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I know that Clifford gates can be efficiently simulated on classical computers using tableaux. How are non-clifford gates handled? Can simulators handle 100 qubit non-clifford gates?

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    $\begingroup$ Stim doesn't simulate non-clifford gates. $\endgroup$ Jun 4, 2022 at 2:40
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    $\begingroup$ @CraigGidney I expected this to be hard or slow but not to be handled at all is surprising. Does qiskit have the same limitation? The class of gates I'm interested in now is controlled $A$ where $A$ is a Pauli string applied depending on state of a number of qubits (similar to CCZ, CCCZ, ... but possibly depending on more that one control qubit) $\endgroup$
    – unknown
    Jun 4, 2022 at 15:08
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    $\begingroup$ Craig can correct me if I am wrong, but from my understanding Stim was built to efficiently simulate and study specifically stabilizer circuits (i.e. circuits with only Clifford gates). With qiskit you can simulate any circuit, but as your circuit size grows, the simulation will become exponentially slower. and probably cannot handle 100 qubit non-clifford gates $\endgroup$ Jun 6, 2022 at 9:57

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For all methods that are known today, there is no efficient way to implement most general cases of non-Clifford gates, unless you have some restriction.

  • Full state vector simulation as qiskit, is not able to reach 100 qubits in non-Clifford gates (qiskit also knows to optimize in case of Clifford)
  • There is a way to simulate in polynomial time in a method that converts each qubit to a representation called Majorana, BUT it is able to make rotations just in 1 of the axes (e.g. only X rotations or only Z rotations) on all the qubits that belong to the running simulation. Read more here for example
  • Another way to be efficient is conversion to Tensor Network, which in cases that your circuit is rectangular and long, let you do more efficient simulation, and reach 100 qubits (again, depending on your circuit) you can read more here for example

In case of surface code simulation for example, where you have $d^2$ qubits: enter image description here

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