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

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.

Figure 1

Thermodynamics of locality: Suppose we have two bits $XY$ in a correlated state where $\frac{1}{2}$ probability is in $XY=00$ and $\frac{1}{2}$ probability is in $XY=11$. (a) A thermodynamically irreversible operation can be performed to erase only $X$ (that is, set $X=0$ without changing $Y)$ if we are not allowed to use knowledge about the state of $Y$. (b) A reversible operation can be performed to erase $X$ if we are allowed to use knowledge about $Y$. Both operations have the same outcome given our initial condition, but the nonlocal operation (a) is more thermodynamically costly because it is irreversible. According to Theorem 1, operation (a) is costly since it erases information in $X$ that is correlated with $Y$.

Figure 2

Information ratchet sequentially generating a symbol string on an empty tape: At time step $t,S_t$ is the random variable for the ratchet state. The generated symbols in the generated (output) process are denoted by $X_{t-1},X_{t-2},X_{t-3},...$. The most recently generated symbol $X_t$ (green) is determined by the internal dynamics of the ratchet's memory by using heat $Q$ from the thermal reservoir as well as work $W$ from the work reservoir. (Inside ratchet) Ratchet memory dynamics and symbol emission are governed by the conditional probabilities $Pr(s_{t+1},x_t|s_t)$, where $s_t$ is the current state at time $t,x_t$ is the generated symbol, and $s_{t+1}$ is the new state. Graphically, this is represented by a hidden Markov model, depicted here as a state-transition diagram in which nodes are states $s$ and edges represent transitions $s\to s^{\prime }$ labeled by the generated symbol and associated probability: $x:Pr(s^{\prime },x|s)$. (Reproduced with permission from Ref. [15].)

Figure 3

Performance trade-offs for $q$-machines, whose variety and dependence on phases $\left\{{\varphi }_{xs}\right\}$ is depicted by a torus: Under all ways of quantifying memory, the $q$-machines constructed from a predictor achieve nonnegative memory compression [50], and they also have a smaller dissipation $\mathrm{\Delta }{S}_{\mathrm{loc}}$, rendering them more thermodynamically efficient [15]. However, to achieve positive compression, they must also have a nonzero $\mathrm{\Delta }{S}_{\mathrm{loc}}$, rendering them less efficient than a classical retrodictor.

Figure 4

Performance trade-offs for reverse $q$-machines, whose variety and dependence on $\left\{{\varphi }_{xs}\right\}$ is represented by a torus: Under all quantifications of memory, the reverse $q$-machines constructed from a retrodictor achieve non-negative memory compression. However, to achieve positive compression, they must also have a nonzero dissipation $\mathrm{\Delta }{S}_{\mathrm{loc}}$. The latter renders them less thermodynamically efficient.