Our goals for this site are to provide adaptable resources to support faculty, including those very new to these topics, who wish to add some student-centered, research-based QIS instruction to your classes.
We have materials for two different audiences, depending on the type of course you teach.
QM Course: Are you teaching quantum mechanics in a physics department to (mostly) majors, and want to add a short unit on quantum information (e.g. quantum computing or cryptography) within your course?
And: if you teach undergraduate quantum mechanics to physics students and are looking for a suite of student-centered materials to help you with the entire QM course, please visit our Adaptable curricular materials for Quantum Mechanics site.
Who is this for? An instructor teaching Quantum Mechanics who would like to add some quantum information science topics to their physics course.
(Materials are suitable for faculty new to these topics.)
Topics: We have divided our materials into three main topical areas.
Each is standalone - materials can be taught without any prerequisite material from the other groups.
(Because of this, there are some duplicate materials across the groups.)
We do not intend for the materials to be used wholly as written. Rather, they form a complete set that a faculty member can pick and choose from as it fits their schedule, the material already taught, and the interests of both the faculty member and student.
The “instructional materials” found on this site include: lecture notes for faculty, concept tests, homework questions, online tutorials, and assessments.
Please use and adapt whatever is helpful to you, however it will most benefit your students. Please credit our work if you share your materials beyond your own classes. Please make an effort to keep assessment materials off the open web - alter questions for your students.
Curriculum designers Dr. Steven Pollock and Dr. Gina Passante have each taught this material in their classes. Feel free to download our lecture notes - we are happy for you to use whatever material works for your class.
Please email if you try our materials and have any feedback:
steven.pollock (at) colorado.edu or gpassante (at) fullerton.edu
CUB is a research-intensive public institution. I teach these materials in junior or senior-level undergraduate quantum mechanics courses, which currently all follow a spins-first curriculum (using McIntyre's Quantum Mechanics textbook.)
Each sequence below takes me one week. These are all taught in "large lecture" (75+ student) settings with three 50-minute lectures/week.
Please reach out if you have questions or feedback: steven.pollock (at) colorado.edu
CSUF is a primarily undergraduate, large, Hispanic-serving institution. I teach these materials in several of my quantum mechanics courses at the undergraduate and master's levels.
Feel free to reach out if you have questions (gpassante at fullerton dot edu).
What is colloquially called "quantum cryptography" is more accurately described as "quantum key distribution" (QKD), where a secret key is distributed to two parties in a way that guarantees security. That key can then be used in one-time pad encryption.
These materials are intended for a faculty member to select and adapt relevant pieces to use in their classroom. In our courses, this module takes one lecture to cover.
We have written brief notes on the topics in this module to help bring non-expert faculty up to speed.
View them by expanding the titles below, or download them as a single pdf file (see download section below - coming soon).
Topics include:
The one-time pad is a perfectly secure way to encrypt information. In this protocol, both parties must share a secret key. The one-time pad security relies on the fact that this key is truly secret (no one else has access to it) and that it is only used a single time (hence the name ``one-time''). The secret key must be at least as long as the message (no portion of the key should be repeated).
If Alice wishes to send a message to Bob, Alice will take the message and add the secret key to obtain the cyphertext (the encoded message). In the case of a binary message and a binary key, this would be addition mod 2.
$ \text{message} \oplus \text{key} = \text{cyphertext} $
In mod 2, Bob will just need to do $\text{cyphertext} \oplus \text{ key} = \text{message}$ to decode the cyphertext.
We will often do examples using the english alphabet, in which the addition will be modulo 26. In this case, to decode the cyphertext you will need to ${\it subtract}$ the key (mod 26) to obtain the message.
