We introduce a holographic framework for analyzing the steady states of repeated quantum channels with strong symmetries. Using channel-state duality, we show that the steady state of a $d$-dimensional quantum channel is holographically mapped to the boundary reduced density matrix of a $(d+1)$-dimensional wavefunction generated by a sequential unitary circuit. From this perspective, strong-to-weak spontaneous symmetry breaking (SW-SSB) in the steady state arises from the anyon condensation on the boundary of a topological order in one higher dimension. The conditional mutual information (CMI) associated with SW-SSB is then inherited from the bulk topological entanglement entropy. We make this duality explicit using isometric tensor network states (isoTNS) by identifying the channel’s time evolution with the transfer matrix of a higher-dimensional isoTNS. Built on isoTNS, we further construct continuously tunable quantum channels that exhibit distinct mixed-state phases and transitions in the steady states.

Achieving ultrafast single-qubit gates that approach the limits set by decoherence requires operating in the strong-driving regime, where conventional semi-classical descriptions and single-leakage models break down, and multi-photon transitions emerge as dominant error channels that grow rapidly as the gate time is reduced. This makes it essential to develop systematic methods to characterize and suppress these errors using pulse shapes that remain simple and experimentally straightforward to calibrate. In contrast to standard derivative removal by adiabatic gate (DRAG) approaches for single-qubit gates, which suppress leakage within an effective three-level description, our framework systematically treats multi-photon transitions that arise in the strong-driving regime due to the participation of higher energy levels. In particular, we identify and suppress the two-photon $|1⟩\leftrightarrow |3⟩$ channel, incorporate time-ordering corrections, and derive a recursive analytical construction that captures the dominant higher-order error processes. We term the resulting pulse families R1D and R2D. These constructions extend and unify DRAG-type approaches by incorporating higher-order processes and time-ordering effects, yielding analytical expressions for optimal quadrature corrections, detunings, and prefactors. In addition, we clarify the optimal choice of DRAG prefactors and constant detuning when time-ordering effects are properly taken into account. This provides a systematic prescription for calibrating additional correction terms to further improve gate performance. Using this framework, we numerically demonstrate gate infidelities below ${10}^{-5}$ for a $7\mathrm{ns}$ full $\pi$-rotation under realistic decoherence rates.

Time-dependent electromagnetic drives are fundamental for controlling complex quantum systems, including superconducting Josephson circuits. In these devices, accurate time-dependent Hamiltonian models are imperative for predicting their dynamics and designing high-fidelity quantum operations. Existing numerical methods, such as black-box quantization (BBQ) and energy-participation ratio (EPR), excel at modeling the static Hamiltonians of Josephson circuits. However, these techniques do not fully capture the behavior of driven circuits stimulated by external microwave drives, nor do they include a generalized approach to account for the inevitable noise and dissipation that enter through microwave ports. Here, we introduce numerical techniques that leverage classical microwave simulations, efficiently executable in finite-element solvers, to obtain the time-dependent Hamiltonian of microwave-driven superconducting circuits with arbitrary geometries under charge, flux, or mixed electromagnetic modulation. Importantly, our techniques do not rely on a lumped-element description of the superconducting circuit, in contrast to previous approaches to tackling this problem. We demonstrate the versatility of our approach by characterizing the driven properties of realistic circuit devices in complex electromagnetic environments, including coherent dynamics due to charge and flux modulation, as well as drive-induced relaxation and dephasing. Our techniques offer a powerful toolbox for optimizing circuit designs and advancing practical applications in superconducting quantum computing.

Quantum networks are a backbone of future quantum technologies thanks to their role in communication and scalable quantum computing. However, their performance is challenged by noise and decoherence. We propose a self-configuring approach that integrates superposed quantum paths with variational quantum optimization techniques. By estimating a cost function via repeated network uses, our approach dynamically optimizes the superposition of noisy paths across multiple nodes to establish high-fidelity connections between different parties. Our framework is in principle capable of adapting to unknown noise without assuming parametric noise models or performing full process tomography. We also discuss the role of vacuum coherence, a quantum effect central to path superposition that impacts protocol performance. Additionally, we demonstrate that our approach remains beneficial even in the presence of imperfections in the generation of path superposition.

