steve brierley's


Efficient implementation of Quantum circuits with limited qubit interactions.

S. Brierley, arXiv:1507.04263

The quantum circuit model allows gates between any pair of qubits yet physical instantiations allow only limited interactions. We address this problem by providing an interaction graph together with an efficient method for compiling quantum circuits so that gates are applied only locally. The graph requires each qubit to interact with 4 other qubits and yet the time-overhead for implementing any n-qubit quantum circuit is 6 log n. Building a network of quantum computing nodes according to this graph enables the network to emulate a single monolithic device with minimal overhead.

On properties of Karlsson Hadamards and sets of Mutually Unbiased Bases in dimension six.

A. Maxwell, S. Brierley, arXiv:1402.4070

The complete classification of all 6x6 complex Hadamard matrices is an open problem. The 3-parameter Karlsson family encapsulates all Hadamards that have been parametrised explicitly. We prove that such matrices satisfy a non-trivial constraint conjectured to hold for (almost) all 6x6 Hadamard matrices. Our result imposes additional conditions in the linear programming approach to the mutually unbiased bases problem recently proposed by Matolcsi et al. Unfortunately running the linear programs we were unable to conclude that a complete set of mutually unbiased bases cannot be constructed from Karlsson Hadamards alone.

Non-classicality of temporal correlations.

S. Brierley, A. Kosowski, M. Markiewicz, T. Paterek, A. Przysiezna, arXiv:1501.03505

The results of space-like separated measurements are independent of distant measurement settings, a property one might call two-way no-signalling. In contrast, time-like separated measurements are only one-way no-signalling since the past is independent of the future but not vice-versa. For this reason temporal correlations that are formally identical to non-classical spatial correlations can still be modelled classically. We define non-classical temporal correlations as the ones which cannot be simulated by propagating in time a classical information content of a quantum system. We first show that temporal correlations between results of any projective quantum measurements on a qubit can be simulated classically. Then we present a sequence of POVM measurements on a single m-level quantum system that cannot be explained by propagating in time m-level classical system and using classical computers with unlimited memory.

Genuinely multipoint temporal quantum correlations and universal measurement-based quantum computing.

M. Markiewicz, A. Przysiężna, S. Brierley, T. Paterek, arXiv:1309.7650

We introduce a constructive procedure that maps all spatial correlations of a broad class of d-level states of N parties into temporal correlations between general d-outcome quantum measurements performed on a single d-level system. This allows us to present temporal phenomena analogous to genuinely multipartite nonlocal phenomena, such as Greenberger-Horne-Zeilinger correlations, which do not exist if only projective measurements on a single qubit are considered. The map is applied to certain lattice systems in order to replace one spatial dimension with a temporal one, without affecting measured correlations. We use this map to show how repeated application of a one-dimensional (1D) cluster gate leads to universal one-way quantum computing when supplemented with general two-outcome quantum measurements. In this way, we recover a temporal version of measurement-based quantum computing performed on a sequentially recreated 1D cluster.

Systems of Imprimitivity for the Clifford Group.

D. M. Appleby, I. Bengtsson, S. Brierley, Å. Ericsson, M. Grassl, J.-Å. Larsson, arXiv:1210.1055

It is known that if the dimension is a perfect square the Clifford group can be represented by monomial matrices. Another way of expressing this result is to say that when the dimension is a perfect square the standard representation of the Clifford group has a system of imprimitivity consisting of one dimensional subspaces. We generalize this result to the case of an arbitrary dimension. Let k be the square-free part of the dimension. Then we show that the standard representation of the Clifford group has a system of imprimitivity consisting of k-dimensional subspaces. To illustrate the use of this result we apply it to the calculation of SIC-POVMs (symmetric informationally complete positive operator valued measures), constructing exact solutions in dimensions 8 (hand-calculation) as well as 12 and 28 (machine-calculation).

Efficient Distributed Quantum Computing.

R. Beals, S. Brierley, O. Gray, A. Harrow, S. Kutin, N. Linden, D. Shepherd and M. Stather, Proc Royal Soc. A

We provide algorithms for efficiently addressing quantum memory in parallel. These imply that the standard circuit model can be simulated with low overhead by the more realistic model of a distributed quantum computer. As a result, the circuit model can be used by algorithm designers without worrying whether the underlying architecture supports the connectivity of the circuit. In addition, we apply our results to existing memory intensive quantum algorithms. We present a parallel quantum search algorithm and improve the time-space trade-off for the Element Distinctness and Collision problems.

Entanglement detection via mutually unbiased bases.

