Thermodynamics of computing with circuits

We have four primary contributions.

(1) We derive exact expressions for how the entropy flow (EF) and entropy production (EP) produced by a fixed dynamical system vary as one changes the initial distribution of states of that system. These expressions capture effect (I) described above. (These expressions extend an earlier analysis [30]).

(2) We introduce 'solitary processes'. These are a type of physical process that can implement any particular gate in a circuit while respecting the constraints on what variables in the rest of the circuit that gate is coupled with. We can use the thermodynamic properties of solitary processes to analyze effect (II) described above.

3) We combine our first two contributions to analyze the thermodynamic costs of implementing circuits in a 'serial-reinitializing' manner. This means two things: the gates in the circuit are run one at a time, so each gate is run as a solitary process; after a gate is run its input wires are reinitialized, allowing for subsequent reuse of the circuit. In particular, we derive expressions relating the minimal EP generated by running an SR circuit to information-theoretic quantities associated with the wiring diagram of the circuit.

4) Our last contribution is an expression for the extra EP that arises in running an SR circuit if the initial state distributions at its gates differ from the ones that result in minimal EP for each of those gates. This expression involves an information-theoretic function that we call 'multi-divergence' which appears to be new to the literature.

In section 2.1 we introduce general notation, and then provide a minimal summary of the parts of stochastic thermodynamics, information theory and circuit theory that will be used in this paper. We also introduce the definition of the 'islands' of a stochastic matrix in that section, which will play a central role in our analysis. In section 3 we derive an exact expression for how the EF and EP of an arbitrary process depends on its initial state distribution. In section 4 we introduce solitary processes and then analyze their thermodynamics. In section 5 we introduce SR circuits. In section 6 we use the tools developed in the previous sections to analyze the thermodynamic properties of SR circuits. In section 7 we discuss related earlier work. Section 8 concludes and presents some directions for future work. All proofs that are longer than several lines are collected in the appendices.

We write a Kronecker delta as δ(a, b). We write a random variable with an upper case letter (e.g., X), and the associated set of possible outcomes with the associated calligraphic letter (e.g., $\mathcal{X}$). A particular outcome of a random variable is written with a lower case letter (e.g., x). We also use lower case letters like p, q, etc to indicate probability distributions.

We use ${{\Delta}}_{\mathcal{X}}$ to indicate the set of probability distribution over a set of outcomes $\mathcal{X}$. For any distribution $p\in {{\Delta}}_{\mathcal{X}}$, we use $\mathrm{s}\mathrm{u}\mathrm{p}\mathrm{p}\;p{:=}\left\{x\in \mathcal{X}:p\left(x\right){ >}0\right\}$ to indicate the support of p. Given a distribution p over $\mathcal{X}$ and any $\mathcal{Z}\subseteq \mathcal{X}$, we write $p\left(\mathcal{Z}\right)={\sum }_{x\in \mathcal{Z}}p\left(x\right)$ to indicate the probability that the outcome of X is in $\mathcal{Z}$. Given a function $f:\mathcal{X}\to \mathbb{R}$, we write ${\mathbb{E}}_{p}\left[f\right]$ to indicate ∑_xp(x)f(x), the expectation of f under distribution p.

Given any conditional distribution P(y|x) of $y\in \mathcal{Y}$ given $x\in \mathcal{X}$, and some distribution p over $\mathcal{X}$, we write Pp for the distribution over $\mathcal{Y}$ induced by applying P to p:

$$\begin{equation}\left[Pp\right]\left(y\right){:=}\sum _{x\in \mathcal{X}}P\left(y\vert x\right)p\left(x\right).\end{equation} \tag{ 1 }$$

We will sometimes use the term 'map' to refer to a conditional distribution.

We say that a conditional distribution P is 'logically reversible' if it is deterministic (the entries of P(y|x) are 0/1-valued for all $x\in \mathcal{X}$ and $y\in \mathcal{Y}$) and if there do not exist $x,{x}^{\prime }\in \mathcal{X}$ and $y\in \mathcal{Y}$ such that P(y|x) > 0 and P(y|x') > 0. When $\mathcal{Y}=\mathcal{X}$, a logically reversible P is simply a permutation matrix. Given any subset of states $\mathcal{Z}\subseteq \mathcal{X}$, we also say that P is 'logically reversible over $\mathcal{Z}$' if the entries P(y|x) are 0/1-valued for all $x\in \mathcal{Z}$ and $y\in \mathcal{Y}$, and there do not exist $x,{x}^{\prime }\in \mathcal{Z}$ and $y\in \mathcal{Y}$ such that P(y|x) > 0 and P(y|x') > 0.

We write a multivariate random variable with components V = {1, 2, ..., } as X_V = (X₁, X₂, ..., ), with outcomes x_V. We will also use upper case letters (e.g., A, V, ..., ) to indicate sets of variables. For any subset A ⊆ V we use the random variable X_A (and its outcomes x_A) to refer to the components of X_V indexed by A. Similarly, for a distribution p_V over X_V, we write the marginal distribution over X_A as p_A. For a singleton set {a}, we slightly abuse notation and write X_a instead of X_{a}.

We will consider a circuit to be physical system in contact with one or more thermodynamic reservoirs (heat baths, chemical baths, etc). The system evolves over some time interval (sometimes implicitly taken to be t ∈ [0, 1], where the units of time are arbitrary), possibly while being driven by a work reservoir. We refer to the set of thermodynamic reservoirs and the driving—and, in particular, the stochastic dynamics they induce over the system during t ∈ [0, 1]—as a physical process.

We use $\mathcal{X}$ to indicate the finite state space of the system. Physically, the states $x\in \mathcal{X}$ can either be microstates or they can be coarse-grained macrostates under some additional assumptions (e.g., that all macrostates have the same 'internal entropy' [2, 20, 32]).

While much of our analysis applies more broadly, to make things concrete one may imagine that the system undergoes master equation dynamics, also known as a continuous-time Markov chain (CTMC). This kind of dynamics is the basis of stochastic thermodynamics, which is often used to analyze the thermodynamics of discrete-state physical systems. In this subsection we briefly review stochastic thermodynamics, referring the reader to [33, 34] for more details.

Under a CTMC, the probability distribution over $\mathcal{X}$ at time t, indicated by p^t, evolves according to the master equation

$$\begin{equation}\frac{\mathrm{d}}{\mathrm{d}t}{p}^{t}\left({x}^{\prime }\right)=\sum _{x}{p}^{t}\left(x\right){K}_{t}\left(x\to {x}^{\prime }\right),\end{equation} \tag{ 2 }$$

where K_t is the rate matrix at time t. For any rate matrix K_t, the off-diagonal entries K_t(x → x') (for x ≠ x') indicate the rate at which probability flows from state x to x', while the diagonal entries are fixed by K_t(x → x) = −∑_x'(≠x)K_t(x → x'), which guarantees conservation of probability. If the system is connected to multiple thermodynamic reservoirs indexed by α, the rate matrix can be further decomposed as ${K}_{t}\left(x\to {x}^{\prime }\right)={\sum }_{\alpha }{K}_{t}^{\alpha }\left(x\to {x}^{\prime }\right)$, where ${K}_{t}^{\alpha }$ is the rate matrix at time t corresponding to reservoir α.

The term entropy flow (EF) refers to the increase of entropy in all coupled reservoirs. The instantaneous rate of EF out of the system at time t is defined as

$$\begin{equation}\dot {\mathcal{Q}}\left({p}^{t}\right)=\sum _{\alpha ,x,{x}^{\prime }}{p}^{t}\left(x\right){K}_{t}^{\alpha }\left(x\to {x}^{\prime }\right)\mathrm{ln}\frac{{K}_{t}^{\alpha }\left(x\to {x}^{\prime }\right)}{{K}_{t}^{\alpha }\left({x}^{\prime }\to x\right)}.\end{equation} \tag{ 3 }$$

The overall EF incurred over the course of the entire process is $\mathcal{Q}={\int }_{0}^{1}\dot {\mathcal{Q}}\;\;\mathrm{d}t$.

The term entropy production (EP) refers to the overall increase of entropy, both in the system and in all coupled reservoirs. The instantaneous rate of EP at time t is defined as

$$\begin{equation}\dot {\sigma }\left({p}^{t}\right)=\frac{\mathrm{d}}{\mathrm{d}t}S\left({p}^{t}\right)+\dot {\mathcal{Q}}\left({p}^{t}\right).\end{equation} \tag{ 4 }$$

The overall EP incurred over the course of the entire process is $\sigma ={\int }_{0}^{1}\dot {\sigma }\;\;\mathrm{d}t$.

Note that we use terms like 'EF' and 'EP' to refer to either the associated rate or the associated integral over a non-infinitesimal time interval; the context should always make the precise meaning clear.

Given some initial distribution p, the EF, EP, and the drop in the entropy of the system from the beginning to the end of the process are related according to

$$\begin{equation}\mathcal{Q}\left(p\right)=\left[S\left(p\right)-S\left(Pp\right)\right]+\sigma \left(p\right).\end{equation} \tag{ 5 }$$

In general, the EF can be written as the expectation $\mathcal{Q}\left(p\right)$ = ∑_xp(x)q(x), where q(x) indicates the expected EF arising from trajectories that begin on state x. Given that the drop in entropy is a nonlinear function of p, while the expectation $\mathcal{Q}\left(p\right)$ is a linear function of p, equation (5) tells us that EP will generally be a nonlinear function of p. Note that if P is logically reversible, then S(p) = S(Pp) and therefore EF and EP will be equal for any p.

While the EF can be positive or negative, the log-sum inequality can be used to prove that EP for master equation dynamics is non-negative [15, 35]:

$$\begin{equation}\mathcal{Q}\left(p\right){\geqslant}S\left(p\right)-S\left(Pp\right).\end{equation} \tag{ 6 }$$

This can be viewed as a derivation of the second law of thermodynamics, given the assumption that our system is evolving forward in time as a CTMC.

All of these results are purely mathematical and hold for any CTMC dynamics, even in contexts having nothing to do with physical systems. However, these results can be interpreted in thermodynamic terms when each ${K}_{t}^{\alpha }$ obeys local detailed balance (LDB) with regard to thermodynamic reservoir α [3, 15, 33]. Consider a system with Hamiltonian H_t(⋅) at time t, and let α label a heat bath whose inverse temperature is β_α. Then, ${K}_{t}^{\alpha }$ will obey LDB when for all $x,{x}^{\prime }\in \mathcal{X}$, either ${K}_{t}^{\alpha }\left(x\to {x}^{\prime }\right)={K}_{t}^{\alpha }\left({x}^{\prime }\to x\right)=0$, or

$$\begin{equation}\frac{{K}_{t}^{\alpha }\left(x\to {x}^{\prime }\right)}{{K}_{t}^{\alpha }\left({x}^{\prime }\to x\right)}={\mathrm{e}}^{{\beta }_{\alpha }\left({H}_{t}\left(x\right)-{H}_{t}\left({x}^{\prime }\right)\right)}.\end{equation} \tag{ 7 }$$

If LDB holds, then EF can be written as [34]

$$\begin{equation}\mathcal{Q}\left(p\right)=\sum _{\alpha }{\beta }_{\alpha }{Q}_{\alpha }\left(p\right),\end{equation} \tag{ 8 }$$

where Q_α is the expected amount of heat transferred from the system into bath α during the process.

We end with two caveats concerning the use of stochastic thermodynamics to analyze real-world circuits. First, many of the processes described in this paper require that some transition rates be exactly zero at some moments. In many physical models this implies there are infinite energy barriers at those times. In addition, perfectly carrying out any deterministic map (such as bit erasure) requires the use of infinite energy gaps between some states at some times. Thus, as is conventional (though implicit) in much of the thermodynamics of computation literature, the thermodynamic costs derived in this paper should be understood as limiting values.

Second, there are some conditional distributions that take the system state at time 0 to its state at time 1, P(x₁|x₀), that cannot be implemented by any CTMC [36, 37]. For example, one cannot carry out (or even approximate) a simple bit flip P(x₁|x₀) = 1 − δ(x₁, x₀) with a CTMC. Now we can design a CTMC to implement any given P(x₁|x₀) to arbitrary precision, if the dynamics is expanded to include a set of 'hidden states' in addition to the states in X [21, 22]. However, as we explicitly demonstrate below, SR circuits can be implemented without introducing any such hidden states; this is one of their advantages. (See also example 9 in appendix A).

Given two distributions p and r over random variable X, we use notation like S(p) for Shannon entropy and D(p||r) for Kullback–Leibler (KL) divergence. We write S(Pp) to refer to the entropy of the distribution over Y induced by p(x) and the conditional distribution P, as defined in equation (5), and similarly for other information-theoretic measures. Given two random variables X and Y with joint distribution p, we write S(p(X|Y)) for the conditional entropy of X given Y, and I_p(X; Y) for the mutual information (we drop the subscript p where the distribution is clear from context). All information-theoretic measures are in nats.

