A computer implemented method for searching in a quantum system

Abstract

The present disclosure provides a method for searching at least one locally marked element defined by marking parameters within a 2D lattice of a quantum system, wherein the quantum system is operated in discrete time and wherein the 2D lattice is arranged for supporting quantum random walks which are defined by quantum random walk parameters, the method having a pre-set time comprises the steps of calibrating the quantum system by: initializing a quantum state in the quantum system on at least one node; performing one step of the split-step quantum random walk on the quantum state; obtaining probability density information of the quantum state through overlap with eigenstates of a unitary operator; repeating previous two steps to determine quantum random walk parameters based on the highest probability in the probability density information of the quantum state. Thereafter, the method comprises the steps of searching the quantum system for the at least one marked element by: tuning the marking parameters of the at least one locally marked element; initializing a uniformly distributed quantum state on the entire quantum system; performing a plurality of split-step quantum random walks for the uniformly distributed quantum state; after the pre-set time has lapsed, converting the uniformly distributed quantum state of step vii) into an observable.

Description

[0001] TITLE

[0002] A computer implemented method for searching in a quantum system

[0003] TECHNICAL FIELD

[0004] This disclosure pertains to quantum computing, and specifically to a computer implemented method for searching in a quantum system by adding topological structure to a search problem.

[0005] BACKGROUND OF THE DISCLOSURE

[0006] Quantum search algorithms are pivotal in quantum computing, offering substantial efficiency improvements over classical search methods. Classical algorithms typically search through each database element sequentially, whereas quantum algorithms exploit superposition and entanglement to search faster.

[0007] A well-known quantum search algorithm, the Grover’s algorithm, for instance, is renowned for finding a marked item within an unstructured database in time proportional to the square root of the number of database elements. Despite its advantages over classical methods, the potential for enhancing Grover’s efficiency exists. And although quantum search algorithms such as the Grover’s algorithm advances over other search algorithms, it demonstrates square root dependence on system size. As a result, the search time is growing as the square root of the system size. Accordingly, there is a need for a computer implemented method for searching within a quantum system with improved properties, and in particular, to obtain a search algorithm which achieves constant search time dependency of the system size.

[0008] SUMMARY OF THE DISCLOSURE

[0009] The present disclosure, in a first aspect, provides a method for searching at least one locally marked element defined by marking parameters within a 2D lattice of a quantum system, wherein the quantum system is operated in discrete time and wherein the 2D lattice is arranged for supporting quantum random walks which are defined by quantum random walk parameters, the method having a pre-set time comprises the steps of:

[0010] 1) calibrating the quantum system by: i) initializing a quantum state in the quantum system on at least one node; ii) performing one step of the split-step quantum random walk on the quantum state; iii) obtaining probability density information of the quantum state through overlap with eigenstates of a unitary operator; iv) repeating steps ii) and iii) to determine quantum random walk parameters based on the highest probability in the probability density information of the quantum state;

[0011] 2) searching the quantum system for the at least one marked element by: v) tuning the marking parameters of the at least one locally marked element; vi) initializing a uniformly distributed quantum state on the entire quantum system; vii) performing a plurality of split-step quantum random walks for the uniformly distributed quantum state; viii) after the pre-set time has lapsed, converting the uniformly distributed quantum state of step vii) into an observable.

[0012] A first aspect of the present disclosure relates to a method for searching marked elements within a 2D lattice of a quantum system. A method of such kind may also be understood as a "computer implemented method", since a series of steps are executed by a quantum computer to accomplish a specific task or to solve a problem. However, part of the method relates to the physical implementation of such quantum system and the operations thereon. The term "2D lattice" refers to a two-dimensional structure consisting of points or nodes where each node is connected to adjacent points, suitable for implementing quantum operations. It is provided that the 2D lattice is arranged for supporting quantum random walks. A "quantum random walk" can be considered a quantum analogy of a classical random walk, distinguished by the use of quantum superposition and entanglement to explore multiple paths simultaneously. An effect of arranging the lattice to support quantum random walks is the enhanced efficiency in searching operations compared to classical random walks. This arrangement leverages the quantum properties to explore large databases more swiftly, potentially reducing the time required for search operations and thus optimizing the use of processing power.

