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Quantum Class 9, Mon 2022-09-26

1 Respite not so bad after all

It trapped a message from AEFIS.

2 Caltech - excellent quantum computing sites

Quantum Science and Engineering

Institute for Quantum Information and Matter, a National Science Foundation Physics Frontiers Center

Alliance for Quantum Technologies

https://scienceexchange.caltech.edu/topics/quantum-science-explained/quantum-computing-computers

https://www.quantamagazine.org/entanglement-made-simple-20160428/ - Frank Wilczek, the author, got a nice free meal with the King of Sweden in 2004.

https://www.science.org/content/article/einstein-s-spooky-action-distance-spotted-objects-almost-big-enough-see

https://www.space.com/31933-quantum-entanglement-action-at-a-distance.html

3 Entanglement

Some sites that might help understanding it.

https://magazine.caltech.edu/post/untangling-entanglement

4 Quantum algorithms

We'll learn some famous quantum algorithms from the old qiskit text, starting with

oracles .

Quantum Class 7, Mon 2022-09-19

1 Student presentations

Mon:

  1. Noah Prisament - Bose Einstein condensate quantum computing

  2. Alex Bozeat - NMR

  3. Ahmed Elmenshawi - Spin Qbit

  4. Adam Goines - Neural Atom Quantum Computers

  5. Sanghyun Kim & Charles Chae - Photonics & Orca computing

  6. Oliver Salvaterra - optical lattice

    Thurs:

  7. Richard Pawelkiewicz - Vibrating Atoms

  8. Denzell Dixon - Photonic quantum computing

  9. Steven Laverty - Diamond Quantum Computing

  10. Alice Bibaud - Topological photonic chips

  11. Vansh Reddy Cheguri - Bose Einstein

2 Homework 5 online

Here.

due next Mon.

Quantum Class 6, Thu 2022-09-15

1 Homework 4 ctd

You talk next Mon. See here.

Use https://doodle.com/meeting/participate/id/dBLvyWka

to sign up. Give your topic and enough of your name that I can id you.

There's only one time; the purpose of this is to state your topic in a way that other people can see.

(By posting the link, I'm opening myself to potential spamming, but so far, that hasn't happened.)

2 Old homeworks

To accommodate late adders (students not Viperidae), I've extended the late due dates for the homeworks. Anyone else may also update their answers.

However I'll probably grade quite easily so it isn't necessary.

3 My class conflicts

  1. To accommodate a presidential reception that I'm attending, today will end at 4:45. Last year, I asked Pres Jackson what to do about the conflict. She said to end class early.

  2. Still thinking about what to do for the presidential installation, which might overlap if it runs late.

    I recommend attending the presentations; I am. These are a benefit of being at RPI.

  3. There'll be no classes the first week of Nov because I'll be at ACM SIGSPATIAL. I'll probably assign outside reading and watching.

4 COVID

What shall we do about quarantined students? I could try to webex the class from my iphone. But that quality will probably be bad. So here's a suggestion:

Any quarantined student, and also nonquarantined ones, is welcome to personal webex calls with me to ask questions about quantum computing etc. We can call these office hours.

If you want one, email with some times; I'll reply. When we agree, I'll call then.

5 IBM Quantum

Continuing what was started last class with qiskit etc.

Quantum Class 5, Mon 2022-09-12

1 Homework 4

You talk next Mon. See here.

2 Quantum computing in the news

(or at least on Slashdot).

  1. https://yro.slashdot.org/story/21/09/04/2147245/americas-nsa-isnt-sure-quantum-computers-will-ever-break-public-key-encryption

3 Abstract computation models ctd

  1. Original motivation was to discover an algorithm for proving (or disproving) theorems.

  2. That can be done in some simple cases, like first order predicate calculus with addition over the integers.

    1. and first order predicate calculus with addition and multiplication over the rationals or reals.

  3. This goal failed because it was proved that it is undecidable in some cases.

    1. like first order predicate calculus with addition and multiplication over the integers.

    2. Some theorems truth or falsity depends on the allowable domain of their variables.

    3. in a deep sense, ints are harder than reals.

    4. Long time ago I wrote a paper on this, in the context of computer graphics. Problems with Raster Graphics Algorithms.

4 Complexity classes

  1. Group problems into broad classes of considerably differing difficulty.

  2. P vs NP.

  3. Steve Cooks's paper first describing this was rejected.

  4. Quantum complexity classes.

5 Hardware implementations

  1. Quantum computation was theoretically started decades before actual quantum computers were designed.

  2. Just like classical computers.

  3. Many competing technologies.

  4. Let the strongest win.

5.1 Superconducting qubits

  1. Dilution fridge: cool by mixing He3 into He4.

  2. Cooper pairs of electrons: pairs of electrons in a metal weakly attract each other. It's a quantum effect.

  3. Josephson Junction.

  4. good ref: A Quantum Engineer's Guide to Superconducting Qubits

5.1.1 Transmon qubit

  1. Sutor: Under the hood of IBM Q

  2. The transmon qubit | QuTech Academy 6:03.

  3. Alexandre Blais - Quantum Computing with Superconducting Qubits (Part 1) - CSSQI 2012 45:11.

  4. Control of transmon qubits using a cryogenic CMOS integrated circuit (QuantumCasts) 35:47.

5.2 Trapped Ion

  1. https://en.wikipedia.org/wiki/Trapped_ion_quantum_computer

  2. Proponents say that it's better than transmon qbits.

  3. Trapped-ion qubit, the maglev train of a quantum computer, 9:34, 2021-08-24.

  4. https://ionq.com/technology

    "To date, we’ve run single-qubit gates on a 79 ion chain, and complex algorithms on chains of up to 11 ions."

5.3 Quantum annealing

  1. This is not comparable to quantum gates and circuits like IBM has.

  2. It minimizes a function by testing many solutions in parallel.

  3. See details in the D-Wave section.

  4. Qbit count is not comparable to gate models.

  5. https://en.wikipedia.org/wiki/D-Wave_Systems

  6. They make a different type of quantum computer, called a quantum annealer. They have been in the news lately, e.g.,

  7. https://arstechnica.com/science/2020/09/d-wave-releases-its-next-generation-quantum-annealing-chip/

  8. What is Quantum Annealing? 6:14.

  9. How The Quantum Annealing Process Works 6:09.

  10. Quantum Programming 101: Solving a Problem From End to End | D-Wave Webinar 54:25.

    "Want to learn how to program a quantum computer? In this webinar, we explain how to do so by running through a complete, simple example. We explain how to formulate the problem, how to write it, and how to tune it for best results. "

    "This webinar is intended for those with little or no experience programming on a D-Wave quantum computer. After watching, get free time on Leap, the quantum cloud service at https://cloud.dwavesys.com/leap/signup/ "

  11. Slides from Programming Quantum Computers: A Primer with IBM Q and D-Wave Exercises by Frank Mueller, Patrick Dreher, Greg Byrd held at PPoPP (Feb 2019) ASPLOS'19 (Apr 2019),

    Part 3: D-Wave -- Adiabatic Quantum Programming

  12. D-Wave factoring tutorial and other demos

    including Jupyter notebooks (you have to login for them).

5.4 Photonics

  1. Xanadu Quantum Cloud

  2. https://venturebeat.com/2020/09/02/xanadu-photonics-quantum-cloud-platform/

  3. Uses photonics.

  4. Operates primarily at room temperature.

  5. Up to 24 qbits, gate depth of 12.

  6. Has free SW tools, some of which can compile to other quantum technologies.

  7. Expected good applications: graphs and networks, machine learning, and quantum chemistry.

  8. They expect to scale up better than competing technologies.

  9. Operates at room temperature.

Quantum Homework 4, Mon 2022-09-12

  1. Student presentation to class next Mon 2022-09-19.

  2. Pick an approach to building quantum computers that is not used by a major company like IBM, Google, Microsoft, Amazon, Rigetti or D-wave. Hidary Chapter 5 has lots of suggestions. Describe the idea, and its major advantages and disadvantages.

  3. About 5-7 minutes. I won't be using a stopwatch, but be respectful of the other students. How the Ig Nobel awards handles this.

  4. Suggestion: test your computer with the classroom setup in advance. The room is open.

Total: 10

Quantum Class 4, Thurs 2022-09-08

1 Scientifically literate people and quantum computing

About people who are generally scientifically aware, but not really practitioners of science. What do you think people like that would like to know about Quantum Computing? Many people have heard about it, but most people don’t know what it means. What do you think their “burning questions” about it might be?