Some of our results below are formulated in terms of an extension of mutual information to more than two random variables that is known as 'total correlation' or multi-information [38]. For a random variable X_A = (X₁, X₂, ..., ), the multi-information is defined as

$$\begin{equation}\mathcal{I}\left({p}_{A}\right)=\left[\sum _{v\in A}S\left({p}_{v}\right)\right]-S\left({p}_{A}\right).\end{equation} \tag{ 9 }$$

Some of the other results below are formulated in terms of the multi-divergence between two probability distributions over the same multi-dimensional space. This is a recently introduced information-theoretic measure which can be viewed as an extension of multi-information to include a reference distribution. Given two distributions p_A and r_A over X_A, the multi-divergence is defined as

$$\begin{equation}\mathcal{D}\left({p}_{A}{\Vert}{r}_{A}\right){:=}D\left({p}_{A}{\Vert}{r}_{A}\right)-\sum _{v\in A}D\left({p}_{v}{\Vert}{r}_{v}\right).\end{equation} \tag{ 10 }$$

Multi-divergence measures how much of the divergence between p_A and r_A arises from the correlations among the variables X₁, X₂, ..., rather than from the marginal distributions of each variable considered separately. See appendix A of [3] for a discussion of the elementary properties of multi-divergence and its relation to conventional multi-information. Note that multi-divergence is defined with 'the opposite sign' of multi-information, i.e., by subtracting a sum of terms involving marginal variables from a term involving the joint random variable, rather than vice-versa.

A central part of our analysis will involve the equivalence relation,

$$\begin{equation}x\sim {x}^{\prime }\;{\Leftrightarrow}\;\exists y\;:\;P\left(y\vert x\right){ >}0,\quad P\left(y\vert {x}^{\prime }\right){ >}0.\end{equation} \tag{ 11 }$$

In words, x ∼ x' if there is a non-zero probability of transitioning to some state y from both x and x' under the conditional distribution P(y|x). We define an island of the conditional distribution P(y|x) as any connected subset of $\mathcal{X}$ given by the transitive closure of this equivalence relation. The set of islands of any P(⋅|⋅) form a partition of $\mathcal{X}$, which we write as L(P).

We will also use the notion of the islands of the conditional distribution P restricted to some subset of states $\mathcal{Z}\subseteq \mathcal{X}$. We write ${L}_{\mathcal{Z}}\left(P\right)$ to indicate the partition of $\mathcal{Z}$ generated by the transitive closure of the relation given by equation (11) for $x,{x}^{\prime }\in \mathcal{Z}$. Note that in this notation, $L\left(P\right)={L}_{\mathcal{X}}\left(P\right)$.

As an example, if P(y|x) > 0 for all $x\in \mathcal{X}$ and $y\in \mathcal{Y}$ (i.e., any final state y can be reached from any initial state x with non-zero probability), then L(P) contains only a single island. As another example, if P(y|x) implements a deterministic function $f:\mathcal{X}\to \mathcal{Y}$, then L(P) is the partition of $\mathcal{X}$ given by the pre-images of f, $L\left(P\right)=\left\{{f}^{-1}\left(y\right):y\in \mathcal{Y}\right\}$. For example, the conditional distribution that implements the logical AND operation of two binary variables,

$$\begin{equation}P\left(c\vert a,b\right)=\delta \left(c,a\,b\right)\end{equation} \tag{ 12 }$$

has two islands, corresponding to (a, b) ∈ {(0, 0), (0, 1), (1, 0)} and (a, b) ∈ {(1, 1)}, respectively. As a final example, let P be the following conditional distribution:

$$\begin{equation}P\left(y\vert x\right)=\left[\begin{bmatrix}{cccc}\hfill 0.5\hfill & \hfill 0.5\hfill & \hfill 0\hfill & \hfill 0\hfill \\ \hfill 0\hfill & \hfill 0.5\hfill & \hfill 0.5\hfill & \hfill 0\hfill \\ \hfill 0\hfill & \hfill 0\hfill & \hfill 1\hfill & \hfill 0\hfill \\ \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 1\hfill \end{bmatrix}\right],\end{equation} \tag{ 13 }$$

where the rows and columns corresponds to the ordered states $\mathcal{X}=\mathcal{Y}=\left\{\mathsf{00},\mathsf{01},\mathsf{10},\mathsf{11}\right\}$. The island decomposition for this map is illustrated in figure 2 (left). We also show the island decomposition for this map restricted to subset of states $\mathcal{Z}=\left\{\mathsf{00},\mathsf{01}\right\}$ in figure 2 (right).

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For any distribution p over $\mathcal{X}$, any $\mathcal{Z}\subseteq \mathcal{X}$, and any $c\in {L}_{\mathcal{Z}}\left(P\right)$, p(c) = ∑_x∈cp(x) is the probability that the state of the system is contained in island c. It will be helpful to use the unusual notation p^c(x) to indicate the conditional probability of x within island c. Formally, p^c(x) = p(x)/p(c) if x ∈ c, and p^c(x) = 0 otherwise.

Intuitively, the islands of a conditional distribution are 'firewalled' subsystems, both computationally and thermodynamically isolated from one another for the duration of the process implementing that conditional distribution. In particular, we will show below that the EP of running P(y|x) on an initial distribution p can be written as a weighted sum of the EPs involved in running P on each separate island c ∈ L(P), where the weight for island c is given by p(c).

For the purposes of this paper, a (logical) circuit is a special type of Bayes net [39–41]. Specifically, we define any circuit Φ as a tuple $\left(V,E,F,{\mathcal{X}}_{V}\right)$. The pair (V, E) specifies the vertices and edges of a directed acyclic graph (DAG). (We sometimes call this DAG the wiring diagram of the circuit.) ${\mathcal{X}}_{V}$ is a Cartesian product ${\prod }_{v}{\mathcal{X}}_{v}$, where each ${\mathcal{X}}_{v}$ is the set of possible states associated with node v. F is a set of conditional distributions, indicating the logical maps implemented at the non-root nodes of the DAG.

Following the convention in the Bayes nets literature, we orient edges in the direction of information flow. Thus, the inputs to the circuit are the roots of the associated DAG and the outputs are the leaves of the DAG ⁵. Without loss of generality, we assume that each node v has a special 'initialized state', indicated as ∅.

We use the term gate to refer to any non-root node, input node to refer to any root node, and output node or output gate to refer to a leaf node. For simplicity, we assume that all output nodes are gates, i.e., there is no root node which is also a leaf node. We write IN and x_IN to indicate the set of input nodes and their joint state, and similarly write OUT and x_OUT for the output nodes.

We write the set of all gates in a given circuit as G ⊆ V, and use g ∈ G to indicate a particular gate. We indicate the set of all nodes that are parents of gate g as pa(g). We indicate the set of nodes that includes gate g and all parents of g as n(g) := {g} ∪ pa(g).

As mentioned, F is a set of conditional distributions, indicating the logical maps implemented by each gate of the circuit. The element of F corresponding to gate g is written as π_g(x_g|x_pa(g)). In conventional circuit theory, each π_g is required to be deterministic (i.e., 0/1-valued). However, we make no such restriction in this paper. We write the overall conditional distribution of output gates given input nodes implemented by the circuit Φ as

$$\begin{equation}{\pi }_{{\Phi}}\left({x}_{\mathrm{O}\mathrm{U}\mathrm{T}}\vert {x}_{\mathrm{I}\mathrm{N}}\right)=\sum _{{x}_{G{\backslash}\mathrm{O}\mathrm{U}\mathrm{T}}}\prod _{g\in G}{\pi }_{g}\left({x}_{g}\vert {x}_{\mathrm{p}\mathrm{a}\left(g\right)}\right).\end{equation} \tag{ 14 }$$

We can illustrate this formalism using the parity circuit shown in figure 1. Here, V has 5 nodes, corresponding to the 3 input nodes and the two gates. The circuit operates over bits, so ${\mathcal{X}}_{v}=\left\{0,1\right\}$ for each v ∈ V. Both gates carry out the $\mathsf{X}\mathsf{O}\mathsf{R}$ operation, so both elements of F are given by π_g(x_g|x_pa(g)) = δ(x_g, $\mathsf{X}\mathsf{O}\mathsf{R}$(x_pa(g))) (where $\mathsf{X}\mathsf{O}\mathsf{R}$(x_pa(g)) = 1 when the two parents of gate g are in different states, and $\mathsf{X}\mathsf{O}\mathsf{R}$(x_pa(g)) = 0 otherwise). Finally, E has four elements representing the edges connecting the nodes in V, which are shown as arrows in figure 1.

In the conventional representation of a physical circuit as a (Bayes net) DAG, the wires in the physical circuit are identified with edges in the DAG. However, in order to account for the thermodynamic costs of communication between gates along physical wires, it will be useful to represent the wires themselves as a special kind of gate. This means that the DAG (V, E) we use to represent a particular physical circuit is not the same as the DAG (V', E') that would be used in the conventional computer science representation of that circuit. Rather (V, E) is constructed from (V', E') as follows.

To begin, V = V' and E = E'. Then, for each edge $\left(v\to ~{v}\right)\in {E}^{\prime }$, we first add a wire gate w to V, and then add two edges to E: an edge from v to w and an edge from w to $~{v}$. So a wire gate w has a single parent and a single child, and implements the identity map, π_w(x_w|x_pa(w)) = δ(x_w, x_pa(w)). (This is an idealization of the real world, in which wires have nonzero probability of introducing errors.) We sometimes call (V, E) the wired circuit, to distinguish it from the original logical circuit defined as in computer science theory, (V', E'). We use W ⊂ G to indicate the set of wire gates in a wired circuit.

Every edge in a wired circuit either connects a wire gate to a non-wire gate or vice versa. Physically, the edges of the DAG of a wired circuit do not represent interconnects (e.g., copper wires), as they do in a logical circuit. Rather they only indicate physical identity: an edge e ∈ E going into a wire gate w from a non-wire node v indicates that the same physical variable will be written as either X_v or X_pa(w). Similarly, an edge e ∈ E going into a non-wire gate g from a wire gate w indicates that X_w is the same physical variable (and so always has the same state) as the corresponding component of X_pa(g). However, despite this modified meaning of the nodes in a wired circuit, equation (14) still applies to any wired circuit, as well as applying to the corresponding logical circuit. In figure 3, we demonstrate how to represent the 3-bit parity circuit from figure 1 as a wired circuit.

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We use the word 'circuit' to refer to either an abstract wired (or logical) circuit, or to a physical system that implements that abstraction. Note that there are many details of the physical system that are not specified in the associated abstract circuit. When we need to distinguish the abstraction from its physical implementation, we will refer to the latter as a physical circuit, with the former being the corresponding wired circuit. The context will always make clear whether we are using terms like 'gate', 'circuit', etc, to refer to physical systems or to their formal abstractions.

Even if one fully specifies the distinct physical subsystems of a physical circuit that will be used to implement each gate in a wired circuit, we still do not have enough information concerning the physical circuit to analyze the thermodynamic costs of running it. We still need to specify the initial states of those subsystems (before the circuit begins running), the precise sequence of operations of the gates in the circuit, etc. However, before considering these issues, we need to analyze the general form of the thermodynamic costs of running individual gates in a circuit, isolated from the rest of the circuit. We do that in the next section.

Let p refer to the initial distribution over the joint state of all nodes in the circuit. By equation (5), the total EF incurred by implementing some overall map P which takes the initial joint state of all nodes in the full circuit to the final joint state is

$$\begin{equation}Q\left(p\right)=\mathcal{L}\left(p\right)+\sigma \left(p\right).\end{equation} \tag{ 36 }$$

where

$$\begin{equation}\mathcal{L}\left(p\right){:=}S\left(p\right)-S\left(Pp\right)\end{equation} \tag{ 37 }$$

is the Landauer cost of computing P for initial distribution p.

The first term in equation (37), $\mathcal{L}$, is the minimal EF that must be incurred by any process over ${\mathcal{X}}_{\mathrm{I}\mathrm{N}}{\times}{\mathcal{X}}_{\mathrm{O}\mathrm{U}\mathrm{T}}$ that implements P, without any constraints on how the variables in ${\mathcal{X}}_{\mathrm{I}\mathrm{N}}{\times}{\mathcal{X}}_{\mathrm{O}\mathrm{U}\mathrm{T}}$ are coupled, and without any reference to a set of intermediate subsystems (e.g., gates) that may connect the input and output variables¹⁴.

The second term in equation (44) is the EP, which reflects the thermodynamic irreversibility of the SR circuit. Using equation (16), the EP can be further decomposed as

$$\begin{equation}\sigma \left(p\right)=\left[D\left(p{\Vert}q\right)-D\left(Pp{\Vert}Pq\right)\right]+\sum _{c\in L\left(P\right)}p\left(c\right)\sigma \left({q}^{c}\right).\end{equation} \tag{ 38 }$$

The decrease in KL reflects the mismatch cost, arising from the discrepancy between p(x_V), the actual initial distribution over all nodes of the circuit defined in equation (39), and q(x_V), the optimal prior distribution over the joint state of all the nodes of the circuit which would result in the least EP. The last sum in equation (38) reflects the residual EP, reflecting EP that remains even when the circuit is initialized with the optimal prior distribution.

