Generalized self-testing and the security of the 6-state protocol

Self-tested quantum information processing provides a means for doing useful information processing with untrusted quantum apparatus. Previous work was limited to performing computations and protocols in real Hilbert spaces, which is not a serious obstacle if one is only interested in final measurement statistics being correct (for example, getting the correct factors of a large number after running Shor's factoring algorithm). This limitation was shown by McKague et al. to be fundamental, since there is no way to experimentally distinguish any quantum experiment from a special simulation using states and operators with only real coefficients. In this paper, we show that one can still do a meaningful self-test of quantum apparatus with complex amplitudes. In particular, we define a family of simulations of quantum experiments, based on complex conjugation, with two interesting properties. First, we are able to define a self-test which may be passed only by states and operators that are equivalent to simulations within the family. This extends work of Mayers and Yao and Magniez et al. in self-testing of quantum apparatus, and includes a complex measurement. Second, any of the simulations in the family may be used to implement a secure 6-state QKD protocol, which was previously not known to be implementable in a self-tested framework.Comment: To appear in proceedings of TQC 201

Perspective review of what is needed for molecular-specific fluorescence-guided surgery

Molecular image-guided surgery has the potential for translating the tools of molecular pathology to real-time guidance in surgery. As a whole, there are incredibly positive indicators of growth, including the first United States Food and Drug Administration clearance of an enzyme-biosynthetic-activated probe for surgery guidance, and a growing number of companies producing agents and imaging systems. The strengths and opportunities must be continued but are hampered by important weaknesses and threats within the field. A key issue to solve is the inability of macroscopic imaging tools to resolve microscopic biological disease heterogeneity and the limitations in microscopic systems matching surgery workflow. A related issue is that parsing out true molecular-specific uptake from simple-enhanced permeability and retention is hard and requires extensive pathologic analysis or multiple in vivo tests, comparing fluorescence accumulation with standard histopathology and immunohistochemistry. A related concern in the field is the over-reliance on a finite number of chosen preclinical models, leading to early clinical translation when the probe might not be optimized for high intertumor variation or intratumor heterogeneity. The ultimate potential may require multiple probes, as are used in molecular pathology, and a combination with ultrahigh-resolution imaging and image recognition systems, which capture the data at a finer granularity than is possible by the surgeon. Alternatively, one might choose a more generalized approach by developing the tracer based on generic hallmarks of cancer to create a more "one-size-fits-all" concept, similar to metabolic aberrations as exploited in fluorodeoxyglucose-positron emission tomography (FDG-PET) (i.e., Warburg effect) or tumor acidity. Finally, methods to approach the problem of production cost minimization and regulatory approvals in a manner consistent with the potential revenue of the field will be important. In this area, some solid steps have been demonstrated in the use of fluorescent labeling commercial antibodies and separately in microdosing studies with small molecules. (C) The Authors

Quantum algorithm for the Boolean hidden shift problem

The hidden shift problem is a natural place to look for new separations between classical and quantum models of computation. One advantage of this problem is its flexibility, since it can be defined for a whole range of functions and a whole range of underlying groups. In a way, this distinguishes it from the hidden subgroup problem where more stringent requirements about the existence of a periodic subgroup have to be made. And yet, the hidden shift problem proves to be rich enough to capture interesting features of problems of algebraic, geometric, and combinatorial flavor. We present a quantum algorithm to identify the hidden shift for any Boolean function. Using Fourier analysis for Boolean functions we relate the time and query complexity of the algorithm to an intrinsic property of the function, namely its minimum influence. We show that for randomly chosen functions the time complexity of the algorithm is polynomial. Based on this we show an average case exponential separation between classical and quantum time complexity. A perhaps interesting aspect of this work is that, while the extremal case of the Boolean hidden shift problem over so-called bent functions can be reduced to a hidden subgroup problem over an abelian group, the more general case studied here does not seem to allow such a reduction.Comment: 10 pages, 1 figur

Hamiltonian Oracles

Hamiltonian oracles are the continuum limit of the standard unitary quantum oracles. In this limit, the problem of finding the optimal query algorithm can be mapped into the problem of finding shortest paths on a manifold. The study of these shortest paths leads to lower bounds of the original unitary oracle problem. A number of example Hamiltonian oracles are studied in this paper, including oracle interrogation and the problem of computing the XOR of the hidden bits. Both of these problems are related to the study of geodesics on spheres with non-round metrics. For the case of two hidden bits a complete description of the geodesics is given. For n hidden bits a simple lower bound is proven that shows the problems require a query time proportional to n, even in the continuum limit. Finally, the problem of continuous Grover search is reexamined leading to a modest improvement to the protocol of Farhi and Gutmann.Comment: 16 pages, REVTeX 4 (minor corrections in v2

Scaling of running time of quantum adiabatic algorithm for propositional satisfiability

We numerically study quantum adiabatic algorithm for the propositional satisfiability. A new class of previously unknown hard instances is identified among random problems. We numerically find that the running time for such instances grows exponentially with their size. Worst case complexity of quantum adiabatic algorithm therefore seems to be exponential.Comment: 7 page

Adiabatic quantum computation and quantum phase transitions

We analyze the ground state entanglement in a quantum adiabatic evolution algorithm designed to solve the NP-complete Exact Cover problem. The entropy of entanglement seems to obey linear and universal scaling at the point where the mass gap becomes small, suggesting that the system passes near a quantum phase transition. Such a large scaling of entanglement suggests that the effective connectivity of the system diverges as the number of qubits goes to infinity and that this algorithm cannot be efficiently simulated by classical means. On the other hand, entanglement in Grover's algorithm is bounded by a constant.Comment: 5 pages, 4 figures, accepted for publication in PR

Adiabatic Quantum Computation in Open Systems

We analyze the performance of adiabatic quantum computation (AQC) under the effect of decoherence. To this end, we introduce an inherently open-systems approach, based on a recent generalization of the adiabatic approximation. In contrast to closed systems, we show that a system may initially be in an adiabatic regime, but then undergo a transition to a regime where adiabaticity breaks down. As a consequence, the success of AQC depends sensitively on the competition between various pertinent rates, giving rise to optimality criteria.Comment: v2: 4 pages, 1 figure. Published versio

Exponential complexity of an adiabatic algorithm for an NP-complete problem

We prove an analytical expression for the size of the gap between the ground and the first excited state of quantum adiabatic algorithm for the 3-satisfiability, where the initial Hamiltonian is a projector on the subspace complementary to the ground state. For large problem sizes the gap decreases exponentially and as a consequence the required running time is also exponential.Comment: 5 pages, 2 figures; v3. published versio