Quadratic unconstrained binary optimization

The set of binary vectors of a fixed length ${\displaystyle n>0}$ is denoted by ${\displaystyle \mathbb {B} ^{n}}$, where ${\displaystyle \mathbb {B} =\lbrace 0,1\rbrace }$ is the set of binary values (or bits). We are given a real-valued upper triangular matrix ${\displaystyle Q\in \mathbb {R} ^{n\times n}}$, whose entries ${\displaystyle Q_{ij}}$ define a weight for each pair of indices ${\displaystyle i,j\in \lbrace 1,\dots ,n\rbrace }$ within the binary vector. We can define a function ${\displaystyle f_{Q}:\mathbb {B} ^{n}\rightarrow \mathbb {R} }$ that assigns a value to each binary vector through

${\displaystyle f_{Q}(x)=x^{\top }Qx=\sum _{i=1}^{n}\sum _{j=i}^{n}Q_{ij}x_{i}x_{j}}$

Intuitively, the weight ${\displaystyle Q_{ij}}$ is added if both ${\displaystyle x_{i}}$ and ${\displaystyle x_{j}}$ have value 1. When ${\displaystyle i=j}$, the values ${\displaystyle Q_{ii}}$ are added if ${\displaystyle x_{i}=1}$, as ${\displaystyle x_{i}x_{i}=x_{i}}$ for all ${\displaystyle x_{i}\in \mathbb {B} }$.

The QUBO problem consists of finding a binary vector ${\displaystyle x^{*}}$ that is minimal with respect to ${\displaystyle f_{Q}}$, namely

${\displaystyle x^{*}={\underset {x\in \mathbb {B} ^{n}}{\arg \min }}~f_{Q}(x)}$

In general, ${\displaystyle x^{*}}$ is not unique, meaning there may be a set of minimizing vectors with equal value w.r.t. ${\displaystyle f_{Q}}$. The complexity of QUBO arises from the number of candidate binary vectors to be evaluated, as ${\displaystyle |\mathbb {B} ^{n}|=2^{n}}$ grows exponentially in ${\displaystyle n}$.

Sometimes, QUBO is defined as the problem of maximizing ${\displaystyle f_{Q}}$, which is equivalent to minimizing ${\displaystyle f_{-Q}=-f_{Q}}$.

• Multiplying the coefficients ${\displaystyle Q_{ij}}$ with a positive factor ${\displaystyle \alpha >0}$ scales the output of ${\displaystyle f}$ accordingly, leaving the optimum ${\displaystyle x^{*}}$ unchanged:

${\displaystyle f_{\alpha Q}(x)=\sum _{i\leq j}(\alpha Q_{ij})x_{i}x_{j}=\alpha \sum _{i\leq j}Q_{ij}x_{i}x_{j}=\alpha f_{Q}(x)}$

• Flipping the sign of all coefficients flips the sign of ${\displaystyle f}$'s output, making ${\displaystyle x^{*}}$ the binary vector that maximizes ${\displaystyle f_{-Q}}$:

${\displaystyle f_{-Q}(x)=\sum _{i\leq j}(-Q_{ij})x_{i}x_{j}=-\sum _{i\leq j}Q_{ij}x_{i}x_{j}=-f_{Q}(x)}$

• If all coefficients are positive, the optimum is trivially ${\displaystyle x^{*}=(0,\dots ,0)}$. Similarly, if all coefficients are negative, the optimum is ${\displaystyle x^{*}=(1,\dots ,1)}$.
• If ${\displaystyle \forall i\neq j:~Q_{ij}=0}$, i.e., the bits can be optimized independently, then the corresponding QUBO problem is solvable in ${\displaystyle {\mathcal {O}}(n)}$, the optimal variable assignments ${\displaystyle x_{i}^{*}}$ simply being 1 if ${\displaystyle Q_{ii}<0}$ and 0 otherwise.

QUBO is a structurally simple, yet computationally hard optimization problem. It can be used to encode a wide range of optimization problems from various scientific areas.[6]

Cluster Analysis

Binary Clustering with QUBO

A bad cluster assignment.

