We determine the computational power of isometric tensor-network states (isoTNSs), a variational ansatz originally developed to numerically find and compute properties of gapped ground states and topological states in two dimensions. By mapping two-dimensional isoTNSs to ($1+1$)D unitary quantum circuits, we find that the computation of local expectation values in isoTNSs (including those with translation-invariant bulk) is BQP-complete. We then introduce injective isoTNSs, which are those isoTNSs that are the unique ground states of frustration-free Hamiltonians and which are characterized by an injectivity parameter $\delta \in (0,1/D]$, where $D$ is the bond dimension of the isoTNS. We show that injectivity necessarily adds depolarizing noise to the circuit at a rate $\eta ={\delta }^2{D}^2$. We show that weakly injective isoTNS (small $\delta$) are still BQP-complete but that there exists an efficient classical algorithm to compute local expectation values in strongly injective isoTNSs ($\eta \ge 0.41$). We also show that while weakly injective isoTNSs can support long-range correlations, strongly injective isoTNSs are approximate Markov states with exponentially decaying correlations. Sampling from isoTNSs corresponds to monitored quantum dynamics and we exhibit a family of isoTNSs that undergo a phase transition from a hard regime to an easy phase in which the monitored circuit can be sampled efficiently. Our results can be used to design provable algorithms to contract isoTNSs. Our mapping between ground states of certain frustration-free Hamiltonians to open circuit dynamics in one dimension fewer may be of independent interest.

The ability to sense and control nuclear spins near solid-state defects might enable a range of quantum technologies. Dynamically decoupled radio-frequency (DDrf) control offers a high degree of design flexibility and long electron-spin coherence times. However, previous studies have considered simplified models and little is known about optimal gate design and fundamental limits. Here, we develop a generalized DDrf framework that has important implications for spin sensing and control. Our analytical model, which we corroborate by experiments on a single NV center in diamond, reveals the mechanisms that govern the selectivity of gates and their effective Rabi frequencies, and enables flexible detuned gate designs. We apply these insights to numerically show a $60\times$ sensitivity enhancement for detecting weakly coupled spins and study the optimization of quantum gates in multiqubit registers. These results advance the understanding for a broad class of gates and provide a toolbox for application-specific design, enabling improved quantum control and sensing.

In quantum mechanics, measuring the expectation value of a general observable has an inherent statistical uncertainty that is quantified by variance or mean squared error of measurement outcome. While the uncertainty can be reduced by averaging several samples, the number of samples should be minimized when each sample is very costly. This is especially the case for fault-tolerant quantum computing that involves measurement of multiple observables of nontrivial states in large quantum systems that exceed the capabilities of classical computers. In this work, we provide an adaptive quantum algorithm for estimating the expectation values of $M$ general observables within root mean squared error $\epsilon$ simultaneously, using $\mathcal{O}({\epsilon }^{-1}\sqrt{M}\log M)$ queries to a state preparation oracle of a target state. This remarkably achieves the scaling of Heisenberg limit $1/\epsilon$, a fundamental bound on the estimation precision in terms of mean squared error, together with the sublinear scaling of the number of observables $M$. The proposed method is an adaptive version of the quantum gradient-estimation algorithm and has a resource-efficient implementation due to its adaptiveness. Specifically, the space overhead in the proposed method is $\mathcal{O}(M)$, which is independent from the estimation precision $\epsilon$ unlike noniterative algorithms. In addition, our method can avoid the numerical instability problem for constructing quantum circuits in a large-scale task (e.g., $\epsilon \ll 1$ in our case), which appears in the actual implementation of many algorithms relying on quantum signal processing techniques. Our method paves a new way to precisely understand and predict various physical properties in complicated quantum systems using quantum computers.

The design and benchmarking of quantum computer architectures traditionally rely on practical hardware restrictions, such as gate fidelities, control, and cooling. At the theoretical and software levels, numerous approaches have been proposed for benchmarking quantum devices, ranging from, inter alia, quantum volume to randomized benchmarking. In this work, we utilize the quantum information-theoretic properties of multipartite maximally entangled quantum states, in addition to their correspondence with quantum error-correction codes, permitting us to quantify the entanglement generated on near-term bilinear spin-qubit architectures. For this aim, we introduce four metrics that ascertain the quality of genuine multipartite quantum entanglement, along with circuit-level fidelity measures. As part of the task of executing a quantum circuit on a device, we devise simulations, which combine expected hardware characteristics of spin-qubit devices with appropriate compilation techniques; we then analyze three different architectural choices of varying lattice sizes for bilinear arrays, under three increasingly realistic noise models. We find that if the use of a compiler is assumed, sparsely connected spin-qubit lattices can approach comparable values of our metrics to those of the most highly connected device architecture. Even more surprisingly, by incorporating crosstalk into our last noise model, we find that, as error rates for crosstalk approach realistic values, the benefits of utilizing a bilinear array with advanced connectivity vanish. Our results highlight the limitations of adding local connectivity to near-term spin-qubit devices, and can be readily adapted to other qubit technologies. The framework developed here can be used for analyzing quantum entanglement on a device before fabrication, informing experimentalists on concomitant realistic expectations.

