Information flow in parameterized quantum circuits

For a bipartite quantum system on two subsystems A and B, the state space can be represented as the tensor product of the individual sub-spaces as: $\mathcal{H}^{AB} = \mathcal{H}^A \otimes \mathcal{H}^B$. The quantum mutual information $I(A:B)$ is a measure of the amount of correlation between the two sub-systems and is defined as:

$$\begin{align} I\left(A:B\right) = S\left(\rho^A\right) + S\left(\rho^B\right) - S\left(\rho^{AB}\right), \end{align} \tag{ 1 }$$

where ρ^AB represents the density matrix of the full system on $\mathcal{H}^{AB}$, $\rho^{A} = \textrm{Tr}_B (\rho^{AB})$ and $\rho^{B} = \textrm{Tr}_A (\rho^{AB})$ are the reduced state of ρ^AB on systems A and B respectively, and $S(\rho) = - \textrm{Tr} (\rho \log \rho)$ represents the von Neumann entropy of a density matrix ρ.

A quantum circuit is a sequence of operations that can be used to prepare a state of interest on a quantum device. It typically comprises of simple unitary operations called quantum gates that depend on some parameters. The action of the circuit $U(\theta)$ on an initial state $\vert \psi_0\rangle$ is to prepare the state $\vert \psi(\theta)\rangle = U(\theta) \vert \psi_0\rangle$.

We develop a simple strategy for converting a quantum circuit into a directed graph by representing every quantum gate as edges in the graph and the states at different moments in the circuits as nodes. Using this picture, a single qubit gate is represented as an edge that connects two nodes, while a two-qubit gate is represented as four edges between four nodes. The nodes are then arranged as per the time steps in the quantum circuit. The open edges represent the initial state and the final state of the qubits. A simple illustration of this conversion is shown in figure 1.

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Information flow is defined as the transfer of information from one variable to another in a process. In the context of quantum circuits, this can be regarded as the spread of correlation between qubits as a result of multi-qubit gates. Analogously one can think of it as the spread of correlation between gates in their corresponding causal cones. The spread of correlation has been considered in previous studies [27, 28] to design better ansätze for variational algorithms. In this work, we use the notion of information flow (spread of correlation) to define paths through the circuit, using which we can control the spread of correlation between qubits. We combine the concept of causal cones with paths and utilize the graph representation of a PQC to identify and control the flow of information (correlation) between gates. An illustration of these concepts is depicted in figure 2.

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In VQAs we usually optimize the parameters of states prepared by a PQC, $U(\theta)$, in order to minimize an objective function of the form:

$$\begin{align} f\left(\theta\right) = \langle H \rangle_{\theta} = \sum_{j = 1}^{M} \langle H_j \rangle_{\theta}, \end{align} \tag{ 2 }$$

where H represent a problem Hamiltonian, which is a linear combination on M Pauli terms H_j . The optimization of the objective function is carried out by iteratively updating the parameters of the quantum circuit as:

$$\begin{align} \mathbf{\theta}^{\left(t+1\right)}_i = \mathbf{\theta}^{\left(t\right)}_i - \alpha \frac{\partial f\left(\mathbf{\theta}\right)}{\partial \mathbf{\theta}_i}, \end{align} \tag{ 3 }$$

where α is the learning rate and $\frac{\partial f(\mathbf{\theta})}{\partial \mathbf{\theta}_i}$ denotes the partial derivative of the objective function with respect to the variable θ_i . The analytical gradient can be calculated using a K-term parameter-shift rule [29–32] as:

$$\begin{align} \frac{\partial f\left(\mathbf{\theta}\right)}{\partial \mathbf{\theta}_i} = \sum_{j = 1}^{M} \frac{\partial \langle H_j \rangle_{\theta}}{\partial \mathbf{\theta}_i} = \sum_{j = 1}^{M} \sum_{k = 1}^{K} \gamma_{k,i} \langle H_j \rangle_{\theta_{k,i}}, \end{align} \tag{ 4 }$$

where, $\gamma_{k,i}$ is the $K-$term coefficient in the shift rule. The calculation of the analytical gradient requires a large number of measurements for a single parameter update, as every gradient calculation requires K objective evaluations, which in turn requires a large number of measurements (shots) for accurate estimation of each objective. To overcome this, n-shot stochastic gradient descent was proposed in [33, 34] where one uses n-shot estimators of the gradient instead of the exact ones,

$$\begin{align} \frac{\partial f\left(\mathbf{\theta}\right)}{\partial \mathbf{\theta}_i} = \sum_{j = 1}^{M} \sum_{k = 1}^{K} \gamma_{k,i} \tilde{h}_j^{\left(n\right)}\left({\theta_{k,i}}\right), \end{align} \tag{ 5 }$$

where, $\tilde{h}_j^{(n)}({\theta_{k,i}})$ is the n-sample mean estimator of $\langle H_j \rangle_{\theta_{k,i}}$. The parameters are then updated similarly to equation (3).

