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arXiv:2406.11646 · Physical Review Research

Boosting projective methods for quantum process and detector tomography

Júlia Barberà-Rodríguez, Leonardo Zambrano, Antonio Acín, Donato Farina

Published Feb 26, 2025• 7 citations
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Abstract

We introduce two methods for quantum process and detector tomography. In the quantum process tomography method, we develop an analytical procedure for projecting the linear inversion estimation of a quantum channel onto the set of completely positive trace-preserving matrices. By integrating this method with alternate projection techniques, we achieve a three-order-of-magnitude improvement in approximating the closest quantum channel to an arbitrary Hermitian matrix compared to existing methods without compromising computational efficiency. Our second method extends this approach to quantum detector tomography, demonstrating superior efficiency compared to current techniques. Through numerical simulations, we evaluate our protocols across channels of up to four qubits in quantum process tomography and systems of up to six qubits in quantum detector tomography, showcasing superior precision and efficiency.

DOI
10.1103/physrevresearch.7.013208

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We introduce two methods for quantum process and detector tomography. In the quantum process tomography method, we develop an analytical procedure for projecting the linear inversion estimation of a quantum channel onto the set of completely positive trace-preserving matrices. By integrating this method with alternate projection techniques, we achieve a three-order-of-magnitude improvement in approximating the closest quantum channel to an arbitrary Hermitian matrix compared to existing methods without compromising computational efficiency. Our second method extends this approach to quantum detector tomography, demonstrating superior efficiency compared to current techniques. Through numerical simulations, we evaluate our protocols across channels of up to four qubits in quantum process tomography and systems of up to six qubits in quantum detector tomography, showcasing superior precision and efficiency.
We present an algorithm for projecting superoperators onto the set of completely positive, trace-preserving maps. When combined with gradient descent of a cost function, the procedure results in an algorithm for quantum process tomography: finding the quantum process that best fits a set of sufficient observations. We compare the performance of our algorithm to the diluted iterative algorithm as well as second-order solvers interfaced with the popular cvx package for matlab, and find it to be significantly faster and more accurate while guaranteeing a physical estimate.
Quantum process tomography (QPT) protocols are often designed by lifting quantum state tomography (QST) protocols through the Choi-Jamiolkowski isomorphism, at the cost of an extra step enforcing the trace-preserving (TP) constraint. Existing approaches fall into two classes. The folklore partial normalization, which sandwiches the density estimate between inverse square roots of its input marginal, has a closed form but is used as a heuristic, its only analyses being asymptotic and losing a factor polynomial in the dimension. Projection-based methods are principled but have no closed forms and scale poorly with dimension. We resolve this dichotomy by showing that the partial normalization is the exact projection onto the Choi states of quantum channels in fidelity (equivalently in Bures or purified distance). Consequently, any QST algorithm with guarantees in fidelity lifts to a QPT algorithm that inherits its sample complexity, through a single closed-form congruence that preserves the Choi rank and is computed in Kraus form without any eigendecomposition of Choi dimension. We lift two single-copy protocols and prove that the non-adaptive one (Fidelity Projected Least Squares) recovers the Choi rank and reaches infidelity ε\varepsilonε from a number of channel uses proportional to 1/ε1/\varepsilon1/ε. Numerically, for channels of low Choi rank, it improves on the state-of-the-art projected least squares in accuracy, and by orders of magnitude in the time and memory of the TP-regularization step and of the returned estimate.
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.
Alternatively to the full reconstruction of an unknown quantum process, the so-called selective and efficient quantum process tomography (SEQPT) allows estimating, individually and up to the required accuracy, a given element of the matrix that describes such an operation with a polynomial amount of resources. The implementation of this protocol has been carried out with success to characterize the evolution of a quantum system that is well described by a trace-preserving quantum map. Here, we deal with a more general type of quantum process that does not preserve the trace of the input quantum state, which naturally arises in the presence of imperfect devices and system-environment interactions, in the context of quantum information science or quantum dynamics control. In that case, we show with the aid of a priori information on the losses structure of the quantum channel that the SEQPT reconstruction can be adapted to reconstruct the non-trace-preserving map. We explicitly describe how to implement the reconstruction in an arbitrary Hilbert space of finite dimension ddd. The method is experimentally verified on a superconducting quantum processor provided by IBM Quantum services, by estimating several non-trace-preserving quantum processes in dimensions up to d=6d=6d=6. Our results show that it is possible to efficiently reconstruct non-trace-preserving processes, with high precision, and with significantly higher fidelity than when the process is assumed to be trace preserving.
We present a new method for quantum process tomography. The method enables us to efficiently estimate, with fixed precision, any of the parameters characterizing a quantum channel. It is selective since one can choose to estimate the value of any specific set of matrix elements of the super-operator describing the channel. Also, we show how to efficiently estimate all the average survival probabilities associated with the channel (i.e., all the diagonal elements of its χχχ--matrix.
Quantum process tomography is a powerful tool for understanding quantum channels and characterizing the properties of quantum devices. Inspired by recent advances using classical shadows in quantum state tomography [H.-Y. Huang, R. Kueng, and J. Preskill, .], we have developed ShadowQPT, a classical shadow method for quantum process tomography. We introduce two related formulations with and without ancilla qubits. ShadowQPT stochastically reconstructs the Choi matrix of the device allowing for an classical evaluation of the device on arbitrary inputs with respect to arbitrary outputs. Using shadows, we then show how to compute overlaps, generate all k -weight reduced processes, and perform reconstruction via Hamiltonian learning. These latter two tasks are efficient for large systems as the number of quantum measurements needed scales only logarithmically with the number of qubits. A number of additional approximations and improvements are developed, including the use of a pair-factorized Clifford shadow and a series of postprocessing techniques that significantly enhance the accuracy for recovering the quantum channel. We have implemented ShadowQPT using both Pauli and Clifford measurements on the IonQ trapped ion quantum computer for quantum processes up to n=4 qubits, and we achieved good performance. Published by the American Physical Society 2024