Magic (nonstabilizerness) is a necessary but “expensive” kind of “fuel” to drive universal fault-tolerant quantum computation. To properly study and characterize the origin of quantum “complexity” in computation as well as physics, it is crucial to develop a rigorous understanding of the quantification of magic. Previous studies of magic mostly focused on small systems and largely relied on the discrete Wigner formalism (which is only well behaved in odd prime power dimensions). Here we present an initiatory study of the magic of genuinely many-body quantum states that may be strongly entangled, with focus on the important case of many qubits, at a quantitative level. We first address the basic question of how “magical” a many-body state can be, and show that the maximum magic of an $n$-qubit state is essentially $n$, simultaneously for a range of “good” magic measures. As a corollary, the resource theory of magic has asymptotic golden currency states. We then show that, in fact, almost all $n$-qubit pure states have magic of nearly $n$. In the quest for explicit, scalable cases of highly entangled states whose magic can be understood, we connect the magic of hypergraph states with the second-order nonlinearity of their underlying Boolean functions. Next, we go on and investigate many-body magic in practical and physical contexts. We first consider a variant of measurement-based quantum computation where the client is restricted to Pauli measurements, in which magic is a necessary feature of the initial “resource” state. We show that $n$-qubit states with nearly $n$ magic, or indeed almost all states, cannot supply nontrivial speedups over classical computers. We then present an example of analyzing the magic of “natural” condensed matter systems of physical interest. We apply the Boolean function techniques to derive explicit bounds on the magic of certain representative two-dimensional symmetry-protected topological states, and comment on possible further connections between magic and the quantum complexity of phases of matter.

We introduce a flexible and graphically intuitive framework that constructs complex quantum error correction codes from simple codes or states, generalizing code concatenation. More specifically, we represent the complex code constructions as tensor networks built from the tensors of simple codes or states in a modular fashion. Using a set of local moves known as operator pushing, one can derive properties of the more complex codes, such as transversal non-Clifford gates, by tracing the flow of operators in the network. The framework endows a network geometry to any code it builds and is valid for constructing stabilizer codes as well as nonstabilizer codes over qubits and qudits. For a contractible tensor network, the sequence of contractions also constructs a decoding or encoding circuit. To highlight the framework’s range of capabilities and to provide a tutorial, we lay out some examples where we glue together simple stabilizer codes to construct nontrivial codes. These examples include the toric code and its variants, a holographic code with transversal non-Clifford operators, a three-dimensional stabilizer code, and other stabilizer codes with interesting properties. Surprisingly, we find that the surface code is equivalent to the two-dimensional Bacon-Shor code after “dualizing” its tensor-network encoding map.

Tensor processing units (TPUs) are specialized hardware accelerators developed by Google to support large-scale machine-learning tasks but they can also be leveraged to accelerate and scale other linear-algebra-intensive computations. In this paper, we demonstrate the usage of TPUs for massively parallel classical simulations of quantum many-body dynamics on long time scales. We apply our methods to study the phenomenon of Floquet prethermalization, i.e., exponentially slow heating in quantum spin chains subject to high-frequency periodic driving. We simulate the dynamics of $L=34$ qubits for over ${10}^5$ Floquet periods, corresponding to circuits with $4\times {10}^6$ nearest-neighbor two-qubit gates. The circuits simulated have no additional symmetries and represent a pure-state evolution in the full $2^L$-dimensional Hilbert space. This is achieved by distributing the computation over 128 TPU cores. On that size TPU cluster, we find speed-ups in wall-clock run time of 230 times and 15 times when compared to reference CPU and single-graphics-processing-unit (GPU) simulations, respectively, for shorter-time 30-qubit simulations that can be handled by all three platforms. We study the computational cost of the simulations, as a function of both the number of qubits and the number of TPU cores used, up to our maximum capacity of $L=40$ qubits, which requires a “full pod” of 2048 TPU cores with tens of terabytes of memory in total. For these simulations, an eight-TPU-core machine is comparable to a single A100 GPU and thus the full TPU pod is comparable to a machine with hundreds of top-of-the-line GPUs. However, the TPU pod is more energy and cost efficient and readily accessible (via Google Cloud), unlike such large many-GPU configurations. We also study the accumulation of numerical error as a function of circuit depth in very deep circuits. Our work demonstrates that TPUs can offer significant advantages for state-of-the-art simulations of quantum many-body dynamics.

