The Recursive Quantum Approximate Optimization Algorithm

OBSTACLES TO STATE PREPARATION AND VARIATIONAL OPTIMIZATION FROM SYMMETRY PROTECTION

3.3 The Recursive Quantum Approximate Optimization Algorithm

dist(𝑗 , 𝑘) > 2𝑅, one infers that𝑉^†𝑍_𝑗𝑉 and𝑉^†𝑍_𝑘𝑉 have disjoint support. Thus h+^𝑛|𝑉^†𝑍_𝑗𝑍_𝑘𝑉|+^𝑛i =h+^𝑛| (𝑉^†𝑍_𝑗𝑉) (𝑉^†𝑍_𝑘𝑉) |+^𝑛i

= h+^𝑛|𝑉^†𝑍_𝑗𝑉|+^𝑛i · h+^𝑛|𝑉^†𝑍_𝑘𝑉|+^𝑛i=0.

This proves Eq. (3.9). Suppose one prepares the state𝑉|+^𝑛iand measures a pair of qubits 𝑗 < 𝑘 in the standard basis. Then 𝜖_{𝑗 , 𝑘} is the probability that the measured values on qubits 𝑗 and𝑘 disagree. By the union bound,

𝜖_{𝑗 , 𝑘} ≤

𝑘−1

∑︁

𝑖=𝑗

𝜖_𝑖,𝑖+

1.

Indeed, if qubits 𝑗 and 𝑘 disagree, at least one pair of consecutive qubits located in the interval [𝑗 , 𝑘] must disagree. Set𝑘 = 𝑗 +2𝑅+1. Then𝜖_{𝑗 , 𝑘} =1/2 by Eq. (3.9).

Take the expected value of Eq. (3.10) with respect to random uniform 𝑗 ∈Z^𝑛. This gives

1 2

≤ 2𝑅+1 𝑛

∑︁

𝑖∈Z^𝑛

𝜖_𝑖,𝑖+

1= 2𝑅+1 𝑛

h+^𝑛|𝑉^†𝐻_𝑛𝑉|+^𝑛i

proving Eq. (3.8). In Appendix 3.B, we construct aZ₂-symmetric range-𝑅circuit𝑈 such that𝑈|+^𝑛iis a tensor product of GHZ-like states on consecutive segments of 2𝑅+1 qubits. We show that such circuit saturates the upper bound Eq. (3.6). This completes the proof of Theorem 3.2.3.

First, run the standard QAOA to maximize the expected value of 𝐻_𝑛 on the state

|𝜓i = 𝑈(𝛽, 𝛾) |+^𝑛i. For every edge (𝑗 , 𝑘) ∈ 𝐸, compute 𝑀_{𝑗 , 𝑘} = h𝜓^∗|𝑍_𝑗𝑍_𝑘|𝜓^∗i, where𝜓^∗ is the optimal variational state.

Next, find a pair of qubits (𝑖, 𝑗) ∈ 𝐸 with the largest magnitude of 𝑀_{𝑖, 𝑗} (breaking ties arbitrarily). The corresponding variables 𝑍_𝑖 and 𝑍_𝑗 are correlated if 𝑀_{𝑖, 𝑗} > 0 and anti-correlated if𝑀_{𝑖, 𝑗} < 0. Impose the constraint

𝑍_𝑗 =sgn(𝑀_{𝑖, 𝑗})𝑍_𝑖

and substitute it into the Hamiltonian𝐻_𝑛to eliminate the variable𝑍_𝑗. For example, a term𝑍_𝑗𝑍_𝑘 with𝑘 ∉{𝑖, 𝑗}gets mapped to sgn(𝑀_{𝑖, 𝑗})𝑍_𝑖𝑍_𝑘. The term 𝐽_{𝑖, 𝑗}𝑍_𝑖𝑍_𝑗 gets mapped to a constant energy shift𝐽_{𝑖, 𝑗}sgn(𝑀_{𝑖, 𝑗}). All other terms remain unchanged.