Bilayer graphene is a maturing material platform for gate-defined quantum dots that hosts long-lived spin and valley states. Implementing solid-state qubits in bilayer graphene requires a fundamental understanding of such confined electronic systems. In particular, states of two and three carriers, for which the exchange interaction between particles plays a crucial role, are a cornerstone for qubit readout and manipulation. Here we report on the spectroscopy of few-carrier states in bilayer graphene quantum dots, using circuit quantum electrodynamics (cQED) techniques that offer substantially improved energy resolution compared to standard transport techniques. Measurements of a superconducting high-impedance resonator capacitively coupled to the double quantum dot reveal dispersive features of two- and three-electron states, enabling the detection of Pauli spin and valley blockade and the characterization of the spin-orbit gap at zero magnetic field. The results deepen our understanding of few-carrier spin and valley states in bilayer graphene quantum dots and demonstrate that cQED techniques are a powerful state-selective probe for semiconductor nanostructures.

Can a distributed network of quantum sensors estimate a global parameter while protecting every locally encoded value? We answer this question affirmatively by introducing and analyzing a protocol for distributed quantum sensing in the continuous-variable regime. We consider a multipartite network in which an unknown local phase is imprinted at each node on a shared entangled Gaussian state. We show that the average phase can be estimated with high precision, exhibiting Heisenberg scaling in the total photon number, while individual phases are inaccessible. We further prove a no-go theorem showing that, for three or more parties, no finite-energy Gaussian probe can provide complete privacy of the average phase, meaning that all phase combinations orthogonal to the average remain entirely hidden. This identifies the two-party case as an exceptional Gaussian setting that can achieve complete privacy. We further investigate the impact of displacements and optical losses, revealing trade-offs between estimation accuracy and privacy. Finally, we benchmark the protocol against other continuous-variable resource states.

The expected emergence of cryptographically relevant quantum computers (CRQCs) will represent a singular discontinuity in the history of digital security, with wide ranging impacts. This Perspective seeks to elucidate specific implications that the capabilities of developing quantum architectures have on blockchain vulnerabilities and potential mitigation strategies. First, we provide new resource estimates for breaking the 256-bit Elliptic Curve Discrete Logarithm Problem over the secp256k1 curve, the core of modern blockchain cryptography. We demonstrate that Shor’s algorithm for this problem can execute with either $\le 1200$ logical qubits and $\le 90\mathrm{million}$ Toffoli gates or $\le 1450$ logical qubits and $\le 70\mathrm{million}$ Toffoli gates. In the interest of responsible disclosure, we use a zero-knowledge proof to validate these results without disclosing attack vectors. On superconducting architectures with ${10}^{-3}$ physical error rates and planar connectivity, those circuits can execute in minutes using fewer than half a million physical qubits. We introduce a critical distinction between “fast-clock” (such as superconducting and photonic) and “slow-clock” (such as neutral atom and ion trap) architectures. Our analysis reveals that the first fast-clock CRQCs would enable “on-spend” attacks on public mempool transactions of some cryptocurrencies. We survey major cryptocurrency vulnerabilities through this lens, identifying systemic risks associated with advanced features in some blockchains such as smart contracts, proof-of-stake consensus, and data availability sampling mechanism, as well as the enduring concern of “abandoned” assets. We argue that technical solutions would benefit from accompanying public policy and discuss various frameworks of “digital salvage” to regulate the recovery or destruction of dormant assets while preventing adversarial seizure. We also discuss implications for other digital assets and tokenization as well as challenges and successful examples of the ongoing transition to post-quantum cryptography (PQC). Finally, we urge all vulnerable cryptocurrency communities to join the migration to PQC without delay.

