#### Presentation Title

Understanding the Limits of Bootstrapping in Quantum State Tomography

#### Faculty Mentor

John Phillip Preskill

#### Start Date

18-11-2017 2:15 PM

#### End Date

18-11-2017 2:30 PM

#### Location

9-285

#### Session

Physical Sciences 2

#### Type of Presentation

Oral Talk

#### Subject Area

physical_mathematical_sciences

#### Abstract

Being able to determine states using quantum tomography is crucial to advance quantum computing. The error analysis of the reconstructed state is usually carried out using statistical methods like bootstrapping. While bootstrapping has worked well in practice, it is known to fail in some extreme cases. We try to find and investigate such cases by simulating data from a true state, performing a Maximum Likelihood Estimation, running a bootstrap procedure to generate an estimated error bar on the figure of merit (fidelity to a target state), and comparing them to the true error. We characterize the reliability of the bootstrap by considering the probability that the error bar is a particular fraction of the true error, allowing us to develop a new statistical representation of bootstrapping data which can be used in various subfields of quantum information theory. We perform the bootstrap for a qubit, Bell pair, Greenberger-Horne-Zeilinger state, and W-state, varying the purity, amount of noise in the measurements, and number of measurements. We reproduce two extreme cases: we consider a single qubit and measure only two outcomes in the z-direction, giving us an error bar of zero. We also vary the number of measurements being simulated in the tomography procedure to high values such as n = 500, so the error bar produced by the bootstrap is very close to the true error bar. We find that the bootstrap is reliable, as the error bars generated from the bootstrap are spread evenly around the true error bar even with the addition of noise to the measurements. The statistical techniques and code we have developed are expected to facilitate future extensions of our work, such as probing other areas of state space like three-sections of qutrits, that may give rise to other extreme cases.

Understanding the Limits of Bootstrapping in Quantum State Tomography

9-285

Being able to determine states using quantum tomography is crucial to advance quantum computing. The error analysis of the reconstructed state is usually carried out using statistical methods like bootstrapping. While bootstrapping has worked well in practice, it is known to fail in some extreme cases. We try to find and investigate such cases by simulating data from a true state, performing a Maximum Likelihood Estimation, running a bootstrap procedure to generate an estimated error bar on the figure of merit (fidelity to a target state), and comparing them to the true error. We characterize the reliability of the bootstrap by considering the probability that the error bar is a particular fraction of the true error, allowing us to develop a new statistical representation of bootstrapping data which can be used in various subfields of quantum information theory. We perform the bootstrap for a qubit, Bell pair, Greenberger-Horne-Zeilinger state, and W-state, varying the purity, amount of noise in the measurements, and number of measurements. We reproduce two extreme cases: we consider a single qubit and measure only two outcomes in the z-direction, giving us an error bar of zero. We also vary the number of measurements being simulated in the tomography procedure to high values such as n = 500, so the error bar produced by the bootstrap is very close to the true error bar. We find that the bootstrap is reliable, as the error bars generated from the bootstrap are spread evenly around the true error bar even with the addition of noise to the measurements. The statistical techniques and code we have developed are expected to facilitate future extensions of our work, such as probing other areas of state space like three-sections of qutrits, that may give rise to other extreme cases.