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EPJ Quantum Technology

On the learnability of quantum state fidelity

Norhan Elsayed Amer, Walid E. Gomaa, Keiji Richard Kimura, Kazunori Ueda, Ahmed El-Mahdy

Published Nov 20, 2022• 6 citations
Publisher

Abstract

Abstract Current quantum processing technology is generally noisy with a limited number of qubits, stressing the importance of quantum state fidelity estimation. The complexity of this problem is mainly due to not only accounting for single gates and readout errors but also for interactions among which. Existing methods generally rely on either reconstructing the given circuit state, ideal state, and computing the distance of which; or forcing the system to be on a specific state. Both rely on conducting circuit measurements, in which computational efficiency is traded off with obtained fidelity details, requiring an exponential number of experiments for full information. This paper poses the question: Is the mapping between a given quantum circuit and its state fidelity learnable? If learnable, this would be a step towards an alternative approach that relies on machine learning, providing much more efficient computation. To answer this question, we propose three deep learning models for 1-, 3-, and 5-qubit circuits and experiment on the following real-quantum processors: ibmq_armonk (1-qubit), ibmq_lima (5-qubit) and ibmq_quito (5-qubit) backends, respectively. Our models achieved a mean correlation factor of 0.74, 0.67 and 0.66 for 1-, 3-, and 5-qubit random circuits, respectively, with the exponential state tomography method. Additionally, our 5-qubit model outperforms simple baseline state fidelity estimation method on three quantum benchmarks. Our method, trained on random circuits only, achieved a mean correlation factor of 0.968 while the baseline method achieved 0.738. Furthermore, we investigate the effect of dynamic noise on state fidelity estimation. The correlation factor substantially improved to 0.82 and 0.74 for the 3- and 5-qubit models, respectively. The results show that machine learning is promising for predicting state fidelity from circuit representation and this work may be considered a step towards efficient end-to-end learning.

