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Quantum decoherent noises have significantly influenced the performance of practical quantum processors. Probabilistic error cancellation quantum error mitigation method quasiprobabilistically simulates the noise inverse operations, which are not physical channels, to cancel the noises. Physical implementability is the minimal cost to simulate a non-physical quantum operation with physical channels by the quasiprobabilistic decomposition. However, in practical, this cancellation may also be influenced by noises, and the implementable channels are not all of the physical channels, so the physical implementability is not sufficient to completely depict the practical situation of the probabilistic error cancellation method. Therefore, we generalize the physical implementability to an arbitrary convex set of free quantum resources and discuss several of its properties. We demonstrate the way to optimally cancel the error channel with the noisy Pauli basis. In addition, we also discuss the several properties relevant to this generalization. We expect that its properties and structures will be investigated comprehensively, and it will have more applications in the field of quantum information processing.

We propose a tensor network approach known as the locally purified density operator (LPDO) to investigate the classification and characterization of symmetry-protected topological (SPT) phases in open quantum systems. We extend the concept of injectivity, originally associated with matrix product states and projected entangled pair states, to LPDOs in $(1+1)D$ and $(2+1)D$ systems, unveiling two distinct types of injectivity conditions inherent in short-range entangled density matrices. Within the LPDO framework, we outline a classification scheme for decohered average symmetry-protected topological (ASPT) phases, consistent with earlier results obtained through spectrum sequence techniques. We first illustrate our framework with ASPT phases protected by fermion parity symmetry, then extend the classification of ASPT phases to a general group extension. We demonstrate examples of explicit construction of fixed-point LPDOs for ASPT phases including intrinsic ASPTs in both $(1+1)D$ and $(2+1)D$ systems., Comment: 27 pages, 5 figures

High-order topological phases of matter refer to the systems of $n$-dimensional bulk with the topology of $m$-th order, exhibiting $(n-m)$-dimensional boundary modes and can be characterized by topological pumping. Here, we experimentally demonstrate two types of second-order topological pumps, forming four 0-dimensional corner localized states on a 4$\times$4 square lattice array of 16 superconducting qubits. The initial ground state of the system for half-filling, as a product of four identical entangled 4-qubit states, is prepared using an adiabatic scheme. During the pumping procedure, we adiabatically modulate the superlattice Bose-Hubbard Hamiltonian by precisely controlling both the hopping strengths and on-site potentials. At the half pumping period, the system evolves to a corner-localized state in a quadrupole configuration. The robustness of the second-order topological pump is also investigated by introducing different on-site disorder. Our work studies the topological properties of high-order topological phases from the dynamical transport picture using superconducting qubits, which would inspire further research on high-order topological phases.

We demonstrate that the dynamics of quantum coherence serves as an effective probe for identifying dephasing, which is a distinctive signature of many-body localization (MBL). Quantum coherence can be utilized to measure both the local coherence of specific subsystems and the total coherence of the whole system in a consistent manner. Our results reveal that the local coherence of small subsystems decays over time following a power law in the MBL phase, while it reaches a stable value within the same time window in the Anderson localized (AL) phase. In contrast, the total coherence of the whole system exhibits logarithmic growth during the MBL phase and reaches a stable value in the AL phase. Notably, this dynamic characteristic of quantum coherence remains robust even with weak interactions and displays unbounded behavior in infinite systems. Our results provide insights into understanding many-body dephasing phenomena in MBL systems and propose a novel feasible method for identifying and characterizing MBL phases in experiments.

Thouless pumping, a dynamical version of the integer quantum Hall effect, represents the quantized charge pumped during an adiabatic cyclic evolution. Here we report experimental observations of nontrivial topological pumping that is induced by disorder even during a topologically trivial pumping trajectory. With a 41-qubit superconducting quantum processor, we develop a Floquet engineering technique to realize cycles of adiabatic pumping by simultaneously varying the on-site potentials and the hopping couplings. We demonstrate Thouless pumping in the presence of disorder and show its breakdown as the strength of disorder increases. Moreover, we observe two types of topological pumping that are induced by on-site potential disorder and hopping disorder, respectively. Especially, an intrinsic topological pump that is induced by quasi-periodic hopping disorder has never been experimentally realized before. Our highly controllable system provides a valuable quantum simulating platform for studying various aspects of topological physics in the presence of disorder.