We study a mechanism whereby quantum information present in the initial state of a quantum many-body system can be protected for arbitrary times due to a combination of symmetry and spatial locality. Remarkably, the mechanism is sufficiently generic that the dynamics can be fully ergodic upon resolving the protecting symmetry and fixing the encoded quantum state, resulting in a quantum memory that persists up to infinite temperature. After exemplifying the mechanism in a strongly nonintegrable two-dimensional (2D) spin model inspired by the surface code, we find it has a natural interpretation in the language of noiseless subsystems and stabilizer subsystem codes. This interpretation yields a number of further examples, including a nonintegrable Hamiltonian with quantum memory based on the Bacon-Shor code. The lifetime of the encoded quantum information in these models is infinite provided that the dynamics respect the stabilizer symmetry of the underlying subsystem code. In the presence of symmetry-violating perturbations, we make contact with previous work leveraging the concept of prethermalization to show that the encoded quantum information can acquire a parametrically long lifetime under dynamics with an enlarged continuous symmetry group. The prethermalization mechanism hinges on the application of external fields that are much larger than the perturbations themselves. We identify conditions on the underlying subsystem code that enable such a prethermal enhancement of the memory lifetime.

Engineered dynamical maps combining coherent and dissipative transformations of quantum states with quantum measurements have demonstrated a number of technological applications, and promise to be a crucial tool in quantum thermodynamic processes. Here we exploit the control on the effective open spin qutrit dynamics of a nitrogen-vacancy center to experimentally realize an autonomous feedback process (Maxwell’s demon) with tunable dissipative strength. The feedback is enabled by random measurement events that condition the subsequent dissipative evolution of the qutrit. The efficacy of the autonomous Maxwell's demon is quantified by means of a generalized Sagawa-Ueda-Tasaki relation for dissipative dynamics. To achieve this, we experimentally characterize the fluctuations of the energy exchanged between the system and its the environment. This opens the way to the implementation of a new class of Maxwell’s demons, which could be useful for quantum sensing and quantum thermodynamic devices.

Fermionic linear optics (FLO) is a restricted model of quantum computation, which in its original form is known to be efficiently classically simulable. We show that, when initialized with suitable input states, FLO circuits can be used to demonstrate quantum computational advantage with strong hardness guarantees. Based on this, we propose a quantum advantage scheme, which is a fermionic analog of boson sampling: fermion sampling with magic input states. We consider in parallel two classes of circuits: particle-number conserving (passive) FLO and active FLO that preserves only fermionic parity. Using low-dimensional continuous symmetry groups that underpin these classes of quantum circuits, we prove anticoncentration and robust average-case hardness of computation of output probabilities probabilities. Taken together, these findings provide hardness guarantees comparable to the paradigm of random circuit sampling and boson sampling, the leading candidates for attaining quantum computational advantage. Our scheme is experimentally feasible. FLO circuits are relevant for quantum chemistry and many-body physics, and have been successfully implemented in superconducting architectures. We also argue that due to the structured nature of FLO circuits, they can be efficiently certified using resources scaling polynomially with the system size, with partial trust in the quantum device.

New experimental quantum simulation platforms have recently been implemented with divalent atoms trapped in optical tweezer arrays, with promising performance. The second valence electron also brings about new prospects through the so-called isolated core excitation (ICE). However, autoionization presents a strong limitation to this use. In this study, we propose and demonstrate a new approach to applying a sizable light shift to a Rydberg state with close-to-resonant ICE while avoiding autoionization. In particular, we investigate the ICE of ytterbium atoms in ${}^1{S}_0$ Rydberg states. Spectroscopic studies of the induced autoionization and the light shift imparted to the Rydberg states are well accounted for with multichannel quantum defect theory. Such control over the inner electron without disturbing the Rydberg electron brings about a new tool for the targeted coherent manipulation of Rydberg states in quantum simulation or quantum computing experiments performed with alkaline-earth atoms.

Scalable local control over gate operations is an outstanding challenge in the field of quantum computing and programmable quantum simulation with Rydberg-atom arrays. One approach is to use a global field to excite atoms to the Rydberg state and tune individual atoms in and out of resonance via local light shifts. In this work, we point out that photon-scattering errors from light shifts can be significantly reduced if the light shift is applied to the Rydberg state instead of the ground state, which can be realized in Rydberg states of alkaline-earth atoms using optical transitions in the ion core. As a proof of concept, we experimentally demonstrate global control of Rydberg excitations in an $\mathrm{Yb}$ optical-tweezer array via light shifts induced by a laser tuned near the $\mathrm{Yb}$${}^{+}6s\to 6p_{1/2}$ transition. We also perform detailed spectroscopy of the induced light shift and scattering rates of the $6sns{}^3{\mathrm{S}}_1$ Rydberg states and reveal the existence of satellite lines where losses from autoionization are strongly suppressed. This work can be readily extended to implement local gate operations in Rydberg-atom arrays.

