An expressive ansatz for low-depth quantum approximate optimisation

Many real-world problems can be phrased as combinatorial optimisation problems [64]. Here, we lay emphasis on XQAOA’s application to an archetypal problem known as MaxCut, which has numerous applications in computer science and operations research, including statistical physics and circuit layout design [65], analysis of social networks [66], data clustering [67], semi-supervised learning [68], and more [69, 70]. The (weighted) MaxCut problem is an optimisation problem in which we are given an undirected weighted graph and asked to partition its vertices into two disjoint sets S and $\overline S$ such that the sum of the weights of the edges between the two sets is as large as possible.

Formally, given an undirected graph $G = (V, E)$ and non-negative weights $w_{uv} = w_{vu}$ on the edges $\{u, v\} \in E$, the MaxCut problem is that of finding a set S of vertices that maximises the weight of the edges in the cut $(S, \overline{S})$; that is, the weight of the edges with one endpoint in S and the other in $\overline{S}$. The MaxCut problem can be formulated as a binary quadratic program of the form

$$\begin{align} \begin{array}{ll@{}ll} \textrm{Maximise} & \displaystyle\sum\limits_{\left\{u,v\right\} \in E} \frac{1}{2} w_{uv} \left(1 - y_uy_v \right) &\\[0.5cm] \textrm{s.t}& y_u \in \left\{-1, 1 \right\} \quad \forall u \in V. \end{array} \end{align} \tag{ 1 }$$

The optimisation problem given by equation (1) is $\mathsf{NP}$-hard ⁴ , which suggests that it is highly plausible that no efficient algorithm exists that can solve it.

However, there are approximation algorithms that can find good solutions in polynomial time for many instances of the problem. The GW algorithm holds the current record for an approximation ratio guarantee on generic graphs, achieving an approximation ratio of $r^* \approx 0.87856$ using semidefinite programming [63]. When confined to unweighted three-regular graphs, this lower bound can be increased to $r^* \approx 0.9326$ [72]. ⁵ Assuming the unique games conjecture [74] ⁶ and that $\mathsf{P} \neq \mathsf{NP}$, this is the best possible approximation ratio for MaxCut [75–77] that polynomial-time classical algorithms can achieve. Additionally, it has been proven that it is $\mathsf{NP}$-hard to approximate the MaxCut value with an approximation ratio that is better than $r^* \unicode{x2A7E} 16/17 \approx 0.94117$ [78, 79].

Combinatorial optimisation problems can be formulated using n bits and m clauses, where each clause represents a constraint on a subset of the bits that is satisfied for certain combinations of values for those bits but not for others. We consider the case when each clause µ is associated with a cost $c_\mu \in \mathbb{R}$. The objective function defined on n-bit strings is then given by the sum of the costs of the satisfied clauses:

$$\begin{align} C\left(z\right) = \sum\limits^m_{\mu = 1}\, C_\mu \left(z\right), \end{align} \tag{ 2 }$$

where $z = z_1 z_2\cdots z_n \in \{0,1\}^n$ is an n-bit string and $C_{\mu} (z) = c_\mu$ if z satisfies the clause µ and 0 otherwise. An approximate optimisation algorithm aims to find a string z that achieves a desired approximation ratio $r^\star$, i.e. it seeks a string z that satisfies

$$\begin{align} \frac{C\left(z\right)}{C_{\text{max} }} \unicode{x2A7E} r^{*}, \end{align} \tag{ 3 }$$

where $C_{\text{max} } = \text{max} _{z} C(z)$. The QAOA algorithm consists of two operators (see figure 1): the problem unitary and the mixing unitary, which are generated by the problem Hamiltonian and mixing Hamiltonian, respectively. The problem unitary is defined as the following unitary operator $U (C,\gamma)$ which depends on a real-valued angle $\gamma \in \mathbb{R}$:

$$\begin{align} U \left(C,\gamma\right) = e^{-i\gamma C} = \prod\limits^{m}_{\mu = 1}\, e^{-i\gamma C_\mu }. \end{align} \tag{ 4 }$$

The operators $C = \sum_z C(z) \left|z\right\rangle \!\!\left\langle z \right|$ and $C_\mu = \sum_z C_\mu(z) \left|z\right\rangle \!\!\left\langle z \right|$ are the diagonal operators whose entries are the objective function values. Next, the mixing unitary is defined as the β-dependent product of commuting one-qubit unitaries

$$\begin{align} U \left(B, \beta\right) = e^{-i\beta B} = \prod\limits^n_{\nu = 1}\, e^{-i \beta X_\nu}, \end{align} \tag{ 5 }$$

where $\beta \in [0, \pi)$ and B is the sum of all single-qubit Pauli-X operators

$$\begin{align} B = \sum\limits^n_{\nu = 1} X_\nu. \end{align} \tag{ 6 }$$

For any positive integer $p \unicode{x2A7E} 1$, the QAOA algorithm generates an angle-dependent quantum state using 2p angles, $\boldsymbol{\gamma} = [\gamma_1, \gamma_2, \ldots, \gamma_p]$ and $\boldsymbol{\beta} = [\beta_1, \beta_2, \ldots, \beta_p]$, where the subscripts of γ and β indicate the iterate number of the quantum ansatz. The quantum state has the form

$$\begin{align} \left|\boldsymbol{\gamma}, \boldsymbol{\beta}\right\rangle = U \left(B, \beta_p\right) \, U \left(C, \gamma_p\right) \cdots U \left(B, \beta_1\right) \, U \left(C, \gamma_1\right)\, \left|s\right\rangle, \end{align} \tag{ 7 }$$

where $\left|s\right\rangle$ denotes the uniform superposition over all n-bit strings

$$\begin{align} \left|s\right\rangle = \frac{1}{\sqrt{2^n}} \! \! \! \sum_{z\in \left\{0,1\right\}^n} \! \! \! \! \left|z\right\rangle . \end{align} \tag{ 8 }$$

We then compute the expectation value of C for the variational state described in equation (7)

$$\begin{align} \langle C \rangle = \left\langle \boldsymbol{\gamma}, \boldsymbol{\beta} \right| C \left|\boldsymbol{\gamma}, \boldsymbol{\beta}\right\rangle, \end{align} \tag{ 9 }$$

which is accomplished by repeated measurements of fresh copies of the quantum system in the computational basis. The optimal parameters $(\boldsymbol{\gamma}^*, \boldsymbol{\beta}^*)$ that maximise the expectation value $\langle C \rangle$ are found using a classical computer:

$$\begin{align} \left(\boldsymbol{\gamma}^*, \boldsymbol{\beta}^*\right) = \underset{\boldsymbol{\gamma}, \boldsymbol{\beta}}{\text{arg} \text{max} }\, \langle C \rangle . \end{align} \tag{ 10 }$$

Typically, this is performed by estimating the parameters and then optimising them using simplex or gradient techniques. The approximation ratio $r^*$ is a relevant metric for assessing the performance of QAOA, where

$$\begin{align} r^* = \frac{\langle C \rangle}{C_{\text{max} }}. \end{align} \tag{ 11 }$$

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We will focus on applying QAOA to the MaxCut problem for the rest of this paper. To this end, note that the optimisation problem in equation (1) is equivalent to finding the maximum eigenvalue of the problem Hamiltonian C for MaxCut:

$$\begin{align} C = \! \! \! \! \sum_{\{u,v\} \in E} \! \! \! C_{uv}, \quad \mbox{with } C_{uv} = \frac{1}{2}w_{uv} (1-Z_u Z_v ) , \end{align} \tag{ 12 }$$

where Z_i denotes the Pauli-Z matrix acting on the ith qubit.

Before proceeding with the rest of the section, let us make a few definitions that will be used throughout the paper. For $w \in V$, let $\mathcal{N}(w) = \{x \in V: \{x,w\} \in E \}$ be the set of neighbours of w, i.e. vertices which are adjacent to w. Then, for an edge $\{u, v\} \in E$, we have that

• $e = \mathcal{N}(v) \backslash \{u\}$ is the set of vertices other than u that are connected to v.
• $d = \mathcal{N}(u) \backslash \{v\}$ is the set of vertices other than v that are connected to u.
• $F = \mathcal{N}(u) \cap \mathcal{N}(v)$ is the set of vertices that form a triangle with the edge $\{u,v\}$. In other words, F is the set of vertices that are neighbours of both u and v.

The following theorem can be used to compute the expectation value of the cost function for QAOA at p = 1 (QAOA₁) for MaxCut on arbitrary weighted graphs, thereby allowing us to assess the performance of QAOA₁.

Theorem 1. Consider the $\mathrm{QAOA}_1$ state $\left|\gamma,\beta\right\rangle$ for MaxCut on an arbitrary weighted graph G. Then, the expectation value of C in $\left|\gamma,\beta\right\rangle$ is $\left\langle \gamma,\beta \right| C\left|\gamma,\beta\right\rangle = \sum_{\{u,v\}\in E} \langle C_{uv}\rangle$, where

$$\begin{align} \left\langle C_{u v}\right\rangle & = \frac{w_{uv}}{2} +\frac{w_{uv}}{4}\left[ \text{sin} 4 \beta \text{sin}\gamma_{uv}^{^{\prime}} \left(\prod_{w \in e}\text{cos}\gamma_{wv}^{^{\prime}} + \prod_{w \in d}\text{cos} \gamma_{uw}^{^{\prime}} \right) \right.\nonumber\\ & \quad + \left. \text{sin}^2 2 \beta \prod_{\substack{w \in e \\ w \notin F}}\text{cos}\gamma_{wv}^{^{\prime}} \prod_{\substack{w \in d \\ w \notin F}}\text{cos} \gamma_{uw}^{^{\prime}} \left( \prod_{f \in F} \text{cos}\left(\gamma_{uf}^{^{\prime}} + \gamma_{vf}^{^{\prime}}\right) - \prod_{f \in F} \text{cos}\left(\gamma_{uf}^{^{\prime}} - \gamma_{vf}^{^{\prime}}\right) \right) \right] \end{align} \tag{ 13 }$$

and $\gamma^{^{\prime}}_{ij} = \gamma w_{ij}$.

In appendix D.3, we give a proof of theorem 1, which we show follows as a straightforward corollary of our main theorem (theorem 3). By taking $w_{ij} = 1$ if $\{i,j\}\in E$ and 0 otherwise, equation (13) simplifies for unweighted graphs to:

$$\begin{align} \langle C_{uv} \rangle & = \frac 12 + \frac 14 \bigg\{\text{sin} 4\beta \text{sin}\gamma \left(\text{cos}^{|e|} \gamma +\text{cos}^{|d|} \gamma\right) \nonumber\\[0.5em] & \quad +\text{sin}^2 2\beta \text{cos}^{|e|+|d|-2|F|}\gamma \left(\text{cos}^{|F|} 2 \gamma-1\right) \bigg\}, \end{align} \tag{ 14 }$$

which has previously appeared as equation (14) of [80]. ⁷

From theorem 1, we see that at p = 1, the expectation value $\left\langle C_{u v}\right\rangle$ of any edge in a graph depends on only the nodes and edges adjacent to it. The overall expectation value for QAOA₁ can then be calculated by summing the expectation values over all edges in the graph. For an n-node graph, the right-hand side of equation (13) can be computed in linear time O(n). Since the total number of edges in any graph is at most $\binom{n}{2} = O(n^2)$, computing the expectation value of QAOA would take at most $O(n^3)$ time. However, to find an actual bit string that represents an approximate solution for an arbitrary graph, here we use the QAOA quantum circuit to generate a quantum state on which measurement is performed.

