Parameter estimation, the quantum way: theory and application of quantum estimation

Recorded lecture. Warning: this a 1-hour video!

Context and caveats

The theme of this session of the workshop is the following question: how can a quantum sensor optimally extract information about its environment?

History: the first quantum advantage

Metrology is the scientific study of measurement and sits at the intersection of many disciplines — most notably physics, engineering, and statistics. It’s the job of the physicist to determine what should be measured. It’s the job of the engineer to decide how to measure it. And it’s the job of the statistician to determine what data tell us about the quantity of interest. The latter is, of course, necessary because every measurement has an error. When the physics is simple (which requires a lot of hard work to get to) and when the engineering is precise (also requiring hard work), the statistics is relatively trivial — a first-year textbook will suffice.

Framework: deciding on decision theory

Before we get started on physics, I want to get the statistics out of the way. It is essential because the question at the outset — how can a quantum sensor optimally extract information about its environment — is vague. And when you read papers in this field — or any field in physics for that matter — you will run up against the same unanswered question. What is the exact problem, and how do I tell if it’s been solved?

Deduction versus induction in physics. Credit: own work.
Definition of risk in parameter estimation. Credit: own work.
Definition of Bayes risk. Credit: own work.
A cartoon of a plot in a quantum metrology pattern. Credit: own work.

Goal: the quest for the ultimate bound

This is a great segue to one of the most common themes in metrology: lower bounds. In particular, you will hear a lot about three lower bounds: the standard quantum limit, the Heisenberg limit and the Quantum Cramer-Rao bound. These can be incredibly confusing because they are never introduced in generality, and it’s not obvious what parts of the particular context in which they are being discussed are necessary. Moreover, the language in which they are often discussed implies that these things are the ultimate bounds on accuracy independent of any of the choices made by the researchers, even implicitly.

The standard quantum limit and Heisenberg limit as lower bounds on root mean square error. Credit: own work.
The Cramer-Rao lower bound. Credit: own work.
“Standard” quantum circuit where the parameter x is encoded in individual quantum systems. Credit: own work.
A quantum circuit for metrology where an entangled input state is represented by the application of a multiqubit gate. Credit: own work.
A quantum circuit for metrology where an entangled input state and entangled measurement are represented by the application of multiqubit gates. Credit: own work.
A quantum circuit for metrology where the input state is represented as the output of error-correcting code. Credit: own work.
Figure 1 from Phys. Rev. Lett. 123, 231107 (2019).
A quantum circuit showing feedforward, where some aspect of successive qubits is controlled by the measurement outcomes of previous qubits. Credit: own work.

Summary

I wanted you to take away two things from this talk: decision theory and quantum circuits. But you were also introduced to a few extra players.

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Chris Ferrie

Chris Ferrie

Quantum theorist by day, father by night. Occasionally moonlighting as a author. csferrie.com