Investigating the effect of circuit cutting in QAOA for the MaxCut problem on NISQ devices

QAOA enables solving combinatorial optimization problems such as the MaxCut [7] and the maximum independent set problem [16]. The goal is to find a bitstring $\boldsymbol{z} = (z_{1},\ldots,z_{n})\in\{0,1\}^{n}$ that maximizes an objective function $\mathcal{C}(\boldsymbol{z})$ which is encoded on the diagonal of the cost Hamiltonian H_C such that $H_{C}\vert\boldsymbol{z}\rangle = \mathcal{C}(\boldsymbol{z})\vert\boldsymbol{z}\rangle$. Thus, the optimization problem can be solved by finding the eigenvector $\vert\boldsymbol{z}\rangle$ of H_C with maximal eigenvalue

$$\begin{equation} \max_{\vert\boldsymbol{z}\rangle} \langle\boldsymbol{z}\vert H_{C} \vert\boldsymbol{z}\rangle = \mathcal{C}_{\max}. \end{equation} \tag{ 1 }$$

To this end, QAOA employs an ansatz that applies alternately p times the cost operator $U(H_{C}, \gamma_{l}) = \exp{(-i\gamma_{l}H_{C})}$ and mixing operator $U(H_{B}, \beta_{l}) = \exp{(-i\beta_{l}H_{B})}$ to the initial state $\vert +\rangle^{\otimes n}$ [7]. Herein, $H_{B} = \sum_{i}X_{i}$, where X_i is the Pauli X matrix applied to the ith qubit. Thus, the result state of the QAOA ansatz is:

$$\begin{equation} \vert\psi(\boldsymbol{\beta}, \boldsymbol{\gamma})\rangle = \left(\prod_{l = 1}^{p}U(H_{B}, \beta_{l})U(H_{C}, \gamma_{l}) \right)\vert +\rangle^{\otimes n} \end{equation} \tag{ 2 }$$

where $\boldsymbol{\beta} = (\beta_{1}, \ldots, \beta_{p})$ and $\boldsymbol{\gamma} = (\gamma_{1}, \ldots, \gamma_{p})$ are variational parameters. A classical optimizer updates the parameters $\boldsymbol{\beta}, \boldsymbol{\gamma}$ to maximize the expectation value of the observable H_C on the ansatz $\vert\psi(\boldsymbol{\beta}, \boldsymbol{\gamma})\rangle$:

$$\begin{equation} \langle H_{C}\rangle_{\boldsymbol{\beta}, \boldsymbol{\gamma}} = \langle\psi(\boldsymbol{\beta}, \boldsymbol{\gamma})\vert H_{C}\vert\psi(\boldsymbol{\beta}, \boldsymbol{\gamma})\rangle. \end{equation} \tag{ 3 }$$

In general, higher values of p can result in better approximations of the optimal solution, but at the cost of more computations [7]. For the sake of simplicity, we shall write $\langle H_{C}\rangle_{\beta_{1}, \gamma_{1}} = \langle H_{C}\rangle_{(\beta_{1}), (\gamma_{1})}$ for p = 1.

A commonly studied problem for QAOA is the MaxCut problem [7, 11, 12, 17–19]. It occurs in many application fields, e.g. solid-state physics and integrated circuit design [20] as well as data clustering [21]. Thus, solving the MaxCut problem efficiently helps speeding up computing solutions of these problems. A cut of a graph $G = (V, E)$ splits its set of vertices V into two partitions and can be represented as a bitstring $\boldsymbol{z}\in\{0, 1\}^{|V|}$, where each bit assigns one of the vertices to one of the two partitions. The size of the cut is the number of edges crossing these two partitions. A maximum cut is at least as large as any other cut of the graph. Finding such a cut for a given graph is known as the MaxCut problem, a well-known NP-hard problem [22].

To solve the MaxCut problem with QAOA [7], the cost Hamiltonian for a graph G is defined as

$$\begin{equation} H_{C_{\text{MaxCut}}} = \frac{1}{2}\sum_{(v,w) \in E}(I - Z_{v}Z_{w}) \end{equation} \tag{ 4 }$$

where Z_v is the Pauli Z matrix applied to the vth qubit. Thus, each individual sum-term $I - Z_{v}Z_{w}$ of $H_{C_{\text{MaxCut}}}$ increases the eigenvalue of a solution z if and only if $z_{v} \ne z_{w}$, that is, if and only if the corresponding edge (v, w) crosses the induced partitions of the cut. The corresponding unitary operator for the Hamiltonian $H_{C_{\text{MaxCut}}}$ is

$$\begin{equation} U(H_{C_{\text{MaxCut}}}, \gamma_{l}) = \exp{(-i\gamma_{l}|E|/2)}\prod_{(v,w) \in E}\exp{(i\gamma_{l}Z_{v}Z_{w}/2)} \end{equation} \tag{ 5 }$$

