A family of sure-success quantum algorithms for solving a generalized Grover search problem

This work considers a generalization of Grover's search problem, viz., to find any one element in a set of acceptable choices which constitute a fraction f of the total number of choices in an unsorted data base. An infinite family of sure-success quantum algorithms are introduced here to solve this problem, each member for a different range of f. The nth member of this family involves n queries of the data base, and so the lowest few members of this family should be very convenient algorithms within their ranges of validity. The even member {A}_{2n} of the family covers ever larger range of f for larger n, which is expected to become the full range 0 infinity.Comment: 8 pages, including 4 figures in 4 page

Finding Optimal Flows Efficiently

Among the models of quantum computation, the One-way Quantum Computer is one of the most promising proposals of physical realization, and opens new perspectives for parallelization by taking advantage of quantum entanglement. Since a one-way quantum computation is based on quantum measurement, which is a fundamentally nondeterministic evolution, a sufficient condition of global determinism has been introduced as the existence of a causal flow in a graph that underlies the computation. A O(n^3)-algorithm has been introduced for finding such a causal flow when the numbers of output and input vertices in the graph are equal, otherwise no polynomial time algorithm was known for deciding whether a graph has a causal flow or not. Our main contribution is to introduce a O(n^2)-algorithm for finding a causal flow, if any, whatever the numbers of input and output vertices are. This answers the open question stated by Danos and Kashefi and by de Beaudrap. Moreover, we prove that our algorithm produces an optimal flow (flow of minimal depth.) Whereas the existence of a causal flow is a sufficient condition for determinism, it is not a necessary condition. A weaker version of the causal flow, called gflow (generalized flow) has been introduced and has been proved to be a necessary and sufficient condition for a family of deterministic computations. Moreover the depth of the quantum computation is upper bounded by the depth of the gflow. However, the existence of a polynomial time algorithm that finds a gflow has been stated as an open question. In this paper we answer this positively with a polynomial time algorithm that outputs an optimal gflow of a given graph and thus finds an optimal correction strategy to the nondeterministic evolution due to measurements.Comment: 10 pages, 3 figure

Commuting Quantum Circuits with Few Outputs are Unlikely to be Classically Simulatable

We study the classical simulatability of commuting quantum circuits with n input qubits and O(log n) output qubits, where a quantum circuit is classically simulatable if its output probability distribution can be sampled up to an exponentially small additive error in classical polynomial time. First, we show that there exists a commuting quantum circuit that is not classically simulatable unless the polynomial hierarchy collapses to the third level. This is the first formal evidence that a commuting quantum circuit is not classically simulatable even when the number of output qubits is exponentially small. Then, we consider a generalized version of the circuit and clarify the condition under which it is classically simulatable. Lastly, we apply the argument for the above evidence to Clifford circuits in a similar setting and provide evidence that such a circuit augmented by a depth-1 non-Clifford layer is not classically simulatable. These results reveal subtle differences between quantum and classical computation.Comment: 19 pages, 6 figures; v2: Theorems 1 and 3 improved, proofs modifie

Grover Algorithm with zero theoretical failure rate

In standard Grover's algorithm for quantum searching, the probability of finding the marked item is not exactly 1. In this Letter we present a modified version of Grover's algorithm that searches a marked state with full successful rate. The modification is done by replacing the phase inversion by two phase rotation through angle $\phi$. The rotation angle is given analytically to be $\phi=2 \arcsin(\sin{\pi\over (4J+6)}\over \sin\beta)$, where $\sin\beta={1\over \sqrt{N}}$, $N$ the number of items in the database, and $J$ an integer equal to or greater than the integer part of $({\pi\over 2}-\beta)/(2\beta)$. Upon measurement at $(J+1)$-th iteration, the marked state is obtained with certainty.Comment: 5 pages. Accepted for publication in Physical Review

Organochlorine exposures influence on breast cancer risk and survival according to estrogen receptor status: a Danish cohort-nested case-control study

BACKGROUND: The relationship between breast cancer and organochlorine exposure is controversial and complex. As estrogen receptor positive and negative breast cancer may represent different entities of the disease, this study was undertaken to evaluate organochlorines influence on breast cancer risk and survival according to receptor status. METHODS: The background material stems from the Copenhagen City Heart Study (Denmark 1976-78). The breast cancer risk was investigated in a cohort nested case-control design including 161 cases and twice as many breast cancer free controls. The cases served as a cohort in the survival analysis. Serum organochlorine concentrations were determined by gaschromotography. RESULTS: The observed increased breast cancer risk associated with exposure to dieldrin derived from women who developed an estrogen receptor negative (ERN) tumor (Odds ratio [OR] I vs. IV quartile, 7.6, 95% confidence interval [95% CI] 1.4-46.1, p-value for linear trend 0.01). Tumors in women with the highest dieldrin serum level were larger and more often spread at the time of diagnosis than ERP tumors. The risk of dying was for the remaining evaluated compounds higher among patients with ERP breast cancer when compared to those with ERN. In the highest quartile of polychlorinated biphenyls (ΣPCB) it was more than 2-fold increased (Relative risk [RR] I vs. IV quartile, 2.5, 95% CI 1.1-5.7), but no dose-response relation was apparent. CONCLUSION: The results do not suggest that exposure to potential estrogenic organochlorines leads to development of an ERP breast cancer. A possible adverse effect on prognosis of hormone-responsive breast cancers needs to be clarified

An Algorithmic Argument for Nonadaptive Query Complexity Lower Bounds on Advised Quantum Computation

