Machine Learning of Noise in Single-Qubit Hardware

A talk I gave at the annual March Meeting of the American Physical Society. Released under SAND2018-1895 C.

Transcript

1. Sandia National Laboratories is a multimission laboratory managed and operated by National Technology and Engineering Solutions of Sandia LLC, a wholly owned subsidiary of Honeywell International Inc. for the U.S. Department of Energy’s National Nuclear Security Administration under contract DE-NA0003525. SAND2018-1895 C CCR Center for Computing Research Machine Learning of Noise in Single-Qubit Hardware Travis L Scholten @Travis_Sch Center for Quantum Information and Control, University of New Mexico, Albuquerque, USA Center for Computing Research, Sandia National Laboratories, Albuquerque, USA APS March Meeting 2018 (Collaboration with Robin Blume-Kohout & Kevin Young)
2. Noise affects the outcome probabilities of quantum circuits. How can we learn about noise using the data we get from running quantum circuits? 10
3. Depending on how much we want to learn, and how quickly, machine learning could be useful. M achine Learning? Randomized Benchmarking Gate set tomography State tomography Speed of learning Fast Slow Amount learned Process tomography Full Limited Caveat: “Speed” doesn’t include *training time* 9
4. Gate set tomography (GST) provides a set of structured circuits we can use for learning. GST assumes the device is a black box, described by a gate set. GST prescribes certain circuits to run that collectively amplify all types of noise. Standard use: Outcome probabilities are analyzed by pyGSTi software to estimate the noisy gates (GST) for self-consistently characterizing an entire set of quantum logic gates on a black-box quan- tum device; (2) an explicit closed-form protocol for linear-inversion gate set tomography (LGST), whose reliability is independent of pathologies such as local maxima of the likelihood; and (3) a simple protocol for objectively scoring the accuracy of a tomographic estimate without reference to target gates, based on how well it predicts a set of testing experiments. We use gate set tomography to characterize a set of Cli↵ord-generating gates on a single trapped-ion qubit, and compare the performance of (i) standard process tomography; (ii) linear gate set tomography; and (iii) maximum likelihood gate set tomography. Quantum information processing (QIP) relies upon precise, repeatable quantum logic operations. Exper- iments in multiple QIP technologies [1–5] have imple- mented quantum logic gates with su cient precision to reveal weaknesses in the quantum tomography protocols used to characterize those gates. Conventional tomo- graphic methods assume and rely upon a precalibrated reference frame, comprising (1) the measurements per- formed on unknown states, and (2) for quantum process tomography, the test states that are prepared and fed into the process (gate) to be characterized. Standard process tomography on a gate G proceeds by repeating a series of experiments in which state ⇢ j is prepared and observable (a.k.a. POVM e↵ect) E k is observed, using the statistics of each such experiment to estimate the corresponding probability p k | j = Tr[E k G[⇢ j ]] (given by Born’s rule), and finally reconstructing G from many such probabilities. But, in most QIP technologies, the various test states (⇢ j ) and measurement outcomes (E k ) are not known ex- actly. Instead, they are implemented using the very same gates that process tomography is supposed to character- ize. The quantum device is e↵ectively a black box, ac- cessible only via classical control and classical outcomes of quantum measurements, and in this scenario standard tomography can be dangerously self-referential. If we do process tomography on gate G under the common assumption that the test states and measurement out- comes are both eigenstates of the Pauli x , y , z opera- tors, then the accuracy of the estimate ˆ G will be limited by the error in this assumption. This is now a critical experimental issue. In plat- forms including (but not limited to) superconducting flux qubits [1], trapped ions [5], and solid-state qubits, quan- tum logic gates are being implemented so precisely that systematic errors in tomography (due to miscalibrated reference frames) are glaringly obvious. Fixes have been proposed [1, 2, 6, 7], but none yet provide a general, comprehensive, reliable scheme