Efficient direct quantum state tomography using fan-out couplings

Abstract

Characterizing quantum states is essential for validating quantum devices, yet conventional quantum state tomography becomes prohibitively expensive as system size grows. Direct tomography offers a distinct route by enabling selective access to individual complex density-matrix elements, with a particular advantage for sparse target states and some verification tasks. Here we introduce a direct quantum state tomography scheme combining strong-measurement estimation with a fan-out coupling architecture. It enables mutually commuting interactions between system qubits and a single meter qubit, thereby achieving constant circuit depth, independent of system size. Notably, the involutory fan-out coupling reduces to the identity under repetition, enabling straightforward noise scaling for quantum error mitigation. We experimentally validate the scheme on a superconducting quantum processor via the IBM Quantum Platform, demonstrating four-qubit state reconstruction and single-circuit GHZ-state fidelity estimation up to 20 qubits with error mitigation. Consistent results with standard tomography and improved efficiency establish our scheme as a promising approach to reconstructing full quantum states and scalable verification tasks.

Figures (3)

• Figure 1: Schematic of DQST.a, Circuit diagram for matrix-element estimation of an $n$-qubit system $\rho_{\mathrm{s}}$. The meter qubit is prepared in $|+\rangle_{\mathrm{m}}$ and coupled to the system via a controlled-$U_{\mathrm{ES}}^{\mathbf{k}}$ gate. The system qubits are measured in the computational basis, while the meter qubit is measured in the $X$ and $Y$ bases to access the real and imaginary parts, respectively (see Eq. \ref{['eq:res']}). b, Example of a controlled-$U_{\mathrm{ES}}^{\mathbf{k}}$ gate with $\mathbf{k}=101$. c, Accessible matrix-element subsets for each $U_{\mathrm{ES}}^{\mathbf{k}}$. The subset corresponding to b is highlighted in yellow.
• Figure 2: Density matrix reconstruction results. Four-qubit density matrices of the $|\mathrm{GHZ}_4\rangle$, $|0\rangle^{\otimes 4}$, and $|+\rangle^{\otimes 4}$ states are reconstructed via DQST (purple) and standard QST (gray). Data were collected on the ibm_aachen device with 10,000 shots per circuit, using 31 circuits for DQST and 81 for standard QST. Quantum readout mitigation and physical state-space projection were applied. Fidelities are reported in Table \ref{['tab:pdf_table']}.
• Figure 3: GHZ-state fidelity estimation results.a, Circuit diagram for direct fidelity estimation of an $n$-qubit GHZ state. To apply zero-noise extrapolation (ZNE) of the controlled-$U_{\mathrm{ES}}^{\mathbf{1}}$ gate, Pauli twirling is applied to render the noise incoherent, followed by digital gate folding with $N=1,3,5$. The zero-noise fidelity is obtained via extrapolation. b,c, GHZ-state fidelity as a function of qubit number under different error mitigation methods, including ZNE and quantum readout error mitigation (QREM). Each fidelity is estimated from 100,000 shots, obtained from 100 random Pauli-twirling instances with 1,000 shots each. Statistical uncertainties due to Pauli twirling are estimated via bootstrapping by resampling the 100 instances with replacement over 50 trials and are smaller than the marker size.