90(3):032001, mar 2021. ISSN 0031-9015. doi: 10.7566/JPSJ.90.032001. URL http://arxiv.org/abs/2011.01382https: //journals.jps.jp/doi/10.7566/JPSJ.90.032001 .

Jarrod R. McClean, Sergio Boixo, Vadim N. Smelyanskiy, Ryan Babbush, and Hartmut Neven. Barren plateaus in quantum neural network training landscapes. Nature Communications , 9(1):1-7, mar 2018. ISSN 20411723. doi: 10.1038/s41467-018-07090-4. URL http://arxiv.org/abs/1803.11173http: //dx.doi.org/10.1038/s41467-018-07090-4 .

Andrew Arrasmith, M. Cerezo, Piotr Czarnik, Lukasz Cincio, and Patrick J. Coles. Effect of barren plateaus on gradient-free optimization. Quantum , 5:1-9, nov 2020. ISSN 2521327X. doi: 10.22331/q-2021-10-05-558. URL http://arxiv.org/abs/2011.12245http://dx.doi.org/10. 22331/q-2021-10-05-558 .

M. Cerezo and Patrick J. Coles. Higher order derivatives of quantum neural networks with barren plateaus. Quantum Science and Technology , 6(3):035006, jul 2021. ISSN 2058-9565. doi: 10.1088/2058-9565/ abf51a. URL https://iopscience.iop.org/article/10.1088/2058-9565/abf51a .

Andrew Arrasmith, Zoë Holmes, M. Cerezo, and Patrick J. Coles. Equivalence of quantum barren plateaus to cost concentration and narrow gorges. pages 1-12, apr 2021. URL http://arxiv.org/abs/2104. 05868 .

Samson Wang, Enrico Fontana, M. Cerezo, Kunal Sharma, Akira Sone, Lukasz Cincio, and Patrick J. Coles. Noise-induced barren plateaus in variational quantum algorithms. Nature Communications , 12 (1):6961, dec 2021. ISSN 2041-1723. doi: 10.1038/s41467-021-27045-6. URL https://www.nature. com/articles/s41467-021-27045-6 .

Zoë Holmes, Kunal Sharma, M. Cerezo, and Patrick J. Coles. Connecting ansatz expressibility to gradient magnitudes and barren plateaus. PRX Quantum , 3(1):1-20, jan 2021. doi: 10.1103/prxquantum.3.010313. URL http://arxiv.org/abs/2101.02138 .

Lennart Bittel and Martin Kliesch. Training Variational Quantum Algorithms Is NP-Hard. Physical Review Letters , 127(12):120502, sep 2021. ISSN 0031-9007. doi: 10.1103/PhysRevLett.127.120502. URL http://arxiv.org/abs/2101.07267%0Ahttp://dx.doi.org/10.1103/PhysRevLett.127. 120502https://link.aps.org/doi/10.1103/PhysRevLett.127.120502 .

Carlos Ortiz Marrero, Mária Kieferová, and Nathan Wiebe. Entanglement-Induced Barren Plateaus. PRX Quantum , 2(4):040316, oct 2021. ISSN 2691-3399. doi: 10.1103/PRXQuantum.2.040316. URL https://link.aps.org/doi/10.1103/PRXQuantum.2.040316 .

Daniel Stilck Franca and Raul García-Patrón. Limitations of optimization algorithms on noisy quantum devices. Nature Physics , 17(11):1221-1227, nov 2021. ISSN 1745-2473. doi: 10.1038/s41567-021-01356-3. URL https://www.nature.com/articles/s41567-021-01356-3 .

A. V. Uvarov and J. D. Biamonte. On barren plateaus and cost function locality in variational quantum algorithms. Journal of Physics A: Mathematical and Theoretical , 54(24):245301, jun 2021. ISSN 17518113. doi: 10.1088/1751-8121/abfac7. URL https://iopscience.iop.org/article/10.1088/ 1751-8121/abfac7 .

434

435

436

437

438

439

440

441

442

443

444

445

446

447

448

449

450

451

452

453

454

455

456

457

458

459

460

461

462

463

464

465

466

467

468

469

470

471

472

473

474

475

476

477

478

479

480

481

482

483

484

485

486

Ernesto Campos, Aly Nasrallah, and Jacob Biamonte. Abrupt transitions in variational quantum circuit training. Physical Review A , 103(3):032607, mar 2021. ISSN 2469-9926. doi: 10.1103/PhysRevA.103. 032607. URL https://link.aps.org/doi/10.1103/PhysRevA.103.032607 .