2 Handwritten notes are online

here. The file name is the class number; last time was 03.pdf . These are not really intended to stand alone; my typed blog is primary.

However, would you like me to add more details as I write them in class?

3 Homework 3 is online

(I think).

due next time.

4 Questions on last videos

  1. IBM:

    1. What are the roles of the kernel, algorithm, and model developers?

    2. Tell me about error suppression.

  2. Google:

    1. What are they doing?

    2. How do they control a qbit?

    3. How to they manage errors?

5 Videos to watch for next time

  1. The Map of Quantum Computing | Quantum Computers Explained, 33:27,

  2. Quantum Entanglement: Spooky Action at a Distance 14:41. 2020-02-12

6 Bloch sphere

  1. https://en.m.wikipedia.org/wiki/Bloch_sphere.

  2. The state of a qbit can be represented as a point on or in a sphere of radius 1.

  3. E.g., |1> is the north pole, |0> the south pole.

  4. Many operations are rotations.

  5. I don't think this idea is particularly big, but people like it.

  6. It does not extend to multiple qbits.

7 Matrices

  1. Quantum computing operators are unitary.

    The conjugate transpose is the inverse.

  2. Measurement operators are self-adjoint aka Hermetian.

    The conjugate transpose is the matrix itself.

8 Hidary Chapter 3: Qubits, Operators and Measurement

with additions from other sources.

8.1 Two qubit operators

  1. Now, let $q$ be a system with two qubits, i.e., a 2-vector of qubits.

  2. $q$ is now a linear combo of 4 basis values, $ | 00\!\!>$, $ | 01\!\!>$, $ | 10\!\!>$, $ | 11\!\!>$.

  3. $q = a_0 | 00\!\!> + a_1 | 01\!\!> + a_2 | 10\!\!> + a_3 | 11\!\!> $

  4. where $a_i$ are complex and $ \sum | a_i | ^2 = 1$.

  5. $q$ exists in all 4 states simultaneously.

  6. If $q$ is a vector with n component qubits, then it exists in $2^n$ states simultaneously.

  7. This is part of the reason that quantum computation is powerful.

  8. A measurement operator applied to $q$ will rotate it to a basis {00, 01, 10, 11}, so that it will be observed in one of those four cases, with probabilities $ | a_i | ^2$.

  9. You operate on $q$ by multiplying it by a 4x4 matrix operator.

  10. The matrices are all invertible (except for measurement matrices), and all leave $ | q | = 1$.

  11. You set the initial value of $q$ by setting its two qubits each to 0 or 1.

  12. How this is done depends on the particular hw.

  13. I.e., initially, $q_1 = \begin{pmatrix}a_1 | 0\!\!> \\b_1 | 1\!\!> \end{pmatrix}$ and $q_2 = \begin{pmatrix}a_2 | 0\!\!> \\b_2 | 1\!\!> \end{pmatrix}$, and so

    $$q = \begin{pmatrix} q_1 \\ q_2 \end{pmatrix} = \begin{pmatrix} a_1 a_2 | 00 \!\!> \\ a_1 b_2 | 01 \!\!> \\ a_2 b_1 | 10 \!\!> \\ b_1 b_2 | 11 \!\!> \end{pmatrix}$$.

  14. The combined state is the tensor product of the individual qubits.

  15. In this case, you could separate out the individual qubits again.

  16. However, sometimes after operating on the combo (i.e., multiplying by a matrix), you cannot any more separate out the result into a tensor product of individual qubits.

  17. For $n$ qubits, the tensor product is a vector with $2^n$ elements, one element for each possible value of each qubit.

  18. Each element of the tensor product has a complex weight.

  19. You transform a state by multiplying it by a matrix.

  20. The matrix is invertible.

  21. The transformation doesn't destroy information.

  22. For some sets of weights, particularly after a transformation, the combined state cannot be separated into a tensor product of individual qubits. In this case, the individual qubits are entangled.

  23. That is the next part of why quantum computation is powerful.

  24. Entanglement means that if you measure one qubit then what you observe restricts what would be observed when you measure the other qubit.

  25. However that does not let you communicate.

  26. The current limitations are that IBM does only a few qubits and that the operation is noisy.

8.2 Common operators

  1. Common operations (aka gates):

    https://en.wikipedia.org/wiki/Quantum_logic_gate

    1. Swap

    2. controlled not CNOT cX

      When input is general, it's more sophisticated that it looks.

    3. Hadamard.

      1-qbit creates a superposition.

      https://cs.stackexchange.com/questions/63859/intuition-behind-the-hadamard-gate

      2-qbit creates a uniform superposition

      https://en.wikipedia.org/wiki/Quantum_logic_gate

    4. Toffoli, aka CCNOT.

      Universal for classical boolean functions.

      (a,b,c) -> (a,b, c xor (a and b))

      https://en.wikipedia.org/wiki/Toffoli_gate

    5. Fredkin, aka CSWAP.

      https://en.wikipedia.org/wiki/Fredkin_gate

      3 inputs. Swaps last 2 if first is true.

      sample app: 5 gates make a 3-bit full adder.

  2. Bell state

    1. Maximally entangled.

    2. Hadamard then CNOT.

8.3 Entanglement of 2-qbit system

State is a 4-vector of length 1.

It was originally created as the exterior product of two 2-vectors, the states of two separate 1-qbit systems.

Originally the separate 1-qbit systems didn't affect each other. Either could be transformed and measured.

Then 2-qbit system was rotated with a transformation matrix.

Now, perhaps it can be decomposed into the exterior product of two 2-vectors. Perhaps not.

  1. Case 1: The 4-vector representing the new state can be decomposed.

    1. Then it's still really two separate 1-qbit systems.

    2. They can still either be transformed and measured.

    3. Measuring one qbit does not affect the other qbit.

  2. Case 2: The 4-vector representing the new state cannot be decomposed.

    1. So the 2 qbits are now entangled.

    2. That means that measuring one qbit affects what you will see when you measure the other.

    3. It might just bias the probabilities of measuring the other qbit as 0 or 1.

    4. Or, it might totally control what you will see.

8.4 Entangling with Toffoli

  1. Here's another way to look at this. Review:

  2. $x'=x \\ y'=y \\ z'= z \oplus xy$

  3. Use those equations only with classical bits, otherwise use the matrix multiplication.

  4. Let x=1, z=0. Then, x'= 1, y'= y, z'= y.

  5. So if y=0, then x'= 1, y'= 0, z'= 0.

  6. and if y=1, then x'= 1, y'= 1, z'= 1.

  7. If y is a superposition of 0 and 1, then the output will be a superposition of the above 2 cases.

  8. That is, 50% of the time, we measure y'= 0, z'= 0 and 50% of the time we measure y'= 1, z'= 1.

  9. We always measure y' and z' the same.

  10. Even if we transport z' a long distance away first.

9 Quantum properties - No Cloning

I made some supplementary material on cloning (copying), which works in the classical world but not in the quantum world.

Classically, you can easily clone a bit. Consider a 2-bit system, $x, y$. Each bit can be 0 or 1. All 4 combos are possible. Represent the state with a 4-vector

$s=\begin{vmatrix} a_0\\a_1\\a_2\\a_3\end{vmatrix}$

where exactly one $a_i=1$ and the other three are 0. E.g., if $a_3=1$ then $x=y=1$.

Here's a 2-input, 2-output circuit that clones the first bit.

$x'=x\\y'=x$

Its truth table is

x y | x'y'
0 0 | 0 0
0 1 | 0 0
1 0 | 1 1
1 1 | 1 1

Represent the operation by a matrix multiplication on s. The matrix M is:

1 1 0 0
0 0 0 0
0 0 0 0
0 0 1 1

That is, the final state is $s'_i = \sum_j M_{ij} s_j$

However, M is singular. For quantum operations, M has to be unitary. So this is not a legal quantum circuit. Let's try again.