Variational quantum classifiers (VQC) are quantum circuits that are trained for supervised learning tasks [7, 8]. There exist several strategies to design a quantum classifier, including generalizations of well-known classical machine learning techniques such as artificial neural networks [7,35] or kernel methods [6, 36].

In this study, the circuit-centric architecture in figure 3 is used to study a binary classification problem [8]. The general objective is to train the VQC on a data set $\{\boldsymbol x_i, y_i\}_\mathrm{train}$ to find a mapping between input $\boldsymbol x_i$ and label y_i . The trained PQC can then be used as a black box to predict labels $\hat{y}_i$ for a given set of test inputs $\{\boldsymbol x_i, y_i\}_\mathrm{test}$.

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The circuit used for this classification task consists of three distinct parts: the state preparation circuit, the model circuit, and a measurement scheme. Preparing a quantum state that embeds some classical data is achieved by applying a static quantum routine on the initial ground state, $\vert \Phi(\boldsymbol x)\rangle = E(\boldsymbol x)\vert 0\rangle$. Here, we use the so-called basis encoding method

$$\begin{align} \vert \Phi\left(\boldsymbol x\right)\rangle = \bigotimes_{i = 0}^n \vert \boldsymbol x_i\rangle \qquad\qquad \boldsymbol x\in \mathbb{B}^n. \end{align} \tag{ 6 }$$

To encode the input data $\boldsymbol x$ as computational basis states. The prepared state $\vert \Phi(\boldsymbol x)\rangle$ is then further processed with a given parametrized quantum circuit $U(\theta)$, resulting in a state $\vert \psi_\mathrm{vqc}(\boldsymbol x,\theta)\rangle = U(\theta)E(\boldsymbol x)\vert 0\rangle$. The ansatz structure used in this study contains parameterized single and two-qubit gates with trainable parameters. The circuit is shown in figure 4. The output state $\vert \psi_\mathrm{vqc}(\boldsymbol x,\theta)\rangle$ is finally measured using the Pauli-Z operator on a qubit (we chose the first qubit) to obtain the expectation value $\mathbb{E}(\hat{Z})$. This yields the predicted label $(\hat{y}_i)$.

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The parameters of the variational block are trained by minimizing the square loss cost function

$$\begin{align} f\left(\theta\right) & = \frac{1}{n^{^{\prime}}}\sum_{i = 0}^{n^{^{\prime}}} \left( y_i - \hat{y}_i\left(\theta\right) \right)^2 \end{align} \tag{ 7 }$$

with n′ as the training set size and the predicted label output of the VQC as

$$\begin{align} \hat{y}_i\left(\theta\right) & = \langle \psi_\mathrm{vqc}\left(\boldsymbol x_i,\theta\right)\vert \hat{Z} \otimes \mathbb{I}^{\otimes \left(n-1\right)}\vert \psi_\mathrm{vqc}\left(\boldsymbol x_i,\theta\right)\rangle. \end{align} \tag{ 8 }$$

The binary classification problem studied in this paper is the n-bit parity problem. The corresponding dataset contains 2ⁿ distinct binary vectors, where each label indicates whether the sum of the n components of the binary vector is odd or even. The Boolean n-bit parity function to be modeled is

$$\begin{align} f : \left\{0,1\right\}^{\otimes n} \xrightarrow{} \left\{0,1\right\}, \end{align} \tag{ 9 }$$

with the property that $f(\boldsymbol x) = 1$ if the number of ones in the vector $\boldsymbol x\in \{0,1\}^{n}$ is odd and zero otherwise.

In what follows, we describe the details of how to quantify information flow for designing algorithms to optimize variatonal algorithms.

