Escaping from the Barren Plateau via Gaussian Initializations in Deep Variational Quantum Circuits
Authors: Kaining Zhang
Liu Liu
Min-Hsiu Hsieh
Dacheng Tao
Venue: NeurIPS 2022 Accept (2022)
Keywords: quantum algorithm
vanishing gradient problem
optimization
gaussian initialization

Abstract: Variational quantum circuits have been widely employed in quantum simulation and quantum machine learning in recent years. 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. This result leads to a general standpoint that deep quantum circuits would not be feasible for practical tasks. In this work, we propose an initialization strategy with theoretical guarantees for the vanishing gradient problem in general deep quantum circuits. 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. 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. Experimental results verify our theoretical findings in quantum simulation and quantum chemistry.

Escaping from the Barren Plateau via Gaussian Initializations in Deep Variational Quantum Circuits
Kaining Zhang", Liu Liu", Min-Hsiu Hsieht, and Dacheng Tao*
"School of Computer Science, Faculty of Engineering, The University of Sydney, Sydney, Australia Hon Hai Quantum Computing Research Center, Taipei, Taiwan
Abstract
Variational quantum circuits have been widely employed in quantum simulation and quantum machine learning in recent years. 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. This result leads to a general standpoint that deep quantum circuits would not be feasible for practical tasks. In this work, we propose an initialization strategy with theoretical guarantees for the vanishing gradient problem in general deep quantum circuits. Specifically, we prove that under proper Gaussian initialized parameters, the norm of the gradient decays most polynomially when the qubit number and the circuit depth increase. Our theoretical results hold for both the local and the global observable cases, where the latter was believed have vanishing gradients even for very shallow circuits. Experimental results verify our theoretical findings in quantum simulation and quantum chemistry.
Introduction
Quantum computing has attracted great attention in recent years, especially since the realization of quantum supremacy 212 with noisy intermediate-scat quantum (NISQ) devices [3]. 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. Specifically, different VQAs have been proposed for many topics, e.g., quantum chemistry [5] 13). quantum simulations [14 machine learning [24] 31]. numerical analysis (6] and linear algebra problems [37-39]. 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 255, generative learning (30 and reinforcement learning (29
Typical variational quantum algorithms a trainable quantum-classical hybrid framework based parameterized quantum circuits (PQCs) 44). Similar to classical counterparts such 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. 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
Roughly speaking, the barren platcau describes the phenomenon that the value of the loss function and its gradients concentrate around their expectation values with exponentially small variances. We remark that gradient-bar based methods could hardly handle trainings with the barren plateau phe- nomenon
45 Both the machine noise of the quantum channel and the statistical noise induced by measurements could severely degrade the estimation of gradients. Moreover, the optimization of the loss with flat surface takes much more time using inaccurate gradients than ideal cases. Thus, solving the barren plateau problem is imperative for achieving practical quantum advantages with
36th Conference on Neural Information Processing Systems (NcurIPS 2022).

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VQAs. In this paper, we propose Gaussian initializations for VQAs which have theoretical guarantees on the trainability. 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. 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. To summarize, our contributions are fourfold:
We propose Gaussian initialization strategy for deep variational quantum circuits. By setting the variance x² = =(2) for N --qubit L-depth circuits with independent parameters and local observables, we lower bound the expectation of the gradient norm by poly L) as provided in Theorem 4.1 which outperforms previous 2-((2) results.
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 cireuits. Moreover, our bound holds for correlated parameterized gates, which are widely employed in practical tasks like quantum chemistry and quantum simulations.
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. The result is presented in Corollary 4.3 which proves that ( 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. Experiment results show that Gaussian initializations outperform uniform initializations, which verify proposed theorems.
1.1
Related work
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. Subsequently, several positive results are proved for shallow quantum circuits such as the alternating-layere
circuit (45 47 and the quantum convolutional neural network [48] when the observable is constrained in small number of qubits (local observable). For shallow circuits with N qubits and (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. Later, several works prove an inherent relationship between the barren plateau phenomenon and the complexity of states generated from the circuit. Specifically, circuit states that satisfy the volume law could lead to the barren plateau problem [49]. Expressive quantum circuits, which is measured by the distance between the Haar distribution and the distribution of circuit states, could have vanishing gradients (50). 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.
The parameterization of quantum eircuits is achieved by tuning the time of Hamiltonian simulations, so the gradient of the circuit satisfies the parameter-shift rule [5T]. Thus, the variance of the loss in VQAs and that of its gradient have similar behaviors for uniform distributions 44 corollary of the parameter-shin 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. Another corollary is that both gradient-free (54) and higher-order methods (55) could not solve the barren plateau problem. 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. 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. Mcanwhile, 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. Contrary to shallow cases, deep VQAs have vanishing gaps between local minima and global minima (58). In practice, experiments show that overparameterized VQAs [59 can be optimized towards the global minima. 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.
One
Inspired by various advantages of deep VQAs, some approaches have been proposed recently for solving the related barren plateau problem in practice (63-65). For example, the block-identity
2

