Related papers: On physical problems that are slightly more diffic…
In this paper we give an overview of the quantum computational complexity class QMA and a description of known QMA-complete problems to date. Such problems are believed to be difficult to solve, even with a quantum computer, but have the…
The quest for quantum computers is motivated by their potential for solving problems that defy existing, classical, computers. The theory of computational complexity, one of the crown jewels of computer science, provides a rigorous…
This article surveys quantum computational complexity, with a focus on three fundamental notions: polynomial-time quantum computations, the efficient verification of quantum proofs, and quantum interactive proof systems. Properties of…
Complexity theory as practiced by physicists and computational complexity theory as practiced by computer scientists both characterize how difficult it is to solve complex problems. Here it is shown that the parameters of a specific model…
I offer a case that quantum query complexity still has loads of enticing and fundamental open problems -- from relativized QMA versus QCMA and BQP versus IP, to time/space tradeoffs for collision and element distinctness, to polynomial…
The complexity of quantum computation remains poorly understood. While physicists attempt to find ways to create quantum computers, we still do not have much evidence one way or the other as to how useful these machines will be. The tools…
Most continuous mathematical formulations arising in science and engineering can only be solved numerically and therefore approximately. We shall always assume that we're dealing with a numerical approximation to the solution. There are two…
Testing the symmetries of quantum states and channels provides a way to assess their usefulness for different physical, computational, and communication tasks. Here, we establish several complexity-theoretic results that classify the…
The polynomial hierarchy plays a central role in classical complexity theory. Here, we define a quantum generalization of the polynomial hierarchy, and initiate its study. We show that not only are there natural complete problems for the…
We obtain the strongest separation between quantum and classical query complexity known to date -- specifically, we define a black-box problem that requires exponentially many queries in the classical bounded-error case, but can be solved…
Quantum computing involving physical systems with continuous degrees of freedom, such as the quantum states of light, has recently attracted significant interest. However, a well-defined quantum complexity theory for these bosonic…
Can the computational complexity theory of computer science and mathematics say something new about unresolved problems in quantum physics? Particularly, can the P versus NP question in the computational complexity theory be a factor in the…
Quantum computing is gaining increased attention as a potential way to speed up simulations of physical systems, and it is also of interest to apply it to simulations of classical plasmas. However, quantum information science is…
The k-local Hamiltonian problem is a natural complete problem for the complexity class QMA, the quantum analog of NP. It is similar in spirit to MAX-k-SAT, which is NP-complete for k<=2. It was known that the problem is QMA-complete for any…
When it comes to NP, its natural definition, its wide applicability across scientific disciplines, and its timeless relevance, the writing is on the wall: There can be only one. Quantum NP, on the other hand, is clearly the apple that fell…
This chapter delves into the realm of computational complexity, exploring the world of challenging combinatorial problems and their ties with statistical physics. Our exploration starts by delving deep into the foundations of combinatorial…
Quantum computational complexity estimates the difficulty of constructing quantum states from elementary operations, a problem of prime importance for quantum computation. Surprisingly, this quantity can also serve to study a completely…
Since its inception at the beginning of the twentieth century, quantum mechanics has challenged our conceptions of how the universe ought to work; however, the equations of quantum mechanics can be too computationally difficult to solve…
These lecture notes are an informal introduction to the theory of computational complexity and its links to quantum computing and statistical mechanics.
QMA and QCMA are possible quantum analogues of the complexity class NP. In QCMA the verifier is a quantum program and the proof is classical. In contrast, in QMA the proof is also a quantum state. We show that two known QMA-complete…