arXiv subjects

Bujiao Wu

Publications and source records attributed to Bujiao Wu.

At least 19 recordsLinked to original sources

arXiv2026

From Bits to Qubits: The Theory and Practice of Quantum Data Encoding

Encoding classical data into quantum systems is a foundational step in the execution of nearly all quantum algorithms, and a critical bottleneck in realizing practical quantum advantage. This review provides a comprehensive account of the concepts, algorithms, and practical considerations associated with quantum data encoding. We trace the development from its early conceptual foundations to recent advances, considering commonly used access models, such as quantum state preparation, unitary synthesis, QRAM and block encoding. We survey the circuit size, depth, space-time tradeoffs, as well as non-Clifford resources required for fault-tolerant implementation. We also discuss the roles of different access models in quantum algorithms. Special attention is given to structured data, such as sparse data, Boolean functions and data represented by tensor networks. This review bridges theory and applications, serving both as a pedagogical guide for newcomers and as a reference for active researchers. We also highlight the pivotal role of quantum data encoding in quantum computing and provide insights into future directions that will enable quantum advantage.

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arXiv2026

Barren Plateaus Beyond Observable Concentration

Parameterized quantum circuits (PQCs) are central to quantum machine learning and near-term quantum simulation, but their scalability is often hindered by barren plateaus (BPs), where gradients decay exponentially with system size. Prior explanations, including expressivity, entanglement, locality, and noise, are often presented in ways that conflate two distinct issues: concentration of the measured observable and loss of parameter sensitivity caused by circuit dynamics.We develop a unified statistical framework that separates these mechanisms. We show that several standard BP explanations, including locality- and entanglement-related effects, can be understood through a single phenomenon that we term observable concentration (OC). Importantly, we prove that avoiding OC is necessary but not sufficient for trainability. Beyond OC, we identify two distinct mid-circuit sources of gradient suppression. First, in circuits with effectively independent forward and backward blocks, parameter perturbations can propagate outside the measurement light cone and become inaccessible to the final observable, yielding information-loss-induced BPs. Second, BPs can also arise in circuits with highly correlated forward and backward blocks, as we demonstrate through an echo-type circuit model that is reminiscent of information scrambling.

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arXiv2025

State Similarity in Modular Superconducting Quantum Processors with Classical Communications

As quantum devices continue to scale, distributed quantum computing emerges as a promising strategy for executing large-scale tasks across modular quantum processors. A central challenge in this paradigm is verifying the correctness of computational outcomes when subcircuits are executed independently following circuit cutting. Here we propose a cross-platform fidelity estimation algorithm tailored for modular architectures. Our method achieves substantial reductions in sample complexity compared to previous approaches designed for single-processor systems. We experimentally implement the protocol on modular superconducting quantum processors with up to 6 qubits to verify the similarity of two 11-qubit GHZ states. Beyond verification, we show that our algorithm enables a federated quantum kernel method that preserves data privacy. As a proof of concept, we apply it to a 5-qubit quantum phase learning task using six 3-qubit modules, successfully extracting phase information with just eight training samples. These results establish a practical path for scalable verification and trustworthy quantum machine learning of modular quantum processors.

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arXiv2025

Adaptive-depth randomized measurement for fermionic observables

Accurate estimation of fermionic observables is essential for advancing quantum physics and chemistry. The fermionic classical shadow (FCS) method offers an efficient framework for estimating these observables without requiring a transformation into a Pauli basis. However, the random matchgate circuits in FCS require polynomial-depth circuits with a brickwork structure, which presents significant challenges for near-term quantum devices with limited computational resources. To address this limitation, we introduce an adaptive-depth fermionic classical shadow (ADFCS) protocol designed to reduce the circuit depth while maintaining the estimation accuracy and the order of sample complexity. Through theoretical analysis and numerical fitting, we establish that the required depth for approximating a fermionic observable $H$ scales as $\max\{d^2_{\text{int}}(H)/\log n, d_{\text{int}}(H)\}$ where $d_{\text{int}}$ is the interaction distance of $H$. We demonstrate the effectiveness of the ADFCS protocol through numerical experiments, which show similar accuracy to the traditional FCS method while requiring significantly fewer resources. Additionally, we apply ADFCS to compute the expectation value of the Kitaev chain Hamiltonian, further validating its performance in practical scenarios. Our findings suggest that ADFCS can enable more efficient quantum simulations, reducing circuit depth while preserving the fidelity of quantum state estimations, offering a viable solution for near-term quantum devices.

