Source-linked AI summary
Quantum circuit architecture search for variational quantum algorithms
Yuxuan Du, Tao Huang, Shan You, Min-Hsiu Hsieh, Dacheng Tao
TL;DR
The study addresses how to design effective VQA ansatze while controlling resource demands and unresolved design challenges. It devises QAS to dynamically and automatically design ansatze, with simulation and experimental results validating its effectiveness and comparable computational resources to fixed-ansatz VQAs.
Problem
QAS still has critical unresolved questions, including how to sample ansatze better and shrink the parameter space without degrading learning performance.
Method
QAS dynamically and automatically designs an ansatz for a VQA learning task, while Pro-QAS progressively searches gate arrangements to narrow the ansatz pool.
Results
Simulation and experimental results validate QAS's effectiveness, while its computational-resource requirements remain similar to conventional VQAs with fixed ansatze.
Takeaways & Limitations
QAS is compatible with all quantum systems and may support further NISQ applications when combined with error-mitigation and trainability-enhancement techniques.
Takeaways & Limitations
The study leaves critical questions about improved ansatz sampling and parameter-space reduction for future work.
Abstract
from arXiv · showhide
Variational quantum algorithms (VQAs) are expected to be a path to quantum advantages on noisy intermediate-scale quantum devices. However, both empirical and theoretical results exhibit that the deployed ansatz heavily affects the performance of VQAs such that an ansatz with a larger number of quantum gates enables a stronger expressivity, while the accumulated noise may render a poor trainability. To maximally improve the robustness and trainability of VQAs, here we devise a resource and runtime efficient scheme termed quantum architecture search (QAS). In particular, given a learning task, QAS automatically seeks a near-optimal ansatz (i.e., circuit architecture) to balance benefits and side-effects brought by adding more noisy quantum gates to achieve a good performance. We implement QAS on both the numerical simulator and real quantum hardware, via the IBM cloud, to accomplish data classification and quantum chemistry tasks. In the problems studied, numerical and experimental results show that QAS can not only alleviate the influence of quantum noise and barren plateaus, but also outperforms VQAs with pre-selected ansatze.
DISCUSSION
QAS dynamically designs ansatze for VQAs, with simulation and experimental results validating its effectiveness while using resources similar to fixed-ansatz VQAs. The study identifies several unresolved directions for improving sampling, parameter reduction, prior-information use, post-processing, theory, and applications beyond VQAs.
- QAS dynamically and automatically designs ansatze for VQAs, with simulation and experimental results validating its effectiveness.
- QAS requests computational resources similar to conventional VQAs with fixed ansatze and is compatible with all quantum systems.
- Future work includes better ansatz sampling, parameter-space reduction, learning-problem priors, redundant-gate removal, theoretical analysis, and applications beyond VQAs.
Methods
QAS uses supernets, indexing, and weight sharing to represent and evaluate a large ansatz pool while reducing parameterization and retaining computational efficiency. Its complexity depends on the ansatz-space dimensions, sampled candidates, and number of supernets.
- QAS differs from classical neural architecture search because quantum gates, entanglement, barren plateaus, noise, and hardware restrictions complicate quantum circuit architecture search.
- Weight sharing correlates parameters across ansatze, enabling joint optimization while reducing the parameter space from O(dQNL) to O(dLQN).
- Supernets define the ansatz pool and parameterize candidate ansatze, which inherit shared weights instead of being trained separately from scratch.
- Index lists encode candidate gate choices across qubits and layers, including single-qubit and two-qubit gate arrangements.
- QAS runtime complexity is O(dT + K), covering parameter optimization and ranking K sampled ansatze.
- With T = O(QNL) and K = O(QNL), runtime and memory scale as O(dQNL) and O(dQNL + QN^2L^2), while W supernets add linear overhead.
A. The ansatz assignment task
QAS frames ansatz assignment as adversarial bandit learning, where supernets are arms and sampled ansatz losses are observed costs. The objective is to allocate ansatze across iterations to minimize cumulative loss.
- In adversarial bandit learning, a player selects among W arms over T iterations while an adversary assigns costs and reveals the selected arm’s cost.
- The goal is to minimize total cost over T iterations, measured by regret relative to the best arm in hindsight.
- QAS maps each arm to a supernet, each cost to the sampled ansatz objective, and ansatz assignment to selecting a supernet sequence minimizing cumulative cost.
2. The comparison between the strategy used in QAS and all bandit algorithms
The paper compares QAS’s supernet-assignment strategy with adversarial bandit algorithms using regret. Under the stated theorem conditions, QAS achieves a nonpositive regret bound while bandit algorithms have a linear lower bound.
- The theorem states that QAS’s assignment strategy outperforms all adversarial bandit algorithms in regret.
