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Though a lot of papers talk about the largest factored number by Shor's Algorithm, I want to know the largest one factored by the IBM Quantum, for example, the simulator_mps. I am able to use it to factor 143, but fail at 323.

The simulator does offer 100 qubits, and I use only 38 of them. The trouble is that the execution will time out always.

My circuit has about 1 million gates, 600,000 of which are CNOT gates.

Could you please set my expectation straight? What is the biggest circuit you have ever run successfully?

I understand the classical computers may never be able to simulate 100 qubits. It is easy to understand why IBM focuses on increasing the qubits of the real quantum hardware. But, it is unclear to me why they would offer 100 simulated qubits without backing them up with massive classical parallel computing.

According to the roadmap of IBM and this article, the year 2023 will be a critical year:

If IBM can meet the 1,121-qubit goal in the next three years, the company says it is confident it will make its 2030 deadline for a 1 million-qubit machine.

One million physical qubits could produce 1,000 logical qubits.

This study shows:

... technological change is exponential, contrary to the common-sense 'intuitive linear' view.

So, I'd like to know how well the state-of-the-art IBM quantum computers/simulators are doing. That may help understand the future trends.

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    $\begingroup$ According to qiskit.org/documentation/tutorials/simulators/… the MPS simulator is a more involved representation of quantum states that allows to significantly reduce memory requirements when the state is not too entangled. In the worst case it degrades to an exponential memory like the state vector simulator. The 100-qubit limit has probably been set with "low-entanglement" circuits in mind, which is not your case. Moreover, 1 million gates is huge, this is probably the reason you get a timeout. $\endgroup$ Mar 29 at 8:32
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I think I can partially answer my own question. I am able to use ibmq_qasm_simulator to factor 667 with 23 qubits of a pure QASM circuit under 6,055 seconds. If you can do better, please share your results. I use the simulator to do research on how effective the algorithm really is.

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    $\begingroup$ Actually IBM roadmap says nothing about cracking RSA in 2023. The roadmap talks about 1121 "physical" qubits quantum computer. Breaking RSA-2048 requires 4000+ "logical" qubits. $\endgroup$ Apr 2 at 7:51
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    $\begingroup$ Also, the time needed for a simulator to run some algorithm is not a "good" measure for algorithm effectiveness. $\endgroup$ Apr 2 at 7:57

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