Lloyd et al. [Nat. Commun. 7, 10138 (2016)] were first to demonstrate the promise of quantum algorithms for computing Betti numbers, a way to characterize topological features of data sets. Here, we propose, analyze, and optimize an improved quantum algorithm for topological data analysis (TDA) with reduced scaling, including a method for preparing Dicke states based on inequality testing, a more efficient amplitude estimation algorithm using Kaiser windows, and an optimal implementation of eigenvalue projectors based on Chebyshev polynomials. We compile our approach to a fault-tolerant gate set and estimate constant factors in the Toffoli complexity. Our analysis reveals that superquadratic quantum speedups are only possible for this problem when targeting a multiplicative error approximation and the Betti number grows asymptotically. Further, we propose a dequantization of the quantum TDA algorithm that shows that having exponentially large dimension and Betti number are necessary, but insufficient conditions, for superpolynomial advantage. We then introduce and analyze specific problem examples which have parameters in the regime where superpolynomial advantages may be achieved, and argue that quantum circuits with tens of billions of Toffoli gates can solve seemingly classically intractable instances.

In a plethora of physical situations, one can distinguish a mediator—a system that couples other, noninteracting, systems. Often, the mediator itself is not directly accessible to experimentation, yet it is interesting and sometimes crucial to understand if it admits nonclassical properties. An example of this sort that has recently been enjoying considerable attention is that of two quantum masses coupled via a gravitational field. It has been argued that the gain of quantum entanglement between the masses indicates nonclassicality of the states of the whole tripartite system. Here, we focus on the nonclassical properties of the involved interactions rather than the states. We derive inequalities the violation of which indicates noncommutativity and nondecomposability (open-system generalization of noncommuting unitaries) of interactions through the mediators. The derivations are based on properties of general quantum formalism and make minimalistic assumptions about the studied systems; in particular, the interactions can remain uncharacterized throughout the assessment. Furthermore, we also present conditions that solely use correlations between the coupled systems, excluding the need to measure the mediator. Next, we show that the amount of violation places a lower bound on suitably defined degree of nondecomposability. This makes the methods quantitative and at the same time experiment ready. We give applications of these techniques in two different fields: for detecting the nonclassicality of gravitational interaction and in bounding the Trotter error in quantum simulations.

Integrable models are characterized by the existence of stable excitations that can propagate indefinitely without decaying. This includes multimagnon bound states in the celebrated $XXZ$ spin-chain model and its integrable Floquet counterpart. A recent Google Quantum AI experiment [A. Morvan et al., Nature 612, 240 (2022)] realizing the Floquet model has demonstrated the persistence of such collective excitations even when the integrability is broken: this observation is at odds with the expectation of ergodic dynamics in generic nonintegrable systems. Here, we study the spectrum of the model realized in the experiment using exact diagonalization and physical arguments. We find that isolated bands corresponding to the descendants of the exact bound states of the integrable model are clearly observable in the spectrum for a large range of system sizes. However, our numerical analysis of the localization properties of the eigenstates suggests that the bound states become unstable in the thermodynamic limit. A perturbative estimate of the decay rate agrees with the prediction of an eventual instability for large system sizes.

Recent quantum simulation by Google [Nature 612, 240 (2022)] has demonstrated the formation of bound states of interacting photons in a quantum-circuit version of the XXZ spin chain. While such bound states are protected by integrability in a one-dimensional chain, the experiment found the bound states to be unexpectedly robust when integrability was broken by decorating the circuit with additional qubits, at least for small numbers of qubits ($\le 24$) within the experimental capability. Here we scrutinize this result by state-of-the-art classical simulations, which greatly exceed the experimental system sizes and provide a benchmark for future studies in larger circuits. We find that the bound states consisting of a finite number of photons are indeed robust in the nonintegrable regime, even after scaling to the infinite-time and infinite-system size limit. Moreover, we show that such systems possess unusual spectral properties, with level statistics that deviates from the random matrix theory expectation. On the other hand, for low but finite density of photons, we find a much faster onset of thermalization and significantly weaker signatures of bound states, suggesting that anomalous dynamics may only be a property of dilute systems with zero density of photons in the thermodynamic limit. The robustness of the bound states is also influenced by the number of decoration qubits and, to a lesser degree, by the regularity of their spatial arrangement.

