www.nature.com/npjqi ARTICLE OPEN Spectral quantum tomography Jonas Helsen 1, Francesco Battistel1 and Barbara M. Terhal1,2 We introduce spectral quantum tomography, a simple method to extract the eigenvalues of a noisy few-qubit gate, represented by a trace-preserving superoperator, in a SPAM-resistant fashion, using low resources in terms of gate sequence length. The eigenvalues provide detailed gate information, supplementary to known gate-quality measures such as the gate fidelity, and can be used as a gate diagnostic tool. We apply our method to one- and two-qubit gates on two different superconducting systems available in the cloud, namely the QuTech Quantum Infinity and the IBM Quantum Experience. We discuss how cross-talk, leakage and non-Markovian errors affect the eigenvalue data. npj Quantum Information (2019) 5:74 ; https://doi.org/10.1038/s41534-019-0189-0 INTRODUCTION of the form λ = exp(−γ)exp(iϕ), contain information about the A central challenge on the path towards large-scale quantum quality of the implemented gate. Intuitively, the parameter γ computing is the engineering of high-quality quantum gates. To captures how much the noisy gate deviates from unitarity due to achieve this goal, many methods that accurately and reliably entanglement with an environment, while the angle ϕ can be characterize quantum gates have been developed. Some of these compared to the rotation angles of the targeted gate U. Hence ϕ methods are scalable, meaning that they require an effort which gives information about how much one over- or under-rotates. scales polynomially in the number of qubits on which the gates The spectrum of S can also be related to familiar gate-quality 1–8 act. Scalable protocols, such as randomized benchmarking, measures such as the average gate fidelity and the unitarity. necessarily give a partial characterization of the gate quality, for Moreover, in the case of a noisy process modeled by a Lindblad fi example, an average gate delity. Other protocols such as robust equation, the spectrum can be easily related to the more familiar tomography9 or gate-set tomography10,11 trade scalability for a notions of relaxation and dephasing times. more detailed characterization of the gate. A desirable feature of The main advantage of spectral quantum tomography is its all the above protocols is that they are resistant to state- simplicity, requiring only the (repeated) application of a single preparation and measurement (SPAM) errors. The price of using noisy gate , as opposed to the application of a large set of gates SPAM-resistant (scalable) methods is that these protocols have S significant experimental complexity and/or require assumptions as in randomized benchmarking, gate-set tomography, and robust on the underlying hardware to properly interpret their results. tomography. Naturally, simplicity and low cost come with some In this work, we present spectral quantum tomography, a drawback, namely, the method does not give information about simple non-scalable method that extracts spectral information the eigenvectors of the noisy gate, such as the axis around which from noisy gates in a SPAM-resistant manner. To process the one is rotating. However, information about the eigenvectors is tomographic data and obtain the spectrum of the noisy gate, we intrinsically hard to extract in a SPAM-resistant fashion since SPAM rely on the matrix-pencil technique, a well-known classical signal errors can lead to additional rotations.16 Another feature of processing method. This technique has been advocated in ref. 8 in spectral quantum tomography is that it can be used to extract the context of randomized benchmarking and has also been used signatures of non-Markovianity, namely, the phenomenon where 12 in ref. for processing data in the algorithm of quantum phase the noisy gate S depends on the context in which it is applied estimation. It has also been used, under the phrase “linear systems (e.g., time of application, whether any gates have been applied fi ” 13 identi cation, in ref. to predict the time evolution of quantum before it). As we show in this paper, our method can be used to systems. While the matrix pencil technique leads to explicitly detect various types of non-Markovian effects, such as coherent useful estimates of eigenvalues and their amplitudes, we note that revivals, parameter drifts, and Gaussian-distributed time-corre- the same underlying idea is used in