It has been a long-standing open problem to construct a general framework for relating the spectra of dual theories to each other. Here, we solve this problem for the case of one-dimensional quantum lattice models with symmetry-twisted boundary conditions. In Ref. [PRX Quantum 4, 020357], dualities are defined between (categorically) symmetric models that only differ in a choice of module category. Using matrix product operators, we construct from the data of module functors explicit symmetry operators preserving boundary conditions as well as intertwiners mapping topological sectors of dual models onto one another. We illustrate our construction with a family of examples that are in the duality class of the spin-$\frac{1}{2}$ Heisenberg $XXZ$ model. One model has symmetry operators forming the fusion category $\mathsf{R}\mathsf{e}\mathsf{p}({\mathcal{S}}_3)$ of representations of the group ${\mathcal{S}}_3$. We find that the mapping between its topological sectors and those of the $XXZ$ model is associated with the nontrivial braided autoequivalence of the Drinfel’d center of $\mathsf{R}\mathsf{e}\mathsf{p}({\mathcal{S}}_3)$.

Quantum computers are expected to drastically accelerate certain computing tasks versus classical computers. Noisy intermediate-scale quantum (NISQ) devices, which have tens to hundreds of noisy physical qubits, are gradually becoming available, but it is still challenging to achieve useful quantum advantages in meaningful tasks. On the other hand, full fault-tolerant quantum computing (FTQC) based on quantum error correction code remains far beyond realization due to its extremely large requirement of high-precision physical qubits. In this study, we propose a quantum computing architecture to close the gap between NISQ and FTQC architectures. Our architecture is based on erroneous arbitrary rotation gates and error-corrected Clifford gates implemented by lattice surgery. We omit the typical distillation protocol to achieve direct analog rotations and small qubit requirements, and minimize the remnant errors of the rotations by a carefully designed state injection protocol. Our estimation based on numerical simulations shows that for early-FTQC devices that consist of ${10}^4$ physical qubits with physical error probability $p={10}^{-4}$, we can perform roughly $1.72\times {10}^7$ Clifford operations and $3.75\times {10}^4$ arbitrary rotations on 64 logical qubits. Such computations cannot be realized by the existing NISQ and FTQC architectures on the same device, as well as classical computers. We hope that our proposal and the corresponding development of quantum algorithms based on it will bring new insights into the realization of practical quantum computers in the future.

The efficient computation of observables beyond the total energy is a key challenge and opportunity for fault-tolerant quantum computing approaches in quantum chemistry. Here, we consider the symmetry-adapted perturbation-theory (SAPT) components of the interaction energy as a prototypical example of such an observable. We provide a guide for calculating this observable on a fault-tolerant quantum computer while optimizing the required computational resources. Specifically, we present a quantum algorithm that estimates interaction energies at the first-order SAPT level with a Heisenberg-limited scaling. To this end, we exploit a high-order tensor-factorization and block-encoding technique that efficiently represents each SAPT observable. To quantify the computational cost of our methodology, we provide resource estimates in terms of the required number of logical qubits and Toffoli gates to execute our algorithm for a range of benchmark molecules, also taking into account the cost of the eigenstate preparation and the cost of block encoding the SAPT observables. Finally, we perform the resource estimation for a heme and artemisinin complex as a representative large-scale system encountered in drug design, highlighting the performance of our algorithm in this new benchmark study and discussing possible bottlenecks that may be improved in future work.

Diagrammatic summation is a common bottleneck in modern applications of projected entangled-pair states, especially in computing low-energy excitations of a two-dimensional quantum many-body system. To solve this problem, here we extend the generating-function approach for tensor-network diagrammatic summation, a scheme previously proposed in the context of matrix product states. Taking the form of a one-particle excitation, we show that the excited state can be computed efficiently in the generating-function formalism, which can further be used in evaluating the dynamical structure factor of the system. Our benchmark results for the spin-$1/2$ transverse-field Ising model and Heisenberg model on the square lattice provide a desirable accuracy, showing good agreement with known results. We then study the spin-$1/2J_1$-$J_2$ model on the same lattice and investigate the dynamical properties of the putative gapless spin liquid phase. We conclude with a discussion on generalizations to multiparticle excitations.