We present a scheme for universal topological quantum computation based on Clifford-complete braiding and fusion of symmetry defects in the 3-fermion anyon theory, supplemented with magic state injection. We formulate a fault-tolerant measurement-based realization of this computational scheme on the lattice using ground states of the Walker-Wang model for the 3-fermion anyon theory with symmetry defects. The Walker-Wang measurement-based topological quantum computation paradigm that we introduce provides a general construction of computational resource states with thermally stable symmetry-protected topological order. We also demonstrate how symmetry defects of the 3-fermion anyon theory can be realized in a two-dimensional subsystem code due to Bombín—making contact with an alternative implementation of the 3-fermion defect-computation scheme via code deformations.

Chaotic systems are highly sensitive to a small perturbation and are ubiquitous throughout the biological sciences, the physical sciences, and even the social sciences. Taking this as the underlying principle, we construct an operational notion for quantum chaos. Namely, we demand that the future state of a many-body isolated quantum system is sensitive to past multitime operations on a small subpart of that system. By “sensitive,” we mean that the resultant states from two different perturbations cannot easily be transformed into each other. That is, the pertinent quantity is the complexity of the effect of the perturbation within the final state. From this intuitive metric, which we call the butterfly-flutter fidelity, we use the language of multitime quantum processes to identify a series of operational conditions on chaos; in particular, the scaling of the spatiotemporal entanglement. Our criteria already contain the routine notions, as well as the well-known diagnostics for quantum chaos. This includes the Peres-Loschmidt echo, dynamical entropy, tripartite mutual information, and local-operator entanglement. We hence present a unified framework for these existing diagnostics within a single structure. We also go on to quantify how several mechanisms, such as evolution generated from random circuits, lead to quantum chaos. Our work paves the way to systematically study many-body dynamical phenomena such as many-body localization, measurement-induced phase transitions, and Floquet dynamics.

Despite its fully unitary dynamics, the bosonic Kitaev chain (BKC) displays key hallmarks of non-Hermitian physics, including nonreciprocal transport and the non-Hermitian skin effect. Here, we demonstrate another remarkable phenomena: the existence of an entanglement phase transition (EPT) in a variant of the BKC that occurs as a function of a Hamiltonian parameter $g$ and which coincides with a transition from a reciprocal to a nonreciprocal phase. As $g$ is reduced below a critical value, the postquench entanglement entropy of a subsystem of size $l$ goes from a volume-law phase, where it scales as $l$, to a supervolume-law phase, where it scales like $lN$, where $N$ is the total system size. This EPT occurs for a system undergoing purely unitary evolution and does not involve measurements, postselection, disorder, or dissipation. We derive analytically the entanglement entropy out of and at the critical point for the cases of $l=1$ and $l/N\ll 1$.

We present a proposal for a tunable source of single photons operating in the terahertz (THz) regime. This scheme transforms incident visible photons into quantum THz radiation by driving a single polar quantum emitter with an optical laser, with its permanent dipole enabling dressed THz transitions enhanced by the resonant coupling to a cavity. This mechanism offers optical tunability of properties such as the frequency of the emission or its quantum statistics (ranging from antibunching to entangled multiphoton states) by modifying the intensity and frequency of the drive. We show that the implementation of this proposal is feasible with state-of-the-art photonics technology.

