Session B44: Quantum Characterization, Verification and Validation I

Towards a Model Selection Rule for Quantum State Tomography

Quantum tomography on large and/or complex systems will rely heavily on model selection techniques, which permit on-the-fly selection of small efficient statistical models (e.g. small Hilbert spaces) that accurately fit the data. Many model selection tools, such as hypothesis testing or Akaike's AIC, rely implicitly or explicitly on the Wilks Theorem, which predicts the behavior of the loglikelihood ratio statistic (LLRS) used to choose between models. We used Monte Carlo simulations to study the behavior of the LLRS in quantum state tomography, and found that it disagrees dramatically with Wilks' prediction. We propose a simple explanation for this behavior; namely, that boundaries (in state space and between models) play a significant role in determining the distribution of the LLRS. The resulting distribution is very complex, depending strongly both on the true state and the nature of the data. We consider a simplified model that neglects anistropy in the Fisher information, derive an almost analytic prediction for the mean value of the LLRS, and compare it to numerical experiments. While our simplified model outperforms the Wilks Theorem, it still does not predict the LLRS accurately, implying that alternative methods may be necessary for tomographic model selection.   

Experimental Demonstration of Self-Guided Quantum Tomography

Robust and precise quantum state characterization is critical for future quantum experiments and technologies, and yet is a fundamentally challenging task. Standard and adaptive quantum tomography procedures are impractical for systems being prepared today due to the exponential scaling of quantum state space. These techniques are sensitive to statistical noise and require highly precise measurement settings. We present an experimental demonstration of autonomous and robust self-guided quantum tomography. Self-guided quantum tomography iteratively learns a quantum state using a stochastic gradient ascent algorithm. As a result it is robust against statistical noise and measurement errors. In addition, self-guided quantum tomography does not require any computationally expensive optimization, necessary for adaptive quantum tomography, or post-processing, required for standard quantum tomography. We demonstrate the robustness of self-guided quantum tomography by engineering the level of statistical noise and experimental errors, achieving measurement fidelities greater than standard quantum tomography in a range of one- and two-qubit experiments. Our demonstration opens pathways towards robust quantum state characterization in current and near-future experiments, where standard techniques are already impractical.   

Improved precision-guaranteed quantum tomography

Quantum tomography is one of the standard tool in current quantum information experiments for verifying that a state/process/measurement prepared in the lab is close to an ideal target. Precision-guaranteed quantum tomography (Sugiyama, Turner, Murao, PRL 111, 160406 2013) gives rigorous error bars on a result estimated from arbitrary finite data sets from any given informationally complete tomography experiments. The rigorous error bars were derived with a real-valued concentration inequality called Hoeffding's inequality. In this talk, with a vector-valued concentration inequality, we provide an improved version of the error bars of precision-guaranteed quantum tomography. We examine the new error bars for specific cases of multi-qubit systems and numerically show that the degree of improvement becomes large as the dimension of the system increases.   

Gate Set Tomography on two qubits

Gate set tomography (GST) is a method for characterizing quantum gates that does not require pre-calibrated operations, and has been used to both certify and improve the operation of single qubits. We analyze the performance of GST applied to a simulated two-qubit system, and show that Heisenberg scaling is achieved in this case. We present a GST analysis of preliminary two-qubit experimental data, and draw comparisons with the simulated data case. Finally, we will discuss recent theoretical developments that have improved the efficiency of GST estimation procedures, and which are particularly beneficial when characterizing two qubit systems. Sandia is a multiprogram laboratory operated by Sandia Corporation, a Lockheed Martin Company, for the United States Department of Energy's National Nuclear Security Administration under Contract DE-AC04-94AL85000.   

Hamiltonian tomography for quantum many-body systems with arbitrary couplings

Characterization of qubit couplings in many-body quantum systems is essential for benchmarking quantum computation and simulation. We propose a tomographic measurement scheme to determine all the coupling terms in a general many-body Hamiltonian with arbitrary long-range interactions, provided the energy density of the Hamiltonian remains finite. Different from quantum process tomography, our scheme is fully scalable with the number of qubits as the required rounds of measurements increase only linearly with the number of coupling terms in the Hamiltonian. The scheme makes use of synchronized dynamical decoupling pulses to simplify the many-body dynamics so that the unknown parameters in the Hamiltonian can be retrieved one by one. We simulate the performance of the scheme under the influence of various pulse errors and show that it is robust to typical noise and experimental imperfections.   

