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Physical Review Applied

Characterizing Midcircuit Measurements on a Superconducting Qubit Using Gate Set Tomography

Kenneth Rudinger, Guilhem J. Ribeill, Luke C. G. Govia, Matthew Ware, Erik Nielsen, Kevin C Young, Thomas Ohki, Robin Blume-Kohout, Timothy J Proctor

Published Jan 12, 2022• 30 citations
Publisher

Abstract

Key applications of quantum computing, including error correction, rely critically on the capability to measure (read out) the state of selected quantum bits, without disturbing other qubits or terminating the computation. Such $m\phantom{\rule{0}{0ex}}i\phantom{\rule{0}{0ex}}d\phantom{\rule{0}{0ex}}c\phantom{\rule{0}{0ex}}i\phantom{\rule{0}{0ex}}r\phantom{\rule{0}{0ex}}c\phantom{\rule{0}{0ex}}u\phantom{\rule{0}{0ex}}i\phantom{\rule{0}{0ex}}t$ $m\phantom{\rule{0}{0ex}}e\phantom{\rule{0}{0ex}}a\phantom{\rule{0}{0ex}}s\phantom{\rule{0}{0ex}}u\phantom{\rule{0}{0ex}}r\phantom{\rule{0}{0ex}}e\phantom{\rule{0}{0ex}}m\phantom{\rule{0}{0ex}}e\phantom{\rule{0}{0ex}}n\phantom{\rule{0}{0ex}}t\phantom{\rule{0}{0ex}}s$ are now being implemented in a few cutting-edge platforms, but their development is hindered by lack of a means to measure their performance and characterize the errors that they produce. The authors show how to extend a popular tomography method for precise characterization of midcircuit measurements, and using their QILGST protocol discover a pernicious kind of non-Markovian error, which once identified can be handled.