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strategy 633 initializes gate blocks in pairs and sets parameters inversely, such that the initial circuit equivalent the identity circuit with zero depth. Since shallow eircuits have no vanishing gradient problem, the corresponding VQA is trainable with guarantees at the first step. However, we remark that the block block-identity condition would not hold after the first step, and the structure of the circuit needs to be designed properly. The layerwise training method [64] trains parameters in the circuit layers by layers, such that the depth of trainable part is limited. However, this method implements circuits with larger depth than that of the origin circuit, and parameters in the first few layers are not optimized. A recent work provides theoretical guarantees on the trainability of deep circuits with certain structures [65]. 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] [6] apply global observables.
2 Notations and quantum computing basics
We denote by [N] the set N.)....N... The form || ll2 represents the E2 norm for the vector and the spectral norm for the matrix, respectively. We denote by ay the j-th component of the vector a. The tensor product operation is denoted as "0"". The conjugate transpose of matrix A is denoted At The trace of a matrix A is denoted as Tr Tr[A]. We denote Vef as the gradient of the function f with respect the variable 0. We employ notations O to describe complexity notions.
Now we introduce quantum computing knowledge and notations. The pure state of qubit could be written (Ø) a|0) + b[1). where a, b C satisfy |a|² + =b|² 1, and 0) = (1, 0) 11] (0, 1)² The N -uquii space is formed by the tensor product of N single-qubit spaces. For pure states, the corresponding density matrix is defined p = (Ø) (o), in which (dl (lo))= . We the density matrix to represent general mixed quantum states, i.c., = Σx ca Φx) (@x where C E IR and Σ. Ck 1. 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. Specifically, single-qubit operations often used as Rx (f) e
-8x e-8x R0( = - and Rz c 102 where
( 0) . x (
1 0 == - ! -.
X
Pauli matrices will be referred to {I, = fou. T1. 02, J3} for the convenience. Moreover, two-qubit operations, such the CZ gate and the ViSWAP gate, are employed for generating quantum entanglement:
0
0
0
0
1
0
0
0
1/v2
1/V2
0
CZ
=
0
ViSWAPP =
0 i/V2
1/v2
0
0
0
0
-1
0
We could obtain information from the quantum system by performing measurements, for example, measuring the state (Ø) a 0) + b|1} generates 0 and 1 with probability p(0) ja|2 and p(1) [b]2 respectively. Such a measurement operation could be mathematically referred to as calculating the average of the observable O 0g under the state (Ø):
(0) Tr(as|o) (o1) = |a| - 1b/2 p(0) p(1).
Mathematically, quantum observables are Hermitian matrices. Specifically, the average of a unitary observable under arbitrary states is bounded by 11 We remark that ( times of measurements could provide an <||O le-error estimation to the value Tr[Op].
3 Framework of general VQAs
In this section, we introduce the framework of general VQAs and corresponding notations. A typical variational quantum algorithm can be viewed the optimization of the function f. which is defined as the expectation of observables. The expectation varies for different initial states and different parameters 0 used in quantum circuits. Throughout this paper, define
f(0) Tr (0) (0)+
(I)
3