Deep defects in silicon carbide ($\mathrm{Si}\mathrm{C}$) possess atomlike electronic, spin, and optical properties, making them ideal for quantum computing and sensing applications. In these applications, deep defects are often placed within fabricated nanostructures that modify defect properties due to surface and quantum confinement effects. Thus far, theoretical studies exploring deep defects in $\mathrm{Si}\mathrm{C}$ have ignored these effects. Using density-functional theory, this work demonstrates site dependence of properties of bright negatively charged silicon monovacancies within a $\mathrm{Si}\mathrm{C}$ nanowire. It is shown that the optical properties of defects depend strongly on the hybridization of the defect states with the surface states and on the structural changes allowed by proximity to the surfaces. Additionally, analysis of the first-principles results indicates that the charge-state conversion and/or migration to thermodynamically favorable undercoordinated surface sites can deteriorate deep-defect properties. These results illustrate the importance of considering how finite-size effects tune defect properties and of creating mitigating protocols to ensure the charge-state stability of a defect within nanostructured hosts.

Tracking the dynamics of physical systems in real time is a prime application of quantum computers. Using a trapped-ion system with up to six qubits, we simulate the real-time dynamics of a lattice gauge theory in $1+1$ dimensions, i.e., the lattice Schwinger model, and demonstrate nonperturbative effects such as pair creation for times much longer than previously accessible. We study the gate requirement of two formulations of the model using the Suzuki-Trotter product formula, as well as the trade-off between errors from the ordering of the Hamiltonian terms, the Trotter step size, and experimental imperfections. To mitigate experimental errors, a recent symmetry-protection protocol for suppressing coherent errors and a symmetry-inspired postselection scheme are applied. This work demonstrates the integrated theoretical, algorithmic, and experimental approach that is essential for efficient simulation of lattice gauge theories and other complex physical systems.

In this work we present a detailed analysis of variational quantum phase estimation (VQPE), a method based on real-time evolution for ground- and excited-state estimation on near-term hardware. We derive the theoretical ground on which the approach stands, and demonstrate that it provides one of the most compact variational expansions to date for solving strongly correlated Hamiltonians, when starting from an appropriate reference state. At the center of VQPE lies a set of equations, with a simple geometrical interpretation, which provides conditions for the time evolution grid in order to decouple eigenstates out of the set of time-evolved expansion states, and connects the method to the classical filter-diagonalization algorithm. Furthermore, we introduce what we call the unitary formulation of VQPE, in which the number of matrix elements that need to be measured scales linearly with the number of expansion states, and we provide an analysis of the effects of noise that substantially improves previous considerations. The unitary formulation allows for a direct comparison to iterative phase estimation. Our results mark VQPE as both a natural and highly efficient quantum algorithm for ground- and excited-state calculations of general many-body systems. We demonstrate a hardware implementation of VQPE for the transverse field Ising model. Furthermore, we illustrate its power on a paradigmatic example of strong correlation (${\mathrm{Cr}}_2$ in the def2-SVP basis set), and show that it is possible to reach chemical accuracy with as few as approximately $50$ time steps.

Finding the ground-state energy of electrons subject to an external electric field is a fundamental problem in computational chemistry. While the theory of QMA-completeness has been instrumental in understanding the complexity of finding ground states in many-body quantum systems, prior to this work it has been unknown whether or not the special form of the Hamiltonian for the electronic structure of molecules can be exploited to find ground states efficiently or whether the problem remains hard for this special case. We prove that the electronic-structure problem, when restricted to a fixed single-particle basis and a fixed number of electrons, is QMA-complete. In our proof, the local Hamiltonian is encoded in the choice of spatial orbitals used to discretize the electronic-structure Hamiltonian. In contrast, Schuch and Verstraete have proved hardness for the electronic-structure problem with an additional site-specific external magnetic field, but without the restriction to a fixed basis, by encoding a local Hamiltonian on qubits in the site-specific magnetic field. We also show that estimation of the energy of the lowest-energy Slater-determinant state (i.e., the Hartree-Fock state) is nondeterministic polynomial time (NP)-complete for the electronic-structure Hamiltonian in a fixed basis.

