scholarly journals Space-efficient classical and quantum algorithms for the shortest vector problem

2018 ◽  
Vol 18 (3&4) ◽  
pp. 283-305
Author(s):  
Yanlin Chen ◽  
Kai-Min Chung ◽  
Ching-Yi Lai
Keyword(s):  
Efficient Algorithm ◽  
Coding Theory ◽  
Quantum Computer ◽  
Quantum Algorithms ◽  
Nonzero Vector ◽  
Vector Problem ◽  
Shortest Vector Problem ◽  
Linearly Independent ◽  
Classical Algorithm ◽  
Quantum Version

A lattice is the integer span of some linearly independent vectors. Lattice problems have many significant applications in coding theory and cryptographic systems for their conjectured hardness. The Shortest Vector Problem (SVP), which asks to find a shortest nonzero vector in a lattice, is one of the well-known problems that are believed to be hard to solve, even with a quantum computer. In this paper we propose space-efficient classical and quantum algorithms for solving SVP. Currently the best time-efficient algorithm for solving SVP takes 2^{n+o(n)} time and 2^{n+o(n)} space. Our classical algorithm takes 2^{2.05n+o(n)} time to solve SVP and it requires only 2^{0.5n+o(n)} space. We then adapt our classical algorithm to a quantum version, which can solve SVP in time 2^{1.2553n+o(n)} with 2^{0.5n+o(n)} classical space and only poly(n) qubits.

scholarly journals An Efficient Algorithm for the Shortest Vector Problem

IEEE Access ◽  
2018 ◽  
Vol 6 ◽  
pp. 61478-61487 ◽  
Author(s):  
Yu-Lun Chuang ◽  
Chun-I Fan ◽  
Yi-Fan Tseng
Keyword(s):  
Efficient Algorithm ◽  
Vector Problem ◽  
Shortest Vector Problem

scholarly journals A random walk approach to quantum algorithms

2006 ◽  
Vol 364 (1849) ◽  
pp. 3407-3422 ◽  
Author(s):  
Vivien M Kendon
Keyword(s):  
Random Walk ◽  
Quantum Dynamics ◽  
Quantum Computer ◽  
Quantum Algorithms ◽  
Quantum Walk ◽  
Quantum Random Walk ◽  
Small Quantum ◽  
Starting Point ◽  
Quantum Version ◽  
Speed Up

The development of quantum algorithms based on quantum versions of random walks is placed in the context of the emerging field of quantum computing. Constructing a suitable quantum version of a random walk is not trivial; pure quantum dynamics is deterministic, so randomness only enters during the measurement phase, i.e. when converting the quantum information into classical information. The outcome of a quantum random walk is very different from the corresponding classical random walk owing to the interference between the different possible paths. The upshot is that quantum walkers find themselves further from their starting point than a classical walker on average, and this forms the basis of a quantum speed up, which can be exploited to solve problems faster. Surprisingly, the effect of making the walk slightly less than perfectly quantum can optimize the properties of the quantum walk for algorithmic applications. Looking to the future, even with a small quantum computer available, the development of quantum walk algorithms might proceed more rapidly than it has, especially for solving real problems.

scholarly journals Parameterized Intractability of Even Set and Shortest Vector Problem

2021 ◽  
Vol 68 (3) ◽  
pp. 1-40
Author(s):  
Arnab Bhattacharyya ◽  
Édouard Bonnet ◽  
László Egri ◽  
Suprovat Ghoshal ◽  
Karthik C. S. ◽  
...  
Keyword(s):  
Linear Codes ◽  
Constant Factor ◽  
Nonzero Vector ◽  
Fixed Parameter Tractable ◽  
Vector Problem ◽  
Shortest Vector Problem ◽  
Open Questions ◽  
Fixed Parameter ◽  
Distance Problem ◽  
Open Question

The -Even Set problem is a parameterized variant of the Minimum Distance Problem of linear codes over , which can be stated as follows: given a generator matrix and an integer , determine whether the code generated by has distance at most , or, in other words, whether there is a nonzero vector such that has at most nonzero coordinates. The question of whether -Even Set is fixed parameter tractable (FPT) parameterized by the distance has been repeatedly raised in the literature; in fact, it is one of the few remaining open questions from the seminal book of Downey and Fellows [1999]. In this work, we show that -Even Set is W [1]-hard under randomized reductions. We also consider the parameterized -Shortest Vector Problem (SVP) , in which we are given a lattice whose basis vectors are integral and an integer , and the goal is to determine whether the norm of the shortest vector (in the norm for some fixed ) is at most . Similar to -Even Set, understanding the complexity of this problem is also a long-standing open question in the field of Parameterized Complexity. We show that, for any , -SVP is W [1]-hard to approximate (under randomized reductions) to some constant factor.

