We present shadow spectroscopy as a simulator-agnostic quantum algorithm for estimating energy gaps using very few circuit repetitions (shots) and no extra resources (ancilla qubits) beyond performing time evolution and measurements. The approach builds on the fundamental feature that every observable property of a quantum system must evolve according to the same harmonic components: we can reveal them by postprocessing classical shadows of time-evolved quantum states to extract a large number of time-periodic signals $N_o\propto {10}^8$, whose frequencies correspond to Hamiltonian energy differences with precision limited as $ϵ\propto 1/T$ for simulation time $T$. We provide strong analytical guarantees that (a) quantum resources scale as $O(\log N_o)$, while the classical computational complexity is linear $O(N_o)$, (b) the signal-to-noise ratio increases with the number of processed signals as $\propto \sqrt{N_o}$, and (c) spectral peak positions are immune to reasonable levels of noise. We demonstrate our approach on model spin systems and the excited-state conical intersection of molecular ${\mathrm{CH}}_2$ and verify that our method is indeed intuitively easy to use in practice, robust against gate noise, amiable to a new type of algorithmic-error mitigation technique, and uses relatively few shots given a reasonable initial state is supplied—we demonstrate that even 10 shots per time step can be sufficient. Finally, we measured a high-quality, experimental shadow spectrum of a spin chain on readily available IBM quantum computers, achieving the same precision as in noise-free simulations without using any advanced error mitigation, and verified scalability in tensor-network simulations of up to 100-qubit systems.

Adaptive quantum circuits—where a quantum many-body state is controlled using measurements and conditional unitary operations—are a powerful paradigm for state preparation and quantum error-correction tasks. They can support two types of nonequilibrium quantum phase transitions: measurement-induced transitions between volume- and area-law-entangled steady states and control-induced transitions where the system falls into an absorbing state or, more generally, an orbit visiting several absorbing states. Within this context, nonlocal conditional operations can alter the critical properties of the two transitions and the topology of the phase diagram. Here, we consider the scenario where the measurements are nonlocal, in order to engineer efficient control onto dynamical trajectories. Motivated by Rydberg-atom arrays, we consider a locally constrained model with global sublattice magnetization measurements and local correction operations to steer the system’s dynamics onto a many-body orbit with finite recurrence time. The model has a well-defined classical limit, which we leverage to aid our analysis of the control transition. As a function of the density of local correction operations, we find control and entanglement transitions with continuously varying critical exponents. For sufficiently high densities of local correction operations, we find that both transitions acquire a dynamical critical exponent $z<1$, reminiscent of criticality in long-range power-law interacting systems. At low correction densities, we find that the criticality reverts to a short-range nature with $z\gtrsim 1$. In the long-range regime, the control and entanglement transitions are indistinguishable to within the resolution of our finite-size numerics, while in the short-range regime we find evidence that the transitions become distinct. We conjecture that the effective long-range criticality mediated by collective measurements is essential in driving the two transitions together.

We explore the relationship between renormalization-group (RG) flow and error correction by constructing quantum algorithms that exactly recognize one-dimensional symmetry-protected topological (SPT) phases protected by finite internal Abelian symmetries. For each SPT phase, our algorithm runs a quantum circuit, which emulates RG flow: an arbitrary input ground-state wave function in the phase is mapped to a unique minimally entangled reference state, thereby allowing for efficient phase identification. This construction is enabled by viewing a generic input state in the phase as a collection of coherent “errors” applied to the reference state, and engineering a quantum circuit to efficiently detect and correct such errors. Importantly, the error-correction threshold is proven to coincide exactly with the phase boundary. We discuss the implications of our results in the context of condensed-matter physics, machine learning, and near-term quantum algorithms.

Fluxonium qubit is a promising elementary building block for quantum information processing due to its long coherence time combined with a strong anharmonicity. In this paper, we realize a 60-ns direct cnot gate on two inductively coupled fluxoniums, which behave almost exactly as a pair of transversely coupled spin-$1/2$ systems. The cnot-gate fidelity, estimated using randomized benchmarking, was as high as 99.94%. Furthermore, the fidelity remains above 99.9% for 24 days without any recalibration between measurements. Compared with the 99.96% fidelity of a 60-ns identity gate, our data brings the investigation of the nondecoherence-related errors during logical operations down to $2\times {10}^{-4}$. The present result adds a simple and robust two-qubit gate into the still relatively small family of the “beyond three nines” gates on superconducting qubits.

