Related papers: Efficient Unitary T-designs from Random Sums
These lectures provide an informal introduction into the notions and tools used to analyze statistical properties of eigenvalues of large random Hermitian matrices. After developing the general machinery of orthogonal polynomial method, we…
We clarify the mathematical structure underlying unitary $t$-designs. These are sets of unitary matrices, evenly distributed in the sense that the average of any $t$-th order polynomial over the design equals the average over the entire…
A fundamental question is understanding the rate at which random quantum circuits converge to the Haar measure. One quantity which is important in establishing this rate is the spectral gap of a random quantum ensemble. In this work we…
Combinatorial optimization is of general interest for both theoretical study and real-world applications. Fast-developing quantum algorithms provide a different perspective on solving combinatorial optimization problems. In this paper, we…
In recent years simulations of chemistry and condensed materials has emerged as one of the preeminent applications of quantum computing, offering an exponential speedup for the solution of the electronic structure for certain strongly…
This work focuses on optimizing the gates of a quantum circuit with a given topology to approximate the unitary time evolution governed by a Hamiltonian. Recognizing that unitary matrices form a mathematical manifold, we employ Riemannian…
We present a general framework for the generation of random unitaries based on random quenches in atomic Hubbard and spin models, forming approximate unitary $n$-designs, and their application to the measurement of R\'enyi entropies. We…
Recently we introduced the hypergraph matrix model (HMM), a Hermitian matrix model generalizing the classical Gaussian Unitary Ensemble (GUE). In this model the Gaussians of the GUE, whose moments count partitions of finite sets into pairs,…
Quantum circuits for mathematical functions such as division are necessary to use quantum computers for scientific computing. Quantum circuits based on Clifford+T gates can easily be made fault-tolerant but the T gate is very costly to…
We prove a Carbery-Wright style anti-concentration inequality for the unitary Haar measure, by showing that the probability of a polynomial in the entries of a random unitary falling into an $\varepsilon$ range is at most a polynomial in…
One of the principal obstacles on the way to quantum computers is the lack of distinguished basis in the space of unitary evolutions and thus the lack of the commonly accepted set of basic operations (universal gates). A natural choice,…
Recent years have enjoyed a strong interest in exploring properties and applications of random quantum circuits. In this work, we explore the ensemble of $t$-doped Clifford circuits on $n$ qubits, consisting of Clifford circuits…
We construct and extensively study a Brownian generalization of the Gaussian Unitary Ensemble (BGUE). Our analysis begins with the non-equilibrium dynamics of BGUE, where we derive explicit analytical expressions for various one-replica and…
We give three new algorithms for efficient in-place estimation, without using ancilla qubits, of average fidelity of a quantum logic gate acting on a d-dimensional system using much fewer random bits than what was known so far. Previous…
Just how fast does the brickwork circuit form an approximate 2-design? Is there any difference between anticoncentration and being a 2-design? Does geometry matter? How deep a circuit will I need in practice? We tell you everything you…
While quantum state tomography is notoriously hard, most states hold little interest to practically-minded tomographers. Given that states and unitaries appearing in Nature are of bounded gate complexity, it is natural to ask if efficient…
We consider one dimensional quantum circuits of the brickwork type, where the fundamental quantum gate is dual unitary. Such models are solvable: the dynamical correlation functions of the infinite temperature ensemble can be computed…
We show how to efficiently simulate continuous-time quantum query algorithms that run in time T in a manner that preserves the query complexity (within a polylogarithmic factor) while also incurring a small overhead cost in the total number…
Unitary decomposition is a widely used method to map quantum algorithms to an arbitrary set of quantum gates. Efficient implementation of this decomposition allows for translation of bigger unitary gates into elementary quantum operations,…
Future quantum devices often rely on favourable scaling with respect to the system components. To achieve desirable scaling, it is therefore crucial to implement unitary transformations in an efficient manner. We develop an upper bound for…