Suppose we know that the dynamics is actually implemented with an SR circuit, but do not know the precise wiring diagram. Then we know that the initial joint distribution over all the nodes is

$$\begin{equation}p\left({x}_{V}\right)={p}_{\mathrm{I}\mathrm{N}}\left({x}_{\mathrm{I}\mathrm{N}}\right)\prod _{v\in V{\backslash}\mathrm{I}\mathrm{N}}\delta \left({x}_{v},\varnothing \right),\end{equation} \tag{ 39 }$$

and the ending joint distribution is

$$\begin{equation}\left[Pp\right]\left({x}_{V}\right)={p}_{\mathrm{O}\mathrm{U}\mathrm{T}}\left({x}_{\mathrm{O}\mathrm{U}\mathrm{T}}\right)\prod _{v\in V{\backslash}\mathrm{O}\mathrm{U}\mathrm{T}}\delta \left({x}_{v},\varnothing \right),\end{equation} \tag{ 40 }$$

So S(p) = S(p_IN) and S(Pp) = S(p_OUT) = S(π_Φp_IN), where π_Φ is the conditional distribution of the final joint state of the output gates given the initial joint state of the input nodes, defined in equation (14). Combining gives

$$\begin{equation}\mathcal{L}\left(p\right)=S\left({p}_{\mathrm{I}\mathrm{N}}\right)-S\left({\pi }_{{\Phi}}{p}_{\mathrm{I}\mathrm{N}}\right)\end{equation} \tag{ 41 }$$

Similarly, the EP becomes

$$\begin{align}\hfill \sigma \left(p\right)& =\left[D\left({p}_{\mathrm{I}\mathrm{N}}{\Vert}{q}_{\mathrm{I}\mathrm{N}}\right)-D\left({\pi }_{{\Phi}}{p}_{\mathrm{I}\mathrm{N}}{\Vert}{\pi }_{{\Phi}}{q}_{\mathrm{I}\mathrm{N}}\right)\right]+\sum _{c\in L\left({\pi }_{{\Phi}}\right)}p\left(c\right)\sigma \left({q}_{\mathrm{I}\mathrm{N}}^{c}\right).\hfill \end{align} \tag{ 42 }$$

While the expressions in equations (38) and (42) for EP must be equal, how they decompose that EP among a mismatch cost term and a residual EP differ. The two decompositions differ because they define the 'optimal initial distribution' relative to differ sets of possible distributions, resulting in different prior distributions (which are also defined over different sets of outcomes). Also note that the residual EP terms in equation (42) are defined in terms of a more constrained minimization problem than the residual EP terms in equation (38). Thus, given the same initial distribution p, the residual EP in equation (42) will generally be larger than the residual EP in equation (38), while the mismatch cost in equation (38) will generally be larger than the mismatch cost in equation (42). We also emphasize that the island decompositions appearing in the two expressions are different.

The decompositions of EF and EP given in (equations (36), (38) and (42)) do not involve the wiring diagram of the SR circuit. As an alternative, we can exploit that wiring diagram to formulate a decomposition of EF and EP which separates the contributions from different gates. In general, such circuit-based decompositions allow for a finer-grained analysis of the EP in SR circuits than do the decompositions proposed in the last section. In particular, they allow us to derive some novel connections between nonequilibrium statistical physics, computer science theory, and information theory, as discussed in the next two subsections.

Before discussing these circuit-based decompositions, we introduce some new notation. We write p_pa(g)(x_pa(g)) and p_n(g)(x_n(g)) = p_pa(g)(x_pa(g))δ(x_g, ∅) for the distributions over x_pa(g) and x_n(g), respectively, at the beginning of the solitary process that implements gate g. We write the EF function of the solitary process of gate g as ${\mathcal{Q}}_{g}\left({p}_{n\left(g\right)}\right)$, and its subsystem EP as

$$\begin{equation}{\hat{\sigma }}_{g}\left({p}_{n\left(g\right)}\right){:=}{\mathcal{Q}}_{g}\left({p}_{n\left(g\right)}\right)-\left[S\left({p}_{n\left(g\right)}\right)-S\left({P}_{g}{p}_{n\left(g\right)}\right)\right].\end{equation} \tag{ 43 }$$

We also write p^beg(g) and p^end(g) to indicate the joint distribution over all circuit nodes at the beginning and end, respectively, of the solitary process that runs gate g. As an illustration of this notation, p^beg(g)(x_pa(g)) = p_pa(g)(x_pa(g)). On the other hand, p^end(g)(x_g) = (π_gp_pa(g))(x_g) is the distribution over x_g after gate g runs, and p^end(g)(x_pa(g)) is a delta function about the joint state of the parents of g in which they are all initialized, by equation (29). Note that since we are considering solitary processes, p^beg(g)(x_V\n(g)) = p^end(g)(x_V\n(g)).

We now present our first circuit-based decomposition, and then we explain what its terms mean in detail:

Theorem 2. The total EF incurred by running an SR circuit where p is the initial distribution over the joint state of all nodes in the circuit is

$$\begin{equation}\mathcal{Q}\left(p\right)=\mathcal{L}\left(p\right)+\underset{\sigma \left(p\right)}{\underbrace{{\mathcal{L}}^{\mathrm{l}\mathrm{o}\mathrm{s}\mathrm{s}}\left(p\right)+\mathcal{M}\left(p\right)+\mathcal{R}\left(p\right)}}.\end{equation} \tag{ 44 }$$

(1) The first term in equation (44), $\mathcal{L}\left(p\right)$, is the Landauer cost of the circuit, as described in equation (37). This Landauer cost can be further decomposed into contributions from the individual gates. Specifically, write ${\mathcal{L}}_{g}\left(p\right)$ for the drop in the entropy of the entire circuit during the time that the solitary process for gate g runs, given that the input distribution over the entire circuit is p:

$$\begin{equation}{\mathcal{L}}_{g}\left(p\right){:=}S\left({p}^{\mathrm{b}\mathrm{e}\mathrm{g}\left(g\right)}\right)-S\left({p}^{\mathrm{e}\mathrm{n}\mathrm{d}\left(g\right)}\right).\end{equation} \tag{ 45 }$$

Note that the distribution over the states of the entire physical circuit at the end of the running of any gate is the same as the distribution at the beginning of the running of the next gate. So by canceling terms, and using the fact that entropy does not change when a wire gate runs, we can expand $\mathcal{L}$ as

$$\begin{equation}\mathcal{L}\left(p\right)=\sum _{g\in G{\backslash}W}{\mathcal{L}}_{g}\left(p\right).\end{equation} \tag{ 46 }$$

(Recall from section 2.5 that W is the set of wire gates in the circuit.) This decomposition will be useful below.

(2) The second term in equation (44), ${\mathcal{L}}^{\mathrm{l}\mathrm{o}\mathrm{s}\mathrm{s}}\left(p\right)$, is the unavoidable additional EF that is incurred by any SR implementation of the SR circuit on initial distribution p, above and beyond $\mathcal{L}$, the Landauer cost of running the map π_Φ on initial distribution p. We refer to this unavoidable extra EF as the circuit Landauer loss. It equals the sum of the subsystem Landauer losses incurred by each non-wire gate's solitary process,

$$\begin{equation}{\mathcal{L}}^{\mathrm{l}\mathrm{o}\mathrm{s}\mathrm{s}}\left(p\right)=\sum _{g\in G{\backslash}W}{\mathcal{L}}_{g}^{\mathrm{l}\mathrm{o}\mathrm{s}\mathrm{s}}\left(p\right),\end{equation} \tag{ 47 }$$

where ${\mathcal{L}}_{g}^{\mathrm{l}\mathrm{o}\mathrm{s}\mathrm{s}}\left(p\right)=S\left({p}_{\mathrm{p}\mathrm{a}\left(g\right)}\right)-S\left({\pi }_{g}{p}_{\mathrm{p}\mathrm{a}\left(g\right)}\right)-{\mathcal{L}}_{g}\left(p\right)$.

Each term ${\mathcal{L}}_{g}^{\mathrm{l}\mathrm{o}\mathrm{s}\mathrm{s}}\left(p\right)$ in this sum is non-negative (see end of section 4), and so ${\mathcal{L}}^{\mathrm{l}\mathrm{o}\mathrm{s}\mathrm{s}}\left(p\right){\geqslant}0$. Note that we can omit wires from the sum in equation (47) because π_g is logically reversible for any wire gate g, which means that ${\mathcal{L}}_{g}^{\mathrm{l}\mathrm{o}\mathrm{s}\mathrm{s}}\left(p\right)=0$ for such gates.

We define circuit Landauer cost to be the minimal EF incurred by running any SR implementation of the circuit, i.e.,

$$\begin{equation}{\mathcal{L}}^{\mathrm{c}\mathrm{i}\mathrm{r}\mathrm{c}}\left(p\right)=\mathcal{L}\left(p\right)+{\mathcal{L}}^{\mathrm{l}\mathrm{o}\mathrm{s}\mathrm{s}}\left(p\right)\end{equation} \tag{ 48 }$$

$$\begin{equation}=\sum _{g\in G{\backslash}W}\left[S\left({p}_{\mathrm{p}\mathrm{a}\left(g\right)}\right)-S\left({\pi }_{g}{p}_{\mathrm{p}\mathrm{a}\left(g\right)}\right)\right].\end{equation} \tag{ 49 }$$

Recall that $\mathcal{L}\left(p\right)$ is the minimal EF that must be generated by any physical process that carries out the map P on initial distribution p. So by equation (48), ${\mathcal{L}}^{\mathrm{l}\mathrm{o}\mathrm{s}\mathrm{s}}\left(p\right)$ is the minimal additional EF that must be generated if we use an SR circuit to carry out P on p, no matter how efficient the gates in the circuit are. In this sense, equation (48) can be viewed as an extension of the generalized Landauer bound, to concern SR circuits.

(3) The third term in equation (44), $\mathcal{M}$, reflects the EF incurred because the actual initial distribution of each gate g is not the optimal one for that gate (i.e., not one that minimizes subsystem EP within each island of the conditional distribution P_g, defined in equation (29)). We refer to this cost as the circuit mismatch cost, and write it as

$$\begin{equation}\;\;\mathcal{M}\left(p\right)\;=\;\sum _{g\in G}\left[D\left(\right.{p}_{n\left(g\right)}{\Vert}{q}_{n\left(g\right)}\left.\right)\;-\;D\left(\right.{P}_{g}{p}_{n\left(g\right)}{\Vert}{P}_{g}{q}_{n\left(g\right)}\left.\right)\right]\end{equation} \tag{ 50 }$$

where the prior q_n(g) is a distribution over ${\mathcal{X}}_{n\left(g\right)}$ whose conditional distributions over the islands c ∈ L(P_g) all obey ${\hat{\sigma }}_{g}\left({q}_{n\left(g\right)}^{c}\right)={\mathrm{min}}_{r:\mathrm{s}\mathrm{u}\mathrm{p}\mathrm{p}\;r\subseteq c}{\hat{\sigma }}_{g}\left(r\right)$. Note that we must include wire gates g in the sum in equation (50) even though π_g for a wire gate is logically reversible. This is because the associated overall map over n(g), equation (29), is not logically reversible over n(g)¹⁵.

$\mathcal{M}$ is non-negative, since each gate's subsystem mismatch cost is non-negative. Moreover, $\mathcal{M}$ achieves its minimum value of 0 when ${p}_{n\left(g\right)}^{{c}_{g}}={q}_{n\left(g\right)}^{{c}_{g}}$ for all g ∈ G and all islands c_g ∈ L(P_g). (Recall that subsystem priors like ${q}_{n\left(g\right)}^{{c}_{g}}$ reflect the specific details of the underlying physical process that implements the gate g, such as how its energy spectrum evolves as it runs.)

Suppose that one wishes to construct a physical system to implement some circuit, and can vary the associated subsystem priors ${q}_{n\left(g\right)}^{{c}_{g}}$ arbitrarily. Then in order to minimize mismatch cost one should choose priors ${q}_{n\left(g\right)}^{{c}_{g}}$ that equal the actual associated initial distributions ${p}_{n\left(g\right)}^{c}$. Moreover, those actual initial distributions ${p}_{n\left(g\right)}^{{c}_{g}}$ can be calculated from the circuit's wiring diagram, together with the input distribution of the entire circuit, p_IN, by 'propagating' p_IN through the transformations specified by the wiring diagram. As a result, given knowledge of the wiring diagram and the input distribution of the entire circuit, in principle the priors can be set so that mismatch cost vanishes.

(4) The fourth term in equation (44), $\mathcal{R}$, reflects the remaining EF incurred by running the SR circuit, and so we call it circuit residual EP. Concretely, it equals the subsystem EP that would be incurred even if the initial distribution within each island of each gate were optimal:

$$\begin{equation}\mathcal{R}\left(p\right)=\sum _{g\in G}\sum _{c\in L\left({P}_{g}\right)}{p}_{n\left(g\right)}\left(c\right)\;{\hat{\sigma }}_{g}\left({q}_{n\left(g\right)}^{c}\right).\end{equation} \tag{ 51 }$$

Circuit residual EP is non-negative, since each ${\hat{\sigma }}_{g}$ is non-negative. Since for every gate g, p_n(g)(c) is a linear function of the initial distribution to the circuit as a whole, circuit residual EP also depends linearly on the initial distribution. Like the priors of the gates, the residual EP terms $\left\{{\hat{\sigma }}_{g}\left({q}_{n\left(g\right)}^{c}\right)\right\}$ reflect the 'nitty-gritty' details of how the gates run.