[0013] It is provided that the 2D lattice comprises at least one marked element. In quantum systems in general and the present disclosure in particular, a "marked element" may refer to a deliberate alteration or irregularity in the lattice structure that localizes certain quantum properties, such as wavefunctions, energy states or interactions. By tuning marking parameters, which quantum mechanically define the marked element and by tuning the split-step quantum random walk parameters defining the split-step quantum random walks, topological localizations arise in the parameters space of the quantum random walk parameters, where timely detection of the at least one marked element is enhanced. These localizations in the topological phase (read specific quantum random walk parameters) provide effective pathways for finding target elements more efficiently and faster existing quantum search algorithms, thereby making better use of computational resources.

[0014] It may be provided that the method includes calibrating the quantum system, “calibrating the quantum system” may be understood as tuning the quantum random walk parameters such that a desirable set of quantum random walk parameters is obtained. These quantum random walk parameters give properties to the quantum random walks in such a way that the at least one marked element can be found faster than existing quantum search algorithms.

[0015] It may be provided that the method includes initializing a quantum state in the quantum system on at least one node. An "at least one node" in a quantum system may refer to one single node or a small number of nodes representing the overall quantum system. Ideally, the at least one node is chosen to comprise a marked element, such that the quantum random walk parameter can be refined. When the at least one node comprises only a handful of nodes, the advantage is that the quantum system is small, and thus only little processing power is needed to obtain the desired quantum random walk parameters. When the at least one node comprises a plurality of nodes, the advantage is that the quantum system is larger, such that many nodes can be initialized simultaneously, as a result of which a large random walk parameter space can be covered from the start.

[0016] It may be provided that the one split-step quantum random walk is performed on the quantum state. "Split-step quantum random walk" involves alternating operations that manipulate the phase and position of quantum states, adapted to the lattice’s topological characteristics. This arrangement allows for a detailed probing of the quantum system without knowing the local of the at least one marked element, providing the advantage of optimizing the path of the quantum random walk. Furthermore, here topological properties of the quantum system may even be exploited to enhance the search efficiency and effectiveness.

[0017] It may be provided that probability density information of the quantum state is obtained through overlap with eigenstates of a unitary operator. An "eigenstate of a unitary operator" represents a state in which the system remains unchanged by the quantum operation described by the operator, except for a phase factor. Comparing overlaps allows for measuring how closely the quantum states resulting from the random walk align with the eigenstates of the system, offering the advantage of precisely identifying when and where the quantum walk converges towards potential solutions, which enhances the predictability and reliability of search outcomes.

[0018] During the calibration steps of the method, the overlap criteria can be explored on a small-scale lattice of at least one node up to a few tens of nodes in each direction of the 2D lattice. This is because the same overlap criteria for smaller quantum system hold for larger quantum systems as well. This behavior starts from an order of magnitude of 100 nodes. On classical computers, it might be impossible to compare the eigenstate overlap for systems of 1000x1000 nodes, because matrix diagonalization is limited. However, sparse matrix techniques can be used to obtain the overlap by time evolution of the system and a precise estimate for the corresponding search time to be lapsed before measuring the quantum state in the quantum system.

[0019] It may be provided that steps of the method are repeated or iterated to determine quantum random walk parameters based on the highest probability in the probability density information of the quantum state. It may be understood that with every step of the quantum random walk the quantum state moves further away for a uniform quantum state towards one which resembles the marked element. For particular quantum random walk parameters, the quantum state may become localized around its initial node, which has a similar analogy as a trapped state in other quantum systems.

[0020] It may be provided that the quantum system for the at least one marked element is searched after calibrating the quantum system. After calibration of the quantum system, the quantum random walk parameters are obtained, which would guide the quantum state in the quantum system towards a localized solution of the search problem in a more efficient manner and in a way which achieves constant search time independent of the system size. A speed-up of such kind, especially for large quantum systems, comprising many nodes, is beneficial since quantum systems nowadays are becoming larger and larger, and key to their success depends on obtaining the desired answers / marked elements in a fast and efficient way.

[0021] It may be provided that marking parameters of the at least one locally marked element are tuned. Tuning involves adjusting the parameters defining the at least one locally marked element based on feedback or intermediate results. This step offers the advantage of dynamically adapting the search strategy to real-time conditions within the lattice, which can significantly enhance the search precision and reduce unnecessary computational overhead.