Quantum Compressed Sensing Using 2-Designs

We develop a method for quantum process tomography that combines the efficiency of compressed sensing with the robustness of randomized benchmarking. Our method is robust to state preparation and measurement errors, and it achieves a quadratic speedup over conventional tomography when the unknown process is a generic unitary evolution. Our method is based on PhaseLift, a convex programming technique for phase retrieval. We show that this method achieves approximate recovery of almost all signals, using measurements sampled from spherical or unitary 2-designs. This is the first positive result on PhaseLift using 2-designs. We also show that exact recovery of all signals is possible using measurements sampled from unitary 4-designs. Previous positive results for PhaseLift required spherical 4-designs, while PhaseLift was known to fail in certain cases when using spherical 2-designs.   

Scalable randomized benchmarking of non-Clifford gates

Randomized benchmarking is a widely used experimental technique to characterize the average error of quantum operations. Benchmarking procedures that scale to enable characterization of $n$-qubit circuits rely on efficient procedures for manipulating those circuits and, as such, have been limited to subgroups of the Clifford group. However, universal quantum computers require additional, non-Clifford gates to approximate arbitrary unitary transformations. We define a scalable randomized benchmarking procedure over $n$-qubit unitary matrices that correspond to protected non-Clifford gates for a class of stabilizer codes. We present efficient methods for representing and composing group elements, sampling them uniformly, and synthesizing corresponding $\mathrm{poly}(n)$-sized circuits. The procedure provides experimental access to two independent parameters that together characterize the average gate fidelity of a group element.   

Benchmarking of Quantum Control in ESR

Quantum error correction is essential for realizing scalable quantum computation. Key ingredients for quantum error correction are highly polarized ancilla qubits and high-fidelity quantum control. While NMR quantum processors have demonstrated high control fidelity, the requirement to prepare highly polarized spin qubits on demand is a major challenge. Electron-nuclear hyperfine coupled spin systems provide a possible solution: electrons can be fully polarized at accessible fields and temperatures, and their polarization is typically reset much faster than nuclei by spin relaxation. This makes open system cooling methods, such as heat bath algorithm cooling, possible. In this talk, I will describe our recent efforts to improve the precision of microwave control in a custom electron spin resonance spectrometer. In particular, we use randomized benchmarking of quantum gates to quantify control errors, and carefully take into account the resonator transfer function in correcting pulses. Moreover, we implement a protocol that distinguishes coherent and incoherent errors, which gives deeper insight into the nature of the remaining control imperfections and how to address them.   

Informational completeness in bounded-rank quantum-state tomography

Quantum-state tomography is a demanding task, however, it can be made more efficient by applying prior information about the system. A common prior assumption is that the state being measured is pure, or close to pure, since most quantum information protocols require pure states. Measurements of pure states can be constructed to be more efficiently than measurements of an arbitrary state, and for these types of measurements, there exists two different notions of informational completeness. One notion, called strict-completeness, is more useful for practical applications since it is compatible with convex optimization and is robust to noise. We present a unified framework for both notions of completeness for a certain type of measurements. These are measurements that allow algebraic reconstruction of a few density matrix elements. The framework also aids in the construction of new strictly-complete measurements. Moreover, the results are easily generalized to the case when the prior information is the state has bounded rank.   

The power of being positive: Robust state estimation made possible by quantum mechanics

Quantum-state tomography (QST) is generally expensive to implement experimentally. Nevertheless, in state-of-the-art experiments in quantum information science the goal is not to produce arbitrary states but states that have very high purity. Including this prior information in QST results in more manageable tomography protocols. In the context of pure-state tomography, and more generally, of bounded-rank states (states with rank $\leq r$) tomography, a natural notion of informational completeness emerges, {\em rank-$r$ completeness}. The purpose of this contribution is two fold. First, to prove and emphasize the significance of a less intuitive, yet more powerful, notion of completeness for practical QST, {\em rank-$r$ strict-completeness}. This notion is made possible due to the positive semidefinite property of density matrices. Strictly-complete quantum measurements ensure a robust estimation of the state of the system, regardless of the convex estimator we use. Thus, pragmatically, quantum state tomography should be done using these kind of measurements. Second, to argue, based on strong numerical indication, that it is fairly straightforward to experimentally implement such measurements by measuring only few random orthonormal bases. For example, in our numerical experi   

