Fault-tolerant quantum computation traditionally incurs substantial resource overhead, with both qubit and time overheads scaling polylogarithmically with the size of the computation. While prior work by Gottesman has shown that constant qubit overhead is achievable under stochastic noise using quantum low-density parity-check (QLDPC) codes, it has remained an open question whether similar guarantees hold under more general, nonstochastic noise models. In this work, we address this question by considering a general circuit-level noise model defined via the diamond norm, which captures both stochastic and nonstochastic noise, including coherent and amplitude damping noise. We prove that constant qubit overhead fault-tolerant quantum computation is achievable in this general setting, using QLDPC codes with constant rate and linear minimum distance. To establish our result, we develop a fault-tolerant error correction scheme and a method for implementing logic gates under general circuit noise. These results extend the theoretical foundations of fault-tolerant quantum computation and offer new directions for fault-tolerant architectures under realistic noise models.

Describing general quantum many-body dynamics is a challenging task due to the exponential growth of the Hilbert space with system size. The time-dependent variational principle (TDVP) provides a powerful tool to tackle this task by projecting quantum evolution onto a classical dynamical system within a variational manifold. In classical systems, periodic orbits play a crucial role in understanding the structure of the phase space and the long-term behavior of the system. However, finding periodic orbits is generally difficult, and their existence and properties in generic TDVP dynamics over matrix product states have remained largely unexplored. In this work, we develop an algorithm to systematically identify and characterize periodic orbits in TDVP dynamics. Applying our method to the periodically kicked Ising model, we uncover both stable and unstable periodic orbits. We characterize the Kolmogorov-Arnold-Moser tori in the vicinity of stable periodic orbits and track the change of the periodic orbits as we modify the Hamiltonian parameters. We observe that periodic orbits exist at any value of the coupling constant of the kicked Ising model between prethermal and fully thermalizing regimes, but their relevance to quantum dynamics and imprint on quantum eigenstates diminishes as the system leaves the prethermal regime. Our results demonstrate that periodic orbits provide valuable insights into the TDVP approximation of quantum many-body evolution and establish a closer connection between quantum and classical chaos.

Mechanisms that give rise to coherent quantum dynamics, such as quantum many-body scars, have recently attracted much interest as a way of controlling quantum chaos. However, identifying the presence of quantum scars in general many-body Hamiltonians remains an outstanding challenge. Here we introduce ScarFinder, a variational framework that reveals possible scar-like dynamics without prior knowledge of scar states or their algebraic structure, assuming only that such dynamics remain low in entanglement. By iteratively evolving and projecting states within a variational manifold, ScarFinder isolates scarred trajectories by suppressing thermal contributions. We validate the method on the analytically tractable spin-1 $XY$ model, recovering the known scar dynamics, as well as the mixed-field Ising model, where we capture and generalize the initial conditions previously associated with “weak thermalization.” We then apply the method to the PXP model of Rydberg atom arrays, finding a previously unknown trajectory with nearly-perfect revival dynamics in the thermodynamic limit. We also demonstrate that ScarFinder can efficiently identify the centers of stable islands in Poincaré sections of the mixed phase space that results from the projection of many-body quantum dynamics to a variational manifold. Our results establish ScarFinder as a powerful, model-agnostic tool for identifying and optimizing coherent dynamics in quantum many-body systems.

Implementations of quantum key distribution (QKD) need vulnerability assessment against loopholes in their optical scheme. Most of the optical attacks involve injecting or receiving extraneous light via the communication channel. An eavesdropper can choose her attack wavelengths arbitrarily within the quantum channel passband to maximize the attack performance, exploiting spectral transparency windows of system components. Here we propose a wide-spectrum security evaluation methodology to achieve full optical spectrum safety for QKD systems. This technique requires transmittance characterization in a wide spectral band with a high sensitivity. We report a testbench that characterizes insertion loss of fiber-optic components in a wide spectral range of 400–2300 nm and up to 70 dB dynamic range. To illustrate practical application of the proposed methodology, we give a full Trojan-horse attack analysis for some typical QKD system configurations and discuss briefly induced-photorefraction and detector-backflash attacks. Our methodology can be used for certification of QKD systems.

