We consider the task of performing quantum state tomography on a $d$-level spin qudit, using only measurements of spin projection onto different quantization axes. After introducing a basis of operators closely related to the spherical harmonics, which obey the rotational symmetries of spin qudits, we map our quantum tomography task onto the classical problem of signal recovery on the sphere. We then provide algorithms with $O(r{d}^{3})$ serial runtime, parallelizable down to $O(r{d}^{2})$ for: (i) computing a priori upper bounds on the expected error with which spin projection measurements along $r$ given axes can reconstruct an unknown qudit state, and (ii) estimating a posteriori the statistical error in a reconstructed state. Our algorithms motivate a simple randomized tomography protocol for which we find that using more measurement axes can yield substantial benefits that plateau after $r\ensuremath{\approx}3d$.
We consider the task of performing quantum state tomography on a d-level spin qudit, using only measurements of spin projection onto different quantization axes. After introducing a basis of operators closely related to the spherical harmonics, which obey the rotational symmetries of spin qudits, we map our quantum tomography task onto the classical problem of signal recovery on the sphere. We then provide algorithms with O(rd3) serial runtime, parallelizable down to O(rd2) for: (i) computing a priori upper bounds on the expected error with which spin projection measurements along r given axes can reconstruct an unknown qudit state, and (ii) estimating a posteriori the statistical error in a reconstructed state. Our algorithms motivate a simple randomized tomography protocol for which we find that using more measurement axes can yield substantial benefits that plateau after r\ensuremath≈3d.
We consider the task of performing quantum state tomography on a d-state spin qudit, using only measurements of spin projection onto different quantization axes. By an exact mapping onto the classical problem of signal recovery on the sphere, we prove that full reconstruction of arbitrary qudit states requires a minimal number of measurement axes, rdmin, that is bounded by 2d−1≤rdmin≤d2. We conjecture that rdmin=2d−1, which we verify numerically for all d≤200. We then provide algorithms with O(rd3) serial runtime, parallelizable down to O(rd2), for (i) computing a priori upper bounds on the expected error with which spin projection measurements along r given axes can reconstruct an unknown qudit state, and (ii) estimating a posteriori the statistical error in a reconstructed state. Our algorithms motivate a simple randomized tomography protocol, for which we find that using more measurement axes can yield substantial benefits that plateau after r≈3d.
We propose and demonstrate a method for quantum-state tomography of qudits encoded in the quantum polarization of N-photon states. This is achieved by distributing N photons nondeterministically into three paths and their subsequent projection, which for N=1 is equivalent to measuring the Stokes (or Pauli) operators. The statistics of the recorded N-fold coincidences determines the unknown N-photon polarization state uniquely. The proposed, fixed setup manifestly rules out any systematic measurement errors due to moving components and allows for simple switching between tomography of different states, which makes it ideal for adaptive tomography schemes.
The reconstruction of quantum states from experimental measurements, often achieved using quantum state tomography (QST), is crucial for the verification and benchmarking of quantum devices. However, performing QST for a generic unstructured quantum state requires an enormous number of state copies that growsexponentiallywith the number of individual quanta in the system, even for the most optimal measurement settings. Fortunately, many physical quantum states, such as states generated by noisy, intermediate-scale quantum computers, are usually structured. In one dimension, such states are expected to be well approximated by matrix product operators (MPOs) with a matrix/bond dimension independent of the number of qubits, therefore enabling efficient state representation. Nevertheless, it is still unclear whether efficient QST can be performed for these states in general. In other words, there exist no rigorous bounds on the number of state copies required for reconstructing MPO states that scales polynomially with the number of qubits. In this paper, we attempt to bridge this gap and establish theoretical guarantees for the stable recovery of MPOs using tools from compressive sensing and the theory of empirical processes. We begin by studying two types of random measurement settings: Gaussian measurements and Haar random projective measurements. We show that the information contained in an MPO with a constant bond dimension can be preserved using a number of random measurements that depends onlylinearlyon the number of qubits, assuming no statistical error of the measurements. We then study MPO-based QST with Haar random projective measurements that can in principle be implemented on quantum computers. We prove that only apolynomialnumber of state copies in the number of qubits is required to guarantee bounded recovery error of an MPO state. Remarkably, such recovery can be achieved by measuring the state in each random basis only once, despite the large statistical error associated with the outcome of each measurement. Our work may be generalized to accommodate random local or t-design measurements that are more practical to implement on current quantum computers. It may also facilitate the discovery of efficient QST methods for other structured quantum states.
