The field of quantum Hamiltonian complexity lies at the intersection of quantum many-body physics and computational complexity theory, with deep implications to both fields. The main object of study is the LocalHamiltonian problem, which is concerned with estimating the ground-state energy of a local Hamiltonian and is complete for the class $\mathsf{Q}\mathsf{M}\mathsf{A}$, a quantum generalization of the class $\mathsf{N}\mathsf{P}$. A major challenge in the field is to understand the complexity of the LocalHamiltonian problem in more physically natural parameter regimes. One crucial parameter in understanding the ground space of any Hamiltonian in many-body physics is the spectral gap, which is the difference between the smallest two eigenvalues. Despite its importance in quantum many-body physics, the role played by the spectral gap in the complexity of LocalHamiltonian is less well understood. In this work, we make progress on this issue by considering the precise regime, in which one estimates the ground-state energy to within inverse exponential precision. Computing ground-state energies precisely is a task that is important for quantum chemistry and quantum many-body physics. In the setting of inverse-exponential precision, there is a surprising result by Fefferman and Lin that the complexity of LocalHamiltonian is magnified from $\mathsf{Q}\mathsf{M}\mathsf{A}$ to $\mathsf{P}\mathsf{S}\mathsf{P}\mathsf{A}\mathsf{C}\mathsf{E}$, the class of problems solvable in polynomial space (but possibly exponential time). We clarify the reason behind this boost in complexity. Specifically, we show that the full complexity of the high-precision case only comes about when the spectral gap is exponentially small. As a consequence of the proof techniques developed to show our results, we uncover important implications for the representability and circuit complexity of ground states of local Hamiltonians, the theory of uniqueness of quantum witnesses, and techniques for the amplification of quantum witnesses in the presence of postselection.

We propose a random circuit model that attempts to capture the behavior of noisy intermediate-scale quantum devices when used for variationally solving classical optimization problems. Our model accounts for the propagation of arbitrary single-qubit errors through the circuit. We find that, even with a small noise rate, the quality of the obtained optima implies that a single-qubit error rate of $1/(nD)$ (where $n$ is the number of qubits and $D$ is the circuit depth) is needed for the possibility of a quantum advantage. We estimate that this translates to an error rate lower than ${10}^{-6}$ using the quantum approximate optimization algorithm for classical optimization problems with two-dimensional circuits.

Measurement incompatibility and quantum nonlocality are two key features of quantum theory. Violations of Bell inequalities require quantum entanglement and incompatibility of the measurements used by the two parties involved in the protocol. We analyze the converse question: for which Bell inequalities is the incompatibility of measurements enough to ensure a quantum violation? We relate the two questions by comparing two tensor norms on the space of dichotomic quantum measurements, one characterizing measurement compatibility and the other characterizing violations of a given Bell inequality. We provide sufficient conditions for the equivalence of the two notions in terms of the matrix describing the correlation Bell inequality. We show that the Clauser-Horne-Shimony-Holt inequality and its variants are the only ones that satisfy it.

Two-dimensional $p+ip$ superconductors and superfluids are systems that feature chiral behavior emerging from the Cooper pairing of electrons or neutral fermionic atoms with nonzero angular momentum. Their realization has been a longstanding goal because they offer great potential utility for quantum computation and memory. However, they have so far eluded experimental observation both in solid-state systems as well as in ultracold quantum gases. Here, we propose to leverage the tremendous control offered by rotating two-dimensional trapped-ion crystals in a Penning trap to simulate the dynamical phases of two-dimensional $p+ip$ superfluids. This is accomplished by mapping the presence or absence of a Cooper pair into an effective spin-1/2 system encoded in the ions’ electronic levels. We show how to infer the topological properties of the dynamical phases, and discuss the role of beyond mean-field corrections. More broadly, our work opens the door to use trapped-ion systems to explore exotic models of topological superconductivity and also paves the way to generate and manipulate skyrmionic spin textures in these platforms.

