Fitting quantum noise models to tomography data

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 unfavorable 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 time-independent Markovian (or nearly Markovian) dynamics, 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.Comment: 51 pages, 8 figures. Code available at: https://gitlab.com/TamaraKohler/non-markovianity Version 2: minor modifications to time series algorith

Fitting quantum noise models to tomography data

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

Universal Hamiltonians for quantum simulation and their applications to holography

Recent work has demonstrated the existence of universal Hamiltonians – simple spin lattice models that can simulate any other quantum many body system. These universal Hamiltonians have applications for developing quantum simulators, as well as for Hamiltonian complexity, quantum computation, and fundamental physics. In this thesis we extend the theory of universal Hamiltonians. We begin by developing a new method for proving that a given family of Hamiltonians is indeed universal. We then use this method to construct two new universal models – both of which consist of translationally invariant interactions acting on a 1D spin chain. But the benefit of our method doesn’t just lie in the simple universal models it allows us to construct. It also gives deeper insight into the origins of universality – and demonstrates a link between the universality and complexity. We make this insight rigorous, and derive a complexity theoretic classification of universal Hamiltonians which encompasses all known universal models. This classification provides a new, simplified route to checking whether a particular family of Hamiltonians meets the conditions to be a universal simulator. We also consider the practical use of analogue Hamiltonian simulation. Under- standing the effect of noise on Hamiltonian simulation is a key issue in practical implementations. The first step to tackling this issue is characterising the noise processes affecting near term quantum devices. Motivated by this, we develop and numerically benchmark an algorithm which fits noise models to tomographic data from quantum devices to enable this process. This algorithm has applicability beyond analogue simulators, and could be used to investigate the physical noise processes in any quantum computing device. Finally, we apply the theory of universal Hamiltonians to high energy physics by using them to construct toy models of holographic duality which capture more of the expected features of the AdS/CFT correspondence

Dynamic message-passing approach for kinetic spin models with reversible dynamics

A method to approximately close the dynamic cavity equations for synchronous reversible dynamics on a locally tree-like topology is presented. The method builds on $(a)$ a graph expansion to eliminate loops from the normalizations of each step in the dynamics, and $(b)$ an assumption that a set of auxilary probability distributions on histories of pairs of spins mainly have dependencies that are local in time. The closure is then effectuated by projecting these probability distributions on $n$-step Markov processes. The method is shown in detail on the level of ordinary Markov processes ($n=1$), and outlined for higher-order approximations ($n>1$). Numerical validations of the technique are provided for the reconstruction of the transient and equilibrium dynamics of the kinetic Ising model on a random graph with arbitrary connectivity symmetry.Comment: 6 pages, 4 figure

Non-Markovian dynamics of open quantum systems

This thesis is centred around the striking phenomenon of non-Markovianity which emanates from exact dynamical descriptions of open quantum systems. Non- Markovianity is associated with the existence of memory effects in the environment and leads to a partial recovery of information of the system, temporarily counteracting the deleterious effect of the surrounding environment. We devote this thesis to addressing two fundamental questions surrounding the topic of non-Markovianity. The first is concerned with how to evaluate the extent to which a specific dynamics is non-Markovian, in terms of a physically meaningful and easily computable measure. In literature, the desire to quantify non-Markovianity has motivated a plethora of measures which provide unique, albeit potentially contradicting, interpretations of memory effects. In an attempt to consolidate the literature, we introduce and critically compare several recently proposed non-Markovianity measures for single qubit and two qubit systems in both pure dephasing and dissipative scenarios. The second question explores the natural optimism of the usefulness of non-Markovianity as a resource in quantum information protocols. In more detail, we study whether memory effects combined with external control techniques offer a possibility to exploit non-Markovianity for an overall superior technique to combat decoherence. The standard approach for Markovian dynamics involves the critical assumption of dissipative dynamics which are fixed in the presence of control. We expose the serious pitfalls in experimentally implementing such a strategy in realistic non-Markovian scenarios and accentuate the importance of using exact approaches in non-Markovian control theory. Using an exact description of a pure dephasing system subject to dynamical decoupling protocols, we demonstrate that contrary to intuitive reasoning, non-Markovianity is not trivially a resource

Non-Markovian Quantum Process Tomography

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. 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 the pathway to diagnostics and informed control of correlated noise. As an example application of the technique, we improve multi-time circuit fidelities on IBM Quantum devices for both standalone qubits and in the presence of crosstalk to a level comparable with the fault-tolerant noise threshold in a variety of different noise conditions. Our methods could form the core for carefully developed software that may help hardware consistently pass the fault-tolerant noise threshold