The better way is the 3-input Toffoli gate. The function is

$x'=x \\ y'=y \\ z'= z \oplus xy$

The truth table is

x y z | x'y'z'
0 0 0 | 0 0 0
0 0 1 | 0 0 1
0 1 0 | 0 1 0
0 1 1 | 0 1 1
1 0 0 | 1 0 0
1 0 1 | 1 0 1
1 1 0 | 1 1 1
1 1 1 | 1 1 0

The matrix is Eqn 5.62 on page 155.:

1 0 0 0 0 0 0 0
0 1 0 0 0 0 0 0
0 0 1 0 0 0 0 0
0 0 0 1 0 0 0 0
0 0 0 0 1 0 0 0
0 0 0 0 0 1 0 0
0 0 0 0 0 0 0 1
0 0 0 0 0 0 1 0

Let x=1, z=0. Then:

x'= 1
y'= y
z'= y

and we've cloned y in the classical case, where the inputs are each 0 or 1.

This matrix is nonsingular and so is also legal in a quantum circuit.

(Looking ahead a little), try $y= \frac{1}{\sqrt{2}} | 0> + \frac{1}{\sqrt{2}} | 1>$, which is an equal superposition of 0 and 1. Using x=1, z=0, the input will be

$\frac{ | 1, 0, 0> + | 1, 1, 0>}{\sqrt{2}}$

and the state is

$(0, 0, 0, 0, \frac{1}{\sqrt{2}} , 0, \frac{1}{\sqrt{2}} , 0)^T$

That's an equal superposition of 2 of the possible 8 classical input states.

Multiplying that by the matrix gives

$\frac{ | 1, 0, 0> + | 1, 1, 1>}{\sqrt{2}}$

Instead of cloning y into z, this entangled y and z.

An operation that is simple on classical input can be more complicated on quantum inputs.

This isn't new; we already know how to entangle two qbits.

Engangling isn't the same as cloning. The point of cloning is that we could operate on the two bits separately. If they're engangled, operating on one affects the other.

Quantum Homework 3, Thurs 2022-09-08

Due 2022-09-12 4pm in gradescope. Groups of 2 are ok; submit one answer set.

(These questions are from Hidary.)

  1. (10 points) Investigate the Stern-Gerlach experiment of 1921. What was the expectation from classical theory for the outcome and what actually occurred?

  2. (10 points) What does non-separable mean for two states and what does that tell us about these states?

  3. (10 points) For the CZ gate, does it matter which qubit is the control qubit and which is the target?

  4. (10 points) What is the final state of the following circuit, in Dirac notation?

/files/hidary-hw-3-1b.png

Total: 40

Quantum Misc notes, Thurs 2022-09-08

Misc notes that I draw from for lectures.

1 Complexity theory - Hidary chap 4

  1. Problems vs algorithms.

  2. Interesting types of resources: time, space, ...

  3. Worst case time, best case time.

  4. We want to group problems and algorithms into classes in a way that captures their important properties and ignores the others.

  5. competing formal ways to describe algorithms: Turing machine, Church's lambda calculus, ...

    1. different ones had different power (could describe different classes).

    2. the ones listed above seemed to all have the same power.

  6. Universal Turing machine.

  7. Church-Turing Thesis (CTT): If an algorithm can be performed on any piece of hardware (say, a modern personal computer), then there is an equivalent algorithm for a Universal Turing Machine (UTM) which performs exactly the same algorithm [161, p. 5].

  8. Strong Church-Turing Thesis (SCTT) Any algorithmic process can be simulated efficiently using a Universal Turing Machine (UTM)

  9. randomness can sometimes lead to a faster algorithm.

  10. Extended Church-Turing Thesis (ECTT) Any algorithmic process can be simulated efficiently using a Probabilistic Turing Machine (PTM) [161, p. 6].

  11. Quantum Extended Church-Turing Thesis (QECTT) Any realistic physical computing device can be efficiently simulated by a fault-tolerant quantum computer.

2 Hardware implementations

  1. Quantum computation was theoretically started decades before actual quantum computers were designed.

  2. Many competing technologies.

  3. Let the strongest win.

3 New material

Various things designed to solidify your understanding of the fundamentals.

3.1 Math review from Quantum Computing for Computer Scientists

  1. Chapter 2:

    1. Complex vector space, page 34.

      1. n-dim vector $\begin{vmatrix} c_0 \\ \cdots \\ c_{n-1}\end{vmatrix}$

      2. Add 2 vectors.

      3. Multiply vector by scalar.

      4. etc.

    2. Set of $m\times n$ complex matrices, $\mathbb{C}^{m\times n}$, is also a complex vector space.

      1. Transpose, conjugate, adjoint.

      2. Matrix mult is $\star$ in book.

    3. $\mathbb{C}^{m\times n}$ is a complex algebra.

    4. Set of polynomials in one variable of degree $le n$ is a complex vector space.

    5. State of a quantum system is a complex vector.

    6. You can make new vector spaces from combos of old ones.

      1. Cartesian product or direct sum.

      2. Just an ordered pair.

      3. $(v_1, v_2)$.

    7. Set of basis vectors for the vector space.

      1. Every vector $v$ in the space is a linear combo of the basis vectors.

      2. Represent $v$ as the list of weights.

      3. There are many possible basis sets.

      4. Each different basis set causes a different representation for the vectors.

      5. Convert: change of basis.

      6. Cartography example: NAD27, WGS84.

        A bridge between Switzerland and Germany across the Rhine River at Laufenberg had its two ends at different elevations because of a conversion error between two different basis systems.

        https://www.science20.com/news_articles/what_happens_bridge_when_one_side_uses_mediterranean_sea_level_and_another_north_sea-121600

      7. Hadamard matrix is an example.

    8. Section 2.4 Inner product, etc, p 53

    9. Section 2.7 Tensor product of vector spaces, p 66.

      1. $\mathbb{V} \otimes \mathbb{V'}$.

      2. Let $dim(\mathbb{V})=p$ and $dim(\mathbb{V'})=q$ . Then $dim(\mathbb{V} \otimes \mathbb{V'}) = pq$.

      3. This is how quantum systems combine.

      4. Example 2.7.2 p 70.

  2. From end of section 2.3, p.52 to 2.6.

    1. P 52. Transition matrix to convert a vector representation from the canonical basis this another basis is the Hadamark matrix.

    2. Section 2.4. Add an inner product operator to the complex vector space. Note the conjugates in the rules; you don't see them with a real vector space.

    3. Norm, aka length.

    4. We won't need limits etc.

    5. Section 2.5 Eigenvalues and eigenvectors

      1. Eigenvalues don't depend on the representation. Converting to a new basis set doesn't change them.

      2. Geometrically, in 2D, in you transform a circle centered at origin, eigenvalues are lengths of the axes. Eigenvectors are the axes.

    6. Section 2.6 Hermitian and unitary matrices

    7. P 39. Adjoint: conjugate transpose of matrix.

    8. If you use its eigenvectors as a basis, then the matrix diagonalizes to a list of its eigenvalues.

    9. P 62. Hermitian matrix: it is its adjoint.

    10. Unitary: its inverse is its adjoint.

    11. Geometrically, they are rotations since they preserve distances.

    12. Notation confusion: the books uses capital letters both for matrices and vectors.

    13. P 71, tensor product of matrices.

      1. Chapter 3 through 3.2, p 74-88.

      1. p 80, doubly stochastic matrix.

      2. Multiplying it by a vector of probabilities gives a vector of probabilities (i.e., they sum to 1 and $0\le a_i \le 1$ ).

      3. P 88, Chapter 3, section 3.3 Quantum systems,

      4. Real probabilities add.

      5. When probabilities are norms of complex numbers, they might cancel.

    14. p 91, Ex 3.2.2

    15. p 93 double slit experiment

    16. p 97 Section 3.4, Assembling systems

3.2 2 slit experiment compared with real vs complex transitions

4 Misc intro to quantum computing stuff

This is misc stuff that you might find interesting, which I'm drawing from.

4.1 Quantum properties - Phase

  1. You cannot measure the phase of qbit.

  2. You can measure the relative phase of 2 qbits.

  3. Many algorithms encode the answer as a phase shift of a qbit.

  4. Phase kickback means that a gate that runs one way, e.g., the control bit affects an output bit, can be made to run the other way, e.g., the control bit is changed, by making the other bit a Hadamad basis.

