Quantum computers are expected to achieve a significant speed-up over classical computers in solving a range of computational problems. Chains of ions held in a linear Paul trap are a promising platform for constructing such quantum computers, due to their long coherence times and high quality of control. Here, we report on the construction of a small five-qubit universal quantum computer using ${}^{88}$$\mathrm{Sr}$${}^{+}$ ions in a radio-frequency (rf) trap. All basic operations, including initialization, quantum logic operations, and readout, are performed with high fidelity. Selective two-qubit and single-qubit gates, implemented using a narrow-line-width laser, comprise a universal gate set, allowing realization of any unitary on the quantum register. We review the main experimental tools and describe in detail unique aspects of the computer: the use of robust entangling gates and the development of a quantum coherent feedback system through electron-multiplying CCD camera acquisition. The latter is necessary for carrying out quantum error-correction protocols in future experiments.

We propose a divide-and-conquer method for the quantum-classical hybrid algorithm to solve larger problems with small-scale quantum computers. Specifically, we concatenate a variational quantum eigensolver (VQE) with a reduction in the system dimension, where the interactions between divided subsystems are taken as an effective Hamiltonian expanded by the reduced basis. Then the effective Hamiltonian is further solved by the VQE, which we call deep VQE. Deep VQE allows us to apply quantum-classical hybrid algorithms on small-scale quantum computers to large systems with strong intrasubsystem interactions and weak intersubsystem interactions, or strongly correlated spin models on large regular lattices. As proof-of-principle numerical demonstrations, we use the proposed method for quasi-one-dimensional models, including one-dimensionally coupled 12-qubit Heisenberg antiferromagnetic models on kagome lattices as well as two-dimensional Heisenberg antiferromagnetic models on square lattices. The largest problem size of 64 qubits is solved by simulating 20-qubit quantum computers with a reasonably good accuracy approximately a few $\mathrm{\%}$. The proposed scheme enables us to handle the problems of $>1000$ qubits by concatenating VQEs with a few tens of qubits. While it is unclear how accurate ground-state energy can be obtained for such a large system, our numerical results on a 64-qubit system suggest that deep VQE provides a good approximation (discrepancy within a few percent) and has room for further improvement. Therefore, deep VQE provides us a promising pathway to solve practically important problems on noisy intermediate-scale quantum computers.

In the early years of fault-tolerant quantum computing (FTQC), it is expected that the available code distance and the number of magic states will be restricted due to the limited scalability of quantum devices and the insufficient computational power of classical decoding units. Here, we integrate quantum error correction and quantum error mitigation into an efficient FTQC architecture that effectively increases the code distance and $T$-gate count at the cost of constant sampling overheads in a wide range of quantum computing regimes. For example, while we need ${10}^4$ to ${10}^{10}$ logical operations for demonstrating quantum advantages from optimistic and pessimistic points of view, we show that we can reduce the required number of physical qubits by 80% and 45% in each regime. From another perspective, when the achievable code distance is up to about 11, our scheme allows executing ${10}^3$ times more logical operations. This scheme will dramatically alleviate the required computational overheads and hasten the arrival of the FTQC era.

We propose a hardware architecture and protocol for connecting many local quantum processors contained within an optical cavity. The scheme is compatible with trapped ions or Rydberg arrays, and realizes teleported gates between any two qubits by distributing entanglement via single-photon transfers through a cavity. Heralding enables high-fidelity entanglement even for a cavity of moderate quality. For processors composed of trapped ions in a linear chain, a single cavity with realistic parameters successfully transfers photons every few $\mu \mathrm{s}$, increasing the interchain entanglement rate over 2 orders of magnitude beyond current methods and eliminating a major bottleneck for scaling trapped-ion systems. For one realistic scenario, we outline how to achieve the any-to-any entanglement of 20 ion chains containing a total of 500 qubits in $200\mu \mathrm{s}$, with both fidelities and rates limited only by local operations and ion readout. For processors composed of Rydberg atoms, our method fully connects a large array of thousands of neutral atoms. The connectivity afforded by our architecture is extendable to tens of thousands of qubits using multiple overlapping cavities, expanding capabilities for noisy intermediate-scale quantum era algorithms and Hamiltonian simulations, as well as enabling more robust high-dimensional error-correcting schemes.