In all QKD protocols, it is an important assumption that an eavesdropper cannot copy (or clone) an unknown quantum state. The proof (by contradiction) is relatively straightforward:
Assume you have a 'cloning machine'. Such a machine would take an unknown state $\ket{\psi}$ and a blank state (let's call it $\ket{b}$) and turn it into two copies of $\ket{\psi}$. Mathematically, we write it as: $U_{clone} (\ket{\psi} \otimes \ket{b}) = \ket{\psi} \otimes \ket{\psi} $
Since we want to be able to clone an arbitrary state, the cloning machine should also work if we input the state $\ket{\phi}$: $U_{clone} (\ket{\phi} \otimes \ket{b}) = \ket{\phi} \otimes \ket{\phi} $
We want to take the inner product of the two equations:
$$ [ ( \bra{\psi} \otimes \ket{b} ) U_{clone}^\dagger ] [U_{clone} ( \ket{\psi} \otimes \ket{b} ) ] = [ \bra{\phi} \otimes \bra{\phi} ] [ \ket{\psi} \otimes \ket{\psi} ] \label{cloneeqn}$$
Then we have that:
$ \ip{\phi}{\psi} \ip{b}{b} = \ip{\phi}{\psi} \ip{\phi}{\psi} $
$ \ip{\phi}{\psi} = \ip{\phi}{\psi}^2$
This is only true if $\ip{\phi}{\psi}$ is equal to 1 or 0. Meaning the cloning machine will only work for states that are identical or orthogonal. In other words, you can build a cloning machine, but it will only clone a set of basis states, not an arbitrary superposition of those basis states.
This is one of the more common QKD protocols and is perfect for use in intro quantum mechanics classes as it only requires knowledge of two-level superposition states and the results of measurements.
In our example, we will use spin-1/2 particles, and we will discuss measurements of spin in the $x$ and $z$ directions.
Players:
Alice - Wants to share a secret key with Bob. She sends qubits to Bob, randomizing what she ends.
Bob - Wants to share a secret key with Alice. Bob receives the qubits sent by Alice and will measure them in a randomly chosen basis.
Eve - An eavesdropper who wants to gain information about the secret key without being detected. She will intercept the qubits that Alice sends to Bob and make measurements before sending the qubit on to Bob.
The Protocol: The following steps can be recorded in the handout (available for download below).
In reality, anything can happen to the qubit as it travels from Alice to Bob. We are going to assume the worst: that a very powerful and quantum-savvy enemy has intercepted the qubit and knows enough of the protocol to make the best measurements they can before sending the qubit to Bob.
Here is the best that an eavesdropper can do:
You can follow along in the handout to see the repercussions of this interception. In short:
If Alice and Bob did not test for the presence of an eavesdropper, even with an error rate of 25%, Eve would certainly gain valuable information if Alice and Bob tried to use their "secret" key to encode a message.
Sometimes we teach this material with lecture and handouts and other times we use a tutorial worksheet (both can be downloaded below). We have only one homework question for this content.
(For more information on how we implement these types of materials in our upper-division classes, see our related AJP paper.)
Quick link to relevant AcePhysics.net Tutorial (which can be done in class, alone or in groups, or at home alone)
Prerequisite knowledge: Basic gates (H), measurements in the computational (Z) basis on the $$|0\rangle, |1\rangle, |+\rangle, |-\rangle$$ states.
If you are assigning this activity in your course, email us at hello[at]acephysics[dot]net to get a course page where you can access student completion information.
Who is this for? These materials are for anyone teaching an introductory (first course in) quantum computing or quantum information science.
Why did we create these materials? Introductory QIS courses often include students from multiple different academic backgrounds (physics, computer science, engineering, math, chemistry, ...). These materials are intended to assist with deep conceptual practice on some of the fundamental ideas in quantum information science. Some students may find these materials too easy, while others may find them too hard. That is exactly the reason for their existence! The goal is to bring all students to the same level of understanding that will set the foundation for future QIS learning.
Note: By design, these materials do not cover complex (or even intermediate) QIS topics. We have chosen to focus on those materials most important at the beginning of a QIS course. These topics happen to be "physics heavy" topics, and ones for which our group's educational and research expertise is particularly well-matched.
This page is currently (July 2024) under construction, and improvements are constantly being made!
For each broad topical area listed below, you will find:
Content: Superposition states, single-qubit gates (X, Z, H, and I), Dirac and matrix notation, measurement probabilities, and single-qubit X, Z, H, and I gates.