  5. Phase Kickback V Abhijith Rao

  6. Qiskit Phase Kickback

4.2 Quantum supremacy

  1. Coined by Preskill in 2012.

  2. Google claimed this in Oct 2019 on a specific (artificial?) problem; see Google section.

  3. IBM disagrees.

4.3 Cloud-based computing

  1. IBM started this.

  2. Alibaba followed.

  3. Then D-Wave Leap, Rigetti, Amazon AWS Braket and Quantum solutions lab, Microsoft Azure.

  4. IBM's intent is to entangle their computers in different sites; exponentially increasing power.

4.4 Technologies

  1. Quantum Materials | QuTech Academy 8:57.

/files/ionq_hws.png

4.4.1 Transmon qubit

  1. Sutor: Under the hood of IBM Q

  2. The transmon qubit | QuTech Academy 6:03.

  3. Alexandre Blais - Quantum Computing with Superconducting Qubits (Part 1) - CSSQI 2012 45:11.

  4. Control of transmon qubits using a cryogenic CMOS integrated circuit (QuantumCasts) 35:47.

4.4.2 Trapped Ion

  1. https://en.wikipedia.org/wiki/Trapped_ion_quantum_computer

  2. Proponents say that it's better than transmon qbits.

  3. https://ionq.com/technology

    "To date, we’ve run single-qubit gates on a 79 ion chain, and complex algorithms on chains of up to 11 ions."

4.4.3 Quantum annealing

  1. This is not comparable to quantum gates and circuits like IBM has.

  2. It minimizes a function by testing many solutions in parallel.

  3. See details in the D-Wave section.

  4. Qbit count is not comparable to gate models.

4.4.4 Photonics

  1. Operates at room temperature.

  2. See Xanadu below.

4.5 Companies - Primary

These companies have their own hardware.

4.5.1 IBM

4.5.1.1 Summary
  1. They have several quantum computers, up to 53 qbits.

  2. The older ones are freely available on the web; see https://quantum-computing.ibm.com/

  3. Note that you can put gates between only adjacent qbits.

  4. You submit a batch job and get emailed when it runs.

  5. IBM github site: https://github.com/Qiskit with

    1. a free simulator.

      It doesn't match all the physical complexity of the real computer, but it's a good start.

    2. and tutorials and presentations.

  6. and a SW development framework. https://qiskit.org/

  7. You can create a quantum computation program either by

    1. designing a circuit, or

    2. using a programming language.

4.5.1.2 Sites
  1. IBM Introduces First Integrated Quantum Computing System for Commercial Use

  2. Go Behind-the-Scenes of a Quantum Experiment (2:10) https://quantumexperience.ng.bluemix.net/qx/community/question?questionId=5ae975690f020500399ed39a&channel=videos

    or

    https://www.youtube.com/watch?v=tfZpJLdkzRU&list=PLOFEBzvs-VvpzQnlazij7cL1mjKvJTAwk&index=7

  3. A Qubit in the Making (2:01)

    https://www.youtube.com/watch?v=2pB87H3_F_c&list=PLOFEBzvs-VvpzQnlazij7cL1mjKvJTAwk&index=10&t=0s

  4. Behold the Mighty Qubit (2:51) https://www.youtube.com/watch?v=_P7K8jUbLU0&list=PLOFEBzvs-VvpzQnlazij7cL1mjKvJTAwk&index=10

  5. Classical and Quantum Randomness (3:39) https://www.youtube.com/watch?v=8kyJfAC4VAo&list=PLOFEBzvs-VvpzQnlazij7cL1mjKvJTAwk&index=6

  6. Quantum Entanglement (2:21) https://www.youtube.com/watch?v=RmXasxLm43k&list=PLOFEBzvs-VvpzQnlazij7cL1mjKvJTAwk&index=5

  7. Benchmarking Quantum Systems (1:58) https://www.youtube.com/watch?v=-7L5o-mzLqU&list=PLOFEBzvs-VvpzQnlazij7cL1mjKvJTAwk&index=8

  8. Quantum Pong — Programming on Quantum Computers Season 1 Ep 1 3:54 by Abraham Asfaw.

  9. IBM’s Roadmap For Scaling Quantum Technology

  10. IBM publishes its quantum roadmap, says it will have a 1,000-qubit machine in 2023

  11. https://quantumcomputing.stackexchange.com/questions/tagged/ibm-q-experience

  12. https://quantumcomputing.stackexchange.com/questions/tagged/qiskit

4.5.1.3 Applications on the IBM Q
  1. Quantum Algorithms for Applications from qiskit

  2. HHL Algorithm

    This is in Huawei HiQ, an open-source software framework for quantum computing.

  3. https://en.wikipedia.org/wiki/Quantum_algorithm_for_linear_systems_of_equations

4.5.1.4 IBM Quantum experience
  1. When you design a circuit here, you don't need to simulate it elsewhere. It shows you the probabilities.

  2. Dual view: you can see and edit both the circuit and the QASM code.

  3. There are some sample programs in https://github.com/Qiskit/openqasm.git

  4. Example circuits in https://quantum-computing.ibm.com/docs/iqx/example-circuits .

4.5.2 Intel

  1. Tangle Lake has 49 superconducting qbits.

  2. Produced in Oregon.

  3. Partners with QuTech in Netherlands.

  4. https://www.intel.com/content/www/us/en/research/quantum-computing.html

4.5.3 Google

  1. Sycamore has 54 (-> 53) transmon qbits in 9x6 array, each coupled to 4 neighbors.

    1. https://www.eenewseurope.com/news/googles-sycamore-quantum-processor-shows-supremacy

    2. https://www.nature.com/articles/d41586-019-03213-z

    3. https://jonathan-hui.medium.com/quantum-supremacy-google-sycamore-processor-6f30073a17fa - detailed. ¹

  2. Google quantum General web site.

  3. Day 1 opening keynote by Hartmut Neven (Quantum Summer Symposium 2020) 29:58.

  4. QuantumCasts link to some videos, including the following.

  5. Quantum supremacy explained (QuantumCasts) 4:26.

  6. Introduction to Quantum Chess (Quantum Summer Symposium 2020) 13:53.

  7. Quantum Money (Quantum Summer Symposium 2020) 15:56. Peter Shor. It's fun to see what Shor is thinking about now.

  8. QSI Seminar: Dr Marissa Giustina, Google Research, Building Google's Quantum Computer, 09/06/2020 55:52.

  9. The Man Who Will Build Google's Elusive Quantum Computer

  10. Programming a quantum computer with Cirq (QuantumCasts) 10:39.

4.5.4 D-Wave

  1. https://en.wikipedia.org/wiki/D-Wave_Systems

  2. They make a different type of quantum computer, called a quantum annealer. They have been in the news lately, e.g.,

  3. https://arstechnica.com/science/2020/09/d-wave-releases-its-next-generation-quantum-annealing-chip/

  4. What is Quantum Annealing? 6:14.

  5. How The Quantum Annealing Process Works 6:09.

  6. Quantum Programming 101: Solving a Problem From End to End | D-Wave Webinar 54:25.

    "Want to learn how to program a quantum computer? In this webinar, we explain how to do so by running through a complete, simple example. We explain how to formulate the problem, how to write it, and how to tune it for best results. "

    "This webinar is intended for those with little or no experience programming on a D-Wave quantum computer. After watching, get free time on Leap, the quantum cloud service at https://cloud.dwavesys.com/leap/signup/ "

  7. Slides from Programming Quantum Computers: A Primer with IBM Q and D-Wave Exercises by Frank Mueller, Patrick Dreher, Greg Byrd held at PPoPP (Feb 2019) ASPLOS'19 (Apr 2019),

    Part 3: D-Wave -- Adiabatic Quantum Programming

  8. D-Wave factoring tutorial and other demos

    including Jupyter notebooks (you have to login for them).

4.5.5 IonQ

  1. founders from Maryland/College Park and Duke.

  2. trapped ion

  3. 32 qbits.

  4. low error rate

  5. excellent quantum volume

  6. will be available from Microsoft Azure and Amazon AWS Braket.

  7. https://ionq.com/

  8. https://en.wikipedia.org/wiki/IonQ

  9. https://ionq.com/posts/october-01-2020-most-powerful-quantum-computer

4.5.6 Honeywell

  1. H1: trapped ion.

  2. 10 qbits,

    1. full connectivity,

    2. can read isolated qbits in mid-computation,

    3. hi-res rotations.

  3. JP Morgan experimenting with it.

4.5.6.1 Sites
  1. The World’s Highest Performing Quantum Computer is Here

  2. Details.

  3. Zdnet: Honeywell's System Model H1 quantum computer available to enterprises

  4. They partner with Microsoft,, Experience quantum impact with Azure Quantum, Cambridge Quantum Computing, Zapata Computing, etc.

