Avoiding Barren Plateaus with Entanglement

Quantum information science, especially quantum computing, has made significant theoretical and experimental progress in recent years. With the advent of the first generation of quantum computers, we have entered the era of noisy intermediate-scale quantum (NISQ) devices [1, 2, 3]. Despite the potential advantages that these early quantum computers have in tackling complex computational problems, their practical application is hindered by several factors, including the quality of the qubits, error rates in the quantum gates, and algorithmic optimization problems such as the phenomenon of barren plateaus [4, 5, 6, 7, 8, 9, 10]. The concept of “barren plateaus” is important for machine quantum learning and optimization. It refers to regions in the parameter space of quantum neural networks [8] whose gradients are minimal, close to zero. This poses a major challenge for gradient-based optimization methods, such as gradient descent, because the gradient information provides little to no guidance, leading to stagnation in the training process. This phenomenon is characterized by extremely flat gradients in high-dimensional quantum parameter spaces. It has proven to be a critical bottleneck in optimizing quantum algorithms and hinders the further development of quantum computing.

The implementation of quantum algorithms can be hindered by barren plateaus, which is especially problematic for quantum machine learning algorithms and variable quantum algorithms (VQAs) [11, 12, 5, 13, 14]. As quantum systems increase in size, the issue of gradient descent becomes more pronounced during the optimization process, resulting in inefficiencies in local search strategies. Moreover, this challenge is closely related to the limitations of quantum hardware, such as the limited number of qubits [15, 16, 17] and errors in quantum gate operations [18, 19, 20, 21]. These challenges affect the practicality of quantum algorithms and limit the potential applications of quantum computing in various fields.

This study explores the causes and possible solutions of the barren plateaus phenomenon and its impact on optimizing quantum algorithms. We focus on developing new algorithms and strategies to address the issue of gradient vanishing in high-dimensional quantum parameter spaces. This includes improving optimization strategies, exploring effective initialization protocols, and investigating the relationship between gradient vanishing and the properties of quantum hardware. Our ultimate goal is to enhance the efficiency and success rate of quantum algorithm optimization, thus enabling the application of quantum computing in practical scenarios. Additionally, we provide insights into the understanding and control of complex quantum systems.

In related works, we review the essential literature and theoretical foundations related to the phenomenon of barren plateaus in variational quantum algorithms (VQAs) [5, 22, 23, 24, 25]. Barren plateaus are characterized by gradient vanishing issues in parameterized quantum circuits and have become a focal point in current quantum computing research. First, we refer to studies on the relationship between entanglement and learning efficiency in quantum neural networks [22], which suggest that excessive quantum entanglement can lead to reduced learning efficiency. Additionally, regarding the importance of initialization strategies in overcoming barren plateaus, we examine technical notes on new initialization techniques [23] that prevent early-stage gradient disappearance by controlling initial parameter selection. Next, we consider research defining weak barren plateaus (WBPs) based on the entropies of reduced local density matrices [24], providing a new perspective for understanding and quantifying barren plateaus. Lastly, we synthesize studies on the impact of observable selection on trainability in VQAs [25], highlighting the differences in defining cost functions using global versus local observables and their effects on gradient vanishing issues. These pieces of literature and theoretical insights form the foundation of our study, paving the way for us to propose new solutions and theoretical insights.

This paper presents an approach to address the barren plateaus phenomenon in quantum circuits by strategically entangling auxiliary qubits into the circuit. The gradient is then maintained while removing the auxiliary qubit. This approach transforms the circuit into a local unitary $1$-design without altering its original structure or functionality. The initial unitary operation is changed into a form similar to $\alpha I+\beta U$, which shifts the distribution from a unitary $t$-design to a unitary $(t-1)$-design. This modification preserves the anticipated gradient value while decreasing the gradient variance’s reliance on the number of qubits. The methodology also involves a structured optimization process in which auxiliary qubits are gradually removed during training sessions. This assimilates the pending layers into the fixed layers with trained parameters. This preserves the circuit’s core functions, ensures efficient operation, and prevents regression to a unitary $t$-design.

The significance of this study lies in its potential impact on the field of quantum computing. Our method simplifies the complexity of the quantum circuit, enhancing the efficiency of parameter training and offering a new perspective on addressing the barren plateaus challenge. This statement has important implications for developing efficient quantum algorithms and advancing quantum computing towards more practical applications. Future research could investigate the application of our method to various types of quantum circuits and parameter training techniques to optimize and expand its applicability.