For solid-state spin systems, unwanted interactions with surrounding spin baths and inhomogeneity are ubiquitous challenges. In defect spin systems, part of this challenge is that the process of generating the desired defect often involves creating numerous unwanted defects that induce dephasing. Here we show that many of these issues can be alleviated with the silicon vacancy in silicon carbide ($\mathrm{SiC}$), which is a simple defect to produce as it does not require the introduction of impurity atoms into the crystal. Previous measurements of the inhomogeneous dephasing time $T_2^{\ast }$ of silicon-vacancy ensembles have thus far been limited to a few 100 ns due to the nuclear spin bath. We perform isotopic purification to minimize the influence of nuclear spins, leading to an order of magnitude improvement in the $T_2^{\ast }$ at room temperature. Further improvements emerge by suppressing the effects of strain inhomogeneity via an informed choice of basis in the spin quartet. Combining these techniques leads to about a factor 50 improvement in the $T_2^{\ast }$ of the defect ensemble even at high defect densities.

Quantum entanglement is commonly assumed to be a central resource for quantum computing and quantum simulation. Nonetheless, the capability to detect it in many-body systems is severely limited by the absence of sufficiently scalable and flexible certification tools. This issue is particularly critical in situations where the structure of entanglement is a priori unknown, and where one cannot rely on existing entanglement witnesses. Here, we implement a scheme in which the knowledge of the mean value of arbitrary observables can be used to probe multipartite entanglement in a scalable, certified, and systematic manner. Specifically, we rely on positive semidefinite conditions, independent of partial-transposition-based criteria, necessarily obeyed if the data can be reproduced by a separable state. The violation of any of these conditions yields a specific entanglement witness, tailored to the data of interest, revealing the salient features of the data which are impossible to reproduce without entanglement. We validate this approach by probing theoretical many-body states of several hundreds of qubits relevant to existing experiments: a single-particle quench in a one-dimensional $XX$ chain; a many-body quench in a two-dimensional $XX$ model with $1/r^3$ interactions; and thermal equilibrium states of Heisenberg and transverse-field Ising chains. In all cases, these investigations have led us to discover new entanglement witnesses, some of which could be characterized analytically, generalizing existing results in the literature. In summary, our paper introduces a flexible data-driven entanglement detection technique for uncharacterized quantum many-body states, of immediate relevance to experiments in a quantum advantage regime.

Continuous-variable quantum key distribution exploits coherent measurements of the electromagnetic field, i.e., homodyne or heterodyne detection. The most advanced security proofs developed so far have relied on idealized mathematical models for such measurements, which assume that the measurement outcomes are continuous and unbounded variables. As physical-measurement devices have a finite range and precision, these mathematical models only serve as an approximation. It is expected that, under suitable conditions, the predictions obtained using these simplified models will be in good agreement with the actual experimental implementations. However, a quantitative analysis of the error introduced by this approximation, and of its impact on composable security, have been lacking so far. Here, we present a theory to rigorously account for the experimental limitations of realistic heterodyne detection. We focus on collective attacks and present security proofs for the asymptotic and finite-size regimes, the latter being within the framework of composable security. In doing this, we establish for the first time the composable security of discrete-modulation continuous-variable quantum key distribution in the finite-size regime. Tight bounds on the key rates are obtained through semidefinite programming and do not rely on a truncation of the Hilbert space.

Many problems in information theory can be reduced to optimizations over matrices, where the rank of the matrices is constrained. We establish a link between rank-constrained optimization and the theory of quantum entanglement. More precisely, we prove that a large class of rank-constrained semidefinite programs can be written as a convex optimization over separable quantum states and, consequently, we construct a complete hierarchy of semidefinite programs for solving the original problem. This hierarchy not only provides a sequence of certified bounds for the rank-constrained optimization problem, but also gives pretty good and often exact values in practice when the lowest level of the hierarchy is considered. We demonstrate that our approach can be used for relevant problems in quantum information processing, such as the optimization over pure states, the characterization of mixed unitary channels and faithful entanglement, and quantum contextuality, as well as in classical information theory including the maximum cut problem, pseudo-Boolean optimization, and the orthonormal representation of graphs. Finally, we show that our ideas can be extended to rank-constrained quadratic and higher-order programming.

We present calculations of both the ground- and excited-state energies of spin defects in solids carried out on a quantum computer, using a hybrid classical-quantum protocol. We focus on the negatively charged nitrogen-vacancy center in diamond and on the double vacancy in $4H$ $\mathrm{Si}\mathrm{C}$, which are of interest for the realization of quantum technologies. We employ a recently developed first-principles quantum embedding theory to describe point defects embedded in a periodic crystal and to derive an effective Hamiltonian, which is then transformed to a qubit Hamiltonian by means of a parity transformation. We use the variational quantum eigensolver (VQE) and quantum subspace expansion methods to obtain the ground and excited states of spin qubits, respectively, and we propose a promising strategy for noise mitigation. We show that by combining zero-noise extrapolation techniques and constraints on electron occupation to overcome the unphysical-state problem of the VQE algorithm, one can obtain reasonably accurate results on near-term-noisy architectures for ground- and excited-state properties of spin defects.