scholarly journals Quantum-assisted quantum compiling

Quantum ◽  
2019 ◽  
Vol 3 ◽  
pp. 140 ◽  
Author(s):  
Sumeet Khatri ◽  
Ryan LaRose ◽  
Alexander Poremba ◽  
Lukasz Cincio ◽  
Andrew T. Sornborger ◽  
...  
Keyword(s):  
Cost Function ◽  
Quantum Dynamics ◽  
Quantum Computer ◽  
Quantum Algorithms ◽  
Quantum Computers ◽  
Problem Size ◽  
Noise Mitigation ◽  
Classical Algorithm ◽  
Noise Resilience ◽  
Near Term

Compiling quantum algorithms for near-term quantum computers (accounting for connectivity and native gate alphabets) is a major challenge that has received significant attention both by industry and academia. Avoiding the exponential overhead of classical simulation of quantum dynamics will allow compilation of larger algorithms, and a strategy for this is to evaluate an algorithm's cost on a quantum computer. To this end, we propose a variational hybrid quantum-classical algorithm called quantum-assisted quantum compiling (QAQC). In QAQC, we use the overlap between a target unitaryUand a trainable unitaryVas the cost function to be evaluated on the quantum computer. More precisely, to ensure that QAQC scales well with problem size, our cost involves not only the global overlapTr(V†U)but also the local overlaps with respect to individual qubits. We introduce novel short-depth quantum circuits to quantify the terms in our cost function, and we prove that our cost cannot be efficiently approximated with a classical algorithm under reasonable complexity assumptions. We present both gradient-free and gradient-based approaches to minimizing this cost. As a demonstration of QAQC, we compile various one-qubit gates on IBM's and Rigetti's quantum computers into their respective native gate alphabets. Furthermore, we successfully simulate QAQC up to a problem size of 9 qubits, and these simulations highlight both the scalability of our cost function as well as the noise resilience of QAQC. Future applications of QAQC include algorithm depth compression, black-box compiling, noise mitigation, and benchmarking.

An Efficient Algorithm for Optimally Solving a Shortest Vector Problem in Compute-and-Forward Design

2016 ◽  
Vol 15 (10) ◽  
pp. 6541-6555 ◽  
Author(s):  
Jinming Wen ◽  
Baojian Zhou ◽  
Wai Ho Mow ◽  
Xiao-Wen Chang
Keyword(s):  
Efficient Algorithm ◽  
Vector Problem ◽  
Shortest Vector Problem

scholarly journals Practical Quantum Computing

2021 ◽  
Vol 2 (1) ◽  
pp. 1-35
Author(s):  
Adrien Suau ◽  
Gabriel Staffelbach ◽  
Henri Calandra
Keyword(s):  
Wave Equation ◽  
Quantum Computer ◽  
Quantum Algorithms ◽  
Quantum Algorithm ◽  
Quantum Circuit ◽  
Equation Solving ◽  
Small Quantum ◽  
Quantum Wave ◽  
Variational Algorithms ◽  
The One

In the last few years, several quantum algorithms that try to address the problem of partial differential equation solving have been devised: on the one hand, “direct” quantum algorithms that aim at encoding the solution of the PDE by executing one large quantum circuit; on the other hand, variational algorithms that approximate the solution of the PDE by executing several small quantum circuits and making profit of classical optimisers. In this work, we propose an experimental study of the costs (in terms of gate number and execution time on a idealised hardware created from realistic gate data) associated with one of the “direct” quantum algorithm: the wave equation solver devised in [32]. We show that our implementation of the quantum wave equation solver agrees with the theoretical big-O complexity of the algorithm. We also explain in great detail the implementation steps and discuss some possibilities of improvements. Finally, our implementation proves experimentally that some PDE can be solved on a quantum computer, even if the direct quantum algorithm chosen will require error-corrected quantum chips, which are not believed to be available in the short-term.

Quantum Algorithms of the Vertex Cover Problem on a Quantum Computer

2009 ◽  
Author(s):  
Weng-Long Chang ◽  
Ting-Ting Ren ◽  
Mang Feng ◽  
Shu-Chien Huang ◽  
Lai Chin Lu ◽  
...  
Keyword(s):  
Quantum Computer ◽  
Quantum Algorithms ◽  
Vertex Cover ◽  
Vertex Cover Problem ◽  
Cover Problem
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