The interplay of symmetry and topology in quantum many-body mixed states has recently garnered significant interest. In a phenomenon not seen in pure states, mixed states can exhibit average symmetries—symmetries that act on component states while leaving the ensemble invariant. In this work, we systematically characterize symmetry-protected topological (SPT) phases of short-range entangled (SRE) mixed states of spin systems—protected by both average and exact symmetries—by studying their pure Choi states in a doubled Hilbert space, where the familiar notions and tools for SRE and SPT pure states apply. This advantage of the doubled space comes with a price: extra symmetries as well as subtleties around how hermiticity and positivity of the original density matrix constrain the possible SPT invariants. Nevertheless, the doubled-space perspective allows us to obtain a systematic classification of mixed-state SPT (MSPT) phases. We also investigate the robustness of MSPT invariants under symmetric finite-depth quantum channels, the bulk-boundary correspondence for MSPT phases, and the consequences of the MSPT invariants for the separability of mixed states and the symmetry-protected sign problem. In addition to MSPT phases, we study the patterns of spontaneous symmetry breaking (SSB) of mixed states, including the phenomenon of exact-to-average SSB, and the order parameters that detect them. Mixed-state SSB is related to an ingappability constraint on symmetric Lindbladian dynamics.

In this paper, we develop a systematic approach to characterize the ${}^{\prime }$t Hooft anomaly in open quantum systems. Owing to nontrivial couplings to the environment, symmetries in such systems manifest as either strong or weak type. By representing their symmetry transformation through superoperators, we incorporate them in a unified framework that enables a direct calculation of their anomalies. In the case where the full symmetry group $\mathrm{\Gamma }$ arises as a central extension of the strong-symmetry group $K$ with the weak-symmetry group $G$, we find that anomalies of bosonic systems are classified by $H^{d+2}(\mathrm{\Gamma },U(1))/H^{d+2}(G,U(1))$ in $d$ spatial dimensions. To illustrate the power of anomalies in open quantum systems, we generally prove that anomaly must lead to nontrivial mixed-state quantum phases as long as the weak symmetry is imposed. Analogous to the “anomaly matching” condition ensuring nontrivial low-energy physics in closed systems, anomaly also guarantees nontrivial steady states and long-time dynamics for open quantum systems governed by Lindbladians. Notably, we identify a novel $(1+1)\mathrm{D}$ mixed-state quantum phase that has no counterpart in closed systems, where the steady state shows no nontrivial correlation function in the bulk, but displays long-range correlation on the boundary, which is enforced by anomalies. We further establish the general relations between mixed-state anomalies and such unconventional boundary correlation. Moreover, we explore the generalization of the “anomaly inflow” mechanism in open quantum systems. We construct $(1+1)\mathrm{D}$ and $(2+1)\mathrm{D}$ Lindbladians whose steady states have mixed-state symmetry-protected-topological order in the bulk, with corresponding edge theories characterized by nontrivial anomalies. Finally, we generalize our results to the cases including lattice translation symmetries, which serves as the Lieb-Schulz-Mattis (LSM) theorem for mixed states.

Quantum complexity measures the difficulty of realizing a quantum process, such as preparing a state or implementing a unitary. We present an approach to quantifying the thermodynamic resources required to implement a process if the process’s complexity is restricted. We focus on the prototypical task of information erasure, or Landauer erasure, wherein an $n$-qubit memory is reset to the all-zero state. We show that the minimum thermodynamic work required to reset an arbitrary state in our model, via a complexity-constrained process, is quantified by the state’s complexity entropy. The complexity entropy therefore quantifies a trade-off between the work cost and complexity cost of resetting a state. If the qubits have a nontrivial (but product) Hamiltonian, the optimal work cost is determined by the complexity relative entropy. The complexity entropy quantifies the amount of randomness a system appears to have to a computationally limited observer. Similarly, the complexity relative entropy quantifies such an observer’s ability to distinguish two states. We prove elementary properties of the complexity (relative) entropy. In a random circuit—a simple model for quantum chaotic dynamics—the complexity entropy transitions from zero to its maximal value around the time corresponding to the observer’s computational-power limit. Also, we identify information-theoretic applications of the complexity entropy. The complexity entropy quantifies the resources required for data compression if the compression algorithm must use a restricted number of gates. We further introduce a complexity conditional entropy, which arises naturally in a complexity-constrained variant of information-theoretic decoupling. Assuming that this entropy obeys a conjectured chain rule, we show that the entropy bounds the number of qubits that one can decouple from a reference system, as judged by a computationally bounded referee. Overall, our framework extends the resource-theoretic approach to thermodynamics to integrate a notion of time, as quantified by complexity.

