Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Log inSign up
Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Blog
Job Board
Q3AS

© 2026 Aqora Quantum S.A.S.

TermsPrivacyLegal Notice

arXiv:1504.07375 · Journal of Physics A Mathematical and Theoretical

Quantum walk search with time-reversal symmetry breaking

Thomas G. Wong

Published Sep 14, 2015• 17 citations
arXivHTMLPDF

Abstract

We formulate Grover's unstructured search algorithm as a chiral quantum walk, where transitioning in one direction has a phase conjugate to transitioning in the opposite direction. For small phases, this breaking of time-reversal symmetry is too small to significantly affect the evolution: the system still approximately evolves in its ground and first excited states, rotating to the marked vertex in time $π\sqrt{N} / 2$. Increasing the phase does not change the runtime, but rather changes the support for the 2D subspace, so the system evolves in its first and second excited states, or its second and third excited states, and so forth. Apart from the critical phases corresponding to these transitions in the support, which become more frequent as the phase grows, this reveals that our model of quantum search is robust against time-reversal symmetry breaking.

DOI
10.1088/1751-8113/48/40/405303

Related papers

Quantum walk search through potential barriers

Thomas G Wong

Abstract An ideal quantum walk transitions from one vertex to another with perfect fidelity, but in physical systems, the particle may be hindered by potential energy barriers. Then the particle has some amplitude of tunneling through the barriers, and some amplitude of staying put. We investigate the algorithmic consequence of such barriers for the quantum walk formulation of Grover’s algorithm. We prove that the failure amplitude must scale as O ( 1 / N ) for search to retain its quantum O ( N ) runtime; otherwise, it searches in classical O ( N ) time. Thus searching larger ‘databases’ requires increasingly reliable hop operations or error correction. This condition holds for both discrete- and continuous-time quantum walks.

arXiv:1503.06605•Journal of Physics A Mathematical and Theoretical•Nov 9, 2016•11 citations

Global Symmetry is Unnecessary for Fast Quantum Search

Jonatan Janmark, DAVID A. MEYER, Thomas G. Wong

Grover's quantum search algorithm can be formulated as a quantum particle randomly walking on the (highly symmetric) complete graph, with one vertex marked by a nonzero potential. From an initial equal superposition, the state evolves in a two-dimensional subspace. Strongly regular graphs have a local symmetry that ensures that the state evolves in a three-dimensional subspace but most have no global symmetry. Using degenerate perturbation theory, we show that quantum random walk search on known families of strongly regular graphs, nevertheless, achieves the full quantum speed-up of $\mathrm{\ensuremath{\Theta}}(\sqrt{N})$, disproving the intuition that fast quantum search requires global symmetry.

arXiv:1403.2228•Physical Review Letters•May 28, 2014•87 citations

Correcting for Potential Barriers in Quantum Walk Search

Andris Ambainis, Thomas G. Wong

A randomly walking quantum particle searches in Grover's $Θ(\sqrt{N})$ iterations for a marked vertex on the complete graph of $N$ vertices by repeatedly querying an oracle that flips the amplitude at the marked vertex, scattering by a "coin" flip, and hopping. Physically, however, potential energy barriers can hinder the hop and cause the search to fail, even when the amplitude of not hopping decreases with $N$. We correct for these errors by interpreting the quantum walk search as an amplitude amplification algorithm and modifying the phases applied by the coin flip and oracle such that the amplification recovers the $Θ(\sqrt{N})$ runtime.

arXiv:1505.02035•May 8, 2015•1 citation

Grover search with lackadaisical quantum walks

Thomas G. Wong

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.

arXiv:1502.04567•Journal of Physics A Mathematical and Theoretical•Oct 7, 2015•94 citations

Correcting for potential barriers in quantum walk search

Andris Ambainis, Thomas G. Wong

A randomly walking quantum particle searches in Grover’s Θ(√ N) iterations for a marked vertex on the complete graph of N vertices by repeatedly querying an oracle that flips the amplitude at the marked vertex, scattering by a “coin” flip, and hopping. Physically, however, potential energy barriers can hinder the hop and cause the search to fail, even when the amplitude of not hopping decreases with N. We correct for these errors by interpreting the quantum walk search as an amplitude amplification algorithm and modifying the phases applied by the coin flip and oracle such that the amplification recovers the Θ(√ N) runtime.

