Neural predictor based quantum architecture search

NAS [35–38] is a recently emerging and rapidly developing field. It aims at automatically searching for optimal neural networks for some given tasks without relying on handcrafted design and expert guidance. An effective NAS workflow is composed of several ingredients, in which the sampling strategy and evaluation protocol are among the most important. In short, sampling strategy refers to a customized recommendation of candidate neural networks, whose fitness for a specific task should be evaluated by training the network and checking accuracy on some validation or test sets. Since the search space is exponentially large, we need an efficient method to traverse and sample candidate architectures. The sampling strategies that have been previously attempted include random search [70], local and greedy search [71–73], evolutionary and genetic type search [74–78], RL based search [79–82], Bayesian optimization search [59, 61, 62] and so on. While sampling is essential for NAS, when it comes to the computational cost, the bottleneck lies in the evaluation part as it is time consuming to train individual networks (from scratch) on a large dataset. Indeed, early NAS works [79] using such plain evaluation methods with RL or evolutionary engines often took thousands of GPU hours in the training process.

There are two approaches to lower the cost of individual evaluations. The first one is weight sharing. In this setup, all candidate networks are organized within a super network, which can be trained either by training the entire network or by training sampled subnetworks in each epoch. After training the super network, one can evaluate each candidate subnetwork by regarding it as a child of the super network, and the weights of such candidates are inherited from the super network directly without further tuning. Therefore, the evaluation on the candidate network is as simple as a forward pass for inference. Such parameter sharing setup is inspired by one-shot NAS [52–54] and has also become popular for DARTS [55, 83–85]. Parameter sharing is not perfect, though. As there is actually no theoretical guarantee on the accuracy correlation between subnetworks with inherited weights and optimal weights from individual training.

There is a second strategy to improve the evaluation methodology which our work focuses on. Instead of training a candidate network and evaluating its performance on some validation dataset, one directly builds ML models to predict the network performance based on the network structure alone, i.e. without specifying trainable weights. If the prediction accuracy manifests a non-trivial correlation with the ground truth, then such predictor based evaluation method may greatly ameliorate NAS efficiency. We call works along this line predictor based NAS, in which the so-called 'predictor' is a regression model [57–69]. Some works take a further step by constructing variational autoencoder (VAE) [86] for neural architectures and trains a predictor with input from the latent space of VAE [87–93]. Such predictor can also be transferred to infer other metrics of the network such as latency or FLOPs [59, 68]. It is worth noting that predictor based NAS can be combined with weight sharing tricks, where the predictor is trained with fitness labels obtained from one-shot setup [62, 65, 87, 88].

The paradigm of variational quantum algorithms [1, 2] is widely speculated to deliver practical quantum advantages with NISQ devices [3] in the near term. The hybrid quantum–classical implementation of VQAs bears many similarities to deep learning in classical computations. Various VQAs have been designed for problems in a vast array of fields. Some typical examples include determining the ground and low excited states of a quantum Hamiltonian [4, 6–8, 94–97], simulating quantum dynamics [98–107], preparing Gibbs thermal states [108–110], supervised and unsupervised ML [15–20, 20–27], compiling quantum circuits [111–113], solving classical combinatorial optimizations [10–14], factoring integers [114–116], and solving linear or differential equations [117–121], etc. A VQA requires a fixed PQC ansatz at the beginning stage. Such fixed ansatz is usually designed to accommodate limitations of current hardware [28] or are inspired by well-established theoretical techniques, such as quantum annealing, unitary coupled-cluster, etc [4, 5, 10, 29, 122].

Despite a wide application range and initial successes, pushing for quantum advantages with VQA faces many challenges such as noise-induced decoherence, barren plateaus [123, 124] that derail the training of parameters, and reachability deficits with certain fixed ansatz [30–32, 34]. Although there are various proposals [98, 125–128] on alleviating these issues, it is extremely difficult to fully resolve them as long as the circuit ansatz is fixed at the beginning.

