Sample-Optimal Quantum State Tomography for Structured Quantum States in One Dimension

Quantum state tomography (QST) remains the gold standard for benchmarking and verifying quantum devices. A recent study has proved that, with Haar random projective measurements, only a $O(n^3)$ number of state copies is required to guarantee bounded recovery error of an matrix product operator (MPO) state of qubits $n$. While this result provides a formal evidence that quantum states with an efficient classical representation can be reconstructed with an efficient number of state copies, the number of state copies required is still significantly larger than the number of independent parameters in the classical representation. In this paper, we attempt to narrow this gap and study whether the number of state copies can saturate the information theoretic bound (i.e., $O(n)$, the number of parameters in the MPOs) using physical quantum measurements. We answer this question affirmatively by using a class of Informationally Complete Positive Operator-Valued Measures (IC-POVMs), including symmetric IC-POVMs (SIC-POVMs) and spherical $t$-designs. For SIC-POVMs and (approximate) spherical 2-designs, we show that the number of state copies to guarantee bounded recovery error of an MPO state with a constrained least-squares estimator depends on the probability distribution of the MPO under the POVM but scales only linearly with $n$ when the distribution is approximately uniform. For spherical $t$-designs with $t\ge3$, we prove that only a number of state copies proportional to the number of independent parameters in the MPO is needed for a guaranteed recovery of any state represented by an MPO. Moreover, we propose a projected gradient descent (PGD) algorithm to solve the constrained least-squares problem and show that it can efficiently find an estimate with bounded recovery error when appropriately initialized.