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as the loss function of VQAs, where V(0) denotes the parameterized quantum circuit, the hermitian matrix O denotes the observable, and Pin denotes the density matrix of the input state. Next, we explain observables, input states, and parameterized quantum circuits in detail.
Both the observable and the density matrix could be decomposed under the Pauli basis. 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. 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.
The observable and the input state in VQAs could have various formulations for specific tasks. For the quantum simulation or the quantum chemistry scenario, observables are constrained to be the system Hamiltonians, while input states are usually prepared as computational basis states. For example, (0) 8N is used frequently in quantum simulations [17 18 Hartree-Fock (HF) states ( 10). which are prepared by the tensor product of (10), [T)]; ). serve good initial states in quantum chemistry tasks 2 [3] For quantum machine learning (QML) tasks, initial states encode the information of the training data, which could have a complex form. Many encoding strategies have been introduced in the literature [24] 66, (69. In contrary with the complex initial states, observables employed in QML are quite simple. For example, To = 10) (0) serves as the observable in most QML tasks related with the classification (24)26] or the dimensional reduction 770].
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 (31-73). Specifically, the ansatz in the VQA denotes the initial guess the circuit structure. For example, sepermaiinn-lyyeeee ansatzes 71 74 are proposed for approximating the Hamiltonian evolution. Recently, hardware efficient ansatzes 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, For quantum chemistry are preferred since they preserve the number of
tasks, unitary coupled cluster ansatzes 78 79 electrons corresponding circuit states.
In practice, ansatz is deployed as the sequence of single-qubit rotations {e {e-i0m k E (1,2,3) }} and two-qubit gates. We remark that the gradient of the VQA satisfies the parameter-shif rule [5]] 80 81; namely, for independently deployed parameters fj. the corresponding partial derivative is
0f 00,
f(e -
(2)
where 0+ and 0_ are different from 0 only at the j-th parameter: 0j + + Thus, the gradient of could be estimated efficiently, which allows optimizing VQAs with gradient- methods [82
4 Theoretical results about Gaussian initialized VQAs
In this section, we provide our theoretical guarantees on the trainability of deep quantum circuits through proper designs for the initial parameter distribution. In short, we prove that the gradient of the L-layer N -uqubit circuit is upper bounded by 1/poly(L4 (L, .V), if initial parameters are sampled from a Gaussian distribution with (1/L)) variance. Our bounds significantly improve existing results of the gradients of VQAs, which have the order 2 O())) for shallow circuits and the order 2 O(N) for deep circuits. We prove different results for the local and global observable cases in Section 4.1 and Section 4.2 respectively.
4.1 Independent parameters with local observables
First, we introduce the Gaussian initialization of parameters for the local observable case. We use the quantum circuit illustrated in Figure D as the ansatz in this section. The circuit in Figure [ performs T. layers of single qubit rotations and CZ. gates on the input state Pmn followed by a Rx layer and a Ry layer. We denote the single-qubit gate on the n-th qubit of the f-th layer as e -9 Ge,.. 2e E (I..... I. + 2} and n E (1..... , N). where iü,, is the corresponding parameter and Ge,, is Hermitian unitary. To eliminate degenerate parameters, we require that single-qubit gates in the first L layers do not commute with the CZ gate. After gates operations, we measure the observable
0 (11 .x) Tsy X Dig Jen!
(3)

---

[M] where L times
1)
Rx (x++1+11
Ry (01+21,
R(0e.2)
Rx (0
Ry (0L+2,2)
CLe
Pin
R(
Rx (0L+1,N)
Ry (0 L+2,N)
Figure The quantum circuit framework for the local observable case. The circuit performs layers of single qubit rotations and CZ. layers on the input state Pin. followed by Rx layer and Ry layer. In the ě-th single qubit layer, we employ the gate GG, for all qubits Ge. is a Hermitian unitary, which anti-commutes with d3 for & [L]. In each CZe layer, CZ gates are employed between arbitrary qubit pairs. The measurement performed on S qubits where the observable acts nontrivially on these qubits.
where E 1,,1, Vj [1,... .). and contains non-zero elements. Figure I provides a general framework of VQAs with local observables, which covers various ansatzes proposed in the literature [65] 85 61 64). The bound of the gradient norm of the Gaussian initialized variational quantum circuit provided in Theorem 4.1 with the proof in Appendix.
Theorem 4.1. Consider the L-layer N-qubit variational quantum circuit (O) defined Figure ! and the cost function f()) Tr (0)V(0) (0))), where the observable as follows the defini- tion 3 Then,
L S 2)
E [Voll
Tr [osppn]]
(4)
where is the number of non-zero elements in and the index j (2,,2, iNN such that Im 0, Vim 0 and Jm 3, Vim f O. The expectation is taken with the Gaussian distribution
N (0, (5((+2) for the parameters
Compared to existing works gradient norm, which improves the complexity exponentially with the depth of trainable circuits. Different from unitary 2-design distributions 44
47
48
65
Theorem
4. provides a larger lower bound of the
or the uniform distribution in the parameter space [52 86 651 that were employed in existing works, we analyze the expectation of the gradient norm under depth-induced Gaussian distribution. This change follows a natural idea that the trainability not required in the whole parameter space or the entire circuit space, only on the parameter trajectory during the training. Moreover, large norm of gradients could only guarantee the trainability in the beginning stage, instead of the whole optimization, since large gradient for trained parameters corresponds to non-convergence. Thus, the barren plateau problem could be crucial if initial parameters have vanishing gradients, which has been proved for deep VQAs with uniform initializations. In contrary, we could solve the barren plateau problem if parameters are initialized properly with large gradients, as provided in Theorem 4.1 Finally, Gaussian initialized circuits converge to benign values if optima appear around 0 O, which holds in many cases. For example, over-parametent quantum circuits have benign local minima (58) the number of parameters exceeds the aer-paaameterizztionn threshold. Moreover, wver-paaaeeterized circuits have exponential convergence rates [8 88 on tasks like quantum machine learning and the quantum eigensolver. These works indicate that quantum circuits with sufficient depths could find good optimums near the
45
47
48
initial points, which is similar to the classical wide neural network case 89
4.2 Correlated parameters with global observables
Next, we extend the Gaussian initialization framework to general quantum circuits with correlated pa- rameters and global observables. Quantum circuits with correlated parameters have wide applications
5

