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.

• The paper introduces a DQST protocol using a single fan-out controlled-X gate for selective density matrix element measurement.
• It demonstrates practical scalability on up to 20 qubits with measured fidelities exceeding 96%, improved to over 98% with error mitigation techniques.
• The approach reduces classical and quantum overhead compared to conventional QST, offering hardware-agnostic implementation and efficiency benefits.

Efficient Direct Quantum State Tomography Using Fan-Out Couplings

Introduction

Direct quantum state tomography (DQST) is positioned as a scalable, element-selective alternative to conventional quantum state tomography (QST), which becomes intractable for large Hilbert spaces due to exponential scaling in both measurement and computational resources. This work introduces a scheme for DQST leveraging strong, mutually commuting system-meter qubit couplings mediated by a single fan-out gate. The proposal integrates several recent advances including programmable selection of density matrix elements, constant-depth circuit design, and built-in compatibility with quantum error mitigation routines based on involutory gate structures.

Methodology

The core technical innovation in this scheme is the use of a single meter qubit coherently coupled to an arbitrary number of system qubits via a multi-target controlled-X (fan-out) gate. The controlled-unitary structure enables selective addressing of arbitrary matrix elements of the $n$-qubit density operator by measurement of the meter qubit in the Pauli-X and Pauli-Y bases, conditioned on the computational outcome of the system register. The involutory property of the fan-out coupling allows for repetition-based noise scaling crucial to error mitigation protocols.

Element selection is facilitated by parameterizing the fan-out unitary by a binary string $k$ which identifies the subset of system qubits participating in the coupling. For a given $k$, the expectation values of the meter in the X/Y basis yield, for each computational state $|a\rangle$, the real/imaginary components of the off-diagonal element $\langle a | \rho | a+k\rangle$. Diagonal elements are accessed with $k=0$ and X-basis measurements. Full reconstruction requires $2^{n+1}-1$ configurations, an exponential but substantially reduced count relative to overcomplete QST.

Implementation and Experimental Benchmarking

The scheme was implemented on the IBM Quantum Platform superconducting device for up to 20 physical qubits. For state reconstruction, three canonical 4-qubit states (GHZ, computational, and equal superposition) were prepared and measured using both DQST and standard QST. Quantum readout error mitigation (QREM) and positivity projection were applied to enforce physicality. Fidelities with the ideal states, even before error mitigation, consistently exceeded 96% and improved beyond 98% with QREM. When measurement shot budgets were matched, statistical uncertainties of DQST and QST were comparable, demonstrating that the DQST reduction in circuit count translates to hardware efficiency, particularly on platforms where shot repetition is less burdensome than circuit reconfiguration.

For quantum state verification tasks, the protocol allows direct estimation of GHZ-state fidelity with a single circuit configuration independent of $n$. In contrast, all competing protocols for GHZ-type stabilizer state verification require an $n$-dependent number of measurement settings. Utilizing the involutory fan-out, zero-noise extrapolation (ZNE) and Pauli twirling were employed to further mitigate gate errors. The scheme demonstrated robust multipartite entanglement certification for up to 20 qubits; with both ZNE and QREM, fidelities at $n=20$ exceeded the genuine multipartite entanglement threshold, a marked improvement over non-mitigated values that fall below this critical value.

Comparative Efficiency and Error Mitigation

DQST exhibits significantly lower classical and quantum overhead relative to both overcomplete QST and compressed-sensing-based approaches, the latter typically requiring $k$0 settings for rank-$k$1 states. For sparse or structured states and verification tasks requiring access to a limited set of density matrix entries, DQST can scale polynomially in relevant parameters. The inherent compatibility of the fan-out structure with hardware error mitigation is crucial: involutory gates allow for systematic noise amplification and extrapolation without altering the logical measurement protocol, maximizing the benefits of ZNE and similar methods. Experimental results underscore the critical role of QREM and ZNE, particularly as system size increases and gate/readout error rates accumulate.

Hardware Agnosticism and Scalability

Despite superconducting qubits with nearest-neighbor connectivity being used for experimental realization, the protocol is fundamentally hardware-agnostic. Architectures with native all-to-all or long-range connectivity, such as ion traps or Rydberg arrays, can realize the full fan-out interaction in a single physical layer. Even in hardware with limited connectivity, mid-circuit measurement and feedforward protocols enable effective constant-depth realization leveraging recent advances in quantum control.

Implications and Future Prospects

This work consolidates several emerging trends in quantum state characterization: (i) a move toward direct-access, element-selective measurement protocols that integrate smoothly with error mitigation and noise characterization, (ii) deployment of involutory gate structures for practical, hardware-friendly zero-noise protocols, and (iii) hardware-aware circuit compilation for scalable QST and verification. The flexible nature of the matrix-element selection operator in DQST also suggests prospects for adaptive strategies where knowledge of state sparsity, structure, or symmetries can further decrease sample complexity, and for hybridization with classical machine-learning-based post-processing for large-scale quantum devices.

Further research directions include generalization to process tomography, exploration of adaptive DQST protocols using real-time feedback, and integration with quantum shadow tomography and randomized measurement protocols for enhanced property prediction.

Conclusion

Efficient direct quantum state tomography via strong fan-out couplings offers a substantive reduction in measurement complexity and circuit depth for both state reconstruction and verification. The demonstrated scalability, error mitigation compatibility, and hardware-agnostic design position this protocol as a viable tool for near-term and future quantum devices, particularly as system sizes approach the practical limits of conventional QST.