- QAS promises regret R_T ≤ 0, whereas the best bandit algorithms have regret lower bounded by R_T = Ω(T).
- QAS assigns each sampled ansatz to the supernet producing the lowest objective among the W supernets.
- With high probability, no bandit learning algorithm achieves lower regret than QAS’s strategy under the theorem’s assumptions.
3. Applying bandit learning algorithms to the ansatz assignment task
The paper considers adversarial bandit learning to reduce the runtime cost of assigning ansatze when QAS uses multiple supernets.
- Bandit learning is proposed to improve ansatz assignment in terms of runtime cost.The approach targets the expense of processing sampled ansatze through multiple supernets.
- QAS separately evaluates each sampled ansatz with W supernets before comparing their objective values.
- The runtime complexity becomes expensive when the number of supernets W is large.
B. The synthetic dataset classification task
The synthetic classification task uses separate training, validation, and test datasets to optimize parameters, select the best validation checkpoint, and benchmark generalization.
- The synthetic dataset is split into training, validation, and test datasets.
- Training data optimize the trainable parameters, while validation data estimate classifier training quality.
- During T epochs, the parameters achieving the highest validation accuracy are selected as the output parameters.
- The selected output parameters are applied to the test dataset to benchmark the trained classifier.
2. Implementation of the synthetic dataset
The synthetic dataset is constructed by sampling three-dimensional points, labeling them through an optimal circuit and measurement rule, and rejecting ambiguous measurement outcomes.
- Data points x^(i) are generated in R^3 to form the synthetic dataset.
- The labeling procedure uses an optimal circuit together with a measurement operator Π = I4 ⊗|0⟩⟨0|.
- Points with measured results in (0.25, 0.75) are discarded and resampled.
3. Simulation results of the synthetic dataset classification and the fierce competition phenomenon
QAS replaces a fixed ansatz with supernet-controlled circuit architectures and is evaluated on synthetic classification under noiseless conditions. The simulations show perfect classification performance, while correlation analysis examines when QAS rankings agree with independently trained ansatze.
- QAS classifier: QAS makes the trainable unitary depend on the sampled ansatz, with parameters controlled by the supernet.
- Noiseless simulation: The conventional quantum kernel classifier reaches 100% training, validation, and test accuracy after 80 epochs under noiseless simulation.Its loss decreases to 0.24.
- Noiseless simulation: QAS outputs an ansatz with fewer CNOT gates than the conventional classifier, and its retraining reaches 100% training and test accuracy after one epoch.
- Fierce competition: The Spearman and Kendall coefficients compare validation-accuracy rankings from QAS with rankings obtained through independent training.
- Fierce competition: With W = 1, the correlation is low at ρK = 0.113, whereas W = 5 yields ρS = 0.723 and ρK = 0.536.
- Fierce competition: With W = 10 and T = 1000, the correlations increase to ρS = 0.774 and ρK = 0.591.
4. The performance of QAS towards the noise model extracted from the real quantum devices
QAS is evaluated for quantum classification under a noise model extracted from IBMQ Lima, comparing searched circuits with a hardware-efficient baseline. The experiments vary optimization resources and assess validation and test accuracy.
- Experimental setting: QAS is evaluated against a hardware-efficient ansatz under noise extracted from IBMQ Lima.The study uses IBMQ Lima’s connectivity and system parameters to define the experimental setting.
- Results: The baseline hardware-efficient classifier achieves 68% test accuracy.This value is used as the baseline for quantifying QAS performance.
- Results: 19 of 500 sampled ansatze exceed the baseline with T = 5 and W = 1, while 58 of 500 do so with T = 400 and W = 5.The second setting also has better average performance across the sampled ansatze.
- Results: After retraining the searched ansatz for 8 epochs, QAS reaches 81% test accuracy.The reported result comes from the second setting, using T = 400 and W = 5 during optimization.
C. Experimental Details of the ground state energy estimation
The ground-state energy task estimates molecular hydrogen’s lowest energy by preparing parameterized wavefunctions and measuring Hamiltonian Pauli terms. VQE optimization uses a quantum natural gradient optimizer alongside QAS.
- Task: The task is to estimate the ground-state energy of a chemical system described by a Hamiltonian.Molecular hydrogen is represented by a linear combination of products of local Pauli operators.
- Method: VQE generates a parameterized wavefunction |Ψ(θ)⟩ to estimate the molecular hydrogen ground-state energy.The Hamiltonian expectation value is obtained by repeatedly measuring the state with the Pauli operators appearing in the Hamiltonian.
- Reference value: The exact molecular hydrogen ground-state energy is Em = −1.136 Ha.Ha denotes Hartree, with 1 Ha = 627.5 kcal/mol; the exact value comes from full configuration-interaction calculation.