This yields a new Ising Hamiltonian 𝐻_𝑛−

1 that depends on 𝑛 −1 variables. By construction, the maximum energy of𝐻_𝑛−

1coincides with the maximum energy of 𝐻_𝑛over the subset of assignments satisfying the constraint Eq. (3.11).

Finally, call RQAOA recursively to maximize the expected value of 𝐻_𝑛−

1. Each recursion step eliminates one variable from the cost function. The recursion stops when the number of variables reaches some specified threshold value𝑛_𝑐 𝑛. The remaining instance of the problem with 𝑛_𝑐 variables is then solved by a purely classical algorithm (for example, by a brute force method). Thus the value of 𝑛_𝑐 controls how the workload is distributed between quantum and classical computers.

We describe a generalization of RQAOA applicable to Ising-like cost functions with multi-spin interactions in Appendix 3.C.

Imposing a constraint of the form (3.11) can be viewed as rounding correlations among the variables 𝑍_𝑖 and 𝑍_𝑗. Indeed, the constraint demands that these vari- ables must be perfectly correlated or anti-correlated. This is analogous to rounding fractional solutions obtained by solving linear programming relaxations of combi- natorial optimization problems. We note that reducing the size of a problem to the point that it can be solved optimally by brute force is a widely used and effective approach in combinatorial optimization.

We compare the performance of the standard QAOA, RQAOA, and local classical algorithms by considering the Ising Hamiltonians in Eq. (3.3) with couplings𝐽_𝑝,𝑞 =

±1 defined on the cycle graph. In Appendix 3.D, we prove:

Theorem 3.3.1. For each integer𝑛divisible by6, there exists a family of2^𝑛/3Ising Hamiltonians of the form 𝐻_𝑛 = Í

𝑘∈Z^𝑛𝐽_𝑘𝑍_𝑘𝑍_𝑘+

1 with 𝐽_𝑘 ∈ {1,−1} such that the

(a) (b)

Figure 3.1: Comparisons of level-1 RQAOA and the Goemans-Williamson Al- gorithm. (a) Approximation ratios achieved by level-1 RQAOA (blue) and the Goemans-Williamson (GW) algorithm [21] (red) for 15 instances of the Ising cost function with random ±1 couplings defined on the 2D toric grid of size 16× 16.

In case (b) the Ising Hamiltonian also includes random ±1 external fields. The RQAOA threshold value is 𝑛_𝑐 = 20. We found that the standard level-1 QAOA achieves approximation ratios below 1/2 for all considered instances (not shown).

The GW algorithm was implemented with𝑛 =256 rounding attempts and the best found solution was selected. The exact maximum energy was computed using integer linear programming.

following holds for all Hamiltonians in the family:

(i) There is a local classical algorithm which achieves the approximation ratio1. (ii) Level-𝑝QAOA achieves an approximation ratio of at most 𝑝/(𝑝+1).

(iii) Level-1RQAOA achieves the approximation ratio1.

Our definition of local classical algorithms follows [20]. We also show that the level-1 RQAOA achieves the optimal approximation ratio for any 1D Ising model with coupling coefficients𝐽_𝑘 ∈ {1,−1}. This proves that, in certain cases, RQAOA is strictly more powerful than QAOA.

Finally, we report a numerical comparison between the level-1 RQAOA and the Goemans-Williamson algorithm [21] for the Ising cost function Eq. (3.3) with ran- dom coefficients𝐽_{𝑗 , 𝑘} =±1. Two graphs are considered: (a) the 2D grid, and (b) the 2D grid with one extra vertex connected to all grid points. The latter is equivalent to the 2D Ising model with random ±1 external fields. As shown in [23], the problem of maximizing the energy 𝐶(𝑥) admits an efficient algorithm in case (a) while case (b) is NP-hard. To compute the mean values h𝜓(𝛽, 𝛾) |𝑍_𝑗𝑍_𝑘|𝜓(𝛽, 𝛾)i,

we used a version of the algorithm by Wang et al [24], as detailed in Appendix 3.C.