Giacomo De Palma, Milad Marvian, Cambyse Rouzé, and Daniel Stilck Franca. Limitations of variational quantum algorithms: a quantum optimal transport approach. arXiv:2204.03455 , pages 1-30, 2022. URL http://arxiv.org/abs/2204.03455 .

Emanuel Knill, Gerardo Ortiz, and Rolando D. Somma. Optimal quantum measurements of expectation values of observables. Phys. Rev. A , 75:012328, Jan 2007. doi: 10.1103/PhysRevA.75.012328. URL https://link.aps.org/doi/10.1103/PhysRevA.75.012328 .

Patrick Huembeli and Alexandre Dauphin. Characterizing the loss landscape of variational quantum circuits. Quantum Science and Technology , 6(2):025011, apr 2021. ISSN 2058-9565. doi: 10.1088/2058-9565/abdbc9. URL http://arxiv.org/abs/2008.02785http://dx.doi.org/10. 1088/2058-9565/abdbc9 .

Ken M. Nakanishi, Keisuke Fujii, and Synge Todo. Sequential minimal optimization for quantumclassical hybrid algorithms. Physical Review Research , 2(4):1-11, mar 2019. ISSN 26431564. doi: 10. 1103/PhysRevResearch.2.043158. URL http://arxiv.org/abs/1903.12166http://dx.doi.org/ 10.1103/PhysRevResearch.2.043158 .

Mateusz Ostaszewski, Edward Grant, and Marcello Benedetti. Structure optimization for parameterized quantum circuits. Quantum , 5:1-13, may 2019. ISSN 2521327X. doi: 10.22331/q-2021-01-28-391. URL http://arxiv.org/abs/1905.09692http://dx.doi.org/10.22331/q-2021-01-28-391 .

Edward Grant, Leonard Wossnig, Mateusz Ostaszewski, and Marcello Benedetti. An initialization strategy for addressing barren plateaus in parametrized quantum circuits. Quantum , 3, mar 2019. ISSN 2521327X. doi: 10.22331/q-2019-12-09-214. URL http://arxiv.org/abs/1903.05076http://dx.doi.org/ 10.22331/q-2019-12-09-214 .

Guillaume Verdon, Michael Broughton, Jarrod R. McClean, Kevin J. Sung, Ryan Babbush, Zhang Jiang, Hartmut Neven, and Masoud Mohseni. Learning to learn with quantum neural networks via classical neural networks. pages 1-12, jul 2019. URL http://arxiv.org/abs/1907.05415 .

Harper R. Grimsley, Sophia E. Economou, Edwin Barnes, and Nicholas J. Mayhall. An adaptive variational algorithm for exact molecular simulations on a quantum computer. Nature Communications , 10(1):3007, dec 2019. ISSN 2041-1723. doi: 10.1038/s41467-019-10988-2. URL http://www.nature.com/ articles/s41467-019-10988-2 .

Feng Zhang, Niladri Gomes, Yongxin Yao, Peter P. Orth, and Thomas Iadecola. Adaptive variational quantum eigensolvers for highly excited states. Physical Review B , 104(7):1-10, apr 2021. ISSN 24699969. doi: 10.1103/PhysRevB.104.075159. URL http://arxiv.org/abs/2104.12636http: //dx.doi.org/10.1103/PhysRevB.104.075159 .

Andrea Skolik, Jarrod R. McClean, Masoud Mohseni, Patrick van der Smagt, and Martin Leib. Layerwise learning for quantum neural networks. Quantum Machine Intelligence , 3(1):5, jun 2021. ISSN 25244906. doi: 10.1007/s42484-020-00036-4. URL http://arxiv.org/abs/2006.14904https://link. springer.com/10.1007/s42484-020-00036-4 .

Harper R. Grimsley, George S. Barron, Edwin Barnes, Sophia E. Economou, and Nicholas J. Mayhall. ADAPT-VQE is insensitive to rough parameter landscapes and barren plateaus. 2022. URL http: //arxiv.org/abs/2204.07179 .

Arthur Pesah, M Cerezo, Samson Wang, Tyler Volkoff, Andrew T Sornborger, and Patrick J Coles. Absence of Barren Plateaus in Quantum Convolutional Neural Networks. Physical Review X , 11(4):041011, oct 2021. ISSN 21603308. doi: 10.1103/PhysRevX.11.041011. URL https://doi.org/10.1103/PhysRevX. 11.041011https://link.aps.org/doi/10.1103/PhysRevX.11.041011 .

