Memory-efficient tracking of complex temporal and symbolic dynamics with quantum simulators

Abstract

Tracking the behaviour of stochastic systems is a crucial task in the statistical sciences. It has recently been shown that quantum models can faithfully simulate such processes whilst retaining less information about the past behaviour of the system than the optimal classical models. We extend these results to general temporal and symbolic dynamics. Our systematic protocol for quantum model construction relies only on an elementary description of the dynamics of the process. This circumvents restrictions on corresponding classical construction protocols, and allows for a broader range of processes to be modelled efficiently. We illustrate our method with an example exhibiting an apparent unbounded memory advantage of the quantum model compared to its optimal classical counterpart.

Memory compression and thermal efficiency of quantum implementations of non-deterministic hidden Markov models

“…In taking quantum memory and thermodynamical advantages in stochastic modelling to a more general medium, we open up a number of future research avenues, some paralleling developments in quantum implementations of predictive generators, some unique to the wider spectrum. For example, our results can be further extended to encompass input-output [37,38] and continuous-time processes [30,32,50,66,70,71] as well as inference protocols [24], and the trade-off between memory-and thermal-efficiency can be explored to determine the generator that best compromises the two -and whether classical and quantum implementations agree on which generator this should be. This latter direction will also require developments even in the purely classical setting: while the optimal predictive generators can be systematically found, this remains an open question across all generators in general.…”

“…The corresponding measures D µ and C µ are known as the topological and statistical complexity. Recently, a growing body of work has established that quantum implementations of ε-machines (with memory costs D q and C q ) can generally undercut this minimality [27][28][29][30][31][32][33][34]: D q ≤ D µ and C q ≤ C µ , with the compression advantage sometimes able to grow unboundedly large [30,32,46,[48][49][50]. The present state-of-the-art quantum implementations [33] are defined implicitly through a unitary interaction:…”

“…Two key areas where quantum HMMs manifest advantages are in stochastic simulation and information engines. Quantum implementations of deterministic HMMs offer memory compression in the former [27][28][29][30][31][32][33][34], and greater thermal efficiency in the latter [35], compared to corresponding classical implementations. The restriction of determinism is sometimes a necessity: nondeterministic consumers of information reservoirs have been shown to be capable of violating the second law of information thermodynamics [36]; and adaptive systems must typically act in a causal manner [37,38].…”

Quantum adaptive agents with efficient long-term memories

“…less memory -and provide a systematic procedure for their design -using insights from quantum stochastic modelling [12][13][14][15][16][17][18][19][20]. The resulting agents can display extreme scaling advantages over provably minimal classical counterparts [21].…”

Quantum-inspired identification of complex cellular automata

“…Further developments have shown that even when modelling classical data, the most efficient models are quantum mechanical [24][25][26][27][28][29][30][31]. This has led to a quantum analogue of the statistical complexity -the quantum statistical memory -with distinct qualitative and quantitative behaviour [27,29,[32][33][34][35][36][37] that demonstrates even better alignment with our intuitive notions of what is complex [32,38], and a greater degree of robustness [39].…”