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arXiv2024

Expectation value estimation with parametrized quantum circuits

Estimating properties of quantum states, such as fidelities, molecular energies, and correlation functions, is a fundamental task in quantum information science. Due to the limitation of practical quantum devices, including limited circuit depth and connectivity, estimating even linear properties encounters high sample complexity. To address this inefficiency, we propose a framework that optimizes sample complexity for estimating the expectation value of any observable using a shallow parameterized quantum circuit. Within this framework, we introduce two decomposition algorithms, a tensor network approach and a greedy projection approach that decompose the target observable into a linear combination of multiple observables, each of which can be diagonalized with the shallow circuit. Using this decomposition, we then apply an importance sampling algorithm to estimate the expectation value of the target observable. We numerically demonstrate the performance of our algorithm by estimating the expectation values of some specific Hamiltonians and inner product of a Slater determinant with a pure state, highlighting advantages compared to some conventional methods. Additionally, we derive the fundamental lower bound for the sample complexity required to estimate a target observable using a given shallow quantum circuit, thereby enhancing our understanding of the capabilities of shallow circuits in quantum learning tasks.

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arXiv2024

Error-mitigated fermionic classical shadows on noisy quantum devices

Efficiently estimating fermionic Hamiltonian expectation values is vital for simulating various physical systems. Classical shadow (CS) algorithms offer a solution by reducing the number of quantum state copies needed, but noise in quantum devices poses challenges. We propose an error-mitigated CS algorithm assuming gate-independent, time-stationary, and Markovian (GTM) noise. For $n$-qubit systems, our algorithm, which employs the easily prepared initial state $|0^n\rangle\!\langle 0^n|$ assumed to be noiseless, efficiently estimates $k$-RDMs with $\widetilde{\mathcal O}(kn^k)$ state copies and $\widetilde{\mathcal O}(\sqrt{n})$ calibration measurements for GTM noise with constant fidelities. We show that our algorithm is robust against noise types like depolarizing, damping, and $X$-rotation noise with constant strengths, showing scalings akin to prior CS algorithms for fermions but with better noise resilience. Numerical simulations confirm our algorithm's efficacy in noisy settings, suggesting its viability for near-term quantum devices.

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arXiv2024

Leakage Benchmarking for Universal Gate Sets

Errors are common issues in quantum computing platforms, among which leakage is one of the most challenging to address. This is because leakage, i.e., the loss of information stored in the computational subspace to undesired subspaces in a larger Hilbert space, is more difficult to detect and correct than errors that preserve the computational subspace. As a result, leakage presents a significant obstacle to the development of fault-tolerant quantum computation. In this paper, we propose an efficient and accurate benchmarking framework called leakage randomized benchmarking (LRB) for measuring leakage rates on multi-qubit quantum systems. Our approach is more insensitive to state preparation and measurement (SPAM) noise than existing leakage benchmarking protocols, requires fewer assumptions about the gate set itself, and can be used to benchmark multi-qubit leakages, which was not done previously. We also extend the LRB protocol to an interleaved variant called interleaved LRB (iLRB), which can benchmark the average leakage rate of generic $n$-site quantum gates with reasonable noise assumptions. We demonstrate the iLRB protocol on benchmarking generic two-qubit gates realized using flux tuning, and analyze the behavior of iLRB under corresponding leakage models. Our numerical experiments show good agreement with theoretical estimations, indicating the feasibility of both the LRB and iLRB protocols. Keywords: quantum computing; randomized benchmarking; leakage error; quantum gates

