Martin Rötteler

Graphs, quadratic forms, and quantum codes

We show that any stabilizer code over a finite field is equivalent to a graphical quantum code. Furthermore we prove that a graphical quantum code over a finite field is a stabilizer code. The technique used in the proof establishes a new connection between quantum codes and quadratic forms.

Discrete cosine transforms on quantum computers

A classical computer does not allow the calculation of 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 This trivial lower bound is no longer valid for a computer that takes advantage of quantum mechanical superposition, entanglement, and interference principles.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 N/spl times/N and types I, II, III and IV with as little as O(log/sup 2/N) operations on a quantum computer; whereas the known fast algorithms on a classical computer need O(N logN) operations.

Asymmetric quantum LDPC codes

Recently, quantum error-correcting codes were proposed that capitalize on the fact that many physical error models lead to a significant asymmetry between the probabilities for bit flip and phase flip errors. An example for a channel which exhibits such asymmetry is the combined amplitude damping and dephasing channel, where the probabilities of bit flips and phase flips can be related to relaxation and dephasing time, respectively. We give systematic constructions of asymmetric quantum stabilizer codes that exploit this asymmetry. Our approach is based on a CSS construction that combines BCH and finite geometry LDPC codes.

Non-catastrophic Encoders and Encoder Inverses for Quantum Convolutional Codes

We present an algorithm to construct quantum circuits for encoding and inverse encoding of quantum convolutional codes. We show that any quantum convolutional code contains a subcode of finite index which has a non-catastrophic encoding circuit. Our work generalizes the conditions for non-catastrophic encoders derived in a paper by Oliver and Tillich (quant-ph/0401134) which are applicable only for a restricted class of quantum convolutional codes We also show that the encodes and their inverse constructed by our method naturally can be applied online, i.e., qubits can be sent and received with constant delay

Quantum Algorithms: Applicable Algebra and Quantum Physics

Classical computer science relies on the concept of Turing machines as a unifying model of universal computation. According to the modern Church-Turing Thesis, this concept is interpreted in the form that every physically reasonable model of computation can be efficiently simulated on a probabilistic Turing machine. Recently this understanding, which was taken for granted for a long time, has required a severe reorientation because of the emergence of new computers that do not rely on classical physics but, rather, use effects predicted by quantum mechanics.

Constructions of Quantum Convolutional Codes

We address the problems of constructing quantum convolutional codes (QCCs) and of encoding them. The first construction is a CSS-type construction which allows us to find QCCs of rate 2/4. The second construction yields a quantum convolutional code by applying a product code construction to an arbitrary classical convolutional code and an arbitrary quantum block code. We show that the resulting codes have highly structured and efficient encoders. Furthermore, we show that the resulting quantum circuits have finite depth, independent of the lengths of the input stream, and show that this depth is polynomial in the degree and frame size of the code.

General Scheme for Perfect Quantum Network Coding with Free Classical Communication

This paper considers the problem of efficiently transmitting quantum states through a network. It has been known for some time that without additional assumptions it is impossible to achieve this task perfectly in general -- indeed, it is impossible even for the simple butterfly network. As additional resource we allow free classical communication between any pair of network nodes. It is shown that perfect quantum network coding is achievable in this model whenever classical network coding is possible over the same network when replacing all quantum capacities by classical capacities. More precisely, it is proved that perfect quantum network coding using free classical communication is possible over a network with k source-target pairs if there exists a classical linear (or even vector-linear) coding scheme over a finite ring. Our proof is constructive in that we give explicit quantum coding operations for each network node. This paper also gives an upper bound on the number of classical communication required in terms of k , the maximal fan-in of any network node, and the size of the network.

Quantum Convolutional BCH Codes

Quantum convolutional codes can be used to protect a sequence of qubits of arbitrary length against decoherence. We introduce two new families of quantum convolutional codes. Our construction is based on an algebraic method which allows to construct classical convolutional codes from block codes, in particular BCH codes. These codes have the property that they contain their Euclidean, respectively Hermitian, dual codes. Hence, they can be used to define quantum convolutional codes by the stabilizer code construction. We compute BCH-like bounds on the free distances which can be controlled as in the case of block codes, and establish that the codes have non-catastrophic encoders.

Using hardware transactional memory for data race detection

Widespread emergence of multicore processors will spur development of parallel applications, exposing programmers to degrees of hardware concurrency hitherto unavailable. Dependable multithreaded software will have to rely on the ability to dynamically detect non-deterministic and notoriously hard to reproduce synchronization bugs manifested through data races. Previous solutions to dynamic data race detection have required specialized hardware, at additional power, design and area costs. We propose RaceTM, a novel approach to data race detection that exploits hardware that will likely be present in future multiprocessors, albeit for a different purpose. In particular, we show how emerging hardware support for transactional memory can be leveraged to aid data race detection. We propose the concept of lightweight debug transactions that exploit the conflict detection mechanisms of transactional memory systems to perform data race detection. We present a proof-of-concept simulation prototype, and evaluate it on data races injected into applications from the SPLASH-2 suite. Our experiments show that this technique is effective at discovering data races and has low performance overhead.

On the power of random bases in fourier sampling: hidden subgroup problem in the heisenberg group

The hidden subgroup problem (HSP) offers a unified framework to study problems of group-theoretical nature in quantum computing such as order finding and the discrete logarithm problem. While it is known that Fourier sampling provides an efficient solution in the abelian case, not much is known for general non-abelian groups. Recently, some authors raised the question as to whether post-processing the Fourier spectrum by measuring in a random orthonormal basis helps for solving the HSP. Several negative results on the shortcomings of this random strong method are known. In this paper however, we show that the random strong method can be quite powerful under certain conditions on the group G. We define a parameter r(G) and show that O((log |G| / r(G))2) iterations of the random strong method give enough classical information to solve the HSP. We illustrate the power of the random strong method via a concrete example of the HSP over finite Heisenberg groups. We show that r(G) = Ω(1) for these groups; hence the HSP can be solved using polynomially many random strong Fourier samplings followed by a possibly exponential classical post-processing without further queries. The quantum part of our algorithm consists of a polynomial computation followed by measuring in a random orthonormal basis. As an interesting by-product of our work, we get an algorithm for solving the state identification problem for a set of nearly orthogonal pure quantum states.