Comment by westurner | Hacker News Reader

Comment by westurner | Hacker News Reader

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> one can literally write down a matrix of numbers in a program to define a new mathematical function, and later use that function to manipulate the contents of the computer's RAM[1].

This "Quantum Computing for Computer Scientists" video https://youtu.be/F_Riqjdh2oM explains classical and quantum operators as just matrices. What are other good references?

Quantum state: https://en.wikipedia.org/wiki/Quantum_state

Quantum logic; quantum logical operators: https://en.wikipedia.org/wiki/Quantum_logic

> All this business about energy levels, evolution, etc. are distractions, just as the electrodynamics of a transistor are distractions from what it means to program a computer.

But a classical simulator - like e.g. qiskit - for a quantum circuit/experiment/function must run the experiment very^very^very many times to even probabilistically approximate a sufficient quantum system; because of the combinatorial probabilistic explosion that results from adding just one more basis state.

What are the fundamental limitations of quantum simulators? Maybe it's possible.

Quantum simulator: https://en.wikipedia.org/wiki/Quantum_simulator

- [ ] Maybe Twistor theory has insight into a classical geometrical formulation that could be run on a non-QC?

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

[Photon] wave-particle constructive superpositions approximate which operators, which may form a neat topology like this:

- [ ] > A research question for a new school year:

> The classical logical operators form a neat topology. Should we expect there to be such symmetry and structure amongst the quantum operators as well? https://commons.m.wikimedia.org/wiki/File:Logical_connective...

Your assessment of how a quantum simulator works is not quite right. These simulators represent the entire probability distribution of basis states succinctly as an array. This array grows very large (exponentially) in the number of qubits.

A simulator only needs to run a computation once (which is multiplication of matrices in a tensor product space) and look at the resulting state. You don't need to run anything multiple times to approximate a quantum state.

The questions you're asking are the whole point of the field of quantum information science. On an ordinary quantum computer where quantum state will collapse to a basis state upon readout, indeed you might need to gather statistics to determine the answer to whatever you've asked your computer. However, "very many times" is mathematically bounded in some way for an ideal quantum computer. It's like saying "we need to do many^many^many^many comparisons to do quicksort". Well yes, but we have a relationship between the size of the input (N) and the average number of comparisons needed (N log N), which makes the algorithm feasible in practice. This is the same with quantum algorithms.

There are also different kinds of quantum algorithms. Some are more probabilistic in nature. Others are—again in purely ideal circumstances—give you the right answer in one go.

As a side note: It is very hard for me to read, understand, and respond to your comments. They seem like random buzzword soups and aren't very coherently put together, mixed with random links and references.

Re: error in simulators and actual QC hardware, which we do need for a reason: Quantum Error Correction # General_codes https://en.wikipedia.org/wiki/Quantum_error_correction#Gener...

How to best quantize reals into matrices (~= tensors)? https://en.wikipedia.org/wiki/Quantization_(signal_processin...

> Peter Shor first discovered this method of formulating a quantum error correcting code by storing the information of one qubit onto a highly entangled state of nine qubits. A quantum error correcting code protects quantum information against errors of a limited form.

Here's "Quantum Algorithm Zoo" by Microsoft Quantum: https://quantumalgorithmzoo.org/

And "Timeline of quantum computing and communication" https://en.wikipedia.org/wiki/Timeline_of_quantum_computing_...

I have a hard time with the idea that the outcome of the ultimate quantum simulation is a collapsed float.

Quantum Monte Carlo: https://en.wikipedia.org/wiki/Quantum_Monte_Carlo :

> Quantum Monte Carlo encompasses a large family of computational methods whose common aim is the study of complex quantum systems. One of the major goals of these approaches is to provide a reliable solution (or an accurate approximation) of the quantum many-body problem. [...] The difficulty is however that solving the Schrödinger equation requires the knowledge of the many-body wave function in the many-body Hilbert space, which typically has an exponentially large size in the number of particles. Its solution for a reasonably large number of particles is therefore typically impossible,

What sorts of independent states can or should we map onto error-corrected qubits in an approximating system?

Propagation of Uncertainty ... Numerical stability ... Chaotic convergence, ultimately, apparently: https://en.wikipedia.org/wiki/Propagation_of_uncertainty

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