Demonstrating quantum advantage requires experimental implementation of a computational task that is hard to achieve using state-of-the-art classical systems. One approach is to perform sampling from a probability distribution associated with a certain class of highly entangled many-body wave functions. It has been suggested that such a quantum advantage can be certified with the linear cross-entropy benchmark (XEB). We critically examine this notion. First, we consider a “benign” setting, where an honest implementation of a noisy quantum circuit is assumed, and characterize the conditions under which the XEB approximates the fidelity of quantum dynamics. Second, we assume an “adversarial” setting, where all possible classical algorithms are considered for comparisons, and show that achieving relatively high XEB values does not imply faithful simulation of quantum dynamics. Specifically, we present an efficient classical algorithm that achieves high XEB values, namely 5–12% of those obtained in the state-of-the-art experiments, within just a few seconds using a single GPU machine. This is made possible by identifying and exploiting several vulnerabilities of the XEB, which allows us to achieve high XEB values without simulating a full quantum circuit. Remarkably, our algorithm features better scaling with the system size than a noisy quantum device for commonly studied random circuit ensembles in various architecture. We quantitatively explain the success of our algorithm and the limitations of the XEB by using a theoretical framework, in which the dynamics of the average XEB and fidelity are mapped to classical statistical mechanics models. Using this framework, we illustrate the relation between the XEB and the fidelity for quantum circuits in various architectures, with different choices of gate sets, and in the presence of noise. Taken together, our results demonstrate that XEB’s utility as a proxy for fidelity hinges on several conditions, which should be independently checked in the benign setting, but cannot be assumed in the general adversarial setting. Therefore, the XEB on its own has a limited utility as a benchmark for quantum advantage. We briefly discuss potential ways to overcome these limitations.

Logical qubits can be protected from decoherence by performing quantum error-correction (QEC) cycles repeatedly. Algorithms for fault-tolerant QEC must be compiled to the specific hardware platform under consideration in order to practically realize a quantum memory that operates for in principle arbitrary long times. All circuit components must be assumed as noisy unless specific assumptions about the form of the noise are made. Modern QEC schemes are challenging to implement experimentally in physical architectures where in-sequence measurements and feed forward of classical information cannot be reliably executed fast enough or even at all. Here we provide a novel scheme to perform QEC cycles without the need of measuring qubits that is fully fault-tolerant with respect to all components used in the circuit. Our scheme can be used for any low-distance CSS code since its only requirement towards the underlying code is a transversal cnot gate. Similarly to Steane-type EC, we coherently copy errors to a logical auxiliary qubit but then apply a coherent feedback operation from the auxiliary system to the logical data qubit. The logical auxiliary qubit is prepared fault tolerantly without measurements, too. We benchmark logical failure rates of the scheme in comparison to a flag-qubit-based EC cycle. We map out a parameter region where our scheme is feasible and estimate physical error rates necessary to achieve the break-even point of beneficial QEC with our scheme. We outline how our scheme could be implemented in ion traps and with neutral atoms in a tweezer array. For recently demonstrated capabilities of atom shuttling and native multiatom Rydberg gates, we achieve moderate circuit depths and beneficial performance of our scheme while not breaking fault tolerance. These results thereby enable practical fault-tolerant QEC in hardware architectures that do not support midcircuit measurements.

Irreversibility, crucial in both thermodynamics and information theory, is naturally studied by comparing the evolution—the (forward) channel—with an associated reverse—the reverse channel. There are two natural ways to define this reverse channel. Using logical inference, the reverse channel is the Bayesian retrodiction (the Petz recovery map in the quantum formalism) of the original one. Alternatively, we know from physics that every irreversible process can be modeled as an open system: one can then define the corresponding closed system by adding a bath (“dilation”), trivially reverse the global reversible process, and finally remove the bath again. We prove that the two recipes are strictly identical, both in the classical and in the quantum formalism, once one accounts for correlations formed between system and the bath. Having established this, we define and study special classes of maps: product-preserving maps (including generalized thermal maps), for which no such system-bath correlations are formed for some states; and tabletop time-reversible maps, when the reverse channel can be implemented with the same devices as the original one. We establish several general results connecting these classes, and a very detailed characterization when both the system and the bath are one qubit. In particular, we show that, when reverse channels are well defined, product preservation is a sufficient but not necessary condition for tabletop reversibility; and that the preservation of local energy spectra is a necessary and sufficient condition to generalized thermal operations.