Nonstabilizerness, or “magic,” is a critical quantum resource that, together with entanglement, characterizes the nonclassical complexity of quantum states. Here, we address the problem of quantifying the average nonstabilizerness of random matrix product states (RMPSs). RMPSs represent a generalization of random product states featuring bounded entanglement that scales logarithmically with the bond dimension $\chi$. We demonstrate that the stabilizer Rényi entropies converge to that of Haar-random states as $N/{\chi }^{\alpha }$, where $N$ is the system size and the $\alpha$ are integer exponents. This indicates that MPSs with a modest bond dimension are as magical as generic states. Subsequently, we introduce the ensemble of Clifford-enhanced matrix product states ($\mathcal{C}\mathrm{MP}\mathrm{Ss}$), built by the action of Clifford unitaries on RMPSs. Leveraging our previous result, we show that $\mathcal{C}\mathrm{MP}\mathrm{Ss}$ can approximate quantum state 4-designs with arbitrary accuracy. Specifically, for a constant $N$, $\mathcal{C}\mathrm{MP}\mathrm{Ss}$ become close to 4-designs, with a scaling as ${\chi }^{-2}$. Our findings indicate that combining Clifford unitaries with polynomially complex tensor-network states can generate highly nontrivial quantum states.

Among all the inventions that quantum physics has produced, the laser holds a particularly important place, both for the rich story of successive discoveries that led to its birth, and for the role it has played in fundamental and applied research. I recall here the lineage of theoretical discoveries and experiments that have marked this history, restricting myself to the contribution of lasers to blue sky science and leaving apart its well-known role in various domains of technology. This story started from advances in the old quantum theory, from Einstein’s theoretical description of stimulated emission to O. Stern’s experimental discovery of the spatial quantization of the electron spin. Nuclear magnetic resonance, atomic clocks, optical pumping, and masers followed and the pace of discoveries accelerated with the appearance of the laser in 1960. This extraordinary light source has since enabled breakthroughs in fundamental physics and opened up fields of research that could not even have been imagined at the time of its birth. I was fortunate to begin my career in physics at this crossroad of atomic physics and optics. I give in this article my personal view of the great adventures in fundamental research in which I participated as an actor or spectator, from the cooling and trapping of atoms by light, to the physics of quantum gases of bosons and fermions, the manipulation of individual quantum particles and quantum simulations. Many other areas of fundamental physics, which I will only mention briefly, owe their development to lasers and further advances are still to be expected in the years to come.

Symmetry in mixed quantum states can manifest in two distinct forms: strong symmetry, where each individual pure state in the quantum ensemble is symmetric with the same charge, and weak symmetry, which applies only to the entire ensemble. This paper explores a novel type of spontaneous symmetry breaking (SSB) where a strong symmetry is broken to a weak one. While the SSB of a weak symmetry is measured by the long-ranged two-point correlation function, the strong-to-weak SSB (SWSSB) is measured by the fidelity correlator. We prove that SWSSB is a universal property of mixed-state quantum phases, in the sense that the phenomenon of SWSSB is robust against symmetric low-depth local quantum channels. We also show that the symmetry breaking is “spontaneous” in the sense that the effect of a local symmetry-breaking measurement cannot be recovered locally. We argue that a thermal state at a nonzero temperature in the canonical ensemble (with fixed symmetry charge) should have spontaneously broken strong symmetry. Additionally, we study nonthermal scenarios where decoherence induces SWSSB, leading to phase transitions described by classical statistical models with bond randomness. In particular, the SWSSB transition of a decohered Ising model can be viewed as the “ungauged” version of the celebrated toric-code decodability transition. We confirm that, in the decohered Ising model, the SWSSB transition defined by the fidelity correlator is the only physical transition in terms of channel recoverability. We also comment on other (inequivalent) definitions of SWSSB, through correlation functions with higher Rényi indices.

The precision and response of trajectory observables offer valuable insights into the behavior of nonequilibrium systems. For classical systems, trade-offs between these characteristics and thermodynamic costs, such as entropy production and dynamical activity, have been established through uncertainty relations. Quantum systems, however, present unique challenges, where quantum coherence can enhance precision and violate classical uncertainty relations. In this study, we derive trade-off relations for stochastic observables in Markovian open quantum systems. Specifically, we present three key results: (i) a quantum generalization of the thermokinetic uncertainty relation, which bounds the relative fluctuations of currents in terms of entropy production and dynamical activity; (ii) a quantum inverse uncertainty relation, which constrains the relative fluctuations of arbitrary counting observables based on their instantaneous fluctuations and the spectral gap of the symmetrized Liouvillian; and (iii) a quantum response kinetic uncertainty relation, which bounds the response of general observables to kinetic perturbations in terms of dynamical activity. These fundamental bounds, validated numerically using a three-level maser and a boundary-driven $XXZ$ spin chain, provide a comprehensive framework for understanding the interplay between precision, response, and thermodynamic costs in quantum systems.