Quantum adiabatic optimization seeks to solve combinatorial problems using quantum dynamics, requiring the Hamiltonian of the system to align with the problem of interest. However, these Hamiltonians are often incompatible with the native constraints of quantum hardware, necessitating encoding strategies to map the original problem into a hardware-conformant form. While the classical overhead associated with such mappings is easily quantifiable and typically polynomial in problem size, it is much harder to quantify their overhead on the quantum algorithm, e.g., the transformation of the adiabatic timescale. In this work, we address this challenge on the concrete example of the encoding scheme proposed in [Nguyen et al., PRX Quantum 4, 010316 (2023)], which is designed to map optimization problems on arbitrarily connected graphs into Rydberg atom arrays. We consider the fundamental building blocks underlying this encoding scheme and determine the scaling of the minimum gap with system size along adiabatic protocols. Even when the original problem is trivially solvable, we find that the encoded problem can exhibit an exponentially closing minimum gap. We show that this originates from a quantum coherent effect, which gives rise to an unfavorable localization of the ground-state wave function. On the QuEra Aquila neutral atom machine, we observe such localization and its effect on the success probability of finding the correct solution to the encoded optimization problem. Finally, we propose quantum-aware modifications of the encoding scheme that avoid this quantum bottleneck and lead to an exponential improvement in the adiabatic performance. This highlights the crucial importance of accounting for quantum effects when designing strategies to encode classical problems onto quantum platforms.

Randomized benchmarking (RB) is the most commonly employed protocol for the characterization of unitary operations in quantum circuits due to its reasonable experimental requirements and robustness against state preparation and measurement (SPAM) errors. So far, the protocol has been limited to discrete or fermionic systems, whereas extensions to bosonic systems have been unclear for a long time due to challenges arising from the underlying bosonic Hilbert space. In this work, we close the gap for bosonic systems and develop an RB protocol to benchmark passive Gaussian transformations on any particle-number subspace, which we call passive bosonic RB. The protocol is built on top of the recently developed filtered RB framework [J. Helsen et al., PRX Quantum 3, 020357 (2022), M. Heinrich et al., Randomized benchmarking with random quantum circuits, arxiv:2212.06181 [quant-ph]] and is designed to isolate the multitude of exponential decays arising for passive bosonic transformations. We give explicit formulas and a Julia implementation for the necessary postprocessing of the experimental data. We also analyze the sampling complexity of passive bosonic RB by deriving analytical expressions for the variance. They show a mild scaling with the number of modes, suggesting that passive bosonic RB is experimentally feasible for a moderate number of modes. We focus on experimental settings involving Fock states and particle-number-resolving measurements, but also discuss Gaussian settings, deriving the first results for heterodyne measurements.

Fusion-based quantum computation is a promising quantum computing model where small-sized photonic resource states are simultaneously entangled and measured by fusion gates. Such operations can be readily implemented with scalable photonic hardware: resource states can be deterministically generated by quantum emitters and fusions require only shallow linear-optical circuits. Here, we propose fusion-based architectures tailored to the capabilities and noise models in quantum emitters. We show that high tolerance to dominant physical error mechanisms can be achieved, with fault-tolerance thresholds of 8% for photon loss, 4% for photon distinguishability between emitters, and spin noise thresholds well above memory-induced errors for typical spin-photon interfaces. Our construction and analysis provide guidelines for the development of photonic quantum hardware targeting fault-tolerant applications with quantum emitters.