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in quantum simulations and quantum chemistry 9
13 One example is the Givens rotation
SSWAP
Rz
2
ASSNAP
R Givens (0)
0
0
sin
cos 0
cos
sin
0
0
0
(5)
0
0
1
Rz e+
R2 (2)
which preserves the number of electrons in parameterized quantum states (1]
To analyze VQAs with correlated parameterized gates, we consider the ansatz V(Ø) II, Vi
which consists of parameterized gates (V, Denote by hy the number of unitary gates that share the same parameter fz. Thus, the parameterized gate Vy(0,) consists of list of fixed and parameterized unitary operations
"1 Gzk
II
W
(6)
with the term aj E [R] where the Hamiltonian Gja and the fixed gate Wjx are unitary Ak E [by]] Moreover, we consider the objective function
f(0) I o II
V Pm Iv
j=L
where Pin and O denote the input state and the observable, respectively. In practical tasks of quantum chemistry, the molecule Hamiltonian H serves as the observable O. Minimizing the funetion 7 provides the ground energy and the corresponding ground state of the molecule. We provide the bound of the gradient norm the Gaussian initialized variational quantum circuit Theorem 4.2 with the proof in Appendix. Similar to the local observable case, we could bound the norm of the gradient of Eq. (7) parameters are initialized with (++) variance. Theorem 4.2 provides nontrivial bounds when the gradient at the point is large. This condition holds when the mean mean-field theory provides a good initial guess to the corresponding problems, c.g. the ground energy task in quantum chemistry and quantum many-body problems [90].
Theorem 4.2. Consider the N N-qubit variational quantum algorithms with the functionn Then the following formula holds for any l E {1, L}.
0f
2
0f a0;
2
2 2 e1
(8)
002
0=0
where 0 E R is the zero vector. The expectation is taken with Gaussian distributions (O, 73) for
parameters in 0 {0,}/=n where the variance +3 (h, OI
agc
(af) 36
0
We remark that Theorem 4.2 not only provides an initialization strategy, but also guarantees the update direction during the training. 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. A guide on the size of acceptable measurement noise could be useful for the complexity analysis of VQAs. Specifically, define ((--1) as the parameter at the t - 1-th iteration. Denote by (()) and ((t) the parameter updated from (t--1) for noiseless and noisy cases, respectively. Then (1)) differs from ((() by a Gaussian error term due to the measurement noise. We expect to derive the gradient norm bound for ((0), as provided in Corollary 4.3 Thus,
J = ( number of measurements is sufficient to guarantee a large gradient.
Corollary 4.3. Consider the N N-qubit variational quantum algorithms with the objective function Then the following formula holds for any l E (1, I.}.
0f 30,
0f 2
E
2 -
00:
0-0
(=)())
The expectation is taken with Gaussian distributions N(0, ?3) ) for parameters 5 18, (6 where
the variance 7 S
a3e
af 882
2
Vj E [T.].
(h
0=0

---

My beraoo
Taring Saat WO1
aa)
b
c
(d)
g
(h)
Figure 2: Numerical results of finding the ground energy of the Heisenberg model. The first row shows training results with the gradient descent optimizer, where Figures 2(a) and 2(b) illustrate the loss function corresponding to (I0) during the optimization with accurate and noisy gradients, respectively. Figures 2(c) and 2(d) show the l2 norm of corresponding gradients. The second row shows training results with the Adam optimizer, where Figures 2 and 2(1) illustrate the loss function with accurate and noisy gradients, respectively. Figures 2(g) and 2(h show the Eg norm of corresponding gradients. Eachh line denotes the average rounds of optimizations.
Corollary 4.3 is derived by analyzing the gradient of the function g( f + ((9) via Theorem 4.2 For any number of measurements such that the corresponding Gaussian noise satisfies the condition in Corollary 4.3 the trainability at the updated point is guaranteed.
Experiments
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. All numerical experiments are provided using the Pennylane package (9])
5.1 Heisenberg model
In the first task, we aim to find the ground state and the ground energy of the Heisenberg model 92 The corresponding Hamiltonian matrix is
N-1 H H= Σ X,X Y. + Z2Zi+11
(10)
where N the number of qubit, X; = 70(1-1) s s 18(N-1) Y; 70(1-1) 60 Y & 7 and
2, /((r-1) @ Z I ( We employ the loss function defined by Eq. 1 with the input state (0) and the observable (10). Thus, by minimizing the function I we can obtain the least eigenvalue of the observable (TO). which is the ground energy. We adopt the ansatz with N 15 qubits, which consists of L, = 10 layers of Ry Rx CZ blocks. In each block, we first employ the CZ gate to neighboring qubits pairs (1, ... (N, 1)}, followed by Rx and Ry rotations for all qubits. Overall, the quantum circuit has 300 parameters. We consider three initialization methods for comparison, i.e., initializations with the Gaussian distribution (0 >2) and the uniform distribution in [0, 2x. respectively, and the zero initialization (all parameters equal to 0 at the initial point). We remark that each term in the observable (TO) contains most S non-identity Pauli matrices, which is consistent with the (S, L) (2, 18) case of Theorem 4.1 Thus, we expect that the Gaussian initialization with the variance 72 4S(L+22 = T60 could provide trainable initial parameters.
In the experiment, we train VQAs with gradient descent (GD) 93 and Adam optimizers [94], respectively. The learning rate is 0.01 and 0.01 for both GD and Adam cases. Since the estimation
7

