Minimizing minor embedding energy: an application in quantum annealing

A significant challenge in quantum annealing is to map a real-world problem onto a hardware graph of limited connectivity. If the maximum degree of the problem graph exceeds the maximum degree of the hardware graph, one employs minor embedding in which each logical qubit is mapped to a tree of physical qubits. Pairwise interactions between physical qubits in the tree are set to be ferromagnetic with some coupling strength $F<0$. Here we address the question of what value $F$ should take in order to maximise the probability that the annealer finds the correct ground-state of an Ising problem. The sum of $|F|$ for each logical qubit is defined as minor embedding energy. We confirm experimentally that the ground-state probability is maximised when the minor embedding energy is minimised, subject to the constraint that no domain walls appear in every tree of physical qubits associated with each embedded logical qubit. We further develop an analytical lower bound on $|F|$ which satisfies this constraint and show that it is a tighter bound than that previously derived by Choi (Quantum Inf. Proc. 7 193 (2008)).


Introduction
Quantum annealing is a widely-used tool for solving quadratic optimization problems [Harris2018, King2018]. The problem is mapped to a Hamiltonian, H P , whose ground-state encodes the opti- where 0 ≤ s ≡ t t f ≤ 1, t is time, t f is the duration of the anneal, A(0) B(0) and A(1) B (1).
The origin of quantum annealing goes back to the quantum adiabatic theorem with a gap condition, which was first shown by Born and Fock [BornFock1928] in 1928, then Kato [Kato1950] simplified the proof of the theorem and extended it to allow degenerate eigenstates and eigenvalue crossings.
For closed quantum systems, Farhi et al. [Farhi2000, Farhi2001] proposed adiabatic quantum computation as an alternative to tackle NP-complete problems. For a recent review of the quantum adiabatic theorem, see for exmaple Albash and Lidar [Lidar2018].
In view of the computational complexity of modelling interacting quantum systems using classical computational resources, a potentially efficient way to find the ground-state of H P is to engineer a physical system whose dynamics follow that of equation (1.1). One such physical system is based on a system of superconducting flux qubits with tunable inductive interactions [Kafri2017]. In this implementation the problem Hamiltonian is of the Ising form: Here σ z i is the quasi-spin of qubit i (corresponding to its flux state) and G is a graph describing all possible two-qubit interactions. The total Hamiltonian is exactly the transverse Ising model introduced by Kadowaki One problem for hardware implementation of quantum annealing now becomes immediately apparent: for a system of N qubits it is at best very difficult to engineer direct interactions between all 1 2 N (N − 1) pairs. In current implementations of flux-qubit quantum annealers the maximum degree of the hardware graph is 6 -i.e. each qubit is directly coupled to at most six other qubits 1 .
It is therefore necessary to employ minor embedding -i.e. to embed an Ising problem Hamiltonian 1 experiments are currently underway on a flux-qubit annealer with degree 15 whose connectivity graph has degree D P onto physical hardware with connectivity graph of degree D H , where D H < D P ≤ N . The requirement of this embedding is that the ground-state of the embedded Hamiltonian of degree D H encodes the same solution as the ground-state of the problem Hamiltonian of degree D P .
Choi [Choi2008] first proposed a method for minor embedding in which each logical qubit is replaced by a tree of physical qubits. All the physical qubits within each tree are constrained to be in the same spin state (which in turn is the spin state of the logical qubit) by the implementation of ferromagnetic interactions of magnituede |F | at each edge of the tree. In practice it is usual to use a one-dimensional chain of physical qubits as the tree for minor embedding. A logical qubit consisting of a chain of L physical qubits in a hardware graph of degree D H can now be directly coupled to L(D H − 2) + 2 other logical qubits, thereby greatly increasing the connectivity. Figure 1 shows an example of a minor embedding.  Figure 1: An illustrative example of (a) a logical graph of maximum degree 6 and (b) a physical graph of maximum degree 3. Logical qubit 1 (coloured in green in (a)) is mapped onto four physical qubits (all labelled by 1 and coloured in green in (b)). J 6,1 in (a) denotes the coupling between the sixth logical qubit and the first qubit, which is mapped identically onto (b). h 1 in (a) is the local field on the first logical qubit, which is mapped onto h 1(a) , h 1(b) , h 1(c) &h 1(d) in (b). Other couplers and local fields are omitted for clarity.
If |F | is sufficiently large, for a closed-system quantum annealer it can be assumed that the In this paper we revisit minor embedding in order to determine the optimum ferromagnetic strength |F | for embedding trees in quantum annealers at finite temperature. We will give a mathematical criterion for the best bound on the value of |F |. As a consequence, the first two theorems ). In order to achieve multi-body interactions via two-body Ising models, one has to couple logical qubits with ancilla qubits, which certainly increases the (vertex) degree of the corresponding two-body Hamiltonian. Minor embedding is the key tool to convert graphs with higher degrees to graphs with lower degrees. Therefore, our paper will also be useful for generating multi-body interactions. G to a sub-graph of another graph U . The pair of mappings satisfies the following properties: is mapped to a set of vertices (denoted byι(i)) of a connected sub-tree of U , fulfilling τ (i, j)τ (j, i) ∈ E(U ). Note that τ induces the mapping of edges, which we also denote by τ .
Note that given graphs G and U , there may be no minor embedding of G into U or there may exist many (ι, τ )'s that embed G into U . For instance, by Kuratowskis theorem the complete bipartite graph K 3,3 cannot be minor embedded into any planar graph. Figure 1 illustrates how to embed a highly connected graph into a less connected graph.
Let G be the logical graph corresponding to expression (1.2). To show its dependence on G, we suppress the subscript P and rewrite the expression as

