Algorithmic Complexity and Entanglement of Quantum States

This paper identifies a formal connection between physical problems related to entanglement detection and complexity classes in theoretical computer science. In particular, we show that to nearly every quantum interactive proof complexity class (including BQP, QMA, QMA(2), QSZK, and QIP), there corresponds a natural entanglement or correlation detection problem that is complete for that class. In this sense, we can say that an entanglement or correlation detection problem captures the expressive power of each quantum interactive proof complexity class. The only promise problem of this flavor that is not known to be complete for a quantum interactive proof complexity class is a variation of the original quantum separability problem in which the goal is to decide if a quantum state output by a quantum circuit is separable or entangled. We also show that the difficulty of entanglement detection depends on whether the distance measure used is the trace distance or the one-way LOCC distance. 1 For example, it is known that if clauses of the Boolean satisfiability problem are limited to two variables each, the resulting problem (2SAT) is NL-complete [Pap94], while if one allows only Horn clauses the resulting problem (HORNSAT) is P-complete [CN10], and if one removes any such limitations on clauses the problem (SAT) is NPcomplete [Coo71].

International Journal of Quantum Information, 2003

We introduce new communication complexity problems whose quantum solution exploits entanglement between higher-dimensional systems. We show that the quantum solution is more efficient than the broad class of classical ones. The difference between the efficiencies for the quantum and classical protocols grows with the dimensionality of the entangled systems.

SIAM Journal on Computing, 2001

We consider a variation of the multi-party communication complexity scenario where the parties are supplied with an extra resource: particles in an entangled quantum state. We show that, although a prior quantum entanglement cannot be used to simulate a communication channel, it can reduce the communication complexity of functions in some cases. Specifically, we show that, for a particular function among three parties (each of which possesses part of the function's input), a prior quantum entanglement enables them to learn the value of the function with only three bits of communication occurring among the parties, whereas, without quantum entanglement, four bits of communication are necessary. We also show that, for a particular two-party probabilistic communication complexity problem, quantum entanglement results in less communication than is required with only classical random correlations (instead of quantum entanglement). These results are a noteworthy contrast to the well-known fact that quantum entanglement cannot be used to actually simulate communication among remote parties.

International Journal of Quantum Information, 2007

Kolmogorov complexity is a measure of the information contained in a binary string. We investigate here the notion of quantum Kolmogorov complexity, a measure of the information required to describe a quantum state. We show that for any definition of quantum Kolmogorov complexity measuring the number of classical bits required to describe a pure quantum state, there exists a pure n-qubit state which requires exponentially many bits of description. This is shown by relating the classical communication complexity to the quantum Kolmogorov complexity. Furthermore, we give some examples of how quantum Kolmogorov complexity can be applied to prove results in different fields, such as quantum computation and thermodynamics, and we generalize it to the case of mixed quantum states.

World Scintific, 2013

It is usually calimed that quantum computer can outperform classical computer. Is this statement true? We discuss this issue, not in general, but in the context of the most famous algorithm of quantum computation: Shor's algorithm.

Mathematics

In 1982, Richard Feynman stated that in order to simulate quantum systems, we would rather go for a sort of brand-new powered quantum processor instead of a classical one [...]

2006

Algorithms have been proposed in the framework of quantum computing which are capable of solving particular problems much more efficiently than any classical computer. The most prominent example is the celebrated polynomial-time algorithm of P. Shor for finding the prime factors of large integers , an algorithm that astonished the scientific community in 1995. To date, no classical polynomial-time algorithm is known for this problem. L. Grover's quantum algorithm for search in a database offers a square root speedup compared to any classical algorithm . It has become obvious that in order to be able to perform quantum information processing in the presence of noise, elaborate protection methods would be necessary. Such methods were developed under the names quantum error correction, fault tolerant quantum computing, quantum error correcting codes, and stabilizer codes, pioneered by the works of B.overview of recent developments is given in the excellent comprehensive introductions in Refs. .

Journal of Computer and System Sciences, 1999

We use the powerful tools of counting complexity and generic oracles to help understand the limitations of the complexity of quantum computation. We show several results for the probabilistic quantum class BQP. • BQP is low for PP, i.e., PP BQP = PP. • There exists a relativized world where P = BQP and the polynomial-time hierarchy is infinite. • There exists a relativized world where BQP does not have complete sets. • There exists a relativized world where P = BQP but P = UP ∩ coUP and one-way functions exist. This gives a relativized answer to an open question of Simon.

1999

We consider a variation of the multi-party communication complexity scenario where the parties are supplied with an extra resource: particles in an entangled quantum state. We show that, although a prior quantum entanglement cannot be used to simulate a communication channel, it can reduce the communication complexity of functions in some cases. Specifically, we show that, for a particular function among three parties (each of which possesses part of the function's input), a prior quantum entanglement enables them to learn the value of the function with only three bits of communication occurring among the parties, whereas, without quantum entanglement, four bits of communication are necessary. We also show that, for a particular two-party probabilistic communication complexity problem, quantum entanglement results in less communication than is required with only classical random correlations (instead of quantum entanglement). These results are a noteworthy contrast to the well-known fact that quantum entanglement cannot be used to actually simulate communication among remote parties.

2000

The paper is intended to be a survey of all the important aspects and results that have shaped the field of quantum computation and quantum information. The reader is first familiarized with those features and principles of quantum mechanics providing a more efficient and secure information processing. Their applications to the general theory of information, cryptography, algorithms, computational complexity and error-correction are then discussed. Prospects for building a practical quantum computer are also analyzed.