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of gradients on real quantum computers could be perturbed by statistical measurement noise, we compare optimizations using accurate and noisy gradients. For the latter case, we set the variance of measurement noises to be 0.01. The numerical results of the Heisenberg model are shown in the F The loss during the training with gradient descents shown Figures 2{a) and 2 for the accurate and the noisy gradient cases, respectively. The Gaussian initialization outperforms the other two initializations with faster convergence rates. Figures 2 and verify that Gaussian initialized vQAs have larger gradients in the early stage, compared to that of uniformly initialized VQAs. We notice that zero initialized VQAs cannot be trained with accurate gradient descent, since the initial gradient equals zero. This problem is alleviated in the noisy case, as shown in Figures 2(b) and 2(d) 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. 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) As illustrated in Figure 2(b) the loss function corresponding the zero initialization
decreases quickly after the variance accumulation stage. Results in Figures 2(e) and 2(h) show similar training behaviors using the Adam optimizer.
5.2 Quantum chemistry
In the second task, we aim to find the ground state and the ground energy of the LiH molecule. We follow settings the ansatz in Refs. [3] For the molecule with ne active electrons and n. free spin orbitals, the corresponding VQA contains = no qubits, which employs the HF state 10]
(ØHF) 1)
as the input state. We construct the parameterized quantum circuit with Givens rotation gates [2] where each gate is implemented on 2 or 4 qubits with one parameter. Specifically, for the LiH
molecule, the number of electrons n. = 2, the number of free spin orbitals n. 10, and the number of different Givens rotations is L = 24 [3]. We follow the molecule Hamiltonian HLSH defined in Ref. [13. Thus, the loss function for finding the ground energy of LiH is defined as
f()) Tr |HIH VGivens (0))oH)) ( (Gveen(()"]
(11)
where VGivens(O) = I R.Givene (0;) denotes the product of all parameterized Givens rotations obtain the least eigenvalue of the
i=1
of the LiH molecule. By minimizing the function [T). Hamiltonian HLAH which is the ground energy of the LiH molecule.
In practice, we initialize parameters in the VQA (11) with three distributions for comparison, i.e., the Gaussian distribution N NO,, 22), the zero distribution (all parameters equal to 0), and the uniform distribution in [0, 2T.. For 2-qubit Givens rotations, the term (h, a) = (2, 2) as shown in Eq. (5) For 4-qubit Givens rotations, the term (h, a) = (8, 8) (55. Thus, we set the variance in the Gaussian
distribution 7² x x which matches the (L, e,e) = (24, 8, ()) case of Theorem 4.2
x²x
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 0.001. Moreover, we consider the noisy case with adaptive noises, where the variance of the noise on each partial derivative 8f 86, lo oze
in the t-th iteration
0f a0,
(12)
96 x 24 82
H1m
@=
The variance in Eq. matches the (L, a, = -22, 4) case of Corollary is nearly converged:
4.3 when the VQA
0f 00, 0=0:))
0f 00: -
In the experiment, we train VQAS with gradient descent and Adam optimizers. Learning rates are set to be 0.1 and 0.01 for GD and Adam cases, respectively. The loss (IT) during training iterations shown in Figure 3 Optimization results with gradient descents are shown in Figures 33a) 3(c)
for the accurate gradient case, the adaptive noisy gradient case, and the noisy gradient case with the constant noise variance 0.001, respectively. The variance of the noise in the adaptive noisy gradient
case follows Eq.
12] Figures ((a) and 3(b) show similar performance, where the loss f with the
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199
184
394
gaussian uniform
gaussian uniform
gaussann unifor
28
43
43 tratning Iterationn GD
100
45 training Iteraaion
training Reration
(a)
(b)
(c)
uniform
Uniform
uniform
zero
20ro
2010
10'1
17
d training Adam
189
26 d
124
46
traiting beration with Adam
training Boration with Adam
(e)
(I)
Figure 3: Numerical results of finding the ground energy of the molecule LiH. The first and second rows show training results with the gradient descent and the Adam optimizer, respectively. The left, the middle, and the right columns show results using accurate gradients, noisy gradients with adaptive-distributed noises, and noisy gradients with constant-distributed noises. The variance of noises in the middle line (Figures 3(b) and 3(e)) follows Eq. (12. while the variance of noises in the right line (Figures 3(c) and 3(1) is 0.001. Each line denotes the average of 5 rounds of optimizations.
Gaussian initialization and the zero initialization converge to 10 over the global minimum f.. The loss with the uniform initialization is higher than 10 over the global minimum. Figure 3(c) shows the training with constantly perturbed gradients. 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. Figures 3(d) ((f) show similar training behaviors using the
Adam optimizer. Based on Figures ((1) the Gaussian initialization and the zero initialization outperform the uniform initialization in cases. 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. We remark that the number of measurements T O( (( ). Thus, finite number of measurements with the noise (12 for gradient estimation is enough achieve the performance of accurate gradients, which verifies Theorem 4.2 and Corollary 4.3
6 Conclusions
In this work, we provide a Gaussian initialization strategy for solving the vanishing gradient problem of deep variational quantum algorithms. We prove that the gradient norm of N-qubit quantum circuits
with T. layers could be lower bounded by poly(N, [N, T.)-1, the parameter is sampled independently
from the Gaussian distribution with the variance O ( ). Our results hold for both the local and the
global observable cases, and could be generalized to VQAs employing correlated parameterized gates. Compared to the local case, the bound for the global case depends on the gradient performance at the zero point. Further analysis towards the pero-case-fire free bound could be investigated as future directions. Moreover, we show that the necessary number of measurements, which scales O ((4), suffices for estimating the gradient during the training. We provide numerical experiments on finding the ground energy and state of the Heisenberg model and the LiH molecule, respectively. Experiments show that the proposed Gaussian initialization method outperforms the uniform initialization method with faster convergence rate, and the training using gradients with adaptive noises shows the same convergence compared to the training using noiseless gradients.

