Description:
The QIS key concept is the fact that quantum states are represented by directions or vectors in an abstract space. This QIS key concept matches with the first Common Core math standard addressing vector and matrix quantities, in which students learn what a vector is and how to represent it.

Learning Outcome:
Quantum states are represented by vectors.

Standards Alignment (Common Core State Standard):

• (HS.N.VM.1) Recognize vector quantities as having both magnitude and direction. Represent vector quantities by directed line segments, and use appropriate symbols for vectors and their magnitudes.

Course:

• Pre-calculus

Description:

Quantum states are determined only by the direction, not the magnitude, of their quantum state vector. Therefore, multiplying a quantum state vector by any nonzero scalar results in a vector that represents the same quantum state. For example, the two vectors [1, 1] and [2, 2] represent the same quantum state – their directions are the same, with only their magnitudes being different. Here are two consequences of this fact:

1. A quantum state is meaningfully represented by a unit vector, which is a vector of magnitude 1. (The reason why quantum states are properly represented by unit vectors is discussed in HS.Math.2b.1.) Any vector that does not have magnitude equal to 1 can be transformed to a unit vector that represents the same quantum state by multiplying it by an appropriate scalar. For example, the vector [3, 4] can be transformed to the unit vector [3/5, 4/5] by multiplying the vector by the scalar 1/5.
2. A quantum state has multiple representations as a vector. It is often simpler and more convenient to choose to represent a quantum state by a vector that does not have magnitude 1. For example, there is a commonly used quantum state that is represented by the unit vector [1/√2, 1/√2]. It is very common to omit the 1√2 scalar and represent this quantum state by the vector [1, 1], because this is a simpler way of expressing the state that has two equal components.

The representation of quantum states as vectors provides an application for the mathematical concept of multiplying
vectors with scalars

Learning Outcome:
Multiplying a quantum state by a scalar quantity results in an equivalent quantum state; the meaningful vector length is always 1.

Standards Alignment (Common Core State Standard):

• (HS.N.VM.5) Multiply a vector by a scalar.

Course:

• Pre-calculus

Description:

The direction, not magnitude, of the vector determines its state, as expressed in LG.2.2. The reason this is true is due to the following two facts:

1. Each component in the vector represents the probability amplitude of a single possible measurement
   outcome. The probability of a measurement outcome is given by the square a² of the corresponding probability amplitude, a.
2. The magnitude of the vector is calculated by computing the square root of the sum of the squares of its components. The sum of the probabilities of all possible outcomes of any scenario is always 1, so the meaningful vector magnitude is always 1.

Therefore, the direction of the vector, with the magnitude of the vector scaled to 1, provides the probabilities of all possible outcomes. This combines knowledge of direction and magnitude of vectors (LG.2.1 and LG.2.2) with basic probability concepts.

• Optional for students who are studying complex numbers: A probability amplitude can be a complex number (See learning goal LG.2.6). In this case, it is necessary to modify the first fact listed above: The probability of a measurement outcome is given by the square |a|²
  of the absolute value of the corresponding probability amplitude, a.

Learning Outcome:
The quantum state vector determines the probabilities of all of the possible outcomes of a set of measurements.

Standards Alignment (Common Core State Standard):

•  (HS.S.MD.3) Develop a probability distribution for a random variable defined for a sample space in which theoretical probabilities can be calculated; find the expected value. For example, find the theoretical probability distribution for the number of correct answers obtained by guessing on all five questions of a multiple-choice test where each question has four choices, and find the expected grade under various grading schemes.

Course:

• AP Statistics

Description:

Quantum states are represented as vectors (LG.2.1), whereas quantum gates are represented as matrices. To calculate the effect of applying a particular gate to a particular quantum state, one multiplies the vector by the matrix. The result of the operation is a vector — the resulting quantum state. Calculating the operation of quantum gates applied to quantum states provides an application for the mathematical concept of multiplying a vector by a matrix.

Learning Outcome:
Quantum gates use matrix multiplication to calculate.

Standards Alignment (Common Core State Standard):

•  (HS.N.VM.11) Multiply a vector (regarded as a matrix with one column) by a matrix of suitable dimensions to produce another vector. Work with matrices as transformations of vectors.

Course:

• Pre-calculus

Description:

The probability of a measurement outcome of a two-qubit system can be computed using the same method described in the previous learning goal LG.2.3: Calculate the square of the absolute value of the corresponding probability amplitude. If two qubits are independent, the probabilities of each two-qubit measurement outcome is the product of the probabilities of the measurement outcomes for each individual qubit. When expressing the quantum state vector of an independent two-qubit system, each of the four probability amplitudes will be a product of probability amplitudes from the two individual qubits.

Learning Outcome:
The probability of a combination of independent qubits’ outcomes is the product of their individual probabilities.

Standards Alignment (Common Core State Standard):

• (HS.S.CP.2) Understand that two events A and B are independent if the probability of A and B occurring together is the product of their probabilities, and use this characterization to determine if they are independent.

