We solve the unstructured search problem in constant time by computing with a physically motivated nonlinearity of the Gross–Pitaevskii type. This speedup comes, however, at the novel expense of increasing the time-measurement precision. Jointly optimizing these resource requirements results in an overall scaling of N 1/4 . This is a significant, but not unreasonable, improvement over the N 1/2 scaling of Grover's algorithm. Since the Gross–Pitaevskii equation approximates the multi-particle (linear) Schrödinger equation, for which Grover's algorithm is optimal, our result leads to a quantum information-theoretic lower bound on the number of particles needed for this approximation to hold, asymptotically.
We solve the unstructured search problem in constant time by computing with a physically motivated nonlinearity of the Gross–Pitaevskii type. This speedup comes, however, at the novel expense of increasing the time-measurement precision. Jointly optimizing these resource requirements results in an overall scaling of N 1/4 . This is a significant, but not unreasonable, improvement over the N 1/2 scaling of Grover's algorithm. Since the Gross–Pitaevskii equation approximates the multi-particle (linear) Schrödinger equation, for which Grover's algorithm is optimal, our result leads to a quantum information-theoretic lower bound on the number of particles needed for this approximation to hold, asymptotically.
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
Evolution by the Gross-Pitaevskii equation, which describes Bose-Einstein condensates under certain conditions, solves the unstructured search problem more efficiently than does the Schr\"odinger equation, because it includes a cubic nonlinearity, proportional to ${|\ensuremath{\psi}|}^{2}\ensuremath{\psi}$. This is not the only nonlinearity of the form $f(|\ensuremath{\psi}{|}^{2})\ensuremath{\psi}$ that arises in effective equations for the evolution of real quantum physical systems, however: The cubic-quintic nonlinear Schr\"odinger equation describes light propagation in nonlinear Kerr media with defocusing corrections, and the logarithmic nonlinear Schr\"odinger 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. In each of these cases the nonlinear equation is an effective approximation to a multiparticle Schr\"odinger equation, for search by which Grover's algorithm is optimal. Thus our results lead to quantum information-theoretic bounds on the physical resources required for these effective nonlinear theories to hold, asymptotically.
Although quant mech is linear, there are nevertheless quant sys with multiple interacting particles in which the effective evo of a single particle is governed by a nonlinear eq. This includes Bose-Einstein condensates, which are gov by the Gross-Pitaevskii eq (GPE), which is a cubic nonlin Schrodinger eq (NLSE) with a term propto ∣ψ∣2ψ. Evo by this eq solves the unstruct search prob in const time, but at the novel expense of increasing the time-measurement precision. Jointly optimizing these resources results in an overall scaling of N1/4, which is a significant, but not unreasonable, improvement over the N1/2 scaling of Grover's algo. Since the GPE effectively approx the multi-particle Schrodinger eq, for which Grover's algo is optimal, our result leads to a quant info-theoretic bound on the num of particles needed for this approx to hold, asymp. The GPE is not the only nonlin of the form f(∣ψ∣2)ψ that arises in effective eqs for the evo of real quant phys sys, however: The cubic-quintic NLSE describes light propagation in nonlin Kerr media with defocusing corrections, and the log NLSE describes Bose liquids under certain cond. Analysis of comput with such sys yields some surprising results; e.g., when time-measurement precision is included in the resource accounting, searching a "database" when there is a single correct ans may be easier than searching when there are multiple correct ans. These results lead to quant info-theoretic bounds on the phys resources required for these effective nonlin theories to hold, asymp. Further, strongly reg graphs, which have no global sym, are sufficiently complete for quant search on them to asymp behave like unstruct search. Certain suff complete graphs retain the improved runtime and resource scalings for some nonlin, so our scheme for nonlin, analog quant comput retains its benefits even when some struct 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.
Grover's algorithm is a fundamental quantum algorithm that achieves a quadratic speedup for unstructured search problems of size N. Recent studies have reformulated this task as a maximization problem on the unitary manifold and solved it via linearly convergent Riemannian gradient ascent (RGA) methods, resulting in a complexity of O(N/Mlog(1/ε)), where M denotes the number of target items and ε denotes the success probability error. In this work, we adopt the Riemannian modified Newton (RMN) method to solve the quantum search problem, under the assumption that the ratio $ M/N$ is known. We show that, in this setting, the Riemannian Newton direction is collinear with the Riemannian gradient in the sense that the Riemannian gradient is always an eigenvector of the corresponding Riemannian Hessian. This structure removes the overhead of Hessian inversion and allows the proposed RMN method to retain the local quadratic convergence in terms of the error ε. More precisely, we rigorously prove an overall complexity of O(N/M+loglog(1/ε)). Furthermore, our approach remains Grover-compatible, namely, it relies exclusively on the standard Grover diffusion and oracle operators to ensure algorithmic implementability, and its parameter update process can be efficiently precomputed on classical computers.
Nonlinear variants of quantum mechanics can solve tasks that are impossible in standard quantum theory, such as perfectly distinguishing nonorthogonal states. Here we derive the optimal protocol for distinguishing two states of a qubit using the Gross-Pitaevskii equation, a model of nonlinear quantum mechanics that arises as an effective description of Bose-Einstein condensates. Using this protocol, we present an algorithm for unstructured search in the Gross-Pitaevskii model, obtaining an exponential improvement over a previous algorithm of Meyer and Wong. This result establishes a limitation on the effectiveness of the Gross-Pitaevskii approximation. More generally, we demonstrate similar behavior under a family of related nonlinearities, giving evidence that the ability to quickly discriminate nonorthogonal states and thereby solve unstructured search is a generic feature of nonlinear quantum mechanics.