We find necessary and sufficient conditions to determine the interconvertibility of quantum systems under time-translation covariant evolution, and use it to solve several problems in quantum thermodynamics both in the single-shot and asymptotic regimes. It is well known that the resource theory of quantum athermality is not reversible, but in Brandão et al. [Phys. Rev. Lett. 111, 250404 (2013)] it was claimed that the theory becomes reversible “provided a sublinear amount of coherent superposition over energy levels is available.” Here we show that if a sublinear amount of coherence among energy levels were considered free, then the resource theory of athermality would become trivial. Instead, we show that by considering a sublinear amount of energy to be free, the theory of athermality becomes reversible for the pure-state case. A proof of the same claim for the mixed-state case is still lacking.

The exploration of highly connected networks of qubits is invaluable for implementing various quantum algorithms and simulations, as it allows for entangling qubits with reduced circuit depth. Here, we demonstrate a multiqubit sideband tone-assisted Rabi-driven gate. Our scheme is inspired by the ion-qubit Mølmer-Sørensen gate and is mediated by a shared photonic mode and Rabi-driven superconducting qubits, which relaxes restrictions on qubit frequencies during fabrication and supports scalability. We achieve a two-qubit gate with a maximum state fidelity of 95% in 310 ns, a three-qubit gate with a state fidelity of 90.5% in 217 ns, and a four-qubit gate with a state fidelity of 66% in 200 ns. Furthermore, we develop a model of the gate that shows that the four-qubit gate is limited by shared resonator losses and the spread of qubit-resonator couplings, which must be addressed to reach high-fidelity operations.

We present a theory of magnetoelectric magnon-photon coupling in cavities hosting noncentrosymmetric magnets. Analogously to nonreciprocal phenomena in multiferroics, the magnetoelectric coupling is time-reversal and inversion asymmetric. This asymmetry establishes a means for exceptional tunability of magnon-photon coupling, which can be switched on and off by reversing the magnetization direction. Taking the multiferroic skyrmion host ${\mathrm{Cu}}_2{\mathrm{OSeO}}_3$ with ultralow magnetic damping as an example, we reveal the electrical activity of skyrmion eigenmodes and propose it for magnon-photon splitting of “magnetically dark” elliptic modes. Furthermore, we predict a cavity-induced magnon-magnon coupling between magnetoelectrically active skyrmion excitations. We discuss applications in quantum information processing by proposing protocols for all-electrical magnon-mediated photon quantum gates, and a photon-mediated split operation of magnons. Our study highlights magnetoelectric cavity magnonics as a novel platform for realizing quantum-hybrid systems and the coherent transduction between photons and magnons in topological magnetic textures.

Spin-$\frac{1}{2}{}^{119}$$\mathrm{Sn}$ nuclei in a silicon semiconductor could make excellent qubits. Nuclear spins in silicon are known to have long coherence times. Tin is isoelectronic with silicon, so we expect that electrons can easily shuttle from one $\mathrm{Sn}$ atom to another to propagate quantum information via a hyperfine interaction that we predict, from all-electron linearized augmented plane-wave density-functional-theory calculations, to be roughly 10 times larger than that of intrinsic ${}^{29}$$\mathrm{Si}$. A hyperfine-induced electron-nuclear controlled-phase (e-n-CPhase) gate operation, generated (up to local rotations) by merely holding an electron at a sweet spot of maximum hyperfine strength for a specific duration of time, is predicted to be exceptionally resilient to charge or voltage noise. Diabatic spin flips are suppressed with a modest magnetic field ($>15\mathrm{mT}$ for $<{10}^{-6}$ flip probabilities) and nuclear-spin-bath noise may be avoided via isotopic enrichment or mitigated using dynamical decoupling or through monitoring and compensation. Combined with magnetic resonance control, this operation enables universal quantum computation.