Although high-threshold and low-overhead quantum low-density parity-check (qLDPC) codes, such as bivariate bicycle (BB) codes, can reduce the physical-qubit cost by an order of magnitude compared to the Kitaev toric code, their torus layout remains difficult for physical implementation. In this work, we introduce the first systematic procedure to convert BB codes into fully planar, open-boundary qLDPC codes, preserving their performance. We present planar code families with logical dimensions $6\le k\le 13$, e.g., $\text{⟦}78,6,6\text{⟧}$, $\text{⟦}107,7,7\text{⟧}$, $\text{⟦}268,8,12\text{⟧}$, $\text{⟦}405,9,15\text{⟧}$, $\text{⟦}348,10,13\text{⟧}$, $\text{⟦}450,11,15\text{⟧}$, $\text{⟦}386,12,12\text{⟧}$, and $\text{⟦}362,13,11\text{⟧}$, all with geometrically local weight-6 stabilizers. Allowing weight-8 stabilizers produces a $\text{⟦}282,12,14\text{⟧}$ code, exhibiting an efficiency metric ($k{d}^2/n$) an order of magnitude higher than the surface code. The construction combines boundary-anyon condensation with the “lattice-grafting” optimization, yielding high-performance qLDPC codes natively compatible with planar hardware architectures. It also uncovers Sierpinski-type fractal logical operators the distance of which scales with the fractal area on finite lattices. These planar qLDPC codes provide an implementable route to resource-efficient, high-threshold fault tolerance and a flexible framework for future code design on realistic two-dimensional hardware.

Hyperfine-encoded qubits in alkali atoms have established themselves as robust platforms for quantum computing, while alkaline-earth-like elements expand the state manipulation toolbox through their rich spectrum of optical transitions and metastable states. In this work, we demonstrate that thulium is a viable candidate for quantum computing, combining advantages of hyperfine qubit encoding with a rich energy-level structure of alkaline-earth-like atoms. We describe protocols for the initial state preparation and state-selective readout, and show single-qubit operations on the microwave transition at 1497 MHz. We demonstrate ground-state hyperfine qubit coherence times up to $T_2^{\ast }={22}_{-2}^{+2}$ s and $T_2={55}_{-14}^{+59}$ s, representing record-scale performance for neutral-atom systems. Furthermore, we show operations involving metastable optical states, including shelving for the state-selective readout as well as coherent population transfer of the ground-state qubit with coherence time primarily limited by the metastable level natural lifetime of 112 ms. These results mark the first step toward using thulium for quantum computing applications and highlight its promising characteristics.

Ultracold gases of atoms from main group III (group 13) of the periodic table, also known as “triel elements,” have great potential for a new generation of quantum matter experiments. The first magneto-optical trap of a triel element (indium) was recently realized, but more progress is needed before a triel is ready for modern ultracold quantum science experiments in optical traps. Reaching this regime typically requires atoms that are cooled to the $10\text{μ}\mathrm{K}$ level or below, prepared in pure quantum states, and confined in a laser field. Here we report the achievement of all three of these milestones in atomic indium. First, we perform polarization gradient cooling of an indium gas to $15\text{μ}\mathrm{K}$. Second, we spin polarize the gas into a single hyperfine sublevel of either the $5P_{1/2}$ indium ground state or the $5P_{3/2}$ metastable state. Third, we trap indium in a 1064 nm optical lattice, achieving a 3 s trap lifetime. With these results, indium is now a candidate for a next generation quantum research platform.

