Quantum Complexity Theory

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Computational Complexity of Quantum Algorithms

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Quantum complexity theory, a rather grim corner of computational complexity theory, concerns itself with complexity classes defined by the peculiar logic of quantum computers. These aren't your garden-variety calculating machines; they operate on the principles of quantum mechanics, a realm where certainty is a rare and often unwelcome guest. The core of this field is the study of how difficult computational problems are within these quantum frameworks, and more importantly, how these quantum classes stack up against their classical, decidedly less exotic counterparts.

Among the notable inhabitants of this quantum landscape are the classes BQP and QMA. They represent specific tiers of computational difficulty, defined by the capabilities and limitations of quantum computation.

Background

One might recall, or perhaps one should, the fundamental concept of a complexity class. It's a categorization, a box, if you will, for computational problems that share similar resource requirements for their solution. Take P, for instance. It’s the set of problems a Turing machine can conquer in polynomial time. A predictable, almost comforting constraint. Quantum complexity classes operate under similar principles but are defined by quantum computational models, such as the quantum circuit model or its equivalent, the quantum Turing machine. The grand ambition here is to map these quantum classes against their classical relatives: P, NP, BPP, and PSPACE. It’s an exercise in understanding power, or perhaps the lack thereof.

The allure of quantum complexity theory is also tied to its implications for the modern Church-Turing thesis. This thesis, in its simplified form, posits that any computational model can be simulated in polynomial time by a probabilistic Turing machine. [1][2] However, quantum computation throws a wrench into this neat formulation. It's not at all clear if the thesis holds true when quantum mechanics enters the equation. The prevailing evidence suggests it doesn't. The notion that a probabilistic Turing machine can efficiently simulate quantum computation is, at best, a hopeful assumption, and quite likely, a flawed one. [1]

When discussing the computational complexity of functions, both quantum and classical, we often resort to asymptotic notation. It's a way to describe the behavior of functions as their input size grows, a necessary abstraction when dealing with the vastness of computation. Common forms include:

$O(T(n))$ , $\Omega(T(n))$ , and $\Theta(T(n))$

$O(T(n))$ signifies an upper bound. It tells us that a function's growth is no faster than a constant multiple of $T(n)$. $\Omega(T(n))$, conversely, provides a lower bound, indicating that the function grows at least as fast as a constant multiple of $T(n)$. And $\Theta(T(n))$? That's the tightest embrace, signifying both an upper and a lower bound. These are often referred to as Big O notation, Big Omega notation, and Big Theta notation, respectively.

Overview of Complexity Classes

The landscape of computational difficulty is often navigated using promise problems. These are decision problems where the input is guaranteed to belong to a specific set. A promise problem $A = (A_{\text{yes}}, A_{\text{no}})$ consists of a set of "yes" instances, $A_{\text{yes}}$, and a set of "no" instances, $A_{\text{no}}$, with the crucial condition that $A_{\text{yes}} \cap A_{\text{no}} = \varnothing$. All the complexity classes we're discussing, including the quantum ones, can be understood in terms of these promise problems. [4]

| Complexity Class | Criteria