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arXiv2023

Tensor-network-assisted variational quantum algorithm

Near-term quantum devices generally suffer from shallow circuit depth and hence limited expressivity due to noise and decoherence. To address this, we propose tensor-network-assisted parametrized quantum circuits, which concatenate a classical tensor-network operator with a quantum circuit to effectively increase the circuit's expressivity without requiring a physically deeper circuit. We present a framework for tensor-network-assisted variational quantum algorithms that can solve quantum many-body problems using shallower quantum circuits. We demonstrate the efficiency of this approach by considering two examples of unitary matrix-product operators and unitary tree tensor networks, showing that they can both be implemented efficiently. Through numerical simulations, we show that the expressivity of these circuits is greatly enhanced with the assistance of tensor networks. We apply our method to two-dimensional Ising models and one-dimensional time-crystal Hamiltonian models with up to 16 qubits and demonstrate that our approach consistently outperforms conventional methods using shallow quantum circuits.

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arXiv2023

Efficient measurement schemes for bosonic systems

Boson is one of the most basic types of particles and preserves the commutation relation. An efficient way to measure a bosonic system is important not only for simulating complex physics phenomena of bosons (such as nuclei) on a qubit based quantum computer, but for extracting classical information from a quantum simulator/computer that itself is built with bosons (such as a continuous variable quantum computer). Extending the recently proposed measurement schemes for qubits, such as shadow tomography and other local measurement schemes, here we study efficient measurement approaches for bosonic systems. We consider truncated qudit and continuous variable systems, corresponding to simulated bosons on a discrete quantum computer and an inherent boson system, respectively, and propose different measurement schemes with theoretical analyses of the variances for these two cases. We numerically test the schemes for measuring nuclei vibrations simulated using a discrete quantum computer and a continuous variable Gaussian state, and the simulation results show great improvement of the performance of the proposed method compared to conventional ones.

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arXiv2023

Quantum Phase Recognition via Quantum Kernel Methods

The application of quantum computation to accelerate machine learning algorithms is one of the most promising areas of research in quantum algorithms. In this paper, we explore the power of quantum learning algorithms in solving an important class of Quantum Phase Recognition (QPR) problems, which are crucially important in understanding many-particle quantum systems. We prove that, under widely believed complexity theory assumptions, there exists a wide range of QPR problems that cannot be efficiently solved by classical learning algorithms with classical resources. Whereas using a quantum computer, we prove the efficiency and robustness of quantum kernel methods in solving QPR problems through Linear order parameter Observables. We numerically benchmark our algorithm for a variety of problems, including recognizing symmetry-protected topological phases and symmetry-broken phases. Our results highlight the capability of quantum machine learning in predicting such quantum phase transitions in many-particle systems.

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arXiv2023

Learning the Complexity of Weakly Noisy Quantum States

Quantifying the complexity of quantum states is a longstanding key problem in various subfields of science, ranging from quantum computing to the black-hole theory. The lower bound on quantum pure state complexity has been shown to grow linearly with system size [Haferkamp et al., 2022]. However, extending this result to noisy circuit environments, which better reflect real quantum devices, remains an open challenge. In this paper, we explore the complexity of weakly noisy quantum states via the quantum learning method. We present an efficient learning algorithm, that leverages the classical shadow representation of target quantum states, to predict the circuit complexity of weakly noisy quantum states. Our algorithm is proved to be optimal in terms of sample complexity accompanied with polynomial classical processing time. Our result builds a bridge between the learning algorithm and quantum state complexity, meanwhile highlighting the power of learning algorithm in characterizing intrinsic properties of quantum states.