which can be implemented as a product of $R_{ZZ}(-\gamma_{l}) = \exp{(i\gamma_{l}Z_{v}Z_{w}/2)}$ gates up to a global factor $\exp{(-i\gamma_{l}|E|/2)}$. The corresponding unitary operator for the Hamiltonian H_B is

$$\begin{equation} U(H_{B}, \beta_{l}) = \prod_{v \in V} \exp{(-i\beta_{l}X_{v})} \end{equation} \tag{ 6 }$$

which is a product of $R_{X}(2\beta_{l}) = \exp{(-i\beta_{l}X_{v})}$ gates. The $R_{ZZ}(\gamma_{l})$ gate implements a two-qubit rotation about $Z\otimes Z$ with angle $\gamma_l$ and the $R_{X}(2\beta_{l})$ gate implements a single qubit rotation about X with angle $2\beta_{l}$. Thus, the problem instance of the MaxCut problem determines the ansatz structure since each edge of the graph translates to a two-qubit R_ZZ gate in the ansatz. However, problem structures of practical interest often cannot be trivially mapped to a planar architecture of current NISQ devices with restricted qubit connectivity [11]. Instead, their quantum circuits have to be adapted by inserting additional SWAP operations to permute qubits and adapt the circuit to the connectivity of the quantum device [23].

Quantum circuit cutting is a set of techniques that enables to split a quantum circuit into multiple smaller circuits with fewer qubits and gates such that the result of executing the collection of the smaller circuits is the same as the result of executing the original circuit by exploiting subsequent classical postprocessing [14, 15]. The primary focus of quantum circuit cutting is on reducing the number of required qubits, but the subcircuits also consist of fewer gates such that cutting may also reduce the circuit depth of the executed circuits. The circuit cutting process consists of three steps: (i) cutting the circuit into a set of smaller subcircuits, (ii) execution of these subcircuits, and (iii) the classical recombination of the subcircuit results. There are two different approaches to cut a circuit: (i) gate cutting replaces two-qubit gates by sets of local operations [15, 24] and (ii) wire cutting divides circuit wires carrying quantum information in the form of a qubit into sets of measurement and state-preparation operations [14]. In this work, gate cutting is used, which is presented in figure 1 and described in more detail below.

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Let $\mathcal{H}^{(1)}\otimes \mathcal{H}^{(2)}$ denote a bipartite n-qubit quantum system consisting of Hilbert spaces $\mathcal{H}^{(1)}$ and $\mathcal{H}^{(2)}$. The quantum state of this system can be described by a density operator $\rho$ , which is a positive Hermitian matrix of size $2^{n} \times 2^{n}$ with trace equal to one. Moreover, consider an n-qubit gate represented by a unitary U. The evolution of the quantum state under the action of gate U can be described by a superoperator $\mathcal{S}(U)$, which is a linear operator that acts on the space of density operators. The new state resulting from the application of U on the state described by $\rho$ is given by $\mathcal{S}(U)\rho = U\rho U^{\dagger}$.

A gate cut of the unitary operator U is its quasiprobability decomposition (QPD) into a set of operators $V_{i} = (V_{i}^{(1)}\otimes V_{i}^{(2)})$ with associated complex factors $c_i$ given by $\{(V_{i}, c_{i})\}$ such that

$$\begin{equation} \mathcal{S}(U) = \sum_{i}c_{i}\mathcal{S}\left(V_{i}\right) = \sum_{i}c_{i} \mathcal{S}\left(V_{i}^{(1)}\right)\otimes \mathcal{S}\left(V_{i}^{(2)}\right) \end{equation} \tag{ 7 }$$

where $V_{i}^{(1)}$ and $V_{i}^{(2)}$ are operators acting on subsystem $\mathcal{H}^{(1)}$ and $\mathcal{H}^{(2)}$, respectively [15]. Each operator $V_{i}^{(j)}$ has to be physically realizable, that is, its superoperator $\mathcal{S}(V_{i}^{(j)})$ is a completely positive linear map that is trace-nonincreasing, i.e. $0 \leqslant \mathrm{tr}[\mathcal{S}(V_{i}^{(j)})\rho] \leqslant 1$ for any density operator $\rho$ [25]. Therefore, this includes non-unitary transformations such as projections. The decomposition of the gate U is depicted in the first step of figure 1.