Minimizing minor embedding energy
Suppose that there is another graph U , which we can interpret as the hardware graph. Moreover, we assume that graph G can be minor embedded onto graph U . Then Definition 2.1 induces a series of problem Hamiltonians associated with graph I(G) ⊂ U : and the ferromagnetic coupling strength (also called internal coupling strength) within each sub-tree In order to match the ground-state of Hamiltonian (2.1) and that of Hamiltonian (2.2), we can set We also require that M i be sufficiently large that all spins in the ground-state of the embedded tree are aligned.
A natural question to ask is: How small can M i be?
Let E G be the energy corresponding to Hamiltonian (2.1) and E I(G) for Hamiltonian (2.2). Then we have and Minimizing minor embedding energy Definition 2.2 (Minor embedding energy). Let I = (ι, τ ) be a minor embedding. Then its minor embedding energy (MEE) is defined by Note that minimizing M i for each logical qubit i is equivalent to minimizing the minor embedding energy.

Main theorem
Our task is to find the mathematical criteria for all the bounds that preserve the ground-state configuration of Hamilton (2.1). Now we will focus on the criteria for tree ι(i).

Definition 2.3 (Boundary operator)
. Let X be a graph and 2 X denote the power set of V (X). The boundary operator is defined as that for any W ⊂ V (X), ∂W gives the boundary edges of W . That is the cut(s) between W and X\W . Moreover, the boundary operator ∂ annihilates both the empty set and the total set V (X).
We will see later that the boundary operator has a strong relationship with the ferromagnetic coupling strength. For a graph with assignments (local h-field) on each vertex, we define the following integral operator.
Similarly, we can define the J-integral operator for other non-negative external field.
Minimizing minor embedding energy Definition 2.5 (J-integral operator). Let X be a graph. The J-integral operator At least one domain wall is present when there is the presence of an inhomogeneous spin configuration in ι(i) or equivalently the presence of an anisotropic magnetization.
Definition 2.6 (Domain wall). If all particles have the same spin in W i ⊂ ι(i) but opposite spin in We say a domain wall ∂W i is positive (negative), if the spins are positive (negative) within W i .
Let us denote Onbh(i(k)) the original neighbourhood of the pre-embedded vertex i that is connected to the embedded vertex i(k).
Now we are ready to state our main theorem.
where the maximum is taken from all ∅ = W i ι(i). Then we have

8)
and where s * k = s * k(j) , for all j ∈ ι(k). • It gives the necessary condition such that M i will preserve the equivalence of ground-states for E I(G) and E G . Moreover it is the necessary condition for the h i(k) 's and J i(k) 's being predefined. Hence M i depends on h i(k) and J i(k) . In practice, the J i(k) 's are defined for a given minor embedding. However, the h i(k) 's need to be determined. Therefore, the true optimal provided that some conditions are satisfied, see Section 3.
We will see later how this will give the true optimal bound for a simple example.
where nbh(i) means the neighbourhood of vertex i. We have and Proof. It suffices to show that (2.13)

