Fast quantum measurement tomography with dimension-optimal error bounds

TL;DR

The paper tackles detector ( POVM ) tomography by introducing a two-step projected least-squares protocol that decouples data fitting from constraint enforcement, achieving dimension-optimal sample complexity up to logarithmic factors. It provides non-asymptotic guarantees for two natural distance metrics, d_op and d_av, with explicit sample bounds for both global and tensor-product 2-design probe ensembles. The authors establish matching information-theoretic lower bounds for non-adaptive tomography and demonstrate practical applicability through experiments on noisy superconducting qubits, highlighting potential uses in error mitigation and detector calibration. The work advances understanding of QMT efficiency, offering analytic LS forms under 2-designs and concrete guidance for implementing reliable detector tomography in larger quantum systems.

Abstract

We present a two-step protocol for quantum measurement tomography that is light on classical co-processing cost and still achieves optimal sample complexity in the system dimension. Given measurement data from a known probe state ensemble, we first apply least-squares estimation to produce an unconstrained approximation of the POVM, and then project this estimate onto the set of valid quantum measurements. For a POVM with $L$ outcomes acting on a $d$-dimensional system, we show that the protocol requires $\mathcal{O}(d^3 L \ln(d)/ε^2)$ samples to achieve error $ε$ in worst-case distance, and $\mathcal{O}(d^2 L^2 \ln(dL)/ε^2)$ samples in average-case distance. We further establish two almost matching sample complexity lower bounds of $Ω(d^3/ε^2)$ and $Ω(d^2 L/ε^2)$ for any non-adaptive, single-copy POVM tomography protocol. Hence, our projected least squares POVM tomography is sample-optimal in dimension $d$ up to logarithmic factors. Our method admits an analytic form when using global or local 2-designs as probe ensembles and enables rigorous non-asymptotic error guarantees. Finally, we also complement our findings with empirical performance studies carried out on a noisy superconducting quantum computer with flux-tunable transmon qubits.

Figures (6)

• Figure 1: Sketch of the quantum measurement tomography protocol. a) Uniformly sample a set of input states from an IC ensemble $\{\ket{\psi_i}\}$ b) A quantum device is initialized in each randomized input state and measured using the POVM. The resulting frequencies $\hat{f}_{ij}$ for input state $\ket{\psi_i}$ and POVM outcome $E_j$ are stored in a vector $\hat{f}$. c) A least-squares estimator for the POVM is computed using the measured frequencies $\hat{f}$ and the operators $\nu_i$. This step depends on the choice of ensemble, such as a global or local 2-design. d) The raw least squares estimate is projected into the physical subspace of all POVMs via convex optimization. This produces the reconstructed POVM $(E_1^*,\ldots,E_L^*)$ as output. e) Example of the two first elements of a single qubit Symmetrical Informationally Complete (SIC-)POVM reconstructed from an experimental implementation on two flux-tunable transmon qubits.
• Figure 2: Results for the reconstruction of a noisy one-qubit Symmetrical Infomationally Complete (SIC-)POVM using an auxiliary qubit for the generalized measurement, implemented on a two-qubit flux-tunable transmon device using a budget of $1.66\cdot10^5$ random initial states. The reconstructed noisy POVM is compared to its expected noiseless counterpart, with the absolute difference displayed. Though noisy, the reconstruction follows the target POVM closely. The deviations from the expected values are due to the noisy device implementation.
• Figure 3: Comparison of the half-sided noisy measurement channel with the expected channel for a one-qubit SIC-POVM. The channels are represented in Pauli basis. The absolute difference between them is also shown on the right. The half-sided noisy channel only slightly deviates from the expected results, proportional to the depolarizing channel. This reconstruction of the channel actually implemented in the device can be used to improve results executed on the experimental device.
• Figure 4: Results for the reconstruction of the full system implementing a noisy two-qubit Symmetrical Infomationally Complete (SIC-)POVM, implemented on a two-qubit flux-tunable transmon device using a budget of $10^6$ random initial states. The reconstructed noisy POVM is compared to its expected noiseless counterpart, with the absolute difference displayed. Though noisy, the reconstruction follows the target POVM closely. The deviations from the expected values are due to the noisy device implementation.
• Figure 5: Quantum circuit depicting the implementation of the SIC-POVM. When the single gate rotation parameters are $\theta=\arccos(1/\sqrt{3})$ and $\phi=3\pi/4$, this results in a Symmetrical Informationally Complete measurement. The choice of a CZ gate as the entangling operation is due to the nate interactions of the experimental device where this protocol is deployed.

Theorems & Definitions (26)

• Definition 1: Operational distance
• Definition 2: Average case distance
• Theorem 1
• Lemma 2
• Theorem 3
• proof
• Theorem 4
• Lemma 5
• Lemma 6
• proof