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Frontiers of Physics

Reconstructing unknown quantum states using variational layerwise method

Junxiang Xiao, Jingwei Wen, Shijie Wei, Guilu Long

Published Apr 1, 2022• 8 citations
Publisher

Abstract

In order to gain comprehensive knowledge of an arbitrary unknown quantum state, one feasible way is to reconstruct it, which can be realized by finding a series of quantum operations that can refactor the unitary evolution producing the unknown state. We design an adaptive framework that can reconstruct unknown quantum states at high fidelities, which utilizes SWAP test, parameterized quantum circuits (PQCs) and layerwise learning strategy. We conduct benchmarking on the framework using numerical simulations and reproduce states of up to six qubits at more than 96% overlaps with original states on average using PQCs trained by our framework, revealing its high applicability to quantum systems of different scales theoretically. Moreover, we perform experiments on a five-qubit IBM Quantum hardware to reconstruct random unknown single qubit states, illustrating the practical performance of our framework. For a certain reconstructing fidelity, our method can effectively construct a PQC of suitable length, avoiding barren plateaus of shadow circuits and overuse of quantum resources by deep circuits, which is of much significance when the scale of the target state is large and there is no a priori information on it. This advantage indicates that it can learn credible information of unknown states with limited quantum resources, giving a boost to quantum algorithms based on parameterized circuits on near-term quantum processors.