Figure 3.1 shows approximation ratios achieved by each algorithm for 15 problem instances with the grid size 16×16. It can be seen that RQAOA outperforms the Goemans-Williamson algorithm for all except for one instance. We found that the standard level-1 QAOA achieves approximation ratio below 1/2 for all considered instances.

Note added: After submission of this work, analogous limitations were estab- lished for random regular graphs by exploiting the locality and spatial uniformity of QAOA [25], [26]. We focus onZ₂-symmetry and locality, and our statements also apply to non-uniform local algorithms.

Acknowledgements

The authors thank Giacomo Nannicini and Kristan Temme for helpful discussions.

SB was partially supported by the IBM Research Frontiers Institute and by the Army Research Office (ARO) under Grant Number W911NF-20-1-0014. ET ac- knowledges the support of the Natural Sciences and Engineering Research Council of Canada (NSERC) and funding provided by the Institute for Quantum Information and Matter, an NSF Physics Frontiers Center (NSF Grant No. PHY–1733907).

RK and AK gratefully acknowledge support by the DFG cluster of excellence 2111 (Munich Center for Quantum Science and Technology) and by IBM.

3.A Proof of Corollary 3.2.2

In this appendix, we give a proof of Corollary 3.2.2 in the main text. Here and below, we will denote the expected approximation ratio achieved by the QAOA with Hamiltonian𝐻as

QAOA𝑝(𝐻) =

max

𝛽,𝛾∈R^𝑝

hΨ𝐻(𝛽, 𝛾) |𝐻|Ψ𝐻(𝛽, 𝛾)i

·

max

𝑥∈{0,1}^𝑛

h𝑥|𝐻|𝑥i −1

, where

|Ψ𝐻(𝛽, 𝛾)i =𝑈_𝐻(𝛽, 𝛾) |+^𝑛i and 𝑈_𝐻(𝛽, 𝛾) =

𝑝

Ö

𝑚=1

𝑒^{𝑖 𝛽𝑚𝐵}𝑒^{𝑖 𝛾𝑚𝐻}

(3.16) for 𝛽, 𝛾 ∈ R^𝑝 and where𝐵 =Í𝑛

𝑗=1𝑋_𝑗. Let us first record a few general features of the QAOA for later use.

Let 𝐺 = (𝑉 , 𝐸) be a graph, 𝑛 = |𝑉|, 𝑚 = |𝐸|, and let 𝐽 = (𝐽_𝑒)_𝑒∈𝐸 ∈ R^𝐸 be an assignment of edge weights on𝐺. Let us define the Hamiltonian𝐻_𝐺(𝐽) as

𝐻_𝐺(𝐽) = ∑︁

{𝑢,𝑣}∈𝐸

𝐽_{𝑢,𝑣}𝑍_𝑢𝑍_𝑣 . (3.17) It will be useful for later to also define

𝐻_𝐺 = ∑︁

{𝑢,𝑣}∈𝐸

𝑍_𝑢𝑍_𝑣, and 𝐻^MaxCut

𝐺 = 1

2

(𝑚 𝐼 −𝐻_𝐺),

where 𝐻^MaxCut

𝐺 is the Hamiltonian used in QAOA for the Maximum Cut problem on the graph𝐺. We will use the following bound on the circuit depth of a QAOA unitary.

Lemma 3.A.1. Let 𝑈 = 𝑈_𝐻(𝛽, 𝛾) with 𝛽, 𝛾 ∈ R^𝑝 be a level-𝑝 QAOA unitary (cf. Eq.(3.A)) for a Hamiltonian𝐻 =𝐻_𝐺(𝐽)on a graph𝐺(cf.(3.A)). Let𝐷be the maximum degree of𝐺. Then𝑈 can be realized by a circuit of depth𝑑 ≤ 𝑝(𝐷+1) consisting of2-qubit gates.

