RQAOA on the Ising Ring

OBSTACLES TO STATE PREPARATION AND VARIATIONAL OPTIMIZATION FROM SYMMETRY PROTECTION

D.2 RQAOA on the Ising Ring

Combined with (D.2) and (D.2), we conclude that

argmax(𝑖, 𝑗):𝑖 < 𝑗 |hΨ𝐻(𝛽^∗, 𝛾^∗) |𝑍_𝑖𝑍_𝑗|Ψ𝐻(𝛽^∗, 𝛾^∗)i| = (𝑖^∗, 𝑖^∗+1) (3.54) for some 𝑖^∗ ∈ Z^𝑛. Without loss of generality, assume that 𝑖^∗ = 𝑛 − 2. Then, according to (D.2), the^RQAOAalgorithm eliminates the variable𝑥_𝑛−

1(i.e.,𝑣=𝑛−1, 𝑓 ={𝑛−2, 𝑛−1}). By (D.2), this is achieved by imposing the constraint

𝑥_𝑛−

1=𝑥_𝑛−

2𝐽_𝑛−

2 (3.55)

i.e., 𝜎 = 𝐽_𝑛−

2. The resulting reduced graph 𝐺⁰ = (𝑉⁰, 𝐸⁰) has vertex set 𝑉⁰ = 𝑉\{𝑛−1}=Z^𝑛−₁and edges

𝐸⁰={{𝑖, 𝑖+1} |𝑖 ∈Z^𝑛\{𝑛−2}} ∪ {{𝑛−2,0}}

={{𝑖, 𝑖+1} |𝑖 ∈Z^𝑛−₂} ,

and it is easy to check that the new cost function takes the form 𝐶⁰(𝑥) =1+ ∑︁

𝑘∈Z^𝑛−1

𝐽⁰

𝑘𝑥_𝑘𝑥_𝑘+

1 (3.56)

with

𝐽⁰

𝑖 =











𝐽_𝑖 when𝑖 ≠𝑛−2 𝐽_𝑛−

2𝐽_𝑛−

1 when𝑖 =𝑛−2

. (3.57)

We note that the transformation (D.2) preserves the parity of the couplings in the sense that

Ö

𝑘∈Z^𝑛

𝐽_𝑘 = Ö

𝑘∈Z^𝑛−1

𝐽⁰

𝑘 . (3.58)

Inductively, the^RQAOAthus eliminates variables𝑥_𝑛−

1, 𝑥_𝑛−

2, . . . , 𝑥_𝑛

𝑐while imposing the constraints (cf. (D.2))

𝑥_𝑛−

1=𝑥_𝑛−

2𝐽_𝑛−

2

𝑥_𝑛−

2=𝑥_𝑛−

3𝐽⁰

𝑛−3

.. . arriving at the cost function𝐶_𝑛

𝑐(𝑥)associated with an Ising chain of length𝑛_𝑐having couplings belonging to {1,−1}. Because of (D.2) and because (D.2) is frustrated

if and only if Î

𝑘∈Z^𝑛𝐽_𝑘 = −1, we conclude that any maximum 𝑥^∗ ∈ {1,−1}^𝑛𝑐 of 𝐶_𝑛

𝑐(𝑥)satisfies

𝐶_𝑛

𝑐(𝑥^∗) =











𝑛_𝑐+1 if Î

𝑘∈Z^𝑛𝐽_𝑘 =1 𝑛_𝑐−2 otherwise.

Because the cost function acquires a constant energy shift in every variable elimi- nation step (cf. (D.2)), the output𝑥 =𝜉(𝑥^∗) of the^RQAOAalgorithm satisfies

𝐶(𝑥) =𝑛−𝑛_𝑐+𝐶_𝑛

𝑐(𝑥^∗)=











𝑛+1 if Î

𝑘∈Z^𝑛𝐽_𝑘 =1 𝑛−2 otherwise. This implies the claim.

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Part II