We implement a scalable platform for quantum sensing comprising hundreds of sites capable of holding individual laser-cooled atoms and demonstrate the applicability of this single-quantum-system sensor array to magnetic field mapping on a two-dimensional grid. With each atom being confined in an optical tweezer within an area of $0.5\text{μ}{\mathrm{m}}^2$ at mutual separations of $7.0(2)\text{μ}\mathrm{m}$, we obtain micrometer-scale spatial resolution and highly parallelized operation. An additional steerable optical tweezer allows rearrangement of atoms within the grid and enables single-atom scanning microscopy with submicron resolution. This individual-atom sensor platform finds an immediate application in mapping an externally applied dc gradient magnetic field. In a Ramsey-type measurement, we obtain a field resolution of 98(29) nT. We estimate the sensitivity to be $25\text{μ}\mathrm{T}/\sqrt{\mathrm{Hz}}$.

Building on the development of momentum state lattices (MSLs) over the past decade, we introduce a simple extension of this technique to higher dimensions. Based on the selective addressing of unique Bragg resonances in matter-wave systems, MSLs have enabled the realization of tight-binding models with tunable disorder, gauge fields, non-Hermiticity, and other features. Here, we examine and outline an experimental approach to building scalable and tunable tight-binding models in two dimensions describing the laser-driven dynamics of atoms in momentum space. Using numerical simulations, we highlight some of the simplest models and types of phenomena this system is well suited to address, including flat-band models with kinetic frustration and flux lattices supporting topological boundary states. Finally, we discuss many of the direct extensions to this model, including the introduction of disorder and non-Hermiticity, which will enable the exploration of new transport and localization phenomena in higher dimensions.

Repeated projective measurements in unitary circuits can lead to an entanglement phase transition as the measurement rate is tuned. In this work, we consider a different setting in which the projective measurements are replaced by dynamically chosen unitary gates that minimize the entanglement. This can be seen as a one-dimensional unitary circuit game in which two players get to place unitary gates on randomly assigned bonds at different rates: the “entangler” applies a random local unitary gate with the aim of generating extensive (volume-law) entanglement. The “disentangler,” based on limited knowledge about the state, chooses a unitary gate to reduce the entanglement entropy on the assigned bond with the goal of limiting to only finite (area-law) entanglement. In order to elucidate the resulting entanglement dynamics, we consider three different scenarios: (i) a classical discrete height model, (ii) a Clifford circuit, and (iii) a general $U(4)$ unitary circuit. We find that both the classical and Clifford circuit models exhibit phase transitions as a function of the rate that the disentangler places a gate, which have similar properties that can be understood through a connection to the stochastic Fredkin chain. In contrast, the entangler always wins when using Haar random unitary gates and we observe extensive, volume-law entanglement for all nonzero rates of entangling.

We report an accurate and efficient classical simulation of a kicked Ising quantum system on the heavy hexagon lattice. A simulation of this system was recently performed on a 127-qubit quantum processor using noise-mitigation techniques to enhance accuracy [Y. Kim et al., Nature, 618, 500–5 (2023)]. Here we show that, by adopting a tensor network approach that reflects the geometry of the lattice and is approximately contracted using belief propagation, we can perform a classical simulation that is significantly more accurate and precise than the results obtained from the quantum processor and many other classical methods. We quantify the treelike correlations of the wave function in order to explain the accuracy of our belief propagation-based approach. We also show how our method allows us to perform simulations of the system to long times in the thermodynamic limit, corresponding to a quantum computer with an infinite number of qubits. Our tensor network approach has broader applications for simulating the dynamics of quantum systems with treelike correlations.

This tutorial article introduces the physics of quantum information scrambling in quantum many-body systems. The goals are to understand how to precisely quantify the spreading of quantum information and how causality emerges in complex quantum systems. We introduce a general framework to study the dynamics of quantum information, including detection and decoding. We show that the dynamics of quantum information is closely related to operator dynamics in the Heisenberg picture, and, under certain circumstances, can be precisely quantified by the so-called out-of-time-ordered correlator (OTOC). The general behavior of the OTOC is discussed based on several toy models, including the Sachdev-Ye-Kitaev model, random circuit models, and Brownian models, in which the OTOC is analytically tractable. We introduce numerical methods, including exact diagonalization and tensor network methods, to calculate the OTOC for generic quantum many-body systems. We also survey current experimental schemes for measuring the OTOC in various quantum simulators.

