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Register a new userHow to efficiently select an arbitrary Clifford group element
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We give an algorithm which produces a unique element of the Clifford group C_n on n qubits from an integer 0le i < |C_n| (the number of elements in the group). The algorithm involves O(n^3) operations. It is a variant of the subgroup algorithm by Diaconis and Shahshahani which is commonly applied to compact Lie groups. We provide an adaption for the symplectic group Sp(2n,F_2) which provides, in addition to a canonical mapping from the integers to group elements g, a factorization of g into a sequence of at most 4n symplectic transvections. The algorithm can be used to efficiently select random elements of C_n which is often useful in quantum information theory and quantum computation. We also give an algorithm for the inverse map, indexing a group element in time O(n^3).
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Quantum key distribution (QKD) promises provably secure cryptography, even to attacks from an all-powerful adversary. However, with quantum computing development lagging behind QKD, the assumption that there exists an adversary equipped with a universal fault-tolerant quantum computer is unrealistic for at least the near future. Here, we explore the effect of restricting the eavesdroppers computational capabilities on the security of QKD, and find that improved secret key rates are possible. Specifically, we show that for a large class of discrete variable protocols higher key rates are possible if the eavesdropper is restricted to a unitary operation from the Clifford group. Further, we consider Clifford-random channels consisting of mixtures of Clifford gates. We numerically calculate a secret key rate lower bound for BB84 with this restriction, and show that in contrast to the case of a single restricted unitary attack, the mixture of Clifford based unitary attacks does not result in an improved key rate.
We construct new families of spin chain Hamiltonians that are local, integrable and translationally invariant. To do so, we make use of the Clifford group that arises in quantum information theory. We consider translation invariant Clifford group transformations that can be described by matrix product operators (MPOs). We classify the translation invariant Clifford group transformations that consist of a shift operator and an MPO of bond dimension two -- this includes transformations that preserve locality of all Hamiltonians; as well as those that lead to non-local images of particular operators but nevertheless preserve locality of certain Hamiltonians. We characterise the translation invariant Clifford group transformations that take single-site Pauli operators to local operators on at most five sites -- examples of Quantum Cellular Automata -- leading to a discrete family of Hamiltonians that are equivalent to the canonical XXZ model under such transformations. For spin chains solvable by algebraic Bethe Ansatz, we explain how conjugating by a matrix product operator affects the underlying integrable structure. This allows us to relate our results to the usual classifications of integrable Hamiltonians. We also treat the case of spin chains solvable by free fermions.
In his 1935 Gedankenexperiment, Erwin Schr{o}dinger imagined a poisonous substance which has a 50% probability of being released, based on the decay of a radioactive atom. As such, the life of the cat and the state of the poison become entangled, and the fate of the cat is determined upon opening the box. We present an experimental technique that keeps the cat alive on any account. This method relies on the time-resolved Hong-Ou-Mandel effect: two long, identical photons impinging on a beam splitter always bunch in either of the outputs. Interpreting the first photon detection as the state of the poison, the second photon is identified as the state of the cat. Even after the collapse of the first photons state, we show their fates are intertwined through quantum interference. We demonstrate this by a sudden phase change between the inputs, administered conditionally on the outcome of the first detection, which steers the second photon to a pre-defined output and ensures that the cat is always observed alive.
We define a general concept of pseudo algebras over theories and 2-theories. A more restrictive such notion was introduced by Hu and Kriz, but as noticed by M. Gould, did not capture the desired examples. The approach taken in this paper corrects the mistake by introducing a more general concept, allowing more flexibility in selecting coherence diagrams for pseudo algebras.
Using condition of relativistic invariance, group theory and Clifford algebra the component Lorentz invariance generalized Dirac equation for a particle with arbitrary mass and spin is suggested, where In the case of half-integral spin particles, this equation is reduced to the sets of two-component independent matrix equations. It is shown that the relativistic scalar and integral spin particles are described by component equation.
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