As discussed in the last section, a parametrized quantum circuit is required as an ansatz for VQA. A badly designed ansatz could possess limited expressive power or entangling capacity, leaving the global minimum for an optimization problem out of reach. Furthermore, such ansatz may be more susceptible to noises [129], waste quantum resource or lead to barren plateaus that frustrate the optimization procedure [123, 124].

Therefore, a systematic approach to search for optimal circuit ansatz is desired, which has been studied in the name of 'quantum architecture search' (QAS) [39]. The aim of the QAS is to recommend tailored quantum circuits for given problems such that it not only minimizes a loss function, but also satisfies a few other constraints imposed by the hardware connectivity among qubits, native quantum gate set, quantum noise model, training loss landscape and other practical issues.

Previous QAS works have heavily borrowed ideas from NAS. More specifically, greedy methods [9, 49–51], evolutionary or genetic methodologies [40–46], RL engine based approaches [47, 48], Bayesian optimization [130], one-shot search [56], and gradient based methods [39] have all been adopted to discover better circuit ansatz for VQA. However, as far as we know, predictor based evaluation strategy has not been applied toward quantum circuit design. Since predictor based NAS has been empirically demonstrated to be highly efficient, one may anticipate predictor based QAS to hold a similar performance boost for VQA in the NISQ era.

We adopt two representations, list representation and image representation, to denote each candidate circuit. The two representations are utilized in the sampling strategy and predictor evaluation of QAS, respectively. The two representations can both represent the quantum circuit structure faithfully, and the choice is from engineering considerations. On the one hand, to generate the possible circuit structure in random search, list representation is simpler to manipulate and yield very diverse circuit structures. On the other hand, the input for the neural predictor must be a tensor, in which scenario, the image representation is more suitable. The list representation is used to generate circuit candidates and thus define the search space. It comes with a strict syntax delineating a quantum circuit in terms of a sequence of applied quantum gates. The list is a set of tuples, and each tuple encodes a quantum gate (referenced to a given gate set) and positional information (i.e. qubits numbered in a certain way). For instance, tuple indicates a two-qubit quantum gate of the 3rd type acting on the first and second qubit in the circuit. To reconstruct a corresponding quantum circuit from a list, the gates should be sequentially placed onto an initially empty circuit, in the order of the given tuple sequence. Note that a circuit ansatz may have multiple list representations. One possibility is that the order of gate placements can be freely exchanged as long as gates encoded by the two tuples commute. This syntax inherently gives a legitimate search space for circuit ansatz. In this work, sampled circuits are generated in this list representation.

We choose different primitives for different problems. Some common examples include non-parameterized gates such as Hadamard gate H; single-qubit gates with trainable parameters such as rotation gate and, similarly, for Ry, Rz, where X, Y, Z are corresponding Pauli matrices. We also include parameterized two-qubit gates such as with counterparts for YY, ZZ gate, as well as parameterized SWAP gate where .

Now, let us elaborate on the sampling strategy considered in this work. A naïve method is by random sampling: after fixing the number of qubits and the total number of quantum gates in a circuit, one then randomly samples gate types and positions to form a list representation of the quantum circuit ansatz. This simple strategy has some undesirable features. Firstly, gate layouts for randomly sampled circuits are highly 'chaotic' and usually incur severe issues of barren plateau since the circuits behave like random unitaries drawn from the Haar measure. Secondly, with the total number of quantum primitive gates fixed, such randomly generated circuit ansatz is often deep as the arrangement of quantum gates is sparse. Lastly, random ansatz is not amenable to further utilization in the sense of generalization and transferability to problems of different sizes since there is no obvious pattern for extraction.

We devise two pipelines for circuit sampling that substantially alleviate the aforementioned concerns, which are gatewise generation and layerwise generation. In the first pipeline, we construct the circuit gate by gate, by specifying their positions and types, while in the second pipeline, we construct the circuit by iteratively adding half-layers. Namely, whenever we pick a type of quantum gate we apply it on the set of even qubits or odd qubits. Both pipelines further incorporate additional techniques such as hierarchical generation and gate correlation enforcement. See supplemental materials (available online at stacks.iop.org/MLST/2/045027/mmedia) for the details of these two sampling pipelines and the consideration on design of circuit search space.