Quantum error correcting (QEC) codes protect quantum information against environmental noise. Computational errors caused by the environment change the quantum state within the qubit subspace, whereas quantum erasures correspond to the loss of qubits at known positions. Correcting either type of error involves different correction mechanisms, which makes studying the interplay between erasure and computational errors particularly challenging. In this work, we propose a framework based on the coherent information (CI) of the mixed-state density operator associated to noisy QEC codes, for treating both types of errors together. We show how to rigorously derive different families of statistical mechanics mappings for generic stabilizer QEC codes in the presence of both types of errors. We observe that the erasure errors enter as a classical average over fully depolarizing channels. Further, we show that computing the CI for erasure errors only can be done efficiently upon sampling over erasure configurations. We then test our approach on the two-dimensional toric and color codes and compute optimal thresholds for erasure errors only, finding a $50\mathrm{\%}$ threshold for both codes. This strengthens the notion that both codes share the same optimal thresholds. When considering both computational and erasure errors, the CI of small-size codes yields thresholds in very accurate agreement with established results that have been obtained in the thermodynamic limit. Next, we perform a similar analysis for a low-density parity-check (LDPC) code—the lift-connected surface code. We find a $50\mathrm{\%}$ threshold under erasure errors alone and, for the first time, derive the exact statistical mechanics mappings in the presence of both computational and erasure errors. We thereby further establish the CI as a practical tool for studying optimal thresholds for code classes beyond topological codes under realistic noise, and as a means for uncovering new relations between QEC codes and statistical physics models.

Quantum computing with superconducting circuits relies on high-fidelity driven nonlinear processes. An ideal quantum nonlinearity would selectively activate desired coherent processes at high strength, without activating parasitic mixing products or introducing additional decoherence. The wide bandwidth of the Josephson nonlinearity makes this difficult, with undesired drive-induced transitions and decoherence limiting qubit readout, gates, couplers, and amplifiers. Significant strides have recently been made into building better “quantum mixers,” with promise being shown by Kerr-free three-wave mixers that suppress driven frequency shifts, and balanced quantum mixers that explicitly forbid a significant fraction of parasitic processes. We propose a novel mixer that combines both these strengths, with engineered selection rules that make it essentially linear (not just Kerr-free) when idle, and activate clean parametric processes even when driven at high strength. Furthermore, its ideal Hamiltonian is simple to analyze analytically, and we show that this ideal behavior is first-order insensitive to dominant experimental imperfections. We expect this mixer to allow significant advances in high-$Q$ control, readout, and amplification.

While stabilizer tableaus have proven useful as a descriptive tool for additive quantum codes, they otherwise offer little guidance for concrete constructions or algorithm analysis. We introduce a representation of stabilizer codes as graphs with certain structures, and prove via the ZX calculus that this representation is related to stabilizer tableaus by an efficiently computable bijection. This gives a new universal recipe for code construction by way of finding graphs with nice properties. The graph representation gives insight into both code construction and algorithms. We construct as examples families of $\text{⟦}n,\mathrm{\Theta }(\frac{n}{\log n}),\mathrm{\Theta }(\log n)\text{⟧}$ and $\text{⟦}n,\mathrm{\Omega }(n^{4/5}),\mathrm{\Theta }(n^{1/5})\text{⟧}$ codes. We use graphs in a probabilistic analysis to extend the quantum Gilbert-Varshamov bound into a three-way distance-rate-weight trade-off. Moreover, code properties such as distance and encoding circuit depth are bounded by simple functions of the graph degree. We prove that key coding algorithms—distance approximation, minimum-weight generator selection, and decoding—are unified as instances of one optimization game on a graph. By studying this game, we construct an efficient greedy decoder and prove that it corrects all recoverable errors for all graphs with cycle lengths no shorter than 13 (reducible to 5 with mild extra constraints); these include the above two families. Our results suggest that graphs are generically useful for the study of stabilizer codes.

