arXiv subjects

Richard Cleve

Publications and source records attributed to Richard Cleve.

48 records · Page 3Linked to original sources

arXiv1999

The cost of exactly simulating quantum entanglement with classical communication

We investigate the amount of communication that must augment classical local hidden variable models in order to simulate the behaviour of entangled quantum systems. We consider the scenario where a bipartite measurement is given from a set of possibilities and the goal is to obtain exactly the same correlations that arise when the actual quantum system is measured. We show that, in the case of a single pair of qubits in a Bell state, a constant number of bits of communication is always sufficient--regardless of the number of measurements under consideration. We also show that, in the case of a system of n Bell states, a constant times 2^n bits of communication are necessary.

quant-ph↗

arXiv1999

How to share a quantum secret

We investigate the concept of quantum secret sharing. In a ((k,n)) threshold scheme, a secret quantum state is divided into n shares such that any k of those shares can be used to reconstruct the secret, but any set of k-1 or fewer shares contains absolutely no information about the secret. We show that the only constraint on the existence of threshold schemes comes from the quantum "no-cloning theorem", which requires that n < 2k, and, in all such cases, we give an efficient construction of a ((k,n)) threshold scheme. We also explore similarities and differences between quantum secret sharing schemes and quantum error-correcting codes. One remarkable difference is that, while most existing quantum codes encode pure states as pure states, quantum secret sharing schemes must use mixed states in some cases. For example, if k <= n < 2k-1 then any ((k,n)) threshold scheme must distribute information that is globally in a mixed state.

quant-ph↗

arXiv1998

Quantum Lower Bounds by Polynomials

We examine the number T of queries that a quantum network requires to compute several Boolean functions on {0,1}^N in the black-box model. We show that, in the black-box model, the exponential quantum speed-up obtained for partial functions (i.e. problems involving a promise on the input) by Deutsch and Jozsa and by Simon cannot be obtained for any total function: if a quantum algorithm computes some total Boolean function f with bounded-error using T black-box queries then there is a classical deterministic algorithm that computes f exactly with O(T^6) queries. We also give asymptotically tight characterizations of T for all symmetric f in the exact, zero-error, and bounded-error settings. Finally, we give new precise bounds for AND, OR, and PARITY. Our results are a quantum extension of the so-called polynomial method, which has been successfully applied in classical complexity theory, and also a quantum extension of results by Nisan about a polynomial relationship between randomized and deterministic decision tree complexity.

quant-ph↗

arXiv1998

Quantum Entanglement and the Communication Complexity of the Inner Product Function

We consider the communication complexity of the binary inner product function in a variation of the two-party scenario where the parties have an a priori supply of particles in an entangled quantum state. We prove linear lower bounds for both exact protocols, as well as for protocols that determine the answer with bounded-error probability. Our proofs employ a novel kind of "quantum" reduction from a quantum information theory problem to the problem of computing the inner product. The communication required for the former problem can then be bounded by an application of Holevo's theorem. We also give a specific example of a probabilistic scenario where entanglement reduces the communication complexity of the inner product function by one bit.

quant-ph↗

arXiv1998

Quantum vs. Classical Communication and Computation

We present a simple and general simulation technique that transforms any black-box quantum algorithm (a la Grover's database search algorithm) to a quantum communication protocol for a related problem, in a way that fully exploits the quantum parallelism. This allows us to obtain new positive and negative results. The positive results are novel quantum communication protocols that are built from nontrivial quantum algorithms via this simulation. These protocols, combined with (old and new) classical lower bounds, are shown to provide the first asymptotic separation results between the quantum and classical (probabilistic) two-party communication complexity models. In particular, we obtain a quadratic separation for the bounded-error model, and an exponential separation for the zero-error model. The negative results transform known quantum communication lower bounds to computational lower bounds in the black-box model. In particular, we show that the quadratic speed-up achieved by Grover for the OR function is impossible for the PARITY function or the MAJORITY function in the bounded-error model, nor is it possible for the OR function itself in the exact case. This dichotomy naturally suggests a study of bounded-depth predicates (i.e. those in the polynomial hierarchy) between OR and MAJORITY. We present black-box algorithms that achieve near quadratic speed up for all such predicates.