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Acknowledgement
This work supported in part by ARC 11-170100117,, IC-19000031, and LE- LE-2001000449
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Checklist
1. For all authors...
(a) Do the main claims made in the abstract and introduction accurately reflect the paper's
contributions and scope? [Yes]
(b) Did you describe the limitations of your work? [Yes]
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(a) Did you include the code, data, and instructions needed to reproduce the main experi- mental results (either in the supplemental material or as URL)? [Yes]
(b) Did you specify all the training details (e.g.. data splits, hyperparameters, how they were chosen)? [Yes]
(c) Did you report error bars (e.g., with respect to the random seed after running experi- ments multiple times)? [Yes] We run each experiment independently for 5 times.
(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)? [No] Each experiment finishes quickly on CPUs with a few hours.
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16


Reviews:
[{"review_id": "_PXEyTGDLDc", "reviewer": "Reviewer_z13w", "summary": "Variational quantum circuits are parametrized models that can be trained to perform mappings using gradient descent methods. In this paper, the authors propose an initialisation strategy to avoid the problem of vanishing gradients that occur when the number of qubits and the circuit depth grow. ", "questions": "- Theoretical analysis: what happens when the training procedure requires many update steps ? in other words, if the convergence is very slow, is the initialisation technique still useful to avoid vanishing gradients ?\n- Experimental analysis: why is the zero initialisation strategy providing competitive results to the Gaussian one ?", "limitations": "No", "rating": 6, "confidence": 4, "soundness": 3, "presentation": 3, "contribution": 3, "strengths": "", "weaknesses": "", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "- Strengths:\n\t- Well-written\n\t- New initialisation: Several initialisation strategies have been studied for classical neural networks but few existing work extend these results to the quantum case. In this work, the authors apply the Gaussian initialisation strategy to variational quantum circuits and study how it may affect the training procedure by providing a theoretical and experimental analysis. \n\t- Theoretical analysis: The authors start by describing the Gaussian initialisation technique and provide theoretical guarantees in different settings. The first setting corresponds to the case when the circuit architecture is made using trainable 1-qubit gates and the output is projected using local observables. These results are then extended to the global observable case and 2-qubit gates. \n\t- Experimental analysis: The authors apply their technique to two quantum machine learning problems where they perform numerical simulations to study experimentally the training behaviour of the parameters. \n\n- Weaknesses:\n\t- The zero initialisation strategy seems to be fine for the performed experiments. ", "was_revised": false, "final_justification": "", "rebuttal": "", "claims_and_evidence": "", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "main_review": "", "summary_of_the_paper": "", "summary_of_the_review": "", "strength_and_weaknesses": "", "limitations_and_societal_impact": "", "review_text": "", "extra_scores": {"ethics_review_area": ""}, "extra_text": {}}, {"review_id": "X_cW57iXlHF", "reviewer": "Reviewer_hWp5", "summary": "The authors demonstrate that over a Gaussian prior of appropriate width, the second moment of the derivatives of a class of variational quantum algorithms (VQAs) is only polynomially small in the problem size $n$ and circuit depth $L$ (for constant problem locality $S$), which is exponentially larger than the traditional barren plateau bounds (taken uniformly over parameters). The authors also give a bound when the locality $S$ grows with the problem size, that lower bounds the expected derivative second moment by (a finite fraction of) the initial squared derivative. The authors then demonstrate numerically that their initialization scheme gives much better optimization performance than uniform initialization in a variety of VQA tasks.", "questions": "* I suggest making the previously discussed limitations of the work more clear (in particular, that these results require staying near $\\vec{\\theta}=\\vec{0}$ to hold).\n* I suggest providing more intuition as to when Theorem 4.2 and Corollary 4.3 give nontrivial bounds.\n* I suggest providing more examples of when the authors' results are expected to hold (e.g. in chemistry where the Hartree--Fock state is a good approximation of the true ground state, etc.).", "limitations": "I previously discussed what limitations of the work I believe should be more explicitly mentioned by the authors; namely, I suggest tempering the claims that Gaussian initialization solves all instances of barren plateaus, and providing more examples or intuition as to situations where one might expect the implicit assumptions on training (i.e. staying near $\\vec{\\theta}=\\vec{0}$) to hold.", "rating": 7, "confidence": 4, "soundness": 3, "presentation": 3, "contribution": 3, "strengths": "", "weaknesses": "", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "The introduced bounds are novel, and though I did not check the proofs in complete detail, the authors' technical methods and proofs seem correct. I also enjoyed that these results are essentially a more rigorous understanding of the intuitive fact that training VQAs while near the identity should be similar to low-depth VQAs, and not experience barren plateaus. I think a couple of weaknesses of the paper, though, are maybe claiming too much from the shown results.