Quantum Oracle Interrogation: Getting all information for almost half the price

Consider a quantum computer in combination with a binary oracle of domain size N. It is shown how N/2+sqrt(N) calls to the oracle are sufficient to guess the whole content of the oracle (being an N bit string) with probability greater than 95%. This contrasts the power of classical computers which would require N calls to achieve the same task. From this result it follows that any function with the N bits of the oracle as input can be calculated using N/2+sqrt(N) queries if we allow a small probability of error. It is also shown that this error probability can be made arbitrary small by using N/2+O(sqrt(N)) oracle queries. In the second part of the article `approximate interrogation' is considered. This is when only a certain fraction of the N oracle bits are requested. Also for this scenario does the quantum algorithm outperform the classical protocols. An example is given where a quantum procedure with N/10 queries returns a string of which 80% of the bits are correct. Any classical protocol would need 6N/10 queries to establish such a correctness ratio.Comment: 11 pages LaTeX2e, 1 postscript figure; error analysis added; new section on approximate interrogation adde

Comment on "Quantum identification schemes with entanglements"

In a recent paper, [Phys. Rev. A 65, 052326 (2002)], Mihara presented several cryptographic protocols that were claimed to be quantum mechanical in nature. In this comment it is pointed out that these protocols can be described in purely classical terms. Hence, the security of these schemes does not rely on the usage of entanglement or any other quantum mechanical property.Comment: 2 pages, revtex

Quantum Algorithms for Weighing Matrices and Quadratic Residues

In this article we investigate how we can employ the structure of combinatorial objects like Hadamard matrices and weighing matrices to device new quantum algorithms. We show how the properties of a weighing matrix can be used to construct a problem for which the quantum query complexity is ignificantly lower than the classical one. It is pointed out that this scheme captures both Bernstein & Vazirani's inner-product protocol, as well as Grover's search algorithm. In the second part of the article we consider Paley's construction of Hadamard matrices, which relies on the properties of quadratic characters over finite fields. We design a query problem that uses the Legendre symbol chi (which indicates if an element of a finite field F_q is a quadratic residue or not). It is shown how for a shifted Legendre function f_s(i)=chi(i+s), the unknown s in F_q can be obtained exactly with only two quantum calls to f_s. This is in sharp contrast with the observation that any classical, probabilistic procedure requires more than log(q) + log((1-e)/2) queries to solve the same problem.Comment: 18 pages, no figures, LaTeX2e, uses packages {amssymb,amsmath}; classical upper bounds added, presentation improve

Two Classical Queries versus One Quantum Query

In this note we study the power of so called query-limited computers. We compare the strength of a classical computer that is allowed to ask two questions to an NP-oracle with the strength of a quantum computer that is allowed only one such query. It is shown that any decision problem that requires two parallel (non-adaptive) SAT-queries on a classical computer can also be solved exactly by a quantum computer using only one SAT-oracle call, where both computations have polynomial time-complexity. Such a simulation is generally believed to be impossible for a one-query classical computer. The reduction also does not hold if we replace the SAT-oracle by a general black-box. This result gives therefore an example of how a quantum computer is probably more powerful than a classical computer. It also highlights the potential differences between quantum complexity results for general oracles when compared to results for more structured tasks like the SAT-problem.Comment: 6 pages, LaTeX2e, no figures, minor changes and correction

Quantum Bounded Query Complexity

We combine the classical notions and techniques for bounded query classes with those developed in quantum computing. We give strong evidence that quantum queries to an oracle in the class NP does indeed reduce the query complexity of decision problems. Under traditional complexity assumptions, we obtain an exponential speedup between the quantum and the classical query complexity of function classes. For decision problems and function classes we obtain the following results: o P_||^NP[2k] is included in EQP_||^NP[k] o P_||^NP[2^(k+1)-2] is included in EQP^NP[k] o FP_||^NP[2^(k+1)-2] is included in FEQP^NP[2k] o FP_||^NP is included in FEQP^NP[O(log n)] For sets A that are many-one complete for PSPACE or EXP we show that FP^A is included in FEQP^A[1]. Sets A that are many-one complete for PP have the property that FP_||^A is included in FEQP^A[1]. In general we prove that for any set A there is a set X such that FP^A is included in FEQP^X[1], establishing that no set is superterse in the quantum setting.Comment: 11 pages LaTeX2e, no figures, accepted for CoCo'9