The generation of multiple entangled qubit pairs between distributed nodes is a prerequisite for a future quantum Internet. To achieve a practicable generation rate, standard protocols based on photonic qubits require multiple long-term quantum memories, which remains a significant experimental challenge. In this paper, we propose a novel protocol based on $2^m$-dimensional time-bin photonic qudits that allows for the simultaneous generation of multiple ($m$) entangled pairs between two distributed qubit registers and we outline a specific implementation of the protocol based on cavity-mediated spin-photon interactions. By adopting the qudit protocol, the required qubit memory time is independent of the transmission loss between the nodes, in contrast to standard qubit approaches. As such, our protocol can significantly boost the performance of near-term quantum networks.

Simulating complex molecules and materials is an anticipated application of quantum devices. With the emergence of hardware designed to target strong quantum advantage in artificial tasks, we examine how the same hardware behaves in modeling physical problems of correlated electronic structure. We simulate static and dynamical electronic structure on a superconducting quantum processor derived from Google’s Sycamore architecture for two representative correlated electron problems: the nitrogenase iron-sulfur molecular clusters and $\alpha$-ruthenium trichloride, a proximate spin-liquid material. To do so, we simplify the electronic structure into low-energy spin models that fit on the device. With extensive error mitigation and assistance from classical recompilation and simulated data, we achieve quantitatively meaningful results deploying about one fifth of the gate resources used in artificial quantum advantage experiments on a similar architecture. This increases to over half of the gate resources when choosing a model that suits the hardware. Our work serves to convert artificial measures of quantum advantage into a physically relevant setting.

The one-dimensional lattice Schwinger model has recently been realized by using bosons in optical lattices. This model contains both confinement and deconfinement phases, the phase diagram of which is controlled by the mass of the matter field and the topological angle. Since varying the mass of the matter field is straightforward experimentally, we propose how to tune the topological angle, allowing exploration of the phase diagram from both directions. We propose that direct experimental evidence of confinement and deconfinement can be obtained by measuring whether a physical charge is localized around a fixed gauge charge to screen it. We also discuss the PXP model realized in the Rydberg-atom array, which is equivalent to the lattice Schwinger model when all local gauge charges are fixed as zero. Although the gauge charges are fixed, we can alternatively probe confinement and deconfinement in the PXP model by studying the relative motion of a pair of a physical charge and an anticharge. Our scheme can be directly implemented in these two relevant experimental platforms of ultracold-atom quantum simulators.

The topological $\theta$-angle in gauge theories engenders a series of fundamental phenomena, including violations of charge-parity ($CP$) symmetry, dynamical topological transitions, and confinement-deconfinement transitions. At the same time, it poses major challenges for theoretical studies, as it implies a sign problem in numerical simulations. Analog quantum simulators open the promising prospect of treating quantum many-body systems with such topological terms, but, contrary to their digital counterparts, they have not yet demonstrated the capacity to control the $\theta$-angle. Here, we demonstrate how a tunable topological $\theta$-term can be added to a prototype theory with $U(1)$ gauge symmetry, a discretized version of quantum electrodynamics in one spatial dimension. As we show, the model can be realized experimentally in a single-species Bose-Hubbard model in an optical superlattice with three different spatial periods, thus requiring only standard experimental resources. Through numerical calculations obtained from the time-dependent density-matrix renormalization group method and exact diagonalization, we benchmark the model system, and illustrate how salient effects due to the $\theta$-term can be observed. These include charge confinement, the weakening of quantum many-body scarring, as well as the disappearance of Coleman’s phase transition due to explicit breaking of $CP$ symmetry. This work opens the door towards studying the rich physics of topological gauge-theory terms in large-scale cold-atom quantum simulators.