Consider for the partition of the quantum system $\mathcal{H}^{(1)}\otimes \mathcal{H}^{(2)}$ an observable $O = O^{(1)}\otimes O^{(2)}$ that acts independently on each subsystem, and a separable initial state $\rho_{0} = \rho^{(1)}_{0}\otimes\rho^{(2)}_{0}$. Thus, the subsystems are independent of each other concerning their initialization and measurement. Then, the evolution of density operator _{$\rho_0$} according to the unitary U and subsequent measurement with observable O can now be reproduced by applying the operations V_i and weighting their measurement with O according to $c_i$:

$$\begin{align} \mathrm{tr}\left[O\mathcal{S}(U)\rho_{0}\right] & = \sum_{i}c_{i} \mathrm{tr}\left[O\mathcal{S}(V_{i})\rho_{0}\right] \end{align} \tag{ 8 }$$

$$\begin{align} & = \sum_{i}c_{i} \underbrace{\mathrm{tr}\left[O^{(1)}\mathcal{S}\left(V_{i}^{(1)}\right)\rho^{(1)}_{0}\right]}_{: = R_{i}^{(1)}} \underbrace{\mathrm{tr}\left[O^{(2)}\mathcal{S}\left(V_{i}^{(2)}\right)\rho^{(2)}_{0}\right]}_{: = R_{i}^{(2)}}. \end{align} \tag{ 9 }$$

This allows us to evaluate each of the expectation values $R_{i}^{(1)}$ and $R_{i}^{(2)}$ individually and then recombine them to produce the expectation value of the original circuit as shown in step two and three of figure 1. The computational overhead of this procedure can be measured by the number of additional shots needed to approximate the expectation value of the original circuit [26]. Although the expectation value of the original circuit, computed from the subcircuit results, remains unchanged, additional shots are necessary due to the increased variance resulting from sampling the subcircuits. This increase in variance is characterized by the factor $\kappa : = \sum_{i}|c_i|$ with $\kappa \geqslant 1$. Previous studies have demonstrated that achieving a fixed statistical accuracy requires an additional $\mathcal{O}(\kappa^2/\epsilon^2)$ shots for estimating the original expectation value within the error [27]. It is worth noting that the computational cost of the classical recombination step is limited by the number of shots on the subcircuits, as only the sampled results need to be recombined through multiplication.

Given cuts for gates U and $\tilde{U}$ with QPDs $\{(V_{i}, c_{i})\}$ and $\{(\tilde{V_{i}}, \tilde{c}_{i})\}$, respectively, the cut of their product $U\tilde{U}$ can be constructed as following

$$\begin{align} \mathcal{S}(U)\mathcal{S}(\tilde{U}) & = \sum_{i} c_{i} \mathcal{S}(V_{i}) \sum_{j}\tilde{c}_{j}\mathcal{S}(\tilde{V_{j}}) \end{align} \tag{ 10 }$$

$$\begin{align} & = \sum_{i,j} c_{i}\tilde{c}_{j} \mathcal{S}(V_{i})\mathcal{S}(\tilde{V_{j}}). \end{align} \tag{ 11 }$$

Thus, a quantum circuit constructed of multiple subsequent gates can be cut by cutting its individual gates. However, the overhead, measured in terms of the number of additional shots, grows in the worst case exponentially as each cut introduces a multiplicative factor $\kappa$ [26].

The only multi-qubit gates in the MaxCut ansatz for QAOA are R_ZZ gates. Thus, the ansatz for a given graph can be cut by partitioning its qubits into separate parts and then cutting all R_ZZ gates between these qubit partitions. To cut a single R_ZZ gate, we use the QPD for cutting arbitrary two-qubit gates introduced by Mitarai and Fujii [15, 24]:

$$\begin{equation} \begin{split} \mathcal{S}\!\left(R_{ZZ}(\gamma)\right)\! = \cos^{2}\left(\frac{\gamma}{2}\right) \mathcal{S}\!\left(I\otimes I\right)+ \sin^{2}\left(\frac{\gamma}{2}\right) \mathcal{S}\!\left(Z\otimes Z\right) \quad+ \cos\left(\frac{\gamma}{2}\right)\sin\left(\frac{\gamma}{2}\right)\left(A\otimes B + B\otimes A\right) \end{split} \end{equation} \tag{ 12 }$$

with

$$\begin{equation} A : = \mathcal{S}\!\left(R_Z\!\left(-\frac{\pi}{2}\right)\!\right) - \mathcal{S}\!\left(R_Z\left(\frac{\pi}{2}\right)\!\right) \end{equation} \tag{ 13 }$$

and

$$\begin{equation} B : = \mathcal{S}\!\left(\dfrac{I - Z}{2}\right) - \mathcal{S}\!\left(\dfrac{I + Z}{2}\right). \end{equation} \tag{ 14 }$$