DOI
10.1103/physrevapplied.17.014014

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Key applications of quantum computing, including error correction, rely critically on the capability to measure (read out) the state of selected quantum bits, without disturbing other qubits or terminating the computation. Such m\phantom{\rule{0}{0ex}}i\phantom{\rule{0}{0ex}}d\phantom{\rule{0}{0ex}}c\phantom{\rule{0}{0ex}}i\phantom{\rule{0}{0ex}}r\phantom{\rule{0}{0ex}}c\phantom{\rule{0}{0ex}}u\phantom{\rule{0}{0ex}}i\phantom{\rule{0}{0ex}}t m\phantom{\rule{0}{0ex}}e\phantom{\rule{0}{0ex}}a\phantom{\rule{0}{0ex}}s\phantom{\rule{0}{0ex}}u\phantom{\rule{0}{0ex}}r\phantom{\rule{0}{0ex}}e\phantom{\rule{0}{0ex}}m\phantom{\rule{0}{0ex}}e\phantom{\rule{0}{0ex}}n\phantom{\rule{0}{0ex}}t\phantom{\rule{0}{0ex}}s are now being implemented in a few cutting-edge platforms, but their development is hindered by lack of a means to measure their performance and characterize the errors that they produce. The authors show how to extend a popular tomography method for precise characterization of midcircuit measurements, and using their QILGST protocol discover a pernicious kind of non-Markovian error, which once identified can be handled.
Measurements that occur within the internal layers of a quantum circuit -- mid-circuit measurements -- are an important quantum computing primitive, most notably for quantum error correction. Mid-circuit measurements have both classical and quantum outputs, so they can be subject to error modes that do not exist for measurements that terminate quantum circuits. Here we show how to characterize mid-circuit measurements, modelled by quantum instruments, using a technique that we call quantum instrument linear gate set tomography (QILGST). We then apply this technique to characterize a dispersive measurement on a superconducting transmon qubit within a multiqubit system. By varying the delay time between the measurement pulse and subsequent gates, we explore the impact of residual cavity photon population on measurement error. QILGST can resolve different error modes and quantify the total error from a measurement; in our experiment, for delay times above 1000 ns we measured a total error rate (i.e., half diamond distance) of ε⋄=8.1±1.4%ε_{\diamond} = 8.1 \pm 1.4 \%ε⋄​=8.1±1.4%, a readout fidelity of 97.0±0.3%97.0 \pm 0.3\%97.0±0.3%, and output quantum state fidelities of 96.7±0.6%96.7 \pm 0.6\%96.7±0.6% and 93.7±0.7%93.7 \pm 0.7\%93.7±0.7% when measuring 000 and 111, respectively.
Error mitigation has enabled quantum computing applications with more than a hundred qubits and deep circuits. Many error mitigation methods are noise-aware, relying on a faithful characterization of the noise channels of the hardware. However, fundamental limitations lead to unlearnable degrees of freedom of the underlying noise models when considering qubits. Here, we show how to leverage noncomputational states as an additional resource to learn state-preparation errors in superconducting qubits. This allows one to fully constrain the noise models. We can thus independently and accurately mitigate state-preparation errors, gate errors, and measurement errors. Our proposed method is also applicable to dynamic circuits with midcircuit measurements. This work opens the door to improved error mitigation for measurements, both at the end of the circuit and midcircuit.
Characterizing errors on many-qubit quantum computers remains a key challenge to understanding and improving the performance of these devices. Current characterization methods either don't scale beyond a few qubits, or make simplifying assumptions (such as assuming stochastic Pauli error) that obscure the underlying physical error mechanisms. In this work, we present a scalable extension to gate set tomography-linearized gate set tomography-that enables characterization of many-qubit systems. Linearized gate set tomography relies on sparse error models, a linear approximation to enable efficient data fitting, and data from shallow circuits-so that the systematic error in the linear approximation is small. We demonstrate the accuracy of our technique using simulations of a ten-qubit system with coherent and stochastic errors, including coherent crosstalk, and we demonstrate that it is robust in presence of additional errors that are not included within the sparse error model ansatz.
As progress is made towards the first generation of error-corrected quantum computers, robust characterization and validation protocols are required to assess the noise environments of physical quantum processors. While standard coherence metrics and characterization protocols such as T1{T}_{1}T1​ and T2{T}_{2}T2​, process tomography, and randomized benchmarking are now ubiquitous, these techniques provide only partial information about the dynamic multiqubit loss channels responsible for processor errors, which can be described more fully by a Lindblad operator in the master equation formalism. Here, we introduce and experimentally demonstrate Lindblad tomography, a hardware-agnostic characterization protocol for tomographically reconstructing the Hamiltonian and Lindblad operators of a quantum noise environment from an ensemble of time-domain measurements. Performing Lindblad tomography on a small superconducting quantum processor, we show that this technique naturally builds on standard process tomography and T1/T2{T}_{1}/{T}_{2}T1​/T2​ measurement protocols, characterizes and accounts for state-preparation and measurement errors, and allows one to place bounds on the fit to a Markovian model. Comparing the results of single- and two-qubit measurements on a superconducting quantum processor, we demonstrate that Lindblad tomography can also be used to identify and quantify sources of crosstalk on quantum processors, such as the presence of always-on qubit-qubit interactions.
Mid-circuit measurements (MCMs) are critical components of the quantum error correction protocols expected to enable utility-scale quantum computing. MCMs can be modeled by quantum instruments (a type of quantum operation or process), which can be characterized self-consistently using gate set tomography. However, experimentally estimated quantum instruments are often hard to interpret or relate to device physics. We address this challenge by adapting the error generator formalism -- previously used to interpret noisy quantum gates by decomposing their error processes into physically meaningful sums of "elementary errors" -- to MCMs. We deploy our new analysis on a transmon qubit device to tease out and quantify error mechanisms including amplitude damping, readout error, and imperfect collapse. We examine in detail how the magnitudes of these errors vary with the readout pulse amplitude, recover the key features of dispersive readout predicted by theory, and show that these features can be modeled parsimoniously using a reduced model with just a few parameters.
In circuit-based quantum computing the available gate set typically consists of single-qubit gates acting on each individual qubit and at least one entangling gate between pairs of qubits.In certain physical architectures, however, some qubits may be "hidden" and lacking direct addressability through dedicated control and readout lines, for instance, because of limited on-chip routing capabilities, or because the number of control lines becomes a limiting factor for many-qubit systems.In this case, no single-qubit operations can be applied to the hidden qubits and their state cannot be measured directly.Instead, they may be controlled and read out only via single-qubit operations on connected "control" qubits and a suitable set of two-qubit gates.We first discuss the impact of such restricted control capabilities on the performance of specific qubit coupling networks.We then experimentally demonstrate full control and measurement capabilities in a superconducting two-qubit device with local single-qubit control and iSWAP and controlledphase two-qubit interactions enabled by a tunable coupler.We further introduce an iterative tune-up process required to completely characterize the gate set used for quantum process tomography and evaluate the resulting gate fidelities.