Ciro Santilli @cirosantilli  40

TODO why is it so hard to find anything non perturbative :-(

On a quantum computer...:

TODO concrete example, please...

The different only shows up for field, not with particles. For fields, there are two types of changes that we can make that can keep the Lagrangian unchanged as mentioned at Physics from Symmetry by Jakob Schwichtenberg (2015) chapter "4.5.2 Noether's Theorem for Field Theories - Spacetime":

From defining properties of elementary particles:

From the spacetime theory alone, we can derive the Lagrangian for the free theories for each spin:Then the internal symmetries are what add the interaction part of the Lagrangian, which then completes the Standard Model Lagrangian.

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Mentioned at: Video "Quantum Computing with Light by Quantum Light University of Sheffield (2015)" youtu.be/nyK-vhoOBpE?t=185.

One possibly interesting and possibly obvious point of view, is that a quantum computer is an experimental device that executes a quantum probabilistic experiment for which the probabilities cannot be calculated theoretically efficiently by a nuclear weapon.

This is how quantum computing was originally theorized by the likes of Richard Feynman: they noticed that "Hey, here's a well formulated quantum mechanics problem, which I know the algorithm to solve (calculate the probability of outcomes), but it would take exponential time on the problem size".

The converse is then of course that if you were able to encode useful problems in such an experiment, then you have a computer that allows for exponential speedups.

This can be seen very directly by studying one specific quantum computer implementation. E.g. if you take the simplest to understand one, photonic quantum computer, you can make systems for which you need exponential time to calculate the probabilities that photons will exit through certain holes and not others.

The obvious aspect of this idea is by coming from quantum logic gates are needed because you can't compute the matrix explicitly as it grows exponentially: knowing the full explicit matrix is impossible in practice, and knowing the matrix is equivalent to knowing the probabilities of every outcome.

Mentioned e.g. at:

These are two conflicting constraints:

It takes time for the quantum state to evolve. So in order to have a deep quantum circuit, we need longer coherence times.

Technique that uses multiple non-ideal qubits (physical qubits) to simulate/produce one perfect qubit (logical).

One is philosophically reminded of classical error correction codes, where we also have multiple input bits per actual information bit.

TODO understand in detail. This appears to be a fundamental technique since all physical systems we can manufacture are imperfect.

Part of the fundamental interest of this technique is due to the quantum threshold theorem.

For example, when PsiQuantum raised 215M in 2020, they announced that they intended to reach 1 million physical qubits, which would achieve between 100 and 300 logical qubits.

Video "Jeremy O'Brien: "Quantum Technologies" by GoogleTechTalks (2014)" youtu.be/7wCBkAQYBZA?t=2778 describes an error correction approach for a photonic quantum computer.

Bibliography:

Only NP-intermediate, which includes notably integer factorization: