Normal Forms for Skew-Symmetric Matrices and Hamiltonian Systems with First Integrals Linear in Momenta

Thompson, G.

doi:10.2307/2046815

Cited by 2 publications

(2 citation statements)

References 4 publications

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“…Since the d × d-matricesĀ i are antisymmetric, one can infer from the results in [66] that there are orthogonal transformations R i ∈ SO(d) such that…”

Section: A Generator Normal Form For All Dimensions D ≥mentioning

confidence: 99%

Quantum computation is the unique reversible circuit model for which bits are balls

Krumm

Müller

2019

npj Quantum Inf

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The computational efficiency of quantum mechanics can be defined in terms of the qubit circuit model, which is characterized by a few simple properties: each computational gate is a reversible transformation in a connected matrix group; single wires carry quantum bits, i.e. states of a threedimensional Bloch ball; states on two or more wires are uniquely determined by local measurement statistics and their correlations. In this paper, we ask whether other types of computation are possible if we relax one of those characteristics (and keep all others), namely, if we allow wires to be described by d-dimensional Bloch balls, where d is different from three. Theories of this kind have previously been proposed as possible generalizations of quantum physics, and it has been conjectured that some of them allow for interesting multipartite reversible transformations that cannot be realized within quantum theory. However, here we show that all such potential beyond-quantum models of computation are trivial: if d is not three, then the set of reversible transformations consists entirely of single-bit gates, and not even classical computation is possible. In this sense, qubit quantum computation is an island in theoryspace.that tomographic locality forces computations to be contained in a class called AWPP [15,16], and that in some theories (satisfying additional axioms) higher-order interference does not lead to a speed-up in Grover's algorithm [17]. Further examples can be found e.g. in [18][19][20][21].

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“…Since the d × d-matricesĀ i are antisymmetric, one can infer from the results in [66] that there are orthogonal transformations R i ∈ SO(d) such that…”

Section: A Generator Normal Form For All Dimensions D ≥mentioning

confidence: 99%

Quantum computation is the unique reversible circuit model for which bits are balls

Krumm

Müller

2019

npj Quantum Inf

View full text Add to dashboard Cite

show abstract

“…By conjugating by an orthogonal matrix O we can map M to a matrix of the form   0 −α 0 α 0 0 0 0 0   for some real α (see e.g. [56] for a proof). By permuting rows and columns, we can assume that O ∈ SO(3) as before.…”

Section: Lemma 8 (Restated) Let H Be a Traceless 2-qubit Hermitian Mmentioning

confidence: 99%

Complexity Classification of Local Hamiltonian Problems

Cubitt

Montanaro

2014

2014 IEEE 55th Annual Symposium on Foundations of Computer Science

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The calculation of ground-state energies of physical systems can be formalised as the k-local Hamiltonian problem, which is a natural quantum analogue of classical constraint satisfaction problems. One way of making the problem more physically meaningful is to restrict the Hamiltonian in question by picking its terms from a fixed set S, and scaling them by arbitrary weights. Examples of such special cases are the Heisenberg and Ising models from condensed-matter physics.In this work we characterise the complexity of this problem for all 2-local qubit Hamiltonians. Depending on the subset S, the problem falls into one of the following categories: in P; NPcomplete; polynomial-time equivalent to the Ising model with transverse magnetic fields; or QMA-complete. The third of these classes has been shown to be StoqMA-complete by Bravyi and Hastings. The characterisation holds even if S does not contain any 1-local terms; for example, we prove for the first time QMA-completeness of the Heisenberg and XY interactions in this setting. If S is assumed to contain all 1-local terms, which is the setting considered by previous work, we have a characterisation that goes beyond 2-local interactions: for any constant k, all k-local qubit Hamiltonians whose terms are picked from a fixed set S correspond to problems either in P; polynomial-time equivalent to the Ising model with transverse magnetic fields; or QMA-complete.These results are a quantum analogue of the maximisation variant of Schaefer's dichotomy theorem for boolean constraint satisfaction problems.

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Normal Forms for Skew-Symmetric Matrices and Hamiltonian Systems with First Integrals Linear in Momenta

Cited by 2 publications

References 4 publications

Quantum computation is the unique reversible circuit model for which bits are balls

Quantum computation is the unique reversible circuit model for which bits are balls

Complexity Classification of Local Hamiltonian Problems

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