A given unitary $\hat{U}$ on two qubits can be conveniently described using a $4 \times 4$ matrix $U_{(a,b),(c,d)}$ with entries:

$$\begin{align} U_{\left(a,b\right),\left(c,d\right)} & = \left< c,d \right| \hat{U} \left| a,b \right> \qquad\qquad a,b,c,d \in \left\{0,1\right\} \; , \end{align} \tag{ 10 }$$

where the indices (a, b) are thought of as a single unified row-index, and (c, d) similarly plays the role of a unified column-index. Using this description, a four qubit state corresponding to the unitary transformation can be defined:

$$\begin{align} \left| \psi_{U} \right> & = \frac{1}{\mathcal{N}} \sum_{a,b,c,d \, \in \, \left\{0,1\right\}} U_{\left(a,b\right),\left(c,d\right)} \left|a,b,c,d\right> , \end{align} \tag{ 11 }$$

where $\mathcal{N}$ is a normalization-constant.

Using this state, we can now define reduced density matrices of different subsystems of qubits. For instance, we can define the single- and two-qubit reduced density-matrices:

$$\begin{align*} \rho^a & = \text{Tr}_{\left\{b,c,d\right\}} \left( \left| \psi_U \right> \left< \psi_U \right| \right)\\ \rho^{ab} & = \text{Tr}_{\left\{c,d\right\}} \left( \left| \psi_U \right> \left< \psi_U \right| \right) , \end{align*}$$

where the subscripts on the trace indicates the degrees of freedom that are traced over.

Using this description of the unitary, [37] proposed the following distance metric across the legs of the unitary (see figure 5):

$$\begin{align} d_{i,j} & = - \log \left( \frac{I\left(i:j\right)}{2 \log\left(2\right)} \right) , \end{align} \tag{ 12 }$$

where $I(i:j)$ is the mutual information between the sites i and j as defined in equation (1).

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In this article we introduce a further modification to the distance metric as:

$$\begin{align} \tilde{d}_{i,j} = \begin{cases} - \log \left( \frac{I\left(i:j\right)}{4 \log\left(2\right)} \right) \quad \text{if } \left(i,j\right) \in \left\{\left(a,c\right), \left(b,d\right)\right\} \\ \\ - \log \left( \frac{I\left(ac:bd\right)}{4 \log\left( 2 \right)} \right) \quad \text{else} \end{cases} \end{align} \tag{ 13 }$$

where $I(ij:kl)$ is defined as:

$$\begin{align} I\left(ij:kl\right) & = S\left(\rho^{ij}\right) + S\left(\rho^{kl}\right) - S\left(\rho^{ijkl}\right) \nonumber \\ & = S\left(\rho^{ij}\right) + S\left(\rho^{kl}\right) . \end{align} \tag{ 14 }$$

This modification was motivated mainly by a need to more accurately treat the small class of gates relevant in a PQC context—see appendix A for details. To develop some intuition about this metric, we present an illustration of the distance between the legs of a two qubit gate, C-Ry(θ), in figure 6. As expected, we observe that the distance (weight of the diagonal edges) approaches infinity when theta equals zero because the two qubit gate at this value acts as identity.

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The objective functions $f(\theta)$ for commonly used quantum algorithms are constructed as a linear combination of expectation values as:

$$\begin{align} f\left(\theta\right) = \sum_{j = 1}^{M} \langle H_j \rangle_{\theta}, \end{align} \tag{ 15 }$$

where H_j represent individual Pauli-strings in the Hamiltonian defining the problem. These individual Pauli-strings usually act on the state of a subset of the full qubits, and their expectation value thus depends on the gate parameter in the causal cone of these qubits [38]. We define different paths in the causal cone of the qubits to be measured by converting the quantum circuits into graphs as described in section 2.2. An illustration of the choices of paths in the causal cone is shown in figure 7.

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We further introduce the notion of shortest paths by adding weights to the edges in the graph representation of the circuit according to the distance measure proposed in section 3.1. This is inspired from [37] where the authors use the length of geodesics for disentangling a quantum state. We use the networkx package [39] to create the graph representation of the quantum circuit and for finding different paths between nodes in the graph representation.

In this section, we present a strategy for optimizing VQAs based on paths in the causal cone of the individual Pauli-strings used to define the objective function. We use two different strategies of choosing sets of parameters by randomly sampling paths from either the set of all possible paths or the set of shortest paths. The overall algorithm is presented in algorithm 1.