The two operations forming B are the projections on the states $\vert 1\rangle$ and $\vert0\rangle$, respectively, i.e. $(I-Z)/2 = \vert 1 \rangle\langle 1 \vert$ and $(I+Z)/2 = \vert 0 \rangle\langle 0 \vert$. Although these operations are not unitary and thus cannot be directly implemented as a gate of a quantum circuit, they can be realized by a post-selective measurement [15]. By applying this QPD, the circuit with the R_ZZ gate can be replaced by circuits that perform at each of the two previously connected qubits one of the following five operations instead: an I, Z, $R_Z\left(-\frac{\pi}{2}\right)$, or $R_Z\left(\frac{\pi}{2}\right)$ gate, or a measurement for the projections. Since the two qubits are now separable, ten different subcircuits consisting of the five operations for the upper and five for the lower qubit must be executed. Appendix A provides a proof of the cutting formula in equation (12).

We adapt the approach described in section 2.3 to reduce the number of subcircuits. Our adjustment of equation (13) is as follows:

$$\begin{equation} A = \mathcal{S}(I) + \mathcal{S}(Z) - 2 \mathcal{S}\!\left(R_Z\left(\frac{\pi}{2}\right)\!\right). \end{equation} \tag{ 15 }$$

The mathematical details are provided in appendix B. By harnessing this equality, the number of different subcircuits of the R_ZZ gate decreases from ten to eight since the two local $R_Z\left(-\frac{\pi}{2}\right)$ gates disappear and A can be expressed as a weighted sum of the I, Z, and $R_Z\left(\frac{\pi}{2}\right)$ gates. The results for the I and Z operations can be reused as they are stored in classical memory. Although this substitution reduces the number of subcircuits, it introduces larger factors c_i in the QPD. This does not change the equality, but it leads to a higher variance, and therefore, more shots are needed for convergence [63]. However, this can be a worthwhile trade-off for a small number of cuts as the factors for them remain relatively small. This technique is applied in all our experiments.

This section describes the steps that all experiments have in common. At a high level, each experiment consists of three parts as depicted in figure 2: (i) the generation of the problem instance, the corresponding circuits, and subcircuits, (ii) the execution of the latter on quantum devices, and (iii) the postprocessing including the recombination of subcircuit results and the evaluation of the objective function. Experiments can repeat steps (ii) and (iii) in a variational optimization loop to refine the circuit parameters with a classical optimizer until a termination condition is met. Hereafter, we will call the QAOA ansatz that uses circuit cutting cut-QAOA. As quantum devices for the execution, we use IBM's superconducting hardware. The experiments are implemented with Qiskit [64]. Wherever a classical optimizer is necessary, COBYLA, in its default configuration from Qiskit, is employed. Since the optimizer only supports minimization, we transformed the maximization of QAOA's expectation value in our implementation into a minimization problem by flipping the sign of the classically computed objective function. Thus, the optimal value corresponds to the minimal objective value. In the following, the individual steps are described in more detail.

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5.2.1. Problem instance generation, circuit generation, and cutting

First, a problem instance has to be created. This is a crucial step since the edges of the graph map to the R_ZZ gates in the QAOA ansatz and therefore, the graph structure determines the number of cuts that are later needed to separate the ansatz. To enable the resulting ansatz to be separable with a specific number of cuts, we first generated two independent connected random subgraphs of equal size with the Erdõs–Rényi $G(n,p)$ model [65], i.e. each possible edge in a graph with n vertices is selected with the probability p. Here, we enforce that each subgraph is connected by repeating the generation procedure until it yields two connected subgraphs. This is illustrated in step a of the generation in figure 2. Afterward, we connect the two subgraphs by randomly inserting two edges between them and thereby fixing the locations of two intended cuts (see step b).

Following this, we generate an ansatz based on the resulting graph where the parameters can be inserted afterward (see step c). In our experiments, only QAOA ansätze with depth p = 1 are considered, meaning that the cost Hamiltonian and mixer Hamiltonian are executed only once. Although the IBM quantum devices used in our experiments do not natively support R_ZZ gates, we use these gates in the circuit generation phase to apply our cutting procedure and leave it to the transpiler to replace the remaining R_ZZ gates with native gates, i.e. two CNOT gates and a R_Z rotation.

