Quantum Circuit Architecture Search on a Superconducting Processor

Abstract

Variational quantum algorithms (VQAs) have shown strong evidence to gain provable computational advantages in diverse fields such as finance, machine learning, and chemistry. However, the heuristic ansatz exploited in modern VQAs is incapable of balancing the trade-off between expressivity and trainability, which may lead to degraded performance when executed on noisy intermediate-scale quantum (NISQ) machines. To address this issue, here, we demonstrate the first proof-of-principle experiment of applying an efficient automatic ansatz design technique, i.e., quantum architecture search (QAS), to enhance VQAs on an 8-qubit superconducting quantum processor. In particular, we apply QAS to tailor the hardware-efficient ansatz toward classification tasks. Compared with heuristic ansätze, the ansatz designed by QAS improves the test accuracy from $31\%$ to $98\%$. We further explain this superior performance by visualizing the loss landscape and analyzing effective parameters of all ansätze. Our work provides concrete guidance for developing variable ansätze to tackle various large-scale quantum learning problems with advantages.

1. Introduction

The successful exhibition of random quantum circuits sampling and Boson sampling over fifty qubits [1,2,3,4] evidences the potential of using current quantum hardware to address classically challenging problems. A leading strategy towards this goal is variational quantum algorithms (VQAs) [5,6], which leverage classical optimizers to train an ansatz that can be implemented on noisy intermediate-scale quantum (NISQ) devices [7]. In the past years, a growing number of theoretical studies have shown the computational superiority of VQAs in the regime of machine learning [8,9,10,11,12,13,14,15,16], quantum many-body physics [17,18,19,20], and quantum information processing [21,22,23]. On par with the achievements, recent studies have recognized some flaws of current VQAs through the lens of the trade-off between expressivity and learning performance [14,24]. That is, an ansatz with very high expressivity may encounter the barren plateau issues [25,26,27,28], while an ansatz with low expressivity could fail to fit the optimal solution [29]. With this regard, designing a problem-specific and hardware-oriented ansatz is of great importance to guarantee the good learning performance of VQAs and the precondition of pursuing quantum advantages.

Pioneered experimental explorations have validated the crucial role of ansatz when applying VQAs to accomplish tasks in different fields such as machine learning [30,31,32,33], quantum chemistry [19,34,35,36,37,38], and combinatorial optimization [39,40,41,42]. On the one side, envisioned by the no-free-lunch theorem [43,44], there does not exist a universal ansatz that can solve all learning tasks with optimal performance. To this end, myriad handcraft ansätze have been designed to address different learning problems [45,46,47]. For instance, the unitary coupled cluster ansatz and its variants attain superior performance in the task of estimating molecular energies [48,49,50,51]. Besides devising the problem-specific ansätze, another indispensable factor to enhance the performance of VQAs is the compatibility between the exploited ansatz and the employed quantum hardware, especially in the NISQ scenario [39]. Concretely, when the circuit layout of ansatz mismatches with the qubit connectivity, additional quantum resources, e.g., SWAP gates, are essential to complete the compilation. Nevertheless, these extra quantum resources may inhibit the performance of VQAs because of the limited coherence time and inevitable gate noise of NISQ machines. Considering that there are countless learning problems and diverse architectures of quantum devices [52,53,54], it is impractical to manually design problem-specific and hardware-oriented ansätze.

To enhance the capability of VQAs, initial studies have been carried out to seek feasible strategies of automatically designing a problem-specific and hardware-oriented ansatz with both good trainability and sufficient expressivity. Conceptually, the corresponding proposals exploit random search [55], evolutionary algorithms [56,57], deep learning techniques [58,59,60,61,62,63,64,65], and adaptive strategies [66,67,68] to tailor a hardware-efficient ansatz [19], i.e., inserting or removing gates, to decrease the cost function. In contrast with conventional VQAs that only adjust parameters, optimizing both parameters and circuit layouts enables the enhanced learning performance of VQAs. Meanwhile, the automatic nature endows the power of these approaches to address broad learning problems. Despite the prospects, little is known about the effectiveness of these approaches executed on real quantum devices.

