Quantum Circuits | Understanding Quantum Information & Computation: Lesson 03

This lesson introduces the quantum circuit model, as well as some mathematical concepts that are important to quantum information including inner products, orthogonality, and projections. Fundamental limitations of quantum information, including the no-cloning theorem, are also discussed.

Timecodes for lesson sections:

0:00 — Introduction
1:06 — Lesson overview
2:28 — Circuits
7:25 — Quantum circuits
18:27 — Inner products
24:15 — Orthogonality and orthonormality
32:35 — Projections
36:22 — Projective measurements
43:47 — Irrelevance of global phases
48:13 — No-cloning theorem
53:37 — Discriminating non-orthogonal states
59:59 — Conclusion

#ibmquantum #learnquantum #qiskit

[AAAE]

These are the best lectures ever made by Qiskit/IBM by far. Please assign more people to the team to get the whole series completed as soon as possible. Thanks for your effort.

[loganfisher]

At 46:35 you mentioned using density matrices in the "third unit of this series". This confuses me a bit, as this is lesson 3, and I don't see any mention of density matrices in any of the other courses either. Was this cut content and the lessons were renumbered? Do you intent to release lessons on the general description using density matrices in the near future?

I've been really enjoying these videos and getting a lot out of them. Thank you for your hard work in producing them.

[isaijeanmary]

I am Isai, I follow this course... really clear and neat teaching of a such complex topic...

[Radical]

I'm glad i could learn this stuff somewhere being in cyber security this kind of information is so important! Thanks!

[hakandoga]

Wonderful series, looking forward to the next video!

[quantumradio]

An important distinction is that matrix multiplication is not the same as the "tensor product". Specifically it helped me to see the definitions in Lecture 1.2 (pp71-72) for: (1) controlled-NOT operation where 1st qubit is the control, and (2) controlled-NOT operation where the 2nd qubit is the control. Once I considered this, the matrix representation at 12:23 came out in the wash.

[aleksybalazinski]

Gotta love "acyclic circuits"!

[komalkumar]

Thank you sir 🙏
Continue this lecture

[sheidalv]

At the 41:52 minute mark in the video, there is an equation that appears to be incorrect. The first right part element should be written as |φ +><φ +|; it seems there is a small error.

[pdelong]

@36:00 - I don't know why, but I didn't make the mental connection between projections and subspaces until they were mentioned in the same breath here. I guess it finally took hearing the topic explained from multiple angles... (no pun intended)

[JorgeGranada-qt]

Happy to discover the "circuit" approach, even when it is strongly emphasized these circuits shouldn't be understood as electric circuits. From an engineering perspective (my background) this aids a lot. Even one dreams with an FPGA-like deployment where complex applications are built on top of mature IP and one does not to start from the scratch when creating end user applications. Just out of personal curiosity, time is assumed "from left to right" but I'm not able to see the mathematical resource representing the advancement of time to synchronized status changes. Maybe is it there, and I just didn't get it, or maybe time evolution is conceived different for quantum information processing.

[salvatoremarino]

Hello, I have a question: regarding what was said around minute 54, what does it mean to initialize a quantum gate? and how do it? Thanks for the excellent lectures Prof. Watrous!

[tatpongkatanyukul]

Very engaging!

BTW, I have a question. Can you give me an example of experimental measurements corresponding to measure a state in basis of |0> and |1> versus a state of |+> and |->?
Like, when we represent spin up and down to be |0> and |1>, what would correspond to |+> or |->?

[asmaa.ali]

Hello. First, thank you very much for these precious lectures. Second, I am curious about the example of representing the projective measurement by quantum circuit at 42:15 . I tried to check it my self and for the projective measurement \pi_1 acting on |00>, to inner products of two qubits and finally get: 1/2{|01><01|00> - |01> <10|00> - |10><01|00> +|10> <10|00>} . and I stopped . how these inner products are evaluated. And I reached by tracking the circuit and the measurement of the lower qubit gives ket zero with probability 1 entering |00?> for the upper qubits. please I would be grateful if you help me on this . Thank you very much

[bvkssameer]

8:42 why does the operation gets reversed. Can anyone explain this, could not get any explanation of this : )

[pdelong]

@29:00 - ah, that's a mild let-down; one of the things I was kind of hoping for was a decent explanation of Graham-Schmidt (without having to go through an entire linear algebra course);

I started going through Linear Algebra Done Right (Axler), which gets to it later, but I've stalled...

That's not to say I'm ungrateful. I'm very much enjoying this (free) course so-far.

[idiosyncraticavenue]

I feel like i am seeing an error at 12:49, the C-NOT gate's matrix seems to have their 2nd, 3rd and 4th row swapped.

[muzamelyahia]

Thanks for the great video. Probably I am asking a question that can be answered later, but here it is: say I know the general description of the quantum circuit i.e. how qubits can interact with each other, but I haven't decided on the basis yet i.e. instead of working on {|0>, |1>}, I would like to work with the basis {|00>, |01>, |10>, |11>} and do the measurements based on this basis using inner product. Is the circuit model general enough for such application?

[byronwatkins]

At 16:00, why doesn't a classical measurement collapse the state? How can it be measured AND continue on?

[TravelHubv]

Thanks for the content, you're doing a lucrative job, but I've got a question. This video uses a CNOT gate that is quite different from what I usually encounter, and I even tried to take the opposite of the matrix, but it didn't work, so if you can explain this to me I would appreciate it. Thanks