Residual noise photons in a readout resonator become a major source of dephasing for a superconducting qubit when the resonator is optimized for a fast, high-fidelity dispersive readout. Here, we propose and demonstrate a nonlinear Purcell filter that suppresses such an undesirable dephasing process without sacrificing the readout performance. When a readout pulse is applied, the filter automatically reduces the effective linewidth of the readout resonator, increasing the sensitivity of the qubit to the input field. The noise tolerance of the device we have fabricated is shown to be enhanced by a factor of 3 relative to a device with a linear filter. The measurement rate is enhanced by another factor of 3 by utilizing the bifurcation of the nonlinear filter. A readout fidelity of 99.4% and a quantum nondemolition fidelity of 99.2% are achieved using a 40-ns readout pulse. The nonlinear Purcell filter will be an effective tool for realizing a fast, high-fidelity readout without compromising the coherence time of the qubit.

As quantum devices make steady progress towards intermediate scale and fault-tolerant quantum computing, it is essential to develop rigorous and efficient measurement protocols that account for known sources of noise. Most existing quantum characterization protocols such as gate-set tomography and randomized benchmarking assume the noise acting on the qubits is Markovian. However, this assumption is often not valid, as for the case of $1/f$ charge noise or hyperfine nuclear spin noise. Here, we present a general framework for quantum process tomography (QPT) in the presence of time-correlated noise. We further introduce fidelity benchmarks that quantify the relative strength of different sources of Markovian and non-Markovian noise. As an application of our method, we perform a comparative theoretical and experimental analysis of silicon spin qubits. We first develop a detailed noise model that accounts for the dominant sources of noise and validate the model against experimental data. Applying our framework for time-correlated QPT, we find that the number of independent parameters needed to characterize one- and two-qubit gates can be compressed by $10\times$ and $100\times$, respectively, when compared to the fully generic case. These compressions reduce the amount of tomographic measurements needed in experiment, while also significantly speeding up numerical simulations of noisy quantum circuit dynamics compared to time-dependent Hamiltonian simulation. Using this compressed noise model, we find good agreement between our theoretically predicted process fidelities and two-qubit interleaved randomized benchmarking fidelities of 99.8% measured in recent experiments on silicon spin qubits. More broadly, our formalism can be directly extended to develop efficient and scalable tuning protocols for high-fidelity control of large arrays of quantum devices with non-Markovian noise.

We establish a general framework for developing approximation algorithms for a class of counting problems. Our framework is based on the cluster expansion of the abstract polymer model formalism of Kotecký and Preiss. We apply our framework to obtain efficient algorithms for (1) approximating probability amplitudes of a class of quantum circuits close to the identity, (2) approximating expectation values of a class of quantum circuits with operators close to the identity, (3) approximating partition functions of a class of quantum spin systems at high temperature, and (4) approximating thermal expectation values of a class of quantum spin systems at high temperature with positive-semidefinite operators. Further, we obtain hardness of approximation results for approximating probability amplitudes of quantum circuits and partition functions of quantum spin systems. This establishes a computational complexity transition for these problems and shows that our algorithmic conditions are optimal under complexity-theoretic assumptions. Finally, we show that our algorithmic condition is almost optimal for expectation values and optimal for thermal expectation values in the sense of zero freeness.

