#qft

[okpalan]

QFT ISO

When quantum fails, physics becomes the firewall. Introducing QFT-ISO: 3 ụzọ states, zero trust, decoherence-only attacks. Build your own reality. github.com/obinexus/qft_iso @obinexus on YouTube 🚀 #QFT #QuantumSecurity #PhysicsNotCode

[admiral-of-the-coffee-squadron]

why the FUCK is Fourier popping up in field theory like some random-ass glitter in your hair from yesterday's party in the neighbouring town

[govindhtech]

Quantum Fourier Transform Applications, Types, & Advantages

The Quantum Fourier Transform, its history, how it works, architecture, types, features, advantages, disadvantages, applications, and challenges are covered in this article.

Quantum Fourier Transform

The Quantum Fourier Transform (QFT) is a fundamental linear transformation in quantum computing and the quantum version of the DFT, a major digital signal processing tool. The classical DFT maps time and frequency representations to analyse periodic functions, while the QFT does the same to a quantum state. A quantum state's computational basis is replaced with a Fourier basis, a superposition of basis states weighted by Fourier coefficients.

QFT overview:

History

Jean-Baptiste Joseph Fourier invented the classical Fourier transform in the early 19th century. In his 1994 breakthrough paper on factoring large numbers, Peter Shor introduced the quantum version. Don Coppersmith also helped develop it. Since its discovery, quantum technology has shown its promise to improve computers, thanks to QFT.

Its Function and Architecture

The QFT is a quantum circuit that uses an n-qubit quantum state. The general procedure:

When a Hadamard gate is used on the first qubit, the |0⟩ and |1⟩ states are superimposed at the start of the operation.

Next, controlled phase shift gates are used. These gates rotate a qubit's phase based on the state of a “control” qubit and the circuit's qubit placement.

The Hadamard gate and controlled phase shift gate sequence is repeated for all n qubits.

After all gates are applied, qubits are swapped to reverse order to obtain the correct output ordering. PennyLane and others do this. Individual qubit states encode the frequency components of the initial quantum state, while the output state is a tensor product of single-qubit states.

Types and Transformations

Discrete Fourier Transform (DFT): A mathematical method called the DFT may transform a finite sequence of equally spaced data points (like signals) from time to frequency. Showing data's numerous frequency components allows signal analysis, filtering, and compression. A frequent digital signal processing application.

Since the QFT is reversible, its inverse, the Hermitian adjoint of the QFT matrix, can be effectively performed by reversing the QFT circuit. Due to an exponent sign convention, DFT and inverse QFT are mathematically identical.

The Fourier transform is the Hadamard transform for n-qubit quantum registers indexed by the Boolean group if a Hadamard gate is applied to each qubit in parallel. A QFT and initial Hadamard transform are employed in Shor's algorithm.

The Fourier transform can be extended to the quantum environment for groups other than the cyclic group, such as the symmetric group or over a finite field. This is Other Groups/Finite Fields QFT.

Features

The QFT converts quantum state amplitudes from computational to Fourier bases, a key feature. QFT is likely to measure states corresponding to frequency multiples of the inverse period when applied to a periodic function, making it useful for periodic structure problems. This allows frequency spectrum inference from time-domain sequences.

Advantages

For some workloads, the QFT is exponentially faster than the Fast Fourier Transform (FFT). A QFT can be implemented in O(n^2) operations on n qubits, while the traditional FFT requires O(N log N) steps (where N = 2^n).

It is a fundamental subroutine and building block for many quantum algorithms, making them faster.

QFT is a unitary transformation, therefore it can be reversed and keep inner products.

Disadvantages

Measurement Issue: The QFT does not immediately yield all Fourier coefficients. Single output state measurements yield only one potential frequency component. To get all results or find the dominant frequency, the process must be performed numerous times.

Hardware Requirements: The QFT circuit requires several quantum gates and all-to-all qubit connection. Limited connection requires more “swap” gates, increasing circuit depth and errors. This is a major issue for loud quantum devices.

[as-if-and-only-if]

I’ve got this basic QFT confusion. Say you’re looking at things in the operator field framework, in the Schrödinger picture. Your Hamiltonian is some integral of the form ∫ d³x [thing involving φ(x) and π(x)] (no time dependence; time dependence is in the state).

According to Tong, the states have time dependence, and evolve according to the Schrodinger equation—we can for example talk about H|ψ⟩. But what space does a state |ψ⟩ actually live in, here?

Intuitively, I would have expected that states also have spatial dependence: |ψ(x)⟩, and maybe the Schrödinger equation is secretly quantified over all x. But if we have e.g. a φ(x)² term in our Hamiltonian, we end up with something like ∫d³x' … φ(x')² |ψ(x)⟩.

This is odd to me. This would mean that we’re acting on the state at x by an operator that lives at some (potentially) other point x'? Which seems strange.