This paper employs a powerful argument, called an algorithmic argument, to prove lower bounds of the quantum query complexity of a multiple-block ordered search problem in which, given a block number i, we are to find a location of a target keyword in an ordered list of the i-th block. Apart from much studied polynomial and adversary methods for quantum query complexity lower bounds, our argument shows that the multiple-block ordered search needs a large number of nonadaptive oracle queries on a black-box model of quantum computation that is also supplemented with advice. Our argument is also applied to the notions of computational complexity theory: quantum truth-table reducibility and quantum truth-table autoreducibility.Comment: 16 pages. An extended abstract will appear in the Proceedings of the 29th International Symposium on Mathematical Foundations of Computer Science, Lecture Notes in Computer Science, Springer-Verlag, Prague, August 22-27, 200

Fast motion-including dose error reconstruction for VMAT with and without MLC tracking

Multileaf collimator (MLC) tracking is a promising and clinically emerging treatment modality for radiotherapy of mobile tumours. Still, new quality assurance (QA) methods are warranted to safely introduce MLC tracking in the clinic. The purpose of this study was to create and experimentally validate a simple model for fast motion-including dose error reconstruction applicable to intrafractional QA of MLC tracking treatments of moving targets.MLC tracking experiments were performed on a standard linear accelerator with prototype MLC tracking software guided by an electromagnetic transponder system. A three-axis motion stage reproduced eight representative tumour trajectories; four lung and four prostate. Low and high modulation 6 MV single-arc volumetric modulated arc therapy treatment plans were delivered for each trajectory with and without MLC tracking, as well as without motion for reference. Temporally resolved doses were measured during all treatments using a biplanar dosimeter. Offline, the dose delivered to each of 1069 diodes in the dosimeter was reconstructed with 500 ms temporal resolution by a motion-including pencil beam convolution algorithm developed in-house. The accuracy of the algorithm for reconstruction of dose and motion-induced dose errors throughout the tracking and non-tracking beam deliveries was quantified. Doses were reconstructed with a mean dose difference relative to the measurements of-0.5% (5.5% standard deviation) for cumulative dose. More importantly, the root-mean-square deviation between reconstructed and measured motion-induced 3%/3 mm γ failure rates (dose error) was 2.6%. The mean computation time for each calculation of dose and dose error was 295 ms. The motion-including dose reconstruction allows accurate temporal and spatial pinpointing of errors in absorbed dose and is adequately fast to be feasible for online use. An online implementation could allow treatment intervention in case of erroneous dose delivery in both tracking and non-tracking treatments

Quantum rejection sampling

Rejection sampling is a well-known method to sample from a target distribution, given the ability to sample from a given distribution. The method has been first formalized by von Neumann (1951) and has many applications in classical computing. We define a quantum analogue of rejection sampling: given a black box producing a coherent superposition of (possibly unknown) quantum states with some amplitudes, the problem is to prepare a coherent superposition of the same states, albeit with different target amplitudes. The main result of this paper is a tight characterization of the query complexity of this quantum state generation problem. We exhibit an algorithm, which we call quantum rejection sampling, and analyze its cost using semidefinite programming. Our proof of a matching lower bound is based on the automorphism principle which allows to symmetrize any algorithm over the automorphism group of the problem. Our main technical innovation is an extension of the automorphism principle to continuous groups that arise for quantum state generation problems where the oracle encodes unknown quantum states, instead of just classical data. Furthermore, we illustrate how quantum rejection sampling may be used as a primitive in designing quantum algorithms, by providing three different applications. We first show that it was implicitly used in the quantum algorithm for linear systems of equations by Harrow, Hassidim and Lloyd. Secondly, we show that it can be used to speed up the main step in the quantum Metropolis sampling algorithm by Temme et al.. Finally, we derive a new quantum algorithm for the hidden shift problem of an arbitrary Boolean function and relate its query complexity to "water-filling" of the Fourier spectrum.Comment: 19 pages, 5 figures, minor changes and a more compact style (to appear in proceedings of ITCS 2012

Time-resolved dose reconstruction by motion encoding of volumetric modulated arc therapy fields delivered with and without dynamic multi-leaf collimator tracking.

BACKGROUND: Organ motion during treatment delivery in radiotherapy (RT) may lead to deterioration of the planned dose, but can be mitigated by dynamic multi-leaf collimator (DMLC) tracking. The purpose of this study was to implement and experimentally validate a method for time-resolved motion including dose reconstruction for volumetric modulated arc therapy (VMAT) treatments delivered with and without DMLC tracking. MATERIAL AND METHODS: Tracking experiments were carried out on a linear accelerator (Trilogy, Varian) with a prototype DMLC tracking system. A motion stage carrying a biplanar dosimeter phantom (Delta4PT, Scandidos) reproduced eight representative clinical tumor trajectories (four lung, four prostate). For each trajectory, two single-arc 6 MV VMAT treatments with low and high modulation were delivered to the moving phantom with and without DMLC tracking. An existing in-house developed program that adds target motion to treatment plans was extended with the ability to split an arc plan into any number of sub-arcs, allowing the calculated dose for different parts of the treatment to be examined individually. For each VMAT sub-arc, reconstructed and measured doses were compared using dose differences and 3%/3 mm γ-tests. RESULTS: For VMAT sub-arcs the reconstructed dose distributions had a mean root-mean-square (rms) dose difference of 2.1% and mean γ failure rate of 2.0% when compared with the measured doses. For final accumulated doses the mean rms dose difference was 1.6% and the γ failure rate was 0.7%. CONCLUSION: The time-resolved motion including dose reconstruction was experimentally validated for complex tracking and non-tracking treatments with patient-measured tumor motion trajectories. The reconstructed dose will be of high value for evaluation of treatment plan robustness facing organ motion and adaptive RT