There are multiple equivalent QAOA ansätze for each problem instance since the R_ZZ gates in the cost unitary $U(H_{C}, \gamma_{l})$ commute, and thus they can be arranged in an arbitrary order. Different orderings of the R_ZZ gates have no impact on a noise-free device and lead to equivalent results. However, this is different for NISQ devices. Depending on the arrangement of operations, the depth of the circuit may vary. In addition, specific arrangements of operations can be mapped to the restricted connectivity of the quantum device with fewer additional SWAP operations and consequently lead to transpiled circuits of a lower depth. Moreover, independent simultaneous operations on different qubits may affect each other unintendedly and hence are another source of noise [66]. All these issues have to be taken into account for optimal circuit transpilation. However, optimal circuit transpilation is NP-complete, and transpilers rely mostly on heuristics [67]. Moreover, the used Qiskit transpiler does not conduct all these optimizations without providing custom transpiler passes [68]. Therefore, to incorporate the effect of different arrangements of operations in our experiments and compare them to the effect of circuit cutting, we consider this in our experiments manually and generate two equivalent ansätze for each problem instance. One ansatz optimizes the number of simultaneous gates in the circuit by executing the gates of the two graph partitions sequentially, and the other ansatz optimizes the depth of the circuit by executing the gates of the two graph partitions in parallel. They are schematically depicted in figure 3. The sequential version of the ansatz starts with all R_ZZ operations of the first partition. Then we apply the operations that get cut. Finally, we add the R_ZZ operations for the second partition. The parallel version of the ansatz applies all R_ZZ operations of both graph partitions in parallel and then the gates that get cut in a final step. We use the same arrangement of R_ZZ operations within the partitions in both versions as well as in the subcircuits of the cut ansatz. The sequential version results in a deeper circuit with fewer parallel operations. In contrast, the parallel ansatz is shallower but with more parallel operations.

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Next, the generated circuit gets cut. To produce its subcircuits that are depicted in step d of figure 2, we use the following procedure:

• (i)  
  Remove all multi-qubit operations between the different qubit partitions corresponding to the two subgraphs such that the circuit decomposes into separable circuit fragments.
• (ii)  
  Generate for each of these circuit fragments their respective subcircuits by inserting all possible combinations of I, Z, $R_Z\left(\frac{\pi}{2}\right)$ and measurement at the positions of the removed gates.

5.2.2. Execution

5.2.3. Postprocessing

In the following, a detailed description of the two experimental setups used for the evaluation is given.

5.3.1. Experiment 1: objective function

5.3.2. Experiment 2: QAOA execution

In this section, we introduce metrics to assess and quantify the objective functions computed on the quantum devices in our first experiment for both QAOA variants, QAOA and cut-QAOA. Therefore, we consider metrics for the objective function themselves and its gradients. First, we assess the mean absolute objective value difference between a QAOA variant on a quantum device and QAOA without cutting on the simulator. The smaller the difference, the closer the computed objective values of the NISQ device are to the noise-free values of the simulator. The mean absolute difference (MAD) between the simulated objective function $\langle H_{C}\rangle_{\beta, \gamma}^{\text{SIM}}$ and the computation on a NISQ device $\langle H_{C}\rangle_{\beta, \gamma}^{\text{QPU}}$ for the finite parameter sets $P_{\beta} \subset [0, \pi]$ and $P_{\gamma} \subset [0, 2\pi]$ is

$$\begin{equation} \operatorname{MAD}_{\text{SIM}} = \frac{1}{|P_{\beta}||P_{\gamma}|}\sum_{\beta \in P_{\beta}, \gamma \in P_{\gamma} }\left|\langle H_{C}\rangle_{\beta, \gamma} ^{\text{QPU}} - \langle H_{C}\rangle_{\beta, \gamma} ^{\text{SIM}}\right|. \end{equation} \tag{ 16 }$$

Secondly, we have examined the mean absolute objective value difference between a QAOA variant on a NISQ device and the maximally mixed state (MMS), i.e. the state with density operator $I/2$. It measures the concentration of the objective function and is an indicator for NIBPs [10]. The larger its distance from the objective value of the MMS, the more the computed objective values of the QAOA variant deviate from the objective value achieved by randomly picking states with equal probability. The MAD from the objective value of the MMS of n qubits is

$$\begin{equation} \operatorname{MAD}_{\text{MMS}} = \frac{1}{|P_{\beta}||P_{\gamma}|}\sum_{\beta \in P_{\beta}, \gamma \in P_{\gamma} }\left|\langle H_{C}\rangle_{\beta, \gamma} - \frac{1}{2^{n}}\mathrm{tr}\left[H_{C}\right]\right|. \end{equation} \tag{ 17 }$$

This metric can be calculated for the NISQ device and the simulator. We compute the objective value of the MMS classically by sampling uniformly at random from the solution space and computing the expected objective value for the sample.