In this study, we demonstrate the first proof-of-principle experiment of applying an efficient automatic ansatz design technique, i.e., quantum architecture search (QAS) scheme [69], to enhance VQAs on an 8-qubit superconducting quantum processor. In particular, we focus on data classification tasks and utilize QAS to pursue a better classification accuracy. To our knowledge, this is the first experimental study of multi-class learning. Moreover, to understand the noise-resilient property of QAS, we fabricate a controllable dephasing noisy channel and integrate it into our quantum processor. Assisted by this technique, we experimentally demonstrate that the ansatz designed by QAS is compatible with the topology of the employed quantum hardware and attains much better performance than the hardware-efficient ansatz [19] when the system noise becomes large. Experimental results indicate that under a certain level of noise, the ansatz designed by QAS achieves the highest test accuracy ($95.6\%$), while other heuristic ansätze only reach $90\%$ accuracy. Additional analyses of loss landscape further explain the advantage of the QAS-based ansatz in both optimization and effective parameter space. These gains in performance suggest the significance of developing QAS and other automatic ansatz design techniques to enhance the learning performance of VQAs.

2. Materials and Methods

2.1. The Mechanism of QAS

The underlying principle of QAS is optimizing the quantum circuit architecture and the trainable parameters simultaneously to minimize an objective function. For elucidating, in the following, we elaborate on how to apply QAS to tailor the hardware-efficient ansatz (HEA). Mathematically, an N-qubit HEA $U\left(\mathit{\theta }\right)={\prod }_{l=1}^L{U}_l\left(\mathit{\theta }\right)\in SU\left(2^N\right)$ yields a multi-layer structure, where the circuit layout of all blocks is identical, the l-th block $U_l\left(\mathit{\theta }\right)$ consists of a sequence of parameterized single-qubit and two-qubits gates, and L denotes the block number. Note that our method can be generalized to prune other ansätze, such as the unitary coupled cluster ansatz [49] and the quantum approximate optimization ansatz [70].

QAS is composed of four steps to tailor HEA and output a problem-dependent and hardware-oriented ansatz as shown in Figure 1. The first step is specifying the ansätze pool $\mathcal{S}$, collecting all candidate ansätze. Suppose that $U_l\left(\mathit{\theta }\right)$ for $\forall l\in \left[L\right]$ can be formed by three types of parameterized single-qubit gates, i.e., rotational gates along three axes, and one type of two-qubits gates, i.e., CNOT gates. When the layout of different blocks can be varied by replacing single-qubit gates or removing two-qubit gates, the ansätze pool $\mathcal{S}$ includes in total $O\left({(3^N+2^N)}^L\right)$ ansatz. Denote the input data as $\mathcal{D}$ and an objective function as $\mathcal{L}$. The goal of QAS is finding the best candidate ansatz $\mathit{a}\in \mathcal{S}$ and its corresponding optimal parameters ${\mathit{\theta }}_{\mathit{a}}^{*}$, i.e.,

$$({\mathit{\theta }}_{\mathit{a}}^{*},{\mathit{a}}^{*})=arg\underset{{\mathit{\theta }}_{\mathit{a}}\in \mathcal{C},\mathit{a}\in \mathcal{S}}{min}\mathcal{L}({\mathit{\theta }}_{\mathit{a}},\mathit{a},\mathcal{D},{\mathcal{E}}_{\mathit{a}}),$$

(1)

where the quantum channel ${\mathcal{E}}_{\mathit{a}}$ simulates the quantum system noise induced by $\mathit{a}$.

