Simulating quantum circuits with classical computers requires resources growing exponentially in terms of system size. Real quantum computer with noise, however, may be simulated polynomially with various methods considering different noise models. In this work, we simulate random quantum circuits in one dimension with matrix product density operators (MPDOs), for different noise models such as dephasing, depolarizing, and amplitude damping. We show that the method based on matrix product states (MPSs) fails to approximate the noisy output quantum states for any of the noise models considered, while the MPDO method approximates them well. Compared with the method of matrix product operators (MPOs), the MPDO method reflects a clear physical picture of noise (with inner indices taking care of the noise simulation) and quantum entanglement (with bond indices taking care of two-qubit gate simulation). Consequently, in case of weak system noise, the resource cost of the MPDO will be significantly less than that of the MPO due to a relatively small inner dimension needed for the simulation. In case of strong system noise, a relatively small bond dimension may be sufficient to simulate the noisy circuits, indicating a regime that the noise is large enough for an “easy” classical simulation, which is further supported by a comparison with the experimental results on an IBM cloud device. Moreover, we propose a more effective tensor updates scheme with optimal truncations for both the inner and the bond dimensions, performed after each layer of the circuit, which enjoys a canonical form of the MPDO for improving simulation accuracy. With truncated inner dimension to a maximum value $\kappa$ and bond dimension to a maximum value $\chi$, the cost of our simulation scales as $\sim ND{\kappa }^3{\chi }^3$, for an $N$-qubit circuit with depth $D$.

The linear intrinsic anomalous Hall effect (IAHE) and second-order IAHE have been intensively investigated in time-reversal broken systems. However, as one of the important members of the nonlinear Hall family, the investigation of third-order IAHE remains absent due to the lack of an appropriate theoretical approach, although the third-order extrinsic AHE has been studied within the framework of first- and second-order semiclassical theory. Herein, we generalize the semiclassical theory for Bloch electrons under the uniform electric field up to the third order using the wave-packet method and based on which we predict that the third-order IAHE can also occur in time-reversal broken systems. Same as the second-order IAHE, we find the band geometric quantity, the second-order field-dependent Berry curvature arising from the second-order field-induced positional shift, plays a pivotal role in the observation of this effect. Moreover, with symmetry analysis, we find that the third-order IAHE, as the leading contribution, is supported by 15 time-reversal broken three-dimensional magnetic point groups, corresponding to a wide class of antiferromagnetic (AFM) materials. Guided by the symmetry arguments, a two-band model is chosen to demonstrate the generalized theory. Furthermore, the generalized third-order semiclassical theory depends only on the properties of Bloch bands, implying that it can also be employed to explore the IAHE in realistic AFM materials, by combining with first-principles calculations.

Quantum error correction is believed to be a necessity for large-scale fault-tolerant quantum computation. In the past two decades, various constructions of quantum error-correcting codes (QECCs) have been developed, leading to many good code families. However, the majority of these codes are not suitable for near-term quantum devices. Here we present VarQEC, a noise-resilient variational quantum algorithm to search for quantum codes with a hardware-efficient encoding circuit. The cost functions are inspired by the most general and fundamental requirements of a QECC, the Knill-Laflamme conditions. Given the target noise channel (or the target code parameters) and the hardware connectivity graph, we optimize a shallow variational quantum circuit to prepare the basis states of an eligible code. In principle, VarQEC can find quantum codes for any error model, whether additive or non-additive, degenerate or non-degenerate, pure or impure. We have verified its effectiveness by (re)discovering some symmetric and asymmetric codes, e.g., ((n,2n−6,3))2 for n from 7 to 14. We also found new ((6,2,3))2 and ((7,2,3))2 codes that are not equivalent to any stabilizer code, and extensive numerical evidence with VarQEC suggests that a ((7,3,3))2 code does not exist. Furthermore, we found many new channel-adaptive codes for error models involving nearest-neighbor correlated errors. Our work sheds new light on the understanding of QECC in general, which may also help to enhance near-term device performance with channel-adaptive error-correcting codes.

