publications

2024

1. Amortized Stabilizer R}’enyi Entropy of Quantum Dynamics

   Chengkai Zhu, Yu-Ao Chen, Zanqiu Shen, Zhiping Liu, and 2 more authors

   Sep 2024
2. Quantum Linear Algebra Is All You Need for Transformer Architectures

   Naixu Guo, Zhan Yu, Matthew Choi, Aman Agrawal, and 3 more authors

   May 2024

   Abs arXiv Bib

   Generative machine learning methods such as large-language models are revolutionizing the creation of text and images. While these models are powerful they also harness a large amount of computational resources. The transformer is a key component in large language models that aims to generate a suitable completion of a given partial sequence. In this work, we investigate transformer architectures under the lens of fault-tolerant quantum computing. The input model is one where pre-trained weight matrices are given as block encodings to construct the query, key, and value matrices for the transformer. As a first step, we show how to prepare a block encoding of the self-attention matrix, with a row-wise application of the softmax function using the Hadamard product. In addition, we combine quantum subroutines to construct important building blocks in the transformer, the residual connection, layer normalization, and the feed-forward neural network. Our subroutines prepare an amplitude encoding of the transformer output, which can be measured to obtain a prediction. We discuss the potential and challenges for obtaining a quantum advantage.

   @misc{guo2024quantum,
     title = {Quantum Linear Algebra Is All You Need for {{Transformer}} Architectures},
     author = {Guo, Naixu and Yu, Zhan and Choi, Matthew and Agrawal, Aman and Nakaji, Kouhei and {Aspuru-Guzik}, Al{\'a}n and Rebentrost, Patrick},
     year = {2024},
     month = may,
     number = {arXiv:2402.16714},
     eprint = {2402.16714},
     primaryclass = {quant-ph},
     publisher = {arXiv},
     doi = {10.48550/arXiv.2402.16714},
     url = {http://arxiv.org/abs/2402.16714},
     archiveprefix = {arXiv},
   }

2023

1. Quantum Phase Processing and Its Applications in Estimating Phase and Entropies

   Youle Wang, Lei Zhang, Zhan Yu, and Xin Wang

   Physical Review A, Dec 2023

   Accepted as a regular talk at AQIS 2023.

   Abs arXiv Bib

   Quantum computing can provide speedups in solving many problems as the evolution of a quantum system is described by a unitary operator in an exponentially large Hilbert space. Such unitary operators change the phase of their eigenstates and make quantum algorithms fundamentally different from their classical counterparts. Based on this unique principle of quantum computing, we develop a new algorithmic toolbox “quantum phase processing” that can directly apply arbitrary trigonometric transformations to eigenphases of a unitary operator. The quantum phase processing circuit is constructed simply, consisting of single-qubit rotations and controlled-unitaries, typically using only one ancilla qubit. Besides the capability of phase transformation, quantum phase processing in particular can extract the eigen-information of quantum systems by simply measuring the ancilla qubit, making it naturally compatible with indirect measurement. Quantum phase processing complements another powerful framework known as quantum singular value transformation and leads to more intuitive and efficient quantum algorithms for solving problems that are particularly phase-related. As a notable application, we propose a new quantum phase estimation algorithm without quantum Fourier transform, which requires the fewest ancilla qubits and matches the best performance so far. We further exploit the power of our method by investigating a plethora of applications in Hamiltonian simulation, entanglement spectroscopy and quantum entropies estimation, demonstrating improvements or optimality for almost all cases.

   @article{wang2023quantum,
     title = {Quantum Phase Processing and Its Applications in Estimating Phase and Entropies},
     author = {Wang, Youle and Zhang, Lei and Yu, Zhan and Wang, Xin},
     year = {2023},
     month = dec,
     journal = {Physical Review A},
     volume = {108},
     number = {6},
     pages = {062413},
     publisher = {American Physical Society},
     doi = {10.1103/PhysRevA.108.062413},
     url = {https://link.aps.org/doi/10.1103/PhysRevA.108.062413},
     note = {Accepted as a regular talk at AQIS 2023.}
   }