Our analysis employs random unitary operations along with $t$-designs. To provide a clear foundation, we will start with an introduction to these key concepts. Let $U(N)$ be the unitary group of degree $N$, and denote the Haar measure on $U(N)$ by $\mathbf{H}$. A Haar random unitary $\mu$ is $U(N)$-valued random variable distributed according to the Haar measure $\mu\sim\mathbf{H}$ [26].

Let $\nu$ be a probability measure on the unitary $U(N)$. A random unitary $\mu$ drawn from $\nu$ is called an $\epsilon$-approximate unitary $t$-designs if it satisfies the condition $\|\mathcal{G}_{\mu\sim\nu}^{(t)}-\mathcal{G}_{\mu\sim\mathbf{H}}^{(t)}\|_{\diamond}\leq\epsilon$, where $\|\bullet\|_{\diamond}$ represents the diamond norm, $\mathcal{G}_{\mu\sim\nu}^{(t)}$ and $\mathcal{G}_{\mu\sim\mathbf{H}}^{(t)}$ represent some specific mathematical expressions or operations involving $\mu$ [26].

Let $X$ be a finite subset of $U(N)$, the group of $N\times N$ unitary matrices. We consider the expression

where $d\mu$ denotes the unit Haar measure on $U(N)$, satisfying $\int_{U(N)}d\mu=1$. The cardinality of $X$, denoted by $\absolutevalue{X}$, refers to the number of subsets contained within $X$.

The subset $X\subseteq U(N)$ is a $t$-design if it satisfies the condition [27]:

where $D$ is defined in appendix A. Equation (2) demonstrates that unitary $t$-design does not hold universally. It necessitates the fulfillment of a specific number of conditions for its validity.

In Random Parameterized Quantum Circuits (RPQCs), comprising $n$ qubits and $L$ layers as demonstrated in Fig. 1, the unitary operator $U(\boldsymbol{\theta})$ is defined as

where $l\in[1,L]$ is for each layer and $i\in[1,n]$ for each qubit. $U_{l}(\boldsymbol{\theta}_{l})$ and $W_{l}$ are unitary operators. Here, $V_{i}$ is a Pauli operator, and $W_{l}$ is a fixed unitary operator that does not depend on the angle $\theta_{l,i}$.

Consider a quantum circuit where the initial state is prepared in $\ket{\mathbf{init}}$. The objective function $E(\boldsymbol{\theta})$ is defined as the expectation value of a Hermitian operator $H$, which is provided externally. This expectation value is obtained by applying a unitary operation $U(\boldsymbol{\theta})$ to the initial state, and then performing a measurement corresponding to the operator $H$. The unitary operation is parameterized by a set of parameters $\boldsymbol{\theta}$. The objective function is given by:

where $U(\boldsymbol{\theta})^{\dagger}$ is the Hermitian conjugate of $U(\boldsymbol{\theta})$. To calculate the gradient of the objective function, we define it as follows: let $\partial_{k}E$ represent the $k$-th partial derivative of the function $E(\boldsymbol{\theta})$ concerning the parameter $\theta_{k}$, which is denoted by $\partial_{k}E\equiv\frac{\partial E(\boldsymbol{\theta})}{\partial\theta_{k}}$. It is important to note that the expected average value of this gradient is determined as

Additionally, the variance of the gradient is given by:

where $\rho=\ket{\mathbf{init}}\bra{\mathbf{init}}$.

As the number of qubits increases, the system’s dimensionality grows exponentially, where $N=2^{n}$ and $n$ represent the number of qubits. In this scenario, the variance gradient decreases with the increase in $N$ and ultimately vanishes. This decrease in variance restricts the expressive potential of random parameterized quantum circuits [4].

As discussed earlier, VQAs are plagued by vanishing gradients, commonly known as the barren plateaus problem. The barren plateaus arise due to the circuit’s unitary $2$-design characteristics. Previous studies [24, 25, 23] have demonstrated that adopting a local unitary $1$-design can mitigate this issue. This paper proposes a method to alleviate the barren plateaus problem by entanglement. It rests on modifying quantum circuits with additional qubits to transform them into a local unitary $1$-design. Our study aims to address the barren plateaus phenomenon in quantum circuits by adding structure to the existing circuit framework. To achieve this, we incorporate auxiliary qubits into the system. These additional elements ensure that the original structure and function of the existing circuit remain unaltered. Instead, their role is primarily to mix additional information, which offers another way to address this quantum computing obstacle without necessitating structural modifications to the circuit.