The finding of physical realizations of topologically ordered states in experimental settings, from condensed matter to artificial quantum systems, has been the main challenge en route to utilizing their unconventional properties. We show how to realize a large class of topologically ordered states and simulate their quasiparticle excitations on a digital quantum computer. To achieve this, we design a set of linear-depth quantum circuits to generate ground states of general string-net models together with unitary open-string operators to simulate the creation and braiding of Abelian and non-Abelian anyons. We show that the Abelian (non-Abelian) unitary string operators can be implemented with a constant- (linear-) depth quantum circuit. Our scheme allows us to directly probe characteristic topological properties, including topological entanglement entropy, braiding statistics, and fusion channels of anyons. Moreover, this set of efficiently prepared topologically ordered states has potential applications in the development of fault-tolerant quantum computers.

Arrays of transmons have proven to be a viable medium for quantum information science and quantum simulations. Despite their widespread popularity as qubit arrays, there remains yet untapped potential beyond the two-level approximation or, equivalently, the hard-core boson model. With the higher excited levels included, coupled transmons naturally realize the attractive Bose-Hubbard model. The dynamics of the full model has been difficult to study due to the unfavorable scaling of the dimensionality of the Hilbert space with the system size. In this work, we present a framework for describing the effective unitary dynamics of highly excited states of coupled transmons based on high-order degenerate perturbation theory. This allows us to describe various collective phenomena—such as bosons stacked onto a single site behaving as a single particle, edge localization, and effective longer-range interactions—in a unified, compact, and accurate manner. A further benefit of our approach is that boson stacks can be naturally interpreted as interacting quasiparticles, enabling transmon arrays to be used to explore and study additional lattice models besides the standard Bose-Hubbard one. While our examples deal with one-dimensional chains of transmons for the sake of clarity, the theory can be readily applied to more general geometries.

In the era of noisy intermediate-scale quantum devices, the number of controllable hardware qubits is insufficient to implement quantum error correction. As an alternative, quantum error mitigation (QEM) can suppress errors in measurement results via repeated experiments and postprocessing of data. Typical techniques for error mitigation, e.g., the quasiprobability decomposition method, ignore correlated errors between different gates. Here, we introduce a QEM method based on the matrix product operator (MPO) representation of a quantum circuit that can characterize the noise channel with polynomial complexity. Our technique is demonstrated on a $\mathrm{depth}=20$ fully parallel quantum circuit of up to $N_q=20$ qubits undergoing local and global noise. The circuit error is reduced by a several-times factor with only a small bond dimension $D^{\mathrm{\prime }}=1$ for the noise channel. The MPO representation increases the accuracy of modeling noise without consuming more experimental resources, which improves the QEM performance and broadens its scope of application. Our method is hopeful of being applied to circuits in higher dimensions with more qubits and deeper depth.

Floquet engineering provides a toolbox for the realization of novel quantum phases without static counterparts, while conventionally the realization may rely on the manipulation of complex temporal evolution. Here, we propose a systematic and high-precision scheme to realize unconventional Floquet topological phases by engineering local band structures in particular momentum subspaces called band-inversion surfaces (BISs). This scheme is based on a new bulk-boundary correspondence that for a class of generic $d$-dimensional periodically driven systems, the local topological structure formed in each BIS uniquely determines the features of gapless boundary modes. By engineering the BIS configuration, we demonstrate a highly efficient approach to realize, manipulate, and detect novel Floquet topological phases. In particular, we predict a two-dimensional (2D) anomalous Floquet valley-Hall phase that carries trivial global bulk topological invariants but features protected counterpropagating edge states in each quasienergy gap. The unconventional nature of this novel 2D phase is further illustrated by the examination of edge-geometry dependence and its robustness to disorder scattering. Anomalous chiral topological phases with valley protection in higher dimensions are also predicted and studied. Our systematic and highly feasible scheme opens up a new route to realizing and engineering unconventional Floquet topological phases for ultracold atoms and other quantum simulators.

