Kerdock Codes Determine Unitary 2-Designs

The non-linear binary Kerdock codes are known to be Gray images of certain extended cyclic codes of length $N = 2^m$ over $\mathbb{Z}_4$. We show that exponentiating these $\mathbb{Z}_4$-valued codewords by $\imath \triangleq \sqrt{-1}$ produces stabilizer states, that are quantum states obtained using only Clifford unitaries. These states are also the common eigenvectors of commuting Hermitian matrices forming maximal commutative subgroups (MCS) of the Pauli group. We use this quantum description to simplify the derivation of the classical weight distribution of Kerdock codes. Next, we organize the stabilizer states to form $N+1$ mutually unbiased bases and prove that automorphisms of the Kerdock code permute their corresponding MCS, thereby forming a subgroup of the Clifford group. When represented as symplectic matrices, this subgroup is isomorphic to the projective special linear group PSL($2,N$). We show that this automorphism group acts transitively on the Pauli matrices, which implies that the ensemble is Pauli mixing and hence forms a unitary $2$-design. The Kerdock design described here was originally discovered by Cleve et al. (arXiv:1501.04592), but the connection to classical codes is new which simplifies its description and translation to circuits significantly. Sampling from the design is straightforward, the translation to circuits uses only Clifford gates, and the process does not require ancillary qubits. Finally, we also develop algorithms for optimizing the synthesis of unitary $2$-designs on encoded qubits, i.e., to construct logical unitary $2$-designs. Software implementations are available at https://github.com/nrenga/symplectic-arxiv18a, which we use to provide empirical gate complexities for up to $16$ qubits.Comment: 16 pages double-column, 4 figures, and some circuits. Accepted to 2019 Intl. Symp. Inf. Theory (ISIT), and PDF of the 5-page ISIT version is included in the arXiv packag

Study of axial strain induced torsion of single wall carbon nanotubes by 2D continuum anharmonic anisotropic elastic model

Recent molecular dynamic simulations have found chiral single wall carbon nanotubes (SWNTs) twist during stretching, which is similar to the motion of a screw. Obviously this phenomenon, as a type of curvature-chirality effect, can not be explained by usual isotropic elastic theory of SWNT. More interestingly, with larger axial strains (before buckling), the axial strain induced torsion (a-SIT) shows asymmetric behaviors for axial tensile and compressing strains, which suggests anharmonic elasticity of SWNTs plays an important role in real a-SIT responses. In order to study the a-SIT of chiral SWNTs with actual sizes, and avoid possible deviations of computer simulation results due to the finite-size effect, we propose a 2D analytical continuum model which can be used to describe the the SWNTs of arbitrary chiralities, curvatures, and lengths, with the concerning of anisotropic and anharmonic elasticity of SWNTs. This elastic energy of present model comes from the continuum limit of lattice energy based on Second Generation Reactive Empirical Bond Order potential (REBO-II), a well-established empirical potential for solid carbons. Our model has no adjustable parameters, except for those presented in REBO-II, and all the coefficients in the model can be calculated analytically. Using our method, we obtain a-SIT responses of chiral SWNTs with arbitrary radius, chiralities and lengthes. Our results are in reasonable agreement with recent molecular dynamic simulations. [Liang {\it et. al}, Phys. Rev. Lett, ${\bf 96}$, 165501 (2006).] Our approach can also be used to calculate other curvature-chirality dependent anharmonic mechanic responses of SWNTs.Comment: 14 pages, 2 figure

Experimental detection of quantum coherent evolution through the violation of Leggett-Garg-type inequalities

We discuss the use of inequalities of the Leggett-Garg type (LGtI) to witness quantum coherence and present the first experimental violation of this type of inequalities using a light-matter interfaced system. By separately benchmarking the Markovian character of the evolution and the translational invariance of the conditional probabilities, the observed violation of a LGtI is attributed to the quantum coherent character of the process. These results provide a general method to benchmark `quantumness' when the absence of memory effects can be independently certified and confirm the persistence of quantum coherent features within systems of increasing complexity.Comment: published version, including supplementary materia

Quantum computation with un-tunable couplings

Most quantum computer realizations require the ability to apply local fields and tune the couplings between qubits, in order to realize single bit and two bit gates which are necessary for universal quantum computation. We present a scheme to remove the necessity of switching the couplings between qubits for two bit gates, which are more costly in many cases. Our strategy is to compute in and out of carefully designed interaction free subspaces analogous to decoherence free subspaces, which allows us to effectively turn off and turn on the interactions between the encoded qubits. We give two examples to show how universal quantum computation is realized in our scheme with local manipulations to physical qubits only, for both diagonal and off diagonal interactions.Comment: 5 pages, 2 figure

Revisiting Bohr's principle of complementarity using a quantum device

Bohr's principle of complementarity lies at the central place of quantum mechanics, according to which the light is chosen to behave as a wave or particles, depending on some exclusive detecting devices. Later, intermediate cases are found, but the total information of the wave-like and particle-like behaviors are limited by some inequalities. One of them is Englert-Greenberger (EG) duality relation. This relation has been demonstrated by many experiments with the classical detecting devices. Here by introducing a quantum detecting device into the experiment, we find the limit of the duality relation is exceeded due to the interference between the photon's wave and particle properties. However, our further results show that this experiment still obey a generalized EG duality relation. The introducing of the quantum device causes the new phenomenon, provides an generalization of the complementarity principle, and opens new insights into our understanding of quantum mechanics.Comment: 5 pages, 4 figure

Linear optical quantum computation with imperfect entangled photon-pair sources and inefficient non-photon-number-resolving detectors

We propose a scheme for efficient cluster state quantum computation by using imperfect polarization-entangled photon-pair sources, linear optical elements and inefficient non-photon-number-resolving detectors. The efficiency threshold for loss tolerance in our scheme requires the product of source and detector efficiencies should be >1/2 - the best known figure. This figure applies to uncorrelated loss. We further find that the loss threshold is unaffected by correlated loss in the photon pair source. Our approach sheds new light on efficient linear optical quantum computation with imperfect experimental conditions.Comment: 5 pages, 2 figure

State estimation from pair of conjugate qudits

We show that, for $N$ parallel input states, an anti-linear map with respect to a specific basis is essentially a classical operator. We also consider the information contained in phase-conjugate pairs $|\phi > |\phi^*>$, and prove that there is more information about a quantum state encoded in phase-conjugate pairs than in parallel pairs.Comment: 4 pages, 1 tabl