Apart from the list representation of quantum circuits, we have alluded to the image representation which is designed for training the neural predictor. To encode the circuit structure as input for an ML predictor model, we need a systematic way to represent circuit structures in the form of tensors. The strategy we invented is to transform a circuit into an image of multiple channels. The shape of the input tensor is . The size of the figure is the number of qubits times the depth of the circuit, where the depth is the number of gate layers in a circuit. A similar representation of quantum circuits has been considered in [131]. For the circuit generation pipelines considered in this work, the total number of quantum gates in a circuit n_t is fixed, but different candidate circuits tend to have incompatible circuit depth. Therefore, we need to set a max depth cutoff as D. All circuits with depth less than D are zero-padded up to D columns, and all circuits with depth more than D are simply retracted. In other words, we only process candidate circuits with n_t quantum gates and limited to a depth of D. Figure 2 gives an example of both the list representation and image representation for an () quantum circuit.

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For simplicity of presentation, we impose the following two assumptions on the search space: (a) the two-qubit gates are restricted to act on adjacent qubits, and (b) all two-qubit gate primitives in our work are symmetric. Under such assumptions, this image representation maintains a one-to-one mapping with the actual quantum circuit architecture. These two assumptions are not intrinsic limits of our general method and the image representation is easy to extend to maintain such one-to-one correspondence while the two assumptions are relaxed (see supplemental materials for the details).

In the proposed QAS workflow, one may further consider different architectures of ML predictors. If an recurrent neural network (RNN) based model is used, then we treat the depth dimension as the time dimension while the dimension of qubits and gate types is flattened out as an input vector for each time slice of such RNN based predictor.

We have tried various canonical neural networks, including multilayer perceptron, convolution neural network (CNN), and long short-term memory (LSTM), as the neural predictors to evaluate circuits in the proposed QAS workflow. In general, we find LSTM based RNN performs better than others in terms of predicting the fitness of circuit ansatz. See figure 3 for a schematic of the RNN neural predictor used in this work.

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In some VQA problems, good circuit ansatzes are dense while bad circuit ansatz has a very wide range of fitness scores as a long tail distribution. Such a distribution is hard to fit with one regression model, and we adopt the strategy of two-stage classification for screening circuits. Firstly, a CNN based binary classification model is trained to differentiate between good and bad circuits for a task. Only good candidates are further fed into an RNN based regression model for a more fine-grained evaluation. Such regression model is only trained with good ansatz.

To train the neural predictor, we need to prepare a dataset composed of a circuit structure and its performance according to a task-specific evaluation metric, e.g. estimated energy in VQE simulation or validation accuracy for QML. It is important that the training of such a predictor model is sample efficient, i.e. a small number of training data points should allow the predictor to deliver an acceptable accuracy. Otherwise, predictor based QAS is not desired as it already takes an enormous amount of time and resources to process the training dataset. In our experiments, several hundreds of data pairs are in general enough to make QAS workflow a success for certain important applications. Of course, more training data can boost the predictor's accuracy and, in turn, elevate the overall efficiency for architecture search. There is certainly a tradeoff between the efforts to prepare the training dataset (for neural predictors) and the search efficiency in a later stage of a QAS workflow.

While preparing the training dataset, it is desired to evaluate each circuit multiple times with different initialization of parameters. This is because of an infamous fact, the energy landscape of typical cost functions for quantum circuits is often decorated with a plethora of local minima. Therefore, one should use the minimum loss of multiple runs as the training label for the neural predictor. There are additional benefits to run the same circuit multiple times. For instance, one may train another regression model to predict the standard deviation of the losses for multiple optimization runs of a quantum circuit. This frustration indicator gives us a hint of whether a candidate circuit can be consistently and easily trained for the task at hand. A prediction of large standard deviation implies that the candidate circuit may suffer from a more ragged energy landscape as well as possible issues of barren plateaus. In principle, we can train a multi-task neural predictor that not only guide us for the circuit with a better potential to perform well but also easier to train from scratch.

Both mean squared loss and mean absolute loss are tried in the training of regression models, and they tend to give similar results. Adam optimizer is utilized for all the training in this work. Batch size is 32 or 64 in most of the training. Dropout layers with high dropout rates are heavily applied to avoid severe overfitting.