When a quantum system is put into an excited state, it will decay back to the ground state through a process termed spontaneous emission. It is generally assumed that spontaneous emissions between different individual emitters will not be coherent with each other; to produce coherent light, one would need population inversion and stimulated emission. In this work, we show that an optically thin ensemble of 11 000 radiating atoms spontaneously organizes to produce spatially coherent light. The reason for this coherence is collective coupling of the individual emitters via Dicke superradiance and subradiance (as opposed to amplification through stimulated emission).

It has been recently shown that there exist universal fundamental limits to the accuracy and efficiency of the transformation from noisy resource states to pure ones (e.g., distillation) in any well-behaved quantum resource theory [Fang and Liu, Phys. Rev. Lett. 125, 060405 (2020)]. Here, we develop a novel and powerful method for analyzing the limitations on quantum resource purification, which not only leads to improved bounds that rule out exact purification for a broader range of noisy states and are tight in certain cases, but also enable us to establish a robust no-purification theory for quantum channel (dynamical) resources. More specifically, we employ the new method to derive universal bounds on the error and cost of transforming generic noisy channels (where multiple instances can be used adaptively, in contrast to the state theory) to some unitary resource channel under any free channel-to-channel map. We address several cases of practical interest in more concrete terms, and discuss the connections and applications of our general results to distillation, quantum error correction, quantum Shannon theory, and quantum circuit synthesis.

Waveguide QED has emerged as a powerful analog quantum simulator due to the possibility of mediating versatile spin-spin interactions with tunable sign, range, and even dimerization. Yet, despite their potential, the many-body phases emerging from these systems have only been scarcely explored. Here, we characterize the many-body phases of a large class of spin models that can be obtained in such waveguide-QED simulators and uncover, importantly, the existence of symmetry-protected topological phases with large-period magnetic orderings with no analog in other state-of-art simulators. We explain that these phases emerge from the unique combination of long-range and dimerized interactions appearing in these platforms and propose several experimental observables to characterize them. Finally, we also develop an adiabatic protocol to prepare such states and analyze its performance with the main decoherence source of these systems.

Encoding logical quantum information in harmonic oscillator modes is a promising and hardware-efficient approach to the realization of a quantum computer. In this work, we propose to encode logical qubits in grid states of an ensemble of harmonic oscillator modes. We first discuss general results about these multimode bosonic codes; how to design them, how to practically implement them in different experimental platforms, and how lattice symmetries can be leveraged to perform logical non-Clifford operations. We then introduce in detail two two-mode grid codes based on the hypercubic and $D_4$ lattices, respectively, showing how to perform a universal set of logical operations. We demonstrate numerically that multimode grid codes have, compared to their single-mode counterpart, increased robustness against propagation of errors from ancillas used for error correction. Finally, we highlight some interesting links between multidimensional lattices and single-mode grid codes concatenated with qubit codes.

We identify five selected open problems in the theory of quantum information, which are rather simple to formulate, are well studied in the literature, but are technically not easy. As these problems enjoy diverse mathematical connections, they offer a huge breakthrough potential. The first four concern existence of certain objects relevant for quantum information, namely a family of symmetric informationally complete generalized measurements in an infinite sequence of dimensions, mutually unbiased bases in dimension six, measurements saturating multiparameter Cramér-Rao bound and bound entangled states with negative partial transpose. The fifth problem requires checking whether a certain state of a two-ququart system is two-copy distillable.

Individual trapped atomic qubits represent one of the most promising technologies to scale quantum computers, owing to their low idle errors and the ability to implement a full set of reconfigurable gate operations via focused optical fields. However, the fidelity of quantum gate operations can be limited by weak confinement of the atoms transverse to the laser. We present measurements of this effect by performing individually addressed entangling gates in chains of up to 25 trapped atomic ions. We present a semiclassical model that accurately describes the observed decoherence from the residual heating of the ions caused by noisy electric fields. We suppress these effects by cotrapping ancilla ions for sympathetically cooling the qubit ions throughout a quantum circuit.

