#### 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.

#### Summary of research results to be presented

We have developed a novel statistical representation of the data retrieved from the bootstrap by asking what the probability is that the error bar resulting from a bootstrapping procedure on a particular figure of merit is a given fraction of the true error bar. This has allowed us to generate cumulative and probability distribution graphs for various canonical quantum systems. Moreover, depending on what the quantum state tomography is being used for, and what the error tolerance is, one can adjust the value of the fraction accordingly. For instance, in a setting such as quantum cryptography, the value chosen would be far closer to one than in a simple laboratory experiment, where the accuracy of the error bar is less critical. Using this procedure, we have found that for pure states that necessarily have all eigenvalues except one equal to zero, such as Bell states, the GHZ state, and the W-state, the distributions of the error bar produced from the bootstrap are spread evenly around the true error bar. In these cases, the bootstrapping procedure almost always overestimates the error. After investigating the addition of noise to both the density matrix itself as well as the positive-operator valued measures, we find that overestimation is exacerbated when smaller amounts of noise were added to the systems. However, this pattern arises when we adjust the noise linearly in the density matrices, and exponentially in the positive-operator valued measures.

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.