• Introduction to Quantum Gates: None. However, previous classroom introduction to basic ideas and notation of quantum states (including Dirac and matrix notation) is useful.
• Quantum Circuit Diagram: Basic gates and quantum states. 
• Tensor Products: Students should be familiar with quantum states, basic single-qubit gates and circuit diagram conventions. 
• CNOT and Entanglement: Students should be familiar with quantum states, basic single-qubit gates and circuit diagram conventions, and the tensor product.

If computers that you build are quantum
then spies everywhere will all want ‘em.
Our codes will all fail
and they’ll read our email,
till we get crypto that’s quantum, and daunt ‘em
- Jennifer and Peter Shor

This first subunit involves a brief introduction to QIS: quantum information science. For physics majors, an introduction to the big ideas of quantum information and quantum computing serves to motivate the upcoming topics. Some discussion of the "hype" vs realities of QIS might be in order here. A little background in classical computers may be relevant (note that some are discussed in the "Parts of a computer" section below)

Classical computers store, process, and manipulate bits (characterized abstractly as 0 or 1). Current optimal algorithms for factoring large numbers take time that grows exponentially with the size of the number to be factored, and current commercial encryption takes advantage of that fact. If one were to have functioning quantum computers that could store and manipulate quantum bits (i.e., superpositions of 0 and 1 - which we will discuss shortly), there exists an algorithm (Shor's) that could "decrypt" current common protocols exponentially faster. This has generated excitement and hype, and there are major efforts in industry and academia to construct such machines and develop new algorithms. A key aspect of quantum computing is the concept of "basis size" - for a 4-bit quantum computer, a given stored qubit can be a superposition of up to $2^4 = 16$ basis states. In some sense, this represents exponentially more information in a single qubit than the 4 bits a classical computer stores. 

The intent of adding QIS to a quantum class for physics students is to see how the tools we have learned (e.g. concepts of superposition and measurement) allow us to understand the basic language and ideas involved in a number of QIS applications, including quantum computing, cryptography, and teleportation. The goal is not a course in these topics, but a brief introduction to motivate further study. 

Reference: "How Media Hype Affects our Physics Teaching: A Case Study on Quantum Computing", Meyer et al, TPT 61, 339 (2023) DOI: https://doi.org/10.1119/5.0117671

This is a (brief) introduction to qubits (including Dirac notation) and superposition. For students in a physics majors quantum class, this is intended to make the quick shift from more general quantum observables to 0's and 1's. 

Classical computers work with bits, denoted by states 0 or 1. The physical implementation of those bits is rather arbitrary (typically low and high voltage on a wire), but for many practical computing purposes also irrelevant. The same will be true for quantum computers, but now the qubit can be in any superposition state $a\ket{0} + b\ket{1}$. 

Classical computing requires gates: operators that act on a bit (one or more) and output a resulting bit. The "Not" gate is the simplest single-bit gate: inputing a 0 yields a 1 and vice versa. This can also be thought of as a logic operation (interpret 0 = "false" and 1 = "true") The "And" gate acts on 2 inputs and returns a single output, the logical "And" operation.

Quantum gates similarly take input states, act on them, and produce output states. The final stage of a set of quantum operations will ultimately be a measurement, but the sequential operations should not make any measurements (indeed, avoiding such collapse of the quantum superposition state is a major engineering challenge at the moment). Schematically, the process can be thought of schematically like this:

Much current physics and engineering research is devoted to studying how to physical implement qubits - examples include spin-1/2 systems where 0 and 1 represent "up" and "down" spin in the Z direction, and certain gates could involve applied magnetic fields (for instance), but the states could also be photon polarization, energy levels of trapped atoms,  current in Josephson junctions, or many more. For discussing gates and algorithms, just as with classical bits, this physical implementation is not essential. 

A quantum gate is an operation performed on qubits.  Mathematically, gates are associated with unitary and linear operators. Linear means e.g. $Z(a\ket{0} + b\ket{1}) = aZ\ket{0} + bZ\ket{1}$. Thus, one way to define them is by their operation on computational basis states.

• $X$-gates: $X\ket{0} = \ket{1}, \ \ X\ket{1}=\ket{0}$. 
  (The $X$-gate is a 'bit flip' or NOT gate: taking $\ket{0}$ to $\ket{1}$, and vice versa. )
• $Z$-gates: $Z\ket{0} = \ket{0}, \ \ Z\ket{1}=-\ket{1}$. 
  The $Z$-gate  is thus sometimes called a "phase gate" (flipping the phase of $\ket{1}$ states.)  
• $Y$-gates: $Y\ket{0} = -i \ket{1}, \ \ Y\ket{1}=i\ket{0}$. 
  The $Y$-gate is thus a combination of the previous two ($Y = iZX$).
• Identity ($I$): $I\ket{0} = \ket{0}, \ \ I\ket{1}=\ket{1}$.
  The identity operator $I$ does not alter input states in any way. 
• Hadamard gates: 
  $H \ket{0} = \frac{1}{\sqrt{2}} ( \ket{0} + \ket{1}) \;\; \text{ and } H\ket{1} = \frac{1}{\sqrt{2}} ( \ket{0} - \ket{1})$.
  The Hadamard is a common and useful gate in quantum computing algorithms -  it converts pure computational states into superposition states.

