21 research outputs found
Generalized Maximum Entropy Methods as Limits of the Average Spectrum Method
We show that in the continuum limit, the average spectrum method (ASM) is
equivalent to maximizing R\'enyi entropies of order , of which Shannon
entropy is the special case . The order of R\'enyi entropy is
determined by the way the spectra are sampled. Our derivation also suggests a
modification of R\'enyi entropy, giving it a non-trivial limit. We
show that the sharper peaks generally obtained in ASM are associated with
entropies of order . Our work provides a generalization of the maximum
entropy method that enables extracting more structure than the traditional
method.Comment: 5 pages, 1 figur
The Average Spectrum Method for Analytic Continuation: Efficient Blocked Modes Sampling and Dependence on Discretization Grid
The average spectrum method is a promising approach for the analytic
continuation of imaginary time or frequency data to the real axis. It
determines the analytic continuation of noisy data from a functional average
over all admissible spectral functions, weighted by how well they fit the data.
Its main advantage is the apparent lack of adjustable parameters and smoothness
constraints, using instead the information on the statistical noise in the
data. Its main disadvantage is the enormous computational cost of performing
the functional integral. Here we introduce an efficient implementation, based
on the singular value decomposition of the integral kernel, eliminating this
problem. It allows us to analyze the behavior of the average spectrum method in
detail. We find that the discretization of the real-frequency grid, on which
the spectral function is represented, biases the results. The distribution of
the grid points plays the role of a default model while the number of grid
points acts as a regularization parameter. We give a quantitative explanation
for this behavior, point out the crucial role of the default model and provide
a practical method for choosing it, making the average spectrum method a
reliable and efficient technique for analytic continuation.Comment: 12 pages, 10 figure
Extending the average spectrum method: Grid points sampling and density averaging
Analytic continuation of imaginary time or frequency data to the real axis is
a crucial step in extracting dynamical properties from quantum Monte Carlo
simulations. The average spectrum method provides an elegant solution by
integrating over all non-negative spectra weighted by how well they fit the
data. In a recent paper, we found that discretizing the functional integral as
in Feynman's path-integrals, does not have a well-defined continuum limit.
Instead, the limit depends on the discretization grid whose choice may strongly
bias the results. In this paper, we demonstrate that sampling the grid points,
instead of keeping them fixed, also changes the functional integral limit and
rather helps to overcome the bias considerably. We provide an efficient
algorithm for doing the sampling and show how the density of the grid points
acts now as a default model with a significantly reduced biasing effect. The
remaining bias depends mainly on the width of the grid density, so we go one
step further and average also over densities of different widths. For a certain
class of densities, including Gaussian and exponential ones, this width
averaging can be done analytically, eliminating the need to specify this
parameter without introducing any computational overhead.Comment: 10 pages, 10 figure
Connecting Tikhonov regularization to the maximum entropy method for the analytic continuation of quantum Monte Carlo data
Analytic continuation is an essential step in extracting information about
the dynamical properties of physical systems from quantum Monte Carlo (QMC)
simulations. Different methods for analytic continuation have been proposed and
are still being developed. This paper explores a regularization method based on
the repeated application of Tikhonov regularization under the discrepancy
principle. The method can be readily implemented in any linear algebra package
and gives results surprisingly close to the maximum entropy method (MaxEnt). We
analyze the method in detail and demonstrate its connection to MaxEnt. In
addition, we provide a straightforward method for estimating the noise level of
QMC data, which is helpful for practical applications of the discrepancy
principle when the noise level is not known reliably.Comment: 12 pages, 10 figure
Robust Extraction of Thermal Observables from State Sampling and Real-Time Dynamics on Quantum Computers
Simulating properties of quantum materials is one of the most promising applications of quantum computation, both near- and long-term. While real-time dynamics can be straightforwardly implemented, the finite temperature ensemble involves non-unitary operators that render an implementation on a near-term quantum computer extremely challenging. Recently, Lu, Bañuls and Cirac \cite{Lu2021} suggested a "time-series quantum Monte Carlo method" which circumvents this problem by extracting finite temperature properties from real-time simulations via Wick's rotation and Monte Carlo sampling of easily preparable states. In this paper, we address the challenges associated with the practical applications of this method, using the two-dimensional transverse field Ising model as a testbed. We demonstrate that estimating Boltzmann weights via Wick's rotation is very sensitive to time-domain truncation and statistical shot noise. To alleviate this problem, we introduce a technique that imposes constraints on the density of states, most notably its non-negativity, and show that this way, we can reliably extract Boltzmann weights from noisy time series. In addition, we show how to reduce the statistical errors of Monte Carlo sampling via a reweighted version of the Wolff cluster algorithm. Our work enables the implementation of the time-series algorithm on present-day quantum computers to study finite temperature properties of many-body quantum systems