Isometric tensor networks in two dimensions enable efficient and accurate study of quantum many-body states, yet the effect of the isometric restriction on the represented quantum states is not fully understood. We address this question in two main contributions. First, we introduce an improved variant of isometric tensor network states (isoTNS) in two dimensions, where the isometric arrows on the columns of the network alternate between pointing upward and downward; hence the name alternating isometric tensor network states. Second, we introduce a numerical tool—the isometric Gaussian fermionic TNS (isoGfTNS)—that incorporates isometric constraints into the framework of Gaussian fermionic tensor network states. We demonstrate in numerous ways that alternating isoTNSs represent many-body ground states of two-dimensional quantum systems significantly better than the original isoTNSs. First, we show that the entanglement in an isoTNS is mediated along the isometric arrows and that alternating isoTNSs mediate entanglement more efficiently than conventional isoTNSs. Second, alternating isoTNSs correspond to a deeper, and thus more representative, sequential-circuit construction of depth $\mathcal{O}(L_x\cdot L_y)$ compared to the original isoTNSs of depth $\mathcal{O}(L_x+L_y)$. Third, using the Gaussian framework and gradient-based energy minimization, we provide numerical evidence of better bond-dimension scaling and variational energy of alternating isoGfTNSs for ground states of various free-fermionic models, including the Fermi surface, the band insulator, and the $p_x+i{p}_y$ mean-field superconductor. Finally, benchmarking on the transverse-field Ising model, we demonstrate that an alternating isoTNS provides substantially improved performance and stability relative to the original isoTNS for the ground-state search algorithm in interacting systems.

Erasure qubits constitute a promising approach for tackling the daunting resources required for fault-tolerant quantum computing. By heralding of erasure errors, both the error-correction threshold and the sub-threshold scaling of the logical error rate are significantly improved. While previous research has focused primarily on fault-tolerant quantum memories, we extend this investigation to magic state injection—a critical yet resource-intensive component of fault-tolerant quantum computation. We show that, after postselection on erasures, the logical error rate of the injected magic state is set by the residual Pauli error, while the space-time overhead is only marginally increased as compared with the case of non-erasure qubits with a similar noise strength. These conclusions hold both for injection into the surface code and for injection and cultivation on the color code. For the former, we show that most of the gains can be achieved by the use of just three strategically placed erasure qubits in the surface code patch, independent of the patch size. For the latter, in contrast, it is beneficial to have all the qubits in the cultivation patch be erasure qubits. Our results for cultivation suggest that algorithmically relevant logical error rates may be within reach without magic state distillation for erasure rates $\lesssim 4\times {10}^{-3}$ and residual Pauli error rates of $\sim {10}^{-4}$.

Lieb-Robinson bounds demonstrate the emergence of locality in many-body quantum systems. Intuitively, Lieb-Robinson bounds state that, with local or exponentially decaying interactions, the correlation that can be built up between two sites separated by distance $r$ after a time $t$ decays as $\exp (vt-r)$, where $v$ is the emergent Lieb-Robinson velocity. In many problems, it is important to also capture how much of an operator grows to act on $r^d$ sites in $d$ spatial dimensions. Perturbation theory and cluster expansion methods suggest that, at short times, these volume-filling operators are suppressed as $\exp (-r^d)$. We confirm this intuition, showing that, for $r>vt$, the volume-filling operator is suppressed by $\exp (-(r-vt{)}^d/(vt{)}^{d-1})$. This closes a conceptual and practical gap between the cluster expansion and the Lieb-Robinson bound. We then present two very different applications of this new bound. Firstly, we obtain improved bounds on the classical computational resources necessary to simulate many-body dynamics with error tolerance $\epsilon$ for any finite time $t$: as $\epsilon$ becomes sufficiently small, only ${\epsilon }^{-\mathrm{O}(t^{d-1})}$ resources are needed. A protocol that likely saturates this bound is given. Secondly, we prove that disorder operators have volume-law suppression near the “solvable (Ising) point” in quantum phases with spontaneous symmetry breaking, which implies a new diagnostic for distinguishing many-body phases of quantum matter.