[0022] It may be provided that the entire quantum system is initialized into a uniformly distributed quantum state, where a " uniformly distributed quantum state " implies a quantum state with a wave function that is distributed over the nodes of the entire quantum system with approximately the same weights. This transition to an entire system concentrates the search efforts on the most promising regions of the search space identified in the initialization phase, providing an advantage by increasing the likelihood of finding the target with fewer steps and less computational time.

[0023] It may be provided that a plurality of split-step quantum random walks is performed for the uniformly distributed quantum state. These final split-step quantum random walks ensure that the search is refined to the highest degree of accuracy, utilizing the full computational capabilities of the system to home in on the marked elements with maximal efficiency and minimal error, capitalizing on the uniformly distributed quantum state’s higher resolution to pinpoint the exact location of targets within the quantum lattice.

[0024] Lastly, it may be provided that steps are repeated until the pre-set time has lapsed, whereafter the obtained quantum state of the quantum system is converted into an observable, thereby offering a systematic approach to exhaust possible solutions within a manageable and efficient timeframe, optimizing the balance between thorough exploration and computational expenditure.

[0025] In an example according to the disclosure calibrating the quantum system may be performed on a processor simulating a physical quantum system and searching the quantum system may be performed on the actual physical quantum system. Because of manufacturing difficulties to isolate at least one node of a physical quantum system, such that the calibration can be performed on a subset of the entire physical quantum system, it may be more beneficial to mimic the physical quantum system in order to obtain the desired quantum random walk search parameters for the actual search problem.

[0026] In another example the marking parameters of are selected based on the highest search probability. Upon tuning of the marking parameters, one might observe that specific marking parameters may obtain higher search probabilities than other. After performing such sweep over the marking parameter space, the optimal marking parameters may be chosen, such that the at least one marked element is found the fastest.

[0027] In an alternative example the marking parameters are selected based on a-priori knowledge of the quantum system. In this example, the tuning of the marking parameters might be much less exploratory since one might already know that the marking parameters would likely fall within a subset of the marking parameter space.

[0028] As an example thereof, the marked element is configured with localized marking parameters as a localized marked lattice node. For instance, the marked element node is characterized by a vacancy, Then the marking of each vacancy can be expected to be approximately the same. With this knowledge the method according to the disclosure can be optimized greatly, since less computational effort has to be put into tuning the marking parameters, since the likelihood of the outcome may already be known.

[0029] In a further example, the localized marking parameters are fixed during steps iii) and iv), preferable the localized marking parameters are also fixed during steps vi) through viii). For the method of the disclosure it is preferable that marking parameters are known from the calibration step or from a-priori knowledge about the quantum system. This way, the method can perform the search of the selected marking parameters for one full run. The search is run for a predetermined fixed time and not until convergence like Grover’s search. In case of a positive hit, steps iv) through viii) could be repeated a number of times to obtain better statistics of the location of the marked element. In case of obtaining zero hits from the search, the method should be rerun with new calibration of the system. It is thus important, that the marking parameters do not dynamically change during the search process, or in other words the marking parameters are not allowed to change during any of the steps of the quantum random walks.

[0030] In yet another example, the localized marking parameters and split-step quantum random walk parameters form distinct topological phases. It may be understood that the topological phases may be related to the 2D lattice, but do not have to be. Furthermore, it should be noted that the two topological phases of the marking parameters and the random walk parameters, respectively, do not have to be identical.

[0031] In an example according to the disclosure, the split-step quantum random walk includes a set of shift and spin flip operations applied sequentially to each quantum state of the system at each time step. For each time step the split-step quantum random walk may perform three rotations and subsequent translations.

[0032] In yet another example the at least one locally marked element comprises a plurality of locally marked elements, wherein the marking parameters further comprise a hyperparameter which indicates the relationship between each of the locally marked elements. In case, a plurality of locally marked elements is present in the quantum system, a hyperparameter may be beneficial to be included in to marking parameters, since it allows to tune the interrelation between the marked elements. For instance, a locally marked element at an intermediate distance, might experience the presence of a tail of a wavefunction of another marked element. By inclusion of a hyperparameter, such interrelation might be captured, such that the quantum random walk parameters can collapse to both locally marked elements, instead of spreading out over both. Furthermore, it should be noted that the hyperparameter may include distance between locally marked elements or the radius of their respective wavefunction, as well as charge ratios. Hyperparameter may in additional or alternatively also include parameters which define the structure, behavior, and characteristics of the quantum system or model, e.g. number of qubits, model complexity, model depth, model layers, entangling gate type, rotation gate parameters, encoding structure, measuring strategy, optimizer type, etc. The skilled person will appreciate which other hyperparameter may also be applicable.