Quantum state tomography (QST) is a fundamental task in quantum information science that aims to reconstruct unknown quantum states from measurement data. However, the exponential growth of Hilbert-space dimension with system size makes full tomography of general quantum states statistically and computationally prohibitive. This challenge has motivated extensive research on structured quantum state tomography, where prior structure, such as low-rankness, tensor-network representations, shallow quantum circuits, and neural quantum states, can substantially reduce the effective degrees of freedom and enable scalable recovery. In this review, we provide a unified perspective on QST for structured quantum states through three closely related themes: compact state representations, measurement design, and computational algorithms. After reviewing common models for structured quantum states, we survey existing work on geometric preservation properties of measurement frameworks, ranging from informationally complete POVMs to randomized measurements, and their implications for sample complexity. On the algorithmic side, we review optimization methods for reconstructing structured quantum states from empirical measurements. By connecting QST with broader principles from compressive sensing, matrix sensing, and structured inverse problems, this survey highlights common theoretical foundations underlying sample complexity, measurement efficiency, and scalable recovery.
The techniques of low-rank matrix recovery were adapted for quantum state tomography (QST) previously by Gross et al. [Phys. Rev. Lett. 105, 150401 (2010)] where they consider the tomography of n spin-1/2 systems. For the density matrix of dimension d=2n and rank r with $r\ensuremath{\ll}{2}^{n}$, it was shown that randomly chosen Pauli measurements of the order O[drlog(d)2] are enough to fully reconstruct the density matrix by running a specific convex optimization algorithm. The result utilized the low operator norm of the Pauli operator basis, which makes it ``incoherent'' to low-rank matrices. For quantum systems of dimension d not a power of two, Pauli measurements are not available, and one may consider using SU(d) measurements. Here, we point out that the SU(d) operators, owing to their high operator norm, do not provide a significant savings in the number of measurement settings required for successful recovery of all rank-r states. We propose an alternative strategy in which the quantum information is swapped into the subspace of a power-two system using only poly[log(d)2] gates at most with QST being implemented, subsequently, by performing O[drlog(d)2] Pauli measurements. We show that, despite the increased dimensionality, this method is more efficient than the one using SU(d) measurements.
Estimating many local expectation values over time is a central measurement bottleneck in quantum simulation and device characterization. We study the task of reconstructing the Pauli-signal matrix Sij=Tr(Oiρ(tj)) for a collection of M low-weight Pauli observables {Oi}i=1M over N timesteps {tj}j=1N, while minimizing the total number of device shots. We propose a Compressed Sensing Shadow Tomography (CSST) protocol that combines two complementary reductions. First, local classical shadows reduce the observable dimension by enabling many Pauli expectation values to be estimated from the same randomized snapshots at a fixed time. Second, compressed sensing reduces the time dimension by exploiting the fact that many expectation-value traces are spectrally sparse or compressible in a unitary (e.g., Fourier) transform basis. Operationally, CSST samples m≪N timesteps uniformly at random, collects shadows only at those times, and then reconstructs each length-N signal via standard ℓ1-based recovery in the unitary transform domain. We provide end-to-end guarantees that explicitly combine shadow estimation error with compressed sensing recovery bounds. For exactly s-sparse signals in a unitary transform basis, we show that m=O(slog2slogN) random timesteps suffice (with high probability), leading to total-shot savings scaling as Θ(N/s) (i.e., up to polylogarithmic factors) relative to collecting shadows at all N timesteps. For approximately sparse signals, the reconstruction error decomposes into a compressibility (tail) term plus a noise term. We present numerical experiments on noisy many-qubit dynamics that support strong Fourier compressibility of Pauli traces and demonstrate substantial shot reductions with accurate reconstruction.