Engineering deterministic photonic gates with simple resources is one of the long-standing challenges in photonic quantum computing. Here, we design a passive conditional gate between copropagating photons using an array of only two-level emitters. The key resource is to harness the effective photon-photon interaction induced by the chiral coupling of the emitter array to two waveguide modes with different resonant momenta at the emitter’s transition frequency. By studying the system’s multiphoton scattering response, we demonstrate that, in certain limits, this configuration induces a nonlinear $\pi$-phase shift between the polariton eigenstates of the system without distorting spectrally the wave packets. Then, we show how to harness this nonlinear phase shift to engineer a conditional, deterministic photonic gate in different qubit encodings, with a fidelity arbitrarily close to 1 in the limit of large number of emitters and coupling efficiency. Our configuration can be implemented in topological photonic setups with multiple chiral-edge modes, opening their use for quantum information processing, or in other setups where such chiral light-matter couplings can be effectively obtained, such as optical fibers or photonic-crystal optical and microwave waveguides.

Existing schemes for demonstrating quantum computational advantage are subject to various practical restrictions, including the hardness of verification and challenges in experimental implementation. Meanwhile, analog quantum simulators have been realized in many experiments to study novel physics. In this work, we propose a quantum advantage protocol based on single-step Feynman-Kitaev verification of an analog quantum simulation, in which the verifier need only run an $O({\lambda }^2)$-time classical computation, and the prover need only prepare $O(1)$ samples of a history state and perform $O({\lambda }^2)$ single-qubit measurements, for a security parameter $\lambda$. We also propose a near-term feasible strategy for honest provers and discuss potential experimental realizations.

Highly entangled multiphoton graph states are a crucial resource in photonic quantum computation and communication. Yet, the lack of photon-photon interactions makes the construction of such graph states especially challenging. Typically, these states are produced through probabilistic single-photon sources and linear-optics entangling operations that require indistinguishable photons. The resulting inefficiency of these methods necessitates a large overhead in the number of sources and operations, creating a major bottleneck in the photonic approach. Here, we show how harnessing single-atom-based photonic operations can enable deterministic generation of photonic graph states, while also lifting the requirement for photon indistinguishability. To this end, we introduce a multigate quantum node comprising a single atom in a $W$-type level scheme coupled to an optical resonator. This configuration provides a versatile toolbox for generating graph states, allowing the operation of both the controlled-$Z$ and swap photon-atom gates, as well as the deterministic generation of single photons. Furthermore, the ability to deterministically entangle photonic qubits enables the expansion of the generated state by stitching together graph states produced by different nodes. We investigate the implementation of this gate-based approach using ${}^{87}\mathrm{Rb}$ atoms and evaluate its performance through numerical simulations.

Entanglement is essential for quantum information processing, but is limited by noise. We address this by developing high-yield entanglement distillation protocols with several advancements. (1) We extend the 2-to-1 recurrence entanglement distillation protocol to higher-rate $n$-to-$(n-1)$ protocols that can correct any single-qubit errors. These protocols are evaluated through numerical simulations focusing on fidelity and yield. We also outline a method to adapt any classical error-correcting code for entanglement distillation, where the code can correct both bit-flip and phase-flip errors by incorporating Hadamard gates. (2) We propose a constant-depth decoder for stabilizer codes that transforms logical states into physical ones using single-qubit measurements. This decoder is applied to entanglement distillation protocols, reducing circuit depth and enabling protocols derived from high-performance quantum error-correcting codes. We demonstrate this by evaluating the circuit complexity for entanglement distillation protocols based on surface codes and quantum convolutional codes. (3) Our stabilizer entanglement distillation techniques advance quantum computing. We propose a fault-tolerant protocol for constant-depth encoding and decoding of arbitrary states in surface codes, with potential extensions to more general quantum low-density parity-check codes. This protocol is feasible with state-of-the-art reconfigurable atom arrays and surpasses the limits of conventional logarithmic depth encoders. Overall, our study integrates stabilizer formalism, measurement-based quantum computing, and entanglement distillation, advancing both quantum communication and computing.