2. Provable Advantage of Parameterized Quantum Circuit in Function Approximation

   Zhan Yu, Qiuhao Chen, Yuling Jiao, Yinan Li, and 3 more authors

   Oct 2023

   Abs arXiv Bib

   Understanding the power of parameterized quantum circuits (PQCs) in accomplishing machine learning tasks is one of the most important questions in quantum machine learning. In this paper, we analyze the expressivity of PQCs through the lens of function approximation. Previously established universal approximation theorems for PQCs are mainly nonconstructive, leading us to the following question: How large do the PQCs need to be to approximate the target function up to a given error? We exhibit explicit constructions of data re-uploading PQCs for approximating continuous and smooth functions and establish quantitative approximation error bounds in terms of the width, the depth and the number of trainable parameters of the PQCs. To achieve this, we utilize techniques from quantum signal processing and linear combinations of unitaries to construct PQCs that implement multivariate polynomials. We implement global and local approximation techniques using Bernstein polynomials and local Taylor expansion and analyze their performances in the quantum setting. We also compare our proposed PQCs to nearly optimal deep neural networks in approximating high-dimensional smooth functions, showing that the ratio between model sizes of PQC and deep neural networks is exponentially small with respect to the input dimension. This suggests a potentially novel avenue for showcasing quantum advantages in quantum machine learning.

   @misc{yu2023provable,
     title = {Provable {{Advantage}} of {{Parameterized Quantum Circuit}} in {{Function Approximation}}},
     author = {Yu, Zhan and Chen, Qiuhao and Jiao, Yuling and Li, Yinan and Lu, Xiliang and Wang, Xin and Yang, Jerry Zhijian},
     year = {2023},
     month = oct,
     number = {arXiv:2310.07528},
     eprint = {2310.07528},
     primaryclass = {quant-ph},
     publisher = {{arXiv}},
     doi = {10.48550/arXiv.2310.07528},
     url = {http://arxiv.org/abs/2310.07528},
   }

3. Efficient Information Recovery from Pauli Noise via Classical Shadow

   Yifei Chen, Zhan Yu, Chenghong Zhu, and Xin Wang

   May 2023

   Abs arXiv Bib

   The rapid advancement of quantum computing has led to an extensive demand for effective techniques to extract classical information from quantum systems, particularly in fields like quantum machine learning and quantum chemistry. However, quantum systems are inherently susceptible to noises, which adversely corrupt the information encoded in quantum systems. In this work, we introduce an efficient algorithm that can recover information from quantum states under Pauli noise. The core idea is to learn the necessary information of the unknown Pauli channel by post-processing the classical shadows of the channel. For a local and bounded-degree observable, only partial knowledge of the channel is required rather than its complete classical description to recover the ideal information, resulting in a polynomial-time algorithm. This contrasts with conventional methods such as probabilistic error cancellation, which requires the full information of the channel and exhibits exponential scaling with the number of qubits. We also prove that this scalable method is optimal on the sample complexity and generalise the algorithm to the weight contracting channel. Furthermore, we demonstrate the validity of the algorithm on the 1D anisotropic Heisenberg-type model via numerical simulations. As a notable application, our method can be severed as a sample-efficient error mitigation scheme for Clifford circuits.

   @misc{chen2023efficient,
     title = {Efficient Information Recovery from {{Pauli}} Noise via Classical Shadow},
     author = {Chen, Yifei and Yu, Zhan and Zhu, Chenghong and Wang, Xin},
     year = {2023},
     month = may,
     number = {arXiv:2305.04148},
     eprint = {2305.04148},
     primaryclass = {math-ph, physics:quant-ph},
     publisher = {{arXiv}},
     doi = {10.48550/arXiv.2305.04148},
     url = {http://arxiv.org/abs/2305.04148},
     archiveprefix = {arxiv},
   }

4. Optimal Quantum Dataset for Learning a Unitary Transformation

   Zhan Yu, Xuanqiang Zhao, Benchi Zhao, and Xin Wang

   Physical Review Applied, Mar 2023

   Abs arXiv Bib

   Unitary transformations formulate the time evolution of quantum states. How to learn a unitary transformation efficiently is a fundamental problem in quantum machine learning. The most natural and leading strategy is to train a quantum machine learning model based on a quantum dataset. Although the presence of more training data results in better models, using too much data reduces the efficiency of training. In this work, we solve the problem on the minimum size of sufficient quantum datasets for learning a unitary transformation exactly, which reveals the power and limitation of quantum data. First, we prove that the minimum size of a dataset with pure states is 2n for learning an n-qubit unitary transformation. To fully explore the capability of quantum data, we introduce a practical quantum dataset consisting of n+1 elementary tensor product states that are sufficient for exact training. The main idea is to simplify the structure utilizing decoupling, which leads to an exponential improvement in the size of the datasets with pure states. Furthermore, we show that the size of the quantum dataset with mixed states can be reduced to a constant, which yields an optimal quantum dataset for learning a unitary. We showcase the applications of our results in oracle compiling and Hamiltonian simulation. Notably, to accurately simulate a three-qubit one-dimensional nearest-neighbor Heisenberg model, our circuit only uses 96 elementary quantum gates, which is significantly less than 4080 gates in the circuit constructed by the Trotter-Suzuki product formula.