C. Spengler, M. Huber, S. Brierley, T. Adaktylos and B. Hiesmayr, Phys. Rev. A 86, 022311 (2012)

We show how to take advantage of mutually unbiased bases (MUBs) for the detection of entanglement in arbitrarily high-dimensional quantum systems. It is shown that their properties can be exploited to construct entanglement criteria which are experimentally implementable with few local measurement settings. The introduced concepts are not restricted to bipartite finite-dimensional systems, but are also applicable to continuous variables and multipartite systems. This is demonstrated by two examples -- the two-mode squeezed state and the Aharonov state. In addition, and more importantly from a theoretical point of view, we find a link between the separability problem and the maximum number of mutually unbiased bases.

The Monomial representations of the Clifford group.

D.M. Appleby, I. Bengtsson, S. Brierley, M. Grassl, D. Gross and J.A Larsson, QIC, vol. 12, 0404.

We show that the Clifford group - the normaliser of the Weyl-Heisenberg group - can be represented by monomial phase-permutation matrices if and only if the dimension is a square number. This simplifies expressions for SIC vectors, and has other applications to SICs and to Mutually Unbiased Bases. Exact solutions for SICs in dimension 16 are presented for the first time.

Mutually Unbiased Bases and Semi-definite Programming.

S. Brierley and S. Weigert, J. Phys.: Conf. Ser. 254 012008 (2010)

A complex Hilbert space of dimension six supports at least three but not more than seven mutually unbiased bases. Two computer-aided analytical methods to tighten these bounds are reviewed, based on a discretization of parameter space and on Grobner bases. A third algorithmic approach is presented: the non-existence of more than three mutually unbiased bases in composite dimensions can be decided by a global optimization method known as semidefinite programming. The method is used to confirm that the spectral matrix cannot be part of a complete set of seven mutually unbiased bases in dimension six.

Complementary observables in low dimensions.

S. Brierley, Conference proceedings, Quantum Theory: Reconsideration of Foundations, Växjö (2010).

We review sets of complementary observables in low dimensions. The qualitatively different structure found in dimension six gives rise to some fundamental questions about the kinematics of quantum systems in composite dimensions.

Quantum key distribution highly sensitive to eavesdropping.

S. Brierley, arXiv 0910.2578

We introduce a natural generalisation of the SARG quantum key distribution protocol that uses d-level quantum systems to encode an alphabet with c letters. It has the property that the error rate introduced by an intercept-and-resend attack tends to one as the numbers c and d increase.

All mutually unbiased bases in dimensions two to five.

S. Brierley, S. Weigert and I. Bengtsson, Quantum Info. and Comp. Vol 10, 0803-0820 (2010)

All complex Hadamard matrices in dimensions two to five are known. We use this fact to derive all inequivalent sets of mutually unbiased (MU) bases in low dimensions. We find a three-parameter family of triples of MU bases in dimension four and two inequivalent classes of MU triples in dimension five. We confirm that the complete sets of (d+1) MU bases are unique (up to equivalence) in dimensions below six, using only elementary arguments for d less than five.

Constructing mutually unbiased bases in dimension Six.

S. Brierley and S. Weigert, Phys. Rev. A 79, 052316 (2009)

The density matrix of a qudit may be reconstructed with optimal efficiency if the expectation values of a specific set of observables are known. In dimension six, the required observables only exist if it is possible to identify six mutually unbiased complex 6x6 Hadamard matrices. Prescribing a first Hadamard matrix, we construct all others mutually unbiased to it, using algebraic computations performed by a computer program. We repeat this calculation many times, sampling all known complex Hadamard matrices, and we never find more than two that are mutually unbiased.

Maximal sets of mutually unbiased quantum states in dimension six.

S. Brierley and S. Weigert, Phys. Rev. A 78, 042312 (2008)

We study sets of pure states in dimension d that are mutually unbiased (MU), that is the squared modulus of their inner products is either zero, one or 1/d. We call these sets MU constellations and if four MU bases were to exist in d=6 they would give rise to 35 different MU constellations. Using a minimisation procedure, we are able to identify only 18 of them in spite of extensive searches. The missing MU constellations provide the strongest numerical evidence so far that no seven MU bases exist in dimension six.

On maximal entanglement between two pairs in four-qubit pure states.

S. Brierley and A. Higuchi, J. Phys. A 40, 8455  (2007)

An obvious definition for maximal entanglement is to require that all one qubit reduced density matrices are maximally mixed. For two and three qubit systems this leads to a unique (up to local unitary operations) state, however this is not true for a system with more than 3 qubits. We therefore ask which states also have maximal two party entanglement.