The MA-QAOA [57] varies from the original QAOA in that it allows each summand of the problem and mixing Hamiltonians to have its own angle, as opposed to these Hamiltonians having a single angle each ⁸ . In this modification for p = 1 (called MA-QAOA₁), the problem and mixing unitaries are defined as

$$\begin{align} U \left(\boldsymbol{C}, \boldsymbol{\gamma}\right) & = e^{-i \sum_\mu \gamma_\mu C_\mu} = \prod\limits^{m}_{\mu = 1}\, e^{-i\gamma_\mu C_\mu }, \ \mbox{and} \end{align} \tag{ 15 }$$

$$\begin{align} U \left(\boldsymbol{B}, \boldsymbol{\beta}\right) & = e^{-i \sum_\nu \beta_\nu B_\nu} = \prod\limits^n_{\nu = 1}\, e^{-i \beta_\nu X_\nu}, \end{align} \tag{ 16 }$$

respectively, where $\boldsymbol{C} = (C_\mu)_{\mu = 1,\ldots,m}$ and $\boldsymbol{B} = (B_\nu)_{\nu = 1,\ldots,n}$ denote collections of operators. Thus, MA-QAOA₁ generates an angle-dependent quantum state of the form

$$\begin{align} \left|\boldsymbol{\gamma}, \boldsymbol{\beta}\right\rangle & = U \left(\boldsymbol{B}, \boldsymbol{\beta}\right) \, U \left(\boldsymbol{C}, \boldsymbol{\gamma}\right)\, \left|s\right\rangle \nonumber\\[0.5em] & = \prod\limits^n_{\nu = 1}\, e^{-i \beta_\nu X_\nu} \prod_{\mu = 1}^m e^{-i \gamma_{\mu} C_{\mu}} \left|s\right\rangle, \end{align} \tag{ 17 }$$

where $\boldsymbol{\gamma} = [\gamma_1, \gamma_2, \dots, \gamma_m]$ and $\boldsymbol{\beta} = [\beta_1, \beta_2, \dots, \beta_n]$. The subscript in γ_µ refers to the µth clause, and the subscript in β_ν refers to the νth qubit. In the context of MaxCut, µ and ν index the edges and vertices, respectively, of the graph involved. The approximation ratio obtained using QAOA lower bounds that of MA-QAOA, and MA-QAOA’s guarantee of convergence to the exact solution as $p \to \infty$ follows immediately from [8, equation (10)] and from noting that MA-QAOA is a generalisation of QAOA.

Herrman et al [57] provide an analytical formula for computing the performance of MA-QAOA₁ on MaxCut for unweighted triangle-free graphs. We generalise their result with the following theorem, where we present an analytical formula for the expectation value of the cost function for MA-QAOA₁ for MaxCut on arbitrary weighted graphs, allowing for the assessment of MA-QAOA₁’s performance on general graphs.

Theorem 2. Consider the $\mathrm{MA\textrm{-}QAOA}_1$ state for MaxCut on an arbitrary graph G. Then, the expectation value of C in $\left|\boldsymbol{\gamma},\boldsymbol{\beta}\right\rangle$ is $\left\langle \boldsymbol{\gamma},\boldsymbol{\beta} \right| C\left|\boldsymbol{\gamma},\boldsymbol{\beta}\right\rangle = \sum_{\{u,v\}\in E} \langle C_{uv}\rangle_{\mathrm{MA}}$, where

$$\begin{align} \left\langle C_{u v}\right\rangle_{\mathrm{MA}} & = \frac{w_{uv}}{2} +\frac{w_{uv}}{2}\left[ \text{cos} 2 \beta_u \text{sin} 2 \beta_v \text{sin}\gamma_{uv}^{^{\prime}} \prod_{w \in e}\text{cos}\gamma_{wv}^{^{\prime}} + \text{sin} 2 \beta_u \text{cos} 2 \beta_v \text{sin} \gamma_{uv}^{^{\prime}} \prod_{w \in d}\text{cos} \gamma_{uw}^{^{\prime}} \right.\nonumber\\[0.5em] & \quad + \left. \frac{1}{2} \text{sin} 2 \beta_u \text{sin} 2 \beta_v \prod_{\substack{w \in e \\ w \notin F}}\text{cos}\gamma_{wv}^{^{\prime}} \prod_{\substack{w \in d \\ w \notin F}}\text{cos} \gamma_{uw}^{^{\prime}} \left( \prod_{f \in F} \text{cos}\left(\gamma_{uf}^{^{\prime}} + \gamma_{vf}^{^{\prime}}\right) - \prod_{f \in F} \text{cos}\left(\gamma_{uf}^{^{\prime}} - \gamma_{vf}^{^{\prime}}\right) \right) \right] \end{align} \tag{ 18 }$$

and $\gamma_{jk}^{^{\prime}} = \gamma_{jk}w_{jk}$.

We present, in appendix D.3, a proof of theorem 2, which again, is a straightforward corollary of our main theorem (theorem 3). Like equation (13), the expectation value $\left\langle C_{u v}\right\rangle_{\mathrm{MA}}$ for any edge in a graph depends on only its neighbouring nodes and edges, and the overall expectation value is the sum of the expectation values over all edges in the graph; hence, computing equation (18) for an arbitrary graph has a time complexity of $O(n^3)$. However, this time complexity has a larger constant prefactor compared to that of computing equation (13). While QAOA₁ involves only two hyperparameters regardless of the size of the problem, MA-QAOA₁ involves $|V| + |E| = O(n) + O(n^2)$ classical hyperparameters.

Our comparative analysis of the three XQAOA variants on three-regular graphs revealed that the $\mathrm{XQAOA}_1^{\mathrm{X = Y}}$ variant performed the best (see figure 3). When benchmarked against MA-QAOA₁, QAOA₁, CR, and the GW algorithm on D-regular graphs with 128 and 256 vertices for $3 \unicode{x2A7D} D \unicode{x2A7D} 10$, $\mathrm{XQAOA}_1^{\mathrm{X = Y}}$ consistently outperformed QAOA₁, MA-QAOA₁, and the CR algorithm on all graph instances. Notably, $\mathrm{XQAOA}_1^{\mathrm{X = Y}}$ demonstrated competitive performance against the GW algorithm for three and four-regular graphs and exceeded it for D-regular graphs with D > 4 (see figure 4) ¹² . The boxplots reveal that the lower, middle and upper quartile values of $\mathrm{XQAOA}_1^{\mathrm{X = Y}}$ are significantly higher than those of the GW algorithm for graphs with D > 4, indicating that $\mathrm{XQAOA}_1^{\mathrm{X = Y}}$ is more likely to produce a better solution irrespective of the parameter initialisation strategy. This robustness of the $\mathrm{XQAOA}_1^{\mathrm{X = Y}}$ to initial parameter choices can be attributed to its overparameterised ansatz, which is further explained in section 5.3.

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Our analysis also highlighted a linear increase in the approximation ratio of QAOA₁ with the degree of the graph. Its performance nears that of MA-QAOA₁ for D > 5, hinting at a possible reachability deficit [44, 62] , limiting MA-QAOA₁’s ability to find an approximate solution close to the optimal. It is also important to note that the QAOA₁ ansatz experiences the barren plateau phenomenon [89], with the size of the plateau increasing with the degree of the graph. To mitigate this, careful selection of initial points for the classical optimiser is crucial, as outlined in appendix C. The results of this strategy are also presented in figure 4 as QAOA$^*$.

In our numerical simulations with the $\mathrm{XQAOA}_1^{\mathrm{X = Y}}$ ansatz, we observed that as the classical optimiser converged to an optimum, the optimal angles $(\boldsymbol{\gamma}^*, \boldsymbol{\beta}^*)$ stabilised at specific values. Specifically, $\gamma_{uv}^*$ converged to values in $\{0, \pi, 2\pi\}$, ¹³ while $\beta_u^*$ converged to values in $\{\pi/4, 3\pi/4\}$ (see figure 5). When $\gamma_{uv}^* \in \{0, 2\pi\}$, the two-qubit R$_{\mathrm{ZZ}}(0)$ and R$_{\mathrm{ZZ}}(2\pi)$ gates act as the identity gate, leaving qubits u and v unchanged. When $\gamma_{uv}^* = \pi$, the two-qubit gate R$_{\mathrm{ZZ}}(\pi) = \left|+,+\right\rangle\left\langle -,- \right| + \left|+,-\right\rangle\left\langle -,+ \right| + \left|-,+\right\rangle\left\langle +,- \right| + \left|-,-\right\rangle\left\langle +,+ \right|$. In other words, it swaps the states $\left|+\right\rangle_u \leftrightarrow \left|-\right\rangle_u$ and $\left|+\right\rangle_v \leftrightarrow \left|-\right\rangle_v$. Since the initial state $\left|s\right\rangle$ is a product state $\left|+\ldots+\right\rangle$, the state after all two-qubit gates will be a product state of $\left|+\right\rangle$s and $\left|-\right\rangle$s. Specifically, qubit u will be in the $\left|+\right\rangle_u$ state if it is swapped an even number of times and in the $\left|-\right\rangle_u$ state if it is swapped an odd number of times. Finally, the action of the mixing gates is to convert the $\left|+\right\rangle$ and $\left|-\right\rangle$ states to $\left|0\right\rangle$ or $\left|1\right\rangle$ depending on the value of $\beta^*$:

$$\begin{align} \textrm{When } \beta^*_i = \pi/4\,, & \begin{cases} \left|+\right\rangle_i \rightarrow \left|1\right\rangle_i\\ \left|-\right\rangle_i \rightarrow \left|0\right\rangle_i\end{cases} \end{align} \tag{ 24 }$$

$$\begin{align} \textrm{and when } \beta_i^* = 3\pi/4\,, & \begin{cases} \left|+\right\rangle_i \rightarrow \left|0\right\rangle_i\\ \left|-\right\rangle_i \rightarrow \left|1\right\rangle_i\end{cases} \,, \end{align} \tag{ 25 }$$

allowing us to read off the classical bit-string. It is important to acknowledge that due to degenerate local optimums, a relatively small subset of angles might not stabilise at the specified values even after convergence. Given the benign characteristics of the $\mathrm{XQAOA}_1^{\mathrm{X = Y}}$ ansatz’s loss landscape, such deviations are highly unlikely. However, in cases where deviations occur, we force $\gamma^*_{uv}$ to take the closest value in $\{0,\pi,2\pi\}$. ¹⁴

Reviewing figure 5 from a different perspective, we observe that the $\mathrm{XQAOA}_1^{\mathrm{X = Y}}$ ansatz, initially set with random angles, creates a highly entangled quantum state with a multitude of superposed states. As optimisation progresses, this entanglement gradually decreases, leading to a marked reduction in the number of superposed states. When the optimum is reached, the entangling layer disappears, leaving the system in a singular definitive state. In essence, what begins as a distinctly quantum state, through the course of optimisation, evolves into a classical state. This transition, marked by the disappearance of the entangling layer, facilitates the extraction of the solution through classical means, negating the need for quantum computers. This quantum-to-classical transition raises a natural question of whether the entangling layer is fundamentally necessary. To answer this, we conducted further numerical simulations by setting $\boldsymbol{\gamma} = \mathbf{0}$ and optimising only the β angles. Our numerical results showed that without any entangling layer, the performance of $\mathrm{XQAOA}_1^{\mathrm{X = Y}}$ was similar to that of CR (see figure 6). In fact, when we set $\boldsymbol{\gamma} = \mathbf{0}$, equation (22) reduces to

$$\begin{align} \left\langle C_{u v}\right\rangle_{\mathrm{X = Y}} = \frac{1}{2} w_{uv} \left(1 - \text{sin} 2\beta_u \text{sin} 2\beta_v\right), \end{align} \tag{ 26 }$$

which is the same as equation (23) with the angles having an additional factor of 2. This demonstrates that the overparameterised entangling layer augments the landscape, making the gradient-based classical optimiser less susceptible to getting trapped in local optima.