Reliably simulating two-dimensional many-body quantum dynamics with projected entangled pair states (PEPS) has long been a difficult challenge. In this work, we overcome this barrier for low-energy quantum dynamics by developing a stable and efficient time-dependent variational Monte Carlo (tVMC) framework for PEPS. By analytically removing all gauge redundancies of the PEPS manifold and exploiting tensor locality, we obtain a numerically well-conditioned tVMC equation, yielding robust solutions using the efficient Cholesky decomposition. This enables long-time evolution in previously inaccessible regimes. We explain how the difficulties in the traditional approach, particularly those associated with gauge redundancies, are resolved within tVMC. We demonstrate the power and generality of the method through five representative real-time local quench dynamics in two dimensions: (I) chiral edge propagation in a free-fermion Chern insulator; (II) vison propagation in a pure ${\mathbb{Z}}_2$ gauge theory; (III) vison confinement dynamics in a ${\mathbb{Z}}_2$ lattice gauge theory coupled to Higgs field; (IV) fractionalized charge transport in a fractional Chern insulator; and (V) superfluidity and critical velocity in interacting bosons. All simulations are performed on $\ge 10\times 10$ lattices with evolution times beyond $T=10$ using modest computational resources. Where exact benchmarks exist (case I), PEPS-tVMC matches free-fermion dynamics with high accuracy up to $T=12$. In addition, we also simulate the paradigmatic dynamics of the Ising model following a global quench at the critical transverse field, and obtain, with modest bond dimension, agreement with previous results. These results demonstrate PEPS-tVMC as a practical and versatile tool for real-time quantum dynamics in two dimensions. The method significantly extends the reach of classical tensor-network simulations for studying elementary excitations in quantum many-body systems in real time and provides a valuable computational counterpart to emerging quantum simulators. As a by-product in the development, we also present a new form of minSR, which is more stable and offers a new perspective on tVMC.

We study the collective emission of a disordered array of $N$ excited two-level atoms into a one-dimensional photonic waveguide. In the perfectly ordered case, where atoms are spaced by exact integer multiples of the wavelength, the system exhibits the characteristic superradiant burst with a peak emission rate scaling as $N^2$. Using large-scale semiclassical simulations, we find that this key signature of superradiance remains asymptotically robust under strong spatial and spectral disorder, but also exhibits subtle finite-size scaling toward this limit. To explain our observations, we provide an analytical variational estimate for the maximal decay rate, which tightly bounds the numerical results and reveals how disorder shapes the collective decay. Specifically, we find that even in the presence of strong disorder, the spins tend to self-organize spontaneously according to their locations, which overall optimizes constructive interference effects and explains the emergence of mirror-asymmetric correlations in superradiant decay. These findings resolve important open questions regarding the existence and nature of superradiance in strongly disordered arrays and offer valuable insights for understanding collective quantum optical phenomena in realistic systems.

Efficient low-energy state preparation is a key objective in quantum computation and quantum simulation. Quantum imaginary-time evolution replaces real-time dynamics with imaginary-time dynamics, exponentially suppressing higher-energy eigenstates. We introduce deterministic unitary protocols that approximate imaginary-time evolution for ground state preparation. The protocols require multiple copies of the system, real-time evolution under the system Hamiltonian, and controlled-swap operations (or more general swap-generated unitaries). Our analysis focuses on two concrete circuit families: a tree architecture with provable polynomial-in-depth convergence but rapidly growing width, and a compact “hedge” architecture that achieves comparable accuracy with only polynomial width in a heuristic construction supported by numerics. Numerical evidence indicates that mid-circuit post-selection can accelerate convergence with practical success probabilities. Separately, we demonstrate that circuit volume can be traded for the shot complexity of post-circuit observable estimation in the ground state preparation setting. Finally, we outline concrete platform-specific implementations in which multi-copy registers and swap-mediated couplings are natural, illustrating how these hybrid analog-digital circuits can complement existing state-preparation methods in the near term.

We report on a cryogenic platform at 4 K incorporating high numerical aperture optics for the generation of large-scale tweezers arrays, and compatible with Rydberg-state manipulation. We achieve trapping lifetimes of around 5000 s, significantly extending the available experimental time for the preparation of large-scale arrays. By combining two trapping lasers at different wavelengths and by minimizing other atom losses during the rearrangement and imaging processes, we demonstrate the preparation of defect-free arrays with up to 1024 atoms. Our cryogenic design opens exciting prospects for analog and digital quantum computing.