4.5.7 Rigetti

  1. Berkeley-based

  2. founder is ex-IBM

  3. https://en.wikipedia.org/wiki/Rigetti_Computing

  4. https://www.rigetti.com/

  5. has lots of tools

  6. available via AWS etc

  7. technical details are sparse

  8. has been absorbing venture capital

  9. https://techcrunch.com/2020/03/05/rigetti-computing-took-a-71-million-down-round-because-quantum-computing-is-hard/

    "Recently, investors are gambling more on the middleware layer of a quantum computing stack. These are companies like Zapata, Q-CTRL, Quantum Machines and Aliro, which improve the performance of quantum computers and create an easier user experience"

  10. makes optimistic promises (8/8/18):

    https://medium.com/rigetti/the-rigetti-128-qubit-chip-and-what-it-means-for-quantum-df757d1b71ea

4.5.8 Xanadu

  1. Xanadu Quantum Cloud

  2. https://venturebeat.com/2020/09/02/xanadu-photonics-quantum-cloud-platform/

  3. Uses photonics.

  4. Operates primarily at room temperature.

  5. Up to 24 qbits, gate depth of 12.

  6. Has free SW tools, some of which can compile to other quantum technologies.

  7. Expected good applications: graphs and networks, machine learning, and quantum chemistry.

  8. They expect to scale up better than competing technologies.

4.5.9 Others

  1. Alibaba

  2. 1QBit

  3. CQC

  4. QC Ware

  5. QSimulate

  6. Quantum Circuits

  7. Rahko

  8. Zapata

See https://www.explainingcomputers.com/quantum.html

4.6 Companies - Aggregators

These companies resell others' computers as a cloud service.

4.6.1 Amazon

  1. https://aws.amazon.com/braket/

    "Amazon Braket is a fully managed quantum computing service that helps researchers and developers get started with the technology to accelerate research and discovery. Amazon Braket provides a development environment for you to explore and build quantum algorithms, test them on quantum circuit simulators, and run them on different quantum hardware technologies."

    "...quantum annealers from D-Wave, and gate-based computers from Rigetti and IonQ."

4.6.2 Microsoft

  1. They offer a cloud service on 3 platforms: Honeywell, IonQ, QCI.

    Microsoft Is Taking Quantum Computers to the Cloud

  2. Microsoft Quantum

    A lot of stuff, with a low S/N.

  3. Microsoft quantum blog

  4. Azure Quantum Developer Workshop. 5:05:25.

    A little diffuse.

  5. For faster quantum computing, Microsoft builds a better qubit 11/7/2019.

  6. Microsoft Quantum Documentation gateway to a lot of stuff.

  7. Microsoft, e.g. Quantum Computing for Computer Scientists 1:28:22.

    This is the same viewpoint as the textbook, but the speaker is different.

    This talk discards hand-wavy pop-science metaphors and answers a simple question: from a computer science perspective, how can a quantum computer outperform a classical computer? Attendees will learn the following:

    • Representing computation with basic linear algebra (matrices and vectors)

    • The computational workings of qbits, superposition, and quantum logic gates

    • Solving the Deutsch oracle problem: the simplest problem where a quantum computer outperforms classical methods

    • Bonus topics: quantum entanglement and teleportation

    The talk concludes with a live demonstration of quantum entanglement on a real-world quantum computer, and a demo of the Deutsch oracle problem implemented in Q# with the Microsoft Quantum Development Kit. This talk assumes no prerequisite knowledge, although comfort with basic linear algebra (matrices, vectors, matrix multiplication) will ease understanding.

    See more at https://www.microsoft.com/en-us/research/video/quantum-computing-computer-scientists/

4.7 Algorithms

  1. Good ref is Chapter 6, Algorithms of Quantum Computing for Computer Scientists, published in 2000. Algorithms don't change fast. It does omit new things like HHL.

  2. For these examples, the quantum algorithm is quite different from the classical algorithm, and is asymptotically faster.

  3. Current research is deciding what algorithms can be made faster.

  4. p 172: Any function can be made invertible by adding a control bit.

  5. Major categories:

    1. Cryptography

    2. Quantum search

    3. Quantum simulation

    4. Quantum annealing and adiabatic optimization

    Nice summary: https://en.wikipedia.org/wiki/Quantum_computing

  6. Algorithm summary:

    1. Some, but not all, are faster.

    2. Bounded-error quantum polynomial time (BQP)

      1. "is the class of decision problems solvable by a quantum computer in polynomial time, with an error probability of at most 1/3 for all instances" - https://en.wikipedia.org/wiki/BQP

      2. Includes integer factorization and discrete log.

      3. Relation to NP is unknown (big unsolved problem).

    3. Searching problems:

      1. Find the answer to a puzzle.

      2. Math examples: factor an integer, solve a polynonial equation.

      3. Testing validity of a putative solution is easy.

      4. Finding that putative solution, naively, requires testing all possibilities.

      5. Quantum computation can solve some searching problems faster.

      6. This is probabilistic or noisy; often the found solution is wrong.

      7. So you repeat the computation enough times that the error rate is acceptably low.

      8. Some classical algorithms are similar. There is an excellent probabilistic primality algorithm.

      9. The quantum algorithms are quite complex. (i.e., I'm still learning them.)

    4. Algorithms, another view

      1. Hadamard matrix rotates the pure state to an entangled superposition.

      2. Then we operate in parallel on each state in the superposition.

      3. Finally we separate out the desired answer.

    5. Grover's algorithm:

      1. https://en.wikipedia.org/wiki/Grover%27s_algorithm

      2. Given a black box with N inputs and 1 output.

      3. Exactly one input makes the output 1.

      4. Problem: which one?

      5. Classical solution: Try each input, T=N.

      6. Quantum: $T=\sqrt(N)$.

      7. Probabilistic.

      8. Apps: mean, median, reverse a crypto hash, find collisions, generate false blocks.

      9. Can extend to quantum partial search.

      10. Grover's algorithm is optimal.

      11. This suggests that NP is not in BQP .

    6. Shor's algorithm:

      1. Factorize an integer.

      2. in BQP.

      3. almost exponentially faster than best classical algorithm.

      4. Largest examples I can find:

        1. 56153 = 233 × 241.

        2. https://medium.com/@aditya.yadav/rsa-2048-cracked-using-shors-algorithm-on-a-quantum-computer-660cb2297a95

4.8 Algorithm details

4.8.1 Deutsch

  1. We have a black box F(x) -> x'.

  2. We're told that F is either balanced or constant.

  3. How to determine which?

  4. We can input any x and see the result.

  5. Classically: eval F(0) and F(1).

  6. That took two evals and some comparisons.

  7. All that matters is the number of evals. We assume that they're slower than everything else.

  8. Quantumly (quantumicly?) we can determine which type F is with only one eval plus some extra matrices.

4.8.2 Deutsch - Jozsa

Now $F: \{0,1\}^n \to \{0,1\}$.

We're told that it's either constant or balanced. It's not neither.

Which is it?

Classically, we need n/2+1 evals of F.

Quantumly, we need only 1.

4.8.3 Simon's periodicity

Blackbox $F: \{0,1\}^n \to \{0,1\}^n$

For some unknown $c$, $F(x\oplus c) = F(x)$.

Determine $c$.

4.8.5 Deutsch-Jozsa

This algorithm is deterministic.

  1. https://www.quantiki.org/wiki/deutsch-jozsa-algorithm

    Quick summary; doesn't say why it works.

  2. https://qiskit.org/textbook/ch-algorithms/deutsch-jozsa.html

    This is an intro to Qiskit. The terminology is confusing. E.g., Register 1 has q0 q1 q2. Register 2 has q3. The run buttons don't seem to work.

  3. https://en.wikipedia.org/wiki/Deutsch%E2%80%93Jozsa_algorithm

    This is a nice detailed description.

4.8.6 Shor's algorithm

  1. Shor on, what is Shor's factoring algorithm? (2:09)

  2. Hacking at Quantum Speed with Shor's Algorithm (16:35)

  3. 43 Quantum Mechanics - Quantum factoring Period finding (19:27)

  4. 44 Quantum Mechanics - Quantum factoring Shor's factoring algorithm (25:42)

  5. Five lectures by Abraham Asfaw in Qiskit's Introduction to Quantum Computing and Quantum Hardware.

4.8.7 HHL algorithm to solve a linear system of equations

  1. Quantum Machine Learning - 37 - Overview of the HHL Algorithm 5:48.

    quick, deep, intro.