Reliable processing of quantum information is a milestone to achieve for the deployment of quantum technologies. Uncontrolled, out-of-equilibrium sources of decoherence need to be characterized in detail for designing the control of quantum devices to mitigate the loss of quantum information. However, quantum sensing of such environments is still a challenge due to their nonstationary nature that in general can generate complex high-order correlations. We introduce a path integral framework to characterize nonstationary environmental fluctuations by a quantum probe. We find the solution for the decoherence decay of nonstationary, generalized Gaussian processes that induce pure dephasing. This dephasing when expressed in a suitable basis, based on the nonstationary noise eigenmodes, is defined by the overlap of a generalized noise spectral density and a filter function that depends on the control fields. This result thus extends the validity of the similar general expression for the dephasing of open quantum systems coupled to stationary noise processes to out-of-equilibrium environments. We show physical insights for a broad subclass of nonstationary noise processes that are local in time, in the sense that the noise correlation functions contain memory based on constraints of the derivatives of the fluctuating noise paths. Spectral and non-Markovian properties are discussed together with implementations of the framework to treat paradigmatic environments that are out of equilibrium, e.g., due to a quench and a pulsed noise. We show that our results provide tools for probing the spectral and time-correlation properties, and for mitigating decoherence effects of out-of-equilibrium—nonstationary—environments.

We propose an implementation of a two-dimensional ${\mathbb{Z}}_2$ lattice gauge theory model on a shallow quantum circuit, involving a number of single- and two-qubit gates comparable to what can be achieved with present-day and near-future technologies. The ground-state preparation is numerically analyzed on a small lattice with a variational quantum algorithm, which requires a small number of parameters to reach high fidelities and can be efficiently scaled up on larger systems. Despite the reduced size of the lattice we consider, a transition between confined and deconfined regimes can be detected by measuring expectation values of Wilson loop operators or the topological entropy. Moreover, if periodic boundary conditions are implemented, the same optimal solution is transferable among all four different topological sectors, without any need for further optimization on the variational parameters. Our work shows that variational quantum algorithms provide a useful technique to be added in the growing toolbox for digital simulations of lattice gauge theories.

Speed of state transitions in macroscopic systems is a crucial concept for foundations of nonequilibrium statistical mechanics as well as various applications in quantum technology represented by optimal quantum control. While extensive studies have made efforts to obtain rigorous constraints on dynamical processes since Mandelstam and Tamm, speed limits that provide tight bounds for macroscopic transitions have remained elusive. Here, by employing the local conservation law of probability, the fundamental principle in physics, we develop a general framework for deriving qualitatively tighter speed limits for macroscopic systems than many conventional ones. We show for the first time that the speed of the expectation value of an observable defined on an arbitrary graph, which can describe general many-body systems, is bounded by the “gradient” of the observable, in contrast with conventional speed limits depending on the entire range of the observable. This framework enables us to derive novel quantum speed limits for macroscopic unitary dynamics. Unlike previous bounds, the speed limit decreases when the expectation value of the transition Hamiltonian increases; this intuitively describes a new trade-off relation between time and the quantum phase difference. Our bound is dependent on instantaneous quantum states and thus can achieve the equality condition, which is conceptually distinct from the Lieb-Robinson bound. We also find that, beyond expectation values of macroscopic observables, the speed of macroscopic quantum coherence can be bounded from above by our general approach. The newly obtained bounds are verified in transport phenomena in particle systems and nonequilibrium dynamics in many-body spin systems. We also demonstrate that our strategy can be applied for finding new speed limits for macroscopic transitions in stochastic systems, including quantum ones, where the bounds are expressed by the entropy production rate. Our work elucidates novel speed limits on the basis of local conservation law, providing fundamental limits to various types of nonequilibrium quantum macroscopic phenomena.

Quantum error correction (QEC) is a procedure by which the quantum state of a system is protected against a known type of noise, by preemptively adding redundancy to that state. Such a procedure is commonly used in quantum computing when thermal noise is present. Interestingly, thermal noise has also been known to play a central role in quantum thermodynamics (QTD). This fact hints at the applicability of certain QTD statements in the QEC of thermal noise, which has been discussed previously in the context of Maxwell’s demon. In this paper, we view QEC as a quantum heat engine with a feedback controller (i.e., a demon). We derive an upper bound on the measurement heat dissipated during the error-identification stage in terms of the Groenewold information gain, thereby also providing the latter with a physical meaning when it is negative. Further, we derive the second law of thermodynamics in the context of this QEC engine, operating with general quantum measurements. Finally, we show that, under a set of physically motivated assumptions, this leads to a fundamental triple-trade-off relation, which implies a trade-off between the maximum achievable fidelity of QEC and the super-Carnot efficiency that heat engines with feedback controllers have been known to possess. A similar trade-off relation occurs for the thermodynamic efficiency of the QEC engine and the efficacy of the quantum measurement used for error identification.

