Classical Simulability of Level- 1 RQAOA for Ising Models

OBSTACLES TO STATE PREPARATION AND VARIATIONAL OPTIMIZATION FROM SYMMETRY PROTECTION

C.4 Classical Simulability of Level- 1 RQAOA for Ising Models

Suppose 𝐽 is a real symmetric matrix of size 𝑛. Here we consider Ising-like cost functions such that the corresponding Hamiltonian is

𝐻 = ∑︁

1≤𝑝 <𝑞≤𝑛

𝐽_𝑝,𝑞𝑍_𝑝𝑍_𝑞 .

The mean values of a Pauli operator𝑍_𝑝𝑍_𝑞on the level-1 QAOA state

|Ψ𝐻(𝛽, 𝛾)i =𝑒^{𝑖 𝛽 𝐵}𝑒^{𝑖 𝛾 𝐻}|+^𝑛i

can be computed in time 𝑂(𝑛) using an explicit analytic formula. Such a formula was derived for the Max-Cut cost function by Wang et al. [24, Theorem 1]. Below we provide a generalization to general Ising Hamiltonians. Since the total number of terms in the cost function is𝑂(𝑛²), simulating each step of^RQAOAtakes time at most𝑂(𝑛³). Assuming that𝑛_𝑐=𝑂(1), the number of steps is roughly𝑛so that the full simulation cost is𝑂(𝑛⁴). Crucially, the simulation cost of this method does not depend on the depth of the variational circuit. This is important because ^RQAOA may potentially increase the depth from𝑂(1)to𝑂(𝑛)since it adds many new terms to the cost function.

Lemma 3.C.1. Fix a pair of qubits 1 ≤ 𝑢 < 𝑣 ≤ 𝑛. Let 𝑐 = cos(2𝛽) and

𝑠 =sin(2𝛽). Then

hΨ𝐻(𝛽,1) |𝑍_𝑢𝑍_𝑣|Ψ𝐻(𝛽,1)i = (𝑠²/2) Ö

𝑝≠𝑢,𝑣

cos[2𝐽_{𝑢, 𝑝}−2𝐽_{𝑣 , 𝑝}] − (𝑠²/2) Ö

𝑝≠𝑢,𝑣

cos[2𝐽_{𝑢, 𝑝}+2𝐽_{𝑣 , 𝑝}] +𝑐 𝑠·sin(2𝐽_𝑢,𝑣)

"

Ö

𝑝≠𝑢,𝑣

cos(2𝐽_{𝑢, 𝑝}) + Ö

𝑝≠𝑢,𝑣

cos(2𝐽_{𝑣 , 𝑝})

# .

Here we only consider the case𝛾 =1 since𝛾can be absorbed into the definition of 𝐽.

Proof. Given a 2-qubit observable𝑂, define the mean value 𝜇(𝑂) =hΨ𝐻(𝛽,1) |𝑂_𝑢,𝑣|Ψ𝐻(𝛽,1)i. We are interested in the observable𝑂 =𝑍 𝑍 ≡ 𝑍 ⊗ 𝑍.

We note that all terms in 𝐻 and 𝐵 that act trivially on {𝑢, 𝑣} do not contribute to 𝜇(𝑂). Such terms can be set to zero. Given a 2-qubit observable𝑂, define a mean value

𝜇⁰(𝑂)= h+^𝑛|𝑒^{𝑖 𝐻}

0

𝑂_𝑢,𝑣𝑒^{−𝑖 𝐻}

0|+^𝑛i, where 𝐻⁰= ∑︁

𝑝≠𝑢,𝑣

(𝐽_{𝑢, 𝑝}𝑍_𝑢+𝐽_{𝑣 , 𝑝}𝑍_𝑣)𝑍_𝑝. Using the identities

𝑒^{𝑖 𝛽(𝑋𝑢+𝑋𝑣)}𝑍_𝑢𝑍_𝑣𝑒^{−𝑖 𝛽(𝑋𝑢+𝑋𝑣)} = 𝑐²𝑍_𝑢𝑍_𝑣+𝑠²𝑌_𝑢𝑌_𝑣+𝑐 𝑠(𝑍_𝑢𝑌_𝑣+𝑌_𝑢𝑍_𝑣), 𝑒^{𝑖 𝐽𝑢 , 𝑣𝑍𝑢𝑍𝑣}𝑍_𝑢𝑍_𝑣𝑒^{−𝑖 𝐽𝑢 , 𝑣𝑍𝑢𝑍𝑣} = 𝑍_𝑢𝑍_𝑣,