Furthermore, data augmentation can be applied to the image representation of quantum circuits. In many VQA problems, the final measurement observable is independent on the order of qubits, where a permutation on the qubits leaves the final result unchanged. Strictly speaking, in our search space there is no qubit permutation symmetry since two-qubit gates are only defined on the neighboring qubits and a random permutation may leave the predefined search space. But if we ignore this subtlety, input permutation on the qubits is still helpful to avoid overfitting as it essentially creates times more data than the original input.

It is worth noting that the trained neural predictors often do not perform well in terms of conventional ML evaluation metrics. The training tends to overfit even with networks of fewer parameters and large dropout ratios. However, as we will show in the Results section, such predictors are actually good enough to greatly improve the search efficiency of QAS.

Once the neural predictor is trained, we randomly sample a large number of quantum circuits according to the generation pipelines we introduce. Only circuits that pass the predictor screening (a tiny fraction of all sampled) will be actually tested to verify their fitness. In the end, one should pick a candidate circuit of best fitness for the current task. The entire workflow of neural predictor based QAS is succinctly summarized in figure 1.

Such neural predictor based evaluation policy can be combined with more sophisticated sampling strategies to further improve search efficiency. For example, the neural predictor can be further trained with additional data collected during the Phase II of QAS workflow. One can then screen more circuits with help from a refined predictor. This fine-tuning of the neural predictor can go a few rounds iteratively and should be beneficial for finding a really strong candidate circuit. Moreover, genetic transformation or Bayesian optimization can be used as sampling strategy in the Phase II of QAS to accelerate the search. Contrary to the plain random sampling utilized in this work, these advanced sampling strategies should help as in high-throughput virtual screenings of molecules and materials. Furthermore, neural predictors combined with a VAE setup can be exploited to directly search for strong candidate circuits with gradient-based method in the continuous latent space. We leave these interesting possibilities to extend predictor based QAS workflow as promising directions to explore in future work.

It is highly desirable that an optimal ansatz identified by QAS supports a transfer capacity since direct QAS on large systems is prohibitively difficult. For each optimal quantum circuit selected from QAS, we then test whether such circuit ansatz can be transferred to similar problems involving a larger number of qubits while maintaining state-of-the-art performance. Quantum circuits generated by the layerwise pipeline can be straightforwardly adapted to work on larger systems by simply extending the range of each gate to cover either all odd or even qubits in a larger circuit. However, such a direct transfer of the quantum ansatz is not rooted in rigorous theoretical analysis and may suffer from a huge performance drop.

In order to find a good ansatz on a large size system with the knowledge gained from QAS on small systems, we develop a beam search based method to accelerate the search for appropriate ansatz for large systems at a low extra overhead. Since quantum circuits generated by the layerwise pipeline always group the same quantum gate acting on either even or odd qubits in one layer, we can first fill the circuit by extending each quantum gate on all qubits (half-layer full layer). Such 'fill-in' quantum circuits are, in general, very good candidates with great fitness for larger size problems. This comes at the price, however, as the new quantum circuit costs twice the quantum resource as the original circuit with half-layers. To reduce the number of gates while maintaining the performance above a threshold, we use a beam search scheme [51] to find simpler circuit structures with respect to a given 'fill-in' circuit. There are three phases at each step of the beam search, reduction phase, evaluation phase, and selection phase. In the reduction phase, we reduce quantum gates of one half-layer and every possible reduction (2 D in total) is generated. In the evaluation phase, these circuit candidates with reduction are evaluated for the fitness. Since these quantum structures can be viewed as subcircuits of the 'fill-in' one, the weights (trainable parameters) can be directly inherited and only some light fine-tuning is required to get a reasonably accurate evaluation. In the selection phase, only top-q circuits in terms of fitness are kept in the queue (q = 1 is reduced to greedy search algorithm). This iterative pruning of circuit structures ends when no further reduction on quantum gates is possible without compromising the expected performance. Finally, we note that this technique of beam search can also incorporate mutations or transformations of quantum circuits in addition to eliminations of quantum gates. In this augmented version of beam search, our transfer protocol operates like an evolutionary algorithm for finding suitable circuits for large-size problems. See figure 4 for the workflow schematic of such 'fill-in + beam search' protocol.