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In previous sections, we showed that $\mathrm{XQAOA}_1^{\mathrm{X = Y}}$ achieves near-optimal solutions and surpasses QAOA₁ and MA-QAOA₁. This led us to compare XQAOA$^{\textrm{X = Y}}_p$’s performance with QAOA_p and MA-QAOA_p for p > 1. Lacking analytical formulas for expectation values for p > 1 and constrained by the computational overhead of large-scale quantum simulation, we performed small-scale simulations using Qiskit [87], benchmarking 20 random three-regular graph instances with 16 vertices each. Given the variability in expectation values due to the limited number of shots (1024), we utilised the Powell optimiser [90] with ten random restarts to optimise the ansatzes’ parameters. The simulation results, presented in figure 7, reveal that the median approximation ratio of XQAOA$^{\textrm{X = Y}}_p$ consistently outperforms QAOA_p and MA-QAOA_p up to p = 4, with $\mathrm{XQAOA}_1^{\mathrm{X = Y}}$’s median nearly reaching 1.0. Interestingly, while QAOA_p shows modest improvement with increasing depth, MA-QAOA_p peaks at p = 3 before declining at $p \unicode{x2A7E} 4$. ¹⁵ In contrast, XQAOA$^{\textrm{X = Y}}_p$ exhibits a gradual decline in performance beyond p = 1, particularly noticeable at $p \unicode{x2A7E} 4$. This decline is attributed to barren plateaus due to the circuit’s overexpressiveness, rather than a lack of effectiveness at higher depths [92, 93].

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We optimise the classical parameters of XQAOA₁, MA-QAOA₁, QAOA₁, and CR algorithms using a gradient-based classical optimiser. We start the optimisation process with a randomly chosen initial point which affects the quality of the solution found, especially if the optimisation landscape is non-convex and has non-trivial features. If the initial point is near a local optimum or barren plateau, the gradient-based optimiser will converge to the local optimum and return a suboptimal solution. As a result, XQAOA₁, MA-QAOA₁, QAOA₁, and CR algorithms have a wide range of approximate solutions for the same problem. The maximum approximation ratio returned by MA-QAOA₁ is 0.82, whereas XQAOA₁ can often achieve an approximation ratio of 1.0 with the right choice of initial parameters. The range of approximation ratios for the CR algorithm is similar to XQAOA₁, but numerical simulations suggest that the cost landscape of the CR algorithm may be difficult to navigate and plagued with local optima, which may require an exponential number of initial points for the CR to match XQAOA₁’s performance, thus negating the benefit of having a polynomial-time approximation algorithm.

The GW algorithm also has randomness in its process. After it solves the relaxed version of the MaxCut problem, it generates an n-dimensional random vector r to perform its hyperplane rounding to find the optimal cut. While GW has an expected approximation ratio of $0.87856$ in the worst case at the asymptotic limit, the distribution of the approximation ratio returned by the GW algorithm for a finite number of randomly generated r vectors and problem instances can vary significantly, which is why we see a distribution for the output of the GW algorithm. While XQAOA₁, MA-QAOA₁, and the CR algorithm each require solving the problem anew for each random initial point, the GW algorithm solves the relaxed MaxCut problem just once and then efficiently generates individual solutions through hyperplane rounding with randomly generated vectors, avoiding repetitive computations.

Computing expectation values of QAOA₁, MA-QAOA₁, and XQAOA₁ for MaxCut on arbitrary graphs all have a time complexity of $O(n^3)$, albeit with varying prefactors. XQAOA₁ is unique in that its entangling layer vanishes at the optimal solution, in contrast to QAOA₁ and MA-QAOA₁, which maintain their entangling layer post-convergence. This entangling layer in QAOA₁ and MA-QAOA₁ requires generating an entangled quantum state through quantum computation followed by measurements to assign values to variables, precluding efficient classical simulation. In contrast, XQAOA₁’s optimal state, characterised by zero gammas, is non-entangled and permits classical bit assignment without quantum computation. However, in cases where XQAOA₁ falls short, a p > 1 XQAOA_p might be required, necessitating quantum computation.

It remains an open question whether a simple analytical formula exists for efficiently computing the mean values of Pauli operators of the XQAOA₁, MA-QAOA₁, and QAOA₁ states when dealing with k-local Ising Hamiltonians for k > 2. Additionally, if the problem assignments take integer-valued arguments, the quantum circuit would require more qubits per variable, further increasing the entangling and non-locality of the circuit, potentially allowing for quantum advantage. The p = 1 recursive QAOA (RQAOA₁) [39] is another quantum algorithm that can solve the MaxCut problem classically in time $O(n^4)$ without requiring any quantum computation or multi-qubit measurements.

The efficacy of XQAOA largely stems from its overparameterised ansatz. Overparameterisation, which involves introducing additional parameters to increase model dimensionality, reshapes the loss landscape to facilitate easier optimisation. To understand the impact of overparameterisation, consider the role of local minima in this context: a local minimum is a point in the loss landscape with a loss value lower or equal to those of its neighbours within an radius. While such minima are prevalent in lower dimensions, they become less likely with increasing dimensionality as it becomes harder for their loss values to remain the lowest across all new dimensions, converting them from minima into saddle points. Saddle points differ from local minima as they are not the lowest points in all directions; thus, optimisers can navigate past them more readily. As overparameterisation turns more potential local minima into saddle points, the path to the global minimum becomes less obstructed. Although the overparameterised models may not be completely devoid of local minima, the remaining local minima tend to be close to the global minimum. This proximity reduces the likelihood of settling for suboptimal solutions and facilitates more efficient convergence to the global minimum [94–98]. This advantageous effect of overparameterisation is evident in figure 4, where the lower quartiles for $\mathrm{XQAOA}_1^{\mathrm{X = Y}}$ consistently surpass 0.92 approximation ratio on all the benchmarked graph instances, strongly hinting at the benign loss landscape devoid of barren plateaus and suboptimal local minima.

While the previous discussion intuitively explains how overparameterisation enhances the efficacy of XQAOA, it is crucial to place this within the wider context of ongoing research on the trainability ¹⁶ of parameterised quantum circuits (PQCs). These studies, which form the basis of quantum landscape theory (QLT) [99], offer a deeper understanding of quantum loss landscapes [37, 48, 53, 89, 92, 93, 100–106]. QLT defines a PQC to be overparameterised when it has sufficiently many parameters to explore all relevant directions of its state space [99]. An essential aspect of this definition is the inherent expressiveness of the PQC, which is characterised by its ability to generate a wide range of unitaries under varied parameter settings [107]. This distinction is particularly evident when comparing $\mathrm{XQAOA}_1^{\mathrm{X = Y}}$ and MA-QAOA₁ for the MaxCut problem; despite having an equal number of parameters, $\mathrm{XQAOA}_1^{\mathrm{X = Y}}$ consistently outperforms MA-QAOA₁ on all problem instances. However, high expressivity also has drawbacks, such as the barren plateau phenomenon, where circuits show vanishingly small gradients due to their expressiveness [92]. XQAOA addresses this by using a problem-specific ansatz, tailoring its circuit design to the task at hand. This approach allows XQAOA to be overparameterised with a quadratic number of parameters, in contrast to generic ansatzes that may require exponentially more parameters and deeper circuits [99, 108]. XQAOA thus achieves an optimal balance, avoiding both the limitations of underparameterisation, such as spurious local minima and reachability deficits [53], and the challenges of high expressivity like barren plateaus. This positions XQAOA in an optimal ‘Goldilocks zone’ of trainability and expressivity.

The decision to use shallow XQAOA versus deep QAOA circuits hinges on their trainability for a given problem. Trainability depends on the loss landscapes of their ansatzes, influenced by factors like parameter count, initialisation strategy, circuit depth, and hardware noise. These factors can adversely affect trainability, hindering optimisation. Thus, the choice between shallow XQAOA and deep QAOA is determined by their relative trainability under these conditions.

It is evident that to surpass the performance of XQAOA₁, QAOA circuits need to be sufficiently deep. For example, it is conjectured that a minimum depth of p = 12 is necessary for QAOA to outperform the GW algorithm on the MaxCut problem on unweighted 3-regular graphs [43]. This conjecture not only posits a depth requirement but also suggests a parameter initialisation strategy of using a set of predetermined angles for warm-starting the optimisation. However, this strategy becomes ineffective in the presence of hardware noise [48]. The noise distorts the loss landscape, leading to barren plateaus, and necessitates deeper, more noise-prone circuits. In contrast, our findings suggest that $\mathrm{XQAOA}_1^{\mathrm{X = Y}}$ is adequate for the MaxCut problem, questioning the need for deeper XQAOA circuits on NISQ devices. While XQAOA may require p > 1 for certain problems, necessitating quantum computation, its relatively shallower depth compared to QAOA makes it less susceptible to noise. With sufficiently low noise levels, XQAOA can still achieve near-optimal solutions, albeit with additional random restarts or parameter initialisation strategies [105].

For problems unlike MaxCut on unweighted three-regular graphs, where patterns are ambiguous [37, 106] and a priori information is limited, training deep QAOA circuits is more challenging [109]. This is due to the difficulty in determining the minimal effective depth [110] and suitable parameter initialisation strategies in scenarios where random initialisation is suboptimal [89]. For such problems, where extracting useful a priori information is challenging, XQAOA may be a preferable choice, regardless of whether the quantum computers are NISQ or fault-tolerant. An example of such a problem could be the MaxCut on randomly weighted regular graphs or randomly generated graphs.

Before proceeding with derivations of the analytical formulas, let us first state and prove some identities that will be used in this paper. We denote the set of positive integers by $\mathbb{Z}^+ = \{1,2,\ldots\}$ and the set of n-bit strings by $\mathbb{F}_2^n = \{0,1\}^n$. When n = 1, we write $\mathbb{F}_2 = \mathbb{F}_2^1 = \{0,1\}$.