Algorithm 1. An outline of the path optimization algorithm.
Input: path selection strategy, problem Hamiltonian and ansatz, learning rate
Output: estimate of the ground state energy of the Hamiltonian
Initialize: generate the circuit graph corresponding to the ansatz
while not converged do
0. Initialize an empty list, $\mathcal{L}$
for term in Hamiltonian do
1. Calculate all paths for that term
2. Select a path based on the strategy (random or shortest)
3. Add all the parameter ids in the path which are not already in the list $\mathcal{L}$ to it
end
4. Update all parameters corresponding to ids in $\mathcal{L}$ using the matching gradients and learning rate by applying equation (3)
end
return Estimate of the energy of the Hamiltonian using the ansatz and optimized parameters

One can also modify the presented algorithm to add more stochasticity by randomly sampling the Hamiltonian terms to optimize in each step [34, 40] or by updating parameters using some terms before calculating paths for further terms. We leave these variations for future studies.

We use the VQE framework to find the ground state energy of the XXZ-Heisenberg model. The Hamiltonian of such a system can be written as follows:

$$\begin{align} \hat{H} = \sum_{\langle i,j\rangle} \left( - J_X \hat{X}_{i} \hat{X}_{j} - J_Y \hat{Y}_{i} \hat{Y}_{j} - J_Z \hat{Z}_{i} \hat{Z}_{j} \right) - h \sum_{k} \hat{Z}_k , \end{align} \tag{ 16 }$$

where $\hat{X}, \hat{Y} , \hat{Z}$ are the Pauli matrices, $\langle i,j\rangle$ denotes all the pairs of adjacent lattice sites, and $J_X, J_Y, J_Z$ are the coupling constants and h on the right represents the external magnetic field. The coupling constants for the XXZ-model follow the relation: $J = J_X = J_Y \ne J_Z \equiv \Delta$. For all the experiments, we fix the values of the different constants to h = 0 (no external magnetic field), $\Delta = -20.0$ and J = 1.0. This corresponds to the model having a ferromagnetic ground state. As illustrated in appendix C, qualitative behavior is similar in other parameter regimes. We use the XXZ-Heisenberg model due to the fact that all the terms in the Hamiltonian depend on only the state of two qubits, e.g. $\hat{X}_i\otimes\hat{X}_j$, and thus our method can potentially be very useful in reducing the cost of parameter updates in every iteration of the optimization, as investigated in more detail below.

We carried out simulations for five different lattice sizes, starting with $3\times 2$ qubits, $4\times 2$ qubits, $5\times 2$ qubits, $6\times 2$ qubits and $7\times 2$ qubits (6, 8, 10, 12, and 14 qubits, respectively). For all the simulations, we use the ansatz shown in figure 1(b) with varying number of layers. The optimization is carried out using the algorithm presented in algorithm 1 and the stochastic gradient descent algorithm, with a fixed learning rate of 0.1. The results from all the simulations are plotted in figure 8. We only plot the first 100 iterations of the trajectories for all the cases, as it is sufficient for comparing the different methods. All the simulations were repeated at least 5 times with random initialization of the gate parameters to collect the statistics for comparison.

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We look at the optimization trajectories from the different simulations using a single layer of the ansatz plotted in figure 8. First, we observe that optimization using algorithm 1 with either a random or shortest path always performs better when compared with stochastic gradient descent. Second, we observe that the optimization trajectories using the shortest path on average have steeper initial convergence, however, the final energies achieved by all optimization methods were very close. This indicates that choosing a set of parameters based on information transfer between qubits via gates can help accelerate the overall convergence of an algorithm. Also, it should be noted that forcing the flow of information along a particular path can be useful in cases where the spread of information can lead to convergence issues [44]. Finally, we point out that the spread in the trajectories corresponding to the runs with the shortest path tends to be smaller, however, all the methods have some runs (particularly in the case of the 12 qubit model) where they converge to a local minimum. This is a common occurrence in stochastic optimization methods and can be mitigated using different methods [45].

As we increase the number of layers of the ansatz used in the numerical simulations, we observe that the rate of convergence for all the methods increases. However, we point out that optimization trajectories with the shortest path are still the fastest converging among all the methods. This implies that using (shortest) paths for optimization of algorithms with objectives depending on only a subset of the qubits might be useful.