Bayesian mean estimation for finite two-photon experiments

Estimations of quantum probabilities are commonly made utilizing frequency based methods to invert Born's rule where $X$ is found $k$ out of $n$ times, $P(X)=|\left<\psi|X\right>|^2\approx k/n$. For an infinite measurement number the maximum likelihood estimation (MLE) represents the true probability. Unfortunately, the number of measurements in any experiment is finite. Given this, better estimates are provided by Bayesian mean estimation (BME). We present a novel method utilizing an experiment-specific probability distribution to make fully informed estimations of any quantum probability, efficiency parameter, or complete density matrix. Our method accounts for the finite measurement number, inter-basis parameter dependence, and estimate physicality. No knowledge of the pathway/detector efficiencies or the photon number generated by the source is required. Only knowledge of the raw singles and coincidence counts is needed. We present our estimation procedure for a single basis experiment, the extension to multiple bases, the application to state tomography to estimate strictly physical quantum states, simulation results comparing MLE and BME estimates, and experimental application of our method using our numerical tomography package TOMOHAK based on slice sampling.   

How to construct the optimal Bayesian measurement in quantum statistical decision theory

Recently, much more attention has been paid to the study aiming at the application of fundamental properties in quantum theory to information processing and technology. In particular, modern statistical methods have been recognized in quantum state tomography (QST), where we have to estimate a density matrix (positive semidefinite matrix of trace one) representing a quantum system from finite data collected in a certain experiment. When the dimension of the density matrix gets large (from a few hundred to millions), it gets a nontrivial problem. While a specific measurement is often given and fixed in QST, we are also able to choose a measurement itself according to the purpose of QST by using qunatum statistical decision theory. Here we propose a practical method to find the best projective measurement in the Bayesian sense. We assume that a prior distribution (e.g., the uniform distribution) and a convex loss function (e.g., the squared error) are given. In many quantum experiments, these assumptions are not so restrictive. We show that the best projective measurement and the best statistical inference based on the measurement outcome exist and that they are obtained explicitly by using the Monte Carlo optimization.   

Experimental Estimation of Average Fidelity of a Clifford Gate on a 7-qubit Quantum Processor

The traditional approach of characterizing a given quantum gate via quantum process tomography (QPT) requires exponential number of experiments. Therefore, estimating the average fidelity of the quantum gate by QPT is not practical for large-scale systems. In this talk, I will discuss about how to certify a Clifford gate within polynomial complexity using a twirling protocol. In particular, we adopted this method in NMR and certified a 7-qubit quantum Clifford gate with only 1600 experiments (in contrast, QPT requires millions of experiments). This Clifford gate is important as it generates maximal coherence from single coherence, and non-trivial for benchmarking the coherent control in experiment. We show that the average fidelity of this gate is over 87\% after accounting for the decoherence effect, and to date this is the largest experimental gate-characterization. This twirling protocol is efficient and scalable, and can also be extended to other systems straightforwardly.   

Crosstalk characterization by eigenvalue estimation: Theory

As qubit systems continue to grow and long coherence times become routine, the dominating sources of error shift away from decoherence and towards control errors. One pervasive source of control errors is crosstalk -- where control fields intended for one qubit leak onto other qubits. In this talk we describe a method to quantify crosstalk by estimating the eigenvalues of the system's evolution using a technique known as ``spectrum estimation''. We discuss the wide applicability of the method, and demonstrate similar accuracy scaling to the robust phase estimation algorithm of Kimmel, Low, and Yoder.   

Crosstalk characterization in superconducting qubits by eigenvalue estimation: Experiment

Superconducting qubit devices offer a promising path towards a scalable quantum computer. As these systems continue to grow in size and complexity, crosstalk errors, which build up during long control sequences, lead to an overall loss in control fidelity. In this talk we explore the use of eigenvalue estimation (a.k.a. "spectrum estimation") in superconducting systems as a high-accuracy method to detect and quantify crosstalk between qubits, and demonstrate how these techniques allow for quick identification and estimation of system crosstalk.