The Gottesman-Kitaev-Preskill (GKP) error-correcting code encodes a finite-dimensional logical space in one or more bosonic modes, and has recently been demonstrated in trapped ions and superconducting microwave cavities. In this work we introduce a new subsystem decomposition for GKP codes that we call the stabilizer subsystem decomposition, analogous to the usual approach to quantum stabilizer codes. The decomposition has the defining property that a partial trace over the nonlogical stabilizer subsystem is equivalent to an ideal decoding of the logical state, distinguishing it from previous GKP subsystem decompositions. We describe how to decompose arbitrary states across the subsystem decomposition using a set of transformations that move between the decompositions of different GKP codes. Besides providing a convenient theoretical view on GKP codes, such a decomposition is also of practical use. We use the stabilizer subsystem decomposition to efficiently simulate noise acting on single-mode GKP codes, and in contrast to more conventional Fock basis simulations, we are able to consider essentially arbitrarily large photon numbers for realistic noise channels, such as loss and dephasing.

Surface acoustic waves (SAWs) are a versatile tool for coherently interfacing with a variety of solid-state quantum systems spanning microwave to optical frequencies, including superconducting qubits, spins, and quantum emitters. Here, we demonstrate SAW cavity optomechanics with quantum emitters in two-dimensional (2D) materials, specifically monolayer ${\mathrm{WSe}}_2$ and $h$-BN, on a planar lithium niobate SAW resonator driven by superconducting electronics. Using steady-state photoluminescence spectroscopy and time-resolved single-photon counting, we map the temporal dynamics of modulated 2D emitters under coupling to different SAW cavity modes, showing energy-level splitting consistent with deformation potential coupling of $35\mathrm{meV}/\mathrm{\%}$ for ${\mathrm{WSe}}_2$ and 12.5 meV/% for $h$-BN visible-light emitters. We leverage the large anisotropic strain from the SAW to modulate the excitonic fine-structure splitting in ${\mathrm{WSe}}_2$ on a nanosecond timescale, which may find applications for on-demand entangled-photon-pair generation from 2D materials. Cavity optomechanics with SAWs and 2D quantum emitters provide opportunities for compact sensors and quantum electro-optomechanics in a multifunctional integrated platform that combines phononic, optical, and superconducting electronic quantum systems.

Whether or not the Kronecker coefficients of the symmetric group count some set of combinatorial objects is a longstanding open question. In this work we show that a given Kronecker coefficient is proportional to the rank of a projector that can be measured efficiently using a quantum computer. In other words a Kronecker coefficient counts the dimension of the vector space spanned by the accepting witnesses of a $\mathrm{QMA}$ verifier, where $\mathrm{QMA}$ is the quantum analogue of $\mathrm{NP}$. This implies that approximating the Kronecker coefficients to within a given relative error is not harder than a certain natural class of quantum approximate counting problems that captures the complexity of estimating thermal properties of quantum many-body systems. A second consequence is that deciding positivity of Kronecker coefficients is contained in $\mathrm{QMA}$, complementing a recent $\mathrm{NP}$-hardness result of Ikenmeyer, Mulmuley, and Walter. We obtain similar results for the related problem of approximating row sums of the character table of the symmetric group. Finally, we discuss an efficient quantum algorithm that approximates normalized Kronecker coefficients to inverse-polynomial additive error.

The demanding experimental access to the ultrafast dynamics of materials challenges our understanding of their electronic response to applied strong laser fields. For this purpose, trapped ultracold atoms with highly controllable potentials have become an enabling tool to describe phenomena in a scenario in which some effects are more easily accessible and 12 orders of magnitude slower. In this work, we introduce a mapping between the parameters of attoscience platforms and atomic cloud simulators and propose an experimental protocol to access the emission spectrum of high-harmonic generation, a regime that has so far been elusive to cold-atom simulation. As we illustrate, the benchmark offered by these simulators can provide new insights into the conversion efficiency of extended and short nuclear potentials, as well as the response to applied elliptical polarized fields or ultrashort few-cycle pulses.

The photon blockade breakdown in a continuously driven cavity QED system has been proposed as a prime example for a first-order driven-dissipative quantum phase transition. However, the predicted scaling from a microscopic behavior—dominated by quantum fluctuations—to a macroscopic one—characterized by stable phases—and the associated exponents and phase diagram have not been observed so far. In this work we couple a single transmon qubit with a fixed coupling strength $g$ to a superconducting cavity that is in situ bandwidth $\kappa$ tunable to controllably approach this thermodynamic limit. Even though the system remains microscopic, we observe its behavior becoming increasingly macroscopic as a function of $g/\kappa$. For the highest realized $g/\kappa$ of approximately $287$, the system switches with a characteristic timescale as long as 6 s between a bright coherent state with approximately $8\times {10}^3$ intracavity photons and the vacuum state. This exceeds the microscopic timescales by 6 orders of magnitude and approaches the perfect hysteresis expected between two macroscopic attractors in the thermodynamic limit. These findings and interpretation are qualitatively supported by neoclassical theory and large-scale quantum-jump Monte Carlo simulations. Besides shedding more light on driven-dissipative physics in the limit of strong light-matter coupling, this system might also find applications in quantum sensing and metrology.

