61,274 research outputs found
Use of Quantum Sampling to Calculate Mean Values of Observables and Partition Function of a Quantum System
We describe an algorithm for using a quantum computer to calculate mean
values of observables and the partition function of a quantum system. Our
algorithm includes two sub-algorithms. The first sub-algorithm is for
calculating, with polynomial efficiency, certain diagonal matrix elements of an
observable. This sub-algorithm is performed on a quantum computer, using
quantum phase estimation and tomography. The second sub-algorithm is for
sampling a probability distribution. This sub-algorithm is not polynomially
efficient. It can be performed either on a classical or a quantum computer, but
a quantum computer can perform it quadratically faster.Comment: V1-12 pages(5 files: 1.tex, 3 .sty, 1 .eps);V2-minor changes;V3-minor
changes and extension of scenario(c
Near-Optimal Quantum Algorithms for Multivariate Mean Estimation
We propose the first near-optimal quantum algorithm for estimating in
Euclidean norm the mean of a vector-valued random variable with finite mean and
covariance. Our result aims at extending the theory of multivariate
sub-Gaussian estimators to the quantum setting. Unlike classically, where any
univariate estimator can be turned into a multivariate estimator with at most a
logarithmic overhead in the dimension, no similar result can be proved in the
quantum setting. Indeed, Heinrich ruled out the existence of a quantum
advantage for the mean estimation problem when the sample complexity is smaller
than the dimension. Our main result is to show that, outside this low-precision
regime, there is a quantum estimator that outperforms any classical estimator.
Our approach is substantially more involved than in the univariate setting,
where most quantum estimators rely only on phase estimation. We exploit a
variety of additional algorithmic techniques such as amplitude amplification,
the Bernstein-Vazirani algorithm, and quantum singular value transformation.
Our analysis also uses concentration inequalities for multivariate truncated
statistics.
We develop our quantum estimators in two different input models that showed
up in the literature before. The first one provides coherent access to the
binary representation of the random variable and it encompasses the classical
setting. In the second model, the random variable is directly encoded into the
phases of quantum registers. This model arises naturally in many quantum
algorithms but it is often incomparable to having classical samples. We adapt
our techniques to these two settings and we show that the second model is
strictly weaker for solving the mean estimation problem. Finally, we describe
several applications of our algorithms, notably in measuring the expectation
values of commuting observables and in the field of machine learning.Comment: 35 pages, 1 figure; v2: minor change
Simulating Quantum Mean Values in Noisy Variational Quantum Algorithms: A Polynomial-Scale Approach
Large-scale variational quantum algorithms possess an expressive capacity
that is beyond the reach of classical computers and is widely regarded as a
potential pathway to achieving practical quantum advantages. However, the
presence of quantum noise might suppress and undermine these advantages, which
blurs the boundaries of classical simulability. To gain further clarity on this
matter, we present a novel polynomial-scale method that efficiently
approximates quantum mean values in variational quantum algorithms with bounded
truncation error in the presence of independent single-qubit depolarizing
noise. Our method is based on path integrals in the Pauli basis. We have
rigorously proved that, for a fixed noise rate , our method's time and
space complexity exhibits a polynomial relationship with the number of qubits
, the circuit depth , the inverse truncation error
, and the inverse success probability
. Furthermore, We also prove that computational complexity
becomes when the noise rate exceeds
and it becomes exponential with when the noise rate
falls below
Predicting Expressibility of Parameterized Quantum Circuits using Graph Neural Network
Parameterized Quantum Circuits (PQCs) are essential to quantum machine
learning and optimization algorithms. The expressibility of PQCs, which
measures their ability to represent a wide range of quantum states, is a
critical factor influencing their efficacy in solving quantum problems.
However, the existing technique for computing expressibility relies on
statistically estimating it through classical simulations, which requires many
samples. In this work, we propose a novel method based on Graph Neural Networks
(GNNs) for predicting the expressibility of PQCs. By leveraging the graph-based
representation of PQCs, our GNN-based model captures intricate relationships
between circuit parameters and their resulting expressibility. We train the GNN
model on a comprehensive dataset of PQCs annotated with their expressibility
values. Experimental evaluation on a four thousand random PQC dataset and IBM
Qiskit's hardware efficient ansatz sets demonstrates the superior performance
of our approach, achieving a root mean square error (RMSE) of 0.03 and 0.06,
respectively
Classical and Quantum Complexity of the Sturm-Liouville Eigenvalue Problem
We study the approximation of the smallest eigenvalue of a Sturm-Liouville
problem in the classical and quantum settings. We consider a univariate
Sturm-Liouville eigenvalue problem with a nonnegative function from the
class and study the minimal number n(\e) of function evaluations
or queries that are necessary to compute an \e-approximation of the smallest
eigenvalue. We prove that n(\e)=\Theta(\e^{-1/2}) in the (deterministic)
worst case setting, and n(\e)=\Theta(\e^{-2/5}) in the randomized setting.
The quantum setting offers a polynomial speedup with {\it bit} queries and an
exponential speedup with {\it power} queries. Bit queries are similar to the
oracle calls used in Grover's algorithm appropriately extended to real valued
functions. Power queries are used for a number of problems including phase
estimation. They are obtained by considering the propagator of the discretized
system at a number of different time moments. They allow us to use powers of
the unitary matrix , where is an
matrix obtained from the standard discretization of the Sturm-Liouville
differential operator. The quantum implementation of power queries by a number
of elementary quantum gates that is polylog in is an open issue.Comment: 33 page
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