If 𝐺 is 𝐷-regular and bipartite, then the circuit depth of 𝑈 can be bounded by 𝑑 ≤ 𝑝 𝐷.

Proof. By Vizing’s theorem [27], there is an edge coloring of𝐺 with at most𝐷+1 colors. Taking such a coloring 𝐸 = 𝐸

1∪ · · · ∪ 𝐸_𝐷+

1, we may apply each level 𝑒^{𝑖 𝛽 𝐵}𝑒^{𝑖 𝛾 𝐻} of𝑈 in depth 𝐷 +1 by applying (Î

𝑣∈𝑉𝑒^{𝑖 𝛽 𝑋𝑣})Î𝐷+1

𝑐=1 𝑉_𝑐(𝛾), where each 𝑉_𝑐(𝛾) =

Î

{𝑢,𝑣}∈𝐸𝑐

𝑒^{𝑖 𝛾 𝐽{𝑢 , 𝑣}𝑍𝑢𝑍𝑣}

is a depth-1-circuit of two-local gates.

If𝐺 is𝐷-regular and bipartite, we may reduce the chromatic number upper bound from 𝐷 +1 to 𝐷 since all bipartite graphs are 𝐷-edge-colorable by Kőnig’s line coloring theorem. We illustrate the construction of the circuit on Figure 3.2 for the case𝐷 =3 and 𝑝=1.

The expected QAOA approximation ratios of suitably related instances are identical:

Lemma 3.A.2. LetL ⊂ 𝑉 be an arbitrary subset of vertices and 𝜕L be the set of edges that have exactly one endpoint inL. Let𝐽 =(𝐽_𝑒)𝑒∈𝐸 ∈R^𝐸 be arbitrary edge weights. Define𝐽˜= (𝐽˜_𝑒)𝑒∈𝐸 ∈R^𝐸 by

˜ 𝐽_𝑒 =











−𝐽_𝑒 if𝑒 ∈𝜕L 𝐽_𝑒 otherwise.

Figure 3.2: Example for the construction of the circuit given in Lemma 3.A.1: a 4-colorable graph with maximum degree 3 alongside its associated depth-5 quantum circuit for the level-1 QAOA unitary.

Then expected QAOA approximation ratios satisfy

QAOA𝑝(𝐻_𝐺(𝐽)) =QAOA𝑝(𝐻_𝐺(𝐽˜)).

Proof. Let us write 𝐻 = 𝐻_𝐺(𝐽) and ˜𝐻 = 𝐻_𝐺(𝐽˜) for brevity. Let 𝑋 = 𝑋[L]

be a tensor product of Pauli-𝑋 operators acting on every qubit in L ⊂ 𝑉. Then

˜

𝐻 = 𝑋 𝐻 𝑋, which implies that

𝑥∈{max0,1}^𝑛h𝑥|𝐻|𝑥i= max

𝑥∈{0,1}^𝑛h𝑥|𝐻˜|𝑥i. (3.18) Let𝛽, 𝛾 ∈R^𝑝be arbitrary. Then we also have

𝑋|Ψ_𝐻˜(𝛽, 𝛾)i =

𝑝

Ö

𝑚=1

(𝑋 𝑒^{𝑖 𝛽𝑚𝐵}𝑒^{𝑖 𝛾𝑚𝐻˜}𝑋) |+^𝑛i=

𝑝

Ö

𝑚=1

(𝑒^{𝑖 𝛽𝑚𝐵}𝑒^{𝑖 𝛾𝑚𝐻}) |+^𝑛i=|Ψ𝐻(𝛽, 𝛾)i,

where identities in the middle follow since|+^𝑛iis stabilized by𝑋, and since[𝑋 , 𝐵] = 0. Therefore we have

hΨ_𝐻˜(𝛽, 𝛾) |𝐻˜|Ψ_𝐻˜(𝛽, 𝛾)i =hΨ_𝐻˜(𝛽, 𝛾) |𝑋 𝐻 𝑋|Ψ_𝐻˜(𝛽, 𝛾)i =hΨ𝐻(𝛽, 𝛾) |𝐻|Ψ𝐻(𝛽, 𝛾)i.