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4.1.1. Setup

We train a PQC as an ML model for a binary classification task on a commonly used benchmark in the ML community, and QAS is utilized to identify a suitable circuit architecture (conforming to specified constraints) that can attain rather high accuracy on the validation set. We choose fashion-MNIST [132], the dataset of clothes images, as the benchmark because it is more challenging than MNIST, the dataset of handwritten digits, commonly tested in QML literature. We only use the data labeled with T-shirt/top(0) and Dress(3) in order to focus on a binary classification problem instead of a multiple categorical classification. As demonstrated in [27], QML could perform better than the classical counterparts when the training dataset is small. We, hence, select only 500 datapoints from fashion-MNIST for the training purpose, and select another 500 datapoints for validation. Each image of clothes is padding and flatten to a 1024-dimensional vector which is further encoded into a 10-qubit wavefunction using amplitude encoding. Our quantum ansatz is hence defined in the search space with qubit number N = 10, the number of quantum primitive gates , and circuit depth cutoff D = 10. We focus only on quantum ansatz sampled from the layerwise pipeline in this QML experiment. The primitives of quantum gates include t = 7 types in total: they are parameterized Rx, Ry, Rz, XX, YY, ZZ, and pSWAP gate.

Inspired from the recent proposal of classical shadow with random measurements [133, 134], we introduce a classical post-processing module to make the actual prediction based on features extracted from random measurements on the proposed quantum circuit. This 'random measurement' is implemented with one extra layer of Rx gate, appended to the end of the ansatz-generation circuit. Classical bits of information are then extracted with measurements of Pauli Z operator for each qubit as . The collected classical information is subsequently fed into a classical dense layer as , where k_i and b are classical trainable parameters and is a scaled sigmoid function. Finally, the mean squared loss between y_pred and y_true (the ground truth) is minimized with a simple gradient descent approach, where the weights of the quantum circuit and the classical post processing module are jointly optimized. Once the training of the QML model converges, we take the classification accuracy on the validation set as the evaluation metric.

Next, we discuss details of preparing the training dataset for neural predictors. For this QML benchmark, we adopt a loose converge condition in the phase I of QAS since the validation accuracy is known to be strongly correlated with the accuracy on the early stop. By using the early stop strategy, we may evaluate a circuit's prediction accuracy (for the fashion-MNIST data) without too many training steps. Furthermore, we decide to train each circuit only once, because it is rather time consuming to independently re-train a QML model multiple times. While we run the risk of getting a suboptimal estimate of a circuit's performance, we remind that our ultimate goal is to run a high-throughput screening of many quantum circuits in the phase II of a QAS workflow. Candidate circuits that pass the predictor-based screening still have to be experimentally trained and verified for their actual performances. As long as some optimal circuit architectures are detected and passed through the screening, a QAS is deemed successful. According to this perspective, building a highly accurate neural predictor is certainly desirable but not strictly indispensable.

We collect 300 datapoints for training the RNN based neural predictor. Each datapoint comprises a quantum circuit and its corresponding validation accuracy on the fashion-MINST as the label. The regression result is shown in figure 5. The performance can be evaluated by R² score of the linear regression which is around 0.71 on the validation dataset.

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4.1.2. QAS result

The ultimate test for the QAS is how well the predictor-based screening performs in phase II. At this stage, we randomly generate quantum circuits from the layerwise pipeline, and predict their accuracy with the trained predictors. We only retain circuits with a predicted accuracy larger than 0.89 for further training and verification of their true accuracy in experiments. For 30 000 random circuits, only 195 () are kept and proceed to the actual testings. The comparisons between the true accuracy of quantum circuits from random search (as the training dataset for the predictor) and the counterpart from QAS filtered by the neural predictor is summarized in figure 6. As shown, the optimal ansatz found by QAS has validation accuracy above 0.92, which is significantly larger than the best result seen in the training set (all lower than 0.9). The optimal quantum circuit recommended by QAS has a layerwise layout as YY, ZZ-odd, Rz-odd, YY-even, Rz-even, ZZ-even, SWAP (see supplemental materials for the layered ansatz notation). It is interesting to observe the generalizability of such neural predictor based approach. As the predictor is only trained with suboptimal quantum circuits with accuracy less than 0.9, it can still single out quantum circuits with accuracy better than any one it has seen during the training.