On the other hand…

I kind of expect the state to (instead) act like a superposition of entire (classical) fields. But then it’s a mystery to me how we can act on it with ɸ(x), or what that even means.

I suppose a superposition of classical fields would give meaning to φ(x)|ψ⟩: multiply each component field in the superposition |ψ⟩ by its value at x. This seems like what the Schrödinger functional is doing (at least according to that wikipedia page; Tong mentions "functional" but does not approach it with the same formalism.)

Notably, though, this is not the same as |ψ⟩ itself having spatial dependence/being fieldlike (i.e. it is not a function from spacetime to some Hilbert space).

But, it’s still very weird to me that we can e.g. integrate φ over x in the first place, and that these operators even live in the same space. I suppose it makes sense if they all act on the “(quantum) state space of classical fields”, so to speak. Is that it?

[superpte]

SCIENCE Through PHILOSOPHY, And Reciprocally

Science is a set of methods and results to ascertain facts. Philosophy is a set of methods and results to ascertain optimal behavior. Facts are part of optimal behavior and optimal behavior is what ascertain facts. So the two laws, science and wisdom, are entangled within love of fate and fate of love. *** Say one took 100 adults living today, selected because they know no science, and one time…

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[averycanadianfilm]

David Wallace's - Reading List in Philosophy of Quantum Field Theory

To anonymous blog viewer: this is not the proper forum to discuss QFT, but you may find David Wallace's reading list a good place to start your investigation of such things. Since, quite likely I won't teach physics or math again I don't have anything to say about the technical aspects of these things. I will, however, probably have to say something about quantum field theory in the context of a theory of mind and consciousness. :-)

Hubert Hugh Burke March 21, 2025 Alberta Canada. 10:00 pm

[averycanadianfilm]

 What Is a Quantum Field Theory? - Michel Talagrand

Okay, so I've looked at a 'preview' of this book. Those interested in quantum field theory may want to take a look for themselves.

Here's what several experts say about Talagrand's book (from Talagrand's webpage https://michel.talagrand.net/endorsements.pdf):

"This book accomplishes the following impossible task. It explains to a mathematician, in a language that a mathematician can understand, what is meant by a quantum field theory from a physicist's point of view. The author is completely and brutally honest in his goal to truly explain the physics rather than filtering out only the mathematics, but is at the same time as mathematically lucid as one can be with this topic. It is a great book by a great mathematician."

-- Sourav Chatterjee, Stanford University

"Talagrand has done an admirable job of making the difficult subject of quantum field theory as concrete and understandable as possible. The book progresses slowly and carefully but still covers an enormous amount of material, culminating in a detailed treatment of renormalization. Although no one can make the subject truly easy, Talagrand has made every effort to assist the reader on a rewarding journey though the world of quantum fields."

-- Brian Hall, University of Notre Dame

"A presentation of the fundamental ideas of QFT in a manner that is both accessible and mathematically accurate seems like an impossible dream. Well, not anymore! This book goes from basic notions to advanced topics with patience and care. It is an absolute delight to anyone looking for a friendly introduction to the beauty of QFT and its mysteries."

-- Shahar Mendelson, Australian National University

"I have been motivated to try and learn about quantum field theories for some time, but struggled to find a presentation in a language that I as a mathematician could understand. This book was perfect for me: I was able to make progress without any initial preparation, and felt very comfortable and reassured by the style of exposition."

-- Ellen Powell, Durham University

"In addition to its success as a physical theory, Quantum Field Theory (QFT) has been a continuous source of inspiration for mathematics. However, mathematicians trying to understand QFT must contend with the fact that some of the most important computations in the theory have no rigorous justification. This has been a considerable obstacle to communication between mathematicians and physicists. It is why despite many fruitful interactions, only very few people would claim to be well versed in both disciplines at the highest level.

There have been many attempts to bridge this gap, each emphasizing different aspects of QFT. Treatments aimed at a mathematical audience often deploy sophisticated mathematics. Michel Talagrand takes a decidedly elementary approach to answering the question in the title of his monograph, assuming little more than basic analysis. In addition to learning what QFT is, the reader will encounter in this book beautiful mathematics that is hard to find anywhere else in such clear pedagogical form, notably the discussion of representations of the Poincaré group and the BPHZ Theorem. The book is especially timely given the recent resurgence of ideas from QFT in probability and partial differential equations. It is sure to remain a reference for many decades."

-- Philippe Sosoe, Cornell University

[averycanadianfilm]

Yang–Mills for mathematicians - Sourav Chatterjee

What is a QFT? — This is an open question, not only in mathematics, but also in physics.

Remarkably, physicists can calculate and make surprisingly accurate predictions using QFTs, without really understanding what these objects are!

The mathematical construction of quantum field theories —more specifically Yang–Mills theories — is one of the seven millennium problems posed by the Clay Institute.