The study of phase transitions and critical phenomena arising in quantum driven-dissipative systems, and whether a correspondence can be drawn to their equilibrium counterparts, is a pressing question in contemporary physics. The development of large-scale superconducting circuits provides an experimental platform for these theoretical models. We report an experimental study of a multimode dissipative first-order phase transition in a 1D Bose-Hubbard chain consisting of 21 superconducting resonators. This phase transition manifests itself as a simultaneous frequency jump in all resonator modes as the frequency or power of a pump tone is swept. By measuring the system’s emission spectrum through the transition, we characterize the dim-to-bright phase transition and construct the full phase diagram. We further perform time-dependent measurements of the switching between the two phases in the transition region, from which we corroborate the transition line and extract transition times ranging from a few ms up to 143 s. Our model, based on single-mode mean-field theory and cross-Kerr interactions, captures the features at moderate pump powers and quantitatively reproduces the transition line. Our results open a new window into nonequilibrium quantum many-body physics and mark a step toward realizing and understanding dissipative phase transitions in the thermodynamic limit using superconducting quantum circuits. In addition, our study opens new possibilities for sensing and cryo-switching applications.

We present an entirely 2D constant-depth realization of topologically protected phase gates at any level of the Clifford hierarchy, and beyond, using non-Abelian surface codes. Our construction encodes a logical qubit in the quantum double $D(G)$ of a non-Abelian group $G$ on a triangular spatial patch. The logical gate is implemented by a constant-depth circuit constructed from stacking on the spatial region a symmetry-protected topological (SPT) phase specified by a group 2-cocycle and boundary counterterms. The Bravyi-König theorem limits the unitary gates implementable by constant-depth quantum circuits on Pauli stabilizer codes in $D$ dimensions to the $D$-th level of the Clifford hierarchy. We bypass this limitation, by constructing constant-depth unitary gates at arbitrary levels of the Clifford hierarchy purely in 2D, without sacrificing locality or fault tolerance, at the cost of using the quantum double of a non-Abelian group $G$. Specifically, for $G=D_{4N}$, the dihedral group of order $8N$, we realize the phase gate $T^{1/N}=\mathrm{diag}(1,e^{i\pi /(4N)})$ in the logical $\overline{Z}$ basis. In this context, we propose a non-Abelian stabilizer group formalism, which we work out for dihedral groups. For $8N=2^n$, the logical gate lies at the $n$-th level of the Clifford hierarchy and, importantly, has a qubit-only realization: we show that it can be constructed in terms of Clifford-hierarchy stabilizers for a code with $n$ physical qubits on each edge of the lattice. We also discuss code-switching to the double surface-code $D({\mathbb{Z}}_2\times {\mathbb{Z}}_2)$, to complete a universal gate set in this setup.

The knowledge of the energy spectrum completely defines the dynamics of a quantum system for a given initial state. This makes spectroscopy a key characterization technique when studying or designing qubits and complex quantum systems. In semiconductor quantum dots, the electronic quantum states can be probed through charge transport or charge sensing, but the technique is practically limited to gate-defined quantum dots. Epitaxial quantum dots benefit from excellent optical properties, but are usually incompatible with charge transport, while alternative spectroscopy techniques provide only limited information. Here we demonstrate a spectroscopy technique which utilizes nuclear spins as a non-invasive probe. By using the cotunneling-assisted transfer of spin angular momentum instead of the charge currents we achieve near-equilibrium probing. Experiments are conducted on low-strain GaAs/AlGaAs epitaxial dots, revealing energy spectra for charge configurations with up to seven electrons and with resolution limited only by the thermal broadening. Beyond the few-electron energy spectroscopy, we apply non-invasive probing via nuclear spins to reveal the subtle many-body physics of the quantized electronic states. We map out the magnetic field dependence of the electron spin lifetimes, revealing an unusual inverse quadratic dependence. The rich variety of further observations includes the long-lived spin-qubit states in the $p$ shell, electron ground-state phase transitions, strong spin-orbit coupling regimes, and an anomalously fast nuclear spin diffusion process. Experiments are backed up by good agreement with the first-principles configuration interaction numerical modeling. Our work uncovers few-electron states as a new operating regime for optically active quantum dots. Accurate control and probing of many-body states offers a test-bed system for fundamental physics studies, while prospective technological applications include electron spin qubits with extended coherence and scalable electrical control.