  2. Quantum algorithm for solving linear equations 36:31.

    quite understandable.

  3. Quantum Algorithms for Systems of Linear Equations (Quantum Summer Symposium 2020) 19:22.

  4. IBM Quantum Algorithms for Applications from qiskit

    E.g., Fourier transform and HHL.

  5. HHL Algorithm

    This is in Huawei HiQ, an open-source software framework for quantum computing.

  6. https://en.wikipedia.org/wiki/Quantum_algorithm_for_linear_systems_of_equations

4.9 Software

4.9.1 General

  1. A difficulty is to compile to the limited connectivity of the machine

  2. Open-Source Quantum Software Projects

  3. ProjectQ open-source software framework for quantum computing.

  4. Programming Quantum Computers: A Primer with IBM Q and D-Wave Exercises . Tutorial given at a few conferences.

4.9.2 Middleware

  1. https://www.hpcwire.com/off-the-wire/quantum-computing-inc-releases-version-1-1-of-mukai-middleware/

  2. https://tbri.com/webinars/middleware-the-quantum-computing-differentiator/ "An integral piece of quantum computing’s success is the middleware bridging existing code and algorithms to the new logical circuitry being established that sits on top of the quantum circuits. This integration and abstraction will allow the technology to process complex algorithms to provide the outcomes the hardware enables."

4.10 More info

4.10.1 QuTech Academy

  1. excellent videos from this Delft research group.

  2. QCI

  3. Per Delsing: Superconducting qubits as artificial atoms 28:59.

    He's describing a different computer from IBM's.

  4. The Taming of the Superconducting Qubit: A Tale of Loss 35:47.

    Presenter: Conal Murray, Research Staff Member, IBM Research

    The potential of quantum computing to enable new ways of solving problems considered intractable on classical computing platforms relies on our understanding of how qubits operate. Qubit scaling follows different metrics than those associated with classical computing, driven by the requirement that the fragile states they possess can be retained for sufficiently long times. After a brief introduction into superconducting transmon qubits, I will discuss how dielectric loss impacts their relaxation times and how we can effectively model such behavior using analytical and computational approaches. The resulting analysis provides guidance into the design aspects associated with such qubits. A secondary issue that follows from manufacturing greater numbers of qubits involves unwanted communication among them. In particular, resonance modes generated in the substrate on which they reside can limit their operating frequencies. It is known that incorporating grounded, through-silicon vias can increase the corresponding cutoff frequency within the substrate. I will show how we can predict the resulting spectrum by considering the array of vias as an effective photonic crystal to arrive at a fundamental frequency dependent on the particulars of the via geometry.

    http://meetings.aps.org/Meeting/MAR20/Session/P28.2

4.10.2 Misc

  1. 8 Best New Quantum Computing Books To Read In 2020

  2. quantikz – Draw quantum circuit diagrams in latex.

  3. Quantum Computing UK nice set of docs and examples.

  4. Ricardo Diaz recommends this book: Something Deeply Hidden: Quantum Worlds and the Emergence of Spacetime .

  5. There are now several business reports on the industry.

4.11 Summary

  1. Quantum computing hasn't solidly proved itself yet. However it's now in the engineering phase - realizing what we basically know how to do.

  2. IMO the physicists have done it again (last time was atomic energy). This will be a fundamental transformation of computing.

  3. Certain searching algorithms will be exponentially faster.

  4. Algorithm design still needs research. Algorithms are quite complex.

  5. Major application areas like drug design.

  6. Several viable technologies competing.

  7. Several HW companies.

  8. Competing toolsets being developed.

  9. Various service platforms to provide simulators and the HW.

  10. Recommendation (remember I'm SW):

    1. Be agnostic wrt platform (we don't know who will win).

    2. Have people use AWS etc to learn and develop apps. No capital investment needed.

    3. Work developing and/or using middleware, which is newer area.

    4. RPI-specific:

      1. Merely playing catchup is a losing game. Need something new.

      2. Assume that IBM etc will make the computers. The big problem with new HW is always how to use it. The ability of customers to use a new tool can determine whether it succeeds.

      3. Include other RPI programs?

        1. Gamify this using Game and Sim?

        2. Work with tetherless world?

      4. I'm still thinking.

Quantum Class 3, Tues 2022-09-06

1 Action at a distance

  1. Two qbits can be entangled tho they are far apart.

  2. How can this be?

  3. Newton faced the same conceptual leap.

  4. How can the earth affect the moon's orbit?

  5. Some people posited invisible whirlpools in space that dragged the planets around.

  6. That was falsified by the existence of retrograde orbits.

  7. Now we say that it happens because, in general relativity, the earth's mass bends space-time, and the moon just follows a geodesic.

  8. My view is that if something is useful but inexplicable, then just use it.

  9. Or, do like Andreas Osiander, Copernicus's editor, "these hypotheses need not be true nor even probable. [I]f they provide a calculus consistent with the observations, that alone is enough." https://en.wikipedia.org/wiki/Nicolaus_Copernicus

    If Giordano Bruno had talked evasively like that, maybe he wouldn't have been burnt at the stake in 1600. However he may have been burnt for other reasons.

2 Questions from Quantum Computing 2022 Update

  1. big application.

  2. what is superposition?

  3. why is it powerful?

  4. change in IBM strategy, or, what is circuit knitting?

  5. What is IBM's quantum parallelization?

  6. What is NIST doing re quantum crypto?

  7. How is Intel advancing quantum HW?

  8. What have some other companies done?

3 Hidary, Quantum Computing: An Applied Approach, 2nd ed, chapter 1

  1. Companion site: https://github.com/jackhidary/quantumcomputingbook

  2. Quantum computer: uses properties of quantum mechanics to compute

    1. world is quantum.

    2. compare to classical computer.

  3. quantum properties

    1. superposition

    2. entanglement

  4. state: complete math description of state.

    1. a complex vector.

    2. classical analog: e.g., position of a particle.

  5. Schrodinger's equation computes future state as a function of current state and stuff.

  6. Compare to Newton: future position depends on force etc.

  7. Analogously to Newton, only simple cases have closed form solutions. 2 body not 3 body. hydrogen atom.

  8. Even if there's a closed form solution, it may be chaotic, and so not as useful.

  9. Must simulate when no closed solution. Unfortunately that's all the good cases. College classes use solvable examples, not realistic ones.

  10. See Wolfram, A New Kind of Science.

  11. superposition: linear combo of states is a legal state.

    1. the weights are complex numbers.

    2. everything in quantum mechanics uses complex numbers.

    3. superposition does not work classically.

  12. A qubit $q$ is a quantum analog to a classical bit.

  13. the quantum analog to classical bits 0 and 1 are $|0\!\!>$ and $|1\!\!>$.

  14. q's state is a superposition (linear combo) of those two basis states:

    1. $q = a|0\!\!> + b|1\!\!>$ ,

    2. where the weights $a$ and $b$ are complex numbers, and $ | a | ^2 + | b | ^2 = 1$.

  15. Note the weird notation (Dirac notation). In $|0\!\!>$, $|$ is like a left bracket and $>$ like a right one.

  16. It is wrong to think that $q$ is really in one of the two states, but you don't know which one. This is the hidden variable theory. It has been proved experimentally to be false.

  17. $q$ is really in both states simultaneously.

  18. You cannot observe its state, unless it is $|0\!\!>$ and $|1\!\!>$, in which case you observe $0$ or $1$. This is the classical case.

  19. measurement of a state $\Psi$, and the Born rule (p 5):

    1. Measurement is an operator or matrix, M, applied to a state $\Psi$.

    2. M changes the qbit irreversibly, see the polarization example in the book.

    3. You cannot reclaim the old value.

    4. M has eigenvalues.

    5. Represent $\Psi$ as a linear combo of M's eigenvalues $\psi_i$, considered as a basis.

    6. $\Psi= \alpha\psi_1+\beta\psi_2$, where $\alpha^2+\beta^2=1$.

    7. you can use different basis systems to represent the same vector, and rotate between them.

    8. M changes $\Psi$ state randomly to one of the basis vectors.

    9. the probability of $\Psi$ changing to a particular basis vector is the modulus squared of the weight of that basis vector.

    10. define $z^c$ to be the complex conjugate of $z$.