Qubit control electronics composed of CMOS circuits are of critical interest for next-generation quantum computing systems. A CMOS-based application-specific integrated circuit (ASIC) fabricated in 14-nm fin field-effect transistor (FinFET) technology was used to generate and sequence qubit control wave forms and demonstrate a two-qubit cross-resonance gate between fixed-frequency transmons. The controller was thermally anchored to the $T=4$ K stage of a dilution refrigerator and the measured power was 23 mW per qubit under active control. The chip generated single-side banded output frequencies between 4.5 and 5.5 GHz, with a maximum power output of $-18$ dBm. Randomized-benchmarking (RB) experiments revealed an average number of 1.71 instructions per Clifford (IPC) for single-qubit gates and 17.51 IPC for two-qubit gates. A single-qubit error per gate of ${ϵ}_{\text{1Q}}=8\times {10}^{-4}$ and a two-qubit error per gate of ${ϵ}_{\text{2Q}}=1.4\times {10}^{-2}$ were shown. A drive-induced $Z$ rotation was observed by way of a rotary-echo experiment; this observation is consistent with the expected qubit behavior given the measured excess local-oscillator (LO) leakage from the CMOS chip. The effect of spurious drive-induced $Z$ errors was numerically evaluated with a two-qubit model Hamiltonian and shown to be in good agreement with the measured RB data. The modeling results suggest that the $Z$ error varies linearly with the pulse amplitude.

Over the last two decades, integrated photonics has profoundly revolutionized the domain of quantum technologies. The ongoing second quantum revolution stands as a timely opportunity for a state-of-the-art review and, most important, an exploration of the directions undertaken by integrated quantum photonics. Within this perspective, based on the recent advances, we discuss the current challenges and future trends related to different technological platforms. Key examples will be considered across various subfields, including quantum communication, quantum metrology, and quantum memories. Our discussion encompasses disruptive concepts, progress, and potential limitations. The main objective of this Perspective is to provide the reader with a forward-looking discussion ranging from state-of-the-art developments to open challenges of the field.

The field of indefinite causal order (ICO) has seen a recent surge in interest. Much of this research has focused on the quantum switch, wherein multiple parties act in a superposition of different orders in a manner transcending the quantum circuit model. This results in a new resource for quantum protocols, and is exciting for its relation to issues in foundational physics. The quantum switch is also an example of a higher-order quantum operation, in that it transforms not only quantum states but also other quantum operations. To date, no quantum process without a definite causal order has been completely experimentally characterized. Indeed, past work on the quantum switch has confirmed its ICO by measuring causal witnesses or demonstrating resource advantages, but the complete process matrix has been described only theoretically. Here we report our performing higher-order quantum process tomography. However, doing so requires exponentially many measurements with a scaling worse than that of standard process tomography. We overcome this challenge by creating a new passively stable fiber-based quantum switch using active optical elements to deterministically generate and manipulate time-bin encoded qubits. Moreover, our new architecture for the quantum switch can be readily scaled to multiple parties. By reconstructing the process matrix, we estimate its fidelity and tailor different causal witnesses directly for our experiment. To achieve this, we measure a set of tomographically complete settings, which also spans the input operation space. Our tomography protocol allows the characterization and debugging of higher-order quantum operations with and without an ICO, while our experimental time-bin techniques could enable the creation of a new realm of higher-order quantum operations with an ICO.

Classical shadows enable us to learn many properties of a quantum state $\rho$ with very few measurements. However, near-term and early fault-tolerant quantum computers will only be able to prepare noisy quantum states $\rho$ and it is thus a considerable challenge to efficiently learn properties of an ideal, noise-free state ${\rho }_{\mathrm{id}}$. We consider error mitigation techniques, such as probabilistic error cancelation (PEC), zero noise extrapolation (ZNE), and symmetry verification (SV), which have been developed for mitigating errors in single expected value measurements and generalize them for mitigating errors in classical shadows. We find that PEC is the most natural candidate and thus develop a thorough theoretical framework for PEC shadows with the following rigorous theoretical guarantees: PEC shadows are an unbiased estimator for the ideal quantum state ${\rho }_{\mathrm{id}}$; the sample complexity for simultaneously predicting many linear properties of ${\rho }_{\mathrm{id}}$ is identical to that of the conventional shadows approach up to a multiplicative factor, which is the sample overhead due to error mitigation. Due to efficient postprocessing of shadows, this overhead does not depend directly on the number of qubits but rather grows exponentially with the number of noisy gates. The broad set of tools introduced in this work may be instrumental in exploiting near-term and early fault-tolerant quantum computers: we demonstrate in detailed numerical simulations a range of practical applications of quantum computers that will significantly benefit from our techniques.