the method of “delayed lated noise. It is also possible to distinguish non-Markovian effects vectors,” which has been proposed in ref. 14 to assess the dimensionality of a quantum system from its dynamics. This from qubit leakage. For these reasons, we believe that spectral “delayed vectors” approach has been applied to assess leakage in quantum tomography adds a useful new tool to the gate- superconducting devices in ref. 15 characterization toolkit. The method could also have future The spectral information of a noisy gate S, which approximates applications in assessing the performance of logical gates in a some target unitary U, is given by the eigenvalues of the so-called manner that is free of logical state preparation and measurement Pauli transfer matrix representing S. These eigenvalues, which are errors, see the “Discussion” section. 1QuTech, Delft University of Technology, P. O. Box 5046, 2600 GA Delft, the Netherlands and 2JARA Institute for Quantum Information, Forschungszentrum Juelich, 52425 Juelich, Germany Correspondence: Jonas Helsen ([email protected]) Received: 10 April 2019 Accepted: 1 August 2019 Published in partnership with The University of New South Wales J. Helsen et al. 2 P P RESULTS Ψ Ψ 1 N y Uy that jihj¼ d μ¼0 Pμ Pμ and UPμU ¼ κ Tμκ Pκ, we can write Eigenvalues of trace-preserving completely positive (TPCP) maps X ÂÃhi 1 y 1 Uy S ; U Tr UPμU Pμ 1 Tr T T : Take a unitary gate U on a d-dimensional space with F entðS Þ¼ 2 Sð Þ ¼ 2 þ ϕ d μ d ψ i j ψ ρ ρ y U j ¼ e j . The corresponding TPCP map SUð Þ¼U U has one trace-full eigenvector, namely, I with eigenvalue 1, as Thus, for the (entanglement) fidelity of a noisy gate S withÀÁ respectP 2 − 2 − = ; 1 λ well as d 1 traceless eigenvectors. In particular, there are d d to the identity channel U I, one has F entðS IÞ¼ 2 1 þ i , ψ 〉〈ψ ≠ d i traceless eigenvectors of the form | j l| for j l with eigenvalues implying a direct relation to the spectrum {λ }ofT S. A more ϕ − ϕ − ψ 〉 i exp(i( j l)) and d 1 traceless eigenvectors of the form | 1 S fi 〈ψ − ψ 〉〈ψ = … interesting relation is how the eigenvalues of T bound the delity 1| | j j| for j 2, , d with eigenvalue 1. with respect to a targeted gate U. In section “Upper bound on the For general TPCP maps, it is convenient to use the Pauli transfer fi ” n entanglement delity with the targeted gate, we prove that the matrix formalism. For an n-qubit system (d = 2 ), consider fi μ = … + entanglement delity can be upper bounded as the normalized set of Paulip matricesffiffiffiffiffi Pμ for 0, , N with N 2 0vffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiP 13 n 2 n u 2 1 = 4 = d , where P0 ¼ I= 2 and the normalization is chosen u λ 6 Bt j jj C7 such that Tr[PμPν] = δμν. For a TPCP map S acting on n qubits, the 1 j F ðS; UÞ 61 þðd2 À 1ÞB 1 À þ ξ C7; (3) Pauli transfer matrix is then defined as ent 2 4 @ 2 maxA5 Âà d d À 1 Sμν ¼ Tr PμSðPνÞ ; μ; ν ¼ 0; ¼ ; N: (1) P 17 ξ 1 λidealλà λideal U The form of the Pauli transfer matrix S is where max ¼ 2 j j with the eigenvalues of T d À1 j j j j 10 withP U the targeted unitary, ordered such that the sum S$S ¼ ; (2) j λidealλÃj is maximal. s TS j j j This upper bound is not particularly tight, but for the case of a S where T is a real N × N matrix and s is a N-dimensional column single qubit we can make a much stronger numerical statement, vector. The 1 and 0s in the top row of the Pauli transfer matrix are see section “Upper bound on the entanglement fidelity with the due to the fact that S is trace-preserving. For a unital S that obeys targeted gate”. S(I) = I, the vector s = 0. Another measure of gate quality, namely, the unitarity or the A few properties are known of the eigenvalue–eigenvector pairs coherence of a channel5 on a d-dimensional system, is defined as of S, i.e., the pairs ðλ; vÞ with Sv = λv: Z hi 1234567890():,; d 0 y 0 S uðSÞ ¼ dϕ Tr ½S ðÞjiϕ hjϕ S ðÞjiϕ hjϕ ; (4) ● The eigenvalues of S, and thus the eigenvalues of T are 1, d À 1 since the solutions of the equation det(S − λI) = 0 are the pffiffiffi 0 ρ : ρ ρ = solutions of the equation (1 − λ)det(T S − λI) = 0. where S ð Þ ¼Sð ÞÀTr½Sð ÞI d. A more convenient but fi ● The eigenvalues of S, and thus the eigenvalues of T S, come in equivalent de nition is – S Âà X complex conjugate pairs. This is true because T is a real 1 Sy S 1 S 2 uðSÞ ¼ Tr T T ¼ σiðT Þ ; (5) matrix. d2 À 1 d2 À 1 ● The eigenvalues of T S (or S for that matter) have modulus <1, i 18 S i.e.