I hope you don't mind me having two questions.

Firstly, I was running a Qiskit HHL simulation on a 12x12 matrix and a 12x1 vector, leading to a 16x16 matrix after expansion, and it resulted in a circuit width of 10 qubits and depth of 198 gates.
What is the maximum depth possible on a quantum computer?

Secondly, on a smaller problem in the HHL of size 2x2 the depth is 326 and width of 7 qubits. Are my results wrong? It seems odd to have a lower depth than such a small problem.

[1] https://qiskit.org/textbook/ch-applications/hhl_tutorial.html#implementationsim


This is quite a broad question. In fact it seems that you have 2 questions:

  1. How can a smaller (in term of size) matrix result in a longer quantum circuit?
  2. What is the maximum depth current quantum computer can execute (reliably)?

About question 1, the number of quantum gates and depth of the quantum circuit generated depends a lot on the matrix $A$ of your linear system, on the method used to implement the evolution $e^{-iAt}$ and on how you "load" the right-hand side $b$ in a quantum register.

Efficient methods to construct the quantum circuit that implements $e^{-iAt}$ exist when $A$ satisfy some properties (like sparsity or locality). But here, efficient does not mean NISQ-compliant, it only means that the circuits generated by the method have a number of quantum gates that scale well with the size of the matrix. Some examples of generic methods can be found here and an example of an hand-craft method for specific matrices has been written here.

Another point that might impact a lot the final depth of the circuit is the encoding of $b$ into a quantum register.

It is not possible to know if your issue is caused by one of the previous points or not without the actual matrix and right-hand side you used.

About your second question, have a look at this answer. Do not use the numbers in it as they are probably outdated, but you can use the method with up-to-date error rates.

The short answer is: in most of the quantum circuits depth is not the important figure, CNOT number and CNOT error rates seems to have a greater impact.

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