The preparation and control of quantum states lie at the heart of quantum information science. Recent advances in solid-state quantum emitters (QEs) and nanophotonics have transformed the landscape of quantum photonic technologies, enabling scalable generation of quantum states of light and matter. A new frontier in solid-state quantum photonics is the engineering of many-body interactions between QEs and photons to achieve robust coherence and controllable many-body entanglement. These entangled states, including photonic graph and cluster states, superradiant emission and emergent quantum phases, are promising for quantum computation, sensing and simulation. However, intrinsic inhomogeneities and decoherence in solid-state platforms pose considerable challenges in realizing such complex entangled states. This Review provides an overview of fundamental many-body interactions and dynamics at the light-matter interfaces of solid-state QEs and discusses recent advances in mitigating decoherence and harnessing robust many-body coherence.

Matrix-product unitaries (MPUs) are many-body unitary operators that, as a consequence of their tensor-network structure, preserve the entanglement area law in 1D systems. However, it is unknown how to implement an MPU as a quantum circuit since the individual tensors describing the MPU are not unitary. In this Letter, we show that a large class of MPUs can be implemented with a polynomial-depth quantum circuit. For an N-site MPU built from a repeated bulk tensor with open boundary, we explicitly construct a quantum circuit of polynomial depth T = O(N alpha) realizing the MPU, where the constant alpha depends only on the bulk and boundary tensor and not the system size N. We show that this class includes nontrivial unitaries that generate long-range entanglement and, in particular, contains a large class of unitaries constructed from representations of C*-weak Hopf algebras. Furthermore, we also adapt our construction to nonuniform translationally varying MPUs and show that they can be implemented by a circuit of depth O(N beta polyD) where beta <= 1 + log2 p /smin, with D being the bond dimension and smin the smallest ffiffiD nonzero Schmidt value of the normalized Choi state corresponding to the MPU.

Tensor network methods have proved to be highly effective in addressing a wide variety of physical scenarios, including those lacking an intrinsic one-dimensional geometry. In such contexts, it is possible for the problem to exhibit a weak form of permutational symmetry, in the sense that entanglement behaves similarly across any arbitrary bipartition. In this paper, we show that translationally-invariant (TI) matrix product states (MPSs) with this property are trivial, meaning that they are either product states or superpositions of a few of them. The results also apply to non-TI generic MPSs, as well as further relevant examples of MPSs including the W state and the Dicke states in an approximate sense. Our findings motivate the usage of Ans & auml,.tze simpler than tensor networks in systems whose structure is invariant under permutations.

We characterize the dynamical state of many-body bosonic and fermionic many-body models with intersite Gaussian couplings, on-site non-Gaussian interactions, and local dissipation comprising incoherent particle loss, particle gain, and dephasing. We first establish that, for fermionic systems, if the dephasing noise is larger than the non-Gaussian interactions, irrespective of the Gaussian coupling strength, the system state is a convex combination of Gaussian states at all times. Furthermore, for bosonic systems, we show that if the particle loss and particle gain rates are larger than the Gaussian intersite couplings, the system remains in a separable state at all times. Building on this characterization, we establish that at noise rates above a threshold, there exists a classical algorithm that can efficiently sample from the system state of both the fermionic and bosonic models. Finally, we show that, unlike fermionic systems, bosonic systems can evolve into states that are not convex Gaussian even when the dissipation is much higher than the on-site non-Gaussianity. Similarly, unlike bosonic systems, fermionic systems can generate entanglement even with noise rates much larger than the intersite couplings.

