Bulletin of the American Physical Society
APS March Meeting 2019
Volume 64, Number 2
Monday–Friday, March 4–8, 2019; Boston, Massachusetts
Session P35: Theory and Scaling of Benchmarking and TomographyFocus

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Sponsoring Units: DQI Chair: Matthew Ware, BBN Technologies Room: BCEC 205B 
Wednesday, March 6, 2019 2:30PM  3:06PM 
P35.00001: A new class of randomized benchmarking protocols: theory and expiriment Invited Speaker: Jonas Helsen In this talk I will discuss a new class of randomized benchmarking protocols called character randomized benchmarking. Character randomized benchmarking allows one to extend standard randomized benchmarking in a principled manner to groups beyond the Clifford group. I will discuss the theory of character randomized benchmarking, some example protocols such as benchmarking a T gate and performing two qubit interleaved randomized benchmarking using only single qubit gates as reference, and I will also precent some recent experimental implementations of character randomized benchmarking. 
Wednesday, March 6, 2019 3:06PM  3:18PM 
P35.00002: Randomized Benchmarking as Convolution Seth Merkel We show that the standard randomized benchmarking protocol can be described as a convolution, and is thus amenable to Fourier analysis. We utilize a form of Fourier transform that maps matrixvalued functions on group elements to matrixvalued functions of the group’s irreducible representations in order to map the average over sequences of Clifford operations to the power of a single matrix. We can then demonstrate that as long as our faulty gateset is close to some representation of the Clifford group, an RB sequence is described by the exponential decay of a process that has exactly two eigenvalues close to one and the rest close to zero, even though the bounds with respect to any particular representation of the Clifford group may not tightly describe the rate of decay. 
Wednesday, March 6, 2019 3:18PM  3:30PM 
P35.00003: Randomized Benchmarking under Different Gatesets Kristine Boone, Arnaud CarignanDugas, Joel Wallman, Joseph Emerson We provide a comprehensive analysis of the differences between two important standards for randomized benchmarking (RB): the Cliffordgroup RB protocol proposed originally in [1] and [2], and a variant of that RB protocol proposed later by the NIST group in [3]. While these two protocols are frequently conflated or presumed equivalent, we prove that they produce distinct exponential fidelity decays leading to differences of up to a factor of 3 in the estimated error rates under experimentally realistic conditions. These differences arise because the NIST RB protocol does not satisfy the unitary twodesign condition for the twirl in the Cliffordgroup protocol and thus the decay rate depends on noninvariant features of the error model. Our analysis provides an important first step towards developing definitive standards for benchmarking quantum gates and a more rigorous theoretical underpinning for the NIST protocol and other RB protocols lacking a groupstructure. We conclude by discussing the potential impact of these differences for estimating faulttolerant overheads. 
Wednesday, March 6, 2019 3:30PM  3:42PM 
P35.00004: On the freedom in representing quantum operations Junan Lin, Brandon Buonacorsi, Raymond Laflamme, Joel Wallman We discuss the effects of a gauge freedom in representing quantum information processing devices, and its implications for characterizing these devices. We demonstrate with experimentally relevant examples that there exists equally valid descriptions of the same experiment which distribute errors differently among objects in a gateset, leading to different error rates. Consequently, it can be misleading to attach a concrete operational meaning to figures of merit for individual gateset elements. We propose an alternative operational figure of merit for a gateset, the mean variation error, and a protocol for measuring this figure. 
Wednesday, March 6, 2019 3:42PM  3:54PM 
P35.00005: Using Fourier Analysis and Maximum Likelihood Estimation to Identify and Model NonMarkovian Noise in Quantum Operations Garrett Simon, Colin Bruzewicz, Kevin Obenland, Isaac Chuang, Richard Rines, Jules Stuart, Robert Niffenegger, John Chiaverini, Jeremy Sage The most wellstudied error models for quantum operations or gates are Markovian, which assume that the error is “memoryless.” Generically, most error sources can lead to gate errors which violate this assumption. One such source includes periodic noise, which causes the amplitude of gate errors to fluctuate at a characteristic frequency, requiring a nonMarkovian error model. Characterizing periodic errors, rather than just identifying them, requires data processing and error modeling beyond the standard procedures used to identify Markovian errors. Using an opensource Fourier transform implementation for qubit measurement data (pyGSTi), we were able to identify the presence of periodic error in quantum operations on trappedion qubits. This information allowed us to hypothesize the sources of periodic noise, and create a timedependent error model, or waveform, whose parameters we optimized to fit the measured data via maximum likelihood estimation. This procedure allows us to quantitatively characterize and predict error sources in an experimental set up, as well as provide a basis for building more generalized nonMarkovian error models. 
Wednesday, March 6, 2019 3:54PM  4:06PM 
P35.00006: Idle tomography: Efficient gate characterization for Nqubit processors Robin BlumeKohout, Erik Nielsen, Kenneth Rudinger, Kevin Young, Mohan Sarovar, Timothy Proctor Quantum process tomography is famously unscalable to many qubits. But the problem is actually the size of the model — arbitrary Nqubit process matrices — rather than the number of qubits. The vast majority of possible Nqubit errors will not occur in real processors. Here, we introduce a concrete reduced model of lowweight (fewqubit) errors on N qubits. It has O(N^{2}) parameters, and captures all the commonly conceived failure modes. Then, we introduce a simple and transparent tomography protocol for measuring the error rates, whose complexity scales very efficiently with N. We demonstrate it with simulations and experimental results. 