To summarize, the EF incurred by a circuit can be decomposed into the Landauer cost (the contribution to the EF that would arise even in a thermodynamically reversible process) plus the EP (the contribution to that EF which is thermodynamically irreversible). In turn, there are three contributions to that EP:

• (a)  
  Circuit Landauer loss, which is independent of how the circuit is physically implemented, but does depend on the wiring diagram of the circuit, the conditional distributions implemented by the gates, and the initial distribution over inputs. It is a nonlinear function of the distribution over inputs, p_IN.
• (b)  
  Circuit mismatch cost, which does depend on how the circuit is physical implemented (via the priors), as well as the wiring diagram. It is also a nonlinear function of p_IN.
• (c)  
  Circuit residual EP, which also depends on how the circuit is physical implemented. It is a linear function of p_IN. However, no matter what the wiring diagram of the circuit is, if we implement each of the gates in a circuit with a quasistatic process, then the associated circuit residual EP is identically zero, independent of p_IN¹⁶.

There are other useful decompositions of the EP incurred by an SR circuit that incorporate the wiring diagram. One such alternative decomposition, which is our second main result, leaves the circuit Landauer loss term in equation (44) unchanged, but modifies the circuit mismatch cost and the circuit residual EP terms.

Theorem 3. The total EF incurred by running an SR circuit where p is the initial distribution over the joint state of all nodes in the circuit is

$$\begin{equation}\mathcal{Q}\left(p\right)=\mathcal{L}\left(p\right)+\underset{\sigma \left(p\right)}{\underbrace{{\mathcal{L}}^{\mathrm{l}\mathrm{o}\mathrm{s}\mathrm{s}}\left(p\right)+{\mathcal{M}}^{\prime }\left(p\right)+{\mathcal{R}}^{\prime }\left(p\right)}}.\end{equation} \tag{ 52 }$$

To present this decomposition, recall from equation (29) that for any gate g, the distribution over ${\mathcal{X}}_{n\left(g\right)}$ has partial support at the beginning of the solitary process that implements P_g, since there is 0 probability that x_g ≠ ∅. We use this fact to apply theorem 1 to equation (18), while taking $\mathcal{Z}=\left\{{x}_{n\left(g\right)}\in {\mathcal{X}}_{n\left(g\right)}:{x}_{g}=\varnothing \right\}$. This allows us to express the modified circuit mismatch cost by replacing all of the map P_g(x'_n(g)|x_n(g)) in the summand in equation (50) with P_g(x'_g|x_pa(g)) = π_g:

$$\begin{equation}{\mathcal{M}}^{\prime }\left(p\right)=\sum _{g\in G{\backslash}W}\left[D\left({p}_{\mathrm{p}\mathrm{a}\left(g\right)}{\Vert}{q}_{\mathrm{p}\mathrm{a}\left(g\right)}\right)-D\left({\pi }_{g}{p}_{\mathrm{p}\mathrm{a}\left(g\right)}{\Vert}{\pi }_{g}{q}_{\mathrm{p}\mathrm{a}\left(g\right)}\right)\right]\end{equation} \tag{ 53 }$$

where the priors q_pa(g) are defined in terms of the island decompositions of the associated conditional distributions π_g, rather than in terms of the island decompositions of the conditional distributions P_g. Note that can exclude the wire gates from the sum in equation (53) because each wire gate's π_g is logically reversible, and so the associated drop in KL divergence are zero. Then, the modified circuit residual EP is

$$\begin{equation}{\mathcal{R}}^{\prime }\left(p\right)=\sum _{g\in G}\sum _{c\in L\left({\pi }_{g}\right)}{p}_{\mathrm{p}\mathrm{a}\left(g\right)}\;{\hat{\sigma }}_{g}\left({q}_{\mathrm{p}\mathrm{a}\left(g\right)}^{c}\right),\end{equation} \tag{ 54 }$$

where each term ${\hat{\sigma }}_{g}\left({q}_{\mathrm{p}\mathrm{a}\left(g\right)}^{c}\right)$ is given by appropriately modifying the arguments in equation (43). In deriving equation (54), we used the fact that ${L}_{\mathcal{Z}}\left({P}_{g}\right)=L\left({\pi }_{g}\right)$ (for $\mathcal{Z}$ as defined above for each gate g).

As with the analogous results in the previous section, theorems 2 and 3 differ, because they define 'optimal initial distribution' relative to different sets of possibilities. In particular, the decomposition in theorem 2 will generally have a larger mismatch cost and smaller residual EP term than the decomposition in theorem 3.

For the rest of this section, we will use the term 'circuit mismatch cost' to refer to the expression in equation (53) rather than the expression in equation (50), and similarly will use the term 'circuit residual EP' to refer to the expression in equation (54) rather than the expression in equation (51).

By combining equations (47) and (26), we can write circuit Landauer loss as

$$\begin{equation}{\mathcal{L}}^{\mathrm{l}\mathrm{o}\mathrm{s}\mathrm{s}}\left(p\right)=\sum _{g\in G{\backslash}W}\left[I\left({X}_{\mathrm{p}\mathrm{a}\left(g\right)};{X}_{V{\backslash}\mathrm{p}\mathrm{a}\left(g\right)}\right)-I\left({X}_{g};{X}_{V{\backslash}g}\right)\right]\end{equation} \tag{ 55 }$$

Any nodes that belong to V\n(g) and that are in their initialized state when gate g starts to run will not contribute to the drop in mutual information terms in equation (55). Keeping track of such nodes and simplifying establishes the following:

Corollary 4. The circuit Landauer loss is

$$\begin{equation*}{\mathcal{L}}^{\mathrm{l}\mathrm{o}\mathrm{s}\mathrm{s}}\left(p\right)=\mathcal{I}\left({p}_{\mathrm{I}\mathrm{N}}\right)-\mathcal{I}\left({\pi }_{{\Phi}}{p}_{\mathrm{I}\mathrm{N}}\right)-\sum _{g\in G{\backslash}W}\mathcal{I}\left({p}_{\mathrm{p}\mathrm{a}\left(g\right)}\right).\end{equation*}$$

(We remind the reader that $\mathcal{I}\left({p}_{A}\right)$ refers to the multi-information between the variables indexed by A.) Corollary 4 suggest a set of novel optimization problems for how to design SR circuits: given some desired computation π_Φ and some given initial distribution p_IN, find the circuit wiring diagram that carries out π_Φ while minimizing the circuit Landauer loss. Presuming we have a fixed input distribution p and map π_Φ, the term $\mathcal{I}\left({p}_{\mathrm{I}\mathrm{N}}\right)-\mathcal{I}\left({\pi }_{{\Phi}}{p}_{\mathrm{I}\mathrm{N}}\right)$ in corollary 4 is an additive constant that does not depend on the particular choice of circuit wiring diagram. So the optimization problem can be reduced to finding which wiring diagram results in a minimal value of ${\sum }_{g\in G{\backslash}W}\mathcal{I}\left({p}_{\mathrm{p}\mathrm{a}\left(g\right)}\right)$. In other words, for fixed p and map π_Φ, to minimize the Landauer loss we should choose the wiring diagram for which the parents of each gate are as strongly correlated among themselves as possible. Intuitively, this ensures that the 'loss of correlations', as information is propagated down the circuit, is as small as possible.

In general, the distributions over the outputs of the gates in any particular layer of the circuit will affect the distribution of the inputs of all of the downstream gates, in the subsequent layers of the circuit. This means that the sum of multi-informations in corollary 4 is an inherently global property of the wiring diagram of a circuit; it cannot be reduced to a sum of properties of each gate considered in isolation, independently of the other gates. This makes the optimization problem particularly challenging. We illustrate this optimization problem in the following example.

Example 6. Consider again the case where we want our circuit to compute the 3-bit parity function using 2-input XOR gates, i.e., it want it to implement the map

$$\begin{equation*}{\pi }_{{\Phi}}\left({x}_{g}\vert {x}_{1},{x}_{2},{x}_{3}\right)=\delta \left({x}_{g},\delta \left({x}_{1}+{x}_{2}+{x}_{3},1\right)+\delta \left({x}_{1}+{x}_{2}+{x}_{3},3\right)\right).\end{equation*}$$

Suppose we happen to know that the input distribution to the circuit will be

$$\begin{equation}{p}_{\mathrm{I}\mathrm{N}}\left({x}_{1},{x}_{2},{x}_{3}\right)=\frac{1}{Z}{\mathrm{e}}^{-\left[\phi \left({x}_{1}\right)\phi \left({x}_{2}\right)/4+\phi \left({x}_{1}\right)\phi \left({x}_{3}\right)/2+\phi \left({x}_{2}\right)\phi \left({x}_{3}\right)\right]},\end{equation} \tag{ 56 }$$

where Z is a normalization constant and ϕ(x) := 2x − 1 is a function that maps bit values x ∈ {0, 1} to spin values, ϕ(x) ∈ {−1, 1}. The distribution equation (56) is a Boltzmann distribution for a pairwise spin model, where inputs 2 and 3 have the strongest coupling strength (1), inputs 1 and 3 have intermediate-strength coupling strength (1/2), and inputs 1 and 2 have the weakest coupling strength (1/4).

We wish to find the wiring diagram connecting our XOR gates that has minimal circuit Landauer cost for this distribution over its three input bits. It turns out that we can restrict our search to three possible wiring diagrams, which are shown in figure 5. We indicate the circuit Landauer loss for the input distribution of equation (56) for each of those three wiring diagrams. So for this input distribution, the right-most wiring diagram results in minimal circuit Landauer loss. Note that this wiring diagram aligns with the correlational structure of the input distribution (given that inputs 2 and 3 have the strongest statistical correlation).

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An interesting variant of the optimization problem described above arises if we model the residual EP terms for the wire gates. In any SR circuit, wire gates carry out a logically reversible operation on their inputs. Thus, by equation (16), all of the EF generated by any wire gates is residual EP. If we allow the physical lengths of wires to vary, then as a simple model we could presume that the residual EP of any wire is proportional to its length. This would allow us to incorporate into our analysis the thermodynamic effect of the geometry with which a circuit is laid out on a two-dimensional circuit boards, in addition to the thermodynamic effect of the topology of that circuit.

Finally, note that for any set of nodes A, multi-information can be bounded as

$$\begin{equation}0{\leqslant}\mathcal{I}\left({p}_{A}\right){\leqslant}{\sum }_{v\in A}^{}S\left({p}_{v}\right){\leqslant}{\sum }_{v\in A}\vert {\mathcal{X}}_{v}\vert .\end{equation}$$

Given this, corollary 4 implies

$$\begin{equation}{\mathcal{L}}^{\mathrm{l}\mathrm{o}\mathrm{s}\mathrm{s}}\left(p\right){\leqslant}\mathcal{I}\left({p}_{\mathrm{I}\mathrm{N}}\right){\leqslant}\mathrm{ln}\vert {\mathcal{X}}_{\mathrm{I}\mathrm{N}}\vert .\end{equation} \tag{ 57 }$$

This means that for a fixed input state space, the circuit Landauer loss cannot grow without bound as we vary the wiring diagram. Interestingly, this bound on Landauer loss only holds for SR circuits that have out-degree 1. If we consider SR circuits that have out-degree greater than 1, then the circuit Landauer cost can be arbitrarily large. This is formalized as the following proposition, which is proved in appendix C.

Proposition 1. For any π_Φ, non-delta function input distribution p_IN, and κ ⩾ 0, there exist an SR circuit with out-degree greater than 1 that implements π_Φ for which ${\mathcal{L}}^{\mathrm{l}\mathrm{o}\mathrm{s}\mathrm{s}}\left(p\right){\geqslant}\kappa$.

Landauer loss captures the gain in minimal EF due to using an SR circuit, which becomes equal to the gain in actual EF if there is no mismatch cost or residual EP. It is harder to make general statements about the gain in actual EF due to using an SR circuit, i.e., when the mismatch cost and/or residual EP is nonzero. In this subsection we make some preliminary remarks about this issue.

Imagine that we wish to build a physical process that implements some computation π_Φ(x_OUT|x_IN) over a space ${\mathcal{X}}_{\mathrm{I}\mathrm{N}}{\times}{\mathcal{X}}_{\mathrm{O}\mathrm{U}\mathrm{T}}$. Suppose we want this process to achieve minimal EP when run with inputs generated by q_IN (e.g., if we expect future inputs to the process to be generated by sampling q_IN), and as usual assume the initial value of x_OUT will be ∅ whenever it is run. Using the decomposition of equation (42) and assuming that the residual EP of the process is zero, the EF that such a process would generate if it is actually run with an input distribution p (initialized like SR circuits are, so that it has the form of equation (39)) is given by the sum of the Landauer cost and the mismatch cost, with no Landauer loss term. We write this as

$$\begin{equation}{\mathcal{Q}}_{\mathrm{A}\mathrm{O}}\left(p\right)=\mathcal{L}\left(p\right)+D\left({p}_{\mathrm{I}\mathrm{N}}{\Vert}{q}_{\mathrm{I}\mathrm{N}}\right)-D\left({\pi }_{{\Phi}}{p}_{\mathrm{I}\mathrm{N}}{\Vert}{\pi }_{{\Phi}}{q}_{\mathrm{I}\mathrm{N}}\right),\end{equation} \tag{ 58 }$$

Note that in order for the EF generated by an actual physical process to be given by equation (58), the prior of that process must be q_IN, and in general this may require that the process couple together arbitrary sets of variables. This means that the EF generated by implementing π_Φ(x_OUT|x_IN) with an SR circuit cannot obey equation (58) in general, due to restrictions on what variables can be coupled in such a circuit. (One can verify, for example, that the prior distribution q_IN of a circuit consisting of two disconnected bit erasing gates must be a product distribution over the two input bits.) To emphasize this distinction, we will refer to a process whose EF is given by equation (58)) as an 'all-at-once' (AO) process (indicated by the subscript 'AO' in equation (58)).

