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arXiv:1405.1454

Scalable extraction of error models from the output of error detection circuits

Austin G. Fowler, D. Sank, J. Kelly, R. Barends, John M. Martinis

Published May 6, 2014• 8 citations
arXivHTMLPDF

Abstract

Accurate methods of assessing the performance of quantum gates are extremely important. Quantum process tomography and randomized benchmarking are the current favored methods. Quantum process tomography gives detailed information, but significant approximations must be made to reduce this information to a form quantum error correction simulations can use. Randomized benchmarking typically outputs just a single number, the fidelity, giving no information on the structure of errors during the gate. Neither method is optimized to assess gate performance within an error detection circuit, where gates will be actually used in a large-scale quantum computer. Specifically, the important issues of error composition and error propagation lie outside the scope of both methods. We present a fast, simple, and scalable method of obtaining exactly the information required to perform effective quantum error correction from the output of continuously running error detection circuits, enabling accurate prediction of large-scale behavior.

DOI
10.48550/arxiv.1405.1454

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Accurate methods of assessing the performance of quantum gates are extremely important. Quantum process tomography and randomized benchmarking are the current favored methods. Quantum process tomography gives detailed information, but significant approximations must be made to reduce this information to a form quantum error correction simulations can use. Randomized benchmarking typically outputs just a single number, the fidelity, giving no information on the structure of errors during the gate. Neither method is optimized to assess gate performance within an error detection circuit, where gates will be actually used in a large-scale quantum computer. Specifically, the important issues of error composition and error propagation lie outside the scope of both methods. We present a fast, simple, and scalable method of obtaining exactly the information required to perform effective quantum error correction from the output of continuously running error detection circuits, enabling accurate prediction of large-scale behavior.
A key requirement for scalable quantum computing is that elementary quantum gates can be implemented with sufficiently low error. One method for determining the error behavior of a gate implementation is to perform process tomography. However, standard process tomography is limited by errors in state preparation, measurement and one-qubit gates. It suffers from inefficient scaling with number of qubits and does not detect adverse error-compounding when gates are composed in long sequences. An additional problem is due to the fact that desirable error probabilities for scalable quantum computing are of the order of 0.0001 or lower. Experimentally proving such low errors is challenging. We describe a randomized benchmarking method that yields estimates of the computationally relevant errors without relying on accurate state preparation and measurement. Since it involves long sequences of randomly chosen gates, it also verifies that error behavior is stable when used in long computations. We implemented randomized benchmarking on trapped atomic ion qubits, establishing a one-qubit error probability per randomized $\ensuremath{\pi}/2$ pulse of 0.00482(17) in a particular experiment. We expect this error probability to be readily improved with straightforward technical modifications.
The characterization of noisy quantum circuits is an important step in the development of large-scale quantum computers. As experimental quantum architectures approach the threshold for fault-tolerant quantum computing, obtaining a benchmark of process fidelities is needed to verify the integrity of the system and ensure functionality. Additionally, the ability to successfully implement a quantum code of choice may depend on the type of noise that is present in the system. It is therefore necessary to characterize not only the fidelity of the process, but also the structure of the noise. \n \nRandomized benchmarking is a characterization protocol which extracts partial information from a noisy quantum process. This thesis gives a detailed review of the randomized benchmarking protocol, and presents the results of three collaborative projects which use techniques similar to randomized benchmarking to obtain previously hidden information. The first protocol is designed to obtain a benchmark on loss errors. This protocol also provides information on the detector efficiency and an additional constraint on estimated parameters from randomized benchmarking. The second protocol is an extension of the first, under the special case of coherent leakage errors, for which we obtain an estimate of the leakage rate. The third protocol provides a method of testing for spatial correlations in noisy processes. This test may be used to determine the correlation structure of errors, and the error correcting codes needed for successful experiments in a particular architecture. All protocols presented in this thesis are scalable, platform-independent and insensitive to state preparation and measurement errors. We also provide results of numerical tests of these protocols for simulated noise, which show their accuracy and robustness.
