Expected speedups for Sz or parity conservation in qc simulations

How do I use this algorithm? What does that parameter do?
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Qottmann
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Expected speedups for Sz or parity conservation in qc simulations

Post by Qottmann »

If I wanted to simulate some quantum computing algorithms using tenpy, can I expect any speedups from using Sz or parity conservation? I know too little of what these actually do.

I.e. one would be using `SpinHalfSites` to simulate qubits, and an MPS on a chain to simulate the current state. The algorithm would be applying unitary operations onto the state, who is finally used to evaluate the expectation value of some Hamiltonian.
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Johannes
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Re: Expected speedups for Sz or parity conservation in qc simulations

Post by Johannes »

In general yes, using Sz or parity conservation can speed up calculations. The actual speedup depends on the number and sizes of symmetry sectors in the tensors - it comes from the fact that the tensors have a block-diagonal form. Parity conservation divides a tensor into 2 subsectors, leading to a speedup of roughly 4: from N^3 for the full matrices to 2 * (N/2)^3 for 2 multiplications of blocks half the size.

In praxis, the real question is whether the circuit you want to apply actually does conserve Sz at each step - it is a severe restriction that for example doesn't allow any X, Y or CNOT gate - you could only use the CZ two-qubit gate. Using only parity conservation, X and Y would still be allowed, but CNOT is again excluded.

Of course, just using MPS can already give a speedup compared to a naive implementation with the full state - and e.g. quiskit actually has a circuit simulator based on MPS. But again, you only get this speedup if your algorithm is not exploiting the full Hilbert space and does not build up too much entanglement - otherwise, you need to grow the bond dimension to avoid too much truncation, and can end up at the same exponential scaling again. And of course, the power of the quantum computer compared to the classical one lies in the fact that it *can* build up entanglement during it's computation - so the standard quantum algorithms also require this.
thunderingcaustic
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Re: Expected speedups for Sz or parity conservation in qc simulations

Post by thunderingcaustic »

Thanks for the insightful overview! It's great to see the potential speedups with Sz or parity conservation in quantum simulations. The trade-offs between gate restrictions and computational efficiency add an interesting layer to the decision-making process. Your explanation about the impact on tensor sizes and entanglement is particularly helpful for understanding the practical considerations. Looking forward to exploring this further in my simulations.
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