   @article{yu2023optimal,
     title = {Optimal {{Quantum Dataset}} for {{Learning}} a {{Unitary Transformation}}},
     author = {Yu, Zhan and Zhao, Xuanqiang and Zhao, Benchi and Wang, Xin},
     year = {2023},
     month = mar,
     journal = {Physical Review Applied},
     volume = {19},
     number = {3},
     pages = {034017},
     publisher = {{American Physical Society}},
     doi = {10.1103/PhysRevApplied.19.034017},
     url = {https://link.aps.org/doi/10.1103/PhysRevApplied.19.034017},
   }

2022

1. Power and Limitations of Single-Qubit Native Quantum Neural Networks

   Zhan Yu, Hongshun Yao, Mujin Li, and Xin Wang

   In Advances in Neural Information Processing Systems, Mar 2022

   Abs arXiv Bib HTML

   Quantum neural networks (QNNs) have emerged as a leading strategy to establish applications in machine learning, chemistry, and optimization. While the applications of QNN have been widely investigated, its theoretical foundation remains less understood. In this paper, we formulate a theoretical framework for the expressive ability of data re-uploading quantum neural networks that consist of interleaved encoding circuit blocks and trainable circuit blocks. First, we prove that single-qubit quantum neural networks can approximate any univariate function by mapping the model to a partial Fourier series. Beyond previous works’ understanding of existence, we in particular establish the exact correlations between the parameters of the trainable gates and the working Fourier coefficients, by exploring connections to quantum signal processing. Second, we discuss the limitations of single-qubit native QNNs on approximating multivariate functions by analyzing the frequency spectrum and the flexibility of Fourier coefficients. We further demonstrate the expressivity and limitations of single-qubit native QNNs via numerical experiments. As applications, we introduce natural extensions to multi-qubit quantum neural networks, which exhibit the capability of classifying real-world multi-dimensional data. We believe these results would improve our understanding of QNNs and provide a helpful guideline for designing powerful QNNs for machine learning tasks.

   @inproceedings{yu2022power,
     title = {Power and Limitations of Single-Qubit Native Quantum Neural Networks},
     booktitle = {Advances in Neural Information Processing Systems},
     author = {Yu, Zhan and Yao, Hongshun and Li, Mujin and Wang, Xin},
     editor = {Koyejo, S. and Mohamed, S. and Agarwal, A. and Belgrave, D. and Cho, K. and Oh, A.},
     year = {2022},
     volume = {35},
     pages = {27810--27823},
     publisher = {{Curran Associates, Inc.}},
     url = {https://proceedings.neurips.cc/paper_files/paper/2022/file/b250de41980b58d34d6aadc3f4aedd4c-Paper-Conference.pdf},
   }

2020

1. Analysis of Lackadaisical Quantum Walks

   Peter Høyer, and Zhan Yu

   Quantum Information and Computation, Nov 2020

   Abs arXiv Bib

   The lackadaisical quantum walk is a quantum analogue of the lazy random walk obtained by adding a self-loop to each vertex in the graph. We analytically prove that lackadaisical quantum walks can find a unique marked vertex on any regular locally arc-transitive graph with constant success probability quadratically faster than the hitting time. This result proves several speculations and numerical findings in previous work, including the conjectures that the lackadaisical quantum walk finds a unique marked vertex with constant success probability on the torus, cycle, Johnson graphs, and other classes of vertex-transitive graphs. Our proof establishes and uses a relationship between lackadaisical quantum walks and quantum interpolated walks for any locally arc-transitive graph.

   @article{hoyer2020analysis,
     title = {Analysis of {{Lackadaisical Quantum Walks}}},
     author = {H{\o}yer, Peter and Yu, Zhan},
     year = {2020},
     month = nov,
     journal = {Quantum Information and Computation},
     volume = {20},
     number = {13\&14},
     eprint = {2002.11234},
     primaryclass = {quant-ph},
     pages = {1138--1153},
     issn = {15337146, 15337146},
     doi = {10.26421/QIC20.13-14-4},
     url = {http://arxiv.org/abs/2002.11234},
   }