Furthermore, we assess the gradients of the objective function's computed parameter landscapes as NIBPs lead to vanishing gradients. The gradient of the objective function is

$$\begin{equation} \nabla \langle H_{C}\rangle_{\beta, \gamma} = \left(\frac{\partial}{\partial\beta}\langle H_{C}\rangle_{\beta, \gamma}, \frac{\partial}{\partial\gamma}\langle H_{C}\rangle_{\beta, \gamma}\right)^{T}. \end{equation} \tag{ 18 }$$

We numerically computed the partial derivatives using second-order accurate central differences [69]. To assess the gradients of the computed parameter landscapes, we consider their size [10] and the mismatch in their direction compared to the simulation. For the size of the gradient, we have taken into account their average size and the variance in size. The average gradient size is defined as

$$\begin{equation} M = \frac{1}{|P_{\beta}||P_{\gamma}|}\sum_{\beta \in P_{\beta}, \gamma \in P_{\gamma} } \left\lVert \nabla \langle H_{C}\rangle_{\beta, \gamma} \right\rVert_{1} \end{equation} \tag{ 19 }$$

where $\left\lVert \cdot \right\rVert_{1}$ is the one-norm. The variance of the gradient size is

$$\begin{equation} \frac{1}{|P_{\beta}||P_{\gamma}|}\sum_{\beta \in P_{\beta}, \gamma \in P_{\gamma} } \left(\left\lVert \nabla \langle H_{C}\rangle_{\beta, \gamma} \right\rVert_{1} - M \right)^{2}. \end{equation} \tag{ 20 }$$

A decreasing variance means that smaller and smaller changes in the gradients occur, which makes it more and more difficult for a classical optimizer to find an optimum. Besides the pure size of the gradients, the gradients must also guide the classical optimizer to the optima. For this, we analyze the direction of the gradients. We compare the gradient direction from the NISQ device with the simulated gradients. To this end, we employ the average pairwise cosine similarity between those gradients [70, 71]. It is defined as follows:

$$\begin{equation} \frac{1}{|P_{\beta}||P_{\gamma}|}\sum_{\beta \in P_{\beta}, \gamma \in P_{\gamma} } \frac{\nabla \langle H_{C}\rangle_{\beta, \gamma}^{\text{QPU}} \cdot \nabla \langle H_{C}\rangle_{\beta, \gamma}^{\text{SIM}}}{\left\lVert \nabla \langle H_{C}\rangle_{\beta, \gamma}^{\text{QPU}}\right\rVert_{2} \left\lVert \nabla \langle H_{C}\rangle_{\beta, \gamma}^{\text{SIM}}\right\rVert_{2}}. \end{equation} \tag{ 21 }$$

Following our experiment design, the first question to consider is how cutting the QAOA ansatz influences the corresponding objective function using NISQ devices. In the following, this is first discussed in detail, for one example graph, and then the results for the whole dataset are presented.

6.1.1. Example graph

In this section, the computed objective functions for an example graph are visualized and evaluated using the introduced metrics for the experiment. Figure 4 shows the ten-node example graph (figure 4(a)) with its corresponding noise-free objective function (figure 4(b)). The cut of edges (0, 8) and (4, 9) separates the graph into two subgraphs of equal size. The objective function is evaluated on a simulator for p = 1 QAOA layers. The desired minima are shown in shades of red in the colorized parameter map. The higher the objective value, the more the color changes to a blue tone.

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Figure 5 visualizes the objective functions as parameter landscapes with 1000 and 10 000 shots for QAOA and cut-QAOA, respectively, executed on the NISQ device ibmq_ehningen, a 27-qubit superconducting quantum device by IBM with their Falcon chip architecture. All parameter landscapes are colorized with the same color scale as used for the simulated result in figure 4(b). It is evident that QAOA and cut-QAOA computed on ibmq_ehningen achieve a smaller range of objective values than the simulated objective function (see figures 4(b) and 5). However, the QAOA ansatz achieves a substantially smaller range of objective values than the cut-QAOA. As seen in figure 5, the maxima and minima of the cut-QAOA emerge more clearly from the plane compared to QAOA. While with QAOA no noticeable change in the objective function can be seen between 1000 and 10 000 shots, with cut-QAOA the objective function becomes noticeably smoother.

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In figure 6, the introduced metrics for the objective functions above are plotted as a function of the number of shots. It can be seen that the objective function of cut-QAOA is closer to the simulated objective function. The MAD between cut-QAOA on ibmq_ehningen and the simulation is significantly lower than for QAOA (figure 6(a)). Moreover, the deviation of cut-QAOA from the objective value of the MMS is substantially higher than the deviation of the QAOA ansatz (figure 6(b)). Furthermore, the more pronounced minima and maxima in the cut-QAOA objective function result in larger gradients with higher variance (figures 6(c) and (d)). The average gradient size of the QAOA ansatz is significantly lower than with cutting, and the variance of gradient size is close to zero, indicating approximately equal size among the gradients. In contrast, the objective function of cut-QAOA has strikingly increased gradients of higher variance. This implies more salient structures in the computed objective function. However, cut-QAOA still does not achieve the same magnitude and variance of gradients as the simulator. Apart from the gradient size, the gradients of the cut-QAOA ansatz have a significantly higher cosine similarity value than the QAOA ansatz on the quantum devices (figure 6(e)). Consequently, the mismatch in the gradient direction is smaller, and thus, the optimum and the optimization path to the optimum deviates less from the noise-free objective function.