Scrambling prevents the access to local information with local operators and therefore can be used to protect quantum information from damage caused by local perturbations. Even though partial quantum information can be recovered if the type of the damage is known, the initial target state cannot be completely recovered, because the obtained state is a mixture of the initial state and a maximally mixed state. Here, we demonstrate an improved scheme to recover damaged quantum information via scrambling in indefinite causal order. We show that scheme with indefinite causal order can record information of the damage and distill the initial state from the damaged state simultaneously. It allows us to retrieve initial information versus any damage. Moreover, by iterating the schemes, the initial quantum state can be completely recovered. In addition, we experimentally demonstrate our schemes on the cloud-based quantum computer, named as Quafu. Our work proposes a feasible scheme to protect whole quantum information from damage, which is also compatible with other techniques such as quantum error corrections and entanglement purification protocols. We expect that our scheme will be useful in the both quantum information recovery from the damage and systems bench-marking.

Characterizing the nature of hydrodynamical transport properties in quantum dynamics provides valuable insights into the fundamental understanding of exotic non-equilibrium phases of matter. Experimentally simulating infinite-temperature transport on large-scale complex quantum systems is of considerable interest. Here, using a controllable and coherent superconducting quantum simulator, we experimentally realize the analog quantum circuit, which can efficiently prepare the Haar-random states, and probe spin transport at infinite temperature. We observe diffusive spin transport during the unitary evolution of the ladder-type quantum simulator with ergodic dynamics. Moreover, we explore the transport properties of the systems subjected to strong disorder or a tilted potential, revealing signatures of anomalous subdiffusion in accompany with the breakdown of thermalization. Our work demonstrates a scalable method of probing infinite-temperature spin transport on analog quantum simulators, which paves the way to study other intriguing out-of-equilibrium phenomena from the perspective of transport., Comment: Main text: 13 pages, 5 figures; Supplementary: 17 pages, 16 figures, 1 table

Efficient preparation of spin-squeezed states is important for quantum-enhanced metrology. Current protocols for generating strong spin squeezing rely on either high dimensionality or long-range interactions. A key challenge is how to generate considerable spin squeezing in one-dimensional systems with only nearest-neighbor interactions. Here, we develop variational spin-squeezing algorithms to solve this problem. We consider both digital and analog quantum circuits for these variational algorithms. After the closed optimization loop of the variational spin-squeezing algorithms, the generated squeezing can be comparable to the strongest squeezing created from two-axis twisting. By analyzing the experimental imperfections, the variational spin-squeezing algorithms proposed in this work are feasible in recent developed noisy intermediate-scale quantum computers.

Effective light cones may emerge in non-relativistic local quantum systems from the Lieb-Robinson bounds, resulting in exponentially decaying commutator norms of two space-time separated operators in the Heisenberg picture. Here, we derive a mechanism for the emergence and consequences of a logarithmic light cone (LLC). As a possible way, the LLC can emerge from a phenomenological model of many-body-localization. We show that the information scrambling is logarithmically slow in the regime of the LLC. We prove that the bipartite entanglement entropy grows logarithmically with time for arbitrary finite space dimensions and arbitrary initial pure states. As an application in quantum information processing, the LLC supports long-lived quantum memory after unitary time evolution: a quantum code with macroscopic code distance and exponentially long lifetime., Comment: 7+3 pages, 2 figures.In version 2, typos were fixed; the proof of theorem 1 was slightly modified

Multipartite entanglement, characterized by the quantum Fisher information (QFI), plays a central role in quantum-enhanced metrology and understanding quantum many-body physics. With a dynamical generalization of the Mazur-Suzuki relations, we provide a rigorous lower bound on the QFI for the thermal Gibbs states in terms of dynamical symmetries, i.e., operators with periodic time dependence. We demonstrate that this bound can be saturated when considering a complete set of dynamical symmetries. Moreover, this lower bound with dynamical symmetries can be generalized to the QFI matrix and to the QFI for the thermal pure states, predicted by the eigenstate thermalization hypothesis. Our results reveal a new perspective to detect multipartite entanglement and other generalized variances in an equilibrium system, from its nonstationary dynamical properties, and is promising for studying emergent nonequilibrium many-body physics., Comment: 6+4 pages, 2+0 figures

