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 minimum-memory requirements on the generator. This extends immediately to quantum computation: Any quantum simulator that employs quantum memory compression cannot be thermodynamically efficient.
Stochastic modeling is an essential component of the quantitative sciences, with hidden Markov models (HMMs) often playing a central role. Concurrently, the rise of quantum technologies promises a host of advantages in computational problems, typically in terms of the scaling of requisite resources such as time and memory. HMMs are no exception to this, with recent results highlighting quantum implementations of deterministic HMMs exhibiting superior memory and thermal efficiency relative to their classical counterparts. In many contexts, however, nondeterministic HMMs are viable alternatives; compared to them, the advantages of current quantum implementations do not always hold. Here, we provide a systematic prescription for constructing quantum implementations of nondeterministic HMMs that reestablish the quantum advantages against this broader class. Crucially, we show that whenever the classical implementation suffers from thermal dissipation due to its need to process information in a time-local manner, our quantum implementations will both mitigate some of this dissipation and achieve an advantage in memory compression.
Stochastic modeling of complex systems plays an essential, yet often computationally intensive, role across the quantitative sciences.Recent advances in quantum information processing have elucidated the potential for quantum simulators to exhibit memory advantages for such tasks.Heretofore, the focus has been on lossless memory compression, wherein the advantage is typically in terms of lessening the amount of information tracked by the model, while-arguably more practical-reductions in memory dimension are not always possible.Here, we address the case of lossy compression for quantum stochastic modeling of continuous-time processes, introducing a method for coarse graining in quantum state space that drastically reduces the requisite memory dimension for modeling temporal dynamics while retaining near-exact statistics.In contrast to classical coarse graining, this compression is not based on sacrificing temporal resolution and brings memory-efficient high-fidelity stochastic modeling within reach of present quantum technologies.
Free classical communication can yield an unbounded single-shot saving in quantum compression, even though it leaves the asymptotic rate of an i.i.d. source unchanged. At every fixed error in
(0,1), we exhibit sources whose unassisted compression cost is arbitrarily large, yet a single qubit suffices with classical communication and no preshared entanglement. We establish this separation by determining the exact optimal source fidelity for compression through an
m-dimensional noiseless quantum channel, with and without free classical communication, while preserving correlations with an inaccessible reference. Without classical assistance, the optimal fidelity is exactly the Ky Fan norm
∥ρ∥(m), the sum of the
m largest eigenvalues of the source state. With classical assistance, we obtain an explicit spectral threshold formula and a protocol attaining it. Classical communication strictly improves fidelity whenever
m is smaller than the source rank. The corresponding compression costs are two smoothed versions of the log-rank entropy, also called max-entropy: trace-distance smoothing under local operations and quadratic smoothing under LOCC. For i.i.d. copies of a fixed source with positive entropy variance, however, the two costs have identical first-, second-, and third-order asymptotic terms. For nonlattice spectra, we determine the constant-order advantage: classical communication saves
log2(e)−1 bits per block asymptotically, and the ratio of assisted to unassisted minimum communication dimensions converges to
2/e.
Thermodynamics is a highly successful macroscopic theory widely used across the natural sciences and for the construction of everyday devices, from car engines to solar cells. With thermodynamics predating quantum theory, research now aims to uncover the thermodynamic laws that govern finite size systems which may in addition host quantum effects. Recent theoretical breakthroughs include the characterisation of the efficiency of quantum thermal engines, the extension of classical non-equilibrium fluctuation theorems to the quantum regime and a new thermodynamic resource theory has led to the discovery of a set of second laws for finite size systems. These results have substantially advanced our understanding of nanoscale thermodynamics, however putting a finger on what is genuinely quantum in quantum thermodynamics has remained a challenge. Here we identify information processing tasks, the so-called projections, that can only be formulated within the framework of quantum mechanics. We show that the physical realisation of such projections can come with a non-trivial thermodynamic work only for quantum states with coherences. This contrasts with information erasure, first investigated by Landauer, for which a thermodynamic work cost applies for classical and quantum erasure alike. Repercussions on quantum work fluctuation relations and thermodynamic single-shot approaches are also discussed.