In order to demonstrate the effectiveness of this proposed method, we perform a numerical experiment. The purpose of this experiment is to compare the performance of three quantum circuit models by using Pennylane [28]. The evaluated models are standard RPQCs, which serve as the unitary $2$-design baseline model, a unitary $1$-design structure incorporating auxiliary qubits, and the proposed structure that optimizes by eliminating these auxiliary qubits. The experimental design involves sampling 100 quantum circuits that are randomly generated according to each model’s configuration. For each circuit, we compute the variance of its gradient for the target operator $H=Z_{1}\otimes Z_{2}$, where $H$ is a single Pauli ZZ operator acting on the first and second qubits [25]. This comparison aims to reveal the impact of different design choices on the performance of quantum circuits, particularly in terms of gradient variance.

Firstly, we are concerned about the impact of the number of qubits on the variance of $\partial_{k}E$. We configure all structures to have 100 layers, with the number of qubits ranging from 2 to 16. The experimental results demonstrate that the variance of quantum circuits with a unitary $2$-design structure decreases exponentially as the number of qubits increases. The slope of the curve is approximately -0.58, as shown in Fig. 4. This trend suggests that as the number of qubits increases, the gradient of the entire quantum circuit becomes easier to zero out, resulting in a significant decline in performance. The variance of quantum circuits with a unitary $1$-design structure also decreases exponentially with the number of qubits. However, the slope of this decrease is only about half that of the unitary $2$-design structure, which is consistent with the Eq. (10).

Subsequently, we evaluate the performance of the gradient to $\theta_{1,1}$, the first parameter in the quantum circuit in Fig. 3, across varying configurations of qubits and layers within the quantum circuit architecture. Initially, the structure adheres to a unitary $2$-design structure. This structure transforms a unitary $1$-design upon entanglement with auxiliary qubits. After setting the pending layers to 20 and following the removal of these auxiliary qubits, the structure transitions into the proposed structure in this study. Our experimental assessment encompasses a range of qubits from 2 to 16 and layers from 5 to 500, providing a comprehensive overview of the impact of these variables on the gradient of the quantum circuit’s performance.

Consistent with previous findings in Fig. 4, the influence of the number of qubits on the variance of a single parameter demonstrates a similar pattern. The variance of quantum circuits with a unitary $2$-design structure exhibits an exponential decline as the number of qubits increases, with a slope of approximately -0.69, as illustrated in Fig. 5. This trend highlights that as the number of qubits increases, the quantum circuit’s gradient becomes more susceptible to nullification on a local variable scale. This indicates the barren plateaus phenomenon, which affects the global landscape and local variational parameters, resulting in a significant performance deterioration. Conversely, quantum circuits with a unitary $1$-design structure display a variance that shows no significant correlation with the number of qubits, with the slope nearing zero. The proposed method inherits the characteristics of the unitary $1$-design and maintains consistent variance as the number of qubits increases.

Regarding the impact of the number of layers on variance, Fig. 6 shows that the variance decreases as the number of layers increases for quantum circuits with a unitary $2$-design structure. Additionally, the decrease becomes more pronounced as the number of qubits increases. There is a smaller range of variance fluctuation for quantum circuits with a unitary $1$-design structure, and the number of layers is not significantly affected. However, it should be noted that under conditions of fewer qubits or shallower layers, the unitary $2$-design structure tends to have an advantage. To utilize both advantages, the proposed structure follows a unitary $2$-design when the pending layers are small. As the layers deepen, the structure transitions to a unitary $1$-design and is more likely to achieve an effective gradient.

Then we investigate the gradient distribution $\partial\theta_{1,1}E$ within a quantum circuit composed of 10 qubits and 500 layers in in Fig. 7. The distribution profile for the proposed structure and a unitary $2$-design are assessed for their respective potentials in gradient optimization. The unitary $2$-design structure exhibits a steep, narrow distribution of gradient values, closely centralized near zero and conforming to a normal distribution expressed as $\mathcal{N}(0,2.738\times 10^{-4})$, where $\mathcal{N}$ denotes the normal distribution with a mean of $0$ and a variance of $2.738\times 10^{-4}$. This highlights the barren plateaus phenomenon, where optimization becomes challenging due to vanishing gradients. In contrast, the proposed structure demonstrates a substantially wider distribution of gradient values, conforming to a normal distribution expressed as $\mathcal{N}(0,1.655\times 10^{-2})$, indicating a decreased propensity for gradients to converge to zero. This increases the likelihood of locating non-trivial gradient values conducive to effective optimization. The proposed architecture outperforms the other structure by avoiding barren plateaus facilitating more robust quantum circuit training and optimization.

