pauli transfer matrix

Pauli channel estimation in quantum systems suffers from severe sensitivity to state preparation and measurement (SPAM) errors, limiting robustness and practical applicability. Method: We propose a novel algorithm combining ensemble recovery with complex analysis, requiring no entangled state preparation or measurement, and enabling accurate estimation of mixed erasure–bit-flip Pauli noise even under strong SPAM errors. Contribution/Results: Our approach achieves the first theoretical guarantee of robustness against arbitrary, arbitrarily severe SPAM errors. It reduces sample complexity to $exp(O(n^{1/3}))$, breaking the prior exponential barrier of $2^n$, and proves near-tightness of this bound under reasonable assumptions. The method is experimentally feasible and provides, to date, the optimal solution for noise characterization on noisy intermediate-scale quantum (NISQ) devices—with provable guarantees approaching fundamental limits.

Estimating eigenvalues of an n-qubit Pauli noise channel under stringent quantum memory constraints, where conventional approaches suffer exponential measurement overhead. Method: We propose a cascaded protocol integrating multi-round channel queries, spectral estimation, and adaptive sampling—requiring only O(log n/ε²) ancillary qubits. Contribution/Results: Our protocol achieves a measurement complexity of Õ(n²/ε²), exponentially improving upon the tight lower bound Ω(2ⁿ/ε²) attainable with zero ancillary qubits. This is the first demonstration that logarithmic quantum memory yields exponential statistical advantage in noise characterization. Moreover, it establishes the optimal sample complexity paradigm for Pauli noise tomography under finite memory constraints, resolving a fundamental question in quantum benchmarking and error mitigation.

This work establishes fundamental lower bounds on the sample complexity of learning an $n$-qubit Pauli channel, under the constraints of no auxiliary entanglement, single-qubit measurements only, non-reusability of the channel, and accuracy measured in diamond norm error $varepsilon$. We analyze both non-adaptive and adaptive strategies. Technically, our approach integrates quantum information theory, diamond norm analysis, structural characterization of Pauli channels, and probabilistic methods. We derive the first tight lower bounds: $Omega(2^{3n}varepsilon^{-2})$ for non-adaptive learning and $Omega(2^{2.5n}varepsilon^{-2})$ for adaptive learning. These results demonstrate that the Flammia–Wallman algorithm is nearly optimal in the non-adaptive setting. Our bounds reveal the intrinsic hardness of Pauli noise learning and establish rigorous theoretical limits for quantum hardware characterization and randomized compiling protocols.

This work introduces the paradigm of *agnostic process tomography*: given query access to an unknown quantum channel Φ—without assuming Φ belongs to any prespecified model class—we select, from a given concept class ℂ, the channel that best approximates Φ. Our core techniques include Pauli spectrum analysis, superoperator spectral estimation, ancilla-enhanced state-tomography transfer, and efficient query sampling. We establish, for the first time, an agnostic learning transfer framework from quantum states to quantum processes, applicable to broad learnable classes including Pauli channels, quantum juntas, and QAC⁰ circuits. We design polynomial-time agnostic learning algorithms for Clifford circuits and circuits with few T-gates. Furthermore, we provide a theoretical characterization of sufficient conditions for agnostic learnability of quantum processes. These results advance foundational tools for quantum machine learning and error mitigation.

This work addresses the lack of efficient numerical methods for computing the quantum channel relative entropy—a fundamental metric in quantum channel discrimination and resource theories. Methodologically, we introduce the first scalable, error-controllable computational framework: we discretize and linearize the integral representation of quantum state relative entropy, thereby reformulating the channel relative entropy maximization problem as a sequence of semidefinite programs (SDPs); tight upper and lower bounds are constructed to enable arbitrarily precise sandwich estimation. Crucially, our approach overcomes prior limitations restricted to minimization settings, enabling rigorous optimization over input states for the first time. Experiments demonstrate high accuracy and low computational complexity, significantly enhancing the tractability and practical utility of channel relative entropy in real-world discrimination tasks and resource quantification.

This work addresses practical quantum communication scenarios—such as satellite-based QKD—where no shared reference frame exists and unknown static unitary noise is present. The authors propose two innovative approaches: first, constructing a reference-frame-independent Pauli transfer matrix (PTM) whose singular vectors identify optimal signal states; second, designing a sequential basis matching (SBM) strategy that adaptively optimizes local measurement bases. For the first time under these conditions, the optimality of both BB84 and the six-state protocol is rigorously proven, and a novel modeling framework is established in which the PTM inherently accounts for the absence of a shared reference frame. Both methods achieve equivalent key rates, significantly enhancing practicality while maintaining information-theoretic security.

This work addresses the longstanding challenges in implementing block-encoding–based quantum linear algebra algorithms, which have been hindered by the complexity of underlying circuit construction and the difficulty of verification. We introduce Unitaria, a NumPy/SciPy-like Python library that provides a composable, array-oriented interface for directly manipulating encoded matrices and vectors without requiring ancilla qubits. By unifying block encoding, quantum singular value transformation, and standard linear algebra operations—augmented with automated circuit synthesis and resource analysis—our approach enables efficient classical validation and scalable simulation. Compared to conventional state-vector simulators, Unitaria significantly extends the attainable problem scale and accurately estimates key resources such as gate count, qubit count, and normalization constants without executing the quantum circuit.

This work addresses the challenge of modeling high-order feature interactions under binarized or quantized inputs by proposing a classically efficient inference method that requires no quantum resources. Leveraging quantum-inspired ideas during training—through learnable Pauli word selection, projection-based encoding, and an sPQC-Parity architecture—the approach constructs parity representations that rely solely on classical computation at inference time. On native binary tasks with 5–10 bits, the method achieves accuracy improvements of 23.9%–41.7% over logistic regression and SVM, and significantly outperforms baselines such as PCA-bin on textual and discrete datasets. Notably, it even surpasses fully continuous models in certain scenarios, marking the first demonstration of a performance advantage for quantum-inspired parity representations within a classically efficient inference framework.

This work addresses the problem of efficiently learning an unknown unitary channel whose Pauli spectrum is approximately sparse, with the goal of achieving high-accuracy approximation under the diamond norm. Given query access to an $(s,\varepsilon)$-approximately sparse unitary operator, the authors propose the first quantum algorithm capable of efficiently estimating large-magnitude Pauli coefficients of arbitrary unitaries and establish a learnability framework under a restricted diamond distance. By integrating techniques from Pauli-sparse recovery, quantum query complexity, and diamond norm analysis, their method outputs an approximating channel using only $\widetilde{O}(s^6/\varepsilon^4)$ queries and polynomial time. This result also yields new learnability guarantees for classes of unitary channels with bounded Pauli $\ell_1$-norm.

Existing quantum machine learning models struggle to explicitly capture the spectral structure—such as eigenvalues and subspaces—of input matrices, limiting their ability to exploit the intrinsic geometry of data. This work proposes the Quantum Spectral Model (QSM), which directly maps input matrices to the generator of a data-encoding unitary operator, thereby embedding spectral information explicitly and constructing a truncated Fourier representation at the output. The model innovatively employs the input-dependent spectral gap as a phase carrier and uses spectral subspaces to determine expansion coefficients, introducing an analytically tractable inductive bias. Three QSM variants are developed based on symmetric, global-block, and local-block Hamiltonian designs, incorporating data re-uploading and input-conditioned frequencies. Experiments demonstrate that QSM achieves state-of-the-art average test accuracy on Pendigits and two synthetic spectral tasks, with the local-block variant excelling on Pendigits and the global-block variant performing best on synthetic tasks.