arXiv subjects

Kaifeng Bu

Publications and source records attributed to Kaifeng Bu.

At least 19 recordsLinked to original sources

arXiv2026

Convex-Gaussianity of fermionic Gibbs states in perturbation theory

We study the structure of Gibbs states in weakly perturbed interacting fermionic systems. First, for a sparse Hamiltonian $H=H_0+V$ with a quadratic term $H_0$ and a non-quadratic perturbation $V$ of scale $ε$, we show that the Gibbs state $ρ_β$ decomposes into a convex combination of Gaussian states whenever the inverse temperature satisfies $β\le O(\log(1/ε))$. Moreover, we prove that this bound is asymptotically tight by establishing that $β\le Θ(\log(1/ε))$ is necessary for certain sparse Hamiltonians. This general framework applies directly to the weak-coupling (small-$\vert{}U\vert{}$) regime of the Fermi--Hubbard model with hopping $t$ and on-site interaction $U$ on any graph of maximum degree $D$. Complementarily, in the strong-coupling (small-$\vert{}t\vert{}$) regime, we show that the Gibbs state remains convex-Gaussian up to $β\le O\big(\vert{}U\vert{}^{-1}\log(\vert{}U\vert{}/(D\vert{}t\vert{}))\big)$, revealing a mechanism for convex-Gaussianity distinct from the weak-coupling setting.

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arXiv2026

Discrete Quantum Gaussians and Central Limit Theorem

We introduce a quantum convolution and a conceptual framework to study states in discrete-variable (DV) quantum systems. All our results suggest that stabilizer states play a role in DV quantum systems similar to the role that Gaussian states play in continuous-variable systems. Hence we suggest the name ``discrete quantum Gaussians'' for stabilizer states. For example, we prove that the convolution of two stabilizer states is another stabilizer state, and that stabilizer states extremize both quantum entropy and Fisher information. We establish a ``maximal entropy principle,'' a ``second law of thermodynamics for quantum convolution,'' and a ``quantum central limit theorem.'' The latter is based on iterating the convolution of a zero-mean quantum state, which we prove converges to a stabilizer state. We bound the exponential rate of convergence of the quantum central limit theorem by the ``magic gap,'' defined in terms of the support of the characteristic function of the state. We elaborate on our general results with a discussion of some examples, as well as extending many of them to quantum channels.

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arXiv2026

Multivariate Decoded Quantum Interferometry for Weighted Optimization

Decoded Quantum Interferometry (DQI) is a recently introduced quantum algorithm that reduces discrete optimization to decoding with potential advantages over the best-known polynomial-time classical algorithms for certain Max-LINSAT problems. In its original formulation, however, DQI treats all constraints uniformly and cannot exploit the weight structure present in most optimization problems of interest. In this work, we develop multivariate Decoded Quantum Interferometry (multivariate DQI) for weighted optimization problems, focusing on the weighted Max-LINSAT problem over a prime field. Grouping constraints into $N$ blocks by distinct weights, we introduce multivariate DQI states built from $N$-variable polynomials of bounded total degree, and derive a closed-form asymptotic expression for both their optimal expectation value and their concentration behavior. We give an explicit preparation circuit using a single decoder call, and extend the analysis to imperfect decoding. We also show that, for certain weighted OPI problems, multivariate DQI outperforms a natural weighted analogue of Prange's algorithm, which serves as the weighted counterpart of the classical benchmark used in the unweighted setting. Finally, we extend the ideas to Hamiltonian DQI, obtaining approximate Gibbs states for commuting Pauli Hamiltonians with block structure.

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arXiv2026

Quantum Ruzsa Divergence to Quantify Magic

In this work, we investigate the behavior of quantum entropy under quantum convolution and its application in quantifying magic. We first establish an entropic, quantum central limit theorem (q-CLT), where the rate of convergence is bounded by the magic gap. We also introduce a new quantum divergence based on quantum convolution, called the quantum Ruzsa divergence, to study the stabilizer structure of quantum states. We conjecture a ``convolutional strong subadditivity'' inequality, which leads to the triangle inequality for the quantum Ruzsa divergence. In addition, we propose two new magic measures, the quantum Ruzsa divergence of magic and quantum-doubling constant, to quantify the amount of magic in quantum states. Finally, by using the quantum convolution, we extend the classical, inverse sumset theory to the quantum case. These results shed new insight into the study of the stabilizer and magic states in quantum information theory.