Lemma 6. Let $f \in \mathbb{Z}^+$ and $x_1, x_2, \dots, x_f, y_1, y_2, \dots, y_f \in \mathbb{R}$ be real numbers. Then,

$$\begin{align} \prod_{i = 1}^{f} \text{cos}\left(x_i - y_i\right) & = \sum_{\mu \in \mathbb{F}_2^f} \prod_{i = 1}^{f} \text{cos}^{1-\mu_i}x_i \text{cos}^{1-\mu_i}y_i \text{sin}^{\mu_i}x_i \text{sin}^{\mu_i}y_i , \end{align} \tag{ D1 }$$

$$\begin{align} \prod_{i = 1}^{f} \text{cos}\left(x_i + y_i\right) & = \sum_{\mu \in \mathbb{F}_2^f} \left(-1\right)^{|\mu|} \prod_{i = 1}^{f} \text{cos}^{1-\mu_i}x_i \text{cos}^{1-\mu_i}y_i \text{sin}^{\mu_i}x_i \text{sin}^{\mu_i}y_i, \end{align} \tag{ D2 }$$

where $|\mu| = \sum_{i = 1}^f \mu_i$.

Proof. The first identity equation (D1) follows from recognising that for any $a,b\in \mathbb{R}$, the sum a + b can be written as the following sum of products:

$$\begin{align} a+b = \sum_{\mu \in \mathbb{F}_2}a^{1-\mu}b^\mu. \end{align} \tag{ D3 }$$

Applying equation (D3) to each term of the sum $\text{cos} x_i \text{cos} y_i + \text{sin} x_i \text{sin} y_i$ gives

$$\begin{align*} \prod_{i = 1}^{f} \text{cos}\left(x_i - y_i\right) & = \prod_{i = 1}^{f} \left(\text{cos} x_i \text{cos} y_i + \text{sin} x_i \text{sin} y_i \right) \\ & = \prod_{i = 1}^{f} \sum_{\mu_i \in \mathbb{F}_2} \text{cos}^{1-\mu_i}x_i \text{cos}^{1-\mu_i}y_i \text{sin}^{\mu_i}x_i \text{sin}^{\mu_i}y_i \\ & = \sum_{\mu \in \mathbb{F}_2^f} \prod_{i = 1}^{f} \text{cos}^{1-\mu_i}x_i \text{cos}^{1-\mu_i}y_i \text{sin}^{\mu_i}x_i \text{sin}^{\mu_i}y_i. \end{align*}$$

The identity in equation (D2) follows immediately from replacing each y_i in equation (D1) with $-y_i$. □

By taking the sum (difference, respectively) of equations (D1) and (D2), only the even (odd, respectively) terms remain. Hence, it follows that

Lemma 7. Let $f \in \mathbb{Z}^+$ and $x_1, x_2, \dots, x_f, y_1, y_2, \dots, y_f \in \mathbb{R}$. Then, we have that

$$\begin{align} \prod_{i = 1}^{f} \text{cos}\left(x_i - y_i\right) + \prod_{i = 1}^{f} \text{cos}\left(x_i + y_i\right) & = 2 \sum_{\substack{\mu \in \mathbb{F}_2^f \\ |\mu| \mathrm{\, even}}} \prod_{i = 1}^{f} \text{cos}^{1-\mu_i}x_i \text{cos}^{1-\mu_i}y_i \text{sin}^{\mu_i}x_i \text{sin}^{\mu_i}y_i, \end{align} \tag{ D4 }$$

$$\begin{align} \prod_{i = 1}^{f} \text{cos}\left(x_i - y_i\right) - \prod_{i = 1}^{f} \text{cos}\left(x_i + y_i\right) & = 2 \sum_{\substack{\mu \in \mathbb{F}_2^f \\ |\mu| \mathrm{\, odd}}} \prod_{i = 1}^{f} \text{cos}^{1-\mu_i}x_i \text{cos}^{1-\mu_i}y_i \text{sin}^{\mu_i}x_i \text{sin}^{\mu_i}y_i. \end{align} \tag{ D5 }$$

We now make a remark about the above identities: while the right-hand sides of equations (D1), (D2), (D4) and (D5) each involves a sum over exponentially (in f) many terms (the cardinality of $\mathbb{F}_2^f$ is 2^f ), their left-hand sides involve just products of polynomially (in f) many terms. Hence, going from the right-hand sides of these identities to their left-hand sides results in exponential savings in computational cost. This will be useful for the expressions that we derive in appendix D.2.

Proof of theorem 3. Consider the XQAOA ansatz applied to MaxCut with $\left|s\right\rangle = \left|+\right\rangle^{\otimes n}$ and $Q = e^{-i \boldsymbol{\alpha}\cdot \boldsymbol{Y}}e^{-i \boldsymbol{\beta}\cdot \boldsymbol{X}}e^{-i \boldsymbol{\gamma}\cdot \boldsymbol{C}}$, where $\boldsymbol{\alpha}\cdot \boldsymbol{Y} = \sum_{i = 1}^n \alpha_i Y_i$, $\boldsymbol{\beta}\cdot \boldsymbol{X} = \sum_{i = 1}^n \beta_i X_i$, and $\boldsymbol{\gamma}\cdot \boldsymbol{C} = \sum_{\{u,v\}\in E} \gamma_{uv} C_{uv} = \frac 12\sum_{\{u,v\}\in E} \gamma_{uv} w_{uv}(I-Z_u Z_v)$.

Observe that

$$\begin{align} \langle C\rangle_{\mathrm{XY}} = \left\langle s\left|Q^{\dagger} C Q\right| s\right\rangle = \sum_{\left\{u,v\right\} \in E} \frac{w_{uv}}{2}-\frac{w_{uv}}{2} \langle s|Q^{\dagger} Z_{u} Z_{v} Q| s\rangle . \end{align} \tag{ D6 }$$

Hence, to compute the expectation value $\langle C \rangle_{\mathrm{XY}}$, it suffices to compute each term

$$\begin{align} \left\langle C_{u v}\right\rangle_{\mathrm{XY}}: = \frac{w_{uv}}{2}-\frac{w_{uv}}{2} \left\langle s \right| Q^{\dagger} Z_{u} Z_{v} Q\left|s\right\rangle \end{align} \tag{ D7 }$$

in the sum separately.

For the rest of this proof, we fix an edge $\{u,v\} \in E$. We shall evaluate the product $Q^{\dagger} Z_{u} Z_{v}Q$ in equation (D7) by conjugating the Pauli operator $Z_{u} Z_{v}$ by the mixing unitary and then by the problem unitary. By the commutation properties of the Pauli matrices Λ and the fact that they satisfy $e^{-i \theta \Lambda} = \text{cos} \theta I-i \text{sin} \theta \Lambda$, it follows that most of the terms in the mixing unitary $e^{-i \boldsymbol{\alpha}\cdot \boldsymbol{Y}}e^{-i \boldsymbol{\beta}\cdot \boldsymbol{X}} = \prod_{j = 1}^{n} e^{-i \alpha_j Y_{j}}e^{-i \beta_j X_{j}}$ commute through $Z_u Z_v$ and annihilate their inverses. Hence,

$$\begin{align} \begin{split} e^{i \boldsymbol{\beta}\cdot \boldsymbol{X}} e^{i \boldsymbol{\alpha}\cdot \boldsymbol{Y}} Z_u Z_v e^{-i \boldsymbol{\alpha}\cdot \boldsymbol{Y}} e^{-i \boldsymbol{\beta}\cdot \boldsymbol{X}} = \left( e^{i \beta_u X_u}e^{2i \alpha_u Y_u}e^{i \beta_u X_u} Z_u \right) \otimes \left(e^{i \beta_v X_v}e^{2i \alpha_v Y_v} e^{i \beta_v X_v} Z_v \right) . \end{split} \end{align} \tag{ D8 }$$

The factors in the above Kronecker product can be expanded as follows. For $a\in\{u,v\}$,

$$\begin{align} \begin{split} e^{i \beta_a X_a}e^{2i \alpha_a Y_a}e^{i \beta_a X_a} Z_a & = \left( \text{cos} \beta_a + i\text{sin} \beta_a X_a \right)\left( \text{cos} 2\alpha_a + i\text{sin} 2\alpha_a Y_u \right)\left( \text{cos} \beta_a + i\text{sin} \beta_a X_a \right)Z_a \\ & = \cos2\alpha_a \cos2\beta_a Z_a + \cos2\alpha_a \sin2\beta_a Y_a - \sin2\alpha_a X_a . \end{split} \end{align} \tag{ D9 }$$

Substituting this into equation (D8) gives

$$\begin{align} \begin{split} e^{i \boldsymbol{\beta}\cdot \boldsymbol{X}} e^{i \boldsymbol{\alpha}\cdot \boldsymbol{Y}} Z_u Z_v e^{-i \boldsymbol{\alpha}\cdot \boldsymbol{Y}} e^{-i \boldsymbol{\beta}\cdot \boldsymbol{X}} & = \cos2\alpha_u \cos2\beta_u \cos2\alpha_v \cos2\beta_v Z_u Z_v + \cos2\alpha_u \cos2\beta_u \cos2\alpha_v \sin2\beta_v Z_u Y_v\\ & \quad - \cos2\alpha_u \cos2\beta_u \text{sin} 2 \alpha_v Z_uX_v + \cos2\alpha_u \text{sin} 2 \beta_u \text{cos} 2\alpha_v \text{cos} 2\beta_v Y_u Z_v\\ & \quad + \text{cos} 2 \alpha_u \text{sin} 2\beta_u \text{cos} 2 \alpha_v \text{sin} 2\beta_v Y_uY_v - \cos2 \alpha_u \text{sin} 2 \beta_u \text{sin} 2 \alpha_v Y_u X_v\\ & \quad - \text{sin} 2 \alpha_u \text{cos} 2 \alpha_v \text{cos} 2 \beta_v X_uZ_v - \text{sin} 2 \alpha_u \text{cos} 2 \alpha_v \text{sin} 2 \beta_v X_u Y_v\\ & \quad + \text{sin} 2\alpha_u \text{sin} 2 \alpha_v X_uX_v. \end{split} \end{align} \tag{ D10 }$$

Hence, by substituting this expression into equation (D7), the expected cost function corresponding to the edge $\{u,v\}\in E$ can be written as

$$\begin{align} \langle C_{uv}\rangle_{\mathrm{XY}} & = \frac{w_{uv}}2 - \frac{w_{uv}}2 \bigg\{ \cos2\alpha_u \cos2\beta_u \cos2\alpha_v \cos2\beta_v \xi\left(Z,Z\right) + \cos2\alpha_u \cos2\beta_u \cos2\alpha_v \sin2\beta_v \xi\left(Z,Y\right) \nonumber\\ & \quad - \cos2\alpha_u \cos2\beta_u \text{sin} 2 \alpha_v \xi\left(Z,X\right) + \cos2\alpha_u \text{sin} 2 \beta_u \text{cos} 2\alpha_v \text{cos} 2\beta_v \xi\left(Y,Z\right)\nonumber\\ & \quad + \text{cos} 2 \alpha_u \text{sin} 2\beta_u \text{cos} 2 \alpha_v \text{sin} 2\beta_v \xi\left(Y,Y\right) - \cos2 \alpha_u \text{sin} 2 \beta_u \text{sin} 2 \alpha_v \xi\left(Y,X\right)\nonumber\\ & \quad - \text{sin} 2 \alpha_u \text{cos} 2 \alpha_v \text{cos} 2 \beta_v \xi\left(X,Z\right) - \text{sin} 2 \alpha_u \text{cos} 2 \alpha_v \text{sin} 2 \beta_v \xi\left(X,Y\right)\nonumber\\ & \quad + \text{sin} 2\alpha_u \text{sin} 2 \alpha_v \xi\left(X,X\right)\bigg\}, \end{align} \tag{ D11 }$$

where for single-qubit Pauli matrices $P,Q \in \{X,Y,Z\}$, we have defined

$$\begin{align} \xi\left(P,Q\right) & = \left\langle s \right| \eta\left(P,Q\right) \left|s\right\rangle, \end{align} \tag{ D12 }$$

$$\begin{align} \eta\left(P,Q\right) & = e^{i \boldsymbol{\gamma}\cdot \boldsymbol{C}} P_u Q_v e^{-i \boldsymbol{\gamma}\cdot \boldsymbol{C}}. \end{align} \tag{ D13 }$$

Moving forward, the approach we take is as follows. Firstly, we shall evaluate the expression for $\eta(P,Q)$ in equation (D13) for all $P,Q \in \{X,Y,Z\}$. Secondly, we shall substitute our expressions for $\eta(P,Q)$ into equation (D12) to derive analytical expressions for $\xi(P,Q)$. Thirdly and also finally, we shall substitute these analytical expressions into equation (D11) to obtain our desired expression equation (22).