\n\nFirst, in their discussion of Theorem 4.2 (and Corollary 4.3), the authors imply that their results are enough to show that Gaussian initialization completely absolves VQAs with global cost functions from barren plateaus. However, these results only lower bound the second moment of the derivative with (a finite fraction of) the initial square of the derivative. This initial derivative can be very small; see, for instance, the global cost function warm-up example in \"Cost function dependent barren plateaus in shallow parametrized quantum circuits\" (Cerezo et al. 2021), where this initial derivative is zero, giving a trivial bound. I would have enjoyed more discussion (or examples) arguing that this bound is typically only polynomially (not superpolynomially) small.\n\nSecond, the authors' results (including now also Theorem 4.1) still rely on assumptions in the training of the model. Namely, once training is far away from the initial $\\vec{\\theta}=\\vec{0}$, it is no longer well-approximated by a Gaussian of polynomially small variance. In fact, for a number of parameters growing superlogarithmically with $n$, roughly the volume of \"allowed\" region where these results are expected to hold is superpolynomially small in the volume of parameter space (polynomially small in diameter). This (that is, superlogarithmically large depth) is the regime previous barren plateau results kick in (i.e. when averaged uniformly over parameters), and I suspect they may be related. I recommend the authors make this limitation of their work more clear.\n\nThese limitations aside, I still find the work a nice, rigorous interpretation of a common approach to circumventing barren plateaus; when one has a good guess for where in parameter space the optimum is (say, near $\\vec{\\theta}=\\vec{0}$), and expect optimization to stay within this region, barren plateaus may be avoided.", "was_revised": false, "final_justification": "", "rebuttal": "", "claims_and_evidence": "", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "main_review": "", "summary_of_the_paper": "", "summary_of_the_review": "", "strength_and_weaknesses": "", "limitations_and_societal_impact": "", "review_text": "", "extra_scores": {"ethics_review_area": ""}, "extra_text": {}}, {"review_id": "EZK8Kr7J79N", "reviewer": "Reviewer_Wz3c", "summary": "The barren plateau phenomenon is a 'vanishing gradient' effect that arises in sufficiently randomly initialized parameterized quantum circuits. Specifically, the norm of the gradient falls exponentially with the number of quantum registers in the circuit. While not a problem for classical neural networks due to efficient gradient estimation procedures, gradients for parameterized quantum circuits are obtained by statistical sampling. Estimating small gradients therefore adds an exponential overhead, eliminating most possible computational advantages.\nIn recent years there has been an effor to handle this problem by suggesting initializations that are not 'fully random' on the space of circuits. This paper takes the following approach: a (practically reasonable) architecture is chosen and the parameters are initialized from a Gaussian distribution. The main technical message is a proof that if the variance of the normal distributions is chosen as $1/L$ where $L$ is the number of layers, the gradient decays polynomially with the number of qubits $n$ and layers $L$.\nModels with the propsoed initialization are evaluated on a variational quantum eigensolver setup to find the ground state of the Heisenberg model and LiH Hamiltonian. On the considered examples, Gaussian initialization appears to outperform the setting where parameters are initialized uniformly.", "questions": "- Can there be a general argument about how the results with Gaussian initialization are maintained during the training? Can there be cases where the gradient starts large but reaches a barren region?\n- As an associated question, can we reason about the bias induced by keeping the initial unitary close to the identity due to the choice of parameter variance?\n- The barren plateau phenomenon was established first for Haar initializations. I did not see a comparison to that setting. Is it similar to the uniformly initialized parameter case.", "limitations": "Yes", "rating": 6, "confidence": 5, "soundness": 3, "presentation": 3, "contribution": 3, "strengths": "", "weaknesses": "", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "The main contribution of the paper is the lower bound on the gradient. This proof is very non-trivial and involves techniques and intermediate results, that I think may be of interest when analyzing properties of Gaussian initializations in quantum circuits in general. I have not checked every statement, but I am overall convinced of correctness of the proofs. On a technical level, I think the contribution of the paper is solid. The experimental results also line up with the conclusions drawn, indicating that the phenomenon described may have applicability beyond the theorem setting.