We study the sampling properties of physical realizations of quantum annealers which are implemented through programmable lattices of superconducting flux qubits. Comprehensive statistical analysis of the data produced by these quantum machines shows that quantum annealers behave as samplers that generate independent configurations from low-temperature noisy Gibbs distributions. We show that the structure of the output distribution probes the intrinsic physical properties of the quantum device such as effective temperature of individual qubits and magnitude of local qubit noise, which result in a nonlinear response function and spurious interactions that are absent in the hardware implementation. We anticipate that our methodology will find widespread use in the characterization of future generations of quantum annealers and other emerging analog computing devices.

We simulate, using nonperturbative methods, the real-time dynamics of small bubbles of “false vacuum” in a quantum spin chain near criticality, where the low-energy physics is described by a relativistic (1+1)-dimensional quantum field theory. We consider bubbles whose walls are kink and antikink quasiparticle excitations, so that wall collisions are kink-antikink scattering events. To construct these bubbles in the presence of strong correlations, we extend a recently proposed matrix product state (MPS) ansatz for quasiparticle wavepackets. We simulate dynamics within a window of about $1000$ spins embedded in an infinite chain at energies of up to about $5$ times the mass gap. By choosing the wavepacket width and the bubble size appropriately, we avoid strong lattice effects and observe relativistic kink-antikink collisions. We use the MPS quasiparticle ansatz to detect scattering outcomes. (i) In the Ising model, with transverse and longitudinal fields, we do not observe particle production despite nonintegrability (supporting recent observations of nonthermalizing states in this model). (ii) Switching on an additional interaction, we see production of confined and unconfined particle pairs. We characterize the amount of entanglement generated as a function of energy and time and conclude that our classical simulation methods will ultimately fail as these increase. We anticipate that kink-antikink scattering in 1+1 dimensions will be an instructive benchmark problem for future quantum computers and analog quantum simulators.

Twin-field quantum key distribution can overcome the secret key capacity of repeaterless quantum key distribution via single-photon interference. However, to compensate for the channel fluctuations and lock the laser fluctuations, the techniques of phase tracking and phase locking are indispensable in experiment, which drastically increase experimental complexity and hinder free-space realization. We herein present an asynchronous measurement-device-independent quantum key distribution protocol that can surpass the secret key capacity even without phase tracking and phase locking. Leveraging the concept of time multiplexing, asynchronous two-photon Bell-state measurement is realized by postmatching two interference detection events. For a 1 GHz system, the new protocol reaches a transmission distance of 450 km without phase tracking. After further removing phase locking, our protocol is still capable of breaking the capacity at 270 km. Intriguingly, when using the same experimental techniques, our protocol has a higher key rate than the phase-matching-type twin-field protocol. In the presence of imperfect intensity modulation, it also has a significant advantage in terms of the transmission distance over the sending-or-not-sending-type twin-field protocol. With high key rates and accessible technology, our work provides a promising candidate for practical scalable quantum communication networks.

Quantum error correction and symmetries play central roles in quantum information science and physics. It is known that quantum error-correcting codes that obey (are covariant with respect to) continuous symmetries in a certain sense cannot correct erasure errors perfectly (a well-known result in this regard being the Eastin-Knill theorem in the context of fault-tolerant quantum computing), in contrast to the case without symmetry constraints. Furthermore, several quantitative fundamental limits on the accuracy of such covariant codes for approximate quantum error correction are known. Here, we consider the quantum error correction capability of uniformly random covariant codes. In particular, we analytically study the most essential cases of U(1) and $\mathrm{SU}(d)$ symmetries, and show that for both symmetry groups the error of the covariant codes generated by Haar-random symmetric unitaries, i.e., unitaries that commute with the group actions, typically scale as $O(n^{-1})$ in terms of both the average- and worst-case purified distances against erasure noise, saturating the fundamental limits to leading order. We note that the results hold for symmetric variants of unitary 2-designs, and comment on the convergence problem of symmetric random circuits. Our results not only indicate (potentially efficient) randomized constructions of optimal $\mathrm{U}(1)$- and $\mathrm{SU}(d)$-covariant codes, but also reveal fundamental properties of random symmetric unitaries, which yield important solvable models of complex quantum systems (including black holes and many-body spin systems) that have attracted great recent interest in quantum gravity and condensed matter physics. We expect our construction and analysis to find broad relevance in both physics and quantum computing.