The second step is optimizing Equation (1) with, in total, T iterations. As discussed in our technical companion paper [69], seeking the optimal solution $({\mathit{\theta }}_{\mathit{a}}^{*},{\mathit{a}}^{*})$ is computationally hard since the optimization of $\mathit{a}$ is discrete and the size of $\mathcal{S}$ and $\mathcal{C}$ exponentially scales with respect to N and L. To conquer this difficulty, QAS exploits the supernet and weight-sharing strategy to ensure a good estimation of $({\mathit{\theta }}_{\mathit{a}}^{*},{\mathit{a}}^{*})$ within a reasonable computational cost. Concisely, the weight-sharing strategy correlates parameters among different ansätze in $\mathcal{S}$ to reduce the parameter space $\mathcal{C}$. As for supernet, it plays two significant roles, i.e., configuring the ansätze pool $\mathcal{S}$ and parameterizing ansatz $\mathit{a}\in \mathcal{S}$ via the specified weight-sharing strategy. In doing so, at each iteration t, QAS randomly samples an ansatz ${\mathit{a}}^{\left(t\right)}\in \mathcal{S}$ and updates its parameters, with ${\mathit{\theta }}_{\mathit{a}}^{(t+1)}={\mathit{\theta }}_{\mathit{a}}^{\left(t\right)}-\eta \nabla \mathcal{L}({\mathit{\theta }}_{\mathit{a}}^{\left(t\right)},{\mathit{a}}^{\left(t\right)},\mathcal{D},{\mathcal{E}}_{{\mathit{a}}^{\left(t\right)}})$, and $\eta$ being the learning rate. Due to the weight-sharing strategy, the parameters of the unsampled ansätze are also updated.

The last two steps are ranking and fine-tuning. Specifically, once the training is completed, QAS ranks a portion of the trained ansätze and chooses the one with the best performance. The ranking strategies are diverse, including random searching and evolutionary searching. Finally, QAS utilizes the selected ansatz to fine-tune the optimized parameters with a few iterations. Refer to Ref. [69] for the omitted technical details of QAS.

2.2. Experimental Implementation

We implement QAS on a quantum superconducting processor to accomplish the classification tasks for the Iris dataset. Namely, the Iris dataset $\mathcal{D}={\{{\mathit{x}}_i,y_i\}}_{i=1}^{150}$ consists of three categories of flowers (i.e., $y_i\in \{0,1,2\}$), and each category includes 50 examples characterized by 4 features (i.e., ${\mathit{x}}_i\in {\mathbb{R}}^4$). In our experiments, we split the Iris dataset into three parts, i.e., the training dataset ${\mathcal{D}}_T=\{\mathit{x},y\}$, the validating dataset ${\mathcal{D}}_V$, and the test dataset ${\mathcal{D}}_E$ with $\mathcal{D}={\mathcal{D}}_T\cup {\mathcal{D}}_V\cup {\mathcal{D}}_E$. The functionality of ${\mathcal{D}}_T$, ${\mathcal{D}}_V$, and ${\mathcal{D}}_E$ is estimating the optimal classifier, preventing the classifier from being over-fitted, and evaluating the generalization property of the trained classifier, respectively.

Our experiments are carried out on a quantum processor, including 8 Xmon superconducting qubits with a one-dimensional chain structure. As shown in Figure 2b, the employed quantum device is fabricated by sputtering an aluminum thin film onto a sapphire substrate. The single-qubit rotation gate ${\mathrm{R}}_X$ (${\mathrm{R}}_Y$) along X-axis (Y-axis) is implemented with a microwave pulse, and the Z rotation gate ${\mathrm{R}}_{\mathrm{z}}$ is realized by virtual Z gate [71]. The construction of the CZ gate is completed by applying the avoided level crossing between the high level states $|11\rangle$ and $|02\rangle$ or $|11\rangle$ and $|20\rangle$. The calibrated readout matrix is shown in Figure 2d and the device parameters is summarized in

Table A1. Device parameters. ${\omega }_i$ and ${\alpha }_i$ represent the qubit frequency and qubit anharmonicity, respectively. $T_1$ and $T_2^{★}$ are the longitudinal and transverse relaxation times, respectively. ${\overline{F}}_s$ and $T_s$ are the average fidelity and length of the single-qubit gates. $g_i$ is the coupling strength between nearby qubits, and $J_{z,i}$ is the effective ZZ coupling strength. ${\overline{F}}_{cz}$ are the fidelity of the CZ gates calibrated by quantum process tomography, and $T_{cz}$ are the length of the CZ gates.

