Can we simulate quantum state and qubits with conventional computers?

Proper simulation of quantum
computer is possible with
an other
quantum system.
Quantum Lattice NIST (Wikipedia)

The short answer is no. 

Certainly, the quantum state isn't merely a vector of complex numbers. It is inherent to the physical system which one observe quantum properties, i.e., the observer can not be removed from the observed. When we try to measure things we affect the result of the measurement. Unfortunately, contrary to popular belief, quantum systems can not be simulated with conventional computers. Numerical solutions to equations representing quantum systems are not simulations both philosophically and physically. 

Similarly, a qubit is not merely a linear combination of two complex vectors or being 1 and 0 at the same time. Qubit is also a property of a physical system and can not be simulated with a classical computer. 

Q.E.D.

Postscript Notes

• This is not new of course. Feynman has expressed the same in his landmark paper from 1982, Simulating physics with computers, here. ".. No! This is called the hidden-variable problem: it is impossible to represent the results of quantum mechanics with a classical universal device..." Richard Feynman  Feynman, R.P. Simulating physics with computers. Int J Theor Phys 21, 467–488 (1982). doi

• It is not about the hardness of simulating qubits, I.e, dynamical evolution of quantum states, the behaviour that would prove quantum advantage is a physical effect not a computational one. Entanglement is a physical process that provides computational advantage over classical computers, if it were to be replicated with numerical procedure we wouldn't have difficulty of building a quantum computer albeit a simulated one.

• Let's ask a similar question: Can we prove or conceptually show that there is a quantum advantage with simulation on the classical hardware? Unfortunately it is the same answer: No, quantum advantage can not be simulated. If we could, then we could have a simulated quantum computer on a classical hardware that solves thing much faster than the host hardware.

• Simulation is not about numerical solutions only, it means the physically intrinsic properties occurring within the simulator: This means one can simulate quantum systems or qubits only with another quantum systems, a recent example is using quantum system of ion traps to simulate another quantum system.

• Difficulty of simulating a quantum state is not about quantum dynamics : One of the hard problems in quantum computing is simulating a quantum mechanical computing device on a classical hardware. However, this is not about solving dynamics of a quantum system rather having a quantum effect on a classical system.

• A misconception in quantum computing frameworks: They don’t mean to simulate qubit as in having its behaviour replicated on a classical hardware. If you see a computational framework that claims that it can simulate a qubit, it doesn’t mean that classical hardware can replicate qubit’s behaviour, even if they solve full quantum hamiltonian dynamic evolution . Simulation in those framework implies given parameter settings and outcome is also set, one could think “simulation” in that setting as validation of already happened quantum measurement.   

• Elusive quantum state simulation : No not possible on “classical machines”
  Even one of the pioneers in quantum computing express his puzzlement of what quantum state implies, i.e., Nielsen (see What does quantum state means?).  Furthermore, current quantum computing libraries presents something called simulation mode or quantum virtual machine. Those novel works do not claim that they can simulate quantum effects on classical machine rather mimics quantum states known behaviour at the time of measurement.

• Quantum Machine Learning models can't be mapped into classical ML models
  A misconception is still repeated that we can somehow simulate or replicate artefacts of quantum computation with a classical counter part with an approximation. This is not possible due to very nature of quantum mechanical process that it can't be replicated with a classical counter part. Quantum states can't be replicated on a classical hardware, as in producing quantum advantage. 
  "..it is impossible to represent the results of quantum mechanics with a classical universal device..." Richard Feynman 
  cf.  Feynman, R.P. Simulating physics with computers. Int J Theor Phys 21, 467–488 (1982).
  More pessimistic interpretation of this statement, unfortunately, that we can't even translate data from classical hardware to quantum hardware or vice versa.

• Quantum states can not be replicated on “classical machines”: Quantum Virtual Machines (QVMs) does not claim to replicate quantum effects on classical hardware. It is a misconception to think otherwise leading to a paradox that we could have a quantum advantage on classical devices albeit a simulated one.

• Simulating quantum computers with LLMs & classical hardware 
  It doesn’t matter if we use LLMs: it isn’t possible to simulate quantum computers on classical hardware. Difficulty is not about exponential computational complexity.  Replication of effects of quantum systems on classical machines is akin to perpetual motion machine. 

• Classical hardware can’t hold qubits : Whether we use LLMs or not, it isn’t possible to simulate quantum states (computer) on a classical hardware. Difficulty is not about exponential computational complexity of quantum Hamiltonian evolution.