𝑒^{𝑖 𝐽𝑢 , 𝑣𝑍𝑢𝑍𝑣}𝑌_𝑢𝑌_𝑣𝑒^{−𝑖 𝐽𝑢 , 𝑣𝑍𝑢𝑍𝑣} = 𝑌_𝑢𝑌_𝑣

𝑒^{𝑖 𝐽𝑢 , 𝑣𝑍𝑢𝑍𝑣}𝑍_𝑢𝑌_𝑣𝑒^{−𝑖 𝐽𝑢 , 𝑣𝑍𝑢𝑍𝑣} = cos(2𝐽_𝑢,𝑣)𝑍_𝑢𝑌_𝑣+sin(2𝐽_𝑢,𝑣)𝑋_𝑣, 𝑒^{𝑖 𝐽𝑢 , 𝑣𝑍𝑢𝑍𝑣}𝑌_𝑢𝑍_𝑣𝑒^{−𝑖 𝐽𝑢 , 𝑣𝑍𝑢𝑍𝑣} = cos(2𝐽_𝑢,𝑣)𝑌_𝑢𝑍_𝑣+sin(2𝐽_𝑢,𝑣)𝑋_𝑢, and noting that𝜇⁰(𝑍 𝑍)=0, one easily gets

𝜇(𝑍 𝑍) =𝑠²·𝜇⁰(𝑌 𝑌) +𝑐 𝑠·cos(2𝐽_𝑢,𝑣) [𝜇⁰(𝑍𝑌) +𝜇⁰(𝑌 𝑍)] +𝑐 𝑠·sin(2𝐽_𝑢,𝑣) [𝜇⁰(𝑋 𝐼) +𝜇⁰(𝐼 𝑋)]. Using the explicit form of𝐻⁰, one gets

𝑒^{−𝑖 𝐻}

0|+^𝑛i= 1 2

∑︁

𝑎,𝑏=0,1

|𝑎, 𝑏i𝑢,𝑣 ⊗ |Φ(𝑎, 𝑏)i_else, where|Φ(𝑎, 𝑏)i is a tensor product state of𝑛−2 qubits defined by

|Φ(𝑎, 𝑏)i = Ì

𝑝≠𝑢,𝑣

|𝐽_{𝑢, 𝑝}(−1)^𝑎+𝐽_{𝑣 , 𝑝}(−1)^𝑏i𝑝 where |𝜃i ≡𝑒^{−𝑖 𝜃 𝑍}|+i.

Combining Eqs. (3.25) and (3.28), one gets 𝜇⁰(𝑂) = (1/4) ∑︁

𝑎,𝑏,𝑎⁰,𝑏⁰=0,1

h𝑎⁰, 𝑏⁰|𝑂|𝑎, 𝑏i · hΦ(𝑎⁰, 𝑏⁰) |Φ(𝑎, 𝑏)i.

Using the tensor product form of the states |Φ(𝑎, 𝑏)i and the identity h𝜃⁰|𝜃i = cos(𝜃−𝜃⁰)gives

hΦ(𝑎⁰, 𝑏⁰) |Φ(𝑎, 𝑏)i = Ö

𝑝≠𝑢,𝑣

cos[𝐽_{𝑢, 𝑝}(−1)^𝑎−𝐽_{𝑢, 𝑝}(−1)^𝑎0+𝐽_{𝑣 , 𝑝}(−1)^𝑏−𝐽_{𝑣 , 𝑝}(−1)^𝑏0]. From Eqs. (3.30) and (3.31), one can easily compute the mean value𝜇⁰(𝑂) for any 2-qubit observable.

Consider first the case𝑂 =𝑌 𝑌. Then the only terms contributing to Eq. (3.30) are those with𝑎⁰=𝑎⊕1 and𝑏⁰=𝑏⊕1. The identityh𝑎⊕1|𝑌|𝑎i=−𝑖(−1)^𝑎gives

𝜇⁰(𝑌 𝑌) =−(1/4) ∑︁

𝑎,𝑏=0,1

(−1)^𝑎+𝑏 Ö

𝑝≠𝑢,𝑣

cos[2𝐽_{𝑢, 𝑝}(−1)^𝑎+2𝐽_{𝑣 , 𝑝}(−1)^𝑏], that is,

𝜇⁰(𝑌 𝑌)= (1/2) Ö

𝑝≠𝑢,𝑣

cos[2𝐽_{𝑢, 𝑝}−2𝐽_{𝑣 , 𝑝}] − (1/2) Ö

𝑝≠𝑢,𝑣

cos[2𝐽_{𝑢, 𝑝}+2𝐽_{𝑣 , 𝑝}].

Next, consider the case𝑂 =𝑌 𝑍. Note that the matrix elementsh𝑎⁰, 𝑏⁰|𝑂|𝑎, 𝑏ihave zero real part. From Eqs. (3.30) and (3.31), one infers that 𝜇⁰(𝑌 𝑍) has zero real part. This implies

𝜇⁰(𝑌 𝑍)= 𝜇⁰(𝑍𝑌) =0.