Courses:

• Algebra
• Geometry

Description:

Quantum states have three components which are expressed using the relative magnitudes of the two vector components, a negative sign, and the complex number i. Introducing this third component matches (the complex number i) corresponds to mathematics standards introducing complex numbers, their representation with i, and the significance of squaring i.

Learning Outcome:
Some quantum states use complex numbers.

Standards Alignment (Common Core State Standard):

• (HS.N.CN.1) Know there is a complex number i such that i²= –1, and every complex number has the form a + bi with a and b real.

Course:

• Algebra 2

Description:

The vectors representing quantum states and the matrices representing gate operations can include complex numbers. Calculating the effects of gate operations requires matrix multiplication, which in turn requires the multiplication of complex numbers.

Learning Outcome:
Perform complex number multiplication to calculate outcomes for gates and/or states with complex numbers.

Standards Alignment (Common Core State Standard):

• (HS.N.CN.2) Use the relation i² = –1 and the commutative, associative, and distributive properties to add, subtract, and multiply complex numbers.

Course:

• Pre-calculus

Description:
For any process which has a probability associated with it, multiple data points are required. Measurements of quantum states are such a process. In 6th grade, students transition from measurements with deterministic outcomes (e.g., the length of the table, one student’s age) to data sets that involve variation (e.g., average height of students in a class). In 7th grade, they further transition to probabilistic events, which matches with quantum systems. Flipping a coin and rolling dice are typical examples, which are also used to explain quantum systems.

Learning Outcome:
A single measurement is not sufficient to determine a quantum state.

Standards Alignment (Common Core State Standard):

• (7.SP.6) Approximate the probability of a chance event by collecting data on the chance process that produces it and observing its long-run relative frequency, and predict the approximate relative frequency given the probability. For example, when rolling a number cube 600 times, predict that a 3 or 6 would be rolled roughly 200 times, but probably not exactly 200 times.

Course:

• Grade 7 Math

Description:
In order to determine the state of a quantum system, many measurements must be taken on the quantum system in the same state. This is because of the statistical nature of quantum states and the fact that states cannot be measured directly. Additionally, since a measurement alters the state, the system needs to be prepared in the same state prior to measurement each time. Likewise, in statistics, determining whether a model matches a known statistical state requires the same methodology. Studying the features of quantum states offer another way of coming at the same statistical concept, and allows students to further apply this concept to a physical example.

Learning Outcome:
The more measurements you make on identically prepared quantum systems, the more accurate your information about the state.

Standards Alignment (Common Core State Standard):

• (HS.S.IC) Decide if a specified model is consistent with results from a given data-generating process, e.g., using simulation. For example, a model says a spinning coin falls heads up with probability 0.5. Would a result of 5 tails in a row cause you to question the model?

Course:

• Algebra 2

Description:
Qubits store information in the form of quantum states. A quantum state expresses the probabilities of every possible measurement outcome. The very beginning of this concept occurs in 7th grade, when students learn how to interpret probabilities between 0 and 1. The idea of superposition of quantum states provides an application where students can explore the concept of probability.

Learning Outcome:
A quantum bit holds information in superposition, with a probability of each outcome.

Standards Alignment (Common Core State Standard):

• (7.SP.5) Understand that the probability of a chance event is a number between 0 and 1 that expresses the likelihood of the event occurring. Larger numbers indicate greater likelihood. A probability near 0 indicates an unlikely event, a probability around 1/2 indicates an event that is neither unlikely nor likely, and a probability near 1 indicates a likely event.

Course:

• Grade 7 Math

Description:
To understand the difference in storage between classical and quantum systems, we first need to be able to calculate how much information a set of classical bits can store (calculating and reasoning about quantum bits is a separate learning goal). An n-bit number has two choices for each bit, giving it 2ⁿ unique values, as addressed by HS.S.CP.8. It only stores one of those values at any point in time.

Learning Outcome:
n classical bits can hold one of 2ⁿ possible values.

Standards Alignment (Common Core State Standard):

• (HS.S.CP.8) Apply the general Multiplication Rule in a uniform probability model, P(A and B) = P(A)P(B|A) = P(B)P(A|B), and interpret the answer in terms of the model.

Course:

• AP Statistics

Description:
A set of n bits has two possible choices for each bit, giving it 2ⁿ unique values, as addressed by HS.S.CP.8. A set of n qubits can store a superposition of any or all of these 2ⁿ possible values, with each value having a given probability of being measured. Bringing in the storage format (vectors) makes this learning goal dependent on the quantum state vector learning goals (LG.2.1 and LG.2.2).

Learning Outcome:
n quantum bits can hold a superposition of 2ⁿ possible states, with each state having a certain probability of being measured.

Standards Alignment (Common Core State Standard):

• (HS.S.CP.9) Use permutations and combinations to compute probabilities of compound events and solve problems.

Course:

• AP Statistics

Description:
When qubits are entangled, the probability of their combined outcomes is no longer always the product of the probability of the independent outcomes. This matches with the definition of independent probabilities as well as determining whether or not probabilities are independent. Independence of probabilities determines whether or not multiple qubits are entangled.

