Academic literature on the topic 'Random unitary circuits'

We study efficient generations of random diagonal-unitary matrices, an ensemble of unitary matrices diagonal in a given basis with randomly distributed phases for their eigenvalues. Despite the simple algebraic structure, they cannot be achieved by quantum circuits composed of a few-qubit diagonal gates. We introduce diagonal-unitaryt-designs and present two quantum circuits that implement diagonal-unitary 2-design with the computational basis in N-qubit systems. One is composed of single-qubit diagonal gates and controlled-phase gates with randomized phases, which achieves an exact diagonal-unitary 2-design after applying the gates on all pairs of qubits. The number of required gates is N(N - 1)/2. If the controlled-Z gates are used instead of the controlled-phase gates, the circuit cannot achieve an exact 2-design, but achieves an ϵ-approximate 2-design by applying gates on randomly selected pairs of qubits. Due to the random choice of pairs, the circuit obtains extra randomness and the required number of gates is at most O(N2(N + log 1/∊)). We also provide an application of the circuits, a protocol of generating an exact 2-design of random states by combining the circuits with a simple classical procedure requiring O(N) random classical bits.

The Hayden-Preskill protocol probes the capability of information recovery from local subsystems after unitary dynamics. As such it resolves the capability of quantum many-body systems to dynamically implement a quantum error-correcting code. The transition to coding behavior has been mostly discussed using effective approaches, such as entanglement membrane theory. Here, we present exact results on the use of Hayden-Preskill recovery as a dynamical probe of scrambling in local quantum many-body systems. We investigate certain classes of unitary circuit models, both structured Floquet (dual-unitary) and Haar-random circuits. We discuss different dynamical signatures corresponding to information transport or scrambling, respectively, that go beyond effective approaches. Surprisingly, certain chaotic circuits transport information with perfect fidelity. In integrable dual-unitary circuits, we relate the information transmission to the propagation and scattering of quasiparticles. Using numerical and analytical insights, we argue that the qualitative features of information recovery extend away from these solvable points. Our results suggest that information recovery protocols can serve to distinguish chaotic and integrable behavior, and that they are sensitive to characteristic dynamical features, such as long-lived quasiparticles or dual-unitarity.

Recent works have investigated the emergence of a new kind of random matrix behaviour in unitary dynamics following a quantum quench. Starting from a time-evolved state, an ensemble of pure states supported on a small subsystem can be generated by performing projective measurements on the remainder of the system, leading to a projected ensemble. In chaotic quantum systems it was conjectured that such projected ensembles become indistinguishable from the uniform Haar-random ensemble and lead to a quantum state design. Exact results were recently presented by Ho and Choi [Phys. Rev. Lett. 128, 060601 (2022)] for the kicked Ising model at the self-dual point. We provide an alternative construction that can be extended to general chaotic dual-unitary circuits with solvable initial states and measurements, highlighting the role of the underlying dual-unitarity and further showing how dual-unitary circuit models exhibit both exact solvability and random matrix behaviour. Building on results from biunitary connections, we show how complex Hadamard matrices and unitary error bases both lead to solvable measurement schemes.

The unitary dynamics of a quantum system initialized in a selected basis state yield, generically, a state that is a superposition of all the basis states. This process, associated with the quantum information scrambling and intimately tied to the resource theory of coherence, may be viewed as a gradual delocalization of the system’s state in the Hilbert space. This work analyzes the Hilbert space delocalization under the dynamics of random quantum circuits, which serve as a minimal model of the chaotic dynamics of quantum many-body systems. We employ analytical methods based on the replica trick and Weingarten calculus to investigate the time evolution of the participation entropies which quantify the Hilbert space delocalization. We demonstrate that the participation entropies approach, up to a fixed accuracy, their long-time saturation value in times that scale logarithmically with the system size. Exact numerical simulations and tensor network techniques corroborate our findings.