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We further compare QAS-screened QML model against a baseline established by a QML built with the conventional hardware efficient ansatz. The details are given in figure 7. As seen, both classical post processing and QAS contribute to the improved accuracy.

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4.2.1. Model

4.2.2. QAOA baseline

4.2.3. Setup

Since obtaining optimal weights for a VQE circuit usually takes less time than that for a QML task requiring a large dataset, we train each quantum circuit for 10 independent runs with different random initializations to avoid local minimum in the estimation of the ground-state energy. These independent training can be cast into the batch dimension with the help of QML framework [135] by clever design which enables fast optimization over independent runs in parallel. The search space for this VQE task is confined to a qubit number N = 6, a total number of quantum primitive gates with depth cutoff D = 10. Candidate circuits are sampled from both layerwise and gatewise generation pipelines with equal contribution. The primitive set of quantum gates include H, Rx, Ry, Rz, XX, YY, and ZZ gate this time. Among these gates, the Hadamard gate H is the only type without a tunable parameter. For the layerwise pipeline, we set of all generated circuits to begin with a layer of the Hadamard gate applied to all qubits, as starting from the state may help VQE to find a better approximation to the ground state.

We adopt the two-stage screening for this VQE investigation. The rationale is that the energy distribution of the TFIM model is not smooth across a wide region in the search space; therefore, a single high-quality predictor is difficult to train. To overcome this problem, we resort to using two predictors as described in Method section. First, we use a CNN based binary classification model to quickly rule out inappropriate circuits covering a wide variety of rather arbitrary circuit structures. The 'good' circuits with limited variety are then screened again with an RNN-based regression model to evaluate their performance more precisely. In this case, we use , the normalized deviation of the estimated ground-state energy, as the predicted label.

We again randomly pick and optimize 300 independent quantum circuits to build the training datasets for the two neural predictors. Figure 8 shows the distribution of converged energy (error ratio ε) for the dataset of 300 circuits. Such a distorted distribution manifests the source of difficulty to rely on a single regression model to directly pick out top-performing circuits from the entire candidate pool. Rather, in our two-stage screening setups, the regression model is only expected to provide an accurate characterization of potentially good circuit candidates with limited variety.

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For the CNN based classification model, we adopt data augmentation by applying random permutations on qubits since TFIM is translationally invariant and the energy is agnostic to the order of qubits. In addition, multiple-layer convolution with dilation, batch normalization on qubit dimension, dropout, ELU activation, and L₂ regularization on weights are all utilized in the classification model.

The training dataset we used only contains 300 quantum circuits and their corresponding error ratios obtained from VQE optimizations. When one is limited to such a small training set, it could be crucial how the circuits are selected to form the circuit pool for a particular problem. Prior knowledge (such as insights from physics) could prove beneficial at this stage. For instance, only circuits conforming to certain symmetry are generated. However, for the present study, we want to emphasize the universality of this predictor-based QAS and simply do a random sampling of circuits according to the two sampling pipelines. With such a scarcity of training data, the neural predictors actually do not perform particularly well according to the traditional metrics for ML model evaluation. Nevertheless, we remind that a complete QAS workflow operates more like a high-throughput screening with the ultimate goal of discovering one or a few top-performing circuits as opposed to making highly accurate predictions for all circuits. Despite this focus being different from the standard ML, we still have to deal with some non-trivial challenges shared by many ML tasks. For the binary classification, we face a highly imbalanced classification with a trade-off between precision and recall. In this case, we prefer a model with higher precision to recall, as the goal is to efficiently filter a large search space. Specifically, our trained CNN based classification model has a precision around 0.7 and a recall around 0.47.

