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arXiv:2102.01327

Experimental characterisation of a non-Markovian quantum process

Kaumudibikash Goswami, Christina Giarmatzi, Christopher Pineda Monterola, Sally Shrapnel, Jacquiline Romero, Fabio Costa

Published Feb 2, 2021• 0 citations
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Abstract

Every quantum system is coupled to an environment. Such system-environment interaction leads to temporal correlation between quantum operations at different times, resulting in non-Markovian noise. In principle, a full characterisation of non-Markovian noise requires tomography of a multi-time processes matrix, which is both computationally and experimentally demanding. In this paper, we propose a more efficient solution. We employ machine learning models to estimate the amount of non-Markovianity, as quantified by an information-theoretic measure, with tomographically incomplete measurement. We test our model on a quantum optical experiment, and we are able to predict the non-Markovianity measure with $90\%$ accuracy. Our experiment paves the way for efficient detection of non-Markovian noise appearing in large scale quantum computers.

DOI
10.48550/arxiv.2102.01327

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Every quantum system is coupled to an environment. Such system-environment interaction leads to temporal correlation between quantum operations at different times, resulting in non-Markovian noise. In principle, a full characterisation of non-Markovian noise requires tomography of a multi-time processes matrix, which is both computationally and experimentally demanding. In this paper, we propose a more efficient solution. We employ machine learning models to estimate the amount of non-Markovianity, as quantified by an information-theoretic measure, with tomographically incomplete measurement. We test our model on a quantum optical experiment, and we are able to predict the non-Markovianity measure with 90%90\%90% accuracy. Our experiment paves the way for efficient detection of non-Markovian noise appearing in large scale quantum computers.
Characterisation protocols have so far played a central role in the development of noisy intermediate-scale quantum (NISQ) computers capable of impressive quantum feats. This trajectory is expected to continue in building the next generation of devices: ones that can surpass classical computers for particular tasks -- but progress in characterisation must keep up with the complexities of intricate device noise. A missing piece in the zoo of characterisation procedures is tomography which can completely describe non-Markovian dynamics over a given time frame. Here, we formally introduce a generalisation of quantum process tomography, which we call process tensor tomography. We detail the experimental requirements, construct the necessary post-processing algorithms for maximum-likelihood estimation, outline the best-practice aspects for accurate results, and make the procedure efficient for low-memory processes. The characterisation is a pathway to diagnostics and informed control of correlated noise. As an example application of the hardware-agnostic technique, we show how its predictive control can be used to substantially improve multi-time circuit fidelities on superconducting quantum devices. Our methods could form the core for carefully developed software that may help hardware consistently pass the fault-tolerant noise threshold.
We show how a tensor-network-based machine learning algorithm can learn the structures of generic, non-Markovian, quantum stochastic processes. First, a process is represented as a matrix product operator (MPO) and trained with a database of local input states at different times and the corresponding time-nonlocal output state. We then apply the algorithm to predict the output state of a process at different times for a system that is coupled to a spin-chain environment. We then reconstruct the full process, and we quantify the non-Markovian memory by means of the bond dimension of the MPO for various properties of the system, of the environment, and of their interaction. Our study paves the way for a possible experimental investigation into the process tensor and its properties, and an effective characterization of noise in quantum devices.
The presence of noise is currently one of the main obstacles to achieving large-scale quantum computation. Strategies to characterise and understand noise processes in quantum hardware are a critical part of mitigating it, especially as the overhead of full error correction and fault-tolerance is beyond the reach of current hardware. Non-Markovian effects are a particularly unfavourable type of noise, being both harder to analyse using standard techniques and more difficult to control using error correction. In this work we develop a set of efficient algorithms, based on the rigorous mathematical theory of Markovian master equations, to analyse and evaluate unknown noise processes. In the case of dynamics consistent with Markovian evolution, our algorithm outputs the best-fit Lindbladian, i.e., the generator of a memoryless quantum channel which best approximates the tomographic data to within the given precision. In the case of non-Markovian dynamics, our algorithm returns a quantitative and operationally meaningful measure of non-Markovianity in terms of isotropic noise addition. We provide a Python implementation of all our algorithms, and benchmark these on a range of 1- and 2-qubit examples of synthesised noisy tomography data, generated using the Cirq platform. The numerical results show that our algorithms succeed both in extracting a full description of the best-fit Lindbladian to the measured dynamics, and in computing accurate values of non-Markovianity that match analytical calculations.
In non-Markovian systems, the current state of the system depends on the full or partial history of its past evolution. Owing to these time correlations, non-Markovian noise violates common assumptions in gate characterization protocols such as randomized benchmarking and gate-set tomography. Here, we perform a case study of the effects of a quantum non-Markovian bath on qubit randomized benchmarking experiments. We consider a model consisting of qubits coupled to a multimode bosonic bath. We apply unitary operations on the qubits, interspersed with brief interactions with the environment governed by a Hamiltonian. Allowing for non-Markovianity in the interactions leads to clear differences in the randomized benchmarking decay curves in this model, which we analyze in detail. The Markovian model’s decay is exponential as expected, whereas the model with non-Markovian interactions displays a much slower, almost polynomial, decay. We develop efficient numerical methods for this problem that we use to support our theoretical results. These results inform efforts on incorporating quantum non-Markovian noise in the characterization and benchmarking of quantum devices.
The complete characterisation of non-Markovian dynamics on quantum devices can be achieved with experiments on the system using a procedure known as process tensor tomography. However, through either hardware or computational restrictions, tomographically complete estimation is usually out of reach. Here, we present methods for bounding any desired facet of multi-time processes only with limited data to arbitrary accuracy that depends on data availability. We then use this method to estimate the strength of non-Markovian memory and construct conditional Markov order models, which are far less complex yet possess high predictive power. Finally, we display the efficacy and utility of our theoretical methods with experimental case studies on IBM Quantum devices.