Determining whether a subspace spanned by certain quantum states is entangled and its entanglement dimensionality remains a fundamental challenge in quantum information science. This paper introduces a geometric measure of $r$-bounded rank, $E_r\left(\mathcal{S}\right)$, for a given subspace $\mathcal{S}$. Derived from the established geometric measure of entanglement, this measure is specifically designed to assess the entanglement within $\mathcal{S}$. It not only serves as a tool for determining the entanglement dimensionality but also illuminates the subspace's capacity to preserve such entanglement. By employing developed nonconvex optimization techniques utilized in machine learning area, we can accurately calculate $E_r\left(\mathcal{S}\right)$ within the manifold optimization framework. Our approach demonstrates notable advantages over existing hierarchical methods, PPT relaxation techniques, and the seesaw strategy, particularly by combining computational efficiency with broad applicability. More importantly, it paves the way for high-dimensional entanglement certification, which is crucial for numerous quantum information tasks. We showcase its effectiveness in validating high-dimensional entangled subspaces in bipartite systems, determining the border rank of multipartite pure states, and identifying genuinely or completely entangled subspaces.

Quantum error correction (QEC) is essential for protecting quantum information against noise, yet understanding the structure of the Knill-Laflamme (KL) coefficients \(\left\{{\lambda }_{ij}\right\}\) from the condition \(P{E}_{i}^{\dagger }{E}_{j}P={\lambda }_{ij}P\) remains challenging, particularly for nonadditive codes. In this work, we introduce the signature vector \(\overrightarrow{\lambda }(P)\), composed of the off-diagonal KL coefficients \(\left\{{\lambda }_{ij}\right\}\), where each coefficient corresponds to equivalence classes of errors counted only once. We define its Euclidean norm λ^*(P) as a scalar measure representing the total strength of error correlations within the code subspace defined by the projector P. We parameterize P on a Stiefel manifold and formulate an optimization problem based on the KL conditions to systematically explore possible values of λ^*. Moreover, we show that, for ((n, K, d)) codes, λ^* is invariant under local unitary transformations. Applying our approach to the ((6, 2, 3)) quantum code, we find that \({\lambda }_{\min }^{* }=\sqrt{0.6}\) and \({\lambda }_{\max }^{* }=1\), with λ^* = 1 corresponding to a known degenerate stabilizer code. We construct continuous families of new nonadditive codes parameterized by vectors in \({{\mathbb{R}}}^{5}\), with λ^* varying over the interval \([\sqrt{0.6},1]\). For the ((7, 2, 3)) code, we identify \({\lambda }_{\min }^{* }=0\) (corresponding to the non-degenerate Steane code) and \({\lambda }_{\max }^{* }=\sqrt{7}\) (corresponding to the permutation-invariant code by Pollatsek and Ruskai), and we demonstrate continuous paths connecting these extremes via cyclic codes characterized solely by λ^*. Our findings provide new insights into the structure of quantum codes, advance the theoretical foundations of QEC, and open new avenues for investigating intricate relationships between code subspaces and error correlations.

In pure-state tomography, the concept of unique determinedness (UD), the ability to uniquely determine pure states from measurement results, is crucial. This study presents a variational approach to examining UD, offering a robust solution to the challenges associated with the construction and certification of UD measurement schemes. We put forward an effective algorithm that minimizes a specially defined loss function, enabling the differentiation between UD and non-UD measurement schemes. This leads to the discovery of numerous optimal pure-state Pauli measurement schemes across a variety of dimensions. Additionally, we discern an alignment between uniquely determined among pure states and uniquely determined among all states in qubit systems when utilizing Pauli measurements, underscoring its intrinsic robustness under pure-state recovery. We further interpret the physical meaning of our loss function, bolstered by a theoretical framework. Our study not only propels the understanding of UD in quantum state tomography forward, but also delivers valuable practical insights for experimental applications, highlighting the need for a balanced approach between mathematical optimality and experimental pragmatism.