For practical reasons, it may be quite difficult to construct an AO process that implements π_Φ, and we must use a circuit implementation instead. In particular, even though the circuit as a whole cannot have prior q_IN, suppose we can set the priors q_pa(g) at its gates by propagating q_IN through the wiring diagram of the circuit. Assuming again that there is zero EP, the EF that must be incurred by any such SR circuit implementation of π_Φ on input distribution p, assuming some particular wiring topology and gate priors, is given by the decomposition of theorem 3,

$$\begin{equation}{\mathcal{Q}}_{\mathrm{c}\mathrm{i}\mathrm{r}\mathrm{c}}\left(p\right)=\mathcal{L}\left(p\right)+{\mathcal{L}}^{\mathrm{l}\mathrm{o}\mathrm{s}\mathrm{s}}\left(p\right)+{\mathcal{M}}^{\prime }\left(p\right).\end{equation} \tag{ 59 }$$

We now ask: how much larger is this EF incurred by the SR circuit implementation, compared to that of the original AO process? Subtracting equation (58) from equation (59) gives

$$\begin{align}\hfill {\Delta}\mathcal{Q}& ={\mathcal{Q}}_{\mathrm{c}\mathrm{i}\mathrm{r}\mathrm{c}}\left({p}_{\mathrm{I}\mathrm{N}}\right)-{\mathcal{Q}}_{\mathrm{A}\mathrm{O}}\left({p}_{\mathrm{I}\mathrm{N}}\right)\hfill \\ \hfill & ={\mathcal{L}}^{\mathrm{l}\mathrm{o}\mathrm{s}\mathrm{s}}\left({p}_{\mathrm{I}\mathrm{N}}\right)+{\mathcal{M}}^{\mathrm{l}\mathrm{o}\mathrm{s}\mathrm{s}}\left({p}_{\mathrm{I}\mathrm{N}}{\Vert}{q}_{\mathrm{I}\mathrm{N}}\right)\hfill \end{align} \tag{ 60 }$$

where we have defined ${\mathcal{M}}^{\mathrm{l}\mathrm{o}\mathrm{s}\mathrm{s}}$ as the difference between the circuit mismatch cost, ${\mathcal{M}}^{\prime }\left(p\right)$, and the mismatch cost of the AO process. We refer to that difference in mismatch costs as the circuit mismatch loss, and use equation (53) to express it as

$$\begin{align}\hfill {\mathcal{M}}^{\mathrm{l}\mathrm{o}\mathrm{s}\mathrm{s}}\left({p}_{\mathrm{I}\mathrm{N}}{\Vert}{q}_{\mathrm{I}\mathrm{N}}\right)& =\mathcal{D}\left({\pi }_{{\Phi}}{p}_{\mathrm{I}\mathrm{N}}{\Vert}{\pi }_{{\Phi}}{q}_{\mathrm{I}\mathrm{N}}\right)-\mathcal{D}\left({p}_{\mathrm{I}\mathrm{N}}{\Vert}{q}_{\mathrm{I}\mathrm{N}}\right)+\sum _{g\in G{\backslash}W}\mathcal{D}\left({p}_{\mathrm{p}\mathrm{a}\left(g\right)}{\Vert}{q}_{\mathrm{p}\mathrm{a}\left(g\right)}\right),\hfill \end{align} \tag{ 61 }$$

where $\mathcal{D}$ refers to the multi-divergence, defined in equation (10). Equation (61) can be compared to corollary 4, which expresses the circuit Landauer cost rather than circuit mismatch cost, and involves multi-informations rather than multi-divergences.

Interestingly, while circuit Landauer loss is non-negative, circuit mismatch loss can either be positive or negative. In fact, depending on the wiring diagram, p_IN and q_IN, the sum of circuit mismatch loss and circuit Landauer loss can be negative. This means that when the actual input distribution p_IN is different from the prior distribution of the AO process, the 'closest equivalent circuit' to the AO process may actually incur less EF than the corresponding AO process. This occurs because an SR circuit cannot implement some of the prior distributions that an AO process can implement, so the two implementations end up having different priors. This is illustrated in the following example.

Example 7. Assume the desired computation is the erasure of two bits, π_Φ(x₃, x₄|x₁, x₂) = δ(x₃, 0)δ(x₄, 0), where x₁ and x₂ refers to the input bits, and x₃ and x₄ refers to the output bits. The prior distribution implemented by an AO process is given by q_IN(0, 0) = q(1, 1) = < 1/2, and q_IN(0, 1) = q(1, 0) = 1/2 − . The actual input distribution is given by a delta function distribution, p_IN(x₁, x₂) = δ(x₁, 0)δ(x₂, 0).

We now implement this computation using an SR circuit which consists of two disconnected erasure gates. The closest equivalent SR circuit has gate priors given by the uniform marginal distributions, q(x₁) = 1/2 and q(x₂) = 1/2. Then the difference between the EF of the AO process and the SR circuit is

$$\begin{align*}\hfill {\Delta}\mathcal{Q}=& -\left[S\left({p}_{\mathrm{I}\mathrm{N}}\right)\;+\;D\left({p}_{\mathrm{I}\mathrm{N}}{\Vert}{q}_{\mathrm{I}\mathrm{N}}\right)\right]+\left[\sum _{g}S\left({p}_{\mathrm{p}\mathrm{a}\left(g\right)}\right)+D\left({p}_{\mathrm{p}\mathrm{a}\left(g\right)}{\Vert}{q}_{\mathrm{p}\mathrm{a}\left(g\right)}\right)\right]\hfill \\ \hfill & =\sum _{{x}_{1},{x}_{2}}{p}_{\mathrm{I}\mathrm{N}}\left({x}_{1},{x}_{2}\right)\mathrm{ln}\;{q}_{\mathrm{I}\mathrm{N}}\left({x}_{1},{x}_{2}\right)-\sum _{{x}_{1}}p\left({x}_{1}\right)\mathrm{ln}\;q\left({x}_{1}\right)-\sum _{{x}_{2}}p\left({x}_{2}\right)\mathrm{ln}\;q\left({x}_{2}\right)\hfill \\ \hfill & =\mathrm{ln}\;{\epsilon}+2\mathrm{ln}\;2\hfill \end{align*}$$

This can be made arbitrarily negative by taking sufficiently close to zero. Thus, the EF of the AO process may be arbitrarily larger than the EF of the closest equivalent SR circuit.

Consider a conditional distribution P(y|x) that specifies the probability of 'output' $y\in \mathcal{Y}$ given 'input' $x\in \mathcal{X}$, where $\mathcal{X}$ and $\mathcal{Y}$ are finite.

Given some $\mathcal{Z}\subseteq \mathcal{X}$, the island decomposition ${L}_{\mathcal{Z}}\left(P\right)$ of P, and any $p\in {{\Delta}}_{\mathcal{X}}$, let p(c) = ∑_x∈cp(x) indicate the total probability within island c, and

$$\begin{equation*}{p}^{c}\left(x\right){:=}\begin{cases}\frac{p\left(x\right)}{p\left(c\right)}\quad \hfill & \mathrm{i}\mathrm{f}\;x\in c\;\mathrm{a}\mathrm{n}\mathrm{d}\;p\left(c\right){ >}0\hfill \\ 0\quad \hfill & \text{otherwise}\hfill \end{cases}\end{equation*}$$

indicate the conditional probability of state x within island c.

In our proofs below, we will make use of the notion of relative interior. Given a linear space V, the relative interior of a subset A ⊆ V is defined as [68]

$$\begin{equation*}\mathrm{r}\mathrm{e}\mathrm{l}\mathrm{i}\mathrm{n}\mathrm{t}\;A{:=}\left\{x\in A:\forall y\in A,\exists {\epsilon}{ >}0\;\text{s.t.}\;x+{\epsilon}\left(x-y\right)\in A\right\}.\end{equation*}$$

Finally, for any g, we use notation

$$\begin{equation*}{\partial }_{x}^{+}g{\vert }_{x=a}{:=}{\mathrm{lim}}_{{\epsilon}\to {0}^{+}}\frac{1}{{\epsilon}}\left(g\left(a+{\epsilon}\right)-g\left(a\right)\right)\end{equation*}$$

to indicate the right-handed derivative of g(x) at x = a.

Given some conditional distribution P(y|x) and function $f:\mathcal{X}\to \mathbb{R}$, we consider the function ${\Gamma}:{{\Delta}}_{\mathcal{X}}\to \mathbb{R}$ as

$$\begin{equation*}{\Gamma}\left(p\right){:=}S\left(Pp\right)-S\left(p\right)+{\mathbb{E}}_{p}\left[f\right].\end{equation*}$$

Note that Γ is continuous on the relative interior of ${{\Delta}}_{\mathcal{X}}$.

Lemma A1. For any $a,b\in {{\Delta}}_{\mathcal{X}}$, the directional derivative of Γ at a toward b is given by

$$\begin{equation*}{\partial }_{{\epsilon}}^{+}{\Gamma}\left(a+{\epsilon}\left(b-a\right)\right){\vert }_{{\epsilon}=0}=D\left(Pb{\Vert}Pa\right)-D\left(b{\Vert}a\right)+{\Gamma}\left(b\right)-{\Gamma}\left(a\right).\end{equation*}$$

Proof. Let a := a + (b − a). Using the definition of Γ, write

$$\begin{equation}{\partial }_{{\epsilon}}^{+}{\Gamma}\left({a}^{{\epsilon}}\right)={\partial }_{{\epsilon}}^{+}\left[S\left(P{a}^{{\epsilon}}\right)-S\left({a}^{{\epsilon}}\right)\right]+{\partial }_{{\epsilon}}^{+}{\mathbb{E}}_{{a}^{{\epsilon}}}\left[f\right].\end{equation} \tag{ A1 }$$

Then, consider the first term on the right-hand side,

$$\begin{align*}\hfill {\partial }_{{\epsilon}}^{+}\left[S\left(P{a}^{{\epsilon}}\right)-S\left({a}^{{\epsilon}}\right)\right]& =-\sum _{y\in \mathcal{Y}}\left[{\partial }_{{\epsilon}}^{+}{a}^{{\epsilon}}\left(y\right)\mathrm{ln}\;{a}^{{\epsilon}}\left(y\right)+{\partial }_{{\epsilon}}^{+}\left[P{a}^{{\epsilon}}\right]\left(y\right)\right]+\sum _{x\in \mathcal{X}}\left[{\partial }_{{\epsilon}}^{+}{a}^{{\epsilon}}\left(x\right)\mathrm{ln}\;{a}^{{\epsilon}}\left(x\right)+{\partial }_{{\epsilon}}^{+}{a}^{{\epsilon}}\left(x\right)\right]\hfill \\ \hfill & =-\sum _{y\in \mathcal{Y}}\left(b\left(y\right)\;-\;a\left(y\right)\right)\mathrm{ln}\;{a}^{{\epsilon}}\left(y\right)+\sum _{x\in \mathcal{X}}\left(b\left(x\right)\;-\;a\left(x\right)\right)\mathrm{ln}\;{a}^{{\epsilon}}\left(x\right)\hfill \end{align*}$$

Evaluated at = 0, the last line can be written as

$$\begin{align*}\hfill & -\sum _{y\in \mathcal{Y}}\left(b\left(y\right)\;-\;a\left(y\right)\right)\mathrm{ln}\;a\left(y\right)+\sum _{x\in \mathcal{X}}\left(b\left(x\right)-a\left(x\right)\right)\mathrm{ln}\;a\left(x\right)\hfill \\ \hfill & =D\left(Pb{\Vert}Pa\right)+S\left(Pb\right)-S\left(Pa\right)-D\left(b{\Vert}a\right)-S\left(b\right)+S\left(a\right)\hfill \end{align*}$$

We next consider the ${\partial }_{{\epsilon}}^{+}{\mathbb{E}}_{{a}^{{\epsilon}}}\left[f\right]$ term,

$$\begin{align*}\hfill {\partial }_{{\epsilon}}^{+}{\mathbb{E}}_{{a}^{{\epsilon}}}\left[f\right]& ={\partial }_{{\epsilon}}^{+}\left[\sum _{x\in \mathcal{X}}\left(a\left(x\right)+{\epsilon}\left(b\left(x\right)-a\left(x\right)\right)\right)f\left(x\right)\right]\hfill \\ \hfill & ={\mathbb{E}}_{b}\left[f\right]-{\mathbb{E}}_{a}\left[f\right].\hfill \end{align*}$$

Combining the above gives

$$\begin{align*}\hfill {\partial }_{{\epsilon}}^{+}{\Gamma}\left({a}^{{\epsilon}}\right){\vert }_{{\epsilon}=0}& =D\left(Pb{\Vert}Pa\right)-D\left(b{\Vert}a\right)+S\left(Pb\right)-S\left(b\right)-\left(S\left(Pa\right)-S\left(a\right)\right)+{\mathbb{E}}_{b}\left[f\right]-{\mathbb{E}}_{a}\left[f\right]\hfill \\ \hfill & =D\left(Pb{\Vert}Pa\right)-D\left(b{\Vert}a\right)+{\Gamma}\left(b\right)-{\Gamma}\left(a\right).\hfill \end{align*}$$

□

Theorem A1. Let V be a convex subset of Δ. Then for any q ∈ argmin_{s ∈ V}Γ(s) and any p ∈ V,

$$\begin{equation}{\Gamma}\left(p\right)-{\Gamma}\left(q\right){\geqslant}D\left(p{\Vert}q\right)-D\left(Pp{\Vert}Pq\right).\end{equation} \tag{ A2 }$$

Equality holds if q is in the relative interior of V.

