Quantum Architectures and Computation Group

We introduce techniques to analyze unitary operations in terms of quadratic form expansions, a form similar to a sum over paths in the computational basis where the phase contributed by each path is described by a quadratic form over ℝ. We show how to relate such a form to an entangled resource akin to that of the one-way measurement model of quantum computing. Using this, we describe various conditions under which it is possible to efficiently implement a unitary operation U, either when provided a quadratic form ...

1/d holds for all b ∈ B and b ∈ B . The size of any set containing pairwise mutually unbiased bases of Cd cannot exceed d + 1. If d is a power of a prime, then extremal sets containing d+ 1 mutually unbiased bases are known to exist. We give a simplified proof of this fact ...

Mutually unbiased bases (MUBs) are a primitive used in quantum information processing to capture the principle of complementarity. While constructions of maximal sets of d + 1 such bases are known for system of prime power dimension d, it is unknown whether this bound can be achieved for any non-prime power dimension. In this paper we demonstrate that maximal sets of MUBs come with a rich combinatorial structure by showing that they actually are the same objects as the complex projective 2-designs with angle set {0, 1/d}. We also give a new and simple proof that symmetric informationally complete POVMs are complex projective 2-designs with angle set {1/(d+1)}.

A classical computer does not allow to calculate a dis-crete 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 This trivial lower bound is no longer valid for a computer that takes advantage of quantum mechanical superposition, entanglement, and inter$erence principles. In fact, ...

The first problem of the King was brought to us by Aharanov, Vaidman, and Albert [19], although they did not dare1 to reveal the tale of the King. The story was later told by Aharanov and Englert [1,2] and we retold the tale. In more modern terms, Alice has the problem to prepare a ...

Quantum computers have the potential to solve certain problems at much higher speed than any classical computer. Some evidence for this statement is given by Shor's algorithm to factor integers in polynomial time on a quan-tum computer. A crucial part of Shor's algorithm ...

The hidden subgroup problem (HSP) provides a unified framework to study problems of grouptheoretical 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) for a group G and show that O((log |G|/r(G)) 2 ) iterations of the random strong method give enough classical information to identify a hidden subgroup in G. 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. This gives the first example of a group where random representation bases do help in solving the HSP and for which no explicit representation bases are known that solve the problem with (log G) O(1) Fourier samplings. 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.

We present a construction of self-orthogonal codes using product codes. From the resulting codes, one can construct both block quantum error-correcting codes and quantum convolutional codes. We show that from the examples of convolutional codes found, we can derive ordinary quantum error-correcting codes using tail-biting with parameters [[42N, 24N, 3]]2. While it is known that the product construction cannot improve the rate in the classical case, we show that this can happen for quantum codes: we show that a code [[15, 7, 3]]2 is obtained by the product of a code [[5, 1, 3]]2 with a suitable code.

Most quantum algorithms that give an exponential speedup over classical algorithms exploit the Fourier transform in some way. In Shor's algorithm, sampling from the quantum Fourier spectrum is used to discover periodicity of the modular exponentiation function. In a generalization of this idea, quantum Fourier sampling can be used to discover hidden subgroup structures of some functions much more efficiently than it is possible classically. Another problem for which the Fourier transform has been recruited successfully on a quantum computer is the hidden shift problem. Quantum algorithms for hidden shift problems usually have a slightly different flavor from hidden subgroup algorithms, as they use the Fourier transform to perform a correlation with a given reference function, instead of sampling from the Fourier spectrum directly. In this paper we show that hidden shifts can be extracted efficiently from Boolean functions that are quadratic forms. We also show how to identify an unknown quadratic form on n variables using a linear number of queries, in contrast to the classical case were this takes Θ(n 2 ) many queries to a black box. What is more, we show that our quantum algorithm is robust in the sense that it can also infer the shift if the function is close to a quadratic, where we consider a Boolean function to be close to a quadratic if it has a large Gowers U 3 norm.

The k-pair problem in network coding theory asks to send k messages simultaneously between k source-target pairs over a directed acyclic graph. In a previous paper [ICALP 2009, Part I, pages 622-633] the present authors showed that if a classical k-pair problem is solvable by means of a linear coding scheme, then the quantum k-pair problem over the same graph is also solvable, provided that classical communication can be sent for free between any pair of nodes of the graph. Here we address the main case that remained open in our previous work, namely whether nonlinear classical network coding schemes can also give rise to quantum network coding schemes. This question is motivated by the fact that there are networks for which there are no linear solutions to the k-pair problem, whereas nonlinear solutions exist. In the present paper we overcome the limitation to linear protocols and describe a new communication protocol for perfect quantum network coding that improves over the previous one as follows: (i) the new protocol does not put any condition on the underlying classical coding scheme, that is, it can simulate nonlinear communication protocols as well, and (ii) the amount of classical communication sent in the protocol is significantly reduced. c x1 : The butterfly network and a classical linear coding protocol. The node s 1 (resp. s 2 ) has for input a bit x 1 (resp. x 2 ). The task is to send x 1 to t 1 and x 2 to t 2 . The capacity of each edge is assumed to be one bit.

