Besides Shor's polynomial-time quantum algorithms for factoring and discrete log, all progress in understanding when quantum algorithms have an exponential advantage over classical algorithms has been through oracle problems. Here we give efficient quantum algorithms for two more non-oracle problems. The first is Pell's equation. Given a positive non-square integer d, Pell's equation is x² - dy² = 1 and the goal is to find its integer solutions. Factoring integers reduces to finding integer solutions of Pell's equation, but a reduction in the other direction is not known and appears more difficult. The second problem is the principal ideal problem in real quadratic number fields. Solving this problem is at least as hard as solving Pell's equation, and is the basis of a cryptosystem which is broken by our algorithm. We also state some related open problems from the area of computational algebraic number theory.

Almost all of the most successful quantum algorithms discovered to date exploit the ability of the Fourier transform to recover subgroup structures of functions, especially periodicity. The fact that Fourier transforms can also be used to capture shift structure has received far less attention in the context of quantum computation. In this paper, we present three examples of “unknown shift” problems that can be solved efficiently on a quantum computer using the quantum Fourier transform. For one of these problems, the shifted Legendre symbol problem, we give evidence that the problem is hard to solve classically, by showing a reduction from breaking algebraically homomorphic cryptosystems. We also define the hidden coset problem, which generalizes the hidden shift problem and the hidden subgroup problem. This framework provides a unified way of viewing the ability of the Fourier transform to capture subgroup and shift structure.

Quantum mechanics requires the operation of quantum computers to be unitary, and thus makes it important to have general techniques for developing fast quantum algorithms for computing unitary transforms. A quantum routine for computing a generalized Kronecker product is given. Applications include re-development of the networks for computing the Walsh-Hadamard and the quantum Fourier transform. New networks for two wavelet transforms are given. Quantum computation of Fourier transforms for non-Abelian groups is defined. A slightly relaxed definition is shown to simplify the analysis and the networks that computes the transforms. Efficient networks for computing such transforms for a class of metacyclic groups are introduced. A novel network for computing a Fourier transform for a group used in quantum errorcorrection is also given. 1

Consider a quantum computer in combination with a binary oracle of domain size N. It is shown how N/2 + √ 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 + √ 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 ( √ 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. ∗ c○1998 IEEE. Published in the Proceedings of FOCS’98, 8-11 November 1998 in Palo Alto, CA. Personal use of this material is permitted. However, permission to reprint/republish this material for advertising or promotional purposes or for creating new collective works for resale or redistribution to servers or lists, or to reuse any copyrighted component of this work in other works, must be obtained from the ieee. Contact: Manager, Copyrights and

Abstract. The purpose of these lecture notes is to provide readers, who have some mathematical background but little or no exposure to quantum mechanics and quantum computation, with enough material to begin reading

We isolate and generalize a technique implicit in many quantum algorithms, including Shor's algorithms for factoring and discrete log. In particular, we show that the distribution sampled after a Fourier transform over Z p can be efficiently approximated by transforming over Z q for any q in a large range. Our result places no restrictions on the superposition to be transformed, generalizing previous applications. In addition, our proof easily generalizes to multi-dimensional transforms for any constant number of dimensions.

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In this paper we give a definition for quantum Kolmogorov complexity. In the classical setting, the Kolmogorov complexity of a string is the length of the shortest program that can produce this string as its output. It is a measure of the amount of innate randomness (or information) contained in the string. We define the quantum Kolmogorov complexity of a qubit string as the length of the shortest quantum input to a universal quantum Turing machine that produces the initial qubit string with high fidelity. The definition of Vitányi [20] measures the amount of classical information, whereas we consider the amount of quantum information in a qubit string. We argue that our definition is natural and is an accurate representation of the amount of quantum information contained in a quantum state. 1

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 significantly 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 to design a more specific problem that uses the Legendre symbol χ (which indicates if an element of a finite field GF(p k) is a quadratic residue or not). It is shown how for a shifted Legendre function fs(x) = χ(x+s), the unknown s ∈ GF(p k) can be obtained exactly with only two quantum calls to fs. This is in sharp contrast with the observation that any classical, probabilistic procedure requires at least k log p queries to solve the same problem. 1

We consider the problem of optimal asymptotically faithful compression for ensembles of mixed quantum states. Although the optimal rate is unknown, we prove upper and lower bounds and describe a series of illustrative examples of compression of mixed states. We also discuss a classical analogue of the problem.