Combined with (3.A), this implies the claim.

In particular, if𝐺 = (𝑉 , 𝐸)is a bipartite graph, then Lemma 3.A.2 implies that QAOA𝑝(𝐻_𝐺) =QAOA𝑝(−𝐻_𝐺)

and

QAOA𝑝(𝐻^MaxCut

𝐺 ) = 1

2

(1+QAOA𝑝(𝐻_𝐺)). (3.19) We now prove Corollary 1. It is a direct consequence of Theorem 1, which we restate here for convenience in the notation of this appendix:

Theorem 3.A.1. Consider a family {𝐺_𝑛 = ( [𝑛], 𝐸_𝑛)}𝑛∈I of graphs with Cheeger constant lower bounded asℎ(𝐺_𝑛) ≥ ℎ > 0for all𝑛∈ I. Then

h𝜑|𝑈^†𝐻_𝐺

𝑛𝑈|𝜑i< |𝐸_𝑛| − ℎ𝑛 3

for any Z₂-symmetric depth-𝑑 circuit 𝑈 composed of two-qubit gates, any Z₂- symmetric product state𝜑, and any𝑛 >48²8^𝑑,𝑛 ∈ I.

Then we have the following:

Corollary 3.A.1. For every integer𝐷 ≥ 3, there exists an infinite family of bipartite 𝐷-regular graphs{𝐺_𝑛}𝑛∈I such that

QAOA𝑝(𝐻^MaxCut

𝐺𝑛

) ≤ 5 6

+

√ 𝐷−1

3𝐷 as long as

𝑝 < (1/3 log₂𝑛−4)𝐷⁻¹ .

Proof. Fix some𝐷 ≥ 3. By the results of [28], [29], there exists an infinite family {𝐺_𝑛}_𝑛∈I of bipartite 𝐷-regular Ramanujan graph with𝑛 vertices for every𝑛 ∈ I. Consider a fixed 𝑛 ∈ I and let 𝑝 = 𝑝(𝑛) be the associated QAOA level. Let 𝑈_𝑛 = 𝑈_𝐻

𝐺𝑛(𝛽^∗, 𝛾^∗) be a level-𝑝 QAOA unitary for the Hamiltonian 𝐻_𝐺

𝑛 on 𝐺_𝑛, and assume that 𝛽^∗, 𝛾^∗ ∈ R^𝑝 are such that the expectation of 𝐻_𝐺

𝑛 is maximized.

Because𝐺_𝑛is𝐷-regular, the circuit depth of𝑈_𝑛can be bounded from above by 𝑝 𝐷 according to Lemma 3.A.1. Condition (3.12) implies that𝑛 >48²8^{𝑝 𝐷}, thus

QAOA𝑝(𝐻_𝐺

𝑛) = 1

|𝐸_𝑛|h+^𝑛|𝑈^†

𝑛𝐻_𝐺

𝑛𝑈_𝑛|+^𝑛i < 1− ℎ

3|𝐸_𝑛|𝑛 =1− 2ℎ 3𝐷

by Theorem 1, where we have used that |𝐸_𝑛| =𝑛 𝐷/2. With (3.A) (using that𝐺_𝑛is bipartite), we conclude that

QAOA𝑝(𝐻^MaxCut

𝐺𝑛

) < 1− ℎ 3𝐷

. The claim then follows from the boundℎ/𝐷 ≥ (𝐷−2

√

𝐷−1)/(2𝐷), valid for all Ramanujan graphs.

