Predicting Many Properties of a Quantum System from Very Few Measurements - Paper Implementation Challenge #938
Description
I will implement the classical shadow algorithm described in this paper. This algorithm enables the recovery of many properties from a quantum system with few measurements. For M observables, the method needs
Algorithm Outline:
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Data Acquisition: Repeat the following N times.
-Select a random unitary transformation from a predefined ensemble (e.g., Clifford group or Pauli measurements).
-Apply the transformation to the quantum state of interest:
$$\rho \rightarrow U\rho U^{\dagger}$$
-Measure the transformed quantum state in the computational basis$\ket b_i \in \set{0,1}^n$
-Store the snapshot of the state by performing the reverse operation$U^{\dagger}\ket b \bra b U$ and storing it in classical memory.
-The average of these snapshots can be viewed as a measurement channel:
$$\mathbb{E}[U^{\dagger}\ket b \bra b U] = \mathcal{M}(\rho)$$
-We can reconstruct the state by inverting the above:
$$\rho = \mathbb{E}[\mathcal{M}^{-1} (U^{\dagger}\ket b \bra b U)]$$ -
Prediction:
-After repeating the above N times, we obtain the classical shadow:
$$S(\rho; N) = n, \quad \hat{\rho}_1 = M^{-1} \left( U_1^\dagger \lvert \hat{b}_1 \rangle \langle \hat{b}_1 \rvert U_1 \right), \ldots, \hat{\rho}_N = M^{-1} \left( U_N^\dagger \lvert \hat{b}_N \rangle \langle \hat{b}_N \rvert U_N \right)$$
-We can estimate any observable using its mean:
- Post-Processing:
-Use Median of Means estimation to more accurately predict observables.