for gate characterization. M ⇢ G1 G2 ... FIG. 1: The GST model of a quantum device. Gate set tomography treats the quantum system of interest as a black box, with strictly limited access. This is a fairly good model for many qubit technologies, especially those based on solid state and/or cryogenic technologies. We do not have direct access to the Hilbert space or any aspect of it. Instead, the device is controlled via buttons that implement various gates (including a preparation gate and a measurement that causes one of two indicator lights to illuminate). Prior information about the gates’ function may be available, and can be used, but should not be relied upon. In this article, we present gate set tomography (GST), a complete scheme for reliably and accurately charac- terizing an entire set of quantum gates. In particular we introduce the first linear-inversion protocol for self- consistent process tomography, linear gate set tomog- raphy (LGST). LGST is a closed-form estimation pro- tocol (inspired in part by [8–10]) that cannot – unlike pure maximum-likelihood (ML) algorithms – run afoul of local maxima in a likelihood function that is gener- } | A i \ bra { B } h B | { B } h A | B i \ op { A }{ B } | A ih B | { j }{ B }{ k } h j | B | k i \ expval { B } h B i SIMPLE QUANTUM CIRCUITS , suppose the reader would like to typeset the mple circuit: | 0 i Y⇡/2 typeset using @C=1em @R=.7em { ate{X} & \qw ive outputs: t { A } | A i \ bra { B } h B | { A }{ B } h A | B i \ op { A }{ B } | A ih B | em { j }{ B }{ k } h j | B | k i \ expval { B } h B i IV. SIMPLE QUANTUM CIRCUITS egin, suppose the reader would like to typeset the ng simple circuit: | 0 i Y⇡/2 Y⇡/2 was typeset using it @C=1em @R=.7em { \gate{X} & \qw grams using standard t and time consuming cro package designed wing quantum circuit an array. In a mat- asic syntax and start se qcircuit from the that they’ve learned the end of § IV, but ose that wish to type- the GNU public license. ndix C. \ ket { A } | A i \ bra { B } \ ip { A }{ B } h A | B i \ op { A }{ B } \ melem { j }{ B }{ k } h j | B | k i \ expval { B } IV. SIMPLE QUANTUM CIRCUIT To begin, suppose the reader would like to ty following simple circuit: | 0 i X⇡/2 Y⇡/2 Z⇡/2 This was typeset using \Qcircuit @C=1em @R=.7em { & \gate{X} & \qw } Blume-Kohout, et. al, Nature Communications 8 (2017) doi:10.1038/ncomms14485 8
5. GST data sets — represented as feature vectors — can be used for ML, as they encode information about noise. ## Columns = minus count, plus count {} 100 0 Gx 44 56 Gy 45 55 GxGx 9 91 GxGxGx 68 32 GyGyGy 70 30 (GST data set) f = (f1, f2, · · · ) 2 Rd We focus on data sets from “linear” GST, where d=92. 0 B B B B B B @ 0 .56 .55 .91 .32 .3 1 C C C C C C A Total number of circuits 7
6. 6 Coherent Noise Ideal Stochastic Noise We studied the feasibility of ML for noise detection using arbitrary realizations of three realistic noise types. L = L0 = i(I ⌦ H0 HT 0 ⌦ I) L = L0 i(I ⌦ e eT ⌦ I) e = aX + bY + cZ a, b, c ⇠ N(0, ⌘2) E[⇢] = eL|⇢) Action of the generator: L = L0 + X jk hjk[A? k (t) ⌦ Aj(t) 1 2 ⇣ I ⌦ A† k (t)Aj(t) + AT j A? k (t) ⌦ I ⌘ h = S 1DS Djk = jk ⇤ N(0, ⌘2)
7. The components of GST feature vectors (i.e., circuits) respond to these noises in different ways. Coherent Noise Ideal Stochastic Noise 5
8. Low-dimensional representations of the data set reveal a certain amount of structure is present. Principal Component Analysis (PCA) Defines a projection: f ! PK j=1 (f · j) j C = PK j=1 j j T j Compute covariance matrix & diagonalize Multidimensional Scaling (MDS) Attempts to preserve pairwise distance:{fj } ! {yj } {yj } = argmin RK P jk ||yj yk ||2 ||fj fk ||2 2 4
9. We investigated the performance of 4 different ML classifiers for learning the noise type. 3 (Toy ML example) Different classifiers draw different decision boundaries; therefore, they may have different accuracies on the same data
10. L=1 GST feature vectors appear to be “quadratically” or “quartically” separable. 2
11. 1 Proof of principle that machine learning can help us characterize qubits. Future work: “intelligent” circuit design & estimating performance metrics. Circuits matter — they encode information about noise.
12. 1 Proof of principle that machine learning can help us characterize qubits. Future work: “intelligent” circuit design & estimating performance metrics. Circuits matter — they encode information about noise. Thank you! @Travis_Sch