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Faster quantum walk algorithm for the two dimensional spatial search
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TL;DR
The paper asks whether quantum-walk search for a marked item on a two-dimensional lattice can improve on the known runtime. It controls the walk with an ancilla qubit, achieving the search without amplitude amplification and improving the runtime.
Problem
Two-dimensional spatial search takes O(sqrt(N) log N) steps with the known quantum-walk algorithms, and whether this can be improved is open.
Source: §I · p. 1Method
The algorithm controls the quantum walk with an ancilla qubit, adding walk eigenstates whose interference brings the final state close to the desired state.
Source: §I · p. 1, §IV · p. 5Results
O(sqrt(N log N)) steps solve the two-dimensional problem, improving on AKR’s algorithm by O(sqrt(log N)).
Source: §IV · p. 5Takeaways & Limitations
The new algorithm reaches the marked state without amplitude amplification, and its abstract-search application gives only a constant-factor improvement in higher dimensions.
Source: §I · p. 1For higher-dimensional spatial search, the improvement is only by a constant factor.
Source: §I · p. 1
Abstract
from arXiv · showhide
We consider the problem of finding a desired item out of $N$ items arranged on the sites of a two-dimensional lattice of size $\sqrt{N} \times \sqrt{N}$. The previous quantum walk based algorithms take $O(\sqrt{N}\log N)$ steps to solve this problem, and it is an open question whether the performance can be improved. We present a new algorithm which solves the problem in $O(\sqrt{N\log N})$ steps, thus giving an $O(\sqrt{\log N})$ improvement over the known algorithms. The improvement is achieved by controlling the quantum walk on the lattice using an ancilla qubit.
I. INTRODUCTION
The introduction frames two-dimensional spatial search as finding a unique marked site when each step is limited to checking the current site or moving to a neighbor. It presents an ancilla-controlled quantum walk that improves the known runtime without amplitude amplification.
I. INTRODUCTION
- The search assumes a unique marked site and allows one step to either examine the current site or move to a neighboring site.A direct Grover search is not faster than brute force under this movement constraint because its global reflection requires traversing the lattice.Source: §I · p. 1
- Earlier two-dimensional quantum-walk algorithms take O(sqrt(N) log N) steps, leaving open whether their performance can be improved.The introduction identifies whether the lower bound can be reached as an open question.Source: §I · p. 1
- The method controls the lattice walk using an ancilla qubit to change the search dynamics.The paper presents the modification in the context of AKR’s discrete-time algorithm and says it also applies to CG’s continuous-time algorithm.Source: §I · p. 1
- O(sqrt(N log N)) steps solve the two-dimensional search problem, an O(sqrt(log N)) improvement over the known quantum-walk algorithms.The new walk reaches a state with significant overlap with the marked state, so it needs no rounds of quantum amplitude amplification.Source: §I · p. 1
- The algorithm applies to instances of abstract search, with only a constant-factor improvement stated for higher-dimensional spatial search.The paper reviews the abstract search algorithm and its analysis before presenting the controlled walk.Source: §I · pp. 1–2
II. BACKGROUND
The background presents abstract search as a generalization of Grover search in which the initial-state reflection is replaced by a more general operator. Its analysis uses the operator’s eigenspectrum and the target state’s expansion in that eigenbasis.
II. BACKGROUND
- Grover search iterates reflections about the initial and target states to rotate the initial state toward the target.The paper describes the resulting evolution in the two-dimensional subspace spanned by those states.Source: §II · p. 2
- Abstract search replaces Grover’s initial-state reflection with a real operator U that has the initial state as its unique eigenstate of eigenvalue 1.The target reflection remains, and the iteration applies U followed by that reflection.Source: §II · p. 2
- The abstract-search analysis depends on U’s eigenspectrum and the target state’s expansion coefficients in U’s eigenbasis.The paper extends the analysis to allow an eigenspace of U with eigenvalue -1.Source: §II · p. 2
- For particular cases, the starting state is almost entirely spanned by two important eigenvectors of the search operator.The algorithm iterates the operator U followed by the target reflection to approach the target state.Source: §II · p. 2
A. Two-dimensional spatial search
The two-dimensional example uses a four-direction coin and a local walk operator, then reviews the AKR search behavior. The section motivates the controlled-walk modification as a way to avoid the amplification rounds required by that baseline.
A. Two-dimensional spatial search
- AKR attaches a four-dimensional coin space whose basis states encode the four possible lattice movement directions.The joint coin-and-lattice walk uses the uniform coin state with a uniform superposition over sites as its initial state.Source: A. Two-dimensional spatial search · p. 2
- The walk combines a coin flip with a move between neighboring sites, while the marked-state reflection is implemented by examining sites with an oracle.The passage states that the walk and marked-state reflection together take two time steps.Source: A. Two-dimensional spatial search · p. 3
- After O(sqrt(N log N)) quantum-walk steps, the final state |α+⟩+|ǫ′⟩, with ∥ǫ′∥=Θ(1/log N), has overlap Θ(1/sqrt(log N)) with |u_c,m⟩.Source: A. Two-dimensional spatial search · p. 3
- AKR’s walk reaches a state with overlap Θ(1/sqrt(log N)) with the marked state, requiring sqrt(log N) rounds of amplitude amplification.The total baseline complexity is O(sqrt(N) log N).Source: A. Two-dimensional spatial search · p. 3
- The controlled-walk approach aims to obtain the marked state in O(sqrt(N log N)) steps without amplitude amplification.The section says the ancilla-controlled step manipulates the search coefficients so amplification rounds are unnecessary.Source: A. Two-dimensional spatial search · p. 3
III. CONTROLLED QUANTUM WALK ALGORITHM
The algorithm controls the lattice walk and target reflection with an ancilla qubit, tuning the walk so its final state has constant overlap with the marked item. With this tuning, it finds the item with constant probability in O(√(N log N)) iterations.