4.1.1. Resource analysis

As mentioned above, an important property of the path-based algorithm is that it selects only a subset of the gate parameters to be updated at each step in the optimization. Since the gradient related to each circuit parameter generally has to be estimated separately using parameter-shift rules, and there would be no need to calculate gradients for parameters that are not going to be updated, fewer parameters to update generally leads to fewer measurements required. Furthermore, since our parameter selection is based on a classical algorithm requiring no additional measurements, there is no additional measurement overhead to consider. Combining these facts, it seems likely that the path-based approach should require fewer measurements than stochastic gradient descent, even if the same number of iterations were required. To investigate this question further, we take the number of updated parameters as a proxy for the required number of measurements, and track it across the 6- and 8-qubit XXZ-Heisenberg runs of appendix C. In figure 9, the number of parameters updated at each step for the two path-selection strategies is compared to each other and to stochastic gradient descent. As can be seen from this figure and the statistics in table 1, using the strategy based on shortest paths allows for a reduction in the number of updated parameters of about 8%, while using random paths allow for an even larger reduction of about 12%. A priori, this would seem to indicate an advantage for the random-path sampling over shortest-path sampling when it comes to minimizing the amount of measurements required. However, as observed in figure 8, the shortest-path strategy sometimes converges in fewer iterations. Thus, the question of the best method with respect to the number of measurements to some extent boils down to the question of using fewer but more expensive shortest-path iterations or more but cheaper random-path iterations. To showcase how these competing effects look in practice, figure 10 shows the XXZ-Heisenberg results of appendix C plotted against the total number of parameter updates. As shown in these figures, the path-based approaches in general tend to outperform stochastic gradient descent, while the best-performing strategy varies from problem to problem and run to run, and depends on whether you value rapid early convergence or better convergence late in the process. This highlights how the metric of updates per iteration in table 1 is not necessarily sufficient to judge relative performances of the methods.

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Table 1. Average number of parameters updated at each step in the optimization for different methods compared in this paper, and their reductions compared to baseline stochastic gradient descent. For a more detailed look at the corresponding distributions, see figure 9.

	SGD	Shortest path	Random path
6 Qubits	24	22.3 (8.3%)	21.3 (12.4%)
8 Qubits	32	29.3 (7.1%)	28.0 (11.2%)

To investigate the use of our framework for a different use-case with a significantly different cost function, we apply it to the VQC framework presented in section 2.5 for the n-bit parity classification problem. The corresponding dataset of such a system consists of 2ⁿ distinct binary vectors with corresponding labels, where each label indicates whether the sum of the n components of the binary vector is odd or even. The Boolean n-bit parity function to be modeled is

$$\begin{align} f : \left\{0,1\right\}^{\otimes n} \xrightarrow{} \left\{0,1\right\} \end{align} \tag{ 17 }$$

with the property that $f(\boldsymbol x) = 1$ if the number of ones in the vector $\boldsymbol x\in \{0,1\}^{n}$ is odd. Binary classification is an interesting choice for this study because there is only one readout qubit, making the problem potentially well-suited for our optimization method.

Simulations were performed on the 4-qubit parity problem using the ansatz shown in figure 4 with $2,3,$ and 4 layers. Again, optimization is performed using the algorithm presented in algorithm 1 and Nesterov momentum with a fixed learning rate of 0.1. We selected the Nesterov optimizer for comparison, as using Adam failed to improve the model accuracies. The simulation results are shown in figure 11. We focus on the first 50 training epochs of optimization for all cases and plot the average of 5 instances with random initialization of the gate parameters to collect statistics for comparison.

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As can be seen from the optimization trajectories in figure 11, the model is able to learn to successfully perform the classification task. We note that the optimization with random paths consistently outperforms the one with the Nesterov momentum method. However, we observe that the optimization using the shortest paths performs poorly. We attribute this to the fact that sampling of the shortest path leads to insufficient exploration of the full circuit, and thus to the model having too little knowledge of the full input to compute a global property like the parity at any given step. To test this hypothesis and help inform the model of the full input state, we perform numerical simulations with paths to all input qubits instead of a single random path. The combined paths in the worst case can correspond to the causal cone of the observable, which has been considered for optimization in previous works [38]. Furthermore we carry out the simulation with the paths alongside the combined paths with a reduced learning rate of 0.05 and plot the results in figure 12. The reason for training with a reduced learning rate is that we observe oscillations in the cost function with the combined paths.

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We note that the trajectories from the optimization with the combined paths outperform the optimization with individual paths consistently. This is the expected behavior as the model has access to the full data for the classification as compared to the case with individual paths. We further point out that the optimization with the random paths on average has a higher convergence rate as compared to the one with the shortest path. This can be due to the fact that it can have access to a larger number of parameters as compared to the shortest path. While this suggests that one needs larger number of parameters here, we note that we can still find the optimal solution using single paths which only depend on a subset of the parameters.

