Distributed quantum computing offers a promising approach to scaling quantum devices by networking multiple quantum processors. We present a quantum state tomography protocol tailored for distributed quantum computers that avoids assuming remote entanglement as a primitive resource. The protocol extends projected least-squares (PLS) tomography based on projective 2-designs to systems composed of multiple quantum processors, using only local operations within each processor and classical communication between nodes. Assuming entanglement within each individual quantum processor is trusted, the protocol can be executed using mutually unbiased bases. We derive rigorous, non-asymptotic trace-norm error bounds for the PLS estimator, with explicit exponential dependence on the number of nodes. In addition, we establish certified error bounds for estimating entanglement negativity from the PLS estimator. Numerical simulations for systems of up to seven qubits distributed across several devices validate the theoretical error bounds.
Distributed quantum computing offers a promising approach to scaling quantum devices by networking multiple quantum processors. We present a quantum state tomography protocol tailored for distributed quantum computers that avoids assuming remote entanglement as a primitive resource. The protocol extends projected least-squares (PLS) tomography based on projective 2-designs to systems composed of multiple quantum processors, using only local operations within each processor and classical communication between nodes. Assuming entanglement within each individual quantum processor is trusted, the protocol can be executed using mutually unbiased bases. We derive rigorous, non-asymptotic trace-norm error bounds for the PLS estimator, with explicit exponential dependence on the number of nodes. In addition, we establish certified error bounds for estimating entanglement negativity from the PLS estimator. Numerical simulations for systems of up to seven qubits distributed across several devices validate the theoretical error bounds.
Abstract Adaptive techniques have great potential for wide application in enhancing the precision of quantum parameter estimation. We present an adaptive quantum state tomography protocol for finite dimensional quantum systems and experimentally implement the adaptive tomography protocol on two-qubit systems. In this adaptive quantum state tomography protocol, an adaptive measurement strategy and a recursive linear regression estimation algorithm are performed. Numerical results show that our adaptive quantum state tomography protocol can outperform tomography protocols using mutually unbiased bases and the two-stage mutually unbiased bases adaptive strategy, even with the simplest product measurements. When nonlocal measurements are available, our adaptive quantum state tomography can beat the Gill–Massar bound for a wide range of quantum states with a modest number of copies. We use only the simplest product measurements to implement two-qubit tomography experiments. In the experiments, we use error-compensation techniques to tackle systematic error due to misalignments and imperfection of wave plates, and achieve about a 100-fold reduction of the systematic error. The experimental results demonstrate that the improvement of adaptive quantum state tomography over nonadaptive tomography is significant for states with a high level of purity. Our results also show that this adaptive tomography method is particularly effective for the reconstruction of maximally entangled states, which are important resources in quantum information.
We propose and investigate a new method of quantum process tomography (QPT) which we call projected least squares (PLS). In short, PLS consists of first computing the least-squares estimator of the Choi matrix of an unknown channel, and subsequently projecting it onto the convex set of Choi matrices. We consider four experimental setups including direct QPT with Pauli eigenvectors as input and Pauli measurements, and ancilla-assisted QPT with mutually unbiased bases (MUB) measurements. In each case, we provide a closed form solution for the least-squares estimator of the Choi matrix. We propose a novel, two-step method for projecting these estimators onto the set of matrices representing physical quantum channels, and a fast numerical implementation in the form of the hyperplane intersection projection algorithm. We provide rigorous, non-asymptotic concentration bounds, sampling complexities and confidence regions for the Frobenius and trace-norm error of the estimators. For the Frobenius error, the bounds are linear in the rank of the Choi matrix, and for low ranks, they improve the error rates of the least squares estimator by a factor d2 , where d is the system dimension. We illustrate the method with numerical experiments involving channels on systems with up to 7 qubits, and find that PLS has highly competitive accuracy and computational tractability.
Quantum state tomography (QST) is a critical technique for characterizing the state of quantum systems, particularly in the context of quantum computing. Traditional QST protocols rely heavily on precise knowledge of the measurement devices used, introducing significant vulnerabilities to systematic errors and device imperfections. This paper proposes a novel measurement device-independent (MDI) quantum state tomography protocol designed specifically for multi-qubit systems. The core principle of this approach is the strategic utilization of entanglement to minimize the impact of device-specific biases, thereby achieving a more robust and accurate reconstruction of the quantum state. The protocol involves a carefully designed series of measurements performed by multiple, potentially imperfect, measurement devices. Through analyzing the correlations generated by these measurements, we can, with a high degree of certainty, deduce the underlying quantum state without requiring perfect knowledge of each individual device. This MDI QST method represents a significant advancement in overcoming the limitations of conventional QST, paving the way for more reliable and scalable quantum state characterization in complex quantum computing architectures. The theoretical framework and key equations are presented, demonstrating the feasibility and advantages of this approach.
We present a two-step protocol for quantum measurement tomography that is light on classical co-processing cost and still achieves optimal sample complexity. 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 O ( ( d 3 + d 2 L ) / ϵ 2 ) samples to achieve error ϵ in worst-case distance, and O ( d 2 L / ϵ 2 ) samples in average-case distance. We further establish two matching sample complexity lower bounds of Ω ( ( d 3 + d 2 L ) / 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 both the dimension and the number of outcomes for both distances. 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.
Quantum computation depends on quantum entanglement, a correlation between subsystems that cannot occur classically. A variety of theoretical measures exist for quantifying the degree entanglement in such schemes, all of which are functions of the system density matrix. How can the entanglement be measured experimentally? Using quantum tomography techniques developed for two photon entangled states, the density matrix can be reconstructed from the appropriate experimental data. In this case the state tomography gives the complete characterization of the physical system (for the relevant degree of freedom, such as spin or polarization). It gives information on both the degree of nonclassical correlation, that is entanglement, as well as the amount of decoherence in the system. In this proceedings we discuss the general state tomography procedure required to characterize a few qubit quantum computer, for any architecture.
Distributed quantum computation is a practical method for large-scale quantum computation on quantum processors with limited size. It can be realized by direct quantum channels in flying qubits. Moreover, the pre-established quantum entanglements can also play the role of quantum channels with local operations and classical channels. However, without quantum correlations like quantum channels and entanglements, the entanglement forging technique allows us to classically forge the entangled states with local operations and classical channels only. In this paper, we demonstrate the methods to implement a nonlocal quantum circuit on two quantum processors without any quantum correlations, which is based on the fact that teleportation with classically forged Bell states is equivalent to quantum state tomography. In compensation, the overhead of single-shot measurement will increase, and several auxiliary qubits are required. Our results extend the possibility of integrating quantum processors. We expect that our methods will complement the toolbox of distributed quantum computation, and facilitate the extension of the scale of quantum computations.