The applications of random quantum circuits range from quantum computing and quantum many-body systems to the physics of black holes. Many of these applications are related to the generation of quantum pseudorandomness: Random quantum circuits are known to approximate unitary t-designs. Unitary t-designs are probability distributions that mimic Haar randomness up to tth moments. In a seminal paper, Brandão, Harrow and Horodecki prove that random quantum circuits on qubits in a brickwork architecture of depth O(nt10.5) are approximate unitary t-designs. In this work, we revisit this argument, which lower bounds the spectral gap of moment operators for local random quantum circuits by Ω(n−1t−9.5). We improve this lower bound to Ω(n−1t−4−o(1)), where the o(1) term goes to 0 as t→∞. A direct consequence of this scaling is that random quantum circuits generate approximate unitary t-designs in depth O(nt5+o(1)). Our techniques involve Gao's quantum union bound and the unreasonable effectiveness of the Clifford group. As an auxiliary result, we prove fast convergence to the Haar measure for random Clifford unitaries interleaved with Haar random single qubit unitaries.

A random quantum circuit is a minimally structured model to study entanglement dynamics of many-body quantum systems. We consider a one-dimensional quantum circuit with noisy Haar-random unitary gates using density matrix operator and tensor contraction methods. It is shown that the entanglement evolution of the random quantum circuits is properly characterized by the logarithmic entanglement negativity. By performing exact numerical calculations, we find that, as the physical error rate is decreased below a critical value p c ≈ 0.056, the logarithmic entanglement negativity changes from the area law to the volume law, giving rise to an entanglement transition. The critical exponent of the correlation length can be determined from the finite-size scaling analysis, revealing the universal dynamic property of the noisy intermediate-scale quantum devices.

One of the main milestones in quantum information science is to realise quantum devices that exhibit an exponential computational advantage over classical ones without being universal quantum computers, a state of affairs dubbed quantum speedup, or sometimes "quantum computational supremacy". The known schemes heavily rely on mathematical assumptions that are plausible but unproven, prominently results on anticoncentration of random prescriptions. In this work, we aim at closing the gap by proving two anticoncentration theorems and accompanying hardness results, one for circuit-based schemes, the other for quantum quench-type schemes for quantum simulations. Compared to the few other known such results, these results give rise to a number of comparably simple, physically meaningful and resource-economical schemes showing a quantum speedup in one and two spatial dimensions. At the heart of the analysis are tools of unitary designs and random circuits that allow us to conclude that universal random circuits anticoncentrate as well as an embedding of known circuit-based schemes in a 2D translation-invariant architecture.

A unitary 2-design can be viewed as a quantum analogue of a 2-universal hash function: it is indistinguishable from a truly random unitary by any procedure that queries it twice. We show that exact unitary 2-designs on n qubits can be implemented by quantum circuits consisting of Oe(n) elementary gates in logarithmic depth. This is essentially a quadratic improvement in size (and in width times depth) over all previous implementations that are exact or approximate (for sufficiently strong approximations).