Finally, consider the case𝑂 =𝑋 𝐼. Then the only terms that contribute to Eq. (3.30) are those with𝑎⁰=𝑎 ⊕1 and 𝑏⁰=𝑏. We get

𝜇⁰(𝑋 𝐼) = Ö

𝑝≠𝑢,𝑣

cos(2𝐽_{𝑢, 𝑝}).

Here we noted that the inner product Eq. (3.31) with𝑎⁰=𝑎⊕1 and𝑏⁰=𝑏does not depend on𝑎, 𝑏. By the same argument,

𝜇⁰(𝐼 𝑋) = Ö

𝑝≠𝑢,𝑣

cos(2𝐽_{𝑣 , 𝑝}).

Combining Eq. (3.27) and Eqs. (3.33),(3.34),(3.35),(3.36), one arrives at Eq. (3.23).

For more general cost functions that include interactions among three or more variables, there are two complications: First, unlike in the Ising case, the variable elimination process will typically increase the degree of non-locality of interactions.

Second, mean values of Pauli operators on the QAOA stateΨ𝐻(𝛽, 𝛾) lack a simple analytic formula (as far as we know). However, one can approximately compute the mean values using the Monte Carlo method due to Van den Nest [30]. A specialization of this method to simulation of the level-1 QAOA is described in [31].

The Monte Carlo simulator has runtime scaling polynomially with the number of qubits, number of terms in the cost function, and the inverse error tolerance, see [31]

for details. This method also requires no restrictions on the depth of the variational circuit.

An important distinction between QAOA and RQAOA lies in the measurement step.

QAOA requires few-qubit measurements to estimate the variational energy as well as the final𝑛-qubit measurement that assigns a value to each individual variable. This last step is what makes QAOA hard to simulate classically and may lead to a quantum advantage [32]. In contrast, RQAOA only needs few-qubit measurements to estimate mean values of individual terms in the cost function. The𝑛-qubit measurement step is replaced by the correlation rounding that eliminates variables one by one. One may ask whether the lack of multi-qubit measurements also precludes a quantum advantage. Indeed, in the special case of level-1 variational circuits and the Ising-like cost function RQAOA can be efficiently simulated classically, see above. However, level-𝑝 RQAOA with 𝑝 > 1 as well as level-1 RQAOA with more general cost functions are not known to be classically simulable in polynomial time, leaving room for a quantum advantage.

3.D Comparison of QAOA, RQAOA, and Classical Algorithms D.1 QAOA versus Classical Local Algorithms

In this section, we discuss another limitation of QAOA which results from its locality and the covariance condition discussed in Lemma 3.A.2: we compare QAOA to a certain very simple classical local algorithm (see Lemma 3.D.1 below). We show that there is an exponential number of problem instances for which the classical local algorithm outperforms QAOA.

Let us briefly sketch the notion of a local classical algorithm. We envision that the tuple(𝐽_𝑒)_𝑒∈𝐸 is given as input. Here we are interested in algorithms which are local

with respect to the underlying graph𝐺. For𝑟 ∈Nand𝑣 ∈𝑉, define 𝐸_𝑟(𝑣)=

𝑟

Ø

ℓ=1

Ø

(𝑒

1,...,𝑒_ℓ) path with𝑣∈𝑒

1

{𝑒

1, . . . , 𝑒_ℓ}

to be the set of edges that belong to a path starting at 𝑣 of length bounded by 𝑟. Consider a classical algorithmAwhich on input{𝐽_𝑒}_𝑒∈𝐸outputs𝑥 = (𝑥

1, . . . , 𝑥_𝑛) ∈ {0,1}^𝑛. We say that A is𝑟-local if there is a family of functions {𝑔_𝑣 : R^{𝐸𝑟(𝑣)} → {0,1}}_𝑣∈𝑉 such that the following holds for every problem instance (𝐽_𝑒)_𝑒∈𝐸 ∈ R^𝐸: We have

𝑥_𝑣 =𝑔_𝑣 {𝐽_𝑒}_{𝑒∈𝐸𝑟(𝑣)}

for every𝑣 ∈𝑉 .

In other words, in an𝑟-local classical algorithm, every output bit𝑥_𝑣only depends on edge weights𝐽_𝑒belonging to paths of length bounded by𝑟starting at𝑣. We note that this definition can easily be generalized to the probabilistic case (e.g., by including local random bits). For the purposes of this section, deterministic functions turn out to be sufficient.

