Thermodynamically-efficient local computation and the inefficiency of quantum memory compression

Abstract

Modularity dissipation identifies how locally-implemented computation entails costs beyond those required by Landauer's bound on thermodynamic computing. We establish a general theorem for efficient local computation, giving the necessary and sufficient conditions for a local operation to have zero modularity cost. Applied to thermodynamically-generating stochastic processes it confirms a conjecture that classical generators are efficient if and only if they satisfy retrodiction, which places minimal memory requirements on the generator. This extends immediately to quantum computation: Any quantum simulator that employs quantum memory compression cannot be thermodynamically efficient.CB is given by W min = k B T ln 2 (H[ρ AB ] − H[ρ CB ]).

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

“…We introduce a systematic approach to constructing quantum implementations of non-deterministic HMMs, showing that memory and thermodynamical advantages arise for almost any HMM. Our findings strengthen, extend, and supersede analogous prior results [33,35,39], which were limited in scope by considering an incomplete suite of possible quantum implementations.…”

“…Theorem 3 declares that this result holds for quantum implementations in general. Our proof mirrors many aspects of that for the deterministic case [39], with appropriate generalisation. Before proceeding with our proof, we introduce a definition and result involved in the proof of the deterministic case.…”

“…The proof is given in Appendix B; it utilises a framework developed in proving a similar statement for quantum implementations of predictive generators [39], effectively condensing and generalising this earlier derivation. The crux of this generalised theorem is that no quantum implementation of any generator can simultaneously achieve an advantage in memory compression and perfect thermal efficiency.…”

“…However, typical generators incur additional dissipative costs beyond this bound -a so-called modularity or locality dissipation -due to the generator acting in a time-local manner to produce the pattern [64]. An implementation of a generator is said to be thermally inefficient if this additional cost is non-zero [39].…”

“…Finally, it has previously been shown that a classical implementation has no thermal inefficiency iff the generator is retrodictive [36,39,64]. Since the quantum implementation of any non-retrodictive generator achieves memory compression, they must also too reduce the thermal inefficiency, by Eq.…”