3.B Optimal Variational Circuit for the Ring of Disagrees

In this section we prove that the upper bound of Theorem 2 in the main text is tight whenever𝑛is a multiple of 2𝑅+1. Let

|GHZ_ni=2^−1/2(|0^𝑛i + |1^𝑛i) be the GHZ state of𝑛qubits.

Lemma 3.B.1. Suppose𝑛=2𝑝+1for some integer𝑝. There exists aZ2-symmetric range-𝑝quantum circuit𝑉 such that

|GHZ_ni=𝑉|+^𝑛i.

Proof. We shall write^CX𝑐,𝑡 for the CNOT gate with a control qubit𝑐 and a target qubit𝑡. Let𝑐= 𝑝+1 be the central qubit. One can easily check that

|GHZ_ni=©

­

«

𝑝

Ö

𝑗=1

CX𝑐,𝑐−𝑗CX𝑐,𝑐+𝑗

ª

®

¬

𝐻_𝑐|0^𝑛i.

All^CXgates in the product pairwise commute, so the order does not matter. Inserting a pair of Hadamards on every qubit 𝑗 ∈ [𝑛] \ {𝑐}before and after the respective^CX gate and using the identity(𝐼 ⊗𝐻)CX(𝐼⊗ 𝐻) =CZ, one gets

|GHZ_ni=©

­

« Ö

𝑗∈[𝑛]\{𝑐}

𝐻_𝑗ª

®

¬

©

­

«

𝑝

Ö

𝑗=1

CZ𝑐,𝑐−𝑗CZ𝑐,𝑐+𝑗ª

®

¬

|+^𝑛i.

Let𝑆=exp[𝑖(𝜋/4)𝑍] be the phase-shift gate. Define the two-qubit Clifford gate RZ=(𝑆⊗ 𝑆)⁻¹CZ=exp(−𝑖 𝜋/4)exp[−𝑖(𝜋/4) (𝑍 ⊗𝑍)].

Expressing^CZin terms of^RZand𝑆 in Eq. (3.14), one gets

|GHZ_ni =𝑆²

𝑝 𝑐

©

­

« Ö

𝑗∈[𝑛]\{𝑐}

𝐻_𝑗𝑆_𝑗ª

®

¬

©

­

«

𝑝

Ö

𝑗=1

RZ𝑐,𝑐−𝑗RZ𝑐,𝑐+𝑗

ª

®

¬

|+^𝑛i.

Multiply both sides of Eq. (3.15) on the left by a product of 𝑆 gates over qubits 𝑗 ∈ [𝑛] \ {𝑐}. Noting that

𝑆 𝐻 𝑆 =𝑖exp[−𝑖(𝜋/4)𝑋], one gets (ignoring an overall phase factor)

Ö

𝑗∈[𝑛]\{𝑐}

𝑆_𝑗|GHZ_ni=𝑆²

𝑝 𝑐

©

­

« Ö

𝑗∈[𝑛]\{𝑐}

exp[−𝑖(𝜋/4)𝑋_𝑗]ª

®

¬

©

­

«

𝑝

Ö

𝑗=1

RZ𝑐,𝑐−𝑗RZ𝑐,𝑐+𝑗ª

®

¬

|+^𝑛i.

Using the identity

Ö

𝑗∈[𝑛]\{𝑐}

𝑆_𝑗|GHZ_ni =𝑆²

𝑝

𝑐 |GHZ_ni, one can cancel𝑆²

𝑝

𝑐 that appears in both sides of Eq. (3.16). We arrive at Eq. (3.13) with

𝑉 =©

­

« Ö

𝑗∈[𝑛]\{𝑐}

exp[−𝑖(𝜋/4)𝑋_𝑗]ª

®

¬

©

­

«

𝑝

Ö

𝑗=1

RZ𝑐,𝑐−𝑗RZ𝑐,𝑐+𝑗ª

®

¬ .