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All metrics as functions of the number of shots show the same pattern in figure 6. They are approaching a plateau. A reasonable explanation for this phenomenon is shot noise which refers to statistical sampling errors that occur when estimating the result distribution of a quantum circuit with a finite number of shots [73]. With an increasing number of shots, the sampled result distribution of a quantum circuit converges to its expected distribution. This behavior manifests itself in decreasing values for all metrics as the number of shots increases except for gradient similarity. There, the values increase as they approach a plateau. The more the curves approach the plateaus, the less shot noise matters, but NISQ errors remain. While this state is reached with QAOA with relatively few shots, cut-QAOA needs significantly more shots and thus is considerably more affected by shot noise. This effect also fits with the fact that cut-QAOA must distribute the number of shots among all subcircuits, and thus, each individual subcircuit receives significantly fewer shots.

6.1.2. Evaluation using a set of random graphs

Our data set includes 61 randomly generated graphs as listed in table 1. The individual graphs can be viewed in the published dataset [72]. Figure 7 shows each metric as a function of the number of shots for the random graphs from the data set. For each metric and QAOA variant, the mean value is plotted as a line. The shaded area around the mean represents a percentile interval of 80% of the data. Thus, the highest and lowest 10% of the values are not visualized. To ensure better comparability of the metrics between problem instances, the MAD to MMS, average gradient size and variance of gradient size were normalized by dividing by the metric value of the simulator. The other two metrics were not normalized further, since they already incorporated the simulator. The averaged values of the data set show the same patterns as observed for the example graph. First, all metrics approach a plateau value with an increasing number of shots. As in the example above, cut-QAOA converges significantly slower on average and starts further away from its plateau value than QAOA for all metrics. Both facts indicate that cut-QAOA is more susceptible to shot noise.

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Table 1. Statistics about the data set: (a) graphs and (b) quantum devices.

	Exp. 1			Exp. 2
Nodes	Count	Min Edges	Max Edges	Count	Min Edges	Max Edges
8	18	8	14	16	8	14
10	25	10	16	15	11	16
12	18	13	21	16	16	21
(a) Statistics about the graphs in the dataset.

Quantum device	Exp. 1	Exp. 2
ibmq_ehningen	39	31
ibmq_mumbai	10	6
ibmq_kolkata	6	5
ibmq_guadalupe	5	5
ibmq_montreal	1	0
(b) Used quantum devices for the experiments.

Moreover, the computed objective function of the cut-QAOA ansatz is closer to the simulated objective function and deviates more from the objective value of the MMS. Furthermore, its gradients are of larger size and have a higher variance in their magnitude. Additionally, the gradients of the objective function of the cut-QAOA are more similar to the simulated gradients. Although figure 7 shows only averaged metrics, the individual results of the experiments also follow these patterns and resemble the results of the example graph above (figure 4(a)). Moreover, while cut-QAOA performs better regarding the metrics, it tends to have a slightly higher spread in metric values across the problem instances except for gradient similarity, where the variance for QAOA is significantly larger.

However, these patterns are more or less pronounced depending on the number of nodes in the generated graph. With an increasing number of nodes, and thus large quantum circuits, the advantage of cut-QAOA in the metrics decreases and approaches the value of QAOA. For this, a probable reason is that with increasing graph size, the QAOA ansätze grow in depth and width, and so do the subcircuits of cut-QAOA. Thus, each subcircuit is more susceptible to the noise of the NISQ devices, and thus the combined result of cut-QAOA gets noisier.

Major observations:

• Compared to QAOA without cutting, the objective function of cut-QAOA computed on NISQ devices is closer to the simulated objective function.
• It deviates more from the objective value of the MMS.
• It has larger gradients with a higher variance that are more similar to the simulated gradients.
• However, cut-QAOA is more affected by shot noise.

Next, we evaluate whether these improvements in the computed objective function lead to better results in QAOA to clarify SRQ 2, i.e. lower expectation values and better MaxCut solutions. We extracted the shot results of the ansatz for the optimized parameters $\beta^{*}, \gamma^{*}$ for each run of QAOA and cut-QAOA and considered the obtained expectation value $\langle H_{C}\rangle_{\beta^{*}, \gamma^{*}}$ and the most frequently sampled state $z^{*}$. To compare these results among different MaxCut problem instances, we normalize the result with the optimal objective value $\mathcal{C}_{\text{opt}}$. This yields the expectation ratio

$$\begin{equation} r_{\text{exp}} = \dfrac{\langle H_{C}\rangle_{\beta^{*}, \gamma^{*}}}{\mathcal{C}_{\text{opt}}} \end{equation} \tag{ 22 }$$

and the approximation ratio

$$\begin{equation} r_{\text{appr}} = \dfrac{\mathcal{C}(z^{*})}{\mathcal{C}_{\text{opt}}}. \end{equation} \tag{ 23 }$$

While QAOA optimizes the expectation value, the state most frequently sampled is usually taken as the solution to the problem.

