Quantum algorithms for highly non-linear Boolean functions

Attempts to separate the power of classical and quantum models of computation have a long history. The ultimate goal is to find exponential separations for computational problems. However, such separations do not come a dime a dozen: while there were some early successes in the form of hidden subgroup problems for abelian groups--which generalize Shor's factoring algorithm perhaps most faithfully--only for a handful of non-abelian groups efficient quantum algorithms were found. Recently, problems have gotten increased attention that seek to identify hidden sub-structures of other combinatorial and algebraic objects besides groups. In this paper we provide new examples for exponential separations by considering hidden shift problems that are defined for several classes of highly non-linear Boolean functions. These so-called bent functions arise in cryptography, where their property of having perfectly flat Fourier spectra on the Boolean hypercube gives them resilience against certain types of attack. We present new quantum algorithms that solve the hidden shift problems for several well-known classes of bent functions in polynomial time and with a constant number of queries, while the classical query complexity is shown to be exponential. Our approach uses a technique that exploits the duality between bent functions and their Fourier transforms.Comment: 15 pages, 1 figure, to appear in Proceedings of the 21st Annual ACM-SIAM Symposium on Discrete Algorithms (SODA'10). This updated version of the paper contains a new exponential separation between classical and quantum query complexit

Efficient Decoupling Schemes Based on Hamilton Cycles

Decoupling the interactions in a spin network governed by a pair-interaction Hamiltonian is a well-studied problem. Combinatorial schemes for decoupling and for manipulating the couplings of Hamiltonians have been developed which use selective pulses. In this paper we consider an additional requirement on these pulse sequences: as few {\em different} control operations as possible should be used. This requirement is motivated by the fact that optimizing each individual selective pulse will be expensive, i. e., it is desirable to use as few different selective pulses as possible. For an arbitrary $d$-dimensional system we show that the ability to implement only two control operations is sufficient to turn off the time evolution. In case of a bipartite system with local control we show that four different control operations are sufficient. Turning to networks consisting of several $d$-dimensional nodes which are governed by a pair-interaction Hamiltonian, we show that decoupling can be achieved if one is able to control a number of different control operations which is logarithmic in the number of nodes.Comment: 4 pages, 1 figure, uses revtex

Non-Additive Quantum Codes from Goethals and Preparata Codes

We extend the stabilizer formalism to a class of non-additive quantum codes which are constructed from non-linear classical codes. As an example, we present infinite families of non-additive codes which are derived from Goethals and Preparata codes.Comment: submitted to the 2008 IEEE Information Theory Workshop (ITW 2008

On the Irresistible Efficiency of Signal Processing Methods in Quantum Computing

We show that many well-known signal transforms allow highly efficient realizations on a quantum computer. We explain some elementary quantum circuits and review the construction of the Quantum Fourier Transform. We derive quantum circuits for the Discrete Cosine and Sine Transforms, and for the Discrete Hartley transform. We show that at most O(log^2 N) elementary quantum gates are necessary to implement any of those transforms for input sequences of length N.Comment: 15 pages, LaTeX 2e. Expanded version of quant-ph/0111038. SPECLOG 2000, Tampere, Finlan

On the Monomiality of Nice Error Bases

Unitary error bases generalize the Pauli matrices to higher dimensional systems. Two basic constructions of unitary error bases are known: An algebraic construction by Knill, which yields nice error bases, and a combinatorial construction by Werner, which yields shift-and-multiply bases. An open problem posed by Schlingemann and Werner (see http://www.imaph.tu-bs.de/qi/problems/6.html) relates these two constructions and asks whether each nice error basis is equivalent to a shift-and-multiply basis. We solve this problem and show that the answer is negative. However, we also show that it is always possible to find a fairly sparse representation of a nice error basis.Comment: 6 page

Discrete Cosine Transforms on Quantum Computers

A classical computer does not allow to calculate a discrete cosine transform on N points in less than linear time. This trivial lower bound is no longer valid for a computer that takes advantage of quantum mechanical superposition, entanglement, and interference principles. In fact, we show that it is possible to realize the discrete cosine transforms and the discrete sine transforms of size NxN and types I,II,III, and IV with as little as O(log^2 N) operations on a quantum computer, whereas the known fast algorithms on a classical computer need O(N log N) operations.Comment: 5 pages, LaTeX 2e, IEEE ISPA01, Pula, Croatia, 200

Beyond Stabilizer Codes I: Nice Error Bases

Nice error bases have been introduced by Knill as a generalization of the Pauli basis. These bases are shown to be projective representations of finite groups. We classify all nice error bases of small degree, and all nice error bases with abelian index groups. We show that in general an index group of a nice error basis is necessarily solvable.Comment: 12 pages, LaTeX2e. Minor changes. Title changed by request of IEEE Trans. I

Quantum MDS Codes over Small Fields

We consider quantum MDS (QMDS) codes for quantum systems of dimension $q$ with lengths up to $q^2+2$ and minimum distances up to $q+1$. We show how starting from QMDS codes of length $q^2+1$ based on cyclic and constacyclic codes, new QMDS codes can be obtained by shortening. We provide numerical evidence for our conjecture that almost all admissible lengths, from a lower bound $n_0(q,d)$ on, are achievable by shortening. Some additional codes that fill gaps in the list of achievable lengths are presented as well along with a construction of a family of QMDS codes of length $q^2+2$, where $q=2^m$, that appears to be new.Comment: 6 pages, 3 figure

Leveraging Automorphisms of Quantum Codes for Fault-Tolerant Quantum Computation

Fault-tolerant quantum computation is a technique that is necessary to build a scalable quantum computer from noisy physical building blocks. Key for the implementation of fault-tolerant computations is the ability to perform a universal set of quantum gates that act on the code space of an underlying quantum code. To implement such a universal gate set fault-tolerantly is an expensive task in terms of physical operations, and any possible shortcut to save operations is potentially beneficial and might lead to a reduction in overhead for fault-tolerant computations. We show how the automorphism group of a quantum code can be used to implement some operators on the encoded quantum states in a fault-tolerant way by merely permuting the physical qubits. We derive conditions that a code has to satisfy in order to have a large group of operations that can be implemented transversally when combining transversal CNOT with automorphisms. We give several examples for quantum codes with large groups, including codes with parameters [[8,3,3]], [[15,7,3]], [[22,8,4]], and [[31,11,5]]