Hi everyone interested,
Since I'm in applied mathematics and not in quantum computing (only having a general Idea), I want to ask for similarities and differences of these two types of linear equation systems:
A) quantum linear equation system A x = b with b a superposition of states for each b_i and with the resulting x also as a superposition of states.
B) non-quantum combinatorical linear equation system A x = b with b consisting of a number of optional discrete values b_i and finding a best fit solution for x.
How are those two systems related in the words of quantum computing? Additional question:
C) Is there research on A x = b with also all A_ij being superpositions of states?
Thanks!
The question is very interesting and intutive. As you know, HHL algorithm can be a hint for the first question. For general linear equations, each X and b can be converted into qubits (This is a quantum embedding process). Then, quantum linear equation algorithm can be applied. Quantum linear equation algorithm is for the general linear system algorithm, as you know.
In this viewpoint, A has roles of "quantum operator". In general, A doesn't need to be superpositioned stated, as a quantum operator. However, A can be converted to Quantum-bit based operators. Then, in the subsequent solving process, superposition states can appear.
Quantum computing linear equations and combinatorial linear equations differ primarily in their problem formulations and underlying computational approaches. Quantum computing linear equations involve solving systems of linear equations using quantum algorithms, such as HHL (Harrow-Hassidim-Lloyd) algorithm, which exploits quantum parallelism and quantum superposition to achieve exponential speedup compared to classical methods. Conversely, combinatorial linear equations pertain to linear equations arising in combinatorial optimization problems, where the objective is typically to minimize or maximize a linear function subject to combinatorial constraints. Despite their differences, both problems share similarities in their reliance on linear algebra principles and their relevance to diverse applications across fields such as optimization, cryptography, and scientific computing. However, while quantum computing linear equations hold promise for revolutionizing computational efficiency, combinatorial linear equations remain predominantly solved using classical algorithms due to the current limitations of quantum hardware and the complexity of combinatorial optimization problems.
Quantum Computing Linear Equations and Combinatorial Linear Equations diverge in their problem formulations and computational methodologies; the former pertains to solving systems of linear equations using quantum algorithms, benefitting from quantum parallelism and exponential speedup, whereas the latter involves linear equations arising in combinatorial optimization problems, notoriously challenging and typically addressed using classical algorithms. However, both domains share a foundation in linear algebra principles, relying on matrix representations and variable manipulations. Additionally, they find application across scientific and engineering disciplines, with Quantum Computing Linear Equations pertinent in quantum chemistry and optimization, while Combinatorial Linear Equations are prevalent in areas such as logistics, telecommunications, and scheduling problems.
- Badges
- Science topic
- Similar topics
- Papers