Xia Liu, Geng Liu, Jiaxin Huang, and Xin Wang. Mitigating barren plateaus of variational quantum eigensolvers. may 2022. URL http://arxiv.org/abs/2205.13539 .

M. Cerezo, Akira Sone, Tyler Volkoff, Lukasz Cincio, and Patrick J. Coles. Cost function dependent barren plateaus in shallow parametrized quantum circuits. Nature Communications , 12(1):1791, dec 2021. ISSN 2041-1723. doi: 10.1038/s41467-021-21728-w. URL http://arxiv.org/abs/2001.00550http://dx.doi.org/10.1038/s41467-021-21728-whttp: //www.nature.com/articles/s41467-021-21728-w .

Maria Kieferova, Ortiz Marrero Carlos, and Nathan Wiebe. Quantum Generative Training Using Rényi 487 Divergences. jun 2021. URL http://arxiv.org/abs/2106.09567 . 488

Shi-Ju Ran. Encoding of matrix product states into quantum circuits of one-and two-qubit gates. Physical 489 Review A , 101(3):032310, 2020. 490

491

492

Iris Cong, Soonwon Choi, and Mikhail D Lukin. Quantum convolutional neural networks. Nature Physics , 15(12):1273-1278, 2019.

493

494

495

496

497

498

499

500

501

502

503

504

505

506

507

508

509

510

Christoph Dankert, Richard Cleve, Joseph Emerson, and Etera Livine. Exact and approximate unitary 2designs and their application to fidelity estimation. Physical Review A , 80(1):012304, jul 2009. ISSN 10502947. doi: 10.1103/PhysRevA.80.012304. URL https://link.aps.org/doi/10.1103/PhysRevA. 80.012304 .

Gian Giacomo Guerreschi and Mikhail Smelyanskiy. Practical optimization for hybrid quantum-classical algorithms. jan 2017. URL http://arxiv.org/abs/1701.01450 .

Kosuke Mitarai, Makoto Negoro, Masahiro Kitagawa, and Keisuke Fujii. Quantum Circuit Learning. mar 2018. doi: 10.1103/PhysRevA.98.032309. URL http://arxiv.org/abs/1803.00745http: //dx.doi.org/10.1103/PhysRevA.98.032309 .

Maria Schuld, Ville Bergholm, Christian Gogolin, Josh Izaac, and Nathan Killoran. Evaluating analytic gradients on quantum hardware. Physical Review A , 99(3):032331, nov 2018. ISSN 24699926. doi: 10.1103/PhysRevA.99.032331. URL https://link.aps.org/doi/10.1103/PhysRevA. 99.032331http://arxiv.org/abs/1811.11184 .

Gavin E. Crooks. Gradients of parameterized quantum gates using the parameter-shift rule and gate decomposition. (2), may 2019. URL http://arxiv.org/abs/1905.13311 .

Andrea Mari, Thomas R. Bromley, and Nathan Killoran. Estimating the gradient and higher-order derivatives on quantum hardware. Physical Review A , 103(1):012405, jan 2021. ISSN 2469-9926. doi: 10. 1103/PhysRevA.103.012405. URL https://link.aps.org/doi/10.1103/PhysRevA.103.012405 .

Artem Kaznatcheev. Unitary t-designs. Talk , 53(1):13-31, 2009. ISSN 0925-1022. URL http://www. 511 springerlink.com/index/10.1007/s10623-009-9290-2 . 512

Alberto Peruzzo, Jarrod McClean, Peter Shadbolt, Man-Hong Yung, Xiao-Qi Zhou, Peter J. Love, Alán 513 Aspuru-Guzik, and Jeremy L. O'Brien. A variational eigenvalue solver on a photonic quantum processor. 514 Nature Communications , 5(1):4213, sep 2014. ISSN 2041-1723. doi: 10.1038/ncomms5213. URL 515 http://www.nature.com/articles/ncomms5213 . 516

517

518

519

520

Abhinav Kandala, Antonio Mezzacapo, Kristan Temme, Maika Takita, Markus Brink, Jerry M. Chow, and Jay M. Gambetta. Hardware-efficient variational quantum eigensolver for small molecules and quantum magnets. Nature , 549(7671):242-246, sep 2017. ISSN 0028-0836. doi: 10.1038/nature23879. URL http://arxiv.org/abs/1704.05018http://www.nature.com/articles/nature23879 .