Proof. Define the convex mixture q := q + (p − q). By lemma A1, the directional derivative of Γ at q in the direction p − q is

$$\begin{equation*}{\partial }_{{\epsilon}}^{+}{\Gamma}\left({q}^{{\epsilon}}\right){\vert }_{{\epsilon}=0}=D\left(Pp{\Vert}Pq\right)-D\left(p{\Vert}q\right)+{\Gamma}\left(p\right)-{\Gamma}\left(q\right).\end{equation*}$$

At the same time, ${\partial }_{{\epsilon}}^{+}{\Gamma}\left({q}^{{\epsilon}}\right){\vert }_{{\epsilon}=0}{\geqslant}0$, since q is a minimizer within a convex set. equation (A2) then follows by rearranging.

When q is in the relative interior of V, q − (p − q) ∈ V for sufficiently small > 0. Then,

$$\begin{align*}\hfill & 0{\leqslant}\underset{{\epsilon}\to {0}^{+}}{\mathrm{lim}}\frac{1}{{\epsilon}}\left({\Gamma}\left(q-{\epsilon}\left(p-q\right)\right)-{\Gamma}\left(q\right)\right)\hfill \\ \hfill & =-\underset{{\epsilon}\to {0}^{-}}{\mathrm{lim}}\frac{1}{{\epsilon}}\left({\Gamma}\left(q+{\epsilon}\left(p-q\right)\right)-{\Gamma}\left(q\right)\right)\hfill \\ \hfill & =-\underset{{\epsilon}\to {0}^{+}}{\mathrm{lim}}\frac{1}{{\epsilon}}\left({\Gamma}\left(q+{\epsilon}\left(p-q\right)\right)-{\Gamma}\left(q\right)\right)\hfill \\ \hfill & =-{\partial }_{{\epsilon}}^{+}{\Gamma}\left({q}^{{\epsilon}}\right){\vert }_{{\epsilon}=0}.\hfill \end{align*}$$

where in the first inequality comes from the fact that q is a minimizer, in the second line we change variables as ↦ −, and the third line we use the continuity of Γ on interior of the simplex. Combining with the above implies

$$\begin{equation*}{\partial }_{{\epsilon}}^{+}{\Gamma}\left({q}^{{\epsilon}}\right)=D\left(Pp{\Vert}Pq\right)-D\left(p{\Vert}q\right)+{\Gamma}\left(p\right)-{\Gamma}\left(q\right)=0.\end{equation*}$$

□

Lemma A2. For any c ∈ L(P) and q ∈ $\underset{s:\mathrm{supp}s\subseteq c}{\mathrm{argmin}}$Γ(s),

$$\begin{equation*}\mathrm{s}\mathrm{u}\mathrm{p}\mathrm{p}\;q=\left\{x\in c:f\left(x\right){< }\infty \right\}.\end{equation*}$$

Proof. We prove the claim by contradiction. Assume that q is a minimizer with supp q ⊂ {x ∈ c: f(x) < ∞}. Note there cannot be any x ∈ supp q and $y\in \mathcal{Y}{\backslash}\mathrm{s}\mathrm{u}\mathrm{p}\mathrm{p}\;Pq$ such that P(y|x) > 0 (if there were such an x, y, then q(y) = ∑_x'P(y|x')q(x') ⩾ P(y|x)q(x) > 0, contradicting the statement that $y\in \mathcal{Y}{\backslash}\mathrm{s}\mathrm{u}\mathrm{p}\mathrm{p}\;Pq$). Thus, by definition of islands, there must be an ∈ c\supp q, ∈ supp Pq such that f() < ∞ and P(|) > 0.

Define the delta-function distribution u(x) := δ(x, ) and the convex mixture q(x) = (1 − )q(x) + u(x) for ∈ [0, 1]. We will also use the notation q(y) = ∑_xP(y|x)q(x).

Since q is a minimizer of Γ, ∂Γ(q)| _{= 0} ⩾ 0. Since Γ is convex, the second derivative ${\partial }_{{\epsilon}}^{2}{\Gamma}\left({q}^{{\epsilon}}\right){\geqslant}0$ and therefore ∂Γ(q) ⩾ 0 for all ⩾ 0. Taking a = q and b = u in lemma A1 and rearranging, we then have

$$\begin{align}\hfill {\Gamma}\left(u\right)& {\geqslant}D\left(u{\Vert}{q}^{{\epsilon}}\right)-D\left(Pu{\Vert}P{q}^{{\epsilon}}\right)+{\Gamma}\left({q}^{{\epsilon}}\right)\hfill \\ \hfill & {\geqslant}D\left(u{\Vert}{q}^{{\epsilon}}\right)-D\left(Pu{\Vert}P{q}^{{\epsilon}}\right)+{\Gamma}\left(q\right),\hfill \end{align} \tag{ A3 }$$

where the second inequality uses that q is a minimizer of Γ. At the same time,

$$\begin{align}\hfill D\left(u{\Vert}{q}^{{\epsilon}}\right)-D\left(Pu{\Vert}P{q}^{{\epsilon}}\right)& =\sum _{y}P\left(y\vert \hat{x}\right)\mathrm{ln}\frac{{q}^{{\epsilon}}\left(y\right)}{{q}^{{\epsilon}}\left(\hat{x}\right)P\left(y\vert \hat{x}\right)}\hfill \\ \hfill & =P\left(\hat{y}\vert \hat{x}\right)\mathrm{ln}\frac{{q}^{{\epsilon}}\left(\hat{y}\right)}{{\epsilon}P\left(\hat{y}\vert \hat{x}\right)}+\sum _{y\ne \hat{j}}P\left(y\vert \hat{i}\right)\mathrm{ln}\frac{{q}^{{\epsilon}}\left(y\right)}{{\epsilon}P\left(y\vert \hat{x}\right)}\hfill \\ \hfill & {\geqslant}P\left(\hat{y}\vert \hat{x}\right)\mathrm{ln}\frac{\left(1-{\epsilon}\right)q\left(\hat{y}\right)}{{\epsilon}P\left(\hat{y}\vert \hat{x}\right)}+\sum _{y\ne \hat{j}}P\left(y\vert \hat{i}\right)\mathrm{ln}\frac{{\epsilon}P\left(y\vert \hat{x}\right)}{{\epsilon}P\left(y\vert \hat{x}\right)}\hfill \\ \hfill & =P\left(\hat{y}\vert \hat{x}\right)\mathrm{ln}\frac{\left(1-{\epsilon}\right)}{{\epsilon}}\frac{q\left(\hat{y}\right)}{P\left(\hat{y}\vert \hat{x}\right)},\hfill \end{align} \tag{ A4 }$$

where in the secondline we have used that q() = , and in the third that q(y) = (1 − )q(y) + P(y|), so q(y) ⩾ (1 − )q(y) and q(y) ⩾ P(y|).

Note that the right-hand side of equation (A4) goes to ∞ as → 0. Combined with equation (A3) and that Γ(q) is finite implies that Γ(u) = ∞. However, ${\Gamma}\left(u\right)=S\left(P\left(Y\vert \hat{x}\right)\right)+f\left(\hat{x}\right){\leqslant}\vert \mathcal{Y}\vert +f\left(\hat{x}\right)$, which is finite. We thus have a contradiction, so q cannot be the minimizer. □

Lemma A3. For any island c ∈ L(P), q ∈ $\underset{s:\mathrm{supp}s\subseteq c}{\mathrm{argmin}}$Γ(p) is unique.

Proof. Consider any two distributions p, q ∈ $\underset{s:\mathrm{supp}s\subseteq c}{\mathrm{argmin}}$Γ(s), and let P' = Pp, q' = Pq. We will prove that p = q.

First, note that by lemma A2, supp q = supp p = c. By theorem A1,

$$\begin{align*}\hfill {\Gamma}\left(p\right)-{\Gamma}\left(q\right)& =D\left(p{\Vert}q\right)-D\left({p}^{\prime }{\Vert}{q}^{\prime }\right)\hfill \\ \hfill & =\sum _{x,y}p\left(x\right)P\left(y\vert x\right)\mathrm{ln}\frac{p\left(x\right){q}^{\prime }\left(y\right)}{q\left(x\right){p}^{\prime }\left(y\right)}\hfill \\ \hfill & =\sum _{x,y}p\left(x\right)P\left(y\vert x\right)\mathrm{ln}\frac{p\left(x\right)P\left(y\vert x\right)}{q\left(x\right){p}^{\prime }\left(y\right)P\left(y\vert x\right)/{q}^{\prime }\left(y\right)}\hfill \\ \hfill & {\geqslant}0\hfill \end{align*}$$

where the last line uses the log-sum inequality. If the inequality is strict, then p and q cannot both be minimizers, i.e., the minimizer must be unique, as claimed.

If instead the inequality is not strict, i.e., Γ(p) − Γ(q) = 0, then there is some constant α such that for all x, y with P(y|x) > 0,

$$\begin{equation}\frac{p\left(x\right)P\left(y\vert x\right)}{q\left(x\right){p}^{\prime }\left(y\right)P\left(y\vert x\right)/{q}^{\prime }\left(y\right)}=\alpha \end{equation} \tag{ A5 }$$

which is the same as

$$\begin{equation}\frac{p\left(x\right)}{q\left(x\right)}=\alpha \frac{{p}^{\prime }\left(y\right)}{{q}^{\prime }\left(y\right)}.\end{equation} \tag{ A6 }$$

Now consider any two different states x, x' ∈ c such that P(y|x) > 0 and P(y|x') > 0 for some y (such states must exist by the definition of islands). For equation (A6) to hold for both x, x' with that same, shared y, it must be that p(x)/q(x) = p(x')/q(x'). Take another state x'' ∈ c such that P(y'|x'') > 0 and P(y'|x') > 0 for some y'. Since this must be true for all pairs x, x' ∈ c, p(x)/q(x) = const for all x ∈ c, and p = q, as claimed. □

Lemma A4. Γ(p) = ∑_c∈L(P)p(c)Γ(p^c).

Proof. First, for any island c ∈ L(P), define

$$\begin{equation*}\phi \left(c\right)=\left\{y\in \mathcal{Y}:\exists x\in c\;\text{s.t.}\;P\left(y\vert x\right){ >}0\right\}.\end{equation*}$$

In words, ϕ(c) is the subset of output states in $\mathcal{Y}$ that receive probability from input states in c. By the definition of the island decomposition, for any y ∈ ϕ(c), P(y|x) > 0 only if y ∈ c. Thus, for any p and any y ∈ ϕ(c), we can write

$$\begin{equation}\frac{p\left(y\right)}{p\left(c\right)}=\frac{{\sum }_{x}P\left(y\vert x\right)p\left(x\right)}{p\left(c\right)}=\sum _{x\in \mathcal{X}}P\left(y\vert x\right){p}^{c}\left(x\right).\end{equation} \tag{ A7 }$$

Using p = ∑_c∈L(P)p(c)p^c and linearity of expectation, write ${\mathbb{E}}_{p}\left[f\right]={\sum }_{c\in L\left(P\right)}p\left(c\right){\mathbb{E}}_{{p}^{c}}\left[f\right]$. Then,

$$\begin{align*}\hfill S\left(Pp\right)-S\left(p\right)& =-\sum _{y}p\left(y\right)\mathrm{ln}\;p\left(y\right)+\sum _{x}p\left(x\right)\mathrm{ln}\;p\left(x\right)\hfill \\ \hfill & =\sum _{c\in L\left(P\right)}p\left(c\right)\left[-\sum _{y\in \phi \left(c\right)}\frac{p\left(y\right)}{p\left(c\right)}\mathrm{ln}\frac{p\left(y\right)}{p\left(c\right)}+\sum _{x\in c}\frac{p\left(x\right)}{p\left(c\right)}\mathrm{ln}\frac{p\left(x\right)}{p\left(c\right)}\right]\hfill \\ \hfill & =\sum _{c\in L\left(P\right)}p\left(c\right)\left[S\left(P{p}^{c}\right)-S\left({p}^{c}\right)\right],\hfill \end{align*}$$

where in the last line we have used equation (A7). Combining gives

$$\begin{align*}\hfill {\Gamma}\left(p\right)& =\sum _{c\in L\left(P\right)}p\left(c\right)\left[S\left(P{p}^{c}\right)-S\left({p}^{c}\right)+{\mathbb{E}}_{{p}^{c}}\left[f\right]\right]\hfill \\ \hfill & =\sum _{c\in L\left(P\right)}p\left(c\right){\Gamma}\left({p}^{c}\right).\hfill \end{align*}$$

□

We are now ready to prove the main result of this appendix.

