Routine 4.3.3: Density matrix exponentiation (in superposition)
5.3 Model complexity
with a mixture of the original qsample and a corrupted qsample weighted byµ.
In the PAC setting with a uniform distribution investigated by Bshouty and Jackson, the quantum algorithm can still learn the function by consuming more examples, while noise is believed to render the classical problem unlearnable [159]. A similar observation is made by Cross, Smith and Smolin [161], who consider the problem of learningnbit parity functions by membership queries and examples, a task that is easy both classically and quantumly, and in the sample as well as time complexity sense. (Parity functions evaluate to 1 if and only if the number of 1s in the input string is odd). However, they find that in the presence of noise, the classical case becomes intractable while the quantum samples required only grow logarithmically. To ensure a fair comparison, the classical oracle is modelled as a dephased quantum channel of the membership oracle and the quantum example oracle respectively, and as a toy model they consider a slight adaptation of the Bernstein-Vazirani algorithm.
These observations are evidence that a fourth category of potential quantum advantages, the robustness against noise, could be a fruitful avenue for further research.
three points four points
Figure 5.3: A model imposing a linear decision boundary in a two-dimensional input space can separate or shatter a dataset of three points in any label configuration, but there are datasets of four points which it cannot separate.
recover. It is therefore interesting how ‘quantum models’ might enhance the space of possible dynamics that can be mimicked. This becomes particularly important when the systems producing the data are quantum systems, which has been termed the ‘QQ’ case in the introduction of this thesis. However, also the ‘CQ’ case, i.e. classically produced data modeled by a quantum model, may reveal interesting facets, and only few attempts have been made in this direction so far.
One model that we understand fairly well from a classical and quantum perspective is the Ising model, whose dynamics are used in Hopfield networks for associative memory recall. A number of studies on “quantum Hopfield models” investigate how the introduction of an inverse field term changes the dynamics of the model significantly. For example, Inoue [162] introduces quantum fluctuations and analyses image restoration via quantum Monte Carlo simulations, finding that quantum fluctuations give finer restoration results than thermal fluctuations. He later performs a similar study on Hopfield networks under ‘quantum noise’ [163] joined by others [164], revealing a few more particulars. Many years previously, a study by Nonomura and Nishimori [165] with a similar aim came to the conclusion that with regards to the ratio of stored patterns and system size, “quantum fluctuations play quite similar roles to thermal fluctuations in neural networks”.
Another advantage can be a drastically increased storage capacity in a quantum annealing framework [166]. Even though the usefulness of these findings to speed up machine learning applications is not necessarily the focus of this kind of work, it raises the interesting point of how the quantum version of a classical model can have different properties.
Another model that lends itself easily to quantum generalisations are hidden Markov models, where a system undergoes a sequence of state transitions under observations, and either the transition/observation probabilities, or the next transition is to be learnt (see Section 2.3.3.2).
State transitions and observation probabilities are central to quantum physics as well, where they can be elegantly described in the formalism of density matrices and open quantum systems.
Monras, Beige and Wiesner [167] therefore introduce a quantum formulation of hidden Markov models, in which the state of the system is a density matrixρ and state transition are formally represented by Kraus operators K with associated probabilities tr{Kρ}. Although the primary idea is to use these quantum models to learn about partially observable quantum systems, they give evidence that the number of internal states necessary for generating some stochastic processes are less than of a comparable classical model.5 The ‘reinforcement learning’ version of hidden Markov models are partially observable Markov decision processes and have been generalised
5In a later paper, Monras and Winter [168] also perform a deep mathematical investigation into the task of learning a model from example process data in the quantum setting, i.e. to ask whether there is there a quantum process that realises the observations.
to a quantum setting by Barry, Barry and Aaronson [169]. The authors show that a certain type of problem, the existence of a sequence of actions that can reach a certain goal state, is undecidable for quantum models. Another variation of a hidden Markov model is provided by [170] and based on Gudder’s quantum Markov processes. Again, the message is that quan- tising a model can enlarge the framework and make it useful for the analysis of quantum dynamics.
A third branch of ‘quantum-extendable’ models are graphical models. Leifer et al. [58] for example give a quantum mechanical verison of belief propagation. A number of studies also look at quantum extensions of the closely related causal models. (Causal models can roughly be imagined as directed graphical models, in which a directed edge stands for a causal influence. An increasingly important goal is to discover causal structure from data, which is possible only to some extend [51] in the classical case.) Quantum systems exhibit very different causal properties, and it is sometimes possible to infer the full causal structure of a model from (quantum) data [171, 172]. Costa and Shrapnel [173] introduce quantum analogues of causal models for the discovery of causal structure and propose a translation for central concepts such as the Markov condition or faithfulness. If such quantum models can have any use for classical data is an interesting question.
Other dimensions of quantum models have been investigated. For example, Stark [174] asks whether computers are able to learn optimal (that is, low-dimensional) quantum models from experimental data, and find that like for classical models the problem is NP-hard. Dunjko et al. [4] investigate a general quantum speed-up for quantum enhanced reinforcement learning by modeling agents as quantum systems.
The contributions mentioned do not answer (and in most cases, they do not even directly address) the question of the complexity or flexibility of quantum models, which is a potential topic for further research. Interesting questions may be what VC dimension a quantum channel has if we interpret it as the model function f(x, w), or if there are dynamics for which a quantum model requires a much lower number of parameters than a classical model. If such quantum models are found, one can think of quantum simulations or special-purpose quantum computers to use as a classification device and investigate potential (classical and quantum) training methods.
In summary, this chapter presented three ways of thinking about quantum-enhanced machine learning, namely the time, sample and model complexity. While under the definitions outlined above, sample complexity does not seem to change in the quantum case, interesting aspects concern the robustness against noise as well as the generalisation properties of quantum models. In the following I will focus on the first dimension and analyse the advantages in terms of time complexity that a quantum algorithm offers. However, as a conclusion to this chapter it should be clear that there is a lot more to quantum machine learning than just speed-ups.
Recent progress in quantum machine learning algorithms
After discussing results for the sample complexity of quantum learning as well as potential avenues to investigate quantum models for machine learning, this chapter will focus on the narrower research question, namely to develop quantum machine learning algorithms that solve supervised pattern recognition tasks with improvements in thetime complexity. With the lack of a large-scale universal quantum computer, most quantum algorithms remain theoretical proposals, although some proof-of-principal experiments have been executed [175, 176, 177] and some proposals have been tested on the D-Wave machine as a special purpose quantum computer.
I divide the literature into four main approaches (see Table 6.1): Those that compute linear algebra routines ‘super-efficiently’ with quantum systems (Section 6.1), approaches based on Grover search or quantum walks (Section 6.2), approaches where samples are drawn from quantum states (Section 6.3), and approaches where the solution to an optimisation problem is encoded in the ground state of a Hamiltonian (Section 6.4). These four classes -to my own surprise - cover most of the proposals for quantum machine learning algorithms found in the literature up to today.