\n\nI have two concerns: firstly, due the Gaussian initialization I am not sure that the proof in the paper suffices to say that the gradients are lower bounded *throughout* training since the distribution has morphed from that at initialization. Note that this was not an issue for the original derivation of 'barren plateau' as Haar distributions are invariant under the shifts induced by training. The second is that the restriction to Gaussians with deviation decaying as 1/L, essentially restricts the initialization to a constant neighborhood of the identity. This assumption seems to put more of a bias on the initialization than most existing approaches, and may create the possibility of adversarial problems where convergence is heavily slowed.\nExperimentally, the advantage over initializing the parameters to zero seems like an artifact of their being a stationary point at identity, and the addition of some noise to perturb the initial state (in the noisy simulations) seems to remove most observed advantage for the proposed scheme.", "was_revised": false, "final_justification": "", "rebuttal": "", "claims_and_evidence": "", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "main_review": "", "summary_of_the_paper": "", "summary_of_the_review": "", "strength_and_weaknesses": "", "limitations_and_societal_impact": "", "review_text": "", "extra_scores": {"ethics_review_area": ""}, "extra_text": {}}, {"review_id": "6NY8puwAtI", "reviewer": "Reviewer_SXHT", "summary": "This paper introduces a new initialization strategy for quantum variational circuits. This gaussian initialization strategy is shown to exponentially increase the upper bound on the gradient, with substantial implications for addressing optimization concerns of medium to large scale quantum machine learning models. Two empirical examples are provided, which show this initialization strategy demonstrates an improvement in performance. \n\n==============================\n\nNote: after author responses score 6 -> 7", "questions": "- The heisenberg gamma squared is said to be 1/160 (line 260), however, in the code gamma is set to 1/(8 * n_layer)**0.5. Since n_layer is 10, gamma = 1/sqrt(80) and gamma squared is thus 1/80. There seems to be a factor of two missing, where does this come from?\n- Do the optimization improvements from this initialization strategy carry over to gradient free optimizers?\n- A variety of gradient based optimizers have been used for QVCs, is there a reason why Adam was chosen over other methods?\n- The zero initialization seems to perform very well on LiH (almost exact overlap with gaussian). Is this just a unique aspect of the problem or is zero initialization comparable in performance on large circuits (when the variance becomes very small)?\n", "limitations": "The authors sufficiently addressed the potential negative societal impact of the work.", "rating": 7, "confidence": 3, "soundness": 2, "presentation": 3, "contribution": 3, "strengths": "", "weaknesses": "", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "Pros:\n- This work provides a very important improvement in gradient bounds. Given the scale of concerns for optimising QVCs, this is a very exciting result. \n- The results for global observables are especially interesting, since previous works have focused on the benefits of local observables to trainability. \n- The paper is generally well written and conveys the point effectively\n- Circuit diagrams are well done and add to the understandability\n- Code is provided in supplementary material, which greatly improves replicability and experimental verification\n- The survey of related work is both useful and extensive\n\nCons:\n- The biggest problem is the empirical results. Although these experiments are just examples and the main result is the theoretical proofs, they don’t add as much as they could. Using shot noise, instead of added measurement noise would improve the realism. It would also be beneficial to add examples with more realistic circuit noise (e.g. depolarizing channels). Additionally, showing the gradient norm in Figure 3 (like in Figure 2) would be beneficial\n- Citations could be condensed, e.g. “quantum simulations [14, 15, 16, 17, 18, 19, 20, 21, 22, 23]” -> “quantum simulations [14-23]”\n- Empirical comparisons to other initialization strategies would be beneficial (e.g. block initialization)\n- The horizontal lines on the graphs don’t aid interpretability\n", "was_revised": false, "final_justification": "", "rebuttal": "", "claims_and_evidence": "", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "main_review": "", "summary_of_the_paper": "", "summary_of_the_review": "", "strength_and_weaknesses": "", "limitations_and_societal_impact": "", "review_text": "", "extra_scores": {"ethics_review_area": ""}, "extra_text": {}}]

Author rebuttal:

Decision: Accept