4.2.4. QAS result

With the two-stage predictor based screening in place, we filter a large number of circuits generated via either gatewise or layerwise pipelines. The filter thresholds for the two models are set to 0.85 and 0.005, respectively. We only keep the most confident candidates for further VQE evaluations. At the first stage, only quantum circuits with predicted value larger than 0.85 instead of 0.5 are kept as promising candidates for further evaluation. At the second stage, only candidates with predicted error ratio less than 0.005 are recommended for experimental verification (i.e. going through standard VQE optimizations). Remarkably, our RNN-based regression model can give outputs with predicted error ratio less than 0.005 when the smallest value in its training dataset 0.00789 is larger than this threshold.

We randomly sample 50 000 quantum circuits, and only 626 () pass the two-stage screening and are evaluated with the VQE optimization. Among these final candidates, five circuits give an energy less than −7.7 and we call them optimal ansatz. This is an interesting result as the training dataset of 300 quantum circuits does not feature such good circuits—the best energy estimation in our training dataset is only −7.61694—yet predictor trained with suboptimal circuits finds optimal ones. This observation demonstrates the effectiveness of our few-data trained predictors to screen unseen structures in a QAS context. In this case, the search efficiency for the predictor-based QAS workflow is around , i.e. one out of every 100 quantum circuits recommended for VQE experiments are optimal (in the sense of VQE energy less than −7.7). Without the neural predictor as the filter, random search efficiency is around following our search space pipelines according to a large number of numerical simulations. In other words, on average, predictor-based QAS only needs to conduct 100 independent VQEs before discovering an optimal candidate while naive random search requires 2000 circuits evaluation before encountering a good candidate in our pre-defined search space. This is a 20 times boost of search efficiency. Figure 9 summarizes the gain brought by the predictor based QAS compared to the random search.

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Since in NAS or QAS, a fair efficiency comparison between different approaches is of high importance, we briefly discuss on the possibility of comparison between our predictor based approach and better QAS alternatives than random search baseline. There are two relatively independent ingredients in QAS, search strategy and evaluation method, that contribute to the final performance of QAS. Most QAS approaches focus on designing a powerful search strategy, such as RL or genetic algorithm based setups. Instead, our approach proposed a new way to improve the evaluation method, i.e. we replace the plain evaluation method (train then evaluate) with neural predictors. Specifically, the comparison of the 20 times acceleration above is from random search + plain evaluation method versus random search + neural predictor evaluation. Note that neural predictor evaluation can be seamlessly combined with other advanced search strategies, as they belong to different aspects of QAS. Therefore, another fair comparison, like RL-QAS + plain evaluation method versus RL-QAS + neural predictor evaluation, should achieve roughly the same level of acceleration since the search strategy keeps consistent. On the contrary, the comparison between better alternative search strategies + plain evaluation method versus random search baseline + neural predictor evaluation is less ideal, as we can easily incorporate these better search strategies into neural predictor based QAS.

4.2.5. Transfer of the predictor

Next, recall that our predictors are trained with circuits limited to quantum gates, but it can be transferred to predict the fitness of quantum circuits consisting of a different number of quantum gates with zero tuning. For example, we know that QAOA-inspired ansatz with p = 3 and gates can give the exact ground state. Although the total gate number is substantially changed compared to the instances in the training set () for the predictors, the screening pipeline still works well. The classification predictor at the first stage gives the output 0.99979 and the regression predictor gives 0.004 as the predicted error ratio for the QAOA-inspired circuit with p = 3. In short, this optimal circuit successfully passes the two-stage screening without having to train these predictors with circuits comprising the same number of quantum gates.

The predictors are not only able to make reasonably accurate predictions for circuits having more quantum gates, but also ones with fewer gates too. For instance, we conduct a large scale search for circuits with gates. 360 out of quantum circuit structures are screened, and three instances of them give VQE energy smaller than −7.7 with the best one being −7.7274. This is a highly nontrivial result, as QAOA ansatz with p = 2 and the same amount of quantum resources only gives energy −7.46. The optimal circuit structure we find by transferred predictor is displayed in figure 10.

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4.2.6. Transferability of the optimal ansatz