LogicalQubits

 Microsoft and Qunatinuum have recently announced a breakthrough in less

error-prone qunatum computing. I tracked this down to a 2012 paper cited below,

and asked Gemini for help on making sense of it:

Quantum computing; can you help me make sense of this paper: 1208.0928 (arxiv.org)

The arxiv paper you refer to,  "Surface codes: Towards practical large-scale quantum computation" by [unknown author(s)],  discusses using surface codes for building large-scale quantum computers. Surface codes are a particular way of arranging and manipulating quantum bits (qubits) to make them more resistant to errors.

Here's a breakdown of the key points:

Introduction to Surface Codes:

The paper starts by estimating the potential size and speed of a quantum computer built with surface codes.

It then dives into the concept of stabilizers, which are error-detecting mechanisms for qubits.

It explains how these stabilizers are applied to a two-dimensional array of physical qubits to form the surface code.

Logical Qubits and Fault Tolerance:

The paper explains how groups of physical qubits within the surface code array can be used to create logical qubits, which are the information-carrying units of a quantum computer.

It explores how fault-tolerant these logical qubits are, meaning how well they resist errors during computations.

Quantum Gates and Operations:

The paper outlines how logical qubits can be moved around within the surface code array.

It describes how sequences of qubit manipulations, called braid transformations, are constructed.

Interestingly, it explains how a braid between two logical qubits can be used to perform a controlled-NOT operation, a fundamental operation in quantum computing.

Finally, it covers single-qubit gates like Hadamard, S, and T gates, which along with controlled-NOT gates, are essential for a universal quantum computer (one that can perform any quantum computation).

Physical Implementations:

The paper briefly touches upon different ways to physically implement surface code quantum computers.

Overall, this paper provides an introduction to surface codes and how they can be used for building large-scale fault-tolerant quantum computers.

If you'd like to delve deeper, you can search for terms like "surface code stabilizer", "fault-tolerant logical qubit", or "braid transformations in surface code" to find more detailed explanations.

What are logical qubits

Logical qubits are a key concept in building robust quantum computers. Here's how they work:

Regular vs. Logical Qubits:

Physical Qubits: These are the actual quantum bits used in a machine. They can be fragile and prone to errors due to their quantum nature.

Logical Qubits: Imagine a group of physical qubits working together. A logical qubit is the information encoded across these multiple physical qubits. By cleverly using entanglement and error correction techniques, the logical qubit becomes more resilient to errors compared to a single physical qubit.

Benefits of Logical Qubits:

Error Correction: Quantum computations are very sensitive to errors. Logical qubits help mitigate this issue. By encoding information across multiple physical qubits, even if one physical qubit flips (changes its state), the overall logical qubit state can be preserved by analyzing the remaining qubits.

Fault Tolerance: Logical qubits allow for building fault-tolerant quantum computers. This means even with some errors in the physical qubits, the computation can still be performed accurately.

Creating Logical Qubits:

Logical qubits are built using multiple physical qubits entangled in a specific way. The entanglement creates a link between the qubits, so their fates are intertwined. This linkage allows error correction to be performed across the group.

Importance of Logical Qubits:

Logical qubits are crucial for achieving large-scale quantum computation. Without them, errors would quickly accumulate, making any meaningful computations impossible. By using logical qubits, researchers hope to build quantum computers that can tackle problems  beyond the reach of classical computers.

In essence, logical qubits are a way to harness the power of multiple physical qubits to create a more robust unit of information for fault-tolerant quantum computing.

Quirk

 Quirk: Quantum Circuit Simulator (algassert.com)

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The Map of Quantum Computing - Quantum Computing Explained (youtube.com)

Euclid's Algorithm

 Been looking at Euclid's algorithm, a procedure on two numbers that yields their

greatest common factor. Euclid lived sometime around 300 BCE. He is credited

with elaborating the standard developement of geometry-Elements - with a text still pertinent

today. (Indeed I remember working through the whole thing in High School, a

wonderful little work). This algorithm is straightforward, but it appears to be the

bugbear of many university students as an instance of 'discrete' mathematics...

The video below takes us through with integers, but one needs to remeber that

at the time, the decimal numbers that we find normal were not in use in Greece. The

mathematical imagination at work had to be visual!!

                                                              

source: Wikipedia

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You

What did numbers look like for Euclid?