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arXiv2026

Hamiltonian Decoded Quantum Interferometry for General Pauli Hamiltonians

In this work, we study the Hamiltonian Decoded Quantum Interferometry (HDQI) for the general Hamiltonians $H=\sum_ic_iP_i$ on an $n$-qubit system, where the coefficients $c_i\in \mathbb{R}$ and $P_i$ are Pauli operators. We show that, given access to an appropriate decoding oracle, there exist efficient quantum algorithms for preparing the state $ρ_{\mathcal P}(H) = \frac{\mathcal P^2(H)}{\text{Tr}[\mathcal P^2(H)]}$, where $\mathcal P(H)$ denotes the matrix function induced by a univariate polynomial $\mathcal P(x)$. Such states can be used to approximate the Gibbs states of $H$ for suitable choices of polynomials. We further demonstrate that the proposed algorithms are robust to imperfections in the decoding procedure. Our results substantially extend the scope of HDQI beyond stabilizer-like Hamiltonians, providing a method for Gibbs-state preparation and Hamiltonian optimization in a broad class of physically and computationally relevant quantum systems.

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arXiv2025

Quantum Higher Order Fourier Analysis and the Clifford Hierarchy

We propose a mathematical framework that we call quantum, higher-order Fourier analysis. This generalizes the classical theory of higher-order Fourier analysis, which led to many advances in number theory and combinatorics. We define a family of quantum measures on a Hilbert space, that reduce in the case of diagonal matrices to the classical uniformity norms. We show that our quantum measures and our related theory of quantum higher-order Fourier analysis characterize the Clifford hierarchy, an important notion of complexity in quantum information. In particular, we give a necessary and sufficient analytic condition that a unitary is an element of the k-th level of the Clifford hierarchy.

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arXiv2025

Decoded Quantum Interferometry Under Noise

Decoded Quantum Interferometry (DQI) is a recently proposed quantum optimization algorithm that exploits sparsity in the Fourier spectrum of objective functions, with the potential for exponential speedups over classical algorithms on suitably structured problems. While highly promising in idealized settings, its resilience to noise has until now been largely unexplored. To address this, we conduct a rigorous analysis of DQI under noise, focusing on local depolarizing noise. For the maximum linear satisfiability problem, we prove that, in the presence of noise, performance is governed by a noise-weighted sparsity parameter of the instance matrix, with solution quality decaying exponentially as sparsity decreases. We demonstrate this decay through numerical simulations on two special cases: the Optimal Polynomial Intersection problem and the Maximum XOR Satisfiability problem. The Fourier-analytic methods we develop can be readily adapted to other classes of random Pauli noise, making our framework applicable to a broad range of noisy quantum settings and offering guidance on preserving DQI's potential quantum advantage under realistic noise.

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arXiv2025

Stabilizer Testing and Magic Entropy via Quantum Fourier Analysis

Quantum Fourier analysis is an important topic in mathematical physics. We introduce a systematic protocol for testing and measuring ``magic'' in quantum states and gates, using a quantum Fourier approach. Magic, as a quantum resource, is necessary to achieve a quantum advantage in computation. Our protocols are based on quantum convolutions and swap tests, implemented via quantum circuits. We describe this for both qubit and qudit systems. Our quantum Fourier approach offers a unified method to quantify magic, in stabilizer circuits, as well as in matchgate and bosonic Gaussian circuits.

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arXiv2025

Graphical Calculus for Fermionic Tensors

We introduce a graphical calculus, consisting of a set of fermionic tensors with tensor-network equations, which can be used to perform various computations in fermionic many-body physics purely diagrammatically. The indices of our tensors primarily correspond to fermionic modes, but also include qubits and fixed odd-parity states. Our graphical calculus extends the ZX calculus for systems involving qubits. We apply the calculus in order to represent various objects, operations, and computations in physics, including fermionic Gaussian states, the partial trace of Majorana modes, purification protocols, fermionization and bosonization maps, and the construction of fermionic codes.

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arXiv2025

Efficient Measurement of Bosonic Non-Gaussianity

Non-Gaussian states are essential resources in quantum information processing. In this work, we investigate methods for quantifying bosonic non-Gaussianity in many-body systems. Building on recent theoretical insights into the self-convolution properties of bosonic pure states, we introduce non-Gaussian entropy as a new measure to characterize non-Gaussianity in bosonic pure states. We further propose a practical protocol for measuring non-Gaussian entropy using three beam splitters and four copies of the input state. In addition, we extend this framework to mixed states, providing a general approach to quantifying non-Gaussianity. Our results offer a convenient and efficient method for characterizing bosonic non-Gaussianity, paving the way for its implementation on near-term experimental platforms.

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arXiv2025

Magic Resource Can Enhance the Quantum Capacity of Channels

We investigate the role of magic resource in the quantum capacity of channels. We consider the quantum channel of the recently proposed discrete beam splitter with the fixed environmental state. We find that if the fixed environmental state is a stabilizer state, then the quantum capacity is zero. Moreover, we find that the quantum capacity is nonzero for some magic states, and the quantum capacity increases linearly with respect to the number of single-qudit magic states in the environment. We also bound the maximal quantum capacity of the discrete beam splitter in terms of the amount of magic resource in the environmental states. These results suggest that magic resource can increase the quantum capacity of channels; it sheds new insight into the role of stabilizer and magic states in quantum communication.