521

522

523

524

525

526

527

528

529

530

531

532

533

Aram W. Harrow, Avinatan Hassidim, and Seth Lloyd. Quantum Algorithm for Linear Systems of Equations. Physical Review Letters , 103(15):150502, oct 2009. ISSN 0031-9007. doi: 10.1103/PhysRevLett.103. 150502. URL https://link.aps.org/doi/10.1103/PhysRevLett.103.150502 .

Fernando G. S. L. Brandão, Aram W. Harrow, and Michał Horodecki. Local Random Quantum Circuits are Approximate Polynomial-Designs. Communications in Mathematical Physics , 346(2):397-434, sep 2016. ISSN 0010-3616. doi: 10.1007/s00220-016-2706-8. URL http://link.springer.com/10. 1007/s00220-016-2706-8 .

Jonathan Romero, Jonathan P. Olson, and Alan Aspuru-Guzik. Quantum autoencoders for efficient compression of quantum data. Quantum Science and Technology , 2(4):1-10, 2017. ISSN 20589565. doi: 10.1088/2058-9565/aa8072.

Chenfeng Cao and Xin Wang. Noise-Assisted Quantum Autoencoder. Physical Review Applied , 15(5): 054012, may 2021. ISSN 2331-7019. doi: 10.1103/PhysRevApplied.15.054012. URL http://arxiv. org/abs/2012.08331https://link.aps.org/doi/10.1103/PhysRevApplied.15.054012 .

Sang Min Lee, Jinhyoung Lee, and Jeongho Bang. Learning unknown pure quantum states. Physical 534 Review A , 98(5):052302, nov 2018. ISSN 2469-9926. doi: 10.1103/PhysRevA.98.052302. URL https: 535 //link.aps.org/doi/10.1103/PhysRevA.98.052302 . 536

Tomonori Shirakawa, Hiroshi Ueda, and Seiji Yunoki. Automatic quantum circuit encoding of a given 537 arbitrary quantum state. pages 1-25, dec 2021. URL http://arxiv.org/abs/2112.14524 . 538

[68] Carlos Bravo-Prieto, Ryan LaRose, M. Cerezo, Yigit Subasi, Lukasz Cincio, and Patrick J. Coles. Vari539 ational Quantum Linear Solver. arXiv:1909.05820 , sep 2019. URL http://arxiv.org/abs/1909. 540 05820 . 541

Peer reviews:
{"IS_META_REVIEW": false, "comments": "SUMMARY:\nIn this work, the authors characterize the problem of the Barren Plateau from different perspectives: (1) local unitary within a QNN on the cost function, particularly the randomness for the generic cost function; (2) quantum information theory; (3) the optimization methods during training. This work discusses those factors impacting the Barren Plateau landscape.\n\nSTRENGTHS:\n(1) The work provides a theoretical understanding of the Barren Plateau problem and determines what factors actually impact the training of VQC, which is very interesting. \n\n(2) Solid mathematical formulation is given and the experiments can corroborate the theoretical analysis.\n\nWEAKNESSES:\n(1) Some latest work on the Barren Plateau problem in the training of VQC should be included, such as Refs. [1], [2], and [3]. Ref. [1] aims at the QNN architecture for dealing with the Barren Plateau problem, Ref. [2] focuses on the initialization strategy, and Ref. [3] puts forth the pre-training method for mitigating the VQC training problem of Barren Plateau. \n\n[1] Jun Qi, Chao-Han Huck Yang, Pin-Yu Chen, Min-Hsiu Hsieh, \"Theoretical Error Performance Analysis for Variational Quantum Circuit Based Functional Regression,\" npj Quantum Information, Vol. 9, no. 4, 2023\n\n[2] Zhang, Kaining, Hsieh, Min-Hsiu, Liu, Liu, and Tao, Dacheng. Gaussian Initializations Help Deep Variational Quantum Circuits Escape From the Barren Plateau. In Neural Information Processing Systems, 2022.\n\n[3] Jun Qi, Chao-Han Huck Yang, Pin-Yu Chen, Min-Hsiu Hsieh, \"Pre-Training Tensor-Train Networks Facilitate Machine Learning with Variational Quantum Circuits,\" arXiv:2306.03741v1\n\nQUESTIONS:\nHow to asscoated with the locality unitary property with other methods for dealing with Barren Plateau problem, such as the DNN architecture and pre-training method?\n\nLIMITATIONS:\nIt is expected to have experimental simulations on real data like the MNIST dataset to demonstrate the effectiveness of the proposed analysis approach.", "TITLE": "Official Review", "RECOMMENDATION": 7, "REVIEWER_CONFIDENCE": 4, "SOUNDNESS_CORRECTNESS": 3, "DATE": "2023-06-30"}

