Global Symmetry is Unnecessary for Fast Quantum Search

Janmark, Jonatan; Meyer, David; Wong, Thomas G.

doi:10.1103/physrevlett.112.210502

Cited by 98 publications

(151 citation statements)

References 6 publications

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“…When γ takes its critical value of 1/N [14,20,21], evolving by Schrödinger's equation with this Hamiltonian yields the success probability shown in figure 3. We see that it reaches a maximum value of 1, and the runtime (which should be Grover's Θ( √ N )) scales better than linear (classical) since doubling N less than doubles the runtime.…”

Section: Grover's Algorithm As a Continuous-time Quantum Walkmentioning

confidence: 99%

Grover search with lackadaisical quantum walks

Wong¹

2015

J. Phys. A: Math. Theor.

106

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Abstract. The lazy random walk, where the walker has some probability of staying put, is a useful tool in classical algorithms. We propose a quantum analogue, the lackadaisical quantum walk, where each vertex is given l self-loops, and we investigate its effects on Grover's algorithm when formulated as search for a marked vertex on the complete graph of N vertices. For the discrete-time quantum walk using the phase flip coin, adding a self-loop to each vertex boosts the success probability from 1/2 to 1. Additional self-loops, however, decrease the success probability. Using instead the Shenvi, Kempe, and Whaley (2003) coin, adding self-loops simply slows down the search. These coins also differ in that the first is faster than classical when l scales less than N , while the second requires that l scale less than N 2 . Finally, continuous-time quantum walks differ from both of these discrete-time examples-the self-loops make no difference at all. These behaviors generalize to multiple marked vertices.

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Section: Grover's Algorithm As a Continuous-time Quantum Walkmentioning

confidence: 99%

Grover search with lackadaisical quantum walks

Wong¹

2015

J. Phys. A: Math. Theor.

106

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“…So the system evolves from |s to |a in time π/∆E = π √ N /2 = Θ( √ N ) [4]. This can also be proved using degenerate perturbation theory [2], as we show rigorously for the next two examples in [9], but in this paper we use the same graphical explanation as above.…”

mentioning

confidence: 93%

“…5 shows the squared overlaps of |s and |a with the eigenstates of H. For large N , γ takes its critical value of γ c = 1/(N/2), at which half of |s is proportional to |ψ 0 + |ψ 2 (with the other half in |ψ 1 ) and |a ∝ |ψ 0 − |ψ 2 with an energy gap of E 2 − E 0 = 2/ N/2 [9]. Comparing this to (2), this is the same as searching on a complete graph with N/2 vertices and total probability 1/2, which proves that the success probability reaches 1/2 in time π N/2/2. This can be seen in Fig.…”

Section: Graphmentioning

confidence: 99%

“…Regarding specific graphs, a randomly walking quantum particle evolving by Schrödinger's equation searches on the complete graph, strongly regular graphs, and the hypercube in optimal Θ( √ N ) time, the first of which is precisely the continous-time analogue of Grover's algorithm [1][2][3][4]. Examples of these graphs are shown in Fig.…”

mentioning

confidence: 99%

“…Nevertheless, much work has been done to further our understanding. For example, we recently showed that global symmetry is unnecessary for fast quantum search [2].…”

mentioning

confidence: 99%

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Connectivity is a Poor Indicator of Fast Quantum Search

Meyer

Wong

2015

Phys. Rev. Lett.

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A randomly walking quantum particle evolving by Schrödinger's equation searches on ddimensional cubic lattices in O( √ N ) time when d ≥ 5, and with progressively slower runtime as d decreases. This suggests that graph connectivity (including vertex, edge, algebraic, and normalized algebraic connectivities) is an indicator of fast quantum search, a belief supported by fast quantum search on complete graphs, strongly regular graphs, and hypercubes, all of which are highly connected. In this paper, we show this intuition to be false by giving two examples of graphs for which the opposite holds true: one with low connectivity but fast search, and one with high connectivity but slow search. The second example is a novel two-stage quantum walk algorithm in which the walking rate must be adjusted to yield high search probability. Introduction.-Despite ten years elapsing since the introduction of continuous-time quantum walk algorithms that search on graphs [1], there is still no comprehensive theory as to which graphs support fast quantum search. Nevertheless, much work has been done to further our understanding. For example, we recently showed that global symmetry is unnecessary for fast quantum search [2].Regarding specific graphs, a randomly walking quantum particle evolving by Schrödinger's equation searches on the complete graph, strongly regular graphs, and the hypercube in optimal Θ( √ N ) time, the first of which is precisely the continous-time analogue of Grover's algorithm [1][2][3][4]. Examples of these graphs are shown in Fig. 1. Additionally, such a particle can search on ddimensional cubic lattices in Θ( √ N ) total time when d ≥ 5, and with progressively slower runtimes as d decreases [1,5,6], as shown in Table I.One might suspect that fast search occurs when graphs are highly connected, as suggested by [1]. In this paper, however, we show this intuition to be false by giving two examples of graphs for which the opposite holds true: one with low connectivity but fast search, and one with high connectivity but slow search; they are shown in Figs. 2 and 3, respectively. To do this, we first introduce four different ways to measure graph connectivity. Then we detail how a randomly walking quantum particle searches on a graph. Finally, we determine the runtimes of our two examples.Measures of Connectivity.-The two most common ways to measure connectivity are vertex connectivity and edge connectivity, which are how many vertices or edges must be removed to make a graph disconnected. For example, Fig. 2 has vertex and edge connectivities of 1 because removing the yellow or green vertex disconnects the graph, and so does removing the edge between them. Note that vertex connectivity is upper bounded by the

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Electric‐Circuit Simulation of Quantum Fast Hitting with Exponential Speedup

et al. 2022

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The ultimate goal of developing quantum algorithms and constructing quantum computers is to achieve faster information processing than using current classical computers. Quantum walks are powerful kernels in quantum computing protocols and possess strong capabilities in speeding up various simulation and optimization tasks. One striking example is provided by quantum walkers evolving on unbalanced trees, which demonstrate faster hitting performances than classical random walk. However, direct experimental construction of unbalanced trees to prove quantum advantage with exponential speedup remains a great challenge due to the highly complex arrangements of the structure. This study attempts to simulate quantum algorithm by classical circuit. Inspired by the quantum algorithm, the classical circuit networks are designed and fabricated with unbalanced tree structures. It is then demonstrated, both theoretically and experimentally, that the quantum algorithm for the fast hitting problem can be simulated in the structure. It is shown that the hitting efficiency of electric signals in the circuit networks with unbalanced tree structures is exponentially faster than the corresponding cases of classical random walks. Because classical circuit networks possess good scalability and stability, the results open up a scalable new path toward quantum speedup in complex problems.

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Global Symmetry is Unnecessary for Fast Quantum Search

Cited by 98 publications

References 6 publications

Grover search with lackadaisical quantum walks

Grover search with lackadaisical quantum walks

Connectivity is a Poor Indicator of Fast Quantum Search

Electric‐Circuit Simulation of Quantum Fast Hitting with Exponential Speedup

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