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arXiv2025

Quantum locally recoverable code with intersecting recovery sets

We introduce the concept of quantum locally recoverable codes (qLRCs) with intersecting recovery sets. We derive a singleton-like bound for these codes by leveraging the additional information provided by the intersecting recovery sets. Furthermore, we provide a construction for qLRCs with intersecting recovery sets by introducing a variation of the hypergraph product. Finally, we apply our qLRC methods to obtain improved results for classical LRCs. These results may provide new insights into the locality of quantum error correction code.

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arXiv2024

Displaced Fermionic Gaussian States and their Classical Simulation

This work explores displaced fermionic Gaussian operators with nonzero linear terms. We first demonstrate equivalence between several characterizations of displaced Gaussian states. We also provide an efficient classical simulation protocol for displaced Gaussian circuits and demonstrate their computational equivalence to circuits composed of nearest-neighbor matchgates augmented by single-qubit gates on the initial line. Finally, we construct a novel Gaussianity-preserving unitary embedding that maps $n$-qubit displaced Gaussian states to $(n+1)$-qubit even Gaussian states. This embedding facilitates the generalization of existing Gaussian testing protocols to displaced Gaussian states and unitaries. Our results provide new tools to analyze fermionic systems beyond the constraints of parity super-selection, extending the theoretical understanding and practical simulation of fermionic quantum computation.

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arXiv2024

Fermionic Gaussian Testing and Non-Gaussian Measures via Convolution

We define fermionic convolution and demonstrate its utility in characterizing fermionic non-Gaussian components, which are essential to the computational advantage of fermionic systems. Using fermionic convolution, we propose an efficient protocol that tests the fermionic Gaussianity of pure states using three copies of the input state. We also introduce "Non-Gaussian Entropy," an experimentally accessible resource measure that quantifies fermionic non-Gaussianity. These results provide new insights into the study of fermionic quantum computation.

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arXiv2024

On the Hardness of Measuring Magic

Quantum computers promise to solve computational problems significantly faster than classical computers. These 'speed-ups' are achieved by utilizing a resource known as magic. Measuring the amount of magic used by a device allows us to quantify its potential computational power. Without this property, quantum computers are no faster than classical computers. Whether magic can be accurately measured on large-scale quantum computers has remained an open problem. To address this question, we introduce Pauli instability as a measure of magic and experimentally measure it on the IBM Eagle quantum processor. We prove that measuring large (i.e., extensive) quantities of magic is intractable. Our results suggest that one may only measure magic when a quantum computer does not provide a speed-up. We support our conclusions with both theoretical and experimental evidence. Our work illustrates the capabilities and limitations of quantum technology in measuring one of the most important resources in quantum computation.

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arXiv2024

Magic of Random Matrix Product States

Magic, or nonstabilizerness, characterizes how far away a state is from the stabilizer states, making it an important resource in quantum computing, under the formalism of the Gotteman-Knill theorem. In this paper, we study the magic of the $1$-dimensional Random Matrix Product States (RMPSs) using the $L_{1}$-norm measure. We firstly relate the $L_{1}$-norm to the $L_{4}$-norm. We then employ a unitary $4$-design to map the $L_{4}$-norm to a $24$-component statistical physics model. By evaluating partition functions of the model, we obtain a lower bound on the expectation values of the $L_{1}$-norm. This bound grows exponentially with respect to the qudit number $n$, indicating that the $1$D RMPS is highly magical. Our numerical results confirm that the magic grows exponentially in the qubit case.

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arXiv2024

Extremality of stabilizer states

We investigate the extremality of stabilizer states to reveal their exceptional role in the space of all $n$-qubit/qudit states. We establish uncertainty principles for the characteristic function and the Wigner function of states, respectively. We find that only stabilizer states achieve saturation in these principles. Furthermore, we prove a general theorem that stabilizer states are extremal for convex information measures invariant under local unitaries. We explore this extremality in the context of various quantum information and correlation measures, including entanglement entropy, conditional entropy and other entanglement measures. Additionally, leveraging the recent discovery that stabilizer states are the limit states under quantum convolution, we establish the monotonicity of the entanglement entropy and conditional entropy under quantum convolution. These results highlight the remarkable information-theoretic properties of stabilizer states. Their extremality provides valuable insights into their ability to capture information content and correlations, paving the way for further exploration of their potential in quantum information processing.

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arXiv2024

Magic Class and the Convolution Group

The classification of many-body quantum states plays a fundamental role in the study of quantum phases of matter. In this work, we propose an approach to classify quantum states by introducing the concept of magic class. In addition, we introduce an efficient coarse-graining procedure to extract the magic feature of states, which we call the ``convolution group (CG).'' We classify quantum states into different magic classes using the fixed points of the CG and circuit equivalence. We also show that magic classes can be characterized by symmetries and the quantum entropy of the CG fixed points. Finally, we discuss the connection between the CG and the renormalization group. These results may provide new insight into the study of the state classification and quantum phases of matter.

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