Quantum Information and Computation•Nov 1, 2015•2 citations

Topological classification of time-asymmetry in unitary quantum processes

Jacob Turner, Jacob Biamonte

Abstract Understanding which physical processes are symmetric with respect to time inversion is a ubiquitous problem in physics. In quantum physics, effective gauge fields allow emulation of matter under strong magnetic fields, realizing the Harper–Hofstadter, the Haldane models, demonstrating one-way waveguides and topologically protected edge states. Central to these discoveries is the chirality induced by time-symmetry breaking. In quantum walk algorithms, recent work has discovered implications time-reversal symmetry breaking has on the transport of quantum states which has enabled a host of new experimental implementations. We provide a full topological classification of Hamiltonian operators that do not exhibit symmetry under time-reversal with respect to the induced transition probabilities between elements in a preferred site-basis, i.e. the nodes of the graph on which the walk takes place. We prove that a quantum process is necessarily time-symmetric for any choice of time-independent Hamiltonian precisely when the underlying support graph is bipartite or no Aharonov–Bohm phases are present in the gauge field. We further prove that certain bipartite graphs exhibit transition probability suppression, but not broken time-reversal symmetry. Furthermore, our development of a general framework characterizes gauge potentials on combinatorial graphs. These results and techniques fill an important missing gap in understanding the role this omnipresent effect has in quantum information and computation.

arXiv:1703.02542•Journal of Physics A Mathematical and Theoretical•Apr 21, 2021•7 citations
We formulate Grover's unstructured search algorithm as a chiral quantum walk, where transitioning in one direction has a phase conjugate to transitioning in the opposite direction. For small phases, this breaking of time-reversal symmetry is too small to significantly affect the evolution: the system still approximately evolves in its ground and first excited states, rotating to the marked vertex in time πN/2π\sqrt{N} / 2πN​/2. Increasing the phase does not change the runtime, but rather changes the support for the 2D subspace, so the system evolves in its first and second excited states, or its second and third excited states, and so forth. Apart from the critical phases corresponding to these transitions in the support, which become more frequent as the phase grows, this reveals that our model of quantum search is robust against time-reversal symmetry breaking.
Abstract An ideal quantum walk transitions from one vertex to another with perfect fidelity, but in physical systems, the particle may be hindered by potential energy barriers. Then the particle has some amplitude of tunneling through the barriers, and some amplitude of staying put. We investigate the algorithmic consequence of such barriers for the quantum walk formulation of Grover’s algorithm. We prove that the failure amplitude must scale as O ( 1 / N ) for search to retain its quantum O ( N ) runtime; otherwise, it searches in classical O ( N ) time. Thus searching larger ‘databases’ requires increasingly reliable hop operations or error correction. This condition holds for both discrete- and continuous-time quantum walks.
Grover's quantum search algorithm can be formulated as a quantum particle randomly walking on the (highly symmetric) complete graph, with one vertex marked by a nonzero potential. From an initial equal superposition, the state evolves in a two-dimensional subspace. Strongly regular graphs have a local symmetry that ensures that the state evolves in a three-dimensional subspace but most have no global symmetry. Using degenerate perturbation theory, we show that quantum random walk search on known families of strongly regular graphs, nevertheless, achieves the full quantum speed-up of $\mathrm{\ensuremath{\Theta}}(\sqrt{N})$, disproving the intuition that fast quantum search requires global symmetry.
A randomly walking quantum particle searches in Grover's Θ(N)Θ(\sqrt{N})Θ(N​) iterations for a marked vertex on the complete graph of NNN vertices by repeatedly querying an oracle that flips the amplitude at the marked vertex, scattering by a "coin" flip, and hopping. Physically, however, potential energy barriers can hinder the hop and cause the search to fail, even when the amplitude of not hopping decreases with NNN. We correct for these errors by interpreting the quantum walk search as an amplitude amplification algorithm and modifying the phases applied by the coin flip and oracle such that the amplification recovers the Θ(N)Θ(\sqrt{N})Θ(N​) runtime.
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.
A randomly walking quantum particle searches in Grover’s Θ(√ N) iterations for a marked vertex on the complete graph of N vertices by repeatedly querying an oracle that flips the amplitude at the marked vertex, scattering by a “coin” flip, and hopping. Physically, however, potential energy barriers can hinder the hop and cause the search to fail, even when the amplitude of not hopping decreases with N. We correct for these errors by interpreting the quantum walk search as an amplitude amplification algorithm and modifying the phases applied by the coin flip and oracle such that the amplification recovers the Θ(√ N) runtime.
Abstract Understanding which physical processes are symmetric with respect to time inversion is a ubiquitous problem in physics. In quantum physics, effective gauge fields allow emulation of matter under strong magnetic fields, realizing the Harper–Hofstadter, the Haldane models, demonstrating one-way waveguides and topologically protected edge states. Central to these discoveries is the chirality induced by time-symmetry breaking. In quantum walk algorithms, recent work has discovered implications time-reversal symmetry breaking has on the transport of quantum states which has enabled a host of new experimental implementations. We provide a full topological classification of Hamiltonian operators that do not exhibit symmetry under time-reversal with respect to the induced transition probabilities between elements in a preferred site-basis, i.e. the nodes of the graph on which the walk takes place. We prove that a quantum process is necessarily time-symmetric for any choice of time-independent Hamiltonian precisely when the underlying support graph is bipartite or no Aharonov–Bohm phases are present in the gauge field. We further prove that certain bipartite graphs exhibit transition probability suppression, but not broken time-reversal symmetry. Furthermore, our development of a general framework characterizes gauge potentials on combinatorial graphs. These results and techniques fill an important missing gap in understanding the role this omnipresent effect has in quantum information and computation.