Note that $ XX = YY = ZZ = H H = I$, and that order matters, e.g. $X$ does not commute with $Z$. 

A natural follow-up is to ask the same question for $H\ket{0}$. Our students scored similarly to the previous one. These questions provide elementary practice and serve as conversation starters. $H$ plays a huge role in algorithms: often the first step is to start with a bunch of $\ket{0}$ states, then act $H$ on each to get a collection of superposition states.

Sample related online Tutorial:
An online introduction/tutorial on quantum gates is available on acephysics.net: 
https://acephysics.net/tutorials/introduction-to-quantum-gates/1-quantum-bits 

Circuit diagrams use lines to indicate individual qubits and boxes to indicate quantum gates.  They are read from left to right.  The following diagram depicts the equation: $\ket{\psi}_{out} = X Z \ket{\psi_{out}}$. 

Note how the equation looks "backwards" compared to the circuit, but it is clear in both that the $Z$ gate acts first.  We can also depict a measurement by using a symbol of a "meter" as shown below (note that different textbooks often have different conventions for a measurement symbol).

When we add additional qubits, they are shown as additional lines where the top line is understood to be the first qubit.  The circuit diagram below shows the output for the first qubit is $ZH\ket{0}$ and the output of the second qubit is $X\ket{1}$.

Sample related online Tutorial:
A very brief online introduction to quantum circuits is available on acephysics.net:
https://acephysics.net/tutorials/quantum-circuit-diagrams

When moving to N-qubit systems, the dimensionality of your basis grows as $2^N$. Here we briefly introduce students to the ideas of combining two single particle states into a  larger-dimensional 2-qubit state, and how to represent this in Dirac (or matrix) notation, using the language of tensor products.  

There is only so much that can be done with a single qubit.  To add a second qubit, we use the tensor product $\otimes$.  If the first qubit is in the state $\ket{0}$ and the second qubit is in the state $\ket{1}$, then the two-qubit state can be written as $\ket{0} \otimes \ket{1}$.  There are several shorthand ways to write this.  Below are all equivalent formulations:

$ \ket{0} \otimes \ket{1} = \ket{0} \ket{1} = \ket{01}$

For brevity, the last one is most commonly used unless it is important to emphasize the states of the individual qubits, in which case the $\otimes$ symbol will be used. 

When dealing with 2-particle systems,  2-qubit operations can (sometimes) be described as a tensor product of operators (e.g., $\hat{A}\otimes\hat{B}$), and can (always) be represented by a 4x4 matrix in the 4-d two-qubit Hilbert space.   

For example, in the circuit below, the first set of gates can be written as a tensor product: $H \otimes Z$, but based on the information we have, the second gate $U$ cannot be broken down into a separate gate on each qubit.

Mathematically, the circuit is represented as:

$ U(H\otimes Z) (\ket{\psi_A} \otimes \ket{\psi_B}) = U ( H\ket{\psi_A} \otimes Z\ket{\psi_B})$

Sample related online Tutorial: Tensor Products - 
An online introduction/tutorial on tensor products is available on acephysics.net: 
https://acephysics.net/tutorials/tensor-products

Two (or more) particle quantum systems can exhibit interesting correlations (entanglement) stronger than any classical system. 

A state is entangled when it cannot be written as a single tensor product of two single-qubit states.  

For example, consider the state $\tfrac{1}{\sqrt{2}}(\ket{00} + \ket{01})$. This state is NOT entangled, because we can 'factor' it into $\ket{0} \otimes \tfrac{1}{\sqrt{2}}(\ket{0} + \ket{1})$.  In this case the first qubit is in the state $\ket{0}$ and the second state is in the state $\tfrac{1}{\sqrt{2}}(\ket{0} + \ket{1})$.  

In contract, we can look at the entangled state $\tfrac{1}{\sqrt{2}}(\ket{00} + \ket{11})$.  There is no way to write a ket for the first or second qubit on their own.  The state must be written together. 

Note: It is possible to describe the first qubit using a reduced density operator.  The state of the first qubit would be the completely mixed state $\rho_1 = \frac{1}{2}\mqty( 1 & 0 \\ 0 & 1)$.  

Second note: Entanglement has some interesting properties and consequences. We won't get into them in this module, but more information can be found in module "Entanglement and EPR".