The circuit diagram of𝑉 in the case 𝑛 = 7 is shown in Figure 3.3. Obviously,𝑉 isZ2-symmetric since any individual gate commutes with 𝑋^⊗𝑛. Let us check that 𝑉 has range-𝑝. Consider any single-qubit observable𝑂_𝑞 acting on the 𝑞-th qubit.

Consider three cases:

Case 1: 𝑞 = 𝑐. Then 𝑉^†𝑂_𝑞𝑉 may be supported on all 𝑛 qubits. However, [𝑐− 𝑝, 𝑐+𝑝] = [1, 𝑛], so the 𝑝-range condition is satisfied trivially.

Case 2: 1 ≤ 𝑞 < 𝑐. Then all gates ^RZ𝑐,𝑐+𝑗 in𝑉 cancel the corresponding gates in𝑉^†, so that𝑉^†𝑂_𝑞𝑉 has support in the interval[1, 𝑐] ⊆ [𝑞− 𝑝, 𝑞+ 𝑝]. Thus the 𝑝-range condition is satisfied.

Case 3: 𝑐 < 𝑞 ≤ 𝑛. This case is equivalent to Case 2 by symmetry.

Recall that we consider the ring of disagrees Hamiltonian 𝐻_𝑛 = 1

2

∑︁

𝑝∈Z^𝑛

(𝐼−𝑍_𝑝𝑍_𝑝+

1).

Lemma 3.B.2. Consider any integers𝑛, 𝑝such that𝑛is even and𝑛is a multiple of 2𝑝+1. Then there exists aZ₂-symmetric range-𝑝circuit𝑈such that

h+^𝑛|𝑈^†𝐻_𝑛𝑈|+^𝑛i= 2𝑝+1/2 2𝑝+1

.

Proof. Let𝑉be theZ₂-symmetric range-𝑝unitary operator preparing the GHZ state on 2𝑝+1 qubits starting from |+^2𝑝+1i, see Lemma 3.B.1. Suppose𝑛 =𝑚(2𝑝+1) for some integer𝑚. Define

𝑈 =𝑈

1𝑈

2,

Figure 3.3: The Z2-symmetric range-3 quantum circuit to prepare the GHZ state

|GHZ_2p+1iof 2𝑝+1=7 qubits (𝑝=3). Here,𝑅_𝑂(𝜃) =exp(−𝑖 𝜃 𝑂). where

𝑈1= (𝑋 ⊗ 𝐼)^⊗𝑛/2 and 𝑈

2 =𝑉^⊗𝑚.

Since each copy of𝑉 acts on a consecutive interval of qubits and has range 𝑝, one infers that𝑈 has range𝑝. We have

𝑈^†

1𝐻_𝑛𝑈

1 = ∑︁

𝑝∈Z^𝑛

𝐺_𝑝, 𝐺_𝑝 = 1 2

(𝐼+𝑍_𝑝𝑍_𝑝+

1). The state𝑈

2|+^𝑛iis a tensor product of GHZ states supported on consecutive tuples of 2𝑝+1 qubits. The expected value of 𝐺_𝑝 on the state𝑈

2|+^𝑛iequals 1 if 𝐺_𝑝 is supported on one of the GHZ states. Otherwise, if𝐺_𝑝crosses the boundary between two GHZ states, the expected value of𝐺_𝑝on the state𝑈

2|+^𝑛iequals 1/2. Thus h+^𝑛|𝑈^†𝐻_𝑛𝑈|+^𝑛i = ∑︁

𝑝∈Z^𝑛

h+^𝑛|𝑈^†

2

𝐺_𝑝𝑈

2|+^𝑛i =𝑚(2𝑝+1/2) =𝑛2𝑝+1/2 2𝑝+1

.

3.C Recursive QAOA

In this appendix, we outline the Recursive QAOA algorithm (RQAOA) for general cost functions.