{"IS_META_REVIEW": false, "comments": "SUMMARY:\nThe paper examines the critical issue of trainability in quantum neural networks (QNNs) by adopting a perspective centered around the locality. Through extensive analysis, the authors convincingly demonstrate that the adjustment of local quantum gates within a diverse range of QNNs results in an exponential decay of the loss function range as the number of qubits scales up. The authors bolster their claims with carefully conducted numerical simulations, providing compelling evidence that locality plays a fundamental role in shaping the behavior of QNNs. Building upon prior research on barren plateaus, the paper makes a technically sound contribution, albeit with an incremental advancement in the field.\n\nSTRENGTHS:\nThe analysis of Theorems and Propositions, which shows the exponential decay of the loss function range by adjusting local quantum gates, is technically sound. Additionally, the ideas, concepts, and results are well presented. The authors effectively communicate their methodology, theoretical framework, and experimental simulations, making it easier for readers to comprehend and follow their arguments.\n\nWEAKNESSES:\nThe main weakness of this paper lies in its limited impact. While the authors conduct a clear and thorough analysis of how the concentration results of random circuits depend on the locality unitary, the technical tools employed bear a striking resemblance to prior literature concerning barren plateaus. The achieved results can be derived from existing works, with the only notable distinction being the introduction of a parameter, m, related to the locality in the derived bound. Additionally, the authors' claim that few rigorous scaling results exist for generic QNNs is contradicted by the abundance of relevant research, as evidenced by references [1], [2], [3], and [4], which address similar theoretical aspects the authors aim to explore. Previous studies have already established that deep ansatz can lead to the concentration of the cost function, rendering the observation regarding the exponential vanishing of the loss function range via the adjustment of local quantum gates less novel.\n\n\n[1] Leone, Lorenzo, et al. \"On the practical usefulness of the Hardware Efficient Ansatz.\" arXiv preprint arXiv:2211.01477 (2022).\n[2] Thanasilp, Supanut, et al. \"Subtleties in the trainability of quantum machine learning models.\" Quantum Machine Intelligence 5.1 (2023): 21.\n[3] Garcia, Roy J., et al. \"Barren plateaus from learning scramblers with local cost functions.\" Journal of High Energy Physics 2023.1 (2023): 1-79.\n[4] Larocca, Martin, et al. \"Diagnosing barren plateaus with tools from quantum optimal control.\" Quantum 6 (2022): 824.\n\nQUESTIONS:\n- The results presented in Theorem 4 are somewhat perplexing. It appears that the upper bound can be independent of the qubit number n when utilizing the global unitary U_A. For example, when m=n, the concentration phenomenon can be avoided. I am not sure if I have overlooked certain details.\n\n- In Remark 2, what is the necessity of decomposing and preserving the local unitary for subsequent analysis of U_A? A more detailed explanation is expected. \n\n- The similar findings discussed in Section 4 can be derived from existing research. Specifically, the concentration phenomenon of the cost function, as outlined in reference [5], in conjunction with the parameter shift rule (for first-order and higher-order) or the finite-difference method (for zero-order), can effectively yield similar outcomes as presented in Section 4. It would be valuable if the authors could acknowledge and discuss these connections with prior findings to provide a clearer context and further highlight the novelty of their results. If the only difference lies in the introduced parameter related to the locality m?\n\n[5] Arrasmith, Andrew, et al. \"Equivalence of quantum barren plateaus to cost concentration and narrow gorges.\" Quantum Science and Technology 7.4 (2022): 045015.\n\nLIMITATIONS:\nNo, the authors did not address the limitations of their work", "TITLE": "Official Review", "RECOMMENDATION": 4, "REVIEWER_CONFIDENCE": 5, "SOUNDNESS_CORRECTNESS": 3, "DATE": "2023-07-02"}

