Multi-dimensional Variables and Expressions

Defining multi-dimensional variables

PyQBPP supports multi-dimensional variables and multi-dimensional integer variables of arbitrary depth using the functions var() and var_int(), respectively. Their basic usage is as follows:

• var("name", s1, s2, ..., sd): Creates an array of Var objects with the given name and shape $s_1\times s_2\times \cdots\times s_d$.
• between(var_int("name", s1, s2, ..., sd), l, u): Creates an array of VarInt objects with the specified range and shape.

The following program creates a binary variable with dimension $2\times 3\times 4$:

import pyqbpp as qbpp

x = qbpp.var("x", 2, 3, 4)
print("x =", x)

Each Var object in x can be accessed as x[i][j][k].

Creating integer variable arrays with individual ranges

When defining a multi-dimensional array of integer variables, all elements created by qbpp.between(qbpp.var_int("name", s1, s2, ...), l, u) share the same range $[l, u]$. In many practical problems, however, each element may need a different range. There are three approaches to achieve this.

Approach 1: Placeholder array

First create a placeholder array using qbpp.var_int("name", s1, s2, ...) == 0, then assign individual ranges to each element using qbpp.between():

import pyqbpp as qbpp

max_vals = [3, 7, 15, 5]
x = qbpp.var_int("x", len(max_vals)) == 0
for i in range(len(max_vals)):
    x[i] = qbpp.between(x[i], 0, max_vals[i])
for i in range(len(max_vals)):
    print(f"x[{i}] = {x[i]}")

Here, qbpp.var_int("x", 4) == 0 creates an array of 4 constant-zero VarInt placeholders. Each element is then reassigned with its own range using qbpp.between(x[i], 0, max_vals[i]). The qbpp.between() function automatically inherits the name from the placeholder, so no explicit name is needed.

NOTE The == 0 syntax creates a VarInt with min_val = max_val = 0 (i.e., a constant zero placeholder). It does not create an equality constraint.

Approach 2: Passing lists to between()

You can pass Python lists as the min_val and max_val arguments to qbpp.between(). Each element of the array will be assigned the corresponding range from the lists:

import pyqbpp as qbpp

max_vals = [3, 7, 15, 5]
x = qbpp.between(qbpp.var_int("x", len(max_vals)), 0, max_vals)
for i in range(len(max_vals)):
    print(f"x[{i}] = {x[i]}")

This is the most concise approach. The var_int("x", n) creates a named array builder, and between() assigns individual ranges from the lists element by element. This is analogous to the C++ syntax lower <= qbpp::var_int("x", n) <= upper.

Approach 3: List comprehension with Array

You can also use a Python list comprehension wrapped with qbpp.Array():

import pyqbpp as qbpp

max_vals = [3, 7, 15, 5]
x = qbpp.Array([qbpp.between(qbpp.var_int(f"x[{i}]"), 0, max_vals[i])
                  for i in range(len(max_vals))])

This approach creates the variables directly without placeholders. Note that an explicit name (e.g., f"x[{i}]") must be provided for each variable, and the result must be wrapped with qbpp.Array() to enable element-wise operations.

Defining multi-dimensional expressions

PyQBPP allows you to define multi-dimensional expressions with arbitrary depth using the function expr():

• expr(s1, s2, ..., sd): Creates a multi-dimensional array of Expr objects with shape $s_1\times s_2\times \cdots\times s_d$.

The following program defines a 3-dimensional array x of Var objects with shape $2\times 3\times 4$ and a 2-dimensional array f of size $2\times 3$. Then, using nested loops, each f[i][j] accumulates the sum of x[i][j][0] through x[i][j][3]:

import pyqbpp as qbpp

x = qbpp.var("x", 2, 3, 4)
f = qbpp.expr(2, 3)
for i in range(2):
    for j in range(3):
        for k in range(4):
            f[i][j] += x[i][j][k]
f.simplify_as_binary()

for i in range(2):
    for j in range(3):
        print(f"f[{i}][{j}] =", f[i][j])