The driven Dicke model, wherein an ensemble of atoms is driven by an external field and undergoes collective spontaneous emission due to coupling to a leaky cavity mode, is a paradigmatic example of a system exhibiting a driven-dissipative phase transition as a function of driving strength. Recently, a similar phenomenon was experimentally observed, not in a cavity setting, but rather in a free-space atomic ensemble. The reason why similar behavior should emerge in free space is not obvious, as the system interacts with a continuum of optical modes, which encodes light-propagation effects. Here, we present and solve a simple model to explain the behavior of the free-space system, based on the one-dimensional Maxwell-Bloch equations. On one hand, we show that a free-space ensemble at a low optical depth can exhibit similar behavior as the cavity system, as spatial propagation effects are negligible. On the other hand, in the thermodynamic limit of large atom number, we show that certain observables such as the transmittance or the atomic excited population exhibit nonanalytic behavior as a function of the driving intensity, reminiscent of a phase transition. However, a closer analysis reveals that the atomic properties are highly inhomogeneous in space, and based on this we argue that the free-space system does not undergo a phase transition but rather a “phase separation,” roughly speaking, between saturated and unsaturated regions.

We study real-time operator evolution using sparse Pauli dynamics, a recently developed method for simulating expectation values of quantum circuits. On the examples of energy and charge diffusion in one-dimensional (1D) spin chains and sudden quench dynamics in the 2D transverse-field Ising model, it is shown that this approach can compete with state-of-the-art tensor network methods. We further demonstrate the flexibility of the approach by studying quench dynamics in the 3D transverse-field Ising model that is highly challenging for tensor network methods. For the simulation of expectation value dynamics starting in a computational basis state, we introduce an extension of sparse Pauli dynamics that truncates the growing sum of Pauli operators by discarding terms with a large number of $X$ and $Y$ matrices. This is validated by our 2D and 3D simulations. Finally, we argue that sparse Pauli dynamics is not only capable of converging challenging observables to high accuracy, but can also serve as a reliable approximate approach even when given only limited computational resources.

Quantum metrology, a cornerstone of quantum technologies, exploits entanglement and superposition to achieve higher precision than classical protocols in parameter-estimation tasks. When combined with critical phenomena such as phase transitions, the divergence of quantum fluctuations is predicted to enhance the performance of quantum sensors. Here, we implement a critical quantum sensor using a superconducting parametric (i.e., two-photon driven) Kerr resonator. The sensor, a linear resonator terminated by a superconducting quantum interference device, operates near the critical point of a finite-component second-order dissipative phase transition obtained by scaling the system parameters. We analyze the performance of a frequency-estimation protocol and show that quadratic precision scaling with respect to the system size can be achieved with finite values of the Kerr nonlinearity. Since each photon emitted from the cavity carries more information about the parameter to be estimated compared to its classical counterpart, our protocol opens up perspectives for faster or more precise metrological protocols. Our results demonstrate that quantum advantage in a sensing protocol can be achieved by exploiting a finite-component phase transition.

Probing correlated states of many-body systems is one of the central tasks for quantum simulators and processors. A promising approach to state preparation is to realize desired correlated states as steady states of engineered dissipative evolution. A recent experiment with a Google superconducting quantum processor [X. Mi et al., Science 383, 1332 (2024)] demonstrated a cooling algorithm utilizing auxiliary degrees of freedom that are periodically reset to remove quasiparticles from the system, thereby driving it towards its ground state. In this work, we develop a kinetic-theory framework to describe quasiparticle cooling dynamics, and employ it to compare the efficiency of different cooling algorithms. In particular, we introduce a protocol where coupling to auxiliaries is modulated in time to minimize heating processes, and demonstrate that it allows a high-fidelity preparation of ground states in different quantum phases. We verify the validity of the kinetic-theory description by an extensive comparison with numerical simulations for the examples of a one-dimensional transverse-field Ising model, the transverse-field Ising model with an additional integrability-breaking field, and a nonintegrable antiferromagnetic Heisenberg spin ladder. In all cases, we are able to efficiently cool into the many-body ground state. The effects of noise, which limits efficiency of variational quantum algorithms in near-term quantum processors, are investigated through the lens of the kinetic theory: we show how the steady-state quasiparticle populations depend on the noise rate, and we establish maximum noise values for achieving high-fidelity ground states. This work establishes quasiparticle cooling algorithms as a practical, robust method for many-body state preparation on near-term quantum processors.