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arXiv2023

Optimization of CNOT circuits on limited connectivity architecture

A CNOT circuit is the key gadget for entangling qubits in quantum computing systems. However, the qubit connectivity of noisy intermediate-scale quantum (NISQ) devices is constrained by their {limited connectivity architecture}. To improve the performance of CNOT circuits on NISQ devices, we investigate the optimization of the size/depth of CNOT circuits under the limited connectivity architecture. We present a method that can optimize the size of any $n$-qubit CNOT circuit $O\left(\frac{n^2}{\log δ}\right)$ on any connected graph with minimum degree $δ$, and prove this bound is optimal for the regular graph. For the near-term sparsely connected structure, we additionally present a method that can optimize the size of any $n$-qubit CNOT circuit to below $2n^2$. The numerical experiment shows that our method performs better than state-of-the-art results. Specifically, we present an example to illustrate the applicability of our algorithm. For the grid structure, which is commonly used in current quantum devices, we demonstrate that the depth of any $n$-qubit CNOT circuit can be optimized to be linear in $n$ with certain ancillary qubits (ancillas). Experimental results indicate that this method has significant improvements compared with all of the existing methods. We additionally test our algorithms on the five-qubit IBMQ devices, and the experiments show that the measurement results of the optimized circuit with our algorithm are more robust to noise compared with the IBM mapping method.

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arXiv2023

Overlapped grouping measurement: A unified framework for measuring quantum states

Quantum algorithms designed for realistic quantum many-body systems, such as chemistry and materials, usually require a large number of measurements of the Hamiltonian. Exploiting different ideas, such as {importance sampling,} observable compatibility, or classical shadows of quantum states, different advanced measurement schemes have been proposed to greatly reduce the large measurement cost. Yet, the underline cost reduction mechanisms seem distinct from each other, and how to systematically find the optimal scheme remains a critical challenge. Here, we address this challenge by proposing a unified framework of quantum measurements, incorporating advanced measurement methods as special cases. Our framework allows us to introduce a general scheme~ -- ~overlapped grouping measurement, which simultaneously exploits the advantages of most existing methods. An intuitive understanding of the scheme is to partition the measurements into overlapped groups with each one consisting of compatible measurements. We provide explicit grouping strategies and numerically verify its performance for different molecular Hamiltonians with up to 16 qubits. Our numerical result shows significant improvements over existing schemes. Our work paves the way for efficient quantum measurement and fast quantum processing with current and near-term quantum devices.

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arXiv2022

Optimal Space-Depth Trade-Off of CNOT Circuits in Quantum Logic Synthesis

Decoherence -- in the current physical implementations of quantum computers -- makes depth reduction a vital task in quantum-circuit design. Moore and Nilsson (SIAM Journal of Computing, 2001) demonstrated that additional qubits -- known as ancillae -- can be used to provide an extended space to parallelize quantum circuits. Specifically, they proved that, with $O(n^2)$ ancillae, any $n$-qubit CNOT circuit can be transformed into an equivalent one of $O(\log n)$ depth. However, the near-term quantum technologies can only support a limited amount of qubits, making space-depth trade-off a fundamental research subject for quantum-circuit synthesis. In this work, we establish an asymptotically optimal space-depth trade-off for CNOT circuits. We prove that any $n$-qubit CNOT circuit can be parallelized to $$O\left(\max\left\{\log n, \frac{n^2}{(n+m)\log (n+m)}\right\}\right)$$ depth with $m$ ancillae. This bound is tight even if the task is expanded from exact synthesis to the approximation of CNOT circuits with arbitrary two-qubit quantum gates. Our result can be extended to stabilizer circuits via the reduction by Aaronson and Gottesman (Physical Review A, 2004). Furthermore, we provide hardness evidence for optimizing CNOT circuits in terms of size or depth. Our result has improved upon two previous papers that motivated our work. Moore-Nilsson's construction (aforementioned) for $O(\log n)$-depth CNOT circuit synthesis: We have reduced their need for ancillae by a factor of $\log^2 n$ by showing that $m= O(n^2/\log^2 n)$ additional qubits -- which is asymptotically optimal -- suffice to build equivalent $O(\log n)$-depth $O(n^2/\log n)$-size CNOT circuits. Patel-Markov-Hayes's construction (Quantum Information & Computation 2008) for $m = 0$: We have reduced their depth by a factor of $n$ and achieved the asymptotically optimal bound of $O(n/\log n)$.