DOI
10.1007/s11467-022-1157-2

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In order to gain comprehensive knowledge of an arbitrary unknown quantum state, one feasible way is to reconstruct it, which can be realized by finding a series of quantum operations that can refactor the unitary evolution producing the unknown state. We design an adaptive framework that can reconstruct unknown quantum states at high fidelities, which utilizes SWAP test, parameterized quantum circuits (PQCs) and layerwise learning strategy. We conduct benchmarking on the framework using numerical simulations and reproduce states of up to six qubits at more than 96% overlaps with original states on average using PQCs trained by our framework, revealing its high applicability to quantum systems of different scales theoretically. Moreover, we perform experiments on a five-qubit IBM Quantum hardware to reconstruct random unknown single qubit states, illustrating the practical performance of our framework. For a certain reconstructing fidelity, our method can effectively construct a PQC of suitable length, avoiding barren plateaus of shadow circuits and overuse of quantum resources by deep circuits, which is of much significance when the scale of the target state is large and there is no a priori information on it. This advantage indicates that it can learn credible information of unknown states with limited quantum resources, giving a boost to quantum algorithms based on parameterized circuits on near-term quantum processors.
Quantum state tomography (QST) is a fundamental technique in quantum information science, aiming to reconstruct the density matrix of an unknown quantum state from experimental measurements. However, traditional QST methods often suffer from significant computational demands and are highly susceptible to measurement noise, hindering their practical application, particularly with near-term quantum computers. This work proposes a novel approach to QST utilizing variational quantum algorithms (VQAs). VQAs iteratively optimize a parameterized quantum circuit to minimize a cost function related to the fidelity between the desired state and the state predicted by the circuit. This approach dramatically reduces the number of required measurements and offers the potential for achieving high-fidelity state reconstruction with lower quantum resource requirements. We detail the theoretical framework, outlining the core algorithms and mathematical formulations. The key advantage lies in the ability to learn the state directly, mitigating the limitations of traditional methods. We explore the optimization landscape and discuss strategies for improving convergence and robustness. The results demonstrate a promising path toward scalable and robust QST on quantum devices.
We present hierarchical learning, a novel variational architecture for efficient training of large-scale variational quantum circuits. We test and benchmark our technique for distribution loading with quantum circuit born machines (QCBMs). With QCBMs, probability distributions are loaded into the squared amplitudes of computational basis vectors represented by bitstrings. Our key insight is to take advantage of the fact that the most significant (qu)bits have a greater effect on the final distribution and can be learned first. One can think of it as a generalization of layerwise learning, where some parameters of the variational circuit are learned first to prevent the phenomena of barren plateaus. We briefly review adjoint methods for computing the gradient, in particular for loss functions that are not expectation values of observables. We first compare the role of connectivity in the variational ansatz for the task of loading a Gaussian distribution on nine qubits, finding that 2D connectivity greatly outperforms qubits arranged on a line. Based on our observations, we then implement this strategy on large-scale numerical experiments with GPUs, training a QCBM to reproduce a 3-dimensional multivariate Gaussian distribution on 27 qubits up to ∼4%\sim4\%∼4% total variation distance. Though barren plateau arguments do not strictly apply here due to the objective function not being tied to an observable, this is to our knowledge the first practical demonstration of variational learning on large numbers of qubits. We also demonstrate hierarchical learning as a resource-efficient way to load distributions for existing quantum hardware (IBM's 7 and 27 qubit devices) in tandem with Fire Opal optimizations.
Quantum combs play a vital role in characterizing and transforming quantum processes, with wide-ranging applications in quantum information processing. However, obtaining the explicit quantum circuit for the desired quantum comb remains a challenging problem. We propose PQComb, a novel framework that employs parameterized quantum circuits (PQCs) or quantum neural networks to harness the full potential of quantum combs for diverse quantum process transformation tasks. This method is well-suited for near-term quantum devices and can be applied to various tasks in quantum machine learning. As a notable application, we present two streamlined protocols for the time-reversal simulation of unknown qubit unitary evolutions, reducing the ancilla qubit overhead from six to three compared to the previous best-known method. We also extend PQComb to solve the problems of qutrit unitary transformation and channel discrimination. Furthermore, we demonstrate the hardware efficiency and robustness of our qubit unitary inversion protocol under realistic noise simulations of IBM-Q superconducting quantum hardware, yielding a significant improvement in average similarity over the previous protocol under practical regimes. PQComb’s versatility and potential for broader applications in quantum machine learning pave the way for more efficient and practical solutions to complex quantum tasks.
Reconstructing quantum states from measurement data represents a formidable challenge in quantum information science, especially as system sizes grow beyond the reach of traditional tomography methods. While recent studies have explored quantum machine learning (QML) for quantum state tomography (QST), nearly all rely on idealized assumptions, such as direct access to the unknown quantum state as quantum data input, which are incompatible with current hardware constraints. In this work, we present a QML-based tomography protocol that operates entirely on classical measurement data and is fully executable on noisy intermediate-scale quantum (NISQ) devices. Our approach employs a variational quantum circuit trained to reconstruct quantum states based solely on measurement outcomes. We test the method in simulation, achieving high-fidelity reconstructions of diverse quantum states, including GHZ states, spin chain ground states, and states generated by random circuits. The protocol is then validated on quantum hardware from IBM and IonQ. Additionally, we demonstrate accurate tomography is possible using incomplete measurement bases, a crucial step towards scaling up our protocol. Our results in various scenarios illustrate successful state reconstruction with fidelity reaching 90% or higher. To our knowledge, this is the first QML-based tomography scheme that has been implemented on real quantum processors using exclusively classical measurements. This work establishes the feasibility of QML-based tomography on current quantum platforms and offers a scalable pathway for practical quantum state reconstruction.
Abstract With the increased focus on quantum circuit learning for near-term applications on quantum devices, in conjunction with unique challenges presented by cost function landscapes of parametrized quantum circuits, strategies for effective training are becoming increasingly important. In order to ameliorate some of these challenges, we investigate a layerwise learning strategy for parametrized quantum circuits. The circuit depth is incrementally grown during optimization, and only subsets of parameters are updated in each training step. We show that when considering sampling noise, this strategy can help avoid the problem of barren plateaus of the error surface due to the low depth of circuits, low number of parameters trained in one step, and larger magnitude of gradients compared to training the full circuit. These properties make our algorithm preferable for execution on noisy intermediate-scale quantum devices. We demonstrate our approach on an image-classification task on handwritten digits, and show that layerwise learning attains an 8% lower generalization error on average in comparison to standard learning schemes for training quantum circuits of the same size. Additionally, the percentage of runs that reach lower test errors is up to 40% larger compared to training the full circuit, which is susceptible to creeping onto a plateau during training.
To efficiently reconstruct noisy quantum states, specifically the normal arbitrary superposition state (NASS), on a quantum computer, we propose a reconstruction scheme based on parameterized quantum circuits (PQCs). First, the NASS image representation method is adopted, encoding images as quantum states, and a PQC is designed for reconstructing NASS. Then, gradient descent is used to train and update the circuit parameters. Finally, experiments validate the reconstruction of the quantum state. Experimental results show that for noisy grayscale image reconstruction, the structural similarity index measure reaches 0.922, and the peak signal-to-noise ratio is 26.74 dB. In terms of algorithmic complexity, the proposed scheme has a complexity of 4n2−4n+1, requiring no ancillary qubits. The proposed reconstruction scheme for noisy NASS quantum states offers the advantages of requiring fewer qubits and lower complexity while achieving effective reconstruction performance.