Identifying stabilizer codes that admit fault tolerant implementations of the full logical Clifford group would significantly advance fault tolerant quantum computation, as such implementations prevent uncontrolled error propagation. Motivated by this goal, we study several classes of fault tolerant gadget constructions consisting of Clifford gates acting on the physical qubits, including transversal gadgets, code automorphisms, and fold-transversal gadgets. While stabilizer codes encoding a single logical qubit—most notably the $[[7,1,3]]$ Steane code—are known to admit transversal implementations of the full logical Clifford group, no analogous examples are known for codes encoding multiple logical qubits. In this work, we prove a no-go theorem establishing that no stabilizer code admits a fully transversal implementation of the Clifford group on more than one logical qubit. We further strengthen this result by showing that fold-transversal implementations of the full logical Clifford group are impossible for stabilizer codes encoding more than two logical qubits. More generally, we introduce the notion of $k$-fold transversal gadgets and prove that implementing the full Clifford group on $k$ logical qubits requires at least $k$-fold transversal gadgets at the physical level. In addition, we analyze code-automorphism-based constructions and demonstrate that they also fail to realize the full Clifford group on multiple logical qubits for any stabilizer code. Together, these results place fundamental constraints on fault tolerant Clifford gadget design and show that stabilizer codes supporting the full logical Clifford group on multiple logical qubits via these architectures do not exist. Since the Clifford group is a core component of universal gate sets, our findings imply that quantum computing with codes encoding multiple logical qubits within a single code block necessarily entails more complex constructions for fault tolerance.

We provide several quantum algorithms for continuous optimization that do not require gradient estimation. Instead, we encode the optimization problem into the dynamics of a physical system and coherently simulate the time evolution. We focus on the setting where the objective function can only be accessed via a phase oracle. Our first two algorithms can find local optima of a differentiable function $f:{\mathbb{R}}^N\to \mathbb{R}$ by simulating either classical or quantum dynamics with friction via a time-dependent Hamiltonian. We show that for the benchmark problem of optimizing a locally quadratic objective function, these methods require a total of $O(N^2{\kappa }^2/h_x^2ϵ)$ queries to a phase oracle to find an $ϵ$-approximate local optimum, where $\kappa$ is the condition number of the Hessian matrix and $h_x$ is the discretization spacing. In contrast, we show that methods based on gradient descent require $O(N^{3/2}(1/ϵ{)}^{\kappa \log (3)/4})$ queries. This corresponds to an exponential separation between the query upper bounds for the benchmark problem. Our third algorithm can find the global optimum of $f$ by preparing a classical low-temperature thermal state via simulation of the classical Liouvillian operator associated with the Nosé Hamiltonian. We use results from the quantum thermodynamics literature to bound the thermalization time for the discrete system. Additionally, we analyze barren plateau effects that commonly plague quantum optimization algorithms and observe that our approach is vastly less sensitive to this problem than standard gradient-based optimization. Our results suggest that these dynamical optimization approaches may be far more scalable for future quantum machine learning, optimization and variational experiments than was widely believed.

Classical methods to simulate quantum systems are not only a key element of the physicist’s toolkit for studying many-body models but are also increasingly important for verifying and challenging upcoming quantum computers. Pauli propagation has recently emerged as a promising new family of classical algorithms for simulating digital quantum systems. Here we provide a comprehensive account of Pauli propagation, tracing its algorithmic structure from its bit-level implementation and formulation as a tree-search problem, all the way to its high-level user applications for simulating quantum circuits and dynamics. Utilizing these observations, we present PauliPropagation.jl, a Julia software package that can perform rapid Pauli propagation simulation straight out-of-the-box and can be used more generally as a building block for novel simulation algorithms.