Precise manipulation of quantum effects at the atomic and nanoscale has become an essential task in ongoing scientific and technological endeavors. Quantum control methods are thus routinely exploited for research in areas such as quantum materials, quantum chemistry, and atomic and molecular physics, as well as in the development of quantum technologies like computing, simulation, and sensing. Here, we present a pedagogical introduction to the basics of quantum control methods in tutorial form, with the aim of providing newcomers to the field with the core concepts and practical tools to use these methods in their research. We focus on three areas: shortcuts to adiabaticity, quantum optimal control, and machine-learning-based control. We lay out the basic theoretical elements of each area in a pedagogical way and describe their application to a series of example cases. For these, we include detailed analytical derivations as well as extensive numerical results. As an outlook, we discuss quantum control methods in the broader context of quantum technologies development and complex quantum systems research, outlining potential connections and synergies between them.

We propose a superconducting qubit based on engineering the first and second harmonics of the Josephson energy and phase relation $E_{J1}\cos \phi$ and $E_{J2}\cos 2\phi$. By constructing a circuit such that $E_{J2}$ is negative and $|E_{J1}|\ll |E_{J2}|$, we create a periodic potential with two nondegenerate minima. The qubit, which we dub “harmonium,” is formed from the lowest-energy states of each minimum. Bit-flip protection of the qubit arises due to the localization of each qubit state to their respective minima, while phase-flip protection can be understood by considering the circuit within the Born-Oppenheimer approximation. We demonstrate with time-domain simulations that single- and two-qubit gates can be performed in approximately 100 ns. Finally, we compute the qubit coherence times using numerical diagonalization of the complete circuit in conjunction with state-of-the-art noise models. We estimate out-of-manifold heating times on the order of milliseconds, which can be treated as erasure errors using conventional dispersive readout. We estimate pure-dephasing times on the order of many tens of milliseconds, and bit-flip times on the order of seconds.

Local counterdiabatic (CD) driving provides a systematic way of constructing a control protocol to approximately suppress the excitations resulting from changing some parameter(s) of a quantum system at a finite rate. However, designing CD protocols typically requires knowledge of the original Hamiltonian a priori. In this work, we design local CD driving protocols in Krylov space using only the characteristic local time scales of the system set by e.g., phonon frequencies in materials or Rabi frequencies in superconducting qubit arrays. Surprisingly, we find that convergence of these universal protocols is controlled by the asymptotic high-frequency tails of the response functions. This finding hints at a deep connection between the long-time, low-frequency response of the system controlling non-adiabatic effects, and the high-frequency response determined by the short-time operator growth and the Krylov complexity. We make this connection concrete by showing how, for a representative integrable model, we may extract long-time universal behavior of the correlation functions from a short-time expansion of the dynamics using a system-independent universal protocol.

Achieving practical quantum speedup with limited resources is a crucial challenge in both academic and industrial communities. To address this, a partially fault-tolerant quantum computing architecture called “space-time efficient analog rotation quantum computing architecture” (STAR architecture) been recently been proposed. This architecture focuses on minimizing resource requirements while maximizing the precision of non-Clifford gates, essential for universal quantum computation. However, nondeterministic processes such as the repeat-until-success (RUS) protocol and state injection can introduce significant computational overhead. Therefore, optimizing the logical circuit to minimize this overhead by using efficient fault-tolerant operations is essential. This paper presents an efficient method for simulating the time evolution of the two-dimensional (2D) Hubbard-model Hamiltonian, a promising application of the STAR architecture. We present two techniques, the parallel-injection protocol and adaptive injection-region updating, to reduce unnecessary time overhead specific to our architecture. By integrating these with the existing fermionic swap (fswap) technique, we develop an efficient Trotter-based time-evolution operation for the 2D Hubbard model. Our analysis reveals an acceleration of over 10 times compared to naive serial compilation. Based on this optimized compilation, we estimate the computational resources required for the quantum phase estimation of the 2D Hubbard model. For devices with a physical error rate of $p_{\mathrm{phys}}={10}^{-4}$, we estimate that approximately $6.2\times {10}^4$ physical qubits are required to achieve faster ground-state energy estimation of the $8\times 8$ Hubbard model compared to classical computation.