Overall, we have presented empirical evidence that our method based on paths (defined using a distance measure) can be used to successfully optimize VQAs. The observed performances seem to indicate that the current form of the metric works well on problems with a degree of locality (here: the Heisenberg model), while modification in the sampling strategy was needed for problems requiring global information (here: parity classification). We conjecture that this may be a result of a bias towards locality inherent in the metric, i.e. a tendency for the metric-based paths to stay close to the readout qubits from which it originates. Strategies to reduce this local focus might include tuning the magnitude of the two terms in equation (13) or using more complicated metric-based sampling than picking the shortest path. Note that, beyond investigating these modifications, further systematic investigations into similarities and differences across a broader range of optimization strategies based on parameter subsets also seems promising, since it could potentially allow for the development of heuristics to determine which strategy to use for a given problem, as well as the construction of hybrid strategies able to combine the strengths of different approaches. We leave investigations of all of the above questions and strategies as future work.

The first observation is that all single-qubit reduced density matrices resulting from the mapping from unitaries to states defined in equation (11) are maximally mixed, and thus have entropies independent from the gate considered. To see this, consider a state resulting from this mapping:

$$\begin{align*} \rho \propto \sum_{\substack{a,b,c,d\\a^{^{\prime}},b^{^{\prime}},c^{^{\prime}},d^{^{\prime}}}} U_{a,b,c,d} \, U_{a^{^{\prime}},b^{^{\prime}},c^{^{\prime}},d^{^{\prime}}}^{{\,}*} \left| a,b,c,d \right> \left< a^{^{\prime}},b^{^{\prime}},c^{^{\prime}},d^{^{\prime}}\right| . \end{align*}$$

Tracing out all other degrees of freedom than a yields a reduced density matrix

$$\begin{align*} \rho^a \propto\sum_{\substack{a,b,c,d\\a^{^{\prime}}}} U_{a,b,c,d} \, U_{a^{^{\prime}},b,c,d}^{{\,}*} \left| a \right> \left< a^{^{\prime}}\right| = \sum_{\substack{a,b,c,d\\a^{^{\prime}}}} U_{a,b,c,d} \, U^\dagger_{c,d,a^{^{\prime}},b} \left| a \right> \left< a^{^{\prime}}\right| . \end{align*}$$

However, from unitarity it follows that

$$\begin{align*} \sum_{c,d} U_{a,b,c,d} \, U^\dagger_{c,d,a^{^{\prime}},b} = \delta_{a,a^{^{\prime}}} , \end{align*}$$

meaning

$$\begin{align*} \rho^a &\propto \sum_{\substack{a,b\\a^{^{\prime}}}} \delta_{a, a^{^{\prime}}} \left| a \right> \left< a^{^{\prime}}\right| \propto \sum_{a} \left| a \right> \left< a\right| . \end{align*}$$

Thus, $\rho^a = \frac{1}{2} \unicode{x1D7D9}_{2\times2}$ after taking normalization into account. Similar methods yield the same form for the other single-qubit reduced density matrices, implying the following property:

$$\begin{align*} S\left(\rho^i\right) = \log\left(2\right) \qquad\qquad \forall i \in \left\{a,b,c,d\right\} . \end{align*}$$

A simple observation based on the result of the previous section and the definitions of the mutual information used in this work is that the sum of the following re-weighted mutual informations across the four links corresponding to a 2-qubit gate fulfills:

$$\begin{align*} I\left(d:b\right) + I\left(c:a\right) + 2\cdot \frac{1}{2} I\left(ac:bd\right) & = S\left(\rho^d \right) + S\left(\rho^b \right) - S\left(\rho^{bd} \right) + S\left(\rho^c \right) + S\left(\rho^a \right) - S\left(\rho^{ac} \right) + S\left(\rho^{ac} \right) + S\left(\rho^{bd} \right) \\ & = S\left(\rho^d \right) + S\left(\rho^b \right)+ S\left(\rho^c \right) + S\left(\rho^a \right) \\ & = 4 \log\left(2\right) . \end{align*}$$

If mutual information is interpreted as a flow of information (as in this work), this represents a conservation-like law stating that the flow of information can be redistributed across different edges, but that the total flow is fixed. While this fact was not explicitly used in the work here, it may represent an independently useful way of reasoning about information flow in quantum circuits. We leave further investigations of this viewpoint as future work.