Reduction of the circuit depth of quantum circuits is a crucial bottleneck to enabling quantum technology. This depth is inversely proportional to the number of available quantum gates that have been synthesized. Moreover, quantum gate-synthesis and control problems exhibit a vast range of external parameter dependencies, both physical and application specific. In this paper, we address the possibility of learning families of optimal-control pulses that depend adaptively on various parameters, in order to obtain a global optimal mapping from the space of potential parameter values to the control space and hence to produce continuous classes of gates. Our proposed method is tested on different experimentally relevant quantum gates and proves capable of producing high-fidelity pulses even in the presence of multiple variables or uncertain parameters with wide ranges.

Quantum computers solve ever more complex tasks using steadily growing system sizes. Characterizing these quantum systems is vital, yet becoming increasingly challenging. The gold-standard method for this task is quantum state tomography (QST), capable of fully reconstructing a quantum state without prior knowledge. The measurement and classical computing costs, however, increase exponentially with the number of constituents (e.g., qubits)—a daunting bottleneck given the scale of existing and near-term quantum devices. Here, we demonstrate a scalable and practical QST approach that only uses a single measurement setting, namely symmetric informationally complete (SIC) positive operator-valued measures (POVMs). We implement these nonorthogonal measurements on an ion trap quantum processor by utilizing additional energy levels within each ion—without requiring ancillary ions to assist in measurements. More precisely, we locally map the SIC POVM to orthogonal states embedded in a higher-dimensional system, which we read out using repeated in-sequence detections, thereby providing full tomographic information in every shot. Combining this SIC tomography with the recently developed randomized measurement toolbox (“classical shadows”) proves to be a powerful combination. SIC tomography alleviates the need for choosing measurement settings at random (“derandomization”), while classical shadows enable the estimation of arbitrary polynomial functions of the density matrix orders of magnitudes faster than standard methods. The latter enables in-depth entanglement characterization, which we experimentally showcase on a five-qubit absolutely maximally entangled state. Moreover, the fact that the full tomography information is available in every shot enables online QST in real time (i.e., while the experiment is running). We demonstrate this on an eight-qubit entangled state (which has $2^8\cdot 2^8-1=65535$ degrees of freedom), as well as for fast state identification. All in all, these features single out SIC-based classical shadow estimation as a highly scalable and convenient tool for quantum state characterization.

The study of confinement in quantum spin chains has seen a large surge of interest in recent years. It is not only important for understanding a range of effective one-dimensional condensed-matter realizations but it also shares some of the nonperturbative physics with quantum chromodynamics (QCD), which makes it a prime target for current quantum simulation efforts. In analogy to QCD, the confinement-induced two-particle bound states that appear in these models are dubbed mesons. Here, we study scattering events due to meson collisions in a quantum spin chain with long-range interactions such that two mesons have an extended interaction. We show how novel hadronic bound states, e.g., with four constituent particles akin to tetraquarks, may form dynamically in fusion events. In a natural collision their signal is weak, as elastic meson scattering dominates. However, we propose two controllable protocols that allow for a clear observation of dynamical hadron formation. We discuss how this physics can be simulated in trapped-ion or Rydberg-atom setups.

We introduce a general setup for the analog quantum simulation of the dynamics of open quantum systems based on semiconductor quantum dots electrically connected to an array of quantum $RLC$ electronic circuits. The dots are chosen to be in the regime of spin-charge hybridization to enhance their sensitivity to the $RLC$ circuits while mitigating the detrimental effects of unwanted noise. In this context, we establish an experimentally realizable map between the hybrid system and a qubit coupled to thermal harmonic environments of arbitrary complexity that enables the analog quantum simulation of open quantum systems. We assess the utility of the simulator by numerically exact emulations that indicate that the experimental setup can faithfully mimic the intended target even in the presence of its natural inherent noise. We further provide a detailed analysis of the physical requirements on the quantum dots and the $RLC$ circuits needed to experimentally realize this proposal that indicates that the simulator can be created with existing technology. The approach can exactly capture the effects of highly structured non-Markovian quantum environments typical of photosynthesis and chemical dynamics and it offers clear potential advantages over conventional and even quantum computation. The proposal opens up a general path for effective quantum dynamics simulations based on semiconductor quantum dots.