Meta AI

Euclid, a Greek mathematician, lived in the 3rd century BCE and wrote the famous book "Elements," which is one of the most influential works in mathematics. In Euclid's time, numbers were represented using Greek letters, with each letter having a specific numerical value. This system was known as the Greek numeral system.

Here's how it worked:

Alpha (α) = 1

Beta (β) = 2

Gamma (γ) = 3

Delta (δ) = 4

Epsilon (ε) = 5

Zeta (ζ) = 6

Eta (η) = 7

Theta (θ) = 8

Iota (ι) = 9

Kappa (κ) = 10

Lambda (λ) = 20

Mu (μ) = 30

Nu (ν) = 40

Xi (ξ) = 50

Omicron (ο) = 60

Pi (π) = 70

Rho (ρ) = 80

Sigma (σ) = 90

Tau (τ) = 100

Upsilon (υ) = 200

Phi (φ) = 300

Chi (χ) = 400

Psi (ψ) = 500

Omega (ω) = 600

Using this system, numbers were represented by combining letters. For example:

11 = ΙΑ (Iota-Alpha)

25 = ΚΕ (Kappa-Epsilon)

100 = Ρ (Rho)

This system was used for basic arithmetic operations like addition and subtraction, but it was not very efficient for more complex calculations. The Greek numeral system was eventually replaced by the Hindu-Arabic numeral system, which is the one we use today (0-9).

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You

What is discrete mathematics?

Copilot

Discrete mathematics is the study of mathematical structures that can be considered “discrete” (in a way analogous to discrete variables, having a bijection with the set of natural numbers) rather than “continuous” (analogously to continuous functions). Objects studied in discrete mathematics include integers, graphs, and statements in logic. By contrast, discrete mathematics excludes topics in “continuous mathematics” such as real numbers, calculus, or Euclidean geometry. It deals with countable sets, which can be either finite or infinite. Research in discrete mathematics has grown due to the development of digital computers, which operate in “discrete” steps and store data in “discrete” bits. Concepts from discrete mathematics are useful in computer science, including computer algorithms, programming languages, cryptography, automated theorem proving, and software development. Although the main objects of study in discrete mathematics are discrete, analytic methods from “continuous” mathematics are often employed as well.

Usingthe web

Discrete mathematics is the study of mathematical structures that can be considered “discrete” (in a way analogous to discrete variables, having a bijection with the set of natural numbers) rather than “continuous” (analogously to continuous functions). Objects studied in discrete mathematics include integers, graphs, and statements in logic. By contrast, discrete mathematics excludes topics in “continuous mathematics” such as real numbers, calculus, or Euclidean geometry. It deals with countable sets, which can be either finite or infinite. Research in discrete mathematics has grown due to the development of digital computers, which operate in “discrete” steps and store data in “discrete” bits. Concepts from discrete mathematics are useful in computer science, including computer algorithms, programming languages, cryptography, automated theorem proving, and software development. Although the main objects of study in discrete mathematics are discrete, analytic methods from “continuous” mathematics are often employed as well123

Aces

 Meta is really aces at giving instructions on working with the computer.

I spent a lot of time yeterday cleaning up my Jupyter notebook opening

page, with a long list of individual notebooks. Couldn't drag'n drop into

folders. So I asked Meta AI for help...

You

Jupyter notebook; how do I move a notebook into a folder

Meta AI

You can move a Jupyter Notebook into a folder by using the Jupyter Notebook interface or command line. Here are the steps for both methods:

Method 1: Using Jupyter Notebook Interface

Open Jupyter Notebook in your web browser.

Navigate to the dashboard, which lists all your notebooks and folders.

Check the checkbox next to the notebook you want to move.

Click the "Move" button (looks like a folder with an arrow) in the toolbar or press m on your keyboard.

Select the destination folder from the dropdown menu or type the path to the folder.

Click "Move" to confirm.

Method 2: Using Command Line

Open a terminal or command prompt.

Navigate to the directory where your notebook is located using the cd command.

Use the mv command to move the notebook file (.ipynb) into the desired folder.

Example:

Bash

mv notebook_name.ipynb path/to/destination/folder

Replace notebook_name.ipynb with the name of your notebook file and path/to/destination/folder with the path to the folder you want to move it to.

Let me know if you have any other questions!

Shor's

 

QC_SoFar

 Been powering through on Qisbit, and the Hello World tutorial. So far:

                                                                       

From Gemini:

The diagram in line 4 shows a quantum circuit created using Qiskit. The circuit consists of two qubits, which are represented by the vertical lines labeled 0 and 1. 