Minimizing minor embedding energy
Since for each W i ⊂ ι(i), we have the inequality (2.13) follows immediately.
In order to get Choi's tighter bound for the ferromagnetic coupler strengths, one needs to introduce the following object.
which defines whether the spin of particle i is locally determinable or non-determinable. When C(i) < 0, the spin of particle i is locally determinable, as the local field h i is dominant, whereas when C(i) ≥ 0, its spin must be determined globally. Without loss of generality, we can assume C(i) ≥ 0. Now, we are ready to state our second corollary.
where Onbh(i(k)) means the original neighbourhood of vertex i(k) ∈ ι(i).Then yields the same result as Corollary 2.9.
• Corollary 2.9 is independent of the values of the C(i)'s and is certainly larger than the bound given in Corollary 2.10. However, Corollary 2.9 does not assign any value to h i(k) , whereas Corollary 2.10 holds only when the h i(k) 's satisfy equations (2.15).
• Corollary 2.10 gives the best bound when C(i) = 0 for all i ∈ V (G).
Minimizing minor embedding energy • The larger (weaker) bound given by Corollary 2.9 does not require any topological information about the minor embedding, while the smaller (stronger) bound given by Corollary 2.10 depends non-trivially on the topology of the minor embedding.
• Both proofs for Corollary 2.9 and Corollary 2.10 are quite different and there is no obvious derivation from Corollary 2.9 to Corollary 2.10.
Now we give a simple proof of Choi's second theorem as a corollary.
Proof. It suffices to show that for h i(k) setting as in equations (2.15) and for all ∅ = W i ι(i). Now we have where L(i) is the set of leaves in ι(i). As |∂W i | ≥ 1 for ∅ = W i ι(i), one can easily verify that Therefore, we have for all ∅ = W i ι(i), which completes the proof.
As it remains open on the tightness of the bound in Corollary 2.10, we will give a simple example in the next subsection, which shows that even for h i(k) 's given as in equation (2.15), the bound is not tight. Furthermore, by relaxing the condition (2.15), one can achieve the best bound.

An example: existence of a tighter bound
In this subsection, we give an example to show the existence of a tighter bound for the ferromagnetic coupling strength compared with Corollary 2.10. Let us consider the minor embedding of a vertex i as in Figure 2. For the sake of this example we set the couplers and local fields such that and According to Corollary 2.10, for this example we have More importantly, the bound for the ferromagnetic coupler strengths according to Corollary 2.10 is given by Our new tighter bound shows that a better bound exists. i.e.
is sufficient for this toy model. See Appendix B for details.
We will show later in Section 3 that the best bound for this example is F i < −5h, if we allow Minimizing minor embedding energy

Proof of the main theorem
In this subsection, we give the full proof of our main theorem.
In order for sufficiently large M i to preserve the homogeneity of spins in ι(i), we need to find a sufficient condition so that the formation of each domain wall is forbidden. Now we have the following lemma.
Lemma 2.12. However, according to equation (2.6), we have Minimizing minor embedding energy Since our assumption also has we then have Minimizing minor embedding energy we have that W i cannot have a positive domain wall ∂W i in the ground-state configuration. Therefore, implies that no positive domain wall can be present in the ground-state configuration. Hence the ground-state configuration has no domain wall in ι(i).

Tightness of the bound
Now we want to show that, if the condition is the best bound for ∅ = W i ι(i). That is for any > 0 and F pq  Before giving the proof of Theorem 3.1, we give some remarks and corollaries.
is satisfied, then M (W i ; h, J) is the tightest bound.
For some s τ (l,i(k)) with i(k) ∈ V (W i ), we have Let us consider the following difference

Minimizing minor embedding energy
For some s τ (l,j(k)) with j(k) ∈ V (W i ), we have Note that we used the fact that h i ≤ |h i | ≤ J(W i ) + J(W i ) in the last step. Therefore, is not a ground-state configuration. Moreover, one can show that Hence W i (+), W i (+), . . . is also not a ground-state configuration.