quant-ph↗

arXiv1997

Quantum Algorithms Revisited

Quantum computers use the quantum interference of different computational paths to enhance correct outcomes and suppress erroneous outcomes of computations. A common pattern underpinning quantum algorithms can be identified when quantum computation is viewed as multi-particle interference. We use this approach to review (and improve) some of the existing quantum algorithms and to show how they are related to different instances of quantum phase estimation. We provide an explicit algorithm for generating any prescribed interference pattern with an arbitrary precision.

quant-ph↗

arXiv1997

Information-theoretic interpretation of quantum error-correcting codes

Quantum error-correcting codes are analyzed from an information-theoretic perspective centered on quantum conditional and mutual entropies. This approach parallels the description of classical error correction in Shannon theory, while clarifying the differences between classical and quantum codes. More specifically, it is shown how quantum information theory accounts for the fact that "redundant" information can be distributed over quantum bits even though this does not violate the quantum "no-cloning" theorem. Such a remarkable feature, which has no counterpart for classical codes, is related to the property that the ternary mutual entropy vanishes for a tripartite system in a pure state. This information-theoretic description of quantum coding is used to derive the quantum analogue of the Singleton bound on the number of logical bits that can be preserved by a code of fixed length which can recover a given number of errors.

quant-ph↗

arXiv1997

Quantum Entanglement and Communication Complexity

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.

quant-ph↗

arXiv1997

Substituting Quantum Entanglement for Communication

We show that quantum entanglement can be used as a substitute for communication when the goal is to compute a function whose input data is distributed among remote parties. 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 one of them to learn the value of the function with only two bits of communication occurring among the parties, whereas, without quantum entanglement, three bits of communication are necessary. This result contrasts the well-known fact that quantum entanglement cannot be used to simulate communication among remote parties.

quant-ph↗

arXiv1996

Quantum Stabilizer Codes and Classical Linear Codes

We show that within any quantum stabilizer code there lurks a classical binary linear code with similar error-correcting capabilities, thereby demonstrating new connections between quantum codes and classical codes. Using this result -- which applies to degenerate as well as nondegenerate codes -- previously established necessary conditions for classical linear codes can be easily translated into necessary conditions for quantum stabilizer codes. Examples of specific consequences are: for a quantum channel subject to a delta-fraction of errors, the best asymptotic capacity attainable by any stabilizer code cannot exceed H(1/2 + sqrt(2*delta*(1-2*delta))); and, for the depolarizing channel with fidelity parameter delta, the best asymptotic capacity attainable by any stabilizer code cannot exceed 1-H(delta).

quant-ph↗

arXiv1996

Efficient Computations of Encodings for Quantum Error Correction

We show how, given any set of generators of the stabilizer of a quantum code, an efficient gate array that computes the codewords can be constructed. For an n-qubit code whose stabilizer has d generators, the resulting gate array consists of O(n d) operations, and converts k-qubit data (where k = n - d) into n-qubit codewords.

quant-ph↗

arXiv1996

Schumacher's quantum data compression as a quantum computation

An explicit algorithm for performing Schumacher's noiseless compression of quantum bits is given. This algorithm is based on a combinatorial expression for a particular bijection among binary strings. The algorithm, which adheres to the rules of reversible programming, is expressed in a high-level pseudocode language. It is implemented using $O(n^3)$ two- and three-bit primitive reversible operations, where $n$ is the length of the qubit strings to be compressed. Also, the algorithm makes use of $O(n)$ auxiliary qubits; however, space-saving techniques based on those proposed by Bennett are developed which reduce this workspace to $O(\sqrt{n})$ while increasing the running time by less than a factor of two.

quant-ph↗