Analyzing and developing new quantum error-correcting (QEC) schemes is one of the most prominent tasks in quantum computing research. In such efforts, introducing time dynamics explicitly in both analysis and design of error-correcting protocols constitutes an important cornerstone. In this work, we present a graphical formalism based on tensor networks to capture the logical action and error-correcting capabilities of any Clifford circuit with Pauli measurements. We showcase the functioning of the formalism on new Floquet codes derived from topological subsystem codes, which we call XYZ ruby codes. Based on the projective symmetries of the building blocks of the tensor network we develop a framework of Pauli flows. Pauli flows allow for a graphical understanding of all quantities entering an error-correction analysis of a circuit, including different types of QEC experiments, such as memory and stability experiments. We lay out how to derive a well-defined decoding problem from the tensor-network representation of a protocol and its Pauli flows alone, independent of any stabilizer code or fixed circuit. Importantly, this framework applies to all Clifford protocols and encompasses both measurement-based and circuit-based approaches to fault tolerance. We apply our method to our new family of dynamical codes, which are in the same topological phase as the $2+1$-dimensional color code, making them a promising candidate for low-overhead logical gates. In contrast to its static counterpart, the dynamical protocol applies a ${\mathbb{Z}}_3$ automorphism to the logical Pauli group every three time steps. We highlight some of its topological properties and comment on the anyon physics behind a planar layout. Lastly, we benchmark the performance of the $\mathrm{XYZ}$ ruby code on a torus by performing both memory and stability experiments and find competitive circuit-level noise thresholds of approximately equal to $0.18\mathrm{\%}$, comparable with other Floquet codes and $2+1$-dimensional color codes.

Trotter and linear combination of unitary (LCU) operations are two popular Hamiltonian simulation methods. The Trotter method is easy to implement and enjoys good system-size dependence endowed by commutator scaling, while the LCU method admits high-accuracy simulation with a smaller gate cost. We propose Hamiltonian simulation algorithms using LCU to compensate Trotter error, which enjoy both of their advantages. By adding few gates after the $K\mathrm{th}$-order Trotter formula, we realize a better time scaling than $2K\mathrm{th}$-order Trotter. Our first algorithm exponentially improves the accuracy scaling of the $K\mathrm{th}$-order Trotter formula. For a generic Hamiltonian, the estimated gate counts of the first algorithm can be 2 orders of magnitude smaller than the best analytical bound of fourth-order Trotter formula. In the second algorithm, we consider the detailed structure of Hamiltonians and construct LCU for Trotter errors with commutator scaling. Consequently, for lattice Hamiltonians, the algorithm enjoys almost linear system-size dependence and quadratically improves the accuracy of the $K\mathrm{th}$-order Trotter. For the lattice system, the second algorithm can achieve 3 to 4 orders of magnitude higher accuracy with the same gate costs as the optimal Trotter algorithm. These algorithms provide an easy-to-implement approach to achieve a low-cost and high-precision Hamiltonian simulation.

We propose the convex-roof extension of quantum conditional mutual information (“co(QCMI)”) as a diagnostic of topological order in a mixed state. We focus primarily on topological states subjected to local decoherence, and employ the Levin-Wen scheme to define co(QCMI), so that for a pure state, co(QCMI) equals topological entanglement entropy (TEE). By construction, co(QCMI) is zero if and only if a mixed state can be decomposed as a convex sum of pure states with zero TEE. We show that co(QCMI) is nonincreasing with increasing decoherence when Kraus operators are proportional to the product of onsite unitaries. This implies that unlike a pure-state transition between a topologically trivial and a nontrivial phase, the long-range entanglement at a decoherence-induced topological phase transition as quantified by co(QCMI) is less than or equal to that in the proximate topological phase. For the two-dimensional toric code decohered by onsite bit- and phase-flip noise, we show that co(QCMI) is nonzero below the error-recovery threshold and zero above it. Relatedly, the decohered state cannot be written as a convex sum of short-range entangled pure states below the threshold. We conjecture and provide evidence that in this example, co(QCMI) equals TEE of a recently introduced pure state. In particular, we develop a tensor-assisted Monte Carlo (TMC) computation method to efficiently evaluate the Rényi TEE for the aforementioned pure state and provide nontrivial consistency checks for our conjecture. We use TMC to also calculate the universal scaling dimension of the anyon-condensation order parameter at this transition.