The gates applied to the qubits are represented by the shapes in the circuit. In this case, there are two gates:

* H gate: This gate is applied to qubit 0 at the beginning of the circuit. The H gate represents a Hadamard gate, which is a fundamental quantum gate that transforms a qubit from the basis state |0> to the superposition state (|0> + |1>)/sqrt(2), and vice versa.

* CNOT gate: This gate is applied to qubits 0 and 1 at the end of the circuit. The CNOT gate is a controlled NOT gate, which is another fundamental quantum gate. It performs a NOT operation on the target qubit (qubit 1 in this case) only if the control qubit (qubit 0 in this case) is in the |1> state.

The output of the circuit is the state of the qubits after the gates have been applied. In this case, the output state is not shown explicitly in the diagram. However, it can be determined by simulating the circuit or by applying the gates mathematically.

 

I hope this explanation is helpful! Let me know if you have any other questions.

                                                                

                                                                     

                                                                          

Copilot

Certainly! Phase inversion is a concept that can be actively manipulated in quantum systems. Let’s dive into it:

Phase Inversion Basics:

Phase inversion involves changing the sign of the quantum state’s phase.

Consider a qubit state |ψ⟩ with a certain phase (e.g., eiϕ|0⟩). Inverting the phase means transforming it to -eiϕ|0⟩.

This operation doesn’t affect the probability of measurement outcomes but can impact interference patterns and quantum algorithms.

How to Achieve Phase Inversion:

Hadamard Gate (H): The Hadamard gate can create superpositions and invert phases. Applying H to |0⟩ results in |+⟩ (with a relative phase of π/2), and applying it again inverts the phase back to |0⟩.

Controlled Phase (CZ) Gate: The CZ gate inverts the phase of the |11⟩ state while leaving other states unchanged.

Phase Kickback: In some algorithms (like the quantum phase estimation algorithm), phase inversion occurs due to entanglement between control and target qubits.

Applications:

Grover’s Algorithm: Grover’s search algorithm uses phase inversion to amplify the amplitude of the correct solution state.

Quantum Fourier Transform: In the quantum phase estimation algorithm, phase inversion helps estimate eigenvalues of unitary operators.

Quantum Error Correction: Stabilizer codes use phase inversion to correct errors.

Experimental Realization:

Quantum gates (like H and CZ) are implemented using physical qubits (e.g., superconducting circuits, trapped ions).

Researchers actively work on improving gate fidelity and minimizing errors to achieve reliable phase manipulation.

In summary, phase inversion is indeed something we can actively control in quantum systems, and it plays a crucial role in quantum algorithms and protocols! 🌌🔀🚀

Quantum computing/ X gate vs CZ gate

Copilot

Using

the web

Certainly! Let’s explore the differences between the X gate and the CZ (Controlled-Z) gate in quantum computing:

X Gate (Pauli-X):

The X gate is a single-qubit operation that maps |0⟩ to |1⟩ and vice versa.

It equates to a rotation around the X-axis of the Bloch sphere by π radians.

Mathematically, it acts as follows:

X|0⟩ = |1⟩

X|1⟩ = |0⟩

CZ Gate (Controlled-Z):

The CZ gate is a multi-qubit gate with a unique behavior.

It performs a Z operation on the target qubit if and only if the control qubit is in the state |1⟩.

A Z operation imparts a phase of -1 to the target qubit when it’s in the state |1⟩, leaving it unchanged when it’s in the state |0⟩.

Mathematically, CZ maps:

α|00⟩ + β|01⟩ + γ|10⟩ + δ|11⟩ to α|00⟩ + β|01⟩ + γ|10⟩ − δ|11⟩

Clearly, the qubit roles are symmetric; it doesn’t matter which qubit is the control or target11.

Summary:

The X gate flips the state of a single qubit, while the CZ gate operates on two qubits.

X gate: |0⟩ ↔ |1⟩

CZ gate: |11⟩ ↔ -|11⟩ (phase inversion for |11⟩ state)

In circuits, CZ gates are often shown as:

───●───

        │

───●───

In summary, the X gate flips qubit states, while the CZ gate introduces phase inversion based on the control qubit’s state! 🚀🔀🔬

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We are ending up with 2 qubit at 1, and 4 at 0.

Qubit

https://www.popsci.com/technology/in-photos-journey-to-the-center-of-a-quantum-computer/

 https://www.youtube.com/watch?v=5kUPZSgHP7A