The (choice of) family{𝑔_𝑣}_𝑣∈𝑉 can be considered as a set of variational parameters for the classical algorithm. To keep the number of variational parameters constant, we consider vertex-transitive graphs 𝐺. Fix 𝑣_∗ ∈ 𝑉. For every 𝑣 ∈ 𝑉, fix an automorphism 𝜋_𝑣 of 𝐺 such that 𝜋_𝑣(𝑣_∗) = 𝑣. Then the sets 𝐸_𝑟(𝑣) for different 𝑣 ∈ 𝑉 can be identified via 𝐸_𝑟(𝑣) = 𝜋_𝑣(𝐸_𝑟(𝑣^∗)). We say that an𝑟-local classical algorithm is uniformif (after this identification)𝑔_𝑣 ≡ 𝑔 for all 𝑣 ∈𝑉, i.e., if there is a single function𝑔 : R^{𝐸𝑟(𝑣∗)} → {0,1} specifying the behavior of the algorithm.

To obtain general-purpose algorithms (applicable to any instance), the function 𝑔 : R^{𝐸𝑟(𝑣∗)} → {0,1} should be chosen adapatively (i.e., potentially depending on the instance). The definition of local classical algorithm sketched here includes e.g., the algorithms considered in Ref. [20], though it is slightly more general as the local functions can be arbitrary.

Let𝑛=6𝑟 be a multiple of 6. Consider𝑛-qubit Hamiltonians (cf. (3.A)) of the form 𝐻(𝐽) = ∑︁

𝑘∈Z^𝑛

𝐽_𝑘𝑍_𝑘𝑍_𝑘+

1 where 𝐽 = (𝐽

0, . . . , 𝐽_𝑛−

1) ∈ {1,−1}^𝑛.

To define locality and uniformity for the cycle graphZ^𝑛, let𝜋_𝑣(𝑤) =𝑣+𝑤 (mod 𝑛) be chosen as translation modulo𝑛for𝑣 ∈Z^𝑛. We show the following:

Lemma 3.D.1. There is a subsetS ⊂ {1,−1}^𝑛of2^𝑛/3problem instances such that the following holds:

(i) QAOA𝑝(𝐻(𝐽)) ≤ 𝑝/(𝑝+1) for every𝑝 ∈Nand every𝐽 ∈ S.

(ii) There is a1-local uniform classical algorithm such that for every 𝐽 ∈ S, the algorithm outputs𝑥 ∈ {0,1}^𝑛such thath𝑥|𝐻(𝐽) |𝑥i=1.

(iii) Level-1^RQAOAachieves the approximation ratio1. Proof. For every𝑠= (𝑠

0, . . . , 𝑠

2𝑟−1) ∈ {0,1}^2𝑟, define𝐽 =𝐽(𝑠) ∈ {1,−1}^𝑛by 𝐽3𝑎 =𝐽

3𝑎+1 =(−1)^𝑠𝑎, and 𝐽

3𝑎+2 =1,

for all𝑎 =0,1, . . . ,2𝑟−1. We claim that the setS ={𝐽(𝑠) | 𝑠 ∈ {0,1}^2𝑟}has the required properties. Consider an instance𝐻(𝐽(𝑠))with𝑠 ∈ S. Define

𝑋(𝑠) =

2𝑟−1

Ö

𝑎=0

𝑋3𝑎+1 . Then𝐻(𝐽(𝑠)) is related to𝐻

Z^𝑛 =Í

𝑗∈Z^𝑛𝑍_𝑗𝑍_𝑗+

1by the gauge transformation 𝐻(𝐽(𝑠))= 𝑋(𝑠)𝐻

Z^𝑛𝑋(𝑠)⁻¹.

Since the QAOA algorithm is invariant under such gauge transformation (see Lemma 3.A.2), we obtain

QAOA𝑝(𝐻(𝐽(𝑠)))=QAOA𝑝(𝐻

Z^𝑛) ≤ 𝑝 𝑝+1 where we use the bound

QAOA𝑝(𝐻^MaxCut

Z^𝑛 ) ≤ 2𝑝+1 2𝑝+2 ,

proven in [22] for even𝑛, in combination with Lemma 3.A.2. This shows (i).

For the proof of (ii), consider the classical algorithm A which on input 𝐽 = (𝐽

0, . . . , 𝐽_𝑛−

1)outputs

𝑥_𝑣 =𝑔(𝐽_𝑣−

1, 𝐽_𝑣) for every𝑣 ∈Z^𝑛, where

𝑔(𝐽 , 𝐽⁰) =











1 if(𝐽 , 𝐽⁰)= (−1,−1) 0 otherwise.

Clearly, the algorithm A is uniform and 1-local, and it is easy to check that the output satisfiesh𝑥|𝐻(𝐽) |𝑥i=1.

The proof of (iii) is given as a part of Lemma 3.D.2.