S.Cette thèse étudie l'émergence du chaos dans la dynamique des systèmes à plusieurs corps, tant classiques que quantiques, à travers trois études interconnectées, aboutissant à plusieurs résultats novateurs. La recherche explore initialement les corrélations dans les circuits symplectiques duaux, fournissant une analyse approfondie des flux hamiltoniens et des systèmes symplectiques. Une contribution significative est l'introduction du modèle Ising-Swap au sein des circuits symplectiques classiques duaux, qui révèle des corrélations dynamiques en utilisant des portes symplectiques et duaux-symplectiques. Une méthode générale est proposée, permettant le calcul exact des fonctions de corrélation dynamique à deux points, qui ne s'annulent que le long des bords des cônes de lumière. Ces résultats sont validés par des simulations de Monte Carlo, montrant une excellente concordance avec les prédictions théoriques pour divers observables.L'étude suivante aborde le chaos et les conceptions unitaires, commençant par un examen des conceptions unitaires, des k-conceptions et de la mesure de Haar, et progressant vers la distribution de Porter-Thomas. Cette recherche fait progresser la compréhension des distributions universelles des chevauchements issus de la dynamique unitaire en utilisant des modèles tels que les circuits en brique et le modèle de phase aléatoire. Notamment, l'étude parvient à la diagonalisation des matrices de Toeplitz généralisées et analyse leur spectre, ce qui permet un calcul exact du potentiel de cadre, essentiel pour comprendre l'universalité de notre théorie.La dernière partie de la thèse se concentre sur la dynamique universelle hors équilibre des systèmes quantiques critiques, en utilisant la théorie des champs conformes (CFT) pour étudier les champs et les fonctions de corrélation. L'étude traite de la dynamique hors équilibre des systèmes quantiques perturbés par le bruit couplé à l'énergie. Les résultats clés incluent des analyses détaillées des corrélations à deux points, des distributions de l'entropie d'intrication et des fluctuations de la densité d'énergie, qui sont montrées être directement liées à un ensemble d'équations différentielles stochastiques (SDE). Il est démontré que l'on peut étudier ces SDE et prouver analytiquement l'existence de distributions stationnaires non triviales avec des queues de −3/2. La validation de ces résultats avec un modèle de fermions libres souligne l'universalité et la robustesse du cadre théorique présenté.Dans l'ensemble, cette thèse intègre des modèles théoriques et des cadres mathématiques pour améliorer la compréhension du chaos dans les systèmes classiques et quantiques. En reliant les résultats des circuits symplectiques, des conceptions unitaires et de la dynamique hors équilibre, elle offre un récit complet qui met en lumière les caractéristiques universelles du comportement chaotique dans la dynamique des systèmes à plusieurs corps
This thesis investigates the emergence of chaos in classical and quantum many-body dynamics through three interconnected studies, yielding several novel results.The research initially explores correlations in dual symplectic circuits, providinga thorough analysis of Hamiltonian flows and symplectic systems. A significantcontribution is the introduction of the Ising-Swap model within dual symplecticclassical circuits, which reveals dynamical correlations using symplectic and dual-symplectic gates. A general method is proposed, which enables the exact compu-tation of two-point dynamical correlation functions, which are shown to be non-vanishing only along the edges of light cones. These findings are validated throughMonte Carlo simulations, displaying excellent agreement with theoretical predic-tions for various observables.The subsequent study addresses chaos and unitary designs, starting with an ex-amination of unitary designs, k-designs, and the Haar measure, progressing to thePorter-Thomas distribution. This research advances the understanding of universaldistributions of overlaps from unitary dynamics by employing models like brick-wall circuits and the Random Phase Model. Notably, the study achieves the di-agonalization of generalized Toeplitz matrices and analyses their spectrum, whichprovides an exact calculation of the Frame Potential, which is essential for under-standing the universality of our theory.The final segment of the thesis focuses on universal out-of-equilibrium dynam-ics of critical quantum systems, utilizing conformal field theory (CFT) to investi-gate fields and correlation functions. The study addresses the out-of-equilibriumdynamics of quantum systems perturbed by noise coupled to energy. Key resultsinclude detailed analyses of two-point correlations, entanglement entropy distribu-tions, and energy density fluctuations, which are shown to be directly related to aset of stochastic differential equation(SDEs). It is shown, that one can study theseSDEs, and analytically prove, the existence of non-trivial stationary distributionswith −3/2 tails. Benchmarking these findings with a free fermion model under-scores the universality and robustness of the presented theoretical framework.Overall, this thesis integrates theoretical models and mathematical frameworksto enhance the understanding of chaos in both classical and quantum systems. Bylinking results from symplectic circuits, unitary designs, and out-of-equilibrium dy-namics, it offers a comprehensive narrative that underscores the universal charac-teristics of chaotic behaviour in many-body dynamics

The effect of dead space on the mean gain and the excess noise factor of double-carrier multiplication avalanche photodiodes has been studied using recurrence relations in the form of coupled integral equations. The dead space is the minimum distance that a newly generated carrier must travel to acquire sufficient energy to become capable of causing an impact ionization. These equations are solved numerically to produce the mean gain and the excess noise factor. We have found that dead space reduces the mean gain since it results in fewer ionizations. The reduction is relatively greater as the hole-to-electron ionization ratio k approaches unity since the growth rate of the branching process is reduced by the inhibiting effect of dead space. We also show that dead space causes a lower excess noise factor since it introduces some orderliness in the random ionization process. In certain conditions the dead space has a beneficial effect on the performance of the optical detector when used in optical receivers with circuit noise. It may therefore be advantageous to select materials for which the dead space is enhanced, without jeopardizing other parameters such as large electron ionization coefficient and small k.