DOI
10.1140/epjqt/s40507-022-00149-8

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Abstract Current quantum processing technology is generally noisy with a limited number of qubits, stressing the importance of quantum state fidelity estimation. The complexity of this problem is mainly due to not only accounting for single gates and readout errors but also for interactions among which. Existing methods generally rely on either reconstructing the given circuit state, ideal state, and computing the distance of which; or forcing the system to be on a specific state. Both rely on conducting circuit measurements, in which computational efficiency is traded off with obtained fidelity details, requiring an exponential number of experiments for full information. This paper poses the question: Is the mapping between a given quantum circuit and its state fidelity learnable? If learnable, this would be a step towards an alternative approach that relies on machine learning, providing much more efficient computation. To answer this question, we propose three deep learning models for 1-, 3-, and 5-qubit circuits and experiment on the following real-quantum processors: ibmq_armonk (1-qubit), ibmq_lima (5-qubit) and ibmq_quito (5-qubit) backends, respectively. Our models achieved a mean correlation factor of 0.74, 0.67 and 0.66 for 1-, 3-, and 5-qubit random circuits, respectively, with the exponential state tomography method. Additionally, our 5-qubit model outperforms simple baseline state fidelity estimation method on three quantum benchmarks. Our method, trained on random circuits only, achieved a mean correlation factor of 0.968 while the baseline method achieved 0.738. Furthermore, we investigate the effect of dynamic noise on state fidelity estimation. The correlation factor substantially improved to 0.82 and 0.74 for the 3- and 5-qubit models, respectively. The results show that machine learning is promising for predicting state fidelity from circuit representation and this work may be considered a step towards efficient end-to-end learning.
Hybridizing machine learning (ML) with quantum computing has become an attractive strategy to increase computational power on Noisy Intermediate Scale Quantum (NISQ) devices. Although these can be potentially quantum advantage devices, their qubit numbers and noise are bounded and there are challenges when it comes to practical quantum circuit realization. This work concentrates on the creation of optimized quantum circuits via ML-based strategies, to enhance the performance of the circuits, to reduce the error rate and to accommodate the particular noise profile of the hardware of NISQ devices. The resulting suboptimal mappings with little or no consideration of noise and hardware topology in traditional quantum compilers can ruin the fidelity of quantum algorithm. Leveraging machine learning, and specifically reinforcement and supervised learning methods, our method learns to optimize compilation strategy for specific quantum hardware architecture. The paper proposes a framework that uses machine learning models to predict qubit mappings, gate decompositions, and transpilation paths that minimize gate depth and error accumulation. We also investigate transfer learning methods for propagating the learned compilation strategies to novel NISQ machines. Such methods are empirically validated on both simulated and real quantum hardware backends, providing evidence of circuit depth, execution time and error reductions. The results show that ML-assisted compilation can achieve better success probabilities for benchmark quantum algorithms like VQE and QAOA than conventional compilers. Furthermore, the work examines the interpretability of ML models providing hardware-aware optimization recommendations and understanding the noise patterns and methods of error correction. Our work highlights the key role of ML for transferring between theoretical quantum algorithms and their practical NISQ implementations. This integration boosts the immediate applicability of existing quantum devices and provides a basis for adaptive and scalable likelihood-based compilation for the era of fault-tolerant quantum computation. Our results motivate further investigation of hybrid classical-quantum workflows, in which ML iteratively improves compilation in accordance with changing hardware noise and capabilities. By improving quantum circuit compilation onNISQ computers, this work is taking steps toward the realization of quantum advantage in application scenarios such as optimization, cryptography, and quantum simulation without needing to wait for fully fault-tolerant large-scale quantum computers.
Given a quantum circuit, a quantum computer can sample the output distribution exponentially faster in the number of bits than classical computers. A similar exponential separation has yet to be established in generative models through quantum sample learning: given samples from an n-qubit computation, can we learn the underlying quantum distribution using models with training parameters that scale polynomial in n under a fixed training time? We study four kinds of generative models: Deep Boltzmann machine (DBM), Generative Adversarial Networks (GANs), Long Short-Term Memory (LSTM) and Autoregressive GAN, on learning quantum data set generated by deep random circuits. We demonstrate the leading performance of LSTM in learning quantum samples, and thus the autoregressive structure present in the underlying quantum distribution from random quantum circuits. Both numerical experiments and a theoretical proof in the case of the DBM show exponentially growing complexity of learning-agent parameters required for achieving a fixed accuracy as n increases. Finally, we establish a connection between learnability and the complexity of generative models by benchmarking learnability against different sets of samples drawn from probability distributions of variable degrees of complexities in their quantum and classical representations.
The vast and complicated many-qubit state space forbids us to comprehensively capture the dynamics of modern quantum computers via classical simulations or quantum tomography. Recent progress in quantum learning theory prompts a crucial question: can linear properties of a many-qubit circuit with d tunable RZ gates and G − d Clifford gates be efficiently learned from measurement data generated by varying classical inputs? In this work, we prove that the sample complexity scaling linearly in d is required to achieve a small prediction error, while the corresponding computational complexity may scale exponentially in d. To address this challenge, we propose a kernel-based method leveraging classical shadows and truncated trigonometric expansions, enabling a controllable trade-off between prediction accuracy and computational overhead. Our results advance two crucial realms in quantum computation: the exploration of quantum algorithms with practical utilities and learning-based quantum system certification. We conduct numerical simulations to validate our proposals across diverse scenarios, encompassing quantum information processing protocols, Hamiltonian simulation, and variational quantum algorithms up to 60 qubits. It is currently unclear whether machine learning model could efficiently learn the dynamics of many-qubit bounded-gate quantum circuits. This work reveals the learnability of predicting linear properties of large- scale quantum circuits and proposes a learning algorithm that enables resource-efficient solutions for many practical quantum applications.
Abstract Quantum circuit optimization is an inevitable task with the current noisy quantum backends. This task is considered non-trivial due to the varying circuits’ complexities in addition to hardware-specific noise, topology, and limited connectivity. The currently available methods either rely on heuristics for circuit optimization tasks or reinforcement learning with complex unscalable neural networks such as transformers. In this paper, we are concerned with optimizing the initial logical-to-physical mapping selection. Specifically, we investigate whether a reinforcement learning agent with simple scalable neural network is capable of finding a near-optimal logical-to-physical mapping, that would decrease as much as possible additional CNOT gates, only from a fixed-length feature vector. To answer this question, we train a Maskable Proximal Policy Optimization agent to progressively take steps towards a near-optimal logical-to-physical mapping on a 20-qubit hardware architecture. Our results show that our agent coupled with a simple routing evaluation is capable of outperforming other available reinforcement learning and heuristics approaches on 12 out of 19 test benchmarks, achieving geometric mean improvements of 2.2% and 15% over the best available related work and two heuristics approaches, respectively. Additionally, our neural network model scales linearly as the number of qubits increases.
Major obstacles remain to the implementation of macroscopic quantum computing: hardware problems of noise, decoherence, and scaling; software problems of error correction; and, most important, algorithm construction. Finding truly quantum algorithms is quite difficult, and many of these genuine quantum algorithms, like Shor's prime factoring or phase estimation, require extremely long circuit depth for any practical application, which necessitates error correction. In contrast, we show that machine learning can be used as a systematic method to construct algorithms, that is, to non-algorithmically “program” quantum computers. Quantum machine learning enables us to perform computations without breaking down an algorithm into its gate “building blocks”, eliminating that difficult step and potentially increasing efficiency by simplifying and reducing unnecessary complexity. In addition, our non-algorithmic machine learning approach is robust to both noise and to decoherence, which is ideal for running on inherently noisy NISQ devices which are limited in the number of qubits available for error correction. We demonstrate this using a fundamentally nonclassical calculation: experimentally estimating the entanglement of an unknown quantum state. Results from this have been successfully ported to the IBM hardware and trained using a hybrid reinforcement learning method.
Quantum computers in the Noisy Intermediate-Scale Quantum (NISQ) era face significant challenges due to inherent noise and limited qubit coherence. Accurate fidelity evaluation of quantum states necessitates multiple repeated measurements to obtain statistical results. But determining the optimal number of measurements remains an open problem due to the dynamic, device-dependent nature of quantum noise. Existing methods either assume prior knowledge of the noise model or inherently employ a fixed measurement strategy, which limits their applicability in practical deployment scenarios. This paper presents AutoFid, an adaptive and noise-aware fidelity measurement framework that automatically determines the number of required tests based on circuit structure and hardware feedback. AutoFid models quantum circuits as Directed Acyclic Graphs and estimates structural complexity via random walks, enabling estimation of measurement effort. It further incorporates transpilation-aware features such as gate fidelity, depth inflation, and crosstalk to refine iteration budgets. During runtime, AutoFid dynamically samples fidelity results and employs an early stopping strategy based on confidence intervals to reduce redundant measurements while preserving accuracy guarantees. We evaluate AutoFid on 18 quantum benchmarks executed on real IBMQ hardware platforms. Experimental results show that AutoFid reduces measurement costs by more than 50% compared to both fixed-shot and learning-based baselines, while consistently maintaining fidelity bias below 0.01. Additional analysis using classical software testing metrics and ablation studies demonstrate its effectiveness, robustness, and adaptability across a wide range of quantum workloads.