Here we describe an approach for simulating electronic structure on quantum computers with significantly lower asymptotic complexity than prior work. The approach uses a real-space first-quantized representation of the molecular Hamiltonian, which we propagate using high-order product formulas. Essential for this low complexity is the use of a technique similar to the fast multipole method for computing the Coulomb operator with $\tilde{\mathcal{O}}(\eta )$ complexity for a simulation with $\eta$ particles. We show how to modify this algorithm so that it can be implemented on a quantum computer. We ultimately demonstrate an approach with $t({\eta }^{4/3}{N}^{1/3}+{\eta }^{1/3}{N}^{2/3})(\eta Nt/ϵ{)}^{o(1)}$ gate complexity, where $N$ is the number of grid points, $ϵ$ is target precision, and $t$ is the duration of time evolution. This is roughly a speedup by $\mathcal{O}(\eta )$ over most prior algorithms. We provide lower complexity than all prior work for $N<{\eta }^7$ (the regime of practical interest), with only first-quantized interaction-picture simulations providing better performance for $N>{\eta }^7$. As with the classical fast multipole method, large particle numbers $\eta \gtrsim {10}^3$ would be needed to realize this advantage.

Advancing quantum information processors and building fault-tolerant architectures rely on the ability to accurately characterize the noise sources and suppress their impact on quantum devices. In practice, noise often drifts over time, whereas conventional noise characterization and decoding methods typically assume stationarity or provide only a time-average behavior of the noise. This treatment can result in suboptimal decoding performance. In this work, we present a rigorous analytical framework to capture time-dependent Pauli noise, which is designed to use only syndrome data obtained from quantum error correction experiments. We validate this framework through numerical simulations. We propose a sliding-window estimation method which allows us to recover the frequency components of the noise, by using optimal window sizes that we derive analytically. We prove the noise-filtering behavior of sliding windows, linking window size to spectral cutoff frequencies, and provide an iterative algorithm that captures multiple drift frequencies. We further introduce an overlapping window algorithm that enables us to capture rapid multi-frequency noise drifts in a single-pass fashion. Simulations for both phenomenological and circuit-level noise models validate our framework, demonstrating robust tracking of multi-frequency drift. The logical error rate obtained from our estimated models consistently aligns with the ground-truth logical error rate, and we find suppression of logical errors compared to static error models. Our window-based estimation methods and adaptive decoding offer new insights into noise spectroscopy and decoder optimization under drift using only syndrome data.

We study measurement-induced entanglement (MIE) generated by column-by-column sampling of noisy 2D random circuits of size $N$ and depth $T$. Focusing primarily on Clifford circuits and using the operator entanglement $S_{\text{op}}$ of the sampling-induced boundary state as a proxy for computational complexity, first, we reproduce in the noiseless limit a finite-depth transition from area- to volume-law scaling at a threshold depth $T_c=6$. In contrast, in the presence of single-qubit depolarizing noise at any constant rate $p>0$, we find that the operator entanglement $S_{\text{op}}$ obeys an area law, with its maximum value scaling approximately linearly with $T/p$ in the regime $T>T_c$. By analyzing the spatial distribution of stabilizer generators, we observe exponential localization of stabilizer generators; these both account for the scaling of the maximal $S_{\text{op}}$ and imply an exponential decay of conditional mutual information across buffered tripartitions, which we also confirm numerically. Together, these results indicate that constant local noise destroys long-range MIE in 2D random Clifford circuits, and that a tensor-network-based algorithm can efficiently sample from noisy 2D random Clifford circuits (i) at sub-logarithmic depths $T=o(\log N)$ for any constant noise rate $p=\mathrm{\Omega }(1)$, and (ii) at constant depths $T=O(1)$ for noise rates $p=\mathrm{\Omega }({\log }^{-1}N)$. Finally, we turn to depth $T=4$ Haar-random and measurement-based quantum computing-type circuits, providing evidence that MIE in noisy 2D Haar-random circuits exhibits the same qualitative behavior as in random Clifford circuits, and that noise destroys the volume-law scaling of MIE in non-Clifford circuits.