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arXiv:2503.06423 · Quantum Information and Computation

Conserved Quantities in Linear and Nonlinear Quantum Search

DAVID A. MEYER, Thomas G. Wong

Published Aug 1, 2025• 1 citation
arXivHTMLPDF

Abstract

Abstract In this tutorial, which contains some original results, we bridge the fields of quantum computing algorithms, conservation laws, and many-body quantum systems by examining three algorithms for searching an unordered database of size N using a continuous-time quantum walk, which is the quantum analogue of a continuous-time random walk. The first algorithm uses a linear quantum walk, and we apply elementary calculus to show that the success probability of the algorithm reaches 1 when the jumping rate of the walk takes some critical value. We show that the expected value of its Hamiltonian H 0 is conserved. The second algorithm uses a nonlinear quantum walk with effective Hamiltonian H ( t ) = H 0 + λ | ψ | 2 , which arises in the Gross-Pitaevskii equation describing Bose-Einstein condensates. When the interactions between the bosons are repulsive, λ > 0, and there exists a range of fixed jumping rates such that the success probability reaches 1 with the same asymptotic runtime of the linear algorithm, but with a larger multiplicative constant. Rather than the effective Hamiltonian, we show that the expected value of H 0 + 1 2 λ | ψ | 2 {H_0} + {1 \over 2}\lambda |\psi {|^2} is conserved. The third algorithm utilizes attractive interactions, corresponding to λ < 0. In this case, there is a time-varying critical function for the jumping rate γ c ( t ) that causes the success probability to reach 1 more quickly than in the other two algorithms, and we show that the expected value of H ( t )/[ γ c ( t ) N ] is conserved.