This program produces the following output:

f[0][0] = x[0][0][0] +x[0][0][1] +x[0][0][2] +x[0][0][3]
f[0][1] = x[0][1][0] +x[0][1][1] +x[0][1][2] +x[0][1][3]
f[0][2] = x[0][2][0] +x[0][2][1] +x[0][2][2] +x[0][2][3]
f[1][0] = x[1][0][0] +x[1][0][1] +x[1][0][2] +x[1][0][3]
f[1][1] = x[1][1][0] +x[1][1][1] +x[1][1][2] +x[1][1][3]
f[1][2] = x[1][2][0] +x[1][2][1] +x[1][2][2] +x[1][2][3]

Creating an array of expressions by operations

An array of Expr objects can be created without explicitly calling expr(). When an arithmetic operation yields an array-shaped result, an array of Expr objects with the same shape is created automatically.

import pyqbpp as qbpp

x = qbpp.var("x", 2, 3)
f = x + 1
f += x - 2
f.simplify_as_binary()
for i in range(2):
    for j in range(3):
        print(f"f[{i}][{j}] =", f[i][j])

This program outputs:

f[0][0] = -1 +2*x[0][0]
f[0][1] = -1 +2*x[0][1]
f[0][2] = -1 +2*x[0][2]
f[1][0] = -1 +2*x[1][0]
f[1][1] = -1 +2*x[1][1]
f[1][2] = -1 +2*x[1][2]

Iterating over multi-dimensional arrays

Since PyQBPP arrays support Python iteration, nested for loops can be used:

import pyqbpp as qbpp

x = qbpp.var("x", 2, 3)
f = x + 1
f += x - 2
f.simplify_as_binary()
for row in f:
    for element in row:
        print(f"({element})", end="")
    print()

This program outputs:

(-1 +2*x[0][0])(-1 +2*x[0][1])(-1 +2*x[0][2])
(-1 +2*x[1][0])(-1 +2*x[1][1])(-1 +2*x[1][2])

Array and Python list

PyQBPP’s Array is an opaque object backed by the QUBO++ shared library (.so). It is not a Python list — it is a specialized data structure optimized for QUBO++ operations.

Creating an Array from a Python list

You can convert a Python list into an Array using qbpp.Array():

w = qbpp.Array([64, 27, 47, 74, 12, 83, 63, 40])

Once converted, the Array supports element-wise arithmetic (+, -, *, /, ~), sum(), sqr(), simplify(), and other QUBO++ functions efficiently.

When you don’t need qbpp.Array()

When a Python list is used in an arithmetic operation with an Array, it is automatically converted. For example:

w = [64, 27, 47, 74, 12, 83, 63, 40]
x = qbpp.var("x", len(w))
f = w * x       # list * Array → element-wise multiplication

In this case, wrapping w with qbpp.Array() is not necessary. However, if w is used repeatedly in multiple operations, wrapping it once with qbpp.Array() can improve performance by avoiding repeated conversions from list to Array.

Example: list vs Array behavior

The following example illustrates the difference between a Python list and an Array:

import pyqbpp as qbpp

x = qbpp.var("x")
u = [x+2, x+3, x+5, x+7]
w = qbpp.Array([x+2, x+3, x+5, x+7])
print(f"2 * u = {2 * u}")
print(f"2 * w = {2 * w}")

Output:

2 * u = [2 +x, 3 +x, 5 +x, 7 +x, 2 +x, 3 +x, 5 +x, 7 +x]
2 * w = [4 +2*x, 6 +2*x, 10 +2*x, 14 +2*x]

With the Python list u, 2 * u produces a repeated list (8 elements). With the Array w, 2 * w produces an element-wise multiplication (each element multiplied by 2).

Key differences from Python list

	Array	Python list
Element-wise +	Element-wise addition	List concatenation
Element-wise *	Element-wise multiplication	List repetition
~x	Element-wise negation	TypeError
sum()	Sum of all elements as Expr	Python built-in sum
sqr()	Element-wise squaring	Not available
append(), pop()	Not available	Available
Slicing	slice(), head(), tail()	x[1:3]

NOTE An Array is a fixed-size, opaque container. Python list operations such as append(), pop(), insert(), and slice assignment are not supported. Use QUBO++ functions like slice(), head(), tail() for extracting sub-arrays.