[0033] In a further example, the observable of the quantum state, is selected from a list not limited to electron spin, magnetic, or photonic. The exact observable depends on the implementation of the quantum system. For some systems is may be beneficial to measure the photonic state of the quantum system, whereas electron spin is utilized more commonly. In another example, the 2D lattice of a quantum system is selected from a list not limited to, triangular, cubic, hexagonal, oblique, rhombic, or rectangular. The exact shape the 2D lattice is defined by the implementation of the quantum system. However, it should be noted that different lattices can be utilized by this method. Note that each lattice will obtain different optimal quantum random walk parameters and will show different topological phases for the quantum random walk parameters and the marking parameters.

[0034] In an example of the method converting the uniformly distributed quantum state into an observable is performed by classical traditional quantum state tomography.

[0035] In an alternative example, converting the uniformly distributed quantum state into an observable is performed by shadow tomography. Through tomography the quantum state can be understood without needing to fully reconstruct it. Through multiple measurements the original quantum state can be characterized. Traditional quantum state tomography may be useful in particular cases, where read-out time is not limited, but when read-out time is limited shadow tomography might be preferable, since only a logarithmic number of measurements is needed.

[0036] In an example, it may be provided that steps are repeated until the probability density information of the quantum state of successive rounds of repeating the steps has changed less than 1%. Instead of or in combination with utilizing a pre-set time for the duration that the quantum state is subjected to a quantum random walk and probability density analysis, one could also track how much the probability density information is changing. When it changes less than 1% the operation might be stopped, thereby offering an approach to find a possible solution faster or with higher fidelity.

[0037] Lastly, in an example, it may be provided what step iv) of repeating steps ii) and iii) is performed until the probability density information of the quantum state of successive rounds of repeating steps ii) and iii) has changed less than 1 %.

[0038] BRIEF DESCRIPTION OF THE DRAWINGS Figure 1 depicts the search probability of a 2D space of random walker parameters.

[0039] Figure 2 shows a flow diagram of the steps according to the method of the disclosure.

[0040] Figure 3 shows a 2D colormap of the search probability of various random walker parameters and their respective topologic position.

[0041] Figure 4 depicts line graphs of the search time needed for specific random walker parameters as a function of quantum system size.

[0042] Figure 5 shows line graphs of the search probability for specific random walker parameters as a function of quantum system size.

[0043] Figure 6 shows line graphs comparing the search probability against two theoretical models for specific quantum random walk parameters.

[0044] Figure 7 depicts the squared product of overlap plotted as function of energy.

[0045] Figure 8 shows line graphs comparing the search probability against two theoretical models for specific quantum random walk parameters different to Figure 6.

[0046] Figure 9 shows a comparison of probability density as a function of time for given quantum random walk parameters.

[0047] Figure 10 shows three different 2D lattice structures of the quantum system.

[0048] DETAILED DESCRIPTION OF THE DISCLOSURE

[0049] A commonly known quantum search algorithm, such as Grover’s algorithm, performs a search of marked elements 1000 in an unstructured database with a given probability achieving an asymptotic search-time scaling as the square root of the database size. Its implementation depends on an oracle operator - a way of distinguishing marking element that applies to a whole quantum state, and an implementation of such oracle operator is a quantum random walk.

[0050] In discrete time, the quantum random walks are defined as a set of shift and spin flip operators applied over the entire quantum system at each time step. A successful search event is then defined as a maximum time to achieve localization of the quantum random walk on a marked element 1000 in that database starting from a uniformly distributed quantum state over the entire system. Although, different kinds of quantum random walks demonstrate a Grover-type convergence of a spatial search in two and higher dimensions, the inventors have found that the choice of the quantum random walk parameters (read choice of spin flip operator and shift operator) in combination with the position of the marked element in the quantum system are crucial and incorrect selection in many cases results in a decreased efficiency of the search algorithm.