- Optimization: A quantum natural gradient optimizer with learning rate 0.2 is used to optimize parameters for both VQE and QAS.The optimizer is employed to accelerate convergence for both methods.
2. The performance of QAS on real quantum devices
QAS-RC is tested against conventional VQE on IBMQ Ourense for molecular hydrogen energy estimation, using a search space constrained by real qubit connectivity. On hardware, QAS-RC produces an energy closer to the exact result and a lower relative deviation than VQE.
- Experimental setting: QAS-RC and VQE are implemented on IBM’s 5-qubit machine, IBMQ Ourense, for estimating molecular hydrogen’s ground-state energy.The experiment uses the machine’s connectivity and system parameters.
- QAS-RC design: QAS-RC restricts its circuit search to the device connectivity, allowing RY and RZ single-qubit gates and conditional CNOT operations.The resulting search space contains 10243 possible circuit architectures.
- Hardware results: On IBMQ Ourense, VQE estimates −0.61 Ha and QAS-RC estimates −0.963 Ha, compared with the exact −1.136 Ha.The QAS-RC estimate is reported as much closer to the exact result.
- Hardware results: Relative deviation is 36.1% for VQE and 6.8% for QAS-RC.The paper attributes the reduction to QAS considering the real qubit connectivities and reports compatibility with weighted gate noise.
- Ranking stage: The QAS-RC ranking stage concentrates most sampled ansatz energies in [−0.6 Ha, −0.4 Ha].The output ansatz and its implementation on IBMQ Ourense are shown in the corresponding panels.
D. Improving the ranking stage of QAS
The paper examines improved QAS ranking, barren-plateau behavior, and progressive search for larger problems. Evolutionary ranking improves the frequency of favorable ansatze, while Pro-QAS narrows the search progressively to address exponential dependence on qubit count.
- Ranking limitation: Uniformly sampling K ansatze can select poor architectures when the poor subset is larger than the good subset.The paper identifies more effective sampling methods as desirable for the ranking stage.
- Evolutionary ranking: NSGA-II finds 143 of 943 ansatze (15.2%) in the energy range [−1 Ha, −1.2 Ha], versus 3 of 500 (0.6%) with uniform sampling.The comparison is conducted for ground-state energy estimation under noiseless simulation.
- Barren plateaus: QAS compares the gradient-norm variance of a hardware-efficient ansatz with that of its ansatze pool to assess trainability and barren plateaus.The simulations sample 2000 parameter settings or ansatze across N = 2 to 10 qubits.
- Barren plateaus: For the hardware-efficient ansatz, gradient-norm variance decreases continuously as N increases, while QAS variance is nearly unchanged for N = 4, 6, 8, 10.The reported contrast is presented as evidence that the QAS ansatze pool alleviates barren-plateau effects.
- Progressive QAS: QAS remains exponentially dependent on N despite reducing parameter-space size to O(dLQN), limiting optimal-ansatz estimation for some large problems.This scope limitation motivates the progressive QAS variant.
- Progressive QAS: Pro-QAS progressively optimizes one layer’s gate arrangement while freezing the remaining layers, then uses the resulting ansatz and parameters to approximate the optimum.The procedure narrows the ansatze pool during construction.
- Progressive QAS: Pro-QAS has at most O(dQNL) memory and O(QNL^2) runtime.These complexities include the extra outer loop for progressive optimization.
2. Numerical simulation results of Pro-QAS
Pro-QAS was evaluated on 7- and 10-qubit binary classification simulations against hardware-efficient ansatz baselines. In both cases, the searched ansatz reduced CNOT gates and achieved higher test accuracy.
- Experimental setup: The simulations used binary classification datasets with 7 and 10 qubits, extending the input feature dimension beyond the 3-qubit task.The dataset contains 300 examples, and the 7- and 10-qubit cases load inputs with N = 7 and N = 10 qubits.
- Experimental setup: For N = 7, the search space contained 240 ansatze using RY, RZ, and CNOT gates with chain connectivity.The simulation used five layers, one supernet, 200 Step 2 epochs, and a depolarization channel.
- 7-qubit results: 65% was the N = 7 hardware-efficient baseline, while Pro-QAS reached 75% test accuracy after optimizing all five layers.Pro-QAS surpassed the baseline after optimization of the third layer.
- 7-qubit results: For N = 7, Pro-QAS reduced the number of CNOT gates from 30 to 14 relative to the hardware-efficient ansatz.The reduction removes unnecessary two-qubit gates to suppress noise.
- 10-qubit results: For N = 10, the hardware-efficient baseline achieved 67% test accuracy, whereas Pro-QAS improved accuracy to 76% and reduced CNOT gates from 27 to 18.The searched ansatz was reported as superior to the hardware-efficient ansatz.