We investigate the meson excitations (particle-antiparticle bound states) in quantum many-body scars of a 1D $\mathbb{Z}_2$ lattice gauge theory coupled to a dynamical spin-$\frac{1}{2}$ chain as a matter field. By introducing a string representation of the physical Hilbert space, we express a scar state $\ket {\Psi_{n,l}}$ as a superposition of all string bases with an identical string number $n$ and a total length $l$. For the small-$l$ scar state $\ket {\Psi_{n,l}}$, the gauge-invariant spin exchange correlation function of the matter field hosts an exponential decay as the distance increases, indicating the existence of stable mesons. However, for large $l$, the correlation function exhibits a power-law decay, signaling the emergence of nonmesonic excitations. Furthermore, we show that this mesonic-nonmesonic crossover can be detected by the quench dynamics, starting from two low-entangled initial states, respectively, which are experimentally feasible in quantum simulators. Our results expand the physics of quantum many-body scars in lattice gauge theories and reveal that the nonmesonic state can also manifest ergodicity breaking., Comment: 7+2 pages, 4+1 figures

We propose a minimal interacting lattice model for two-dimensional class-$D$ higher-order topological superconductors with no free-fermion counterpart. A Lieb-Schultz-Mattis-type constraint is proposed and applied to guide our lattice model construction. Our model exhibits a trivial product ground state in the weakly interacting regime, whereas, increasing electron interactions provoke a novel topological quantum phase transition to a $D_4$-symmetric higher-order topological superconducting state. The symmetry-protected Majorana corner modes are numerically confirmed with the matrix-product-state technique. Our theory paves the way for studying interacting higher-order topology with explicit lattice model constructions., Comment: 12 pages, 6 figures

Quantum simulation of different exotic topological phases of quantum matter on a noisy intermediate-scale quantum (NISQ) processor is attracting growing interest. Here, we develop a one-dimensional 43-qubit superconducting quantum processor, named as Chuang-tzu, to simulate and characterize emergent topological states. By engineering diagonal Aubry-Andr$\acute{\mathrm{e}}$-Harper (AAH) models, we experimentally demonstrate the Hofstadter butterfly energy spectrum. Using Floquet engineering, we verify the existence of the topological zero modes in the commensurate off-diagonal AAH models, which have never been experimentally realized before. Remarkably, the qubit number over 40 in our quantum processor is large enough to capture the substantial topological features of a quantum system from its complex band structure, including Dirac points, the energy gap's closing, the difference between even and odd number of sites, and the distinction between edge and bulk states. Our results establish a versatile hybrid quantum simulation approach to exploring quantum topological systems in the NISQ era., Comment: Main text: 6 pages, 4 figures; Supplementary: 16 pages, 14 figures

Nonequilibrium dynamics of many-body systems is challenging for classical computing, providing opportunities for demonstrating practical quantum computational advantage with analogue quantum simulators. It is proposed to be classically intractable to sample driven thermalized many-body states of Bose-Hubbard systems, and further extract multi-point correlations for characterizing quantum phases. Here, leveraging dedicated precise manipulations and number-resolved detection through a quantum gas microscope, we implement and sample a 32-site driven Hubbard chain in the thermalized phase. Multi-point correlations of up to 14th-order extracted from experimental samples offer clear distinctions between the thermalized and many-body-localized phases. In terms of estimated computational powers, the quantum simulator is comparable to the fastest supercomputer with currently known best algorithms. Our work paves the way towards practical quantum advantage in simulating Floquet dynamics of many-body systems., Comment: 18 pages, 14 figures

The ground state degeneracy of topologically ordered gapped Hamiltonians is the bedrock for a quantum error-correcting code with macroscopic distance, which is unfortunately not stable away from equilibrium. In this work, we show that the presence of a bounded lightcone preserves topological order. As a quantum code, the initial ground space will keep its macroscopic distance during unitary time evolution. We also show how a bounded lightcone can emerge through suitable perturbations in the two-dimensional toric code. Our results suggest that topological quantum memory can be dynamically robust at zero temperature., Comment: 6+3 pages, 4 figures, the Supplemental Material is updated. arXiv admin note: substantial text overlap with arXiv:2102.03046

This note is devoted to inverse spectral problems for canonical Hamiltonian systems on the half-line. An approach to inverse spectral problems based on the use of truncated Toeplitz operators has been especially effective in the case when the spectral measure of the system is a locally finite periodic measure (see \cite{MP}). In this note we extend the periodic algorithm to the case of non-periodic measures by considering periodizations of a spectral measure and showing that the Hamiltonians corresponding to the periodizations converge to the Hamiltonian of the original measure.