[0051] Therefore, it is proposed to analyze the parameter space of the quantum random walks and the marking parameters prior to performing the actual search. This way sets of parameters can be obtained that achieve an efficient search of given marked elements. These regions in the quantum random walk parameter space are shown in Figure 1 as a 3D graph. In this Figure a different quantum random walk parameter is shown on either the X- or Y-axis. Then on the Z-axis the search probability is shown obtained from probability density function analysis of the quantum state with overlap from a unitary operator, more on that later. The graph in Figure 1 shows that specific localizations (sharp peaks) arise for given combinations of quantum search parameters, indicating that with these combinations of parameters the marked elements can be found efficiently and fast.

[0052] An important aspect of the disclosure’s method is that constant search time can be achieved with local marking of the to-be searched element, thus making it more straight-forward to implement such method in experimental quantum systems, like spin, magnetic or photonic qubit systems.

[0053] Therefore, it is the aim of the disclosure to provide a method for searching at least one locally marked element defined by marking parameters within a 2D lattice of a quantum system, wherein the quantum system is operated in discrete time and wherein the 2D lattice is arranged for supporting quantum random walks which are defined by quantum random walk parameters, the method having a pre-set time comprises the steps of:

[0054] 1) calibrating the quantum system by: i) initializing a quantum state in the quantum system on at least one node 100; ii) performing one step of the split-step quantum random walk on the quantum state; iii) obtaining probability density information of the quantum state through overlap with eigenstates of a unitary operator; iv) repeating steps ii) and iii) to determine quantum random walk parameters based on the highest probability in the probability density information of the quantum state;

[0055] 2) searching the quantum system for the at least one marked element by: v) tuning the marking parameters of the at least one locally marked element; vi) initializing a uniformly distributed quantum state on the entire quantum system; vii) performing a plurality of split-step quantum random walks for the uniformly distributed quantum state; viii) after the pre-set time has lapsed, converting the uniformly distributed quantum state of step vii) into an observable.

[0056] The steps of the method are shown in Figure 2. With the calibration step 1) a similar figure is obtained as in Figure 1 . For instance, one could calibrate the quantum system on just one node 100 or a few nodes 100, wherein one is certain that at least one marked element is present. In such case the quantum system is only small, and quantum operations or simulations can be performed with high efficiency. Thereafter, the whole quantum system can be searched with adequately defined quantum random walk parameters based on the probability density or search probability information as obtained during the calibration. This way it is ensured that a locally marked element can actually be found with high fidelity and that the search parameters (quantum random walk parameters) make the quantum state converge to a solution.

[0057] As further shown in Figure 2, as part of step 2) of searching the entire quantum system, the marking parameters of the at least one marked element may be tuned, which is useful to ensure better and / or faster convergence towards the searched element.

[0058] The marking parameters of the at least one marked element may be tuned by applying a phase shift to marked elements within the quantum search system, wherein the phase shift may be controlled by a marking angle parameter 0. Further, the tuning may encompass adjusting the value of the marking angle parameter 0 to control the degree of amplification or suppression of the marked elements. The optimal value of the marking angle parameter 0 may depend on the specific problem instance and the number of marked elements within the search space.

[0059] The tuning during the calibration steps may also comprise tuning a number of iterations parameter k in a quantum search system. The tuning may comprise repeating an amplification process comprising a marking step and an inversion-about- mean step for a number of iterations k. The tuning may further comprise adjusting the value of the number of iterations parameter k to control the number of times the amplification process is repeated. The optimal value of the number of iterations parameter k may depend on factors such as the number of marked elements and the size of the search space. The number of iterations k may be understood and the number of steps in the quantum random walk process, which in the end relate to a predefined time, since every step takes At amount of time.

[0060] It is thus important for the method of the disclosure that the iterations parameter k is tuned only during the calibration step, since the method of the disclosure is not run until convergence, like other method such as Grover’s. Instead it is run over a constant pre-defined amount of time.

[0061] The quantum search system may comprise a marking module configured to apply a phase shift to marked elements within the quantum search system, wherein the phase shift is controlled by a marking angle parameter 0. The system further comprises a parameter tuning module configured to adjust the value of the marking angle parameter 0 to control the degree of amplification or suppression of the marked elements.