For random quantum circuits on n qubits of depth Theta(log n) with depolarizing noise, the task of sampling from the output state can be efficiently performed classically using a Pauli path method [1]. This paper aims to study the performance of this method beyond random circuits. We first consider the classical simulation of local observables in circuits composed of Clifford and T gates - going beyond the average-case analysis, we derive sufficient conditions for simulability in terms of the noise rate and the fraction of gates that are T gates, and show that if noise is introduced at a faster rate than T gates, the simulation becomes classically easy. As an application of this result, we study 2D QAOA circuits that attempt to find low-energy states of classical Ising models on general graphs. There, our results shows that for hard instances of the problem, which correspond to Ising model's graph being geometrically non-local, a QAOA algorithm mapped to a geometrically local circuit architecture using SWAP gates does not have any asymptotic advantage over classical algorithms if depolarized at a constant rate. Finally, we illustrate instances where the Pauli path method fails to give the correct result, and also initiate a study of the trade-off between fragility to noise and classical complexity of simulating a given quantum circuit.

Many-body open quantum systems, described by Lindbladian master equations, are a rich class of physical models that display complex equilibrium and out-of-equilibrium phenomena which remain to be understood. In this paper, we theoretically analyze noisy analog quantum simulation of geometrically local open quantum systems and provide evidence that this problem both is hard to simulate on classical computers and could be approximately solved on near-term quantum devices. First, given a noiseless quantum simulator, we show that the dynamics of local observables and the fixed-point expectation values of rapidly mixing local observables in geometrically local Lindbladians can be obtained to a precision of epsilon in time that is poly(epsilon-1) and uniform in system size. Furthermore, we establish that the quantum simulator would provide a superpolynomial advantage, in run-time scaling with respect to the target precision and either the evolution time (when simulating dynamics) or the Lindbladian's decay rate (when simulating fixed points), over any classical algorithm for these problems, assuming BQP # BPP. We then consider the presence of noise in the quantum simulator in the form of additional geometrically local Lindbladian terms. We show that the simulation tasks considered in this paper are stable to errors,. i.e., they can be solved to a noise-limited, but system-size independent, precision. Finally, we establish that, assuming BQP # BPP, there are stable geometrically local Lindbladian simulation problems such that, as the noise rate on the simulator is reduced, classical algorithms must take time superpolynomially longer in the inverse noise rate to attain the same precision as the analog quantum simulator.

Matrix-product unitaries (MPU) are 1D tensor networks describing time evolution and unitary symmetries of quantum systems, while their action on states by construction preserves the entanglement area law. MPU which are formed by a single repeated tensor are known to coincide with 1D quantum cellular automata (QCA), i.e., unitaries with an exact light cone. However, this correspondence breaks down for MPU with open boundary conditions, even if the resulting operator is translation-invariant. Such unitaries can turn short- to long-range correlations and thus alter the underlying phase of matter. Here we make the first steps towards a theory of MPU with uniform bulk but arbitrary boundary. In particular, we study the structure of a subclass with a directsum form which maximally violates the QCA property. We also consider the general case of MPU formed by site-dependent (nonuniform) tensors and show a correspondence between MPU and locally maximally entanglable states.

Several quantum hardware platforms, while being unable to perform fully fault-tolerant quantum computation, can still be operated as analogue quantum simulators for addressing many-body problems. However, due to the presence of errors, it is not clear to what extent those devices can provide us with an advantage with respect to classical computers. In this work, we make progress on this problem for noisy analogue quantum simulators computing physically relevant properties of many-body systems both in equilibrium and undergoing dynamics. We first formulate a system-size independent notion of stability against extensive errors, which we prove for Gaussian fermion models, as well as for a restricted class of spin systems. Remarkably, for the Gaussian fermion models, our analysis shows the stability of critical models which have long-range correlations. Furthermore, we analyze how this stability may lead to a quantum advantage, for the problem of computing the thermodynamic limit of many-body models, in the presence of a constant error rate and without any explicit error correction. Analogue quantum simulators have looser requirements than digital ones, but rigorous results on their usefulness in the noisy case are few. Here, the authors conclude that analogue quantum simulators are robust to errors and can provide superpolynomial to exponential quantum advantage when used to compute relevant many-body observables.