    11. if $\Psi= \alpha\psi_1+\beta\psi_2$, where $\alpha^c\alpha+\beta^c\beta=1$ then the probability of $\Psi$ changing to $\psi_1$ is $\alpha^c\alpha$. (the Born rule)

    12. $\alpha^c\alpha$ is called the modulus squared.

  20. There are many possible measurement operators available.

    1. You can choose which to apply to $q$. Say, position.

    2. That prevents you from applying the others to $q$, say, momentum, because you don't have $q$ available any more.

    3. Heisenberg uncertainty: measuring, say, position, prevents you from accurately measuring momentum.

  21. $q$, that is, $q$ 's value, can be considered to be a vector of length two: $$\begin{pmatrix} a | 0\!\!> \\ b | 1\!\!> \end{pmatrix} $$ or simply $$\begin{pmatrix}a\\b\end{pmatrix}$$.

  22. You operate on $q$ with a matrix multiplication: $q_2 = M q$.

  23. Unless $M$ is a measurement operator, it is invertible, so you can go backwards.

  24. Contrast to classical operators like and and or.

  25. Examples of 1-qubit gates

    1. not, aka X. page 28.

    2. square root of not

    3. Y, Z

    4. S (rotation by $90^o$), T ($45^o$), phase shift

    5. Hadamark. "it enables us to take a qubit from a definite computational basis state into a superposition of two states"

  26. All operators used in quantum computation other than for measurement must be reversible. - textbook.

  27. No cloning: You cannot copy a qubit, but can move it.

  28. The life cycle of a qubit:

    1. Create a qubit with a classical value, 0 or 1.

    2. Operate on it with matrices, which rotate it to have complex weights.

    3. Measure it by randomly projecting it onto a basis vector.

  29. So far, not very powerful.

  30. a quantum state $\Psi$ usually has many qubits.

    compare to a classical byte with 8 classical bits.

  31. However the different qubits in $\Psi$$ might be entangled.

    1. This is very weird and powerful.

    2. Entanglement means that if you measure one qubit then what you observe restricts what would be observed when you measure the other qubit.

    3. Even if the two qubits are 1000 mi apart. This has been experimentally observed.

    4. However that does not let you communicate.

3.1 Entanglement

  1. Crazy counterintuitive idea that's the basis for quantum speedup.

  2. Classical metaphor for entanglement:

    1. Start with a piece of paper.

    2. Tear it into two halves.

    3. Put each half into an envelope, seal them, and mix them up, so that you can't tell which half is in which envelope.

    4. Address and mail one envelope to a friend in Australia, and the other to a friend in Greenland.

    5. When the Australian opens his envelope, he knows what the Greenlander will find in his.

    6. However that doesn't let the Australian send any info to the Greenlander, or vv.

  3. This has been demonstrated with real qubits transported 1000 miles apart.

  4. Entanglement means that if you measure one qubit then what you observe restricts what would be observed when you measure the other qubit.

  5. However that does not let you communicate.

  6. The preceding metaphor is wrong in that it has a hidden variable, the unobserved half-paper state. That does not happen in quantum physics. With qubits, the states are not fixed until one is observed. I'm trying to get the idea across.

4 Chapter 2: history

  1. Read it on your own, but here are some additions:

  2. The property list on p15 is controversial and seems designed to exclude D-Wave.

  3. Like for classical computation, the main ideas of quantum computing were proposed before actual machines could be built.

5 Videos to watch for next time

  1. IBM Quantum 2022 Updated Development Roadmap, 18:49, 2022-05-10

  2. Google Quantum AI Update 2022, 25:21, 2022-04-14

Quantum Class 2, Thurs 2022-09-01

1 Class 1

I edited the blog to show what we actually did.

2 Homework 2

is online, due Tues.

3 Theory vs Experiment

  1. Sometimes the theoreticians posit something new, then the experimentalists try to build it.

    1. atom bomb

    2. classical computer

    3. quantum computer

  2. Other times the experimentalists (aka hackers) build something new.

    1. bridges

    2. airplanes

    3. medieval cathedrals

    4. steam engines

    5. calculus

  3. Then those may become so successful that bigger and bigger ones are built until those collapse / crash / fall down / blow up / have paradoxes.

  4. Then the theoreticians have to come in and clean up the mess to allow more progress.

  5. They may look for unifying themes in apparently separate problems.

  6. Group theory in abstract (modern) algebra happened that way.

  7. Anything discovered about groups was then valid in all the applied areas.

  8. All theories have limits that may or may not be relevant.

  9. Paradoxes expose limits. "All triangles are isosceles" error exposes something not covered by Euclid's axioms.

  10. Physics Nobels tend to go to theoretical explanations of surprising experiments. Although, there is nothing in common among all Nobel winners, not even intelligence (according to Enrico Fermi).

  11. Very rarely, experimentalists do something that the theoreticians had proven was impossible.

    1. Galileo seeing sunspots.

    2. Marconi radioing across Atlantic.

    3. Michaelson-Morley. (However several other contemporaneous inexplicable observations all proved to have classical explanations.)

    and something new is discovered / invented. Unfortunately the crackpots then seize on these rare examples...

4 Video

  1. Quantum Computing in Under 11 Minutes daytonellwanger https://www.youtube.com/watch?v=TAzZKAdX2Tw 10:56 more technical

5 Intro to quantum computing

  1. This is from

    1. Yanofsky and Mannucci, Quantum Computing for Computer Scientists,

    2. Hidary, 2nd edition, and

    3. my opinions.

  2. Isomorphism between geometry and algebra. Work in whichever domain is easier. Results carry over.

    1. 2D: x+ij -> (x,y)

    2. 3D: quaternions. 3D rotations are not commutative, and so quaternions are not. There's a memorial on a bridge in Trinity College Dublin where William Rowan Hamilton thought of them. Quaternions are equivalent to 3D vector algebra.

    3. qbits - Bloch sphere.

  3. qbits vs qubits? Above my pay grade.

  4. Evidence for quanta:

    1. 1905 photo-electric effect explanation. Einstein Nobel.

    2. Ultraviolet catastrophe.

    3. 1922 Stern Gerlach experiment. Very influential.

      https://en.wikipedia.org/wiki/Stern%E2%80%93Gerlach_experiment

  5. Hidary 1.1 Quantum Computer Definition, p 3

    A quantum computer is a device that leverages specific properties de- scribed by quantum mechanics to perform computation.

  6. State vector

    1. completely describes a system

    2. cannot be directly observed, only measured...

    3. which changes it.

  7. Hidary 1.1 Superposition and Entanglement, p 4

    2 big properties.

  8. bra-ket notation.

  9. can rewrite a state vector in a different basis

  10. Hidary 1.3 The Superposition Principle, p 4

    1. "The linear combination of two or more state vectors is another state vector in the same Hilbert space and describes another state of the system."

    2. Generally false in classical domain.

      Particle can go thru only 1 of 2 slits.

      However vibration states in a string add.

  11. Hidary entanglement p 7

    description is confusing, defer till later.

  12. Yanovsky p 40, vector space review

    1. One qbit is a simple vector space.

    2. Can change basis, e.g., to Hadamard basis. p 51.

  13. Eigenvalues and eigenvectors, p 60

  14. Hermetian matrices, p 63

  15. Classical 2 slit experiment, p 86.

    Probabilities add.

  16. Quantum 2 slit experiment, p 93.

    Complex waves add.

    Probabilities might reduce.

6 Quantum properties - Entanglement

  1. Crazy counterintuitive idea that's the basis for quantum speedup.

  2. Classical metaphor for entanglement:

    1. Start with a piece of paper.

    2. Tear it into two halves.

    3. Put each half into an envelope, seal them, and mix them up, so that you can't tell which half is in which envelope.

    4. Address and mail one envelope to a friend in Australia, and the other to a friend in Greenland.

    5. When the Australian opens his envelope, he knows what the Greenlander will find in his.

    6. However that doesn't let the Australian send any info to the Greenlander, or vv.

    7. This has been demonstrated with real qbits transported 1000 miles apart.

    8. Entanglement means that if you measure one qbit then what you observe restricts what would be observed when you measure the other qbit.

    9. However that does not let you communicate.

  3. The above metaphor is inaccurate in ways that don't affect us now. (It assumes a hidden variable, which is false.)

  4. Entanglement and superposition do funny things to operations. Consider the Controlled NOT gate. It has 2 inputs, x and y.