Wednesday, March 6, 2019 4:06PM  4:18PM 
P35.00007: Randomized benchmarking of manyqubit devices Timothy Proctor, Kenneth Rudinger, Robin BlumeKohout, Arnaud CarignanDugas, Erik Nielsen, Kevin Young Quantum information processors incorporating 5  10s of qubits are now commonplace, but the standard method for benchmarking quantum gates  Clifford randomized benchmarking  is infeasible to implement on more than a few qubits in any nearterm devices. In this talk, we present a series of modifications to Clifford randomized benchmarking that enable truly holistic benchmarking of entire devices. Importantly, these new techniques are adaptable based on experimental goals. They can be made highly robust or more scalable as needed, and they can be used to estimate, e.g., twoqubit gate error rates or the magnitude of crosstalk errors. Moreover, our methods allow for the benchmarking of universal gates, and continuously parameterized gates. We demonstrate our techniques on current systems, with experimental results on up to 16 qubits. 
Wednesday, March 6, 2019 4:18PM  4:30PM 
P35.00008: Perturbative density matrix propagation in Gate Set Tomography Erik Nielsen, Robin BlumeKohout, Timothy Proctor, Kenneth Rudinger, Mohan Sarovar, Kevin Young Modelbased quantum tomography protocols like Gate Set Tomography optimize a noise model with some number of parameters in order to fit experimental data. As the number of qubits increases, two issues emerge: 1) the number of model parameters grows, and 2) the cost of propagating quantum states (density matrices) increases exponentially. The first issue can be addressed by considering reduced models that limit errors to being lowweight and geometrically local. In this talk, we focus on the second issue and present a method for performing approximate density matrix propagation based on perturbative expansions of error generators. The method is tailored to the likelihood optimization problem faced by modelbased tomography protocols. We will discuss the advantages and drawbacks of using this method when characterizing the errors in up to 8qubit systems. 
Wednesday, March 6, 2019 4:30PM  4:42PM 
P35.00009: Benchmarking the quantum processing power of largescale quantum processors to execute specific programs Joseph Emerson, Joel Wallman We propose an experimentally measurable quantity, the circuit quality Q of any hardware implementation of a quantum circuit, which can be efficiently estimated via clock cycle benchmarking, and discuss how this quantity can be applied to two key applications of circuit benchmarking: (i) determining the size of a quantum program that can be run on specific quantum hardware to within a specified tolerance; and (ii) establishing a family of crossplatform benchmarks for overall hardware performance. The first application provides an efficient means of assessing the overall error probability with which hardware can implement a quantum program beyond the horizon of classical computability, which will be tremendously important once quantum processors can outperform their classical counterparts. The second application is to define a practical, evenhanded, and robust crossplatform benchmark of hardware performance for standardized quantum circuits. Our proposal for circuit benchmarking overcomes several problematic limitations of the `quantum volume’ figure of merit and the implicit recipe for measuring it. These circuit quality benchmarks will be essential to assessing and improving hardware performance on the road to practiucal quantum computation that can solve realworld problems. 
Wednesday, March 6, 2019 4:42PM  4:54PM 
P35.00010: Efficient learning of Pauli channels: learning tensor network models. Steven Flammia, Joel Wallman Noise is the central obstacle to building largescale quantum computers. Of crucial importance is the ability to reliably and efficiently characterize quantum noise afflicting a large scale quantum device with high precision. Here we show that where we have a Pauli channel whose errors are have only klocal correlations we can learn the entire nqubit Pauli channel to relative precision ε with only O(ε^{2 }n^{2} log(n)) measurements. This is efficient in the number of qubits and represents a major breakthrough in the characterization of multiqubit devices. These results have proven recovery guarantees for quantum channels to relative precision, representing a qualitative shift in the ability to characterize quantum devices. These results are practical, relevant and immediately applicable to characterizing error rates in current intermediate scale and future largescale quantum devices on hundreds to thousands of qubits. 
Wednesday, March 6, 2019 4:54PM  5:06PM 
P35.00011: Efficient learning of Pauli channels: learning sparse models Joel Wallman, Steven Flammia

Wednesday, March 6, 2019 5:06PM  5:18PM 
P35.00012: Experimental reconstruction of all correlated error rates on a 16 qubit device. Robin Harper, Steven Flammia, Joel Wallman, Joseph Emerson Recent results by Flammia and Wallman have shown how to reliably and efficiently characterize Pauli channels on intermediate and largescale quantum devices. Here we leverage those results to give a complete, efficient, and highprecision, characterization of IBM's online 16 qubit device. Using experimental results from the device we obtain and present complete information about the correlated error rates across the device, comparing device characteristics when qubits are operated in single qubit mode and with qubittoqubit interactions enabled. The protocol we use obtains multiplicative precision and robustness to SPAM errors by using techniques originating from randomized benchmarking, and it can be executed on all current NISQ devices. The protocol is easy to implement, involving just computational basis state preparation and measurement together with only onequbit Clifford gates. We show how this protocol can be scaled up to even larger devices of 100 or even 1000 qubits. 
Wednesday, March 6, 2019 5:18PM  5:30PM 
P35.00013: Efficient Unitarity Randomized Benchmarking of Fewqubit Clifford Gates Bas Dirkse, Jonas Helsen, Stephanie Wehner Unitarity randomized benchmarking (URB) is an experimental procedure for estimating the unitarity of implemented quantum gates independently of state preparation and measurement (SPAM) errors. The unitarity is a measure of coherence of a quantum gate that provides information independent of average fidelity. A central problem in the URB experiment is relating the number of data points to rigorous confidence intervals around the unitarity. 
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