Theorem 1. Consider any function ${\Gamma}:{{\Delta}}_{\mathcal{X}}\to \mathbb{R}$ of the form

$$\begin{equation*}{\Gamma}\left(p\right){:=}S\left(Pp\right)-S\left(p\right)+{\mathbb{E}}_{p}\left[f\right]\end{equation*}$$

where $P\left(y\vert x\right)$ is some conditional distribution of $y\in \mathcal{Y}$ given $x\in \mathcal{X}$ and $f:\mathcal{X}\to \mathbb{R}\cup \left\{\infty \right\}$ is some function. Let $\mathcal{Z}$ be any subset of $\mathcal{X}$ such that $f\left(x\right){< }\infty$ for $x\in \mathcal{Z}$, and let $q\in {{\Delta}}_{\mathcal{Z}}$ be any distribution that obeys

$$\begin{equation*}{q}^{c}\in \underset{r:\mathrm{s}\mathrm{u}\mathrm{p}\mathrm{p}r\subseteq c}{argmin}{\Gamma}\left(r\right)\quad \quad \;\text{for}\;\text{all}\;\;c\in {L}_{\mathcal{Z}}\left(P\right).\end{equation*}$$

Then, each ${q}^{c}$ will be unique, and for any $p$ with $\mathrm{s}\mathrm{u}\mathrm{p}\mathrm{p}p\subseteq \mathcal{Z}$,

$$\begin{equation*}{\Gamma}\left(p\right)=D\left(p{\Vert}q\right)-D\left(Pp{\Vert}Pq\right)+\sum _{c\in {L}_{\mathcal{Z}}\left(P\right)}p\left(c\right){\Gamma}\left({q}^{c}\right).\end{equation*}$$

Proof. We prove the theorem by considering two cases separately.

Case 1: $\mathcal{Z}=\mathcal{X}$. This case can be assumed when f(x) < ∞ for all x, so that ${L}_{\mathcal{Z}}\left(P\right)=L\left(P\right)$. Then, by lemma A4, we have Γ(p) = ∑_c∈L(P)p(c)Γ(p^c). By lemma A2 and theorem A1,

$$\begin{equation*}{\Gamma}\left({p}^{c}\right)-{\Gamma}\left({q}^{c}\right)=D\left({p}^{c}{\Vert}{q}^{c}\right)-D\left(P{p}^{c}{\Vert}P{q}^{c}\right),\end{equation*}$$

where we have used that if some supp q^c = c, then q^c is in the relative interior of the set $\left\{s\in {{\Delta}}_{\mathcal{X}}:\mathrm{s}\mathrm{u}\mathrm{p}\mathrm{p}\;s\subseteq c\right\}$. q^c is unique by lemma A3.

At the same time, observe that for any $p,r\in {{\Delta}}_{\mathcal{X}}$,

$$\begin{align*}\hfill D\left(p{\Vert}r\right)-D\left(Pp{\Vert}Pr\right)& =\sum _{x}p\left(x\right)\mathrm{ln}\frac{p\left(x\right)}{r\left(x\right)}-\sum _{y}p\left(y\right)\mathrm{ln}\frac{p\left(y\right)}{r\left(y\right)}\hfill \\ \hfill & =\sum _{c\in L\left(P\right)}p\left(c\right)\left[\sum _{x\in c}\frac{p\left(x\right)}{p\left(c\right)}\mathrm{ln}\frac{p\left(x\right)/p\left(c\right)}{r\left(x\right)/r\left(c\right)}-\sum _{y\in \phi \left(c\right)}\frac{p\left(y\right)}{p\left(c\right)}\mathrm{ln}\frac{p\left(y\right)/p\left(c\right)}{r\left(y\right)/r\left(c\right)}\right]\hfill \\ \hfill & =\sum _{c\in L\left(P\right)}p\left(c\right)\left[D\left({p}^{c}{\Vert}{r}^{c}\right)-D\left(P{p}^{c}{\Vert}P{r}^{c}\right)\right].\hfill \end{align*}$$

The theorem follows by combining.

Case 2: $\mathcal{Z}\subset \mathcal{X}$. In this case, define a 'restriction' of f and P to domain $\mathcal{Z}$ as follows:

• (a)  
  Define $\tilde {f}:\mathcal{Z}\to \mathbb{R}$ via $\tilde {f}\left(x\right)=f\left(x\right)$ for $x\in \mathcal{Z}$.
• (b)  
  Define the conditional distribution $\tilde {P}\left(y\vert x\right)$ for $y\in \mathcal{Y},x\in \mathcal{Z}$ via $\tilde {P}\left(y\vert x\right)=P\left(y\vert x\right)$ for all $y\in \mathcal{Y},x\in \mathcal{Z}$.

In addition, for any distribution $p\in {{\Delta}}_{\mathcal{X}}$ with $\mathrm{s}\mathrm{u}\mathrm{p}\mathrm{p}\;p\subseteq \mathcal{Z}$, let $\tilde {p}$ be a distribution over $\mathcal{Z}$ defined via $\tilde {p}\left(x\right)=p\left(x\right)$ for $x\in \mathcal{Z}$. Now, by inspection, it can be verified that for any $p\in {{\Delta}}_{\mathcal{X}}$ with $\mathrm{s}\mathrm{u}\mathrm{p}\mathrm{p}\;p\subseteq \mathcal{Z}$,

$$\begin{equation}{\Gamma}\left(p\right)=S\left(\tilde {P}\tilde {p}\right)-S\left(\tilde {p}\right)+{\mathbb{E}}_{\tilde {p}}\left[\tilde {f}\right]{:=}\tilde {{\Gamma}}\left(\tilde {p}\right)\end{equation} \tag{ A8 }$$

We can now apply case 1 of the theorem to the function $\tilde {{\Gamma}}:{{\Delta}}_{\mathcal{Z}}\to \mathbb{R}$, as defined in terms of the tuple $\left(\mathcal{Z},\tilde {f},\tilde {P}\right)$ (rather than the function ${\Gamma}:{{\Delta}}_{\mathcal{X}}\to \mathbb{R}$, as defined in terms of the tuple $\left(\mathcal{X},f,P\right)$). This gives

$$\begin{equation}\tilde {{\Gamma}}\left(\tilde {p}\right)=D\left(\tilde {p}{\Vert}\tilde {q}\right)-D\left(\tilde {P}\tilde {p}{\Vert}\tilde {P}\tilde {q}\right)+\sum _{c\in L\left(\tilde {P}\right)}\tilde {p}\left(c\right)\tilde {{\Gamma}}\left({\tilde {q}}^{c}\right),\end{equation} \tag{ A9 }$$

where, for all $c\in L\left(\tilde {P}\right)$, ${\tilde {q}}^{c}$ is the unique distribution that satisfies ${\tilde {q}}^{c}\in {\mathrm{argmin}}_{r\in {{\Delta}}_{\mathcal{Z}}:\mathrm{s}\mathrm{u}\mathrm{p}\mathrm{p}\;r\subseteq c}\tilde {{\Gamma}}\left(r\right)$.

Now, let q be the natural extension of $\tilde {q}$ from ${{\Delta}}_{\mathcal{Z}}$ to ${{\Delta}}_{\mathcal{X}}$. Clearly, for all $c\in L\left(\tilde {P}\right)$, ${\Gamma}\left({q}^{c}\right)=\tilde {{\Gamma}}\left({\tilde {q}}^{c}\right)$ by equation (A8). In addition, each q^c is the unique distribution that satisfies ${q}^{c}\in {\mathrm{argmin}}_{r\in {{\Delta}}_{\mathcal{X}}:\text{supp}\;r\subseteq c}{\Gamma}\left(r\right)$. Finally, it is easy to verify that $D\left(\tilde {p}{\Vert}\tilde {q}\right)=D\left(p{\Vert}q\right)$, $D\left(\tilde {P}\tilde {p}{\Vert}\tilde {P}\tilde {q}\right)=D\left(Pp{\Vert}Pq\right)$, $L\left(\tilde {P}\right)={L}_{\mathcal{Z}}\left(P\right)$ (recall the definition of ${L}_{\mathcal{Z}}$ from section 2.4). Combining the above results with equation (A8) gives

$$\begin{equation*}{\Gamma}\left(p\right)=\tilde {{\Gamma}}\left(\tilde {p}\right)=D\left(p{\Vert}q\right)-D\left(Pp{\Vert}Pq\right)+\sum _{c\in {L}_{\mathcal{Z}}\left(P\right)}p\left(c\right){\Gamma}\left({q}^{c}\right).\end{equation*}$$

□

Example 8. Suppose we are interested in thermodynamic costs associated with functions f whose image contains the value infinity, i.e., $f:\mathcal{X}\to \mathbb{R}\cup \left\{\infty \right\}$. For such functions, Γ(p) = ∞ for any p which has support over an $x\in \mathcal{X}$ such that f(x) = ∞. In such a case it is not meaningful to consider a prior distribution q (as in theorem 1) which has support over any x with f(x) = ∞. For such functions we also are no longer able to presume that the optimal distribution has full support within each island of c ∈ L(P), because in general the proof of lemma A2 no longer holds when f can take infinite values.

Nonetheless, by equation (A9), for the purposes of analyzing the thermodynamic costs of actual initial distributions p that have finite Γ(p) (and so have zero mass on any x such that f(x) = ∞), we can always carry out our usual analysis if we first reduce the problem to an appropriate 'restriction' of f.

Example 9. Suppose we wish to implement a (discrete-time) dynamics P(x'|x) over $\mathcal{X}$ using a CTMC. Recall from the end of section 2.2 that by appropriately expanding the state space $\mathcal{X}$ to include a set of 'hidden states' $\mathcal{Z}$ in addition to $\mathcal{X}$, and appropriately designing the rate matrices over that expanded state space $\mathcal{X}\cup \mathcal{Z}$, we can ensure that the resultant evolution over $\mathcal{X}$ is arbitrarily close to the desired conditional distribution P. Indeed, one can even design those rates matrices over $\mathcal{X}\cup \mathcal{Z}$ so that not only is the dynamics over $\mathcal{X}$ arbitrarily close to the desired P, but in addition the EF generated in running that CTMC over $\mathcal{X}\cup \mathcal{Z}$ is arbitrarily close to the lower bound of equation (21) [21].

However, in any real-world system that implements some P with a CTMC over an expanded space $\mathcal{X}\cup \mathcal{Z}$, that lower bound will not be achieved, and nonzero EP will be generated. In general, to analyze the EP of such real-world systems one has to consider the mismatch cost and residual EP of the full CTMC over the expanded space $\mathcal{X}\cup \mathcal{Z}$. Fortunately though, we can design the CTMC over $\mathcal{X}\cup \mathcal{Z}$ so that when it begins the implementation of P, there is zero probability mass on any of the states in $\mathcal{Z}$ [21, 22]. If we do that, then we can apply equation (A9), and so restrict our calculations of mismatch cost and residual EP to only involve the dynamics over $\mathcal{X}$, without any concern for the dynamics over $\mathcal{Z}$.

Example 10. Our last example is to derive the alternative decomposition of the EP of an SR circuit which is discussed in section 6.2. Recall that due to equation (29), the initial distribution over any gate in an SR circuit has partial support. This means we can apply equation (29) to decompose the EF, in direct analogy to the use of theorem 1 to derive theorem 2—only with the modification that the spaces X and Y are set to ${\mathcal{X}}_{\mathrm{p}\mathrm{a}\left(g\right)}$ and ${\mathcal{X}}_{g}$, respectively, rather than both set to ${\mathcal{X}}_{n\left(g\right)}$, as was done in deriving theorem 2. (Note that the islands also change when we apply equation (29) rather than theorem 1, from the islands of P_g to the islands of π_g). The end result is a decomposition of EF just like that in theorem 2, in which we have the same circuit Landauer cost and circuit Landauer loss expressions as in that theorem, but now have the modified forms of circuit mismatch cost and of circuit residual EP introduced in section 6.2.