Parameter	Q1		Q2		Q3		Q4		Q5		Q6		Q7		Q8
${\omega }_i/2\pi$ (GHz)	$5.202$		$4.573$		$5.146$		$4.527$		$5.099$		$4.47$		$5.118$		$4.543$
${\alpha }_i/2\pi$ (GHz)	$-0.240$		$-0.240$		$-0.239$		$-0.240$		$-0.239$		$-0.239$		$-0.239$		$-0.242$
$T_1$ ($\mathsf{\mu }\mathrm{s}$)	$10.3$		$13.6$		$14.7$		$16.7$		$14.9$		$13.2$		$12.7$		$8.0$
$T_2^{★}$ ($\mathsf{\mu }\mathrm{s}$)	$11.7$		$2.0$		$15.2$		$1.9$		$15.9$		$1.8$		$12.6$		$1.6$
${\overline{F}}_s$	$99.67\%$		$98.96\%$		$99.76\%$		$99.55\%$		$99.05\%$		$99.69\%$		$98.51\%$		$99.09\%$
$T_s$ ($\mathrm{n}\mathrm{s}$)	37		37		37		35		35		37		37		35
$g_i/2\pi \left(\mathrm{MHz}\right)$		$19.7$		$19.6$		$19.1$		$19.1$		$19.2$		$19.3$		$19.6$
$J_{z,i}/2\pi \left(\mathrm{MHz}\right)$		$0.675$		$0.8$		$0.7$		$0.78$		$0.626$		$0.655$		$0.819$
${\overline{F}}_{cz}$		$97.23\%$		$96.11\%$		$97.48\%$		$93.54\%$		$93.72\%$		$94.95\%$		$95.54\%$
$T_{cz}$ ($\mathrm{n}\mathrm{s}$)		$18.5$		$21.5$		$23$		$19.5$		$23.5$		$21.5$		$18.5$

of Appendix A.

We fabricate the controllable dephasing noise as a measurable disturbance to the quantum evolution. The operators for the noise channel can be written as $E_0=\sqrt{1-\alpha p}[1,0;0,1]$ and $E_1=\sqrt{\alpha p}[1,0;0,-1]$. $\alpha$ is a constant, and the value of p can be tuned in our experiment by changing the average number of the coherent photons on the readout cavity’s steady state. The intensity of coherent photons is represented by the amplitude p of the curve shown on the AWGs.

The experimental implementation of the quantum classifiers is as follows. As illustrated in Figure 2a, the gate encoding method is exploited to load classical data into quantum states. The encoding circuit yields $U_E\left(\mathit{x}\right)={\otimes }_{j=1}^4{\mathrm{R}}_Y\left({\mathit{x}}_{i,j}\right)$. To evaluate the effectiveness of QAS, three types of ansätze $U\left(\mathit{\theta }\right)$ are used to construct the quantum classifier. The first two types are heuristic ansätze, which are the hardware-agnostic ansatz (HAA) and hardware-efficient ansatz (HEA). As depicted in Figure 2a, HAA $U_{\mathrm{HAA}}\left(\mathit{\theta }\right)$ is designed for a general paradigm and ignores the topology of specific quantum hardware platforms; HEA $U_{\mathrm{HEA}}\left(\mathit{\theta }\right)$ adapts to the quantum hardware constraints, where all inefficient two-qubit operators that connect two physically nonadjacent qubits are forbidden. The third type of ansatz refers to the output of QAS, denoted as $U_{\mathrm{QAS}}\left(\mathit{\theta }\right)$. The mean square error between the prediction and real labels is employed as the objective function for all quantum classifiers. The noise rate of the dephasing channel p is set as 0, $0.01$, and $0.015$. We benchmark the test accuracy of these three ansätze HAA, HEA, and QAS, and explore whether QAS attains the highest test accuracy. Refer to Appendix B for more implementation details.

3. Results

To comprehend the importance of the compatibility between quantum hardware and ansätze, we first examine the learning performance of the quantum classifiers with HAA and HEA under different noise rates. The achieved experimental results are demonstrated in Figure 3a. In particular, in the measure of training loss (i.e., the lower the better), the quantum classifier with the HEA significantly outperforms HAA for all noise settings. At the 10-th epoch, the training loss of the quantum classifier with HAA and HEA is $0.06$ and $0.017$ ($0.049$ and $0.012$; $0.049$ and $0.012$) when $p=0.015$ ($p=0$; $p=0.01$), respectively. In addition, the optimization of the quantum classifier with HAA seems to be divergent when $p=0.015$. We further evaluate the test accuracy to compare their learning performance. As shown in Figure 3b, there exists a manifest gap between the two ansätze, highlighted by the blue and yellow colors. For all noise settings, the test accuracy corresponding to HAA is only 31.1%, whereas the test accuracy corresponding to HEA is at least 95.6%. These observations signify the significance of reconciling the topology between the employed quantum hardware and ansatz as the key motivation of this study. Specifically, the performance gap between HAA and HEA drives us to introduce QAS, which focuses on searching for the optimal quantum circuit architecture rather than iteratively updating gate parameters within a single, fixed architecture.