Measuring the Loschmidt amplitude for finite-energy properties of the Fermi-Hubbard model on an ion-trap quantum computer
Calculating the equilibrium properties of condensed matter systems is one of
the promising applications of near-term quantum computing. Recently, hybrid
quantum-classical time-series algorithms have been proposed to efficiently
extract these properties from a measurement of the Loschmidt amplitude from initial states and a
time evolution under the Hamiltonian up to short times . In this
work, we study the operation of this algorithm on a present-day quantum
computer. Specifically, we measure the Loschmidt amplitude for the
Fermi-Hubbard model on a -site ladder geometry (32 orbitals) on the
Quantinuum H2-1 trapped-ion device. We assess the effect of noise on the
Loschmidt amplitude and implement algorithm-specific error mitigation
techniques. By using a thus-motivated error model, we numerically analyze the
influence of noise on the full operation of the quantum-classical algorithm by
measuring expectation values of local observables at finite energies. Finally,
we estimate the resources needed for scaling up the algorithm.Comment: 18 pages, 12 figure
Impact of opioid-free analgesia on pain severity and patient satisfaction after discharge from surgery: multispecialty, prospective cohort study in 25 countries
Background: Balancing opioid stewardship and the need for adequate analgesia following discharge after surgery is challenging. This study aimed to compare the outcomes for patients discharged with opioid versus opioid-free analgesia after common surgical procedures.Methods: This international, multicentre, prospective cohort study collected data from patients undergoing common acute and elective general surgical, urological, gynaecological, and orthopaedic procedures. The primary outcomes were patient-reported time in severe pain measured on a numerical analogue scale from 0 to 100% and patient-reported satisfaction with pain relief during the first week following discharge. Data were collected by in-hospital chart review and patient telephone interview 1 week after discharge.Results: The study recruited 4273 patients from 144 centres in 25 countries; 1311 patients (30.7%) were prescribed opioid analgesia at discharge. Patients reported being in severe pain for 10 (i.q.r. 1-30)% of the first week after discharge and rated satisfaction with analgesia as 90 (i.q.r. 80-100) of 100. After adjustment for confounders, opioid analgesia on discharge was independently associated with increased pain severity (risk ratio 1.52, 95% c.i. 1.31 to 1.76; P < 0.001) and re-presentation to healthcare providers owing to side-effects of medication (OR 2.38, 95% c.i. 1.36 to 4.17; P = 0.004), but not with satisfaction with analgesia (beta coefficient 0.92, 95% c.i. -1.52 to 3.36; P = 0.468) compared with opioid-free analgesia. Although opioid prescribing varied greatly between high-income and low- and middle-income countries, patient-reported outcomes did not.Conclusion: Opioid analgesia prescription on surgical discharge is associated with a higher risk of re-presentation owing to side-effects of medication and increased patient-reported pain, but not with changes in patient-reported satisfaction. Opioid-free discharge analgesia should be adopted routinely
Stochastic Analytic Continuation: A Bayesian Approach
The stochastic sampling method (StochS) is used for the analytic continuation of quantum Monte Carlo data from the imaginary axis to the real axis. Compared to the maximum entropy method, StochS does not have explicit parameters, and one would expect the results to be unbiased. We present a very efficient algorithm for performing StochS and use it to study the effect of the discretization grid. Surprisingly, we find that the grid affects the results of StochS acting as an implicit default model. We provide a recipe for choosing a reliable StochS grid.To reduce the effect of the grid, we extend StochS into a gridless method (gStochS) by sampling the grid points from a default model instead of having them fixed. The effect of the default model is much reduced in gStochS compared to StochS and depends mainly on its width rather than its shape. The proper width can then be chosen using a simple recipe like we did in StochS.Finally, to avoid fixing the width, we go one step further and extend gStochS to sample over a whole class of default models with different widths. The extended method (eStochS) is then able to automatically relocate the grid points and concentrate them in the important region. Test cases show that eStochS gives good results resolving sharp features in the spectrum without the need for fine tuning a default model