In addition to the computed solution with the QAOA variants on the quantum devices, we classically draw samples randomly from a uniform distribution of the solutions, i.e. we classically simulated sampling the MMS. This sample serves as a benchmark to show whether the QAOA variant on the quantum device performs better than random sampling. Furthermore, we executed QAOA on a noise-free simulator to obtain the expectation ratio and approximation ratio in an ideal, noise-free scenario. These noise-free solutions provide a reference point for evaluating the impact of noise and imperfections in the executions of the QAOA variants on the NISQ devices.

We conducted this experiment on a subset of graphs from experiment 1. The data set includes 47 graphs as described in table 1. The obtained expectation values can be seen in figure 8 for the different QAOA variants and graph sizes. Cut-QAOA achieves the highest expectation ratio, while QAOA performs hardly better than random sampling. However, as the graphs get larger, the advantage of cut-QAOA gets smaller, and it also approaches the expectation ratio of QAOA. Considering the results from experiment 1, the computed objective function gets noisier, and thus, it is less accurate and flattens. Therefore, it is harder for the classical optimizer to find optimal parameters, and even for optimal parameters, the sampled noisy solutions yield a lower expectation value in general. The fact that QAOA without cutting is only marginally different from random sampling suggests that noise in its computation prevails, rendering the results impractical. The better expectation values of cut-QAOA also translate to solutions of higher approximation ratio, as can be seen in figure 9. Cut-QAOA follows the previously established pattern and approaches the approximation ratio of QAOA with increasing graph size.

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While our findings offer valuable perspectives and potential avenues for further investigation, their generalizability may be limited due to the influence of various factors, such as the selection of the problem, the problem instances, and the experimental setup. Even though we focus on solving the MaxCut problem with QAOA in our work, the application of circuit cutting is independent of it since it operates at the circuit level and is detached from the underlying problem and algorithm. Moreover, the experiments were only conducted with a limited set of graphs due to the restricted availability of NISQ devices and execution time on these. All investigated graphs have either eight, ten, or twelve nodes since, for graphs with more nodes, the noise dominates even with cut-QAOA. To control the number of cuts per problem instance in the experiments, each graph in the dataset has a particular structure: it consists of two equal-sized subgraphs connected by two edges. The choice of two edges has two reasons. On the one hand, the solution to the MaxCut problem cannot be generated in general by combining the solutions of the two subgraphs. On the other hand, the used circuit cutting approach must execute only 32 subcircuits for the two cuts. Other work focuses on experiments with more than two cuts [33, 34] and also on significantly reducing the number of required subcircuits [41]. Regarding the selection of the NISQ devices, the experiments were conducted on multiple devices of IBM's Falcon chip architecture. Furthermore, the calibration of the devices changes periodically and cannot be controlled by the user. This harms the reproducibility of the experiments. To compensate for this, we have included all calibration data of the quantum devices in the dataset to improve the traceability of results [72]. Despite all these limitations, we obtained comparable results for the different graphs, NISQ devices, and their calibrations that show the same patterns concerning the application of circuit cutting.

The connection between the computed objective function's quality and the quality of the result of the algorithm becomes evident when combining the data from the two conducted experiments. We can see a correlation between the introduced metrics of objective functions and the expectation and approximation ratio of the corresponding QAOA results. QAOA tends to produce better results for objective functions that perform better concerning the metrics. Figure 10 visualizes the correlations between three metrics and the increase in expectation ratio compared to the MMS, i.e. $\left(\langle H_{C}\rangle_{\beta^{*}, \gamma^{*}} - \frac{1}{2^{n}}\mathrm{tr}\left[H_{C}\right]\right) \mathcal{C}_{\text{opt}}^{-1}$. The correlation of other metrics can be found in the data set [72]. While there are always runs where no improvement was achieved, e.g. due to poor initial parameters and local optima, the advantage in expected value increases as the metrics improve. This positive correlation supports the relevance of the metrics used, and in addition, it fortifies the assumption that an improved objective function aids the classical optimizer in finding better solutions. Furthermore, these results were obtained, although the COBYLA optimizer in the default configuration was employed, i.e. without adjusting its hyperparameters to the optimization task. Further tuning of the optimizer to the underlying problem and noise can lead to better results [75].

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