In order to have a quantum computer, we need gates that act not only on one qubit, but gates that work on multiple qubits.  More specifically, we need gates that will "entangle" two qubits.  The most common gate of this type is called the CNOT, or controlled-NOT gate.  It does exactly as the name suggests: performs a NOT gate controlled by another qubit.  The following is the circuit diagram for a CNOT gate:

The solid black dot on the top 'wire' indicates that this is the control qubit. The gate on the second qubit will be performed only if the control qubit is a 1. (This has interesting effects for superposition states, as we see in the clicker questions below!) 

We can define how the CNOT gate acts on the four 2-qubit computational basis states:

$$ CNOT\ket{00} = \ket{00} \;\;\;\; CNOT\ket{01} = \ket{01}$$

$$ CNOT\ket{10} = \ket{11} \;\;\;\; CNOT\ket{11} = \ket{10}$$

We can use the CNOT to entangle two independent qubits as in the circuit in the clicker question example below.

Sample related online Tutorial: 
An online introduction/tutorial on CNOT gates and entanglement is available on acephysics.net: 
https://acephysics.net/tutorials/cnot-entanglement

An alternative set of basis states for 2-particle systems which is convenient when describing entangled systems is the "Bell basis", defined as 

$\ket{\beta_{00}} \equiv \frac{1}{\sqrt{2}}(\ket{00} + \ket{11})$
$\ket{\beta_{01}} \equiv \frac{1}{\sqrt{2}}(\ket{01} + \ket{10})$
$\ket{\beta_{10}} \equiv \frac{1}{\sqrt{2}}(\ket{00} - \ket{11})$
$\ket{\beta_{11}} \equiv \frac{1}{\sqrt{2}}(\ket{01} - \ket{10})$

This basis is complete and orthonormal. The naming convention is not immediately obvious but can be understood when considering a simple conventional "entangling circuit" (the example from the clicker question in the "CNOT" section above).  

Note: For spin-1/2 systems, the last state, $\ket{\beta_{11}}$ is a total spin-0 singlet, with interesting properties for e.g. Bell tests. More discussion of the Bell basis states happens in the "Entanglement and EPR" module.   

More discussion of this type is in the "Entanglement and EPR" module. 

An important part of quantum theory is the fact that you cannot copy an arbitrary 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: 

$$ [(\bra{\phi} \otimes \bra{b}) U_{clone}^\dagger ] [U_{clone} (\ket{\psi} \otimes \ket{b})] = [\bra{\phi} \otimes \bra{\phi} ] [ \ket{\psi} \otimes \ket{\psi}] \label{cloneeqn}$$

• $U_{clone}^\dagger U_{clone} = 1$ since $U$ is unitary
• Make sure to take the inner product of qubit 1 (from eqn 1) with qubit 1 (from eqn 2) and then for qubit 2 (from eqn 1) with qubit 2 (from eqn 2)

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.

Quantum teleportation is an example of using entanglement as a resource. The idea is to take a qubit at one location, and transport all information of that state to a second location (without physically moving the initial particle through the intermediate space.)

Some notes about teleportation:

• We are not making a clone. The initial state is destroyed in the process.
• The person "sending" the qubit does not gain any information about the state they are teleporting.  
• The two parties require a shared entangled system before teleportation can happen.
• During teleportation, two classical bits (00, 01, 10, or 11) are transported.   This means that no information can be sent faster than classical information can be transmitted (thus limited by the speed of light).

A circuit diagram representation of the process is shown below. 

Alice posses a "secret" qubit she wishes to teleport to Bob. This could be for communication, encryption, or e.g. as a means to transport qubits long distances in the context of a quantum computer. Alice need not know what the "secret" state is.

Preparation: Alice and Bob need to share a two-qubit entangled state $\beta_{00}$, such that Alice has the first qubit and Bob has the second. Alice thus possesses two qubits (the secret qubit and her half of the original entangled state), and Bob has the second qubit of the original entangled state.

Algorithm: The circuit above shows the algorithm, which involves the following steps.

• Alice runs her 2 qubits through the circuit shown above, effectively entangling her two bits. Note that this type of circuit was discussed in the preceding sections on "Entangling gates" and "Bell States".
• She follows this by a measurement of the two output bits, which results in one of 4 possibilities: 00, 01, 10, or 11.
• She transmits this information through a classical channel to Bob (thus, limiting the process to speed-of-light communication).
• Bob uses the information to run his single qubit through one of four gate choices (X, Y, Z, or 1) depending on which classical information he received (Details in our downloadable lecture notes).
• The output from this final gate is now the original secret qubit state.   

The probabilities of the 4 measurement outcomes that Alice can get are all 25%, so she has gained no information. Bob has not yet made a measurement; he possesses a qubit in the same state as the original secret one, which can now be used for further purposes. An interesting side note: if the original secret bit was entangled with yet other qubits, the outcome of the process results in Bob's output bit entangled in the same way with those other qubits.