[0062] The quantum search system may comprise an amplification module configured to repeat an amplification process comprising a marking step and an inversion-about- mean step for a number of iterations k. The system further comprises a parameter tuning module configured to adjust the value of the number of iterations parameter k to control the number of times the amplification process is repeated.

[0063] The tuning of marking parameters, such as the marking angle parameter 0 and the number of iterations parameter k, may be achieved through various techniques, including analytical methods, numerical simulations, quantum machine learning, or hybrid quantum-classical approaches. Optimal marking parameters may depend on the specific problem instance, the size of the search space, and the available quantum resources. Proper tuning of these parameters is crucial for achieving efficient and accurate quantum search results, as it directly impacts the amplification of the desired solution states and the overall performance of the quantum search algorithm. Furthermore, it is preferred that the marking parameters and the number of iterations do not change dynamically during the performance of the search method. From the start of performing step 2) of the method, the best marking parameters and number of iterations known from calibration should be used in the physical quantum system. In the case that no positive outcome of search is obtained, either calibration should be performed again, or it means that no elements with that specific marking are present in the quantum system.

[0064] The subset of quantum search parameter that result in efficient and fast searching, as obtained in step iv), can be categorized into two distinct classes based on their parameter space position inside a topological phase 500 or on a phase separation lines 501. This is depicted in Figure 3. Here, the topological phases 500 are inherent to the quantum system that is being used.

[0065] The inventors have found that the search time for the quantum random walk parameters 10, 20 inside both classes demonstrate saturation to a constant search time with growing quantum system size, as shown in Figure 4. This search convergence outperforms Grover’s square root asymptotic behavior for system size. The graph indicated with the star marker in Figure 4 for a specific set of quantum random walk parameters 10 have the quantum random walk parameters 10 as indicated with the star symbol in the quantum random walk parameters space in Figure 3. The same is true for the graph of the set of quantum random walk parameters 20 indicated with the square marker.

[0066] In the following few sections mathematical prove will be given as to why these specific sets of quantum random walk parameters outperform existing search strategies and what the mathematical definitions are that are used in the search method of the disclosure.

[0067] The time evolution of states for a range of quantum walker parameters is analyzed by studying the overlap of the initial quantum state with the eigenstates of a unitary operator corresponding to a single time step of the quantum random walk. The appearance of trapped states of a specific structure near the marked element allow for asymptotic speedup of the search to a constant time independent of system size. More specifically, two pairs of trapped states should simultaneously have a large matrix element of coordinate operator of the marked element and a large overlap with the initial uniformly distributed state. In the mathematical sense the trapped states are defined as a states that have compact support and zero density elsewhere. Their analogy in quantum mechanics would be the strongly localized bound states to impurities in a system.

[0068] Optimizations made to quantum search algorithms has been study for many inventors in the past decades, however the possible efficiency of search algorithm based on exploiting the particular structure of the quantum random walk was not previously addressed to our knowledge. The strong dependence of the trapped states structure on the at least one marked element and the walker parameters also suggests a scenario of protecting the quantum search method against random disorder by an appropriate fine-tuning of marking parameters for the searched element.

[0069] Protection against disorder means that out of all possible random walk parameters only a very small subspace has overlap with the eigenstates of the quantum state of the system. This subspace of random walk parameters is localized close to optimal random walk parameters. In other words, the peaks in Figure 1 and cross-sections of the colored spots in Figure 3 have narrow widths.

[0070] Disorder is represented by a small number of individual nodes with different quantum random walk parameters compared to the other parts of the system, which would not result in localization as the search method is performed, however the search output of these specific nodes would actually be correct when employed.

[0071] In that view, protection against disorder would mean finding the marked elements correctly in a system not having perfectly optimized all the nodes. It would also mean that only “strong” disorder (meaning a large percentage of all the nodes of the system would need to not result in localization) might lead to not finding the marked element correctly, even while the correct quantum random walk parameters are employed.