We consider quantum circuits consisting of randomly chosen two-local gates and study the number of gates needed for the distribution over measurement outcomes for typical circuit instances to be anticoncentrated, roughly meaning that the probability mass is not too concentrated on a small number of measurement outcomes. An understanding of the conditions for anticoncentration is important for determining which quantum circuits are difficult to simulate classically, as anticoncentration has been in some cases an ingredient of mathematical arguments that simulation is hard and in other cases a necessary condition for easy simulation. Our definition of anticoncentration is that the expected collision probability of the distribution—that is, the probability that two independently drawn outcomes will agree—is only a constant factor larger than the collision probability for the uniform distribution. We show that when the two-local gates are each drawn from the Haar measure (or any 2-design), at least $\mathrm{\Omega }(n\log (n))$ gates (and thus $\mathrm{\Omega }(\log (n))$ circuit depth) are needed for this condition to be met on an $n$-qudit circuit. In both the case where the gates are nearest neighbor on a one-dimensional ring and the case where gates are long range, we show that $O(n\log (n))$ gates are also sufficient and we precisely compute the optimal constant prefactor for the $n\log (n)$. The technique we employ relies upon a mapping from the expected collision probability to the partition function of an Ising-like classical statistical-mechanical model, which we manage to bound using stochastic and combinatorial techniques.

The photonic temporal degree of freedom is one of the most promising platforms for quantum communication over fiber networks and free-space channels. In particular, time-bin states of photons are robust to environmental disturbances, support high-rate communication, and can be used in high-dimensional schemes. However, the detection of photonic time-bin states remains a challenging task, particularly for the case of photons that are in a superposition of different time bins. Here, we experimentally demonstrate the feasibility of picosecond time-bin states of light, known as ultrafast time bins, for applications in quantum communications. With the ability to measure time-bin superpositions with excellent phase stability, we enable the use of temporal states in efficient quantum key distribution protocols such as the BB84 protocol.

Lattice-surgery protocols allow for the efficient implementation of universal gate sets with two-dimensional topological codes where qubits are constrained to interact with one another locally. In this work, we first introduce a decoder capable of correcting spacelike and timelike errors during lattice-surgery protocols. Subsequently, we compute the logical failure rates of a lattice-surgery protocol for a biased circuit-level noise model. We then provide a protocol for performing twist-free lattice surgery, where we avoid twist defects in the bulk of the lattice. Our twist-free protocol eliminates the extra circuit components and gate-scheduling complexities associated with the measurement of higher weight stabilizers when using twist defects. We also provide a protocol for temporally encoded lattice surgery that can be used to reduce both the run times and the total space-time costs of quantum algorithms. Lastly, we propose a layout for a quantum processor that is more efficient for rectangular surface codes exploiting noise bias and that is compatible with the other techniques mentioned above.

We propose the use of trapped ions for the detection of millicharged dark matter. Millicharged particles will scatter off the ions, giving a signal either in individual events or in the overall heating rate of the ions. Ion traps have several properties that make them ideal detectors for such a signal. First, ion traps have demonstrated significant isolation of the ions from the environment, greatly reducing the background heating and event rates. Second, ion traps can have low thresholds for the detection of energy deposition, down to approximately the nanoelectronvolt range. Third, since the ions are charged, they naturally have large cross sections for scattering with the millicharged particles, this being further enhanced by the low velocities of the thermalized millicharged particles. Despite ion-trap setups being optimized for other goals, we find that existing measurements put new constraints on millicharged dark matter that are many orders of magnitude beyond previous bounds. For example, for a millicharged dark matter mass $m_Q=10\mathrm{GeV}$ and charge ${10}^{-3}$ of the electron charge, ion traps limit the local density to be $n_Q\lesssim 1{\mathrm{cm}}^{-3}$, a factor of approximately ${10}^8$ better than current constraints. Future dedicated ion-trap experiments could reach even further into unexplored parameter space.

We present a comprehensive architectural analysis for a proposed fault-tolerant quantum computer based on cat codes concatenated with outer quantum error-correcting codes. For the physical hardware, we propose a system of acoustic resonators coupled to superconducting circuits with a two-dimensional layout. Using estimated physical parameters for the hardware, we perform a detailed error analysis of measurements and gates, including cnot and Toffoli gates. Having built a realistic noise model, we numerically simulate quantum error correction when the outer code is either a repetition code or a thin rectangular surface code. Our next step toward universal fault-tolerant quantum computation is a protocol for fault-tolerant Toffoli magic state preparation that significantly improves upon the fidelity of physical Toffoli gates at very low qubit cost. To achieve even lower overheads, we devise a new magic state distillation protocol for Toffoli states. Combining these results together, we obtain realistic full-resource estimates of the physical error rates and overheads needed to run useful fault-tolerant quantum algorithms. We find that with around 1000 superconducting circuit components, one could construct a fault-tolerant quantum computer that can run circuits, which are currently intractable for classical computers. Hardware with 18 000 superconducting circuit components, in turn, could simulate the Hubbard model in a regime beyond the reach of classical computing.