Finally, we conduct four separate experiments in Fig. 8 to evaluate the effectiveness of the unitary $2$-design and proposed structure methodologies in achieving fixed target values of -0.1, -0.05, 0.05, and 0.1 in expectation and cost function. Each experiment is configured with a 10-qubit, 100-layer structure. The proposed structure uses a pending layer setting of 1 and is trained for 10 epochs per pending structure, totaling 1000 epochs. The cost function is $(\mathrm{expectation}-\mathrm{target})^{2}$ and the optimizer is pennylane.AdamOptimizer() [28].

Over the 1000 observed epochs for each experiment in Fig. 8, the proposed structure’s expectations tend to cluster closely around the target line, unlike the unitary $2$-design, whose data points are notably more dispersed. The proposed structure’s convergence towards the target value indicates superior performance, highlighting its effectiveness over the unitary $2$-design in achieving the desired outcomes.

Calculating the probability distribution of expected values across 300 to 1000 epochs allows us to determine the variability of the predicted value. A narrower peak indicates a more consistent approach to the target, while a wider distribution indicates a greater variance. The central peak of each distribution corresponds to the most frequently occurring expected value within 1000 periods. Ideally, these peaks should be the targets. Compared to the unitary $2$-design structure, the proposed structure’s distribution around the target value is sharper and has larger peaks. This indicates that the proposed structure is closer to the target’s expected value.

The cost function is a measure used to evaluate the model’s performance, with lower values indicating better performance. The proposed structure is more likely to obtain effective gradients during training, allowing its cost function maintain at a level less than $10^{-4}$. In contrast, the unitary $2$-design structure cannot achieve this, preventing the cost function from converging to a smaller value.

This paper addressed a central issue in quantum computing – the barren plateaus phenomenon. This phenomenon presents a significant challenge in quantum machine learning and optimization algorithms. Large-scale quantum circuits are characterized by a vanishing gradient variance, which must be addressed. Our approach successfully transforms the quantum circuit from a unitary $2$-design to a unitary $1$-design without changing the original structure, marking a significant stride in gradient optimization. We introduced auxiliary control qubits that can be eliminated to achieve this transformation.

Our experiments systematically demonstrated the advantages of the proposed structure over global unitary $2$-design quantum circuits. The entanglement and elimination of auxiliary qubits facilitate the gradual transition from a unitary $2$-design to a unitary $1$-design. This approach effectively mitigates the challenges posed by barren plateaus. The experimental data consistently indicated that the proposed structure maintains a stable gradient variance, avoiding the exponential decline associated with increased qubits. This is a notable issue in global unitary $2$-design structures. Moreover, the proposed structure showcases a diminished sensitivity to the number of layers, maintaining effective gradients more reliably than the unitary $2$-design, particularly in circuits with more layers.

The analysis of gradient distributions emphasizes the superiority of the proposed structure. The unitary $2$-design is prone to narrow gradient distributions, which centralize near zero and indicate barren plateaus. In contrast, the proposed structure exhibits a broader gradient distribution, increasing the probability of achieving non-trivial gradients and successful optimization.

In achieving fixed target values for the expectation and cost function, the proposed structure demonstrates a pronounced ability to align with target values closely. It outperforms the unitary $2$-design in both consistency and cost function minimization. This is evidenced by the sharper and taller peaks in the probability distribution of expected values and the ability to maintain the cost function consistently low throughout the training process.

Our method has demonstrated significant efficacy in random parameterized quantum circuits (RPQCs). We have experimentally validated our approach to RPQCs, demonstrating its theoretical feasibility and practical applicability. Importantly, our method achieves these benefits without introducing additional variables or significantly altering the existing quantum circuit structure. This substantially reduces implementation complexity and cost in practical applications.

The methodology and results of this study may pave new pathways in the field of quantum computing. Our study will be extended to other types of quantum circuits, such as quantum neural networks (QNNs). Applying the same method in these circuits is expected to yield positive results in gradient optimization. This strategy effectively addresses the barren plateaus problem and opens up new opportunities for designing and optimizing future quantum algorithms.

In summary, this study proposes an innovative and practical method that effectively solves an essential issue in quantum computing. Our study aims to enhance the comprehension and optimization of gradient behaviors in large-scale quantum circuits. It provides valuable guidance for future gradient optimization strategies in quantum machine learning and other advanced quantum computing applications.

The code used for this work is released in [29].