We next experiment on QAS to quantify how it is a problem-specific and hardware-oriented design to enhance the learning performance of quantum classifiers. Concretely, as shown in Figure 3b, for all noise settings, the quantum classifier with the ansatz searched by QAS attains the best test accuracy compared to those of HAA and HEA. That is, when $p=0$ ($p=0.01$ and $p=0.015$), the test accuracy achieved by QAS is $97.8\%$ ($97.8\%$ and $95.6\%$), which is higher than HEA with $96.7\%$ ($92.2\%$ and $90.0\%$). Notably, although the test accuracy is slightly decreased for the increased system noise, the strength of QAS becomes evident over the other two ansätze. In other words, QAS shows the advantages of simultaneously alleviating the effect of quantum noise and searching the optimal ansatz to achieve high accuracy. The superior performance validates the effectiveness of QAS in classification tasks.

We last investigate the potential factors of ensuring the good performance of QAS from two perspectives, i.e., the circuit architecture and the corresponding loss landscape. The searched ansätze under three noise settings, HAA, and HEA are pictured at the top of Figure 4. Compared with HEA and HAA, QAS reduces the number of CZ gates with respect to the increased level of noise. When $p=0.015$, QAS chooses the ansatz containing only one CZ gate. This behavior indicates that QAS can adaptively control the number of quantum gates to balance the expressivity and learning performance. We plot the loss landscape of HAA, HEA, and the ansatz searched by QAS in the middle row of Figure 4. To visualize the high-dimension loss landscape in a 2D plane, the dimension reduction technique, i.e., principal component analysis (PCA) [72] is applied to compress the parameter trajectory corresponding to each optimization step. After dimension reduction, we choose the obtained first two principal components that explain most of the variance as the landscape spanning vector. Refer to [73] and Appendix C for details. For HAA and HEA, the objective function is governed by both the 0-th component ($98.75\%$ of variance for HAA, $98.55\%$ of variance for HEA) and 1-th component ($1.24\%$ of variance for HAA, $1.45\%$ of variance for HEA). By contrast, for the ansätze searched by QAS, their loss landscapes totally depend on the 0-th component. Furthermore, the optimization path for QAS is exactly linear, while the optimization of HAA and HAA experiences a nonlinear curve. This difference reveals that QAS enables a more efficient optimization trajectory. As indicated in the bottom row of Figure 4, there is a major parameter that contributes the most to the 0-th component in the three ansätze searched by QAS, while HAA and HEA have to consider multiple parameters to determine the 0-th component. This phenomenon reflects that ansätze searched by QAS are prone to having a smaller effective parameter space, which leads to less noise accumulation and further stronger noise robustness. These observations can be treated as empirical evidence to explain the superiority of QAS.

4. Discussion

Our experimental results provide the following insights. First, we experimentally verify the feasibility of applying automatically designing a problem-specific and hardware-oriented ansatz to improve the power of quantum classifiers. Second, the analysis related to the loss landscape and the circuit architectures exhibits the potential of applying QAS and other variable ansatz construction techniques to compensate for the caveats incurred by executing variational quantum algorithms on NISQ machines.

Besides classification tasks, it is crucial to benchmark QAS and its variants towards other learning problems in quantum chemistry and quantum many-body physics. In these two areas, the employed ansatz is generally Hamiltonian dependent [48,49,74]. As a result, the way of constructing the ansatz pool should be carefully conceived. In addition, another important research diction is understanding the capabilities of QAS for large-scale problems. How to find the near-optimal ansatz among the exponential candidates is a challenging issue.

We note that although QAS can reconcile the imperfection of quantum systems, a central law to enhance the performance of variational quantum algorithms is promoting the quality of quantum processors. For this purpose, a promising area for future exploration is to delve into carrying out QAS and its variants on more types of quantum machines to accomplish more real-world generation tasks with potential advantages.