A good cluster assignment.

Visual representation of a clustering problem with 20 points: Circles of the same color belong to the same cluster. Each circle can be understood as a binary variable in the corresponding QUBO problem.

As an illustrative example of how QUBO can be used to encode an optimization problem, we consider the problem of cluster analysis. Here, we are given a set of 20 points in 2D space, described by a matrix ${\displaystyle D\in \mathbb {R} ^{20\times 2}}$, where each row contains two cartesian coordinates. We want to assign each point to one of two classes or clusters, such that points in the same cluster are similar to each other. For two clusters, we can assign a binary variable ${\displaystyle x_{i}\in \mathbb {B} }$ to the point corresponding to the ${\displaystyle i}$-th row in ${\displaystyle D}$, indicating whether it belongs to the first (${\displaystyle x_{i}=0}$) or second cluster (${\displaystyle x_{i}=1}$). Consequently, we have 20 binary variables, which form a binary vector ${\displaystyle x\in \mathbb {B} ^{20}}$ that corresponds to a cluster assignment of all points (see figure).

One way to derive a clustering is to consider the pairwise distances between points. Given a cluster assignment ${\displaystyle x}$, the values ${\displaystyle x_{i}x_{j}}$ or ${\displaystyle (1-x_{i})(1-x_{j})}$ evaluate to 1 if points ${\displaystyle i}$ and ${\displaystyle j}$ are in the same cluster. Similarly, ${\displaystyle x_{i}(1-x_{j})}$ or ${\displaystyle (1-x_{i})x_{j}}$ indicate that they are in different clusters. Let ${\displaystyle d_{ij}\geq 0}$ denote the Euclidean distance between points ${\displaystyle i}$ and ${\displaystyle j}$. In order to define a cost function to minimize, when points ${\displaystyle i}$ and ${\displaystyle j}$ are in the same cluster we add their positive distance ${\displaystyle d_{ij}}$, and subtract it when they are in different clusters. This way, an optimal solution tends to place points which are far apart into different clusters, and points that are close into the same cluster. The cost function thus comes down to

${\displaystyle {\begin{aligned}f(x)&=\sum _{i<j}d_{ij}(x_{i}x_{j}+(1-x_{i})(1-x_{j}))-d_{ij}(x_{i}(1-x_{j})+(1-x_{i})x_{j})\\&=\sum _{i<j}\left[4d_{ij}x_{i}x_{j}-2d_{ij}x_{i}-2d_{ij}x_{j}+d_{ij}\right]\end{aligned}}}$

From the second line, the QUBO parameters can be easily found by re-arranging to be:

${\displaystyle {\begin{aligned}Q_{ij}&={\begin{cases}d_{ij}&{\text{if }}i\neq j\\-\left(\sum \limits _{k=1}^{i-1}d_{ki}+\sum \limits _{\ell =i+1}^{n}d_{i\ell }\right)&{\text{if }}i=j\end{cases}}\end{aligned}}}$

Using these parameters, the optimal QUBO solution will correspond to an optimal cluster w.r.t. above cost function.

QUBO is very closely related and computationally equivalent to the Ising model, whose Hamiltonian function is defined as

${\displaystyle H(\sigma )=-\sum _{\langle i~j\rangle }J_{ij}\sigma _{i}\sigma _{j}-\mu \sum _{j}h_{j}\sigma _{j}}$

with real-valued parameters ${\displaystyle h_{j},J_{ij},\mu }$ for all ${\displaystyle i,j}$. The spin variables ${\displaystyle \sigma _{j}}$ are binary with values from ${\displaystyle \lbrace -1,+1\rbrace }$ instead of ${\displaystyle \mathbb {B} }$. Moreover, in the Ising model the variables are typically arranged in a lattice where only neighboring pairs of variables ${\displaystyle \langle i~j\rangle }$ can have non-zero coefficients. Applying the identity ${\displaystyle \sigma \mapsto 2x-1}$ yields an equivalent QUBO problem:[2]