DOI
10.2478/qic-2025-0017

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Abstract In this tutorial, which contains some original results, we bridge the fields of quantum computing algorithms, conservation laws, and many-body quantum systems by examining three algorithms for searching an unordered database of size N using a continuous-time quantum walk, which is the quantum analogue of a continuous-time random walk. The first algorithm uses a linear quantum walk, and we apply elementary calculus to show that the success probability of the algorithm reaches 1 when the jumping rate of the walk takes some critical value. We show that the expected value of its Hamiltonian H 0 is conserved. The second algorithm uses a nonlinear quantum walk with effective Hamiltonian H ( t ) = H 0 + λ | ψ | 2 , which arises in the Gross-Pitaevskii equation describing Bose-Einstein condensates. When the interactions between the bosons are repulsive, λ > 0, and there exists a range of fixed jumping rates such that the success probability reaches 1 with the same asymptotic runtime of the linear algorithm, but with a larger multiplicative constant. Rather than the effective Hamiltonian, we show that the expected value of H 0 + 1 2 λ | ψ | 2 {H_0} + {1 \over 2}\lambda |\psi {|^2} is conserved. The third algorithm utilizes attractive interactions, corresponding to λ < 0. In this case, there is a time-varying critical function for the jumping rate γ c ( t ) that causes the success probability to reach 1 more quickly than in the other two algorithms, and we show that the expected value of H ( t )/[ γ c ( t ) N ] is conserved.
One approach to the development of quantum search algorithms is the quantum walk. A spatial search can be effected by the continuous-time evolution of a single quantum particle on a graph containing a marked site. In many physical implementations, however, one might expect to have multiple particles. In interacting bosonic systems at zero temperature, the dynamics is well described by a discrete nonlinear Schr\"odinger equation. We investigate the role of nonlinearity in determining the efficiency of the spatial search algorithm within the quantum walk model, for the complete graph. The analytical calculations reveal that the nonlinear search time scales with the size of the search space NNN like N\sqrt{N}N​, equivalent to the linear case, though with a different overall constant. The results indicate that interacting Bose-Einstein condensates at zero temperature could be natural systems for implementation of the quantum search algorithm.
Although quantum mechanics is linear, there are nevertheless quantum systems with multiple interacting particles in which the effective evolution of a single particle is governed by a nonlinear equation. This includes Bose-Einstein condensates, which are governed by the Gross-Pitaevskii equation, which is a cubic nonlinear Schrodinger equation with a term proportional to /[psi]/ ²[psi] scaling of Grover's algorithm. Since the Gross- Pitaevskii equation effectively approximates the multi- particle Schrodinger equation, for which Grover's algorithm is optimal, our result leads to a quantum information-theoretic bound on the number of particles needed for this approximation to hold, asymptotically. The Gross-Pitaevskii equation is not the only nonlinearity of the form f(/[psi]/²)[psi] that arises in effective equations for the evolution of real quantum physical systems, however : The cubic-quintic nonlinear Schrodinger equation describes light propagation in nonlinear Kerr media with defocusing corrections, and the logarithmic nonlinear Schrodinger equation describes Bose liquids under certain conditions. Analysis of computation with such systems yields some surprising results; for example, when time-measurement precision is included in the resource accounting, searching a database'' when there is a single correct answer may be easier than searching when there are multiple correct answers. These results lead to quantum information-theoretic bounds on the physical resources required for these effective nonlinear theories to hold, asymptotically. Furthermore, strongly regular graphs, which have no global symmetry, are sufficiently complete for quantum search on them to asymptotically behave like unstructured search. Certain sufficiently complete graphs retain the improved runtime and resource scalings for some nonlinearities, so our scheme for nonlinear, analog quantum computation retains its benefits even when some structure is introduced
The optimal runtime of a quantum computer searching a database is typically cited as the square root of the number of items in the database, which is famously achieved by Grover's algorithm. With parallel oracles, however, it is possible to search faster than this. We consider a many-body quantum system that naturally effects parallel queries, and we show that its parameters can be tuned to search a database in constant time, assuming a sufficient number of interacting particles. In particular, we consider Bose-Einstein condensates with pairwise and three-body interactions in the mean-field limit, which effectively evolve by a nonlinear Schr\"odinger equation with cubic and quintic nonlinearities. We solve the unstructured search problem formulated as a continuous-time quantum walk searching the complete graph in constant time. Depending on the number of marked vertices, however, the success probability can peak sharply, necessitating high precision time measurement to observe the system at this peak. Overcoming this, we prove that the relative coefficients of the cubic and quintic terms can be tuned to eliminate the need for high time-measurement precision by widening the peak in success probability or having it plateau. Finally, we derive a lower bound on the number of atoms needed for the many-body system to evolve by the effective nonlinearity.
One approach to the development of quantum search algorithms is the quantum walk. A spatial search can be effected by the continuous-time evolution of a single quantum particle on a lattice or graph containing a marked site. In most conceivable physical applications, however, one would rather expect to have multiple interacting particles. In bosonic systems at zero temperature, the dynamics would be well-described by a discrete non-linear Schrodinger equation. In this thesis we investigate the role of non-linearity in determining the efficiency of the spatial search algorithm within the quantum walk model, for the complete graph. Our analytical results indicate that the search time for this non-linear quantum search scales with size of the database N like square root of N, equivalent to linear spatial search time. The analytical results will be compared with numerical calculations of multiple interacting quantum walkers.
Many-body quantum systems with effective nonlinearities have been shown to speed up quantum search on the complete graph, \textit{i.e.}, the combinatorial version of Grover's algorithm, at the expense of the number of particles needed for the effective nonlinearity to hold. Physically, however, data may not be arranged in an all-to-all network, and the task of searching incomplete graphs is the spatial search problem. We explore spatial search using a continuous-time nonlinear quantum walk on a variety of graphs. First, we consider incomplete graphs that are ``sufficiently complete'' so as to asymptotically search like the complete graph under a continuous-time (linear) quantum walk, which includes strongly regular graphs such as Paley graphs, regular graphs such as hypercubes, and irregular graphs such as complete bipartite graphs. For these sufficiently complete graphs, we analytically prove nonlinear speedups for Paley graphs and for complete bipartite graphs whose two partite sets both have size Θ(N)Θ(N)Θ(N), for suitable cubic and cubic-quintic nonlinearities, and we give numerical evidence for stronger nonlinearities and for hypercubes. Second, we explore arbitrary-dimensional cubic lattices, and we numerically show that certain nonlinearities speed up search on sufficiently high dimensional lattices. Thus, nonlinear quantum search can remain viable even when the underlying graph is incomplete.
In the first part of the thesis we construct a new, simple framework which amplifies to a constant the success probability of any abstract search algorithm. The total query complexity is given by the quantum hitting time of the resulting operator, which we show that it is of the same order as the quantum hitting time of the original algorithm. As a major application of our framework, we show that for any reversible walk PPP and a single marked state, the quantum walk corresponding to PPP can find the solution using a number of queries that is quadratically smaller than the classical hitting time of PPP. Our algorithm is more general and simpler to implement than the solution known previously in the literature (Krovi, Magniez, Ozols, and Roland, 2015), which was developed specifically for quantum walks; we also prove that, for the particular case of quantum walks, we can embed their algorithm into our framework, thus simulating it exactly. Finally, we show that we can implement amplitude amplification using our tool. In the second part of the thesis we give a new lower bound in the query model which proves that Grover's algorithm for unordered searching is exactly optimal. Similar to existing methods for proving lower bounds, we bound the amount of information we can gain from a single oracle query, but we bound this information in terms of angles. This allows our proof to be simple, self-contained, based on only elementary mathematics, capturing our intuition, while obtaining at the same an exact bound. We then turn our attention to non-adaptive algorithms for the same problem of searching an unordered set. In this model, we obtain a lower bound and we give an algorithm which matches the lower bound, thus showing that the lower bound is exactly optimal.