Integrated photonic microresonators have become an essential resource for generating photonic qubits for quantum information processing, entanglement distribution and networking, and quantum communications. The pair-generation rate is enhanced by reducing the microresonator radius, but this comes at the cost of increasing the frequency-mode spacing and reducing the quantum information spectral density. Here, we circumvent this rate-density trade-off in an $\mathrm{Al}\mathrm{Ga}\mathrm{As}$-on-insulator photonic device by multiplexing an array of 20 small-radius microresonators, each producing a 650-GHz-spaced comb of time-energy entangled-photon pairs. The resonators can be independently tuned via integrated thermo-optic heaters, enabling control of the mode spacing from degeneracy up to a full free spectral range. We demonstrate simultaneous pumping of five resonators with up to $50$-GHz relative comb offsets, where each resonator produces pairs exhibiting time-energy entanglement visibilities up to $95\mathrm{\%}$, coincidence-to-accidental ratios exceeding $5000$, and an on-chip pair rate up to $2.6{\mathrm{G}\mathrm{Hz}/\mathrm{mW}}^2$ per comb line—an improvement over prior work by more than a factor of 40. As a demonstration, we generate frequency-bin qubits in a maximally entangled two-qubit Bell state with fidelity exceeding $87\mathrm{\%}$ ($90\mathrm{\%}$ with background correction) and detected frequency-bin entanglement rates up to 7 kHz (an approximately $70$ MHz on-chip pair rate) using a pump power of approximately $250\text{μ}\mathrm{W}$. Multiplexing small-radius microresonators combines the key capabilities required for programmable and dense photonic qubit encoding while retaining high pair-generation rates, heralded single-photon purity, and entanglement fidelity.

Quantum magic is a necessary resource for quantum computers to be not efficiently simulable by classical computers. Previous results have linked the amount of quantum magic, characterized by the number of $T$ gates or the stabilizer rank, to classical simulability. However, the effect of the distribution of quantum magic on the hardness of simulating a quantum circuit remains open. In this work, we investigate the classical simulability of quantum circuits with alternating Clifford and $T$ layers across three tasks: amplitude estimation, sampling, and evaluating Pauli observables. In the case in which all $T$ gates are distributed in a single layer, performing amplitude estimation and sampling to multiplicative error are already classically intractable under reasonable assumptions, but Pauli observables are easy to evaluate. Surprisingly, with the addition of just one $T$-gate layer or merely replacing all $T$ gates with $T^{1/2}$, the Pauli evaluation task reveals a sharp complexity transition from being in P to being GapP-complete. Nevertheless, when the precision requirement is relaxed to $1/\mathrm{poly}(n)$ additive error, we are able to give a polynomial-time classical algorithm to compute amplitudes, Pauli observables, and sampling from $\log (n)$-sized marginal distributions for any magic-depth-1 circuit that is decomposable into a product of diagonal gates. This rules out certain forms of quantum advantage in these circuits. Our research provides new techniques to simulate highly magical circuits while shedding light on their complexity and their significant dependence on the magic depth.

We investigate the problem of efficiently estimating expectation values of large sets of observables from copies of an unknown many-body quantum state. This task, known as shadow tomography, is crucial for quantum simulation and quantum chemistry, where the relevant observables often include $k$-body fermionic or $k$-local Pauli operators. A key goal is to achieve sample efficiency, computational efficiency, and few-copy measurements. We introduce the notion of triply efficient shadow tomography to formalize these requirements. Prior work has shown that single-copy measurements suffice for $k$-local Pauli operators with constant $k$. However, we prove that for $k$-body fermionic observables and the full set of $n$-qubit Pauli operators, single-copy protocols fail to achieve sample-efficient tomography. We then present new protocols based on measuring two copies of the state at a time that overcome these lower bounds, providing a triply efficient protocol.

Quantum property preservation (QPP) is the problem of maintaining a target property of a quantum system for as long as possible. This problem arises naturally in the context of open quantum systems subject to decoherence. Here, we develop a general theory to formalize and analyze QPP. We characterize properties encoded as scalar functions of the system state that can be preserved time-locally via continuous control using smoothly varying, time-dependent control Hamiltonians. The theory offers an intuitive geometric interpretation involving the level sets of the target property and the stable and unstable points related to the noise channel. We present solutions for various noise channels and target properties, which are classified as trivially controllable, uncontrollable, or controllable. In the controllable scenario, we demonstrate the existence of control Hamiltonian singularities and breakdown times, beyond which property preservation fails. QPP via Hamiltonian control is complementary to quantum error correction, as it does not require ancilla qubits or rely on measurement and feedback. It is also complementary to dynamical decoupling, since it uses only smooth Hamiltonians without pulsing and works in the regime of Markovian open-system dynamics. From the perspective of control theory, this work addresses the challenge of tracking control for open quantum systems.