Before we execute these three steps, we first introduce some notation to help us keep track of the neighbours of the edge $\{u,v\}$: let

$$\begin{align} \mathcal N_{u\backslash\!\!\backslash v} = \mathcal N\left(u\right) \backslash \left( \mathcal N\left(v\right) \cup \left\{v\right\} \right) = \left\{\omega_1, \omega_2,\ldots, \omega_b\right\} \end{align} \tag{ D14 }$$

be the set of neighbours of u that are of distance 2 or greater from v. Similarly, let

$$\begin{align} \mathcal N_{v\backslash\!\!\backslash u} = \mathcal N\left(v\right) \backslash \left( \mathcal N\left(u\right) \cup \left\{u\right\} \right) = \left\{q_1, q_2,\ldots, q_c\right\} \end{align} \tag{ D15 }$$

be the set of neighbours of v that are of distance 2 or greater from u. Next, let

$$\begin{align} \mathcal N_{uv} = \mathcal N\left(u\right) \cap \mathcal N\left(v\right) = \left\{a_1, a_2,\ldots ,a_f\right\} = \left\{\omega_{b+1}, \omega_{b+2},\ldots,\omega_d\right\} = \left\{q_{c+1}, q_{c+2},\ldots,q_e\right\}, \end{align} \tag{ D16 }$$

where $a_i = \omega_{b+i} = q_{c+i}$ for $i = 1,\ldots ,f$, be the set of vertices that are neighbours of both u and v, i.e. $\mathcal N_{uv}$ comprises those nodes in V that form a triangle with both u and v. Finally, let

$$\begin{align} \mathcal N_{u\backslash v} = \mathcal N\left(u\right) \backslash \left\{v\right\} = \mathcal N_{u\backslash\!\!\backslash v} \cup \mathcal N_{uv} = \left\{\omega_1,\ldots, \omega_b,a_1,\ldots, a_f\right\} = \left\{\omega_1,\ldots, \omega_b,\ldots, \omega_d\right\} \end{align} \tag{ D17 }$$

be the set of neighbours of u that are not v and let

$$\begin{align} \mathcal N_{v\backslash u} = \mathcal N_{v\backslash\!\!\backslash u} \cup \mathcal N_{uv} = \left\{q_1,\ldots, q_c,a_1,\ldots, a_f\right\} = \left\{q_1, \ldots q_c,\ldots, q_e\right\} \end{align} \tag{ D18 }$$

be the set of neighbours of v that are not u. We are now ready to execute our aforementioned three steps.

Step 1: evaluation of $\eta(P,Q)$

First, we rewrite equation (D13) as

$$\begin{align} \eta\left(P,Q\right) & = e^{i \boldsymbol{\gamma}\cdot \boldsymbol{C^{^{\prime}}}} P_u Q_v e^{-i \boldsymbol{\gamma}\cdot \boldsymbol{C^{^{\prime}}}}, \end{align} \tag{ D19 }$$

where

$$\begin{align} \boldsymbol{\gamma}\cdot \boldsymbol{C^{^{\prime}}} = \sum_{\left\{x,y\right\}\in E} \gamma_{xy} C^{^{\prime}}_{xy}, \quad C^{^{\prime}}_{xy} = -\frac 12 w_{xy} Z_x Z_y, \end{align} \tag{ D20 }$$

since the term $\frac 12\sum_{\{u,v\}\in E} \gamma_{uv} w_{uv} I$ in C_uv is cancelled by its inverse and does not contribute to equation (D13). Next, we expand $\boldsymbol{\gamma}\cdot \boldsymbol{C^{^{\prime}}}$ as

$$\begin{align} \boldsymbol{\gamma}\cdot \boldsymbol{C^{^{\prime}}} = \gamma_{uv} C_{uv}^{^{\prime}} + \boldsymbol{\gamma}\cdot \boldsymbol{C_u^{^{\prime}}} + \boldsymbol{\gamma}\cdot \boldsymbol{C_v^{^{\prime}}} + \sum_{\left\{x,y\right\}\in E_{\left(uv\right)}} \gamma_{xy} C^{^{\prime}}_{xy}, \end{align} \tag{ D21 }$$

where

$$\begin{align} \boldsymbol{\gamma}\cdot \boldsymbol{C_u^{^{\prime}}} & = \sum_{i = 1}^d \gamma_{u \omega_i} C_{u \omega_i}^{^{\prime}} = \sum_{i = 1}^b \gamma_{u \omega_i} C_{u \omega_i}^{^{\prime}} + \sum_{i = 1}^f \gamma_{u a_i} C_{u a_i}^{^{\prime}} \end{align} \tag{ D22 }$$

contains terms involving u but not v, and

$$\begin{align} \boldsymbol{\gamma}\cdot \boldsymbol{C_v^{^{\prime}}} & = \sum_{i = 1}^e \gamma_{v q_i} C_{v q_i}^{^{\prime}} = \sum_{i = 1}^c \gamma_{v q_i} C_{v q_i}^{^{\prime}} + \sum_{i = 1}^f \gamma_{v a_i} C_{v a_i}^{^{\prime}} \end{align} \tag{ D23 }$$

contains terms involving v but not u, and $E_{(uv)} = \{\{a,b\}\in E : a,b \notin \{u,v\}\}$ denotes the set of edges that do not contain either u or v as an endpoint. By substituting equation (D21) into equation (D19), we obtain

$$\begin{align} \eta\left(P,Q\right) = e^{i \boldsymbol{\gamma}\cdot \boldsymbol{C_u^{^{\prime}}}} e^{i \boldsymbol{\gamma}\cdot \boldsymbol{C_v^{^{\prime}}}}e^{i \gamma_{uv}C^{^{\prime}}_{uv}} P_u Q_v e^{-i \gamma_{uv}C^{^{\prime}}_{uv}} e^{-i \boldsymbol{\gamma}\cdot \boldsymbol{C_v^{^{\prime}}}} e^{-i \boldsymbol{\gamma}\cdot \boldsymbol{C_u^{^{\prime}}}} = \Gamma_{PQ} P_u Q_v, \end{align} \tag{ D24 }$$

where $\Gamma_{PQ} = \eta(P,Q) P_u Q_v$. By using the fact that Pauli operators either commute or anti-commute, evaluating equation (D24) gives

$$\begin{align} \Gamma_{XX} & = \Gamma_{XY} = \Gamma_{YX} = \Gamma_{YY} = e^{2i \boldsymbol{\gamma}\cdot \boldsymbol{C_u^{^{\prime}}}} e^{2i \boldsymbol{\gamma}\cdot \boldsymbol{C_v^{^{\prime}}}}, \end{align} \tag{ D25 }$$

$$\begin{align} \Gamma_{XZ} & = \Gamma_{YZ} = e^{2i \gamma_{uv}C_{uv}^{^{\prime}}} e^{2i \boldsymbol{\gamma}\cdot \boldsymbol{C_u^{^{\prime}}}} , \end{align} \tag{ D26 }$$

$$\begin{align} \Gamma_{ZX} & = \Gamma_{ZY} = e^{2i \gamma_{uv}C_{uv}^{^{\prime}}} e^{2i \boldsymbol{\gamma}\cdot \boldsymbol{C_v^{^{\prime}}}}, \end{align} \tag{ D27 }$$

$$\begin{align} \Gamma_{ZZ} & = I. \end{align} \tag{ D28 }$$

To evaluate equation (D25), we first compute

$$\begin{align} e^{2 i \boldsymbol{\gamma}\cdot \boldsymbol{C_u^{^{\prime}}}} & = \prod_{i = 1}^d \left[ \text{cos}\left(\gamma_{u\omega_i}w_{u \omega_i}\right)I - i \text{sin} \left(\gamma_{u\omega_i}w_{u \omega_i}\right) Z_{u} Z_{\omega_i} \right] \nonumber\\ & = \sum_{x\in \mathbb{F}_2^d} \text{cos}^{1-x_1}\left(\gamma_{u\omega_1}w_{u \omega_1}\right)\cdots \text{cos}^{1-x_d}\left(\gamma_{u\omega_d}w_{u \omega_d}\right) \left[-i \text{sin}\left(\gamma_{u\omega_1}w_{u \omega_1}\right)\right]^{x_1}\cdots \left[-i \text{sin}\left(\gamma_{u\omega_d}w_{u \omega_d}\right)\right]^{x_d} Z_u^{|x|} Z_{\omega_1}^{x_1}\cdots Z_{\omega_d}^{x_d}, \end{align} \tag{ D29 }$$

where we applied the trick in equation (D3) to each term in the above product. In the above expression, $|x| = \sum_{i = 1}^d x_i$ denotes the Hamming weight of the string $x\in \{0,1\}^d$. Similarly,

$$\begin{align} e^{2 i \boldsymbol{\gamma}\cdot \boldsymbol{C_v^{^{\prime}}}} = \sum_{y\in \mathbb{F}_2^e} \text{cos}^{1-y_1}\left(\gamma_{vq_1}w_{v q_1}\right)\cdots \text{cos}^{1-y_e}\left(\gamma_{vq_e}w_{v q_e}\right) \left[-i \text{sin}\left(\gamma_{vq_1}w_{v q_1}\right)\right]^{y_1}\cdots \left[-i \text{sin}\left(\gamma_{vq_e}w_{v q_e}\right)\right]^{y_e} Z_v^{|y|} Z_{q_1}^{y_1}\cdots Z_{q_e}^{y_e}. \end{align} \tag{ D30 }$$