Learning Outcome:
Measurement outcomes of entangled quantum bits are not all independent.

Standards Alignment (Common Core State Standard):

• (HS.S.CP.2) Understand that two events A and B are independent if the probability of A and B occurring together is the product of their probabilities, and use this characterization to determine if they are independent.

Courses:

• Geometry
• Algebra

Description:
The amount of information stored in classical bits and quantum bits is described LG.4.2 and LG.4.3. This learning goal does the final comparison between the amount of information that can be stored in a classical computer and on a quantum computer. Therefore, it engages the same mathematical concepts as the previous two learning goals.

Learning Outcome:
More information can be stored in a quantum computer than a classical computer.

Standards Alignment (Common Core State Standard):

• (HS.S.CP.9) Use permutations and combinations to compute probabilities of compound events and solve problems.

Course:

• AP Statistics

Description:
The amount of time it takes to run programs that solve computational problems on classical computers and quantum computers are compared by plotting linear, polynomial (of 2nd or higher degree), and exponential functions, where the graph displays the problem size vs. the running time of the program. Students can use this to understand the speed advantage of quantum computing. They can see that there are crossover points between classical computation and quantum computation – quantum computation starts at a higher point (slower), but as the problem size increases, its running time grows more slowly. Therefore, at some point, there is a crossover between the running time of quantum and classical computation.

Learning Outcome:
Quantum algorithms can solve certain computational problems in polynomial time that require exponential time for classical computers.

Standards Alignment (Common Core State Standard):

• (HS.F.LE.3) Observe using graphs and tables that a quantity increasing exponentially eventually exceeds a quantity increasing linearly, quadratically, or (more generally) as a polynomial function.

Courses:

• Algebra 1
• Algebra 2

Description:
One of the most prominent algorithms for quantum computers applies to the area of cryptography. Students can explore the basis of cryptography by exploring the difference in runtime between calculating a function and its inverse. When a function is easy to compute, but has an inverse that is hard to compute, it is called a one-way function. One-way functions play a fundamental role in many cryptographic applications. An example of a one-way function is the operation of multiplying prime numbers. On a classical computer, computing a product of two numbers is easy, but factoring a product of prime numbers is extremely difficult. Shor’s algorithm, however, factors much more quickly than a classical computer, allowing fast decryption (once a quantum computer large enough to solve it is built) of commonly used types of public-key cryptography. Students can explore different one-way functions, their runtimes, and how quantum algorithms will even out the encryption vs decryption runtimes.

Learning Outcome:
Quantum algorithms can break certain types of classical cryptography, which rely on the disparity between time to calculate a function’s output vs its inverse.

Standards Alignment (Common Core State Standard):

• (HS.F.LE.3) Observe using graphs and tables that a quantity increasing exponentially eventually exceeds a quantity increasing linearly, quadratically, or (more generally) as a polynomial function.

Courses:

• Algebra 1
• Algebra 2

Description: Because quantum computers have errors so frequently, they require error mitigation in the form of fault tolerance. Without needing to understand exactly how they work, students can get a sense of one of the costs of fault tolerance: extra quantum bits. Depending on the error rate (per gate operation), the number of gates, and the number of logical qubits desired, the calculation for the number of physical qubits necessary to perform error-tolerant computation is the following: \( O(q(log \frac{kq}{p_f})^2) \) where p_f is the overall probability of failure (error rate), k is the number of gates, and q is the number of logical qubits. Students can graph the number of physical qubits necessary either as the error rate changes, the number of gate operations changes, and/or the number of logical qubits changes. Learning Outcome: Fault tolerance requires extra quantum bits to protect computation. Standards Alignment (Common Core State Standard):

• (HS.A-REI.11) Explain why the x-coordinates of the points where the graphs of the equations y = f(x) and y = g(x) intersect are the solutions of the equation f(x) = g(x); find the solutions approximately, e.g., using technology to graph the functions, make tables of values, or find successive approximations. Include cases where f(x) and/or g(x) are linear, polynomial, rational, absolute value, exponential, and logarithmic functions.

Courses:

• Algebra 1
• Algebra 2

Description:
There is an element of randomness in all measurement outcomes (LG.3.1 and LG.3.2). If we measure the position or momentum of a particle in motion, there will be a spread of possible outcomes that can be measured by the standard deviation (HS.SD.ID.2). The Heisenberg uncertainty principle describes mathematically how small we can make the uncertainties in the position and momentum. It says that multiplying the standard deviation for the position measurement and the standard deviation for the momentum measurement is always bigger than a certain fixed number. Students can explore the consequences of the uncertainty principle by interpreting the product of the two uncertainties as the area of a rectangle of fixed area. “Squeezing” can also be discussed by manipulating the length and width of this rectangle.

Learning Outcome:
In quantum systems, there is a tradeoff in measurement precision between two attributes like position and momentum.

Standards Alignment (Common Core State Standard):

•  (HS.SD.ID.2) Use statistics appropriate to the shape of the data distribution to compare center (median, mean) and spread (interquartile range, standard deviation) of two or more different data sets.

Course:

• Algebra 1