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arXiv2022

Power of Quantum Generative Learning

The intrinsic probabilistic nature of quantum mechanics invokes endeavors of designing quantum generative learning models (QGLMs). Despite the empirical achievements, the foundations and the potential advantages of QGLMs remain largely obscure. To narrow this knowledge gap, here we explore the generalization property of QGLMs, the capability to extend the model from learned to unknown data. We consider two prototypical QGLMs, quantum circuit Born machines and quantum generative adversarial networks, and explicitly give their generalization bounds. The result identifies superiorities of QGLMs over classical methods when quantum devices can directly access the target distribution and quantum kernels are employed. We further employ these generalization bounds to exhibit potential advantages in quantum state preparation and Hamiltonian learning. Numerical results of QGLMs in loading Gaussian distribution and estimating ground states of parameterized Hamiltonians accord with the theoretical analysis. Our work opens the avenue for quantitatively understanding the power of quantum generative learning models.

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arXiv2022

Efficient quantum circuit synthesis for SAT-oracle with limited ancillary qubit

How to implement quantum oracle with limited resources raises concerns these days. We design two ancilla-adjustable and efficient algorithms to synthesize SAT-oracle, the key component in solving SAT problems. The previous work takes 2m-1 ancillary qubits and O(m) elementary gates to synthesize an m clauses oracle. The first algorithm reduces the number of ancillary qubits to 2\sqrt{m}, with at most an eightfold increase in circuit size. The number of ancillary qubits can be further reduced to 3 with a quadratic increase in circuit size. The second algorithm aims to reduce the circuit depth. By leveraging of the second algorithm, the circuit depth can be reduced to O(log m) with m ancillary qubits.

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arXiv2020

Quantum-classical algorithms for skewed linear systems with optimized Hadamard test

The solving of linear systems provides a rich area to investigate the use of nearer-term, noisy, intermediate-scale quantum computers. In this work, we discuss hybrid quantum-classical algorithms for skewed linear systems for over-determined and under-determined cases. Our input model is such that the columns or rows of the matrix defining the linear system are given via quantum circuits of poly-logarithmic depth and the number of circuits is much smaller than their Hilbert space dimension. Our algorithms have poly-logarithmic dependence on the dimension and polynomial dependence in other natural quantities. In addition, we present an algorithm for the special case of a factorized linear system with run time poly-logarithmic in the respective dimensions. At the core of these algorithms is the Hadamard test and in the second part of this paper we consider the optimization of the circuit depth of this test. Given an $n$-qubit and $d$-depth quantum circuit $\mathcal{C}$, we can approximate $\langle 0|\mathcal{C}|0\rangle$ using $(n + s)$ qubits and $O\left(\log s + d\log (n/s) + d\right)$-depth quantum circuits, where $s\leq n$. In comparison, the standard implementation requires $n+1$ qubits and $O(dn)$ depth. Lattice geometries underlie recent quantum supremacy experiments with superconducting devices. We also optimize the Hadamard test for an $(l_1\times l_2)$ lattice with $l_1 \times l_2 = n$, and can approximate $\langle 0|\mathcal{C} |0\rangle$ with $(n + 1)$ qubits and $O\left(d \left(l_1 + l_2\right)\right)$-depth circuits. In comparison, the standard depth is $O\left(d n^2\right)$ in this setting. Both of our optimization methods are asymptotically tight in the case of one-depth quantum circuits $\mathcal{C}$.

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arXiv2019

Speedup in Classical Simulation of Gaussian Boson Sampling

Gaussian boson sampling is a promising model for demonstrating quantum computational supremacy, which eases the experimental challenge of the standard boson-sampling proposal. Here by analyzing the computational costs of classical simulation of Gaussian boson sampling,we establish a lower bound for achieving quantum computational supremacy for a class of Gaussian boson-sampling problems, where squeezed states are injected into every input mode. Specifically, we propose a method for simplifying the brute-force calculations for the transition probabilities in Gaussian boson sampling, leading to a significant reduction of the simulation costs. Particularly, our numerical results indicate that we can simulate 18 photons Gaussian boson sampling at the output subspace on a normal laptop, 20 photons on a commercial workstation with 256 cores, and suggest about 30 photons for supercomputers. These numbers are significantly smaller than those in standard boson sampling, suggesting Gaussian boson sampling may be more feasible for demonstrating quantum computational supremacy.

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