State tomography with Pauli basis measurements for a high number of qubits

Question

My end goal is to recover the quantum state in its computational basis or reduced density matrix of a high number qubit circuit in a real QPU. Taking into account that the number of qubits will be high (+16 or +32 qubits) getting the density matrix as it is done in the common quantum state tomography algorithms is unfeasible.

My idea is to try tomography with Pauli basis measurements to get the individual reduced density matrix of each qubit, but due to the high number of qubits, the number of measurements required will be huge too.

I was wondering if there is a feasible way to get this or in case there isn't which option would be the least unfeasible to continue researching on it. Thanks in advance.

Pd: the circuit is a 16 or 32 qubit circuit with some hadamard, cnot and u(rotation) gates.

Answer 1

32 qubits give you a Hilbert space dimension of $d=2^{32} = 4\,294\,967\,296$. A recent paper has done tomography of $d=20$ dimensional state, with the highest infidelities for tomography made so far https://journals.aps.org/prl/abstract/10.1103/PhysRevLett.126.100402, and it was featured in Physics https://physics.aps.org/articles/v14/s34.

Let's say you want to perform full quantum state tomography of a $d$-dimensional state $\rho$ (possibly mixed). Assume that you want to use the trace norm to estimate how close you are from the true state, i.e., letting $\hat{\rho}$ be the estimate you want to get $\varepsilon$ close to $\rho$ with high-probability. Therefore you want $\Vert \rho - \hat{\rho}\Vert_1 < \varepsilon$. The minimum number of samples of the state you'll need to use scales with $d^2/\varepsilon^2$. Check out the results of this reference here. If you know the rank, this becomes $\sim dr/\varepsilon^2$. For 32 qubits, it is a lot!

So, I would suggest you to not do full tomography because Pauli basis tomography will perform much worse than the above optimally efficient quantum algorithms. Pauli basis measurement tomography algorithm requires $O(d^4/\varepsilon^2)$ copies of $\rho$ in total. Instead of doing full tomography, would it be useful to use Shadow Tomography techniques? The number of measurements needed scales much better. Another possibility, if high-precision is not needed, direct estimation methods using weak-measurements is also a possibility (although these do not seem to be well behaved when there is presence of noise).