Apply theorem 1 to equation (18) to give

$$\begin{align}\hfill {\hat{\sigma }}_{g}\left({p}_{n\left(g\right)}\right)& =D\left(\right.{p}_{n\left(g\right)}{\Vert}{q}_{n\left(g\right)}\left.\right)-D\left(\right.{P}_{g}{p}_{n\left(g\right)}{\Vert}{P}_{g}{q}_{n\left(g\right)}\left.\right)+\sum _{c\in L\left({P}_{g}\right)}{p}_{n\left(g\right)}\left(c\right){\hat{\sigma }}_{g}\left({q}_{n\left(g\right)}^{c}\right),\hfill \end{align} \tag{ B2 }$$

where q_n(g) is a distribution that satisfies

$$\begin{equation*}{q}_{n\left(g\right)}^{c}\in \underset{r:\mathrm{s}\mathrm{u}\mathrm{p}\mathrm{p}\;r\subseteq c}{\mathrm{argmin}}{\hat{\sigma }}_{g}\left(r\right)\end{equation*}$$

for all islands c ∈ L(P_g). Next, for convenience use equation (B1) to write $\mathcal{Q}\left(p\right)$ as

$$\begin{equation}\mathcal{Q}\left(p\right)=\mathcal{L}\left(p\right)+\left(\left[\sum _{g\in G}{\mathcal{Q}}_{g}\left({p}_{\mathrm{p}\mathrm{a}\left(g\right)}\right)\right]-\mathcal{L}\left(p\right)\right)\end{equation} \tag{ B3 }$$

The basic decomposition of $\mathcal{Q}\left(p\right)$ given in theorem 2 into a sum of $\mathcal{L}$ (defined in equation (37)), ${\mathcal{L}}^{\mathrm{l}\mathrm{o}\mathrm{s}\mathrm{s}}$ (defined in equation (47)), $\mathcal{M}$ (defined in equation (50)), and $\mathcal{R}$ (defined in equation (51)) comes from combining equations (43), (B2) and (B3) and then grouping and redefining terms.

Next, again use the fact that the solitary processes have no overlap in time to establish that the minimal value of the sum of the EPs of the gates is the sum of the minimal EPs of the gates considered separately of one another. As a result, we can jointly take ${\hat{\sigma }}_{g}\left({p}_{n\left(g\right)}\right)\to 0$ for all gates g in the circuit [21]. We can then use equation (B1) to establish that the minimal EF of the circuit is simply the sum of the minimal EFs of running each of the gates in the circuit, i.e., the sum of the subsystem Landauer costs of running the gates. In other words, the circuit Landauer cost is

$$\begin{equation} {\mathcal{L}}^{\mathrm{c}\mathrm{i}\mathrm{r}\mathrm{c}}\left(p\right)=\sum _{g\in G}\left[S\left({p}_{n\left(g\right)}\right)-S\left({P}_{g}{p}_{n\left(g\right)}\right)\right]\end{equation} \tag{ B4 }$$

$$\begin{equation}=\sum _{g\in G}\left[S\left({p}_{\mathrm{p}\mathrm{a}\left(g\right)}\right)-S\left({\pi }_{g}{p}_{\mathrm{p}\mathrm{a}\left(g\right)}\right)\right]\end{equation} \tag{ B5 }$$

$$\begin{equation}=\sum _{g\in G{\backslash}W}\left[S\left({p}_{\mathrm{p}\mathrm{a}\left(g\right)}\right)-S\left({\pi }_{g}{p}_{\mathrm{p}\mathrm{a}\left(g\right)}\right)\right]\end{equation} \tag{ B6 }$$

To derive the second line, we have used the fact that in an SR circuit, each gate is set to its initialized value at the beginning of its solitary process with probability 1, and that its parents are set to their initialized states with probability 1 at the end of the process. Then to derive the third line we have used the fact that wire gates implement the identity map, and so S(p_pa(g)) − S(π_gp_pa(g)) = 0 for all g ∈ W.

This establishes equation (49).

To derive equation (55), we first write Landauer cost as

$$\begin{equation*}\mathcal{L}\left(p\right)=S\left(p\right)-S\left(Pp\right)=\sum _{g\in G}\left[S\left({p}_{V}^{\mathrm{b}\mathrm{e}\mathrm{g}\left(g\right)}\right)-S\left({p}_{V}^{\mathrm{e}\mathrm{n}\mathrm{d}\left(g\right)}\right)\right].\end{equation*}$$

We then write circuit Landauer loss as

$$\begin{equation} {\mathcal{L}}^{\mathrm{l}\mathrm{o}\mathrm{s}\mathrm{s}}\left(p\right)=\sum _{g\in G}\left[S\left({p}_{n\left(g\right)}\right)-S\left({P}_{g}{p}_{n\left(g\right)}\right)\right]-\mathcal{L}\left(p\right)=\sum _{g\in G}\left[\left(S\left({p}_{n\left(g\right)}\right)-S\left({p}_{V}^{\mathrm{b}\mathrm{e}\mathrm{g}\left(g\right)}\right)\right)-\left(S\left({P}_{g}{p}_{n\left(g\right)}\right)-S\left({p}_{V}^{\mathrm{e}\mathrm{n}\mathrm{d}\left(g\right)}\right)\right)\right]=\sum _{g\in G}\left[\left(S\left({p}_{n\left(g\right)}\right)+S\left({p}_{V{\backslash}n\left(g\right)}^{\mathrm{b}\mathrm{e}\mathrm{g}\left(g\right)}\right)-S\left({p}_{V}^{\mathrm{b}\mathrm{e}\mathrm{g}\left(g\right)}\right)\right)-\left(S\left({P}_{g}{p}_{n\left(g\right)}\right)+S\left({p}_{V{\backslash}n\left(g\right)}^{\mathrm{e}\mathrm{n}\mathrm{d}\left(g\right)}\right)-S\left({p}_{V}^{\mathrm{e}\mathrm{n}\mathrm{d}\left(g\right)}\right)\right)\right]\end{equation} \tag{ B7 }$$

$$\begin{equation}=\sum _{g\in G}\left[{I}_{{p}^{\mathrm{b}\mathrm{e}\mathrm{g}\left(g\right)}}\left({X}_{n\left(g\right)};{X}_{V{\backslash}n\left(g\right)}\right)-{I}_{{p}^{\mathrm{e}\mathrm{n}\mathrm{d}\left(g\right)}}\left({X}_{n\left(g\right)};{X}_{V{\backslash}n\left(g\right)}\right)\right],\end{equation} \tag{ B8 }$$

In equation (B7), we used the fact that a solitary process over n(g) leaves the nodes in V\n(g) unmodified, thus $S\left({p}_{V{\backslash}n\left(g\right)}^{\mathrm{b}\mathrm{e}\mathrm{g}\left(g\right)}\right)=S\left({p}_{V{\backslash}n\left(g\right)}^{\mathrm{e}\mathrm{n}\mathrm{d}\left(g\right)}\right)$.

Given the assumption that x_g = ∅ at the beginning of the solitary process for gate g, we can rewrite

$$\begin{equation}{I}_{{p}^{\mathrm{b}\mathrm{e}\mathrm{g}\left(g\right)}}\left({X}_{n\left(g\right)};{X}_{V{\backslash}n\left(g\right)}\right)={I}_{{p}^{\mathrm{b}\mathrm{e}\mathrm{g}\left(g\right)}}\left({X}_{\mathrm{p}\mathrm{a}\left(g\right)};{X}_{V{\backslash}\mathrm{p}\mathrm{a}\left(g\right)}\right).\end{equation} \tag{ B9 }$$

Similarly, because X_v = ∅ for all v ∈ pa(g) at the end of the solitary process for gate g, we can rewrite

$$\begin{equation}{I}_{{p}^{\mathrm{e}\mathrm{n}\mathrm{d}\left(g\right)}}\left({X}_{n\left(g\right)};{X}_{V{\backslash}n\left(g\right)}\right)={I}_{{p}^{\mathrm{e}\mathrm{n}\mathrm{d}\left(g\right)}}\left({X}_{g};{X}_{V{\backslash}g}\right).\end{equation} \tag{ B10 }$$

Finally, for any wire gate g ∈ W, given the assumption that X_g = ∅ at the beginning of the solitary process, we can write

$$\begin{equation}{I}_{{p}^{\mathrm{b}\mathrm{e}\mathrm{g}\left(g\right)}}\left({X}_{\mathrm{p}\mathrm{a}\left(g\right)};{X}_{V{\backslash}\mathrm{p}\mathrm{a}\left(g\right)}\right)={I}_{{p}^{\mathrm{e}\mathrm{n}\mathrm{d}\left(g\right)}}\left({X}_{g};{X}_{V{\backslash}g}\right).\end{equation} \tag{ B11 }$$

Equation (55) then follows from combining equations (B8)–(B11) and simplifying.

First, write circuit Landauer loss as

$$\begin{equation}{\mathcal{L}}^{\mathrm{l}\mathrm{o}\mathrm{s}\mathrm{s}}\left(p\right)=\sum _{g\in G{\backslash}W}\left[S\left({p}_{\mathrm{p}\mathrm{a}\left(g\right)}\right)-S\left({p}_{g}\right)\right]-\mathcal{L}\left(p\right).\end{equation} \tag{ B12 }$$

Then, rewrite the sum in equation (B12) as

$$\begin{align}\hfill \sum _{g\in G{\backslash}W}\left[S\left({p}_{\mathrm{p}\mathrm{a}\left(g\right)}\right)-S\left({p}_{g}\right)\right]& =\sum _{g\in G{\backslash}W}S\left({p}_{\mathrm{p}\mathrm{a}\left(g\right)}\right)-\sum _{g\in G{\backslash}W}S\left({p}_{g}\right)\hfill \\ \hfill & =\sum _{g\in G{\backslash}W}S\left({p}_{\mathrm{p}\mathrm{a}\left(g\right)}\right)-\sum _{g\in G{\backslash}\left(W\cup \mathrm{O}\mathrm{U}\mathrm{T}\right)}S\left({p}_{g}\right)-\sum _{g\in \mathrm{O}\mathrm{U}\mathrm{T}}S\left({p}_{g}\right)\hfill \\ \hfill & =\sum _{g\in G{\backslash}W}S\left({p}_{\mathrm{p}\mathrm{a}\left(g\right)}\right)-\sum _{v\in V{\backslash}\left(W\cup \mathrm{O}\mathrm{U}\mathrm{T}\right)}S\left({p}_{v}\right)+\sum _{v\in \mathrm{I}\mathrm{N}}S\left({p}_{v}\right)-\sum _{g\in \mathrm{O}\mathrm{U}\mathrm{T}}S\left({p}_{g}\right).\hfill \end{align} \tag{ B13 }$$

Now, notice that for every v ∈ V\(W ∪ OUT) (i.e., every node which is not a wire and not an output), there is a corresponding wire w which transmits v to its child, and which has S(p_w) = S(p_v). This lets us rewrite equation (B13) as

$$\begin{align}\hfill & \sum _{g\in G{\backslash}W}S\left({p}_{\mathrm{p}\mathrm{a}\left(g\right)}\right)-\sum _{w\in W}S\left({p}_{w}\right)+\sum _{v\in \mathrm{I}\mathrm{N}}S\left({p}_{v}\right)-\sum _{g\in \mathrm{O}\mathrm{U}\mathrm{T}}S\left({p}_{g}\right)\hfill \\ \hfill & =\sum _{g\in G{\backslash}W}\left[S\left({p}_{\mathrm{p}\mathrm{a}\left(g\right)}\right)-\sum _{v\in \mathrm{p}\mathrm{a}\left(g\right)}S\left({p}_{v}\right)\right]+\sum _{v\in \mathrm{I}\mathrm{N}}S\left({p}_{v}\right)-\sum _{g\in \mathrm{O}\mathrm{U}\mathrm{T}}S\left({p}_{g}\right)\hfill \\ \hfill & =-\sum _{g\in G}\mathcal{I}\left({p}_{\mathrm{p}\mathrm{a}\left(g\right)}\right)+\sum _{v\in \mathrm{I}\mathrm{N}}S\left({p}_{v}\right)-\sum _{g\in \mathrm{O}\mathrm{U}\mathrm{T}}S\left({p}_{g}\right)\hfill \end{align} \tag{ B14 }$$

where in the second line we have used the fact that every wire belongs to exactly one set pa(g) for a non-wire gate g, and in the last line we used the definition of multi-information. Then, using the definition $\mathcal{L}\left(p\right)=S\left({p}_{\mathrm{I}\mathrm{N}}\right)-S\left({p}_{\mathrm{O}\mathrm{U}\mathrm{T}}\right)$, the definition of multi-information, and by combining equations (B12)–(B14), we have

$$\begin{equation}{\mathcal{L}}^{\mathrm{l}\mathrm{o}\mathrm{s}\mathrm{s}}\left(p\right)=\mathcal{I}\left({p}_{\mathrm{I}\mathrm{N}}\right)-\mathcal{I}\left({\pi }_{{\Phi}}{p}_{\mathrm{I}\mathrm{N}}\right)-\sum _{g\in G}\mathcal{I}\left({p}_{\mathrm{p}\mathrm{a}\left(g\right)}\right).\end{equation} \tag{ B15 }$$

To derive equation (57), note that

$$\begin{align}\hfill \mathcal{I}\left({p}_{\mathrm{I}\mathrm{N}}\right)& =\left[\sum _{v\in \mathrm{I}\mathrm{N}}S\left({p}_{v}\right)\right]-S\left({p}_{\mathrm{I}\mathrm{N}}\right){\leqslant}\sum _{v\in \mathrm{I}\mathrm{N}}S\left({p}_{v}\right){\leqslant}\sum _{v\in \mathrm{I}\mathrm{N}}\mathrm{ln}\;\left\vert {\mathcal{X}}_{v}\right\vert =\mathrm{ln}\left\vert \;,\prod _{v\in \mathrm{I}\mathrm{N}},{\mathcal{X}}_{v}\right\vert =\mathrm{ln}\;\left\vert {\mathcal{X}}_{\mathrm{I}\mathrm{N}}\right\vert .\hfill \end{align} \tag{ B16 }$$