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 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 ddimensional 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.

We present an algorithm for computing depth-optimal decompositions of logical operations, leveraging a meet-in-the-middle technique to provide a significant speed-up over simple brute force algorithms. As an illustration of our method we implemented this algorithm and found factorizations of the commonly used quantum logical operations into elementary gates in the Clifford+T set. In particular, we report a decomposition of the Toffoli gate over the set of Clifford and T gates. Our decomposition achieves a total T -depth of 3, thereby providing a 40% reduction over the previously best known decomposition for the Toffoli gate. Due to the size of the search space the algorithm is only practical for small parameters, such as the number of qubits, and the number of gates in an optimal implementation. arXiv:1206.0758v3 [quant-ph]

We present families of quantum error-correcting codes which are optimal in the sense that the minimum distance is maximal. These maximum distance separable (MDS) codes are defined over q-dimensional quantum systems, where q is an arbitrary prime power. It is shown that codes with parameters [[n, n − 2d + 2, d]]q exist for all 3 ≤ n ≤ q and 1 ≤ d ≤ n/2+1. We also present quantum MDS codes with parameters [[q 2 , q 2 −2d+2, d]]q for 1 ≤ d ≤ q which additionally give rise to shortened codes [[q 2 − s, q 2 − 2d + 2 − s, d]]q for some s.

We address the problem of constructing positive operator-valued measures (POVMs) in finite dimension n consisting of n 2 operators of rank one which have an inner product close to uniform. This is motivated by the related question of constructing symmetric informationally complete POVMs (SIC-POVMs) for which the inner products are perfectly uniform. However, SIC-POVMs are notoriously hard to construct and despite some success of constructing them numerically, there is no analytic construction known. We present two constructions of approximate versions of SIC-POVMs, where a small deviation from uniformity of the inner products is allowed. The first construction is based on selecting vectors from a maximal collection of mutually unbiased bases and works whenever the dimension of the system is a prime power. The second construction is based on perturbing the matrix elements of a subset of mutually unbiased bases.

We introduce a new quantum adversary method to prove lower bounds on the query complexity of the quantum state generation problem. This problem encompasses both, the computation of partial or total functions and the preparation of target quantum states. There has been hope for quite some time that quantum state generation might be a route to tackle the Graph Isomorphism problem. We show that for the related problem of Index Erasure our method leads to a lower bound of Ω( √ N ) which matches an upper bound obtained via reduction to quantum search on N elements. This closes an open problem first raised by Shi [FOCS'02].

Rejection sampling is a well-known method to sample from a target distribution, given the ability to sample from a given distribution. The method has been first formalized by von Neumann (1951) and has many applications in classical computing. We define a quantum analogue of rejection sampling: given a black box producing a coherent superposition of (possibly unknown) quantum states with some amplitudes, the problem is to prepare a coherent superposition of the same states, albeit with different target amplitudes. The main result of this paper is a tight characterization of the query complexity of this quantum state generation problem. We exhibit an algorithm, which we call quantum rejection sampling, and analyze its cost using semidefinite programming. Our proof of a matching lower bound is based on the automorphism principle which allows to symmetrize any algorithm over the automorphism group of the problem. Our main technical innovation is an extension of the automorphism principle to continuous groups that arise for quantum state generation problems where the oracle encodes unknown quantum states, instead of just classical data. Furthermore, we illustrate how quantum rejection sampling may be used as a primitive in designing quantum algorithms. As an example, we derive a new quantum algorithm for the hidden shift problem for an arbitrary Boolean function whose running time is obtained by "water-filling" its Fourier spectrum.

We study the quantum query complexity of the Boolean hidden shift problem. Given oracle access to f (x + s) for a known Boolean function f , the task is to determine the n-bit string s. The quantum query complexity of this problem depends strongly on f . We demonstrate that the easiest instances of this problem correspond to bent functions, in the sense that an exact onequery algorithm exists if and only if the function is bent. We partially characterize the hardest instances, which include delta functions. Moreover, we show that the problem is easy for random functions, since two queries suffice. Our algorithm for random functions is based on performing the pretty good measurement on several copies of a certain state; its analysis relies on the Fourier transform. We also use this approach to improve the quantum rejection sampling approach to the Boolean hidden shift problem. arXiv:1304.4642v1 [quant-ph]