${\displaystyle {\begin{aligned}f(x)&=\sum _{\langle i~j\rangle }-J_{ij}(2x_{i}-1)(2x_{j}-1)+\sum _{j}\mu h_{j}(2x_{j}-1)\\&=\sum _{\langle i~j\rangle }(-4J_{ij}x_{i}x_{j}+2J_{ij}x_{i}+2J_{ij}x_{j}-J_{ij})+\sum _{j}(2\mu h_{j}x_{j}-\mu h_{j})&&{\text{using }}x_{j}=x_{j}x_{j}\\&=\sum _{\langle i~j\rangle }(-4J_{ij}x_{i}x_{j})+\sum _{\langle i~j\rangle }2J_{ij}x_{i}+\sum _{\langle i~j\rangle }2J_{ij}x_{j}+\sum _{j}2\mu h_{j}x_{j}-\sum _{\langle i~j\rangle }J_{ij}-\sum _{j}\mu h_{j}\\&=\sum _{\langle i~j\rangle }(-4J_{ij}x_{i}x_{j})+\sum _{\langle j~i\rangle }2J_{ji}x_{j}+\sum _{\langle i~j\rangle }2J_{ij}x_{j}+\sum _{j}2\mu h_{j}x_{j}-\sum _{\langle i~j\rangle }J_{ij}-\sum _{j}\mu h_{j}&&{\text{using }}\sum _{\langle i~j\rangle }=\sum _{\langle j~i\rangle }\\&=\sum _{\langle i~j\rangle }(-4J_{ij}x_{i}x_{j})+\sum _{j}\sum _{\langle k=j~i\rangle }2J_{ki}x_{j}+\sum _{j}\sum _{\langle i~k=j\rangle }2J_{ik}x_{j}+\sum _{j}2\mu h_{j}x_{j}-\sum _{\langle i~j\rangle }J_{ij}-\sum _{j}\mu h_{j}\\&=\sum _{\langle i~j\rangle }(-4J_{ij}x_{i}x_{j})+\sum _{j}\left(\sum _{\langle i~k=j\rangle }(2J_{ki}+2J_{ik})+2\mu h_{j}\right)x_{j}-\sum _{\langle i~j\rangle }J_{ij}-\sum _{j}\mu h_{j}&&{\text{using }}\sum _{\langle k=j~i\rangle }=\sum _{\langle i~k=j\rangle }\\&=\sum _{i=1}^{n}\sum _{j=1}^{i}Q_{ij}x_{i}x_{j}+C\end{aligned}}}$

where

${\displaystyle {\begin{aligned}Q_{ij}&={\begin{cases}-4J_{ij}&{\text{if }}i\neq j\\\sum _{\langle i~k=j\rangle }(2J_{ki}+2J_{ik})+2\mu h_{j}&{\text{if }}i=j\end{cases}}\\C&=-\sum _{\langle i~j\rangle }J_{ij}-\sum _{j}\mu h_{j}\end{aligned}}}$

and using the fact that for a binary variable ${\displaystyle x_{j}=x_{j}x_{j}}$.

As the constant ${\displaystyle C}$ does not change the position of the optimum ${\displaystyle x^{*}}$, it can be neglected during optimization and is only important for recovering the original Hamiltonian function value.

A. P. Punnen (editor), Quadratic unconstrained binary optimization problem: Theory, Algorithms, and Applications, Springer, Springer, 2022.

• QUBO Benchmark (Benchmark of software packages for the exact solution of QUBOs; part of the well-known Mittelmann benchmark collection)

• Endre Boros, Peter L Hammer & Gabriel Tavares (April 2007). "Local search heuristics for Quadratic Unconstrained Binary Optimization (QUBO)". Journal of Heuristics. Association for Computing Machinery. 13 (2): 99–132. doi:10.1007/s10732-007-9009-3. S2CID 32887708. Retrieved 12 May 2013.
• Di Wang & Robert Kleinberg (November 2009). "Analyzing quadratic unconstrained binary optimization problems via multicommodity flows". Discrete Applied Mathematics. Elsevier. 157 (18): 3746–3753. doi:10.1016/j.dam.2009.07.009. PMC 2808708. PMID 20161596.