Prerequisite knowledge: None. However, previous classroom introduction to basic ideas and notation of quantum states (including Dirac and matrix notation) is useful.
Faculty notes: See the first 4 entries in "Notes for Faculty" in our QIS for QM tab labeled "Qubits, Quantum Gates, ...": (1. Motivation, 2. Qubits, 3. Parts of a Computer, and 4. Quantum Gates)
Materials available:
Content: Practice with single-qubit gates represented as circuit diagrams
Prerequisite knowledge: Basic gates and quantum states. (This can all be found in the previous tutorial, "Introduction to Quantum Gates.")
Faculty notes: See the fifth entry under "Notes for Faculty" in our QIS for QM tab labeled "Qubits, Quantum Gates, ...": (5. Circuit Diagrams)
Materials available:
Content: Helping students describe systems with multiple qubits
Prerequisite knowledge: Students should be familiar with quantum states, basic single-qubit gates and circuit diagram conventions. (i.e., The sections above, on this page.)
Faculty notes: See the sixth entry under "Notes for Faculty" in our QIS for QM tab labeled "Qubits, Quantum Gates, ..." : (6. Tensor products of states and operators)
Materials available:
Content: An introduction to the controlled-NOT (CNOT) gate and the related concept of entanglement.
Prerequisite knowledge: Students should be familiar with quantum states, basic single-qubit gates and circuit diagram conventions, and the tensor product. (i.e., The previous sections on this page.)
Faculty notes: See entries 7-8 under "Notes for Faculty" in our QIS for QM tab labeled "Qubits, Quantum Gates, ...": (7. Definition of Entanglement and 8. Entangling gates/CNOT)
Materials available:
Content: This activity takes students through the BB84 quantum key distribution protocol. It uses quantum circuit notation and includes the effect of an eavesdropper.
Prerequisite knowledge: Basic single qubit gates (in particular, the Hadamard, H gate). Students should know how to predict probabilistic outcomes of measurements in the computational (Z) basis on superpositions of $|0\rangle, |1\rangle, |+\rangle, |-\rangle$ states.
Faculty notes: See the "Notes for Faculty" in our QIS for QM tab: Quantum Cryptography
Materials available:
We are PER research faculty teaching at very different institutions - different in class sizes and setups, student demographics, institutional research-focus - but all interested in helping introduce undergraduates to basic elements of Quantum Information Science. We have all taught a variety of quantum courses for many years.
The materials you will find here are not meant to be taken as givens, this is not a "fixed curriculum" that you are supposed to fully adopt (or reject). We hope that you will be inspired by some of the activities, notes, concept-tests, homeworks and more, and will borrow and adapt them for your own situation and students. We do not all use exactly the same materials ourselves.
We have borrowed where we can from PER literature on Quantum Mechanics (and tried - but apologize up front where we occasionally have failed - to appropriately credit the the hard development work of others!) We do not claim that these materials are "Research-validated" (they are still under development), but cheerfully present sometimes half-baked or partially-tested materials that we might argue are "research-based", a vaguer but perhaps more realistic description.
We welcome feedback and suggestions. If you make significant changes or additions, and particularly if you have classroom evidence that suggests it works well - let us know. We hope someday this site will be flexible enough to allow for community-sharing of new resources, and will work towards making that happen. (See contact information at the bottom of the page)
CONTACT INFORMATION:
This page was built by Steven Pollock (steven.pollock (at) colorado.edu), Gina Passante (gpassante (at) Fullerton.edu), and Bethany Wilcox (bethany.wilcox (at) colorado.edu). Contact us with questions or feedback.
We are funded in part by NSF DUE- 2012147 and 2011958: Collaborative Research: Connecting Spins-First Quantum Mechanics Instruction to Quantum Information Science
PLEASE USE AND ADAPT whatever is helpful to you, however it will most benefit your students. Please credit our work if you share your materials beyond your own classes. Please make an effort to keep assessment materials off the open web - alter questions for your students.
©2022 University of Colorado Boulder
and California State University, Fullerton
Funded by the National Science Foundation
grants 2011958 and 2012147