The quantum Hall effect, fundamental in modern condensed matter physics, continuously inspires new theories and predicts emergent phases of matter. Here we experimentally demonstrate three types of Chern insulators with synthetic dimensions on a programable 30-qubit-ladder superconducting processor. We directly measure the band structures of the 2D Chern insulator along synthetic dimensions with various configurations of Aubry-Andr\'e-Harper chains and observe dynamical localisation of edge excitations. With these two signatures of topology, our experiments implement the bulk-edge correspondence in the synthetic 2D Chern insulator. Moreover, we simulate two different bilayer Chern insulators on the ladder-type superconducting processor. With the same and opposite periodically modulated on-site potentials for two coupled chains, we simulate topologically nontrivial edge states with zero Hall conductivity and a Chern insulator with higher Chern numbers, respectively. Our work shows the potential of using superconducting qubits for investigating different intriguing topological phases of quantum matter., Comment: main text: 18 pages, 9 figures; supplementary information: 25 pages, 26 figures

Superradiant phase transitions (SPTs) are important for understanding light-matter interactions at the quantum level, and play a central role in criticality-enhanced quantum sensing. So far, SPTs have been observed in driven-dissipative systems, but the emergent light fields did not show any nonclassical characteristic due to the presence of strong dissipation. Here we report an experimental demonstration of the SPT featuring the emergence of a highly nonclassical photonic field, realized with a resonator coupled to a superconducting qubit, implementing the quantum Rabi model. We fully characterize the light-matter state by Wigner matrix tomography. The measured matrix elements exhibit quantum interference intrinsic of a photonic mesoscopic superposition, and reveal light-matter entanglement, Comment: 20 pages, 19 figures, 2 tables

We demonstrate that multipartite entanglement, witnessed by the quantum Fisher information (QFI), can characterize topological quantum phase transitions in the spin-$\frac{1}{2}$ toric code model on a square lattice with external fields. We show that the QFI density of the ground state can be written in terms of the expectation values of gauge-invariant Wilson loops for different sizes of square regions and identify $\mathbb{Z}_2$ topological order by its scaling behavior. Furthermore, we use this multipartite entanglement witness to investigate thermalization and disorder-assisted stabilization of topological order after a quantum quench. Moreover, with an upper bound of the QFI, we demonstrate the absence of finite-temperature topological order in the 2D toric code model in the thermodynamic limit. Our results provide insights to topological phases, which are robust against external disturbances, and are candidates for topologically protected quantum computation., Comment: 6+4 pages, 3+0 figures

One-way quantum computing is a promising candidate for fault-tolerant quantum computing. Here, we propose new protocols to realize a deterministic one-way CNOT gate and one-way $X$-rotations on quantum-computing platforms. By applying a delayed-choice scheme, we overcome a limit of most currently available quantum computers, which are unable to implement further operations on measured qubits or operations conditioned on measurement results from other qubits. Moreover, we decrease the error rate of the one-way logic gates, compared to the original protocol using local operations and classical communication (LOCC). In addition, we apply our deterministic one-way CNOT gate in the Deutsch-Jozsa algorithm to show the feasibility of our proposal. We demonstrate all these one-way gates and algorithms by running experiments on the cloud quantum-computing platform IBM Quantum Experience.

Operator spreading, often characterized by out-of-time-order correlators (OTOCs), is one of the central concepts in quantum many-body physics. However, measuring OTOCs is experimentally challenging due to the requirement of reversing the time evolution of systems. Here we apply Floquet engineering to investigate operator spreading in a superconducting 10-qubit chain. Floquet engineering provides an effective way to tune the coupling strength between nearby qubits, which is used to demonstrate quantum walks with tunable couplings, reversed time evolution, and the measurement of OTOCs. A clear light-cone-like operator propagation is observed in the system with multiple excitations, and has a nearly equal velocity as the single-particle quantum walk. For the butterfly operator that is nonlocal (local) under the Jordan-Wigner transformation, the OTOCs show distinct behaviors with (without) a signature of information scrambling in the near integrable system., Comment: 6+12 pages, 4 figures