We show how to realize a general quantum circuit involving gates between arbitrary pairs of qubits by means of geometrically local quantum operations and efficient classical computation. We prove that circuit-level local stochastic noise modeling an imperfect implementation of our derived schemes is equivalent to local stochastic noise in the original circuit. Our constructions incur a constant-factor increase in the quantum circuit depth and a polynomial overhead in the number of qubits. To execute an arbitrary quantum circuit on $n$ qubits, we give a three-dimensional quantum fault-tolerance architecture involving $O(n^{3/2}{\log }^3n)$ qubits and a quasi-two-dimensional architecture using $O(n^2{\log }^3n)$ qubits. Applied to recent fault-tolerance constructions, this gives a fault-tolerance-threshold theorem for universal quantum computations with local operations, a polynomial qubit overhead, and a quasipolylogarithmic depth overhead. More generally, our transformation dispenses with the need for considering the locality of operations when designing schemes for fault-tolerant quantum information processing.

We study random constant-depth quantum circuits in a two-dimensional (2D) architecture. While these circuits only produce entanglement between nearby qubits on the lattice, long-range entanglement can be generated by measuring a subset of the qubits of the output state. It is conjectured that this long-range measurement-induced entanglement (MIE) proliferates when the circuit depth is at least a constant critical value $d^{\ast }$. For circuits composed of Haar-random two-qubit gates, it is also believed that this coincides with a quantum advantage phase transition in the classical hardness of sampling from the output distribution. Here, we provide evidence for a quantum advantage phase transition in the setting of random Clifford circuits. Our work extends the scope of recent separations between the computational power of constant-depth quantum and classical circuits, demonstrating that this kind of advantage is present in canonical random circuit sampling tasks. In particular, we show that in any architecture of random shallow Clifford circuits, the presence of long-range MIE gives rise to an unconditional quantum advantage. In contrast, any depth-$d$ 2D quantum circuit that satisfies a short-range MIE property can be classically simulated efficiently and with depth $O(d)$. Finally, we introduce a 2D depth-$2$ “coarse-grained” circuit architecture, composed of random Clifford gates acting on $O(\log (n))$ qubits, for which we prove the existence of long-range MIE and establish an unconditional quantum advantage.

We propose a quantum algorithm to simulate the dynamics in quantum chemistry problems. It is based on adding fresh qubits at each Trotter step, which enables a simpler implementation of the dynamics in the extended system. After each step, the extra qubits are recycled, so that the whole process accurately approximates the correct unitary evolution. A key ingredient of the approach is an isometry that maps a simple diagonal Hamiltonian in the extended system to the original one, and we give a procedure to compute this isometry. We estimate the error at each time step, as well as the number of gates, which scales as $O(N^2)$, where $N$ is the number of orbitals. We illustrate our results with three examples: the hydrogen chain, small molecules, and the FeMoco ($\mathrm{Fe}$${}_7$$\mathrm{Mo}\mathrm{S}$${}_9$$\mathrm{C}$) molecule. In the hydrogen chain and the hydrogen molecule, we observe that the error scales in the same way as the Trotter error. For FeMoco, we estimate the number of gates in a fault-tolerant setup.

Quantum error mitigation is a promising route to achieving quantum utility, and potentially quantum advantage in the near term. Many state-of-the-art error-mitigation schemes use knowledge of the errors in the quantum processor, which opens up the question of the extent to which inaccuracy in the error model impacts the performance of error mitigation. In this work, we develop a methodology to efficiently compute upper bounds on the impact of error-model inaccuracy in error mitigation. Our protocols require no additional experiments and instead rely on comparisons between the error model and the error-learning data from which the model is generated. We demonstrate the efficacy of our methodology by deploying it on an IBM Quantum superconducting qubit quantum processor, and through numerical simulation of standard error models. We show that our estimated upper bounds are typically close to the worst observed performance of error mitigation on random circuits. Our methodology can also be understood as an operationally meaningful metric to assess the quality of error models and we further extend our methodology to allow for comparison between error models. Finally, contrary to what one might expect, we show that observable error in noisy layered circuits of sufficient depth is not always maximized by a Clifford circuit, which may be of independent interest.

Circular Rydberg atoms (CRAs), i.e., Rydberg atoms with maximal orbital momentum, ideally combine long coherence times and strong interactions, a key property of quantum systems, in particular for the development of quantum technologies. However, the dipole-dipole interaction between CRAs has not been observed so far. We report the measurement and characterization of the resonant dipole-dipole interaction between two CRAs, individually trapped in optical tweezers, and find excellent agreement with theoretical predictions. We demonstrate a dynamic control over the strength of the interaction by tuning the orientation of an electric field. We use the interaction between the CRAs as a meter for the interatomic distance, and record the relative motion between two atoms in their traps. This motion, that we induce through the interaction between Rydberg levels with permanent electric dipoles, transiently populated during the preparation of the circular states, is a signature of spin-motion coupling.