{"IS_META_REVIEW": false, "comments": "SUMMARY:\nThis paper investigates the trainability of random quantum circuits from the perspective of their locality and demonstrates the variation range of the cost function via adjusting any local quantum gate vanishes exponentially in the number of qubits. This theorem unifies the restrictions on gradient-based and gradient-free optimizations. The paper also verifies their theorem on three applications with numerical simulations and deepens the understanding of the role of locality in QNNs.\n\nSTRENGTHS:\n1. The paper is well-written and provides a rigorous analysis of QNN trainability and scalability from the perspective of their locality. \n2. The paper applies the proposed theorem to three representative QNN models, including the VQE, quantum autoencoder, and quantum state learning, and provides the numerical simulation results.\n\nWEAKNESSES:\n1.  Although Line 66-73 provides the advances of the proposed method, the comparison with previous works is not clear enough. It is important to review the previous methods and compare their specific differences. \n2.  The contribution of the paper seems weak, the novelty, comparisons with related works, and guidances for future QNN training or design need to be highlighted and enhanced.\n\nQUESTIONS:\nCompared with previous approaches that prove barren plateaus, what new enlightenment does the proposed method bring to the design and training of QNN? Specifically, barren plateaus are a well-known phenomenon in the field of quantum machine learning, and we usually design QNNs with single-qubit and tow-qubit parametric gates for practicality on NISQ devices, and it seems to have defaulted to the property of barren plateaus w.r.t local unitary. What guidance can the proposed finding give to the design of QNNs? Moreover, based on the finding, what are some specific suggestions to design more effective training strategies?\n\nLIMITATIONS:\nThe paper can be improved by considering the above weaknesses and questions.", "TITLE": "Official Review", "RECOMMENDATION": 5, "REVIEWER_CONFIDENCE": 3, "SOUNDNESS_CORRECTNESS": 3, "DATE": "2023-07-04"}

{"IS_META_REVIEW": false, "comments": "SUMMARY:\nThe paper proof a result on the range of possible values that the cost function of a variational quantum algorithm can take when one optimises over a given unitary that is before or after random gates that form unitary 2 designs. This quantity vanishes exponentially with the number of qubits. This generalises previous results on Barren plateaus, concerned with the vanishing of the gradient of the cost function.\nThe material is presented clearly and the paper also has numerical verification of the scaling in the case of VQE, quantum autoencoder and quantum state learning.\n\nSTRENGTHS:\n- The paper is clearly written, with figures explaining concepts. It presents both theory and numerical checks\n- The problem studied is relevant in scaling up quantum neural networks \n- The main theorem allows the authors to recover and unify previous results on exponentially vanishing gradients and cost function differences\n\nWEAKNESSES:\n- The paper does not comment on recommendations to avoid the exponentially vanishing variation range\n- The comparison to previous works is limited. The authors mention that their work opens a new venue for analysing trainability of QNNs but it is not clear to me what new insights are gained. It would be useful to comment on what is gained wrt previous literature. Also, on whether the methods used to prove their main theorem are similar to those used in the literature or not.\n- The VQE experiments are taken with circuits of depth 10 x n. That depth was chosen so that the hardware aware ansatz approximates a 2-design. However no comment on the required depth to compute the ground state of the Hamiltonian is presented and it is not clear whether the choice of ansatz and depth is something that practitioner would actually do.\n\nMinor\n\n- Sentence Line 71 - 73 does not read very well, you could rephrase it\n- Line 153 - 155: it would be helpful for the reader to have an explanation of the connection between parameter shift rule and $e^{-i\\theta \\Omega}$ with $\\Omega^2=1$. Also why does this imply the existence of $W$ as claimed?\n\nQUESTIONS:\n- Can you comment on your choice of VQE ansatz explaining whether 10x n is needed or smaller depth would be enough to solve this problem? Also, do other ansatz, such as alternating ansatz, suffer from exponentially vanishing variation range?\n- What are the additional insights that you get on top of previous works for quantum neural network trainability?\n- Can you expand on the previous literature? Do the previous results also require approximate 2-design?\n\nLIMITATIONS:\n- The method applies when the $V_1$ or $V_2$ are 2-designs. This limits applicability.\n- Strategies to overcome the exponentially vanishing range are not discussed.", "TITLE": "Official Review", "RECOMMENDATION": 6, "REVIEWER_CONFIDENCE": 3, "SOUNDNESS_CORRECTNESS": 3, "DATE": "2023-07-04"}