    1. Classically, y is negated iff x=1. x doesn't change.

    2. If x and y are superposed with a Hadamard basis, then y can affect x.

7 Video to watch on your own

  1. Quantum Computing 2022 Update, 15:12, July 24 2022.

  2. You prepare discussion points and questions for next class.

8 Misc intro to quantum computing stuff

This is misc stuff that you might find interesting, which I'm drawing from.

8.1 Intro sites - 2

  1. "Spooky" physics | Leo Kouwenhoven | TEDxDelft (18:00)

  2. A beginner's guide to quantum computing | Shohini Ghose https://www.youtube.com/watch?v=QuR969uMICM 10:04

  3. https://towardsdatascience.com/introduction-to-quantum-programming-a19aa0b923a9?gi=69d861e26d80

  4. https://medium.com/@jonathan_hui/qc-programming-with-quantum-gates-8996b667d256

  5. https://medium.com/@jonathan_hui/qc-programming-with-quantum-gates-2-qubit-operator-871528d136db

  6. https://www.cl.cam.ac.uk/teaching/0910/QuantComp/notes.pdf

    1. They have a nice description of measurement starting at slide 10.

    2. Each measurement operator has a basis vector set.

    3. The operator represents the qbit as a linear combo of the basis vectors.

    4. Then it projects the qbit onto one of the basis vectors, with probability being the length of that component.

    5. It is possible for two different qbits to measure the same in some basis, but measure different in a different basis.

  7. Can we make quantum technology work? | Leo Kouwenhoven | TEDxAmsterdam (18:19)

  8. Quantum Computing Concepts – Quantum Hardware (3:22)

  9. Experiment with Basic Quantum Algorithms (Ali Javadi-Abhari, ISCA 2018) (19:05)

  10. Quantum Computing for Babies

8.2 Current status sites

  1. https://www.telegraph.co.uk/technology/2020/09/02/britain-must-act-fast-prevent-brain-drain-quantum-computing/ - good summary of programs around the world.

  2. Quantum startup CEO suggests we are only five years away from a quantum desktop computer

9 No new homework

Finish 1 and enjoy Labor Day.

10 Videos to watch for Tues

  1. Watch A Beginner’s Guide to Quantum Computing, 18 min, by Dr. Talia Gershon, IBM Research.

Quantum Homework 2, Thurs 2022-09-01

Due 2022-09-06 4pm in gradescope. Groups of 2 are ok; submit one answer set.

These are questions about the videos watched or assigned in class.

  1. (10 points) What are the 2 ways that Talia Gershon says that quantum is different?

  2. (10 points) According to David Deutsch, what is the quantum theory of computation?

Total: 20

Quantum Class 1, Mon 2022-08-29

1 Misc

1.1 Syllabus

Read the syllabus, accessible from the top bar.

1.2 Gradescope

We'll use Gradescope for submitting homeworks and possibly projects. Use the entry code that I'll give in class to add yourself.

"If you already have a Gradescope account, log into that account and navigate to your Account Dashboard by clicking the Gradescope logo in the top left, and click Add Course in the bottom right corner. Then enter your course code."

Then, the quick link to the course should be https://www.gradescope.com/courses/432866 . It is available in the top bar of this page.

1.3 My research

I do parallel geometry algorithms on large problems for CAD and GIS. See my home page. I will be retiring this year after 45 years at RPI (including several years at places like the National Science Foundation and UC Berkeley).

1.4 Changes from last year

  1. New textbooks.

  2. More non-IBM material.

  3. No piazza.

  4. Class is in person.

  5. Switch to the new edition of Hidary (2nd ed, 2021).

1.5 My role

  1. My main job is to be a curator selecting the best material for the class.

  2. I try to show the principals themselves describing their work and their opinions. E.g., Peter Shor talking about his algorithm and about quantum computing in general.

  3. I try to leave you wanting to learn more.

  4. There's a lot that I still have to learn about quantum computing. For some specific topics, some of you may know more than me.

1.6 Office hours

  1. After most classes for an hour.

  2. By webex at mutually agreeable times; email me.

1.7 Textbooks

1.7.1 Preferred text

  1. Jack D. Hidary. Quantum Computing: An Applied Approach, 3nd edition, 2021. @ Springer. Also on Amazon.

1.7.2 Optional extra texts

  1. Noson S. Yanofsky and Mirco A. Mannucci. Quantum Computing for Computer Scientists 1st Edition

    I used to use this. This is nice but is 20 years old, and so omits some things. However I'll refer to it a little.

    I encourage you to read several books, and pick and choose.

  2. Abraham Asfaw et al, Learn Quantum Computation using Qiskit, http://community.qiskit.org/textbook, 2020

    There's an old and a new version. The old version was more comprehensive.

1.8 Web sites

  1. IBM's detailed online stuff. Not just qiskit but algorithms etc.

  2. Other universities provided inspiration.

  3. Misc quantum research centers, like Delft

  4. Many videos.

1.9 Course blog

  1. https://wrfranklin.org/Teaching/quantum-f2022

  2. My former web site, wrf.ecse.rpi.edu, now redirects to wrfranklin.org .

1.10 Learning Outcomes

  1. Demonstrate proficiency with the mathematics behind quantum computing.

  2. Understand important quantum computing algorithms.

  3. Understand the three main quantum platforms: transmon qubit, trapped ion, and quantum annealing.

  4. Apply that to write and run programs on those platforms.

2 Intro to quantum computing

2.1 Intro

Alice laughed. "There's no use trying," she said: "one can't believe impossible things." "I daresay you haven't had much practice," said the Queen. "When I was your age, I always did it for half-an-hour a day. Why, sometimes I've believed as many as six impossible things before breakfast." - Through the Looking-Glass, and What Alice Found There (1871), by Lewis Carroll (Charles Lutwidge Dodgson).

2.2 Brief History

The WIRED Guide to Quantum Computing 08.24.2018. Nice non-too-technical summary of the history. The theory preceded the realization. This happens sometimes, e.g., with atomic energy. From there:

1980

Physicist Paul Benioff suggests quantum mechanics could be used for computation.

1981

Nobel-winning physicist Richard Feynman, at Caltech, coins the term quantum computer.

1985

Physicist David Deutsch, at Oxford, maps out how a quantum computer would operate, a blueprint that underpins the nascent industry of today.

1994

Mathematician Peter Shor, at Bell Labs, writes an algorithm that could tap a quantum computer’s power to break widely used forms of encryption.

2007

D-Wave, a Canadian startup, announces a quantum computing chip it says can solve Sudoku puzzles, triggering years of debate over whether the company’s technology really works.

2013 Google teams up with NASA to fund a lab to try out D-Wave’s hardware.

2014

Google hires the professor behind some of the best quantum computer hardware yet to lead its new quantum hardware lab.

2016

IBM puts some of its prototype quantum processors on the internet for anyone to experiment with, saying programmers need to get ready to write quantum code.

2017

Startup Rigetti opens its own quantum computer fabrication facility to build prototype hardware and compete with Google and IBM.

2.3 Intro sites

  1. Feynman was the first to propose the theoretical idea of a quantum computer.

    Richard Feynman - Quantum Mechanics 4:01.

    Extracted from HD Feynman: FUN TO IMAGINE complete 1080p 1:06:49. Recorded in 1983.

  2. David Deutsch - Why is the Quantum so Strange? (8:43)

  3. Quantum Algorithms (2:52)

    Which problems can quantum computers solve exponentially faster than classical computers? David Gosset, IBM quantum computing research scientist, explains why algorithms are key to finding out.

Quantum Homework 1, Mon 2022-08-29

Due 2022-09-01 4pm in gradescope.

  1. (5 points) Compute (5+2i)/(3+4i).

  2. (5 pts) Compute the eigenvalues of $\begin{vmatrix} 5&2\\3&4 \end{vmatrix}$.

  3. (5 pts) Considering complex numbers as points in the 2D plane, what is the geometric effect of multiplying a complex number by (.6-.8i) ?

  4. (5 pts) Let c = 1 + i. Convert it to polar coordinates, calculate its fifth power, and revert the answers to Cartesian coordinates.

  5. (5 pts) Find all the cube roots of c = 1 - i.

  6. (5 pts) Invert the Hadamard matrix $\frac{1}{\sqrt{2}} \begin{vmatrix} 1&1\\1&-1\end{vmatrix}$.

Total: 30