The possibility of engineering artificial Kitaev chains in arrays of quantum dots coupled via narrow superconducting regions has emerged as an attractive way to overcome the disorder issues that complicate the realization and detection of topological superconducting phases in other platforms. Although a true topological phase would require long chains, a two-site chain realized in a double quantum dot can already be tuned to points in parameter space where it hosts zero-energy states that seem identical to the Majorana bound states that characterize a topological phase. These states have been named “poor man’s Majorana bound states” (PMMs) because they lack formal topological protection. In this work, we propose a pathway for next-generation experiments on PMMs. The pathway starts with experiments to characterize a single pair of PMMs by measuring the Majorana quality and then moves on to initialization and readout of the parity of a PMM pair, which allows the measurement of quasiparticle poisoning times. The next step is to couple two PMM systems to form a qubit. We discuss measurements of the coherence time of such a qubit, as well as a test of Majorana fusion rules in the same setup. Finally, we propose and analyze three different types of braidinglike experiments that require more complex device geometries. Our conclusions are supported by calculations based on a realistic model with interacting and spinful quantum dots, as well as by simpler models to gain physical insight. Our calculations show that it is indeed possible to demonstrate non-Abelian physics in minimal two-site Kitaev chains despite the lack of a true topological phase. However, our findings also reveal that doing so requires some extra care, appropriately modified protocols, and awareness of the details of this particular platform.

A quantum impulse is a brief but strong perturbation that produces a sudden change in a wave function $\psi (\mathbf{x})$. We develop a theory of quantum impulses, distinguishing between ordinary and super impulses. An ordinary impulse paints a phase onto $\psi$, while a super impulse—the main focus of this paper—deforms the wave function under an invertible map, $\mu :\mathbf{x}\to {\mathbf{x}}^{\mathrm{\prime }}$. Borrowing tools from optimal-mass-transport theory and shortcuts to adiabaticity, we show how to design a super impulse that deforms a wave function under a desired map $\mu$ and we illustrate our results using solvable examples. We point out a strong connection between quantum and classical super impulses, expressed in terms of the path-integral formulation of quantum mechanics. We briefly discuss hybrid impulses, in which ordinary and super impulses are applied simultaneously. While our central results are derived for evolution under the time-dependent Schrödinger equation, they apply equally well to the time-dependent Gross-Pitaevskii equation and thus may be relevant for the manipulation of Bose-Einstein condensates.

Electronic spin defects in the environment of an optically active spin can be used to increase the size and hence the performance of solid-state quantum registers, especially for applications in quantum metrology and quantum communication. Previous works on multiqubit electronic spin registers in the environment of a nitrogen-vacancy (NV) center in diamond have only included spins directly coupled to the NV. As this direct coupling is limited by the central spin coherence time, it significantly restricts the maximum attainable size of the register. To address this problem, we present a scalable approach to increase the size of electronic spin registers. Our approach exploits a weakly coupled probe spin together with double-resonance control sequences to mediate the transfer of spin polarization between the central NV spin and an environmental spin that is not directly coupled to it. We experimentally realize this approach to demonstrate the detection and coherent control of an unknown electronic spin outside the coherence limit of a central NV. Our work paves the way for engineering larger quantum spin registers with the potential to advance nanoscale sensing, enable correlated noise spectroscopy for error correction, and facilitate the realization of spin-chain quantum wires for quantum communication.

Transmon qubits experience open-system effects that manifest as noise at a broad range of frequencies. We present a model of these effects using the Redfield master equation with a hybrid bath consisting of low- and high-frequency components. We use two-level fluctuators to simulate $1/f$-like noise behavior, which is a dominant source of decoherence for superconducting qubits. By measuring quantum state fidelity under free evolution with and without dynamical decoupling (DD), we can fit the low- and high-frequency noise parameters in our model. We train and test our model using experiments on quantum devices available through IBM quantum experience. Our model accurately predicts the fidelity decay of random initial states, including the effect of DD pulse sequences. We compare our model with two simpler models and confirm the importance of including both high frequency and $1/f$ noise in order to accurately predict transmon behavior.