Related papers: Sample-optimal classical shadows for pure states
We focus on analyzing the classical stochastic projected gradient methods under a general dependent data sampling scheme for constrained smooth nonconvex optimization. We show the worst-case rate of convergence $\tilde{O}(t^{-1/4})$ and…
In the quantum state tomography problem, one wishes to estimate an unknown $d$-dimensional mixed quantum state $\rho$, given few copies. We show that $O(d/\epsilon)$ copies suffice to obtain an estimate $\hat{\rho}$ that satisfies…
We present an optimal quantum algorithm for fidelity estimation between two quantum states when one of them is pure. In particular, the (square root) fidelity of a mixed state to a pure state can be estimated to within additive error…
Shadow estimation is a sample-efficient protocol for learning the properties of a quantum system using randomized measurements, but the current understanding of qudit shadow estimation is quite limited compared with the qubit setting. Here…
Estimating expectation values is a key subroutine in quantum algorithms. Near-term implementations face two major challenges: a limited number of samples required to learn a large collection of observables, and the accumulation of errors in…
We study the complexity of two closely related learning problems, one quantum and one classical. In the quantum setting, we consider agnostic tomography for the natural class of product mixed states. Given $N$ copies of an $n$-qubit state…
We introduce a method to enforce some symmetries starting from a trial wave-function prepared on quantum computers that might not respect these symmetries. The technique eliminates the necessity for performing the projection on the quantum…
We derive exceedingly simple practical procedures revealing the quantum nature of states and measurements by the violation of classical upper bounds on the statistics of arbitrary measurements. Data analysis is minimum and definite…
The optimization of measurements for the state distinction problem has recently been applied to the theory of quantum algorithms with considerable successes, including efficient new quantum algorithms for the non-abelian hidden subgroup…
There has been a surge of progress in recent years in developing algorithms for testing and learning quantum states that achieve optimal copy complexity. Unfortunately, they require the use of entangled measurements across many copies of…
We show that $n = \Omega(rd/\varepsilon^2)$ copies are necessary to learn a rank $r$ mixed state $\rho \in \mathbb{C}^{d \times d}$ up to error $\varepsilon$ in trace distance. This matches the upper bound of $n = O(rd/\varepsilon^2)$ from…
We study the classical complexity of the exact Boson Sampling problem where the objective is to produce provably correct random samples from a particular quantum mechanical distribution. The computational framework was proposed by Aaronson…
We consider the problem of heteroscedastic linear regression, where, given $n$ samples $(\mathbf{x}_i, y_i)$ from $y_i = \langle \mathbf{w}^{*}, \mathbf{x}_i \rangle + \epsilon_i \cdot \langle \mathbf{f}^{*}, \mathbf{x}_i \rangle$ with…
Quantum dynamical simulations of statistical ensembles pose a significant computational challenge due to the fact that mixed states need to be represented. If the underlying dynamics is fully unitary, for example in ultrafast coherent…
Measuring the distinguishability between quantum states is a basic problem in quantum information theory. In this paper, we develop optimal quantum algorithms that estimate both the trace distance and the (square root) fidelity between pure…
We present shadow spectroscopy as a simulator-agnostic quantum algorithm for estimating energy gaps using very few circuit repetitions (shots) and no extra resources (ancilla qubits) beyond performing time evolution and measurements. The…
We study classical shadows protocols based on randomized measurements in $n$-qubit entangled bases, generalizing the random Pauli measurement protocol ($n = 1$). We show that entangled measurements ($n\geq 2$) enable nontrivial and…
Classical shadows are a versatile tool to probe many-body quantum systems, consisting of a combination of randomised measurements and classical post-processing computations. In a recently introduced version of the protocol, the…
We investigate quantum algorithms for classification, a fundamental problem in machine learning, with provable guarantees. Given $n$ $d$-dimensional data points, the state-of-the-art (and optimal) classical algorithm for training…
Quantum machine learning has received significant attention in recent years, and promising progress has been made in the development of quantum algorithms to speed up traditional machine learning tasks. In this work, however, we focus on…