Understanding the performance of realistic noisy encoded circuits is an important task for the development of large-scale practical quantum computers. Specifically, the development of proposals for quantum computation must be well informed by both the qualities of the low-level physical system of choice, and the properties of the high-level quantum error correction and fault-tolerance schemes. Gaining insight into how a particular computation will play out on a physical system is in general a difficult problem, as the classical simulation of arbitrary noisy quantum circuits is inefficient. Nevertheless, important classes of noisy circuits can be simulated efficiently. Such simulations have led to numerical estimates of threshold errors rates and resource estimates in topological codes subject to efficiently simulable error models. \n \nThis thesis describes and analyzes a method that my collaborators and I have introduced for leveraging efficient simulation techniques to understand the performance of large quantum processors that are subject to errors lying outside of the efficient simulation algorithm's applicability. The idea is to approximate an arbitrary gate error with an error from the efficiently simulable set in a way that ``honestly'' represents the original error's ability to preserve or distort quantum information. After introducing and analyzing the individual gate approximation method, its utility as a means for estimating circuit performance is studied. In particular, the method is tested within the use-case for which it was originally conceived; understanding the performance of a hypothetical physical implementation of a quantum error-correction protocol. It is found that the method performs exactly as desired in all cases. That is, the circuits composed of the approximated error models honestly represent the circuits composed of the errors derived from the physical models.
The performance of quantum gates is often assessed using some form of randomized benchmarking. However, the existing methods become infeasible for more than approximately five qubits. Here we show how to use a simple and customizable class of circuits-randomized mirror circuits-to perform scalable, robust, and flexible randomized benchmarking of Clifford gates. We show that this technique approximately estimates the infidelity of an average many-qubit logic layer, and we use simulations of up to 225 qubits with physically realistic error rates in the range 0.1%-1% to demonstrate its scalability. We then use up to 16 physical qubits of a cloud quantum computing platform to demonstrate that our technique can reveal and quantify crosstalk errors in many-qubit circuits.
Abstract There is currently a significant need for robust and efficient methods for characterizing quantum devices. While there has been significant progress in this direction, there remains a crucial need to precisely determine the strength and type of errors on individual gate operations, in order to assess and improve control as well as reliably bound the total error in a quantum circuit given some partial information about the errors on the components. In this work, we first provide an optimal bound on the total fidelity of a circuit in terms of component fidelities, which can be efficiently experimentally estimated via randomized benchmarking (RB). We then derive a tighter bound that applies under additional information about the coherence of the error, namely, the unitarity, which can also be estimated via a related experimental protocol. This improved bound smoothly interpolates between the worst-case quadratic and best-case linear scaling for composite error channels. As an application we show how our analysis substantially improves the achievable precision on estimates of the infidelities of individual gates under interleaved RB, enabling greater precision for current experimental methods to assess and tune-up control over quantum gate operations.
As quantum processors grow, new performance benchmarks are required to capture the full quality of the devices at scale. While quantum volume is an excellent benchmark, it focuses on the highest quality subset of the device and so is unable to indicate the average performance over a large number of connected qubits. Furthermore, it is a discrete pass/fail and so is not reflective of continuous improvements in hardware nor does it provide quantitative direction to large-scale algorithms. For example, there may be value in error mitigated Hamiltonian simulation at scale with devices unable to pass strict quantum volume tests. Here we discuss a scalable benchmark which measures the fidelity of a connecting set of two-qubit gates over NNN qubits by measuring gate errors using simultaneous direct randomized benchmarking in disjoint layers. Our layer fidelity can be easily related to algorithmic run time, via γγγ defined in Ref.\cite{berg2022probabilistic} that can be used to estimate the number of circuits required for error mitigation. The protocol is efficient and obtains all the pair rates in the layered structure. Compared to regular (isolated) RB this approach is sensitive to crosstalk. As an example we measure a N=80 (100)N=80~(100)N=80 (100) qubit layer fidelity on a 127 qubit fixed-coupling "Eagle" processor (ibm\_sherbrooke) of 0.26(0.19) and on the 133 qubit tunable-coupling "Heron" processor (ibm\_montecarlo) of 0.61(0.26). This can easily be expressed as a layer size independent quantity, error per layered gate (EPLG), which is here 1.7×10−2(1.7×10−2)1.7\times10^{-2}(1.7\times10^{-2})1.7×10−2(1.7×10−2) for ibm\_sherbrooke and 6.2×10−3(1.2×10−2)6.2\times10^{-3}(1.2\times10^{-2})6.2×10−3(1.2×10−2) for ibm\_montecarlo.