[0072] The quantum random walk employed in this search algorithm is introduced as a way to simulate topological insulators. Its single time step: is defined via three rotations of spin and subsequent translations:

[0073] The translation operations shift the wavefunction’s spin-up components by +Vj and spin-down by -Vj. The vectors vi,2,3 can for example be defined on a triangular lattice as vi = (1 , 0), V2 = (1 / 2, - 3 / 2) and v3 = (1 / 2, — 3 / 2). The spectrum of quasi-energies of this QRW without marked elements has two particle-hole symmetric bands due to matrix U(0i,2) being real. The corresponding states of ±E energy are related by complex conjugation = (4^)*. Depending on the quantum random walk parameters 01 ,2, such QRW realizes distinct topological phases with Chern numbers of Floquet- type bands C = ±1 , 0. The QRW spectrum is defined as Floquet-type quasi-energies and thus is limited to the (-TT, TT) interval. The gap closing happens either at zero energy or at the ends of the interval on the phase separation lines as shown in Figure 3, and at 02 = 0.

[0074] In this search method, the goal is defined as a problem of finding at least one marked element on a finite 2D lattice of size > / N unit cells starting from an initial quantum state |i). As prove of this search method, a computer simulation is used, wherein periodic boundary conditions are implemented to avoid the appearance of edge states. The marked element node 1000 is marked by parameters ©def 1 ,2 that are different from the surrounding ©1,2. To estimate the possibility of a given quantum walker to perform a search efficiently, the entire parameter range ©1,2 is scanned for a given marked element values ©def 1 ,2. Then the performance of that particular set of quantum random walk parameters is measured by its search probability. This is defined as a peak maxte[o,T] Pdef (t) in the time-dependent probability density of the quantum state at the marked element node 1000:

[0075] Pdef (t) = , where |d) =a=u |x = def, a) is the marked element coordinate operator with both components of spin o, and |i) is the initial quantum state of the system defined as |i) = 1 / ( / (2N)) ^a,x |x, o) with summation over the entire system.

[0076] When scanning the entire quantum random walk parameter space, localizations of efficient search in parameters form positioned in two distinct types of regions: namely, inside topological phases and on phase separation lines. These sets of quantum random walk parameters 10, 20 in both regions show search time behavior that goes from T ~ > / N dependence to T ~ const, and are independent of system size.

[0077] In Figure 5 the search probability dependence for two specific random walker parameters 10, 20 is compared against the average density and Grover’s search algorithm as a function of quantum system size. It is noted that the constant search time regime corresponds to Pdef (t) = O(1 / N) with a large numerical prefactor. This decrease of search probability with system size N is faster than Grover’s search algorithm, where Pdef Grover = O(1 / log N). The search probability dependence for the method of the disclosures takes on the form O(1 / N) due to the trapped states having finite support. The system size growing beyond the trapped state support size corresponds to the asymptotic behavior of the search time to a constant value.

[0078] To check the dependence of the search probability on the marked element parameters for given quantum random walk parameters, relating to the tuning of the at least one marked element 1000 in step v) of the method, a 2D marked element parameter space with the quantum random walk parameters of walker 1 10 are calculated.

[0079] For the example explored here, the search probability is shown as a function of ©def i, depicted in Figure 6, wherein a clear single peak arises.

[0080] To describe the origin of different speed-ups in this system and why the regions of efficient search represent a small fraction of parameter space, we analyze the spectrum and localization properties of eigenvectors of a unitary operator of system with one marked element. The probability evolution at a marked element site at discrete times t can be rewritten through the unitary operator of QRW, The U notation is introduced in order to underline that unitary operator takes into account the marked element node 1000. Substituting the eigenbasis decomposition of the U operator, U|n, ±) = ei±En|n, ±), we have

[0081] F,fe / (t) = (<*l £ e^ ln. AXn. AH) " r A^i

[0082] The asymptotic behavior of the search time towards a constant value, independent of the system size after reaching a specific system size suggests that the properties of the quantum states that mainly contribute to the last equation do not change with system size. This feature directly corresponds to the definition of trapped states staying localized around its initial starting node 100 as time evolves. The contribution of different eigenstates of U to the evolution of the probability P(t) is examined, in Figure 7 by comparing the product of overlap |<d| n, A)(n, A|i>|2. The panels in Figure 7 correspond to different marked element parameters denoted by vertical dashes shown in Figure 6. The scale of the x-axis in each panel of Figure 7 shows that marked element parameters with higher search probability also correspond to higher overlap product for a few states. In this case of the particular example, this corresponds to two pairs of particle-hole symmetric states. To further verify this observation, the maximal value