III. CONTROLLED QUANTUM WALK ALGORITHM
- An ancilla qubit controls the walk and target reflection, each implemented in one step when the ancilla is |1⟩.The algorithm adds the ancilla to the AKR working space; one UC iteration takes two implementation steps.Source: §III · pp. 3–4
- O(√(N log N)) iterations of UC yield the marked item with constant probability.The final state has constant overlap with |m⟩.Source: §III · p. 4
- The controlled operator has a preserved subspace in which it satisfies the conditions for an abstract search algorithm.The ancilla’s |0⟩ states form the −1 eigenspace, and the target state has a nonzero projection onto it, which the text identifies as crucial to the new algorithm.Source: §III · p. 4
- Choosing tan²δ = Θ(log N) gives αδ = Θ(1/√(N log N)), enabling the stated iteration count and constant overlap.The method balances the iteration count against overlap with the marked state.Source: §III · pp. 4–5
IV. DISCUSSION
The ancilla-controlled modification solves two-dimensional spatial search in O(√(N log N)) time steps and improves on AKR by O(√log N). The paper also describes extensions and leaves open whether the complexity can be further improved.
IV. DISCUSSION
- O(√(N log N)) time steps improve on AKR’s algorithm by a factor of O(√log N).The paper presents the method as a modification of the discrete-time AKR quantum walk search algorithm.Source: §IV · p. 5
- The method also extends to Childs and Goldstone’s continuous-time walk by adding an ancilla and evolving under a suitably controlled Hamiltonian.That walk uses a time-independent Hamiltonian coupling only neighboring sites.Source: §IV · p. 5
- The authors leave further improvement open; within their framework, they suggest O(√(N log N)) may be best because the minimum eigenvalue gap requires O(√(N log N)) adiabatic time.The problem has a lower bound of Ω(√N) time steps.Source: §IV · p. 5
- Extra walk eigenstates introduced by the ancilla allow interference that brings the final state close to the desired state.The discussion contrasts this with AKR and CG, whose final states are not close to the desired state despite O(√(N log N)) evolution time.Source: §IV · p. 5
- For multiple marked items, generalizing the algorithm from its unique-marked-item setting incurs O(log N) computational-complexity overhead, giving total complexity O(N log N) time steps.Source: §IV · p. 5
- In higher-dimensional spatial search, the improvement is only a constant factor; when AKR costs c√N, the gain can be √c for c ≫ 1.For c = O(1), the discussion says there is not much improvement.Source: §IV · p. 5
APPENDIX: ABSTRACT SEARCH ALGORITHM
The appendix analyzes the abstract search iteration UA = U·Rt through the spectrum of U and the target state's expansion in U's eigenbasis. It identifies the eigenstates that govern the evolution from |Φ0⟩ toward |α+⟩.
APPENDIX: ABSTRACT SEARCH ALGORITHM
- The analysis studies UA = U·Rt iterated on |Φ0⟩; its relevant eigenspectrum is determined by U's eigenspectrum and the target state's expansion coefficients.U is required to be real, and the analysis includes the possibility of a −1 eigenspace.Source: APPENDIX: ABSTRACT SEARCH ALGORITHM · p. 5
- For real λ, |t⟩ + i|wλ⟩ is an eigenvector of UA with eigenvalue e^iλ when |wλ⟩ is orthogonal to |t⟩.The coefficients of |wλ⟩ in U's eigenbasis are defined using Fλ(θl).Source: APPENDIX: ABSTRACT SEARCH ALGORITHM · pp. 5–6
- Let θmin be the smallest θj; the equation has exactly two solutions, λ = α and λ = −α, with |α| < θmin/2, whose eigenvectors almost completely span |Φ0⟩.Source: APPENDIX: ABSTRACT SEARCH ALGORITHM · p. 6
- For spatial search at large N, Tj and Tk are small compared with T0, supporting the approximation of |Φ0⟩ by |α−⟩.The appendix defines the terms for contributions from l = 0, l ∈ j, and l ∈ k.Source: APPENDIX: ABSTRACT SEARCH ALGORITHM · p. 6
- After T = ⌈π/2α⌉ iterations of UA, the state initially close to |α−⟩ comes close to |α+⟩.The two states are formed from the eigenvectors of UA with eigenvalues e^iα and e^-iα.Source: APPENDIX: ABSTRACT SEARCH ALGORITHM · p. 6