Whether a given target state can be prepared by starting with a simple product state and acting with a finite-depth quantum circuit is a key question in condensed matter physics and quantum information science. It underpins classifications of topological phases, as well as the understanding of topological quantum codes, and has obvious relevance for device implementations. Traditionally, this question assumes that the quantum circuit is made up of unitary gates that are geometrically local. Inspired by the advent of noisy intermediate-scale quantum devices, we reconsider this question with $k$-local gates, i.e., gates that act on no more than $k$ degrees of freedom but are not restricted to be geometrically local. First, we construct explicit finite-depth circuits of symmetric $k$-local gates that create symmetry-protected topological (SPT) states from an initial product state. Our construction applies to SPT states protected by global symmetries or subsystem symmetries but not to those with higher-form symmetries, which we conjecture remain nontrivial. Next, we show how to implement arbitrary translationally invariant quantum cellular automata in any dimension using finite-depth circuits of $k$-local gates. These results imply that the topological classifications of SPT phases and quantum cellular automata both collapse to a single trivial phase in the presence of $k$-local interactions. We furthermore argue that SPT phases are fragile to generic $k$-local symmetric perturbations. We conclude by discussing the implications for other phases, such as fracton phases, and surveying future directions. Our analysis opens a new experimentally motivated conceptual direction examining the stability of phases and the feasibility of state preparation without the assumption of geometric locality.

Practical quantum networks require interfacing quantum memories with existing channels and systems that operate in the telecom band. Here we demonstrate low-noise, bidirectional quantum frequency conversion that enables a solid-state quantum memory to directly interface with telecom-band systems. In particular, we demonstrate conversion of visible-band single photons emitted from a silicon-vacancy ($\mathrm{Si}$$V$) center in diamond to the telecom $\mathrm{O}$ band, maintaining low noise ($g^2(0)<0.1$) and high indistinguishability ($V=89\pm 8\mathrm{\%}$). We further demonstrate the utility of this system for quantum networking by converting telecom-band time-bin pulses, sent across a lossy and noisy 50-km deployed fiber link, to the visible band and entangling them with a diamond quantum memory with fidelity $\mathcal{F}\ge 87\pm 2.5\mathrm{\%}$. These results demonstrate the viability of $\mathrm{Si}$$V$ quantum memories integrated with telecom-band systems for scalable quantum networking applications.

We introduce a framework for fault-tolerant postselection (FTPS) of fault-tolerant codes and channels—such as those based on surface codes—using soft-information metrics based on visible syndrome and erasure information. We introduce several metrics for ranking configurations of syndromes and erasures. In particular, we introduce the logical gap (and variants thereof) as a powerful soft-information metric for predicting logical error rates of fault-tolerant channels based on topological error-correcting codes. The logical gap is roughly the unsigned weight difference between inequivalent logical corrections and is adaptable to any tailored noise model or decoder. We deploy this framework to prepare high-quality surface-code magic states with low overhead under a model of independent and identically distributed (IID) Pauli and erasure errors. Postselection strategies based on the logical gap can suppress the encoding error rate (EER) of a magic state preparation channel to the level of the physical error rate with low overhead. For example, when operating at $60\mathrm{\%}$ of the bulk threshold of the corresponding surface code, an overall reduction of the EER by a factor of $15$ is achievable with a relative overhead factor of $<2$ (approximately $23$ times less than that of simple syndrome-counting rules). We analyze a schematic buffer architecture for implementing postselection rules on magic state factories in the context of magic state distillation. The FTPS framework can be utilized for mitigating errors in more general fault-tolerant logical channels.

High-fidelity qubit readout is critical in order to obtain the thresholds needed to implement quantum error-correction protocols and achieve fault-tolerant quantum computing. Large-scale silicon qubit devices will have densely packed arrays of quantum dots with multiple charge sensors that are, on average, farther away from the quantum dots, entailing a reduction in readout fidelities. Here, we present a readout technique that enhances the readout fidelity in a linear SiMOS four-dot array by amplifying correlations between a pair of single-electron transistors, known as a twin SET. By recording and subsequently correlating the twin SET traces as we modulate the dot detuning across a charge transition, we demonstrate a reduction in the charge readout infidelity by over one order of magnitude compared to traditional readout methods. We also study the spin-to-charge conversion errors introduced by the modulation technique and conclude that faster modulation frequencies avoid relaxation-induced errors without introducing significant spin-flip errors, favoring the use of the technique at short integration times. This method not only allows for faster and higher-fidelity qubit measurements but it also enhances the signal corresponding to charge transitions that take place farther away from the sensors, enabling a way to circumvent the reduction in readout fidelities in large arrays of qubits.