Diagrammatic Reasoning (pt. 1)CSE 490Q: Quantum Computation

• Linear algebra is not always easy to follow or illuminating

• Roger Penrose invented a graphical notation for linear algebra

• Inspired a number of graphical approaches forunderstanding quantum mechanics• see, e.g., “categorical quantum mechanics”

• (Richard Feynman used diagrams in quantumfield theory, but the math is very different.)

Diagrammatic Reasoning

Basic components• States are lines• Operations are boxes• Connected operations are composed• Tensor products drawn side-by-side

• Will draw states vertically to avoidconfusion with circuits

Graphical Notation

• Notation works well because it implies facts that are true• (saw some of this with circuits also)

Graphical Notation

• Conjugate transpose by flipping about the horizontal axis and daggering boxes

Graphical Notation

• Add additional facts about particular operations• e.g., our facts about SWAP from lecture on Quantum Information

Graphical Notation

• Need to support measurements as well• linear but not unitary operations

• One approach is to introduce the following operations:

Measurements

• One approach is to introduce the following operations:

Measurements

Checks that the bits matchFails if they differ• does not have full rank!

Prepares an (un-normalized)equal superposition state

• One approach is to introduce the following operations:

Measurements

• Conjugate transpose are these operations:

Measurement

• These may seem like strange operations to include

• But they have very nice mathematical properties

• They are useful for helping us analyze and understand quantum processes

• Circuit diagrams aim to describe quantum computations• Diagrammatic reasoning aims to understand quantum computations

Measure & Prepare

Spider Theorem: Two different diagrams built from measure, prepare, copy, and erase are equivalent iff they have the same number of inputs and outputs

Spider Theorem

Example

Example

Example

Example

Example

Example

Example

Example

A Useful Shape

Prepare & Copy

• Some combinations are so common that we have special notation for them:

Trace

• That simplifies the trace to just this:

Cup and Cap

• Cup and cap have additional useful properties such as

• This makes it even easier to prove tr(AB) = tr(BA)

More Prepare

• Our prepare state only gives us an (un-normalized) equal superposition

• However, we can prepare any |x> just by applying an appropriate unitary

More Measure

• Conjugate transposes measures & check that we are in state |x>• failing if this is not true

More Measure

• Conjugate transposes measures & check that we are in state |x>• failing if this is not true

• Prepares a properly normalized |x> state but measuring