Minimizing minor embedding energy
For some s τ (l,j(k)) with j(k) ∈ V (W i ), we have is not a ground-state configuration.
which is equivalent to Therefore, following the same as Case 1, we complete the proof.

Admissible minor embeddings
Now we show that conditions (3.1) and (3.3) should be satisfied for any reasonable minor embedding.
We call a minor embedding, say (I, h, J, F ), admissible if the following condition is satisfied.
• (I, h, J, F ) does not exclude any possible spin configuration for any i ∈ G in any embedded Ising problem.
Here F denotes the absolute value of the chain strength. Note that admissible minor embeddings are more suitable for practical purposes, since for general NP-hard problems we do not expect any pre-assignment for any logical qubit in G. It can be shown that the condition for admissible minor embeddings implies conditions (3.1) and (3.3). Verification.

¬condition (3.1) ∨ ¬condition (3.3)
Minimizing minor embedding energy is equivalent to is the tightest bound for admissible minor embeddings.
to be greater or equal to zero for admissible minor embeddings. In other words, we must have . This condition can be easily violated when h i(k) is concentrated in a single physical qubit and F is comparably small. This is the situation when we apply the single distribution method as defined in [Pudenz]. Therefore, there are likely to be some non-admissible minor embeddings in the single distribution method.
Minimizing minor embedding energy where each job jn has L n operations. Each operation O n,k has a non-negative integer execution time τ n,k and has to be executed by an assigned machine m n,k ∈ M . The goal of solving JSP is to find an optimal scheduling that minimises the makespan, i.e. the minimum time to finish all the jobs.
A generalised tabular representation of job shop scheduling problems is shown in Table 1.  Here t is bounded from above by the timespan T , which represents the maximum time we allow for all jobs to be completed. The resulting classical objective function (Hamiltonian) is given by where E problem is the energy scaling parameter and each penalty term is explained briefly as follows.
• h 1 (x) = n,k ( t x n,k;t − 1) 2 , checks that an operation must start once and only once.
• h 2 (x) = n k<n t+τ n,k >t x n,k;t x n,k+1;t , ensures that the order of the operations within a job is preserved.
• h 3 (x) = t+τ nK >T x n,K;t , guarantees that the last operation in each job finishes by time T .
• h 4 (x) = m (n,k;t|n ,k ;t )∈Rm x n,k;t x n ,k ;t , R m consists of two penalty sets given in the following.
-Forbidding operation O n ,k from starting at t if there is another operation O n,k still running.
-Two operations cannot start at the same time, unless at least one of them has an execution time equal to zero .
Minimizing minor embedding energy and the spectral gap is given by Hence, an easy follow-up from Corollary 2.10 can be derived (or see [Choi2008]). i.e. If topological embeddings are chosen to embed the job shop scheduling problem Hamiltonian, we find that |F | ≥  and is given by [Ronnow2014]: where s is the success probability for each embedding and t a is the single-run annealing time, which is equal to 2µs in our experiments. For each instance the minimum TTS for the five embeddings is recorded. The same procedure is conducted for the 200 random instances and then the mean TTS is the data shown in Figure 4. Error bars are obtained by bootstrapping method and the confidence intervals are chosen to be 95%.
We expect that the theoretical optimal bound plays an important role in a general quantum annealer and it is not constrained to JSPs.

Appendix B An example for the existence of a better bound
Here we show that tighter bounds exists then those given in [Choi2008] by continuing the toy example of Figure 2. According to Corollary 2.16, the assignments of local h i(k) are given as in

Case 1 inequality
Now we have the following inequalities. Minimizing minor embedding energy Using the same method, one can derive that

Case 3 inequality
Using the same method, one can derive that

Case 4 inequality
Using the same method, one can derive that

Case 5 inequality
Using the same method, one can derive that and That is

Case 6 inequality
Using the same method, one can derive that only homogeneous configurations within ι(i) (i.e. s 0 = s 1 = s 2 = s 3 ) are possible for the groundstate configuration. Note that this is a better bound that the one (2.21) given by Corollary 2.10.

Minimizing minor embedding energy
Here we show how to derive the best bound on the internal coupling strength using the toy model of Figure 2 as an example. By Remark 3.3, we have that the best bound is given by Now let h i(k) = {a, b, c, d} and we have the example as shown in Figure 6. Now follow the same Minimizing minor embedding energy