As fully fault-tolerant quantum computers capable of solving useful problems remain a distant goal, we anticipate an era of early fault tolerance during which limited error correction is available. We propose a framework for designing early fault-tolerant algorithms by trading between error-correction overhead and residual logical noise, and apply it to quantum phase estimation (QPE). We develop a quantum-Fourier-transform- (QFT) based QPE technique that is robust to global depolarizing noise and outperforms the previous state of the art at low and moderate noise rates. We further introduce the explicitly unbiased maximum-likelihood estimator (EUMLE), a data-processing technique that mitigates arbitrary errors in QFT-based QPE schemes. EUMLE provides consistent asymptotically normal error-mitigated estimates, addressing the open problem of extending error mitigation beyond expectation-value estimation. Applying this scheme to the ground-state problem of the two-dimensional Hubbard model and various molecular Hamiltonians, we find that we can roughly halve the number of physical qubits with a $\sim 10$ times wall-clock time overhead, but that further reduction causes a steep run-time increase. This work provides an end-to-end analysis of early fault-tolerance cost reductions and space-time trade-offs, and identifies which areas can be improved in the future.

Continuous gate sets are a key ingredient for near-term quantum algorithms. Here, we demonstrate a hardware-efficient, continuous set of controlled arbitrary-phase (${\mathrm{CZ}}_{\theta }$) gates acting on flux-tunable transmon qubits. This implementation is robust to control pulse distortions on time scales longer than the duration of the gate, making it suitable for deep quantum circuits. Our calibration procedure makes it possible to parameterize the continuous gate set with a single control parameter, the idle time between the two rectangular halves of the net-zero control pulse. For calibration and characterization, we develop a leakage measurement based on coherent amplification, and a new cycle design for cross-entropy benchmarking. We demonstrate gate errors of $0.7\mathrm{\%}$ and leakage of $4\times {10}^{-4}$ across the entire gate set. This native gate set has the potential to reduce the depth and improve the performance of near-term quantum algorithms compared to decompositions into ${\mathrm{CZ}}_{\pi }$ gates and single-qubit gates. Moreover, we expect the calibration and benchmarking methods to find further possible applications.

We devise tractable models of unitary quantum many-body dynamics on tree graphs, as a first step toward a deeper understanding of dynamics in non-Euclidean spaces. To this end, we first demonstrate how to construct strictly local quantum circuits that preserve the symmetries of trees, such that their dynamical light cones grow isotropically. For trees with coordination number $z$, such circuits can be built from $z$-site gates. We then introduce a family of gates for which the dynamics are exactly solvable; these satisfy a set of constraints that we term “tree-unitarity.” Notably, tree-unitarity reduces to the previously established notion of dual-unitarity for $z=2$, when the tree reduces to a line. Among the unexpected features of tree-unitarity is a trade-off between “maximum butterfly velocity” dynamics of out-of-time-order correlators and the existence of nonvanishing correlation functions in multiple directions, a tension absent in one-dimensional dual-unitary models and their Euclidean generalizations. We connect the existence of (a wide class of) solvable dynamics with nonmaximal butterfly velocity directly to a property of the underlying circuit geometry called $\delta$-hyperbolicity, and argue that such dynamics can only arise in non-Euclidean geometries. We give various examples of tree-unitary gates, discuss dynamical correlations, out-of-time-order correlators, and entanglement growth, and show that the kicked Ising model on a tree is a physically motivated example of maximum-velocity tree-unitary dynamics.