Quantum verification of matrix products

Buhrman, Harry; Špalek, Robert

doi:10.1145/1109557.1109654

Cited by 129 publications

(133 citation statements)

References 21 publications

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“…For examples, in quantum information science, the DTQW is used as the quantum-speedup algorithms [5,8,52,53]. The Grover algorithm, which is the spatially search algorithm, can be taken as the tool of the DTQW [68].…”

Section: Review Of Discrete Time Quantum Walkmentioning

confidence: 99%

From Discrete Time Quantum Walk to Continuous Time Quantum Walk in Limit Distribution

Shikano¹

2013

Jnl of Comp & Theo Nano

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The discrete time quantum walk defined as a quantum-mechanical analogue of the discrete time random walk have recently been attracted from various and interdisciplinary fields. In this review, the weak limit theorem, that is, the asymptotic behavior, of the one-dimensional discrete time quantum walk is analytically shown. From the limit distribution of the discrete time quantum walk, the discrete time quantum walk can be taken as the quantum dynamical simulator of some physical systems.

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Section: Review Of Discrete Time Quantum Walkmentioning

confidence: 99%

From Discrete Time Quantum Walk to Continuous Time Quantum Walk in Limit Distribution

Shikano¹

2013

Jnl of Comp & Theo Nano

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“…Optical multiport interferometers provide an attractive implementation for quantum walks, since optical fields naturally exhibit coherence between pathways and allow multi-walker scenarios using multiple single photons. However, this approach is susceptible to losses, which limit the achievable scale before being overtaken by noise, and which abrogate many of the advantages implied for applications in quantum information processing [4][5][6][7][8][9][10][11][12]. This challenge motivates the development of low-loss, modular, guided-wave interferometer networks for quantum walks.…”

Section: Introductionmentioning

confidence: 99%

Large scale quantum walks by means of optical fiber cavities

Boutari

Feizpour

Barz

et al. 2016

J. Opt.

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We demonstrate a platform for implementing quantum walks that overcomes many of the barriers associated with photonic implementations. We use coupled fiber-optic cavities to implement time-bin encoded walks in an integrated system. We show that this platform can achieve very low losses combined with high-fidelity operation, enabling an unprecedented large number of steps in a passive system, as required for scenarios with multiple walkers. Furthermore the platform is reconfigurable, enabling variation of the coin, and readily extends to multidimensional lattices. We demonstrate variation of the coin bias experimentally for three different values. INTRODUCTIONQuantum walks are the quantum counterparts to random walks. A quantum walker moves by superpositions of possible paths, resulting in a probability amplitude for being observed at a particular position [1][2][3]. Optical multiport interferometers provide an attractive implementation for quantum walks, since optical fields naturally exhibit coherence between pathways and allow multi-walker scenarios using multiple single photons. However, this approach is susceptible to losses, which limit the achievable scale before being overtaken by noise, and which abrogate many of the advantages implied for applications in quantum information processing [4][5][6][7][8][9][10][11][12]. This challenge motivates the development of low-loss, modular, guided-wave interferometer networks for quantum walks.Quantum walks with multiple interacting walkers have been shown to realize universal quantum computation [13]. For the case of multiple non-interacting walkers, quantum walks are also thought to have quantum computational power, as shown by the boson sampling problem [14]. Quantum walks with a single walker can implement quantum computation, though this requires an exponentially large graph [15,16]. A key feature of quantum walks as information processors is that they do not require time-dependent feedforward control. Quantum walks, both discrete and continuous, have been realised using cold atoms [17,18], single optically trapped atoms [19], trapped ions [20][21][22], and photons. Optical systems have been used to implement quantum walks using bulk optics [23,24], photonic chips [25][26][27][28], fiber optics [29], and hybrid bulkfiber optic approaches [30][31][32].In most experimental implementations, the quantum walk takes place over spatial locations arranged in a lattice. The physical size of the lattice then determines the maximum size of the walk. This limitation can be avoided by use of optical cavities, in which case the number of physical elements required is independent of the size of the walk. This approach was first formulated for frequency-encoded quantum walks [33]. From an experimental perspective, a further key advance was the development of optical cavities that implement time-encoded walks [29,30]. In this case, the walker's location is represented by the time at which a pulse completes a round trip of a cavity [29,30]. In practice, the achievable scale of t...

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“…In [7], it was shown that the quantum walk on the so-called "glued trees" graph reaches the final vertex from the initial vertex exponentially faster that a similar classical walk. Other algorithms based on quantum walks include matrix product verification [8], triangle finding [9] and group commutativity testing [10].…”

Section: Introductionmentioning

confidence: 99%

Quantum walks on quotient graphs

2007

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A discrete-time quantum walk on a graph Γ is the repeated application of a unitary evolution operator to a Hilbert space corresponding to the graph. If this unitary evolution operator has an associated group of symmetries, then for certain initial states the walk will be confined to a subspace of the original Hilbert space. Symmetries of the original graph, given by its automorphism group, can be inherited by the evolution operator. We show that a quantum walk confined to the subspace corresponding to this symmetry group can be seen as a different quantum walk on a smaller quotient graph. We give an explicit construction of the quotient graph for any subgroup H of the automorphism group and illustrate it with examples. The automorphisms of the quotient graph which are inherited from the original graph are the original automorphism group modulo the subgroup H used to construct it. The quotient graph is constructed by removing the symmetries of the subgroup H from the original graph. We then analyze the behavior of hitting times on quotient graphs. Hitting time is the average time it takes a walk to reach a given final vertex from a given initial vertex. It has been shown in earlier work [Phys. Rev. A 74, 042334 (2006)] that the hitting time for certain initial states of a quantum walks can be infinite, in contrast to classical random walks. We give a condition which determines whether the quotient graph has infinite hitting times given that they exist in the original graph. We apply this condition for the examples discussed and determine which quotient graphs have infinite hitting times. All known examples of quantum walks with hitting times which are short compared to classical random walks correspond to systems with quotient graphs much smaller than the original graph; we conjecture that the existence of a small quotient graph with finite hitting times is necessary for a walk to exhibit a quantum speed-up.

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Quantum verification of matrix products

Cited by 129 publications

References 21 publications

From Discrete Time Quantum Walk to Continuous Time Quantum Walk in Limit Distribution

From Discrete Time Quantum Walk to Continuous Time Quantum Walk in Limit Distribution

Large scale quantum walks by means of optical fiber cavities

Quantum walks on quotient graphs

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