By taking the product of equations (D29) and (D30), and using equation (D16) to relabel the vertices in $\mathcal N_{uv}$ by a_i ’s, we obtain the following expression for equation (D25):

$$\begin{align} \Gamma_{XX} & = \sum_{\alpha \in \mathbb{F}_2^b} \sum_{\beta \in \mathbb{F}_2^c} \sum_{\mu \in \mathbb{F}_2^f} \sum_{\nu \in \mathbb{F}_2^f} \text{cos}^{1-\alpha_1}\left(\gamma_{u\omega_1}w_{u \omega_1}\right)\cdots \text{cos}^{1-\alpha_b}\left(\gamma_{u\omega_b}w_{u \omega_b}\right) \text{cos}^{1-\mu_1}\left(\gamma_{u a_1}w_{u a_1}\right)\cdots \text{cos}^{1-\mu_f}\left(\gamma_{u a_f}w_{u a_f}\right)\nonumber\\ &\quad\times \text{cos}^{1-\beta_1}\left(\gamma_{vq_1}w_{v q_1}\right)\cdots \text{cos}^{1-\beta_c}\left(\gamma_{vq_c}w_{v q_c}\right) \text{cos}^{1-\nu_1}\left(\gamma_{v a_1}w_{v a_1}\right)\cdots \text{cos}^{1-\nu_f}\left(\gamma_{v a_f}w_{v a_f}\right)\nonumber\\ &\quad\times \left[-i \text{sin}\left(\gamma_{u\omega_1}w_{u \omega_1}\right)\right]^{\alpha_1}\cdots \left[-i \text{sin}\left(\gamma_{u\omega_b}w_{u \omega_b}\right)\right]^{\alpha_b} \left[-i \text{sin}\left(\gamma_{u a_1}w_{u a_1}\right)\right]^{\mu_1}\cdots \left[-i \text{sin}\left(\gamma_{ua_f}w_{u a_f}\right)\right]^{\mu_f}\nonumber\\ &\quad\times \left[-i \text{sin}\left(\gamma_{vq_1}w_{v q_1}\right)\right]^{\beta_1}\cdots \left[-i \text{sin}\left(\gamma_{vq_c}w_{v q_c}\right)\right]^{\beta_c} \left[-i \text{sin}\left(\gamma_{v a_1}w_{v a_1}\right)\right]^{\nu_1}\cdots \left[-i \text{sin}\left(\gamma_{va_f}w_{v a_f}\right)\right]^{\nu_f}\nonumber\\ &\quad\times Z_u^{|\alpha|+|\mu|} \cdot Z_v^{|\beta|+|\nu|} \cdot \prod_{i = 1}^b Z_{\omega_i}^{\alpha_i}\cdot \prod_{i = 1}^c Z_{q_i}^{\beta_i} \cdot \prod_{i = 1}^f Z_{a_i}^{\mu_i+\nu_i}. \end{align} \tag{ D31 }$$

Next, by substituting equation (D29) into equation (D26), we obtain

$$\begin{align} \Gamma_{XZ} & = \sum_{x_0 x_1 \ldots x_d \in \mathbb{F}_2^{d+1}} \text{cos}^{1-x_0}\left(\gamma_{uv}w_{uv}\right) \text{cos}^{1-x_1}\left(\gamma_{u\omega_1}w_{u \omega_1}\right)\cdots \text{cos}^{1-x_d}\left(\gamma_{u\omega_d}w_{u \omega_d}\right) \nonumber\\ &\quad\times \left[-i \text{sin}\left(\gamma_{uv}w_{u v}\right)\right]^{x_0}\left[-i \text{sin}\left(\gamma_{u\omega_1}w_{u \omega_1}\right)\right]^{x_1}\cdots \left[-i \text{sin}\left(\gamma_{u\omega_d}w_{u \omega_d}\right)\right]^{x_d} Z_u^{|x|} Z_v^{x_0} Z_{\omega_1}^{x_1}\cdots Z_{\omega_d}^{x_d}. \end{align} \tag{ D32 }$$

Similarly, by substituting equation (D30) into equation (D27), we obtain

$$\begin{align} \Gamma_{ZX} & = \sum_{y_0 y_1 \ldots y_{e} \in \mathbb{F}_2^{e+1}} \text{cos}^{1-y_0}\left(\gamma_{uv}w_{uv}\right) \text{cos}^{1-y_1}\left(\gamma_{vq_1}w_{v q_1}\right)\cdots \text{cos}^{1-y_e}\left(\gamma_{vq_e}w_{v q_e}\right) \nonumber\\ &\quad\times \left[-i \text{sin}\left(\gamma_{uv}w_{u v}\right)\right]^{y_0}\left[-i \text{sin}\left(\gamma_{vq_1}w_{v q_1}\right)\right]^{y_1}\cdots \left[-i \text{sin}\left(\gamma_{vq_e}w_{v q_e}\right)\right]^{y_e} Z_u^{y_0} Z_v^{|y|} Z_{q_1}^{y_1}\cdots Z_{q_{e}}^{y_{e}}. \end{align} \tag{ D33 }$$

This completes Step 1, where $\eta(P,Q)$ is specified by equations (D24), (D31), (D32), (D33) and (D28).

Step 2: evaluation of $\xi(P,Q)$

There are $3^2 = 9$ different choices of $PQ \in \{X,Y,Z\}^2$. We will split these choices into four cases, as follows:

• Case 1: $PQ = XX, XY, YX, YY$. By substituting equations (D24) and (D31) into equation (D12), we obtain
  $$\begin{align} \xi\left(P,Q\right) & = \sum_{\alpha \in \mathbb{F}_2^b} \sum_{\beta \in \mathbb{F}_2^c} \sum_{\mu \in \mathbb{F}_2^f} \sum_{\nu \in \mathbb{F}_2^f} \text{cos}^{1-\alpha_1}\left(\gamma_{u\omega_1}w_{u \omega_1}\right)\cdots \text{cos}^{1-\alpha_b}\left(\gamma_{u\omega_b}w_{u \omega_b}\right) \text{cos}^{1-\mu_1}\left(\gamma_{u a_1}w_{u a_1}\right)\cdots \text{cos}^{1-\mu_f}\left(\gamma_{u a_f}w_{u a_f}\right)\nonumber\\ &\quad\times \text{cos}^{1-\beta_1}\left(\gamma_{vq_1}w_{v q_1}\right)\cdots \text{cos}^{1-\beta_c}\left(\gamma_{vq_c}w_{v q_c}\right) \text{cos}^{1-\nu_1}\left(\gamma_{v a_1}w_{v a_1}\right)\cdots \text{cos}^{1-v_f}\left(\gamma_{v a_f}w_{v a_f}\right)\nonumber\\ &\quad\times \left[-i \text{sin}\left(\gamma_{u\omega_1}w_{u \omega_1}\right)\right]^{\alpha_1}\cdots \left[-i \text{sin}\left(\gamma_{u\omega_b}w_{u \omega_b}\right)\right]^{\alpha_b} \left[-i \text{sin}\left(\gamma_{u a_1}w_{u a_1}\right)\right]^{\mu_1}\cdots \left[-i \text{sin}\left(\gamma_{ua_f}w_{u a_f}\right)\right]^{\mu_f}\nonumber\\ &\quad\times \left[-i \text{sin}\left(\gamma_{vq_1}w_{v q_1}\right)\right]^{\beta_1}\cdots \left[-i \text{sin}\left(\gamma_{vq_c}w_{v q_c}\right)\right]^{\beta_c} \left[-i \text{sin}\left(\gamma_{v a_1}w_{v a_1}\right)\right]^{\nu_1}\cdots \left[-i \text{sin}\left(\gamma_{va_f}w_{v a_f}\right)\right]^{\nu_f}\nonumber\\ &\quad\times \underbrace{ \mathrm{tr} \left\{\left|s\right\rangle\!\!\left\langle s \right| Z_u^{|\alpha|+|\mu|} P_u \cdot Z_v^{|\beta|+|\nu|} Q_v \cdot \prod_{i = 1}^b Z_{\omega_i}^{\alpha_i}\cdot \prod_{i = 1}^c Z_{q_i}^{\beta_i} \cdot \prod_{i = 1}^f Z_{a_i}^{\mu_i+\nu_i} \right\} }_{\unicode{x2460}}. \end{align} \tag{ D34 }$$
  By writing the initial state as $\left|s\right\rangle\!\!\left\langle s \right| = \bigotimes_{j = 1}^n \frac{1}{2}(I+X_j)$, the last line of equation (D34) can be expanded as
  $$\begin{align} \unicode{x2460} & = \mathrm{tr}\left\{\frac{I+X_u}{2} Z_u^{|\alpha|+|\mu|} P_u\right\} \mathrm{tr}\left\{\frac{I+X_v}{2} Z_v^{|\beta|+|\nu|} Q_v\right\} \nonumber\\ &\quad\times \prod_{i = 1}^b \underbrace{\mathrm{tr}\left\{\frac{I+X_{\omega_i}}{2} Z_{\omega_i}^{\alpha_i} \right\}}_{ = \, \delta_{\alpha_i,0}} \prod_{i = 1}^c \underbrace{\mathrm{tr}\left\{\frac{I+X_{q_i}}{2} Z_{q_i}^{\beta_i} \right\}}_{ = \, \delta_{\beta_i,0}} \prod_{i = 1}^f \underbrace{\mathrm{tr}\left\{\frac{I+X_{a_i}}{2} Z_{a_i}^{\mu_i+\nu_i} \right\}}_{ = \, \delta_{\mu_i, \nu_i}} \nonumber\\ & = \underbrace{\mathrm{tr}\left\{\frac{I+X_{u}}{2} Z_{u}^{|\mu|} P_{u} \right\} }_{\unicode{x2461}} \underbrace{\mathrm{tr}\left\{\frac{I+X_{v}}{2} Z_{v}^{|\nu|} Q_{v}\right\} }_{\unicode{x2462}} \prod_{i = 1}^b \delta_{\alpha_i,0} \prod_{i = 1}^c \delta_{\beta_i,0} \prod_{i = 1}^f \delta_{\mu_i,\nu_i}. \end{align} \tag{ D35 }$$
  Now, for $P \in \{X,Y\}$ and for $|\mu| \in \mathbb{N}$,
  $$\begin{align} \unicode{x2461} = \left[P = X\right]\left[|\mu| \ \mbox{even}\right] - i \left[P = Y\right]\left[|\mu| \ \mbox{odd}\right], \end{align} \tag{ D36 }$$
  where the $[A]$ denotes the Iverson bracket of statement A; i.e. $[A] = 1$ if A is true and 0 otherwise. Similarly,
  $$\begin{align} \unicode{x2462} = \left[Q = X\right]\left[|\mu| \ \mbox{even}\right] - i \left[Q = Y\right]\left[|\mu| \ \mbox{odd}\right]. \end{align} \tag{ D37 }$$
  Hence, the product of equations (D36) and (D37) is
  $$\begin{align} \unicode{x2461}\times \unicode{x2462} = \left[P = Q = X\right]\left[|\mu| \ \mbox{even}\right] - \left[P = Q = Y\right]\left[|\mu| \ \mbox{odd}\right]. \end{align} \tag{ D38 }$$
  Substituting equation (D38) into equation (D35) gives
  $$\begin{align} \unicode{x2460} = \delta_{\alpha,0} \delta_{\beta,0} \delta_{\mu,\nu} \left(\left[P = Q = X\right]\left[|\mu| \ \mbox{even}\right] - \left[P = Q = Y\right]\left[|\mu| \ \mbox{odd}\right]\right). \end{align} \tag{ D39 }$$
  Substituting equation (D39) into equation (D34) and using the Kronecker deltas to eliminate terms in the sum, we get
  $$\begin{align} \xi\left(P,Q\right) & = \sum_{\mu \in \mathbb{F}_2^f} \text{cos}\left(\gamma_{u\omega_1}w_{u \omega_1}\right)\cdots \text{cos}\left(\gamma_{u\omega_b}w_{u \omega_b}\right) \text{cos}^{1-\mu_1}\left(\gamma_{u a_1}w_{u a_1}\right)\cdots \text{cos}^{1-\mu_f}\left(\gamma_{u a_f}w_{u a_f}\right)\nonumber\\ &\quad\times \text{cos}\left(\gamma_{vq_1}w_{v q_1}\right)\cdots \text{cos}\left(\gamma_{vq_c}w_{v q_c}\right) \text{cos}^{1-\mu_1}\left(\gamma_{v a_1}w_{v a_1}\right)\cdots \text{cos}^{1-\mu_f}\left(\gamma_{v a_f}w_{v a_f}\right) \nonumber\\ &\quad \nonumber\\ &\quad\times \left(-i\right)^{2|\mu|} \text{sin}^{\mu_1}\left(\gamma_{ua_1}w_{u a_1}\right)\cdots \text{sin}^{\mu_f}\left(\gamma_{ua_f}w_{u a_f}\right) \text{sin}^{\mu_1}\left(\gamma_{va_1}w_{v a_1}\right)\cdots \text{sin}^{\mu_f}\left(\gamma_{va_f}w_{v a_f}\right) \nonumber\\ &\quad\times \left(\left[P = Q = X\right]\left[|\mu| \ \mbox{even}\right] - \left[P = Q = Y\right]\left[|\mu| \ \mbox{odd}\right]\right) \nonumber\\ & = \prod_{\omega\in \mathcal N_{u\backslash\!\!\backslash v}} \text{cos} \left(\gamma_{u\omega} w_{u\omega}\right) \prod_{q\in \mathcal N_{v\backslash\!\!\backslash u}} \text{cos} \left(\gamma_{vq} w_{vq}\right) \left( \left[P = Q = X\right] \sum_{ \substack{\mu \in \mathbb{F}_2^f \\ |\mu| \ \textrm{even}}} + \left[P = Q = Y\right] \sum_{ \substack{\mu \in \mathbb{F}_2^f \\ |\mu| \ \textrm{odd}}} \right) \nonumber\\ & \quad \prod_{i = 1}^f \text{cos}^{1-\mu_i}\left(\gamma_{ua_i}w_{ua_i}\right) \text{cos}^{1-\mu_i}\left(\gamma_{va_i}w_{va_i}\right) \text{sin}^{\mu_i}\left(\gamma_{ua_i}w_{ua_i}\right) \text{sin}^{\mu_i}\left(\gamma_{va_i}w_{va_i}\right). \end{align} \tag{ D40 }$$
  Since the Iverson brackets in equation (D40) vanish when PQ = XY or YX, it follows that
  $$\begin{align} \xi\left(X,Y\right) = \xi\left(Y,X\right) = 0. \end{align} \tag{ D41 }$$
  When PQ = XX, equation (D40) reduces to
  $$\begin{align} \xi\left(X,X\right) & = \prod_{\omega\in \mathcal N_{u\backslash\!\!\backslash v}} \text{cos} \left(\gamma_{u\omega} w_{u\omega}\right) \prod_{q\in \mathcal N_{v\backslash\!\!\backslash u}} \text{cos} \left(\gamma_{vq} w_{vq}\right) \nonumber\\ &\quad \times \sum_{ \substack{\mu \in \mathbb{F}_2^f \\ |\mu| \ \textrm{even}}}\prod_{i = 1}^f \text{cos}^{1-\mu_i}\left(\gamma_{ua_i}w_{ua_i}\right) \text{cos}^{1-\mu_i}\left(\gamma_{va_i}w_{va_i}\right) \text{sin}^{\mu_i}\left(\gamma_{ua_i}w_{ua_i}\right) \text{sin}^{\mu_i}\left(\gamma_{va_i}w_{va_i}\right) \nonumber\\ & = \frac 12 \prod_{\omega\in \mathcal N_{u\backslash\!\!\backslash v}} \text{cos} \left(\gamma_{u\omega} w_{u\omega}\right) \prod_{q\in \mathcal N_{v\backslash\!\!\backslash u}} \text{cos} \left(\gamma_{vq} w_{vq}\right) \nonumber\\ &\quad \times \left[ \prod_{i = 1}^f \text{cos}\left(\gamma_{u a_i} w_{ua_i}+\gamma_{va_i} w_{va_i}\right)+ \prod_{i = 1}^f \text{cos}\left(\gamma_{u a_i} w_{ua_i}-\gamma_{va_i} w_{va_i}\right) \right] \nonumber\\ & = \frac 12 \prod_{\omega\in \mathcal N_{u\backslash\!\!\backslash v}} \text{cos} \left(\gamma_{u\omega} w_{u\omega}\right) \prod_{q\in \mathcal N_{v\backslash\!\!\backslash u}} \text{cos} \left(\gamma_{vq} w_{vq}\right) \nonumber\\ &\quad \times \left[ \prod_{q\in \mathcal N_{uv}} \text{cos}\left(\gamma_{u q} w_{uq}-\gamma_{vq} w_{vq}\right)+ \prod_{q\in \mathcal N_{uv}} \text{cos}\left(\gamma_{uq} w_{uq}+\gamma_{vq} w_{vq}\right) \right], \end{align} \tag{ D42 }$$
  where the second equality follows from equation (D4).Similarly, for the case PQ = YY, utilising equation (D5) gives
  $$\begin{align} \xi\left(Y,Y\right)& = \frac 12 \prod_{\omega\in \mathcal N_{u\backslash\!\!\backslash v}} \text{cos} \left(\gamma_{u\omega} w_{u\omega}\right) \prod_{q\in \mathcal N_{v\backslash\!\!\backslash u}} \text{cos} \left(\gamma_{vq} w_{vq}\right) \nonumber\\ &\quad \times \left[ \prod_{q\in \mathcal N_{uv}} \text{cos}\left(\gamma_{u q} w_{uq}-\gamma_{vq} w_{vq}\right)- \prod_{q\in \mathcal N_{uv}} \text{cos}\left(\gamma_{u q} w_{u q}+\gamma_{v q} w_{v q}\right) \right], \end{align} \tag{ D43 }$$
  which differs from the expression given by equation (D42) for $\xi(X,X)$ by only a single minus sign. As we mentioned in appendix D.1, the identities equations (D4) and (D5) allowed us to simplify the exponential sums in equation (D40) to get exponential savings in the computational cost in the worst case.
• Case 2: $PQ = XZ, YZ$. By substituting equations (D24) and (D32) into equation (D12), we obtain, for $P = X,Y$,
  $$\begin{align} \xi\left(P,Z\right) & = \sum_{x_0 x_1 \ldots x_d \in \mathbb{F}_2^{d+1}} \text{cos}^{1-x_0}\left(\gamma_{uv}w_{uv}\right) \text{cos}^{1-x_1}\left(\gamma_{u\omega_1}w_{u \omega_1}\right)\cdots \text{cos}^{1-x_d}\left(\gamma_{u\omega_d}w_{u \omega_d}\right) \nonumber\\ &\quad\times \left[-i \text{sin}\left(\gamma_{uv}w_{u v}\right)\right]^{x_0}\left[-i \text{sin}\left(\gamma_{u\omega_1}w_{u \omega_1}\right)\right]^{x_1}\cdots \left[-i \text{sin}\left(\gamma_{u\omega_d}w_{u \omega_d}\right)\right]^{x_d} \nonumber\\ &\quad\times \underbrace{\mathrm{tr}\left\{\left|s\right\rangle\!\!\left\langle s \right| Z_u^{|x|}P_u Z_v^{x_0+1} Z_{\omega_1}^{x_1}\cdots Z_{\omega_d}^{x_d} \right\} }_{\unicode{x2463}}. \end{align} \tag{ D44 }$$
  By writing $\left|s\right\rangle\!\!\left\langle s \right| = \bigotimes_{j = 1}^n \frac{1}{2}(I+X_j)$, the last line of equation (D44) can be expanded as
  $$\begin{align} \unicode{x2463} & = \mathrm{tr}\left\{\frac{I+X_u}{2} Z_u^{x_0+x_1+\cdots+x_d} P_u\right\} \mathrm{tr}\left\{\frac{I+X_v}{2} Z_v^{x_0+1} \right\} \prod_{i = 1}^d \underbrace{\mathrm{tr}\left\{\frac{I+X_{\omega_i}}{2} Z_{\omega_i}^{x_i} \right\}}_{ = \, \delta_{x_i,0}} \nonumber\\ & = \mathrm{tr}\left\{\frac{I+X_u}{2} Z_u P_u\right\} \delta_{x_0,1} \delta_{x_1,0}\ldots\delta_{x_d,0} \nonumber\\ & = -i \left[P = Y\right] \delta_{x_0,1} \delta_{x_1,0}\ldots\delta_{x_d,0} . \end{align} \tag{ D45 }$$
  Hence, by substituting equation (D45) into equation (D44), we obtain
  $$\begin{align} \xi\left(X,Z\right) & = 0, \end{align} \tag{ D46 }$$
  $$\begin{align} \xi\left(Y,Z\right) & = -\text{sin}\left(\gamma_{uv}w_{uv}\right) \prod_{\omega \in \mathcal N_{u\backslash v}}\text{cos}\left(\gamma_{u\omega}w_{u\omega}\right). \end{align} \tag{ D47 }$$
• Case 3: $PQ = ZX, ZY$. This case is identical to Case 2, but with the vertices u and v swapped. Hence, from equations (D46) and (D47), we deduce that
  $$\begin{align} \xi\left(Z,X\right) & = 0, \end{align} \tag{ D48 }$$
  $$\begin{align} \xi\left(Z,Y\right) & = -\text{sin}\left(\gamma_{uv}w_{uv}\right) \prod_{q \in \mathcal N_{v\backslash u}}\text{cos}\left(\gamma_{vq}w_{vq}\right). \end{align} \tag{ D49 }$$
• Case 4: PQ = ZZ. This case is straightforward. One readily computes that $\xi(Z,Z)$ vanishes:
  $$\begin{align} \xi\left(Z,Z\right) & = \mathrm{tr}\left(\bigotimes_{j = 1}^n \frac 12\left(I+X_j\right) Z_u Z_v\right) \nonumber\\ & = \prod_{a\neq u,v} \mathrm{tr}\left(\frac {I+X_a}2\right)\cdot \mathrm{tr}\left(\frac {I+X_u}2 Z_u\right) \cdot \underbrace{\mathrm{tr}\left(\frac {I+X_v}2 Z_v\right)}_{ = 0} \nonumber\\ & = 0. \end{align} \tag{ D50 }$$

From the above calculations, we see that for five of the nine choices of P and Q, $\xi(P,Q)$ vanishes: $\xi(Z,Z) = \xi(Z,X) = \xi(Y,X) = \xi(X,Z) = \xi(X,Y) = 0$. The remaining four non-vanishing ones are given by equations (D42), (D43), (D47) and (D49).

Step 3: derivation of equation (22)

We are now essentially done. By substituting the expressions for $\xi(P,Q)$ that we obtained in Step 2 into equation (D11), and noting that $\mathcal N_{v\backslash u} = e$, $\mathcal N_{u \backslash v} = d$, $\mathcal N_{u\backslash\!\!\backslash v} = d\backslash F$, $\mathcal N_{v\backslash\!\!\backslash u} = e\backslash F$, and $N_{uv} = F$, we obtain equation (22). □

The analytical expressions for the expected cost function of both MA-QAOA and QAOA can be derived from XQAOA’s analytical formula. Setting all α_i to 0 in equation (22) gives equation (18) in theorem 2. Moreover, setting all β_i to β and γ_i to γ in equation (18) and simplifying gives equation (13) in theorem 1.

Proof of corollary 4. By setting $\beta_k = 0$ for all k in equation (22), we obtain the following expected cost function (corresponding to the edge $\{u,v\}$) for $\mathrm{XQAOA}_1^{\mathrm{Y}}$ :

$$\begin{align} \langle C_{uv}\rangle_{\mathrm{Y}} = \frac {w_{uv}}2 - \frac {w_{uv}}4 \text{sin}{2\alpha_u}\text{sin}{2\alpha_v} \prod_{\substack{w \in e \\ w \notin F}}\text{cos}\gamma_{wv}^{^{\prime}} \prod_{\substack{w \in d \\ w \notin F}}\text{cos} \gamma_{uw}^{^{\prime}} \left( \prod_{f \in F} \text{cos}\left(\gamma_{uf}^{^{\prime}} + \gamma_{vf}^{^{\prime}}\right) + \prod_{f \in F} \text{cos}\left(\gamma_{uf}^{^{\prime}} - \gamma_{vf}^{^{\prime}}\right) \right). \end{align} \tag{ D51 }$$

To prove corollary 4, we shall specialise equation (D51) to unweighted graphs with odd edge degrees. First, we show that if every edge degree of a graph $G = (V,E)$ is odd, then the graph is necessarily two-colourable, i.e. there exists a map $\tau:V\rightarrow \{0,1\}$ such that for all edges $\{u,v\}\in E$, $\tau(u)\neq \tau(v)$. Indeed, an instantiation of such a map τ is given by:

$$\begin{align} \tau\left(v\right) = \begin{cases} 0, & \mathrm{deg}\left(v\right) \ \textrm{even,} \\ 1, & \mathrm{deg}\left(v\right) \ \textrm{odd.} \end{cases} \end{align} \tag{ D52 }$$

It now remains for us to prove that τ is indeed a two-colouring: let $\{u,v\}\in E$. Note that the edge degree of an edge $\{i,j\}\in E$ is $\mathrm{deg}(\{i,j\}) = |\mathcal{N}(i)\cup \mathcal{N}(j)|-2 = \mathrm{deg}(i)+\mathrm{deg}(j)-2$. By our assumption, the edge degree $\mathrm{deg}(\{u,v\})$ is odd, and hence $\mathrm{deg}(u) + \mathrm{deg}(v)$ is also odd. This implies that exactly one of $\mathrm{deg}(u)$ and $\mathrm{deg}(v)$ is odd, from which it follows that exactly one of $\tau(u)$ and $\tau(v)$ is equal to 1. Hence, $\tau(u)$ and $\tau(v)$ cannot be equal to each other, i.e. $\tau(u) \neq \tau(v)$. This completes our proof that G is two-colourable.

Now, a graph is two-colourable if and only if it does not contain an odd cycle (see [123, theorem 5.21], for example). Hence, the graph G considered in corollary 4 with all edge degrees being odd must be triangle-free, i.e. G does not contain a three-cycle. This implies that the set F in equation (D51) must be empty.

Therefore, setting $F = \emptyset$, $w_{uv} = 1$, and $\gamma_{ij}^{^{\prime}} = \gamma_{ij}$ for all edges $\{i,j\}\in E$ gives the expected $\mathrm{XQAOA}_1^{\mathrm{Y}}$ cost function (for the edge $\{u,v\}$) for unweighted graphs with all edge degrees being odd:

$$\begin{align} \langle C_{uv}\rangle_{\mathrm{Y}} = \frac 12 - \frac 12 \text{sin}{2\alpha_u}\text{sin}{2\alpha_v} \prod_{w \in e}\text{cos}\gamma_{wv} \prod_{w \in d}\text{cos} \gamma_{uw}. \end{align} \tag{ D53 }$$

Note that if all the mixer unitary angles $\alpha_i = \alpha$ are equal and all the problem unitary angles $\gamma_{jk} = \gamma$ are equal, then equation (D53) simplifies to

$$\begin{align} \left\langle C_{u v}\right\rangle_{\mathrm{Y}} = \frac{1}{2}- \frac 12 \text{sin}^2 2\alpha \text{cos}^{|e|+|d|} \gamma. \end{align} \tag{ D54 }$$

In equation (D54), if one takes $\alpha = \tfrac \pi 4$ and $\gamma = \pi$, one obtains

$$\begin{align} \left\langle C_{u v}\right\rangle_{\mathrm{Y}} = \frac 12-\frac 12 \left(-1\right)^{|e|+|d|}. \end{align} \tag{ D55 }$$

By assumption, G has only odd edge degrees. Therefore, $|e|+|d|$ is odd, which implies that

$$\begin{align} \left\langle C_{u v}\right\rangle_{\mathrm{Y}} = \frac 12+\frac 12 = 1. \end{align} \tag{ D56 }$$

So, the optimal expected cost function is given by

$$\begin{align} \sum_{\left\{u,v\right\}\in E} \left\langle C_{u v}\right\rangle_{\mathrm{Y}} = |E|, \end{align} \tag{ D57 }$$

which coincides with the maximum cut size of two-colourable graphs (the MaxCut of two-colourable graphs is $|E|$ because one could just choose the maximum cut to be the two-colouring).

Therefore, the approximation ratio achieved is 1, i.e. the $\mathrm{XQAOA}_1^{\mathrm{Y}}$ state $\left|\gamma, \alpha\right\rangle$ provides the exact MaxCut solution for G. □

In this appendix, we give a proof of corollary 5, which quantifies the advantage that XQAOA has over QAOA for an unweighted $(k+1)$-vertex star graph $G = S_k$, illustrated in figure 2. To keep our analysis general, we will take k to be an arbitrary positive integer for now and only later specialise to k = 4.

Note that when k is even, all the edge degrees in S_k are odd; hence, corollary 4 implies that $\mathrm{XQAOA}_1^{\mathrm{Y}}$ with optimal angles computes the maximum cut of G with an approximation ratio of 1 whenever k is even. Hence, to complete the proof of corollary 5, it remains to show that QAOA₁ can do no better than achieve an approximation ratio of $3/4$.

To this end, consider the cost function for QAOA₁ given by equation (14). For the star graph S_k , $|e| = |F| = 0$ and $|d| = k-1$. Hence, equation (14) reduces to $\langle C_{uv}\rangle = \frac 12 + \frac 14 \text{sin} 4\beta \text{sin}\gamma(1+\text{cos}^{k-1} \gamma)$, which is independent of the edge $\{u,v\}$. Therefore, the expected value of C can be written as

$$\begin{align} \langle C \rangle = \frac k2 + \frac k4 \text{sin} 4\beta \text{sin}\gamma\left(1+\text{cos}^{k-1} \gamma\right). \end{align} \tag{ D58 }$$

Maximising equation (D58) over all β’s and γ’s gives

$$\begin{align} \langle C\rangle_{\text{max}} & = \frac k2 + \frac k4 \max_{\gamma,\beta\in \mathbb{R}} \left[\text{sin} 4\beta \text{sin}\gamma\left(1+\text{cos}^{k-1} \gamma\right)\right] \nonumber\\ & = \frac k2 + \frac k4 \max_{\beta\in \mathbb{R}} \left[\text{sin} 4\beta\right] \max_{\gamma\in \mathbb{R}} \left[\text{sin}\gamma\left(1+\text{cos}^{k-1} \gamma\right)\right] \nonumber\\ & = \frac k2 + \frac k4 \max_{\gamma \in \mathbb{R}} g_k\left(\gamma\right), \end{align} \tag{ D59 }$$

$$\begin{align} \mbox{where}\quad g_k\left(\gamma\right) & = \text{sin} \gamma\left(1+\text{cos}^{k-1}\gamma\right). \end{align} \tag{ D60 }$$

To find the maximum point(s) of g_k , we start by taking its derivative. For $k\unicode{x2A7E} 2$, differentiating equation (D60) and simplifying gives

$$\begin{align} g_k^{^{\prime}}\left(\gamma\right) = k \text{cos}^k \gamma - \left(k-1\right)\text{cos}^{k-2}\gamma + \text{cos}\gamma, \end{align} \tag{ D61 }$$

which is a degree-k polynomial in $\text{cos} \gamma$. Hence, finding the maximum point(s) of $g_k(\gamma)$ involves finding the roots of this polynomial. At this point, we specialise to k = 4. This allows to factorise the quartic polynomial equation (D61) as

$$\begin{align} g_4^{^{\prime}}\left(\gamma\right) & = 4 \text{cos}^4 \gamma - 3\text{cos}^2\gamma + \text{cos}\gamma = \text{cos} \gamma \left(\text{cos}\gamma+1\right)\left(2\text{cos}\gamma-1\right)^2. \end{align} \tag{ D62 }$$

Hence, at a maximum point $\gamma^* \in \text{arg}\max_{\gamma \in \mathbb{R}} g_k(\gamma)$, solving $g_4^{^{\prime}}(\gamma^*) = 0$ gives $\text{cos}\gamma^* \in \{0,-1,\tfrac 12\}$. When $\text{cos}\gamma^* = 0$, $g_4(\gamma^*) = \pm 1$; when $\text{cos}\gamma^* = -1$, $g_4(\gamma^*) = 0$; and when $\text{cos}\gamma^* = \tfrac 12$, $g_4(\gamma^*) = \pm \tfrac 9{16}\sqrt{3} \approx \pm 0.974 \lt 1$. Taking the maximum of all these values gives $\max_{\gamma \in \mathbb{R}} g_k(\gamma) = 1$. Substituting this into equation (D59) gives an approximation ratio of

$$\begin{align} \tfrac 1k \langle C\rangle_{\text{max}} = \frac 12 + \frac 14 \cdot 1 = \frac 34. \end{align} \tag{ D63 }$$

In conclusion, for the unweighted 5-vertex star graph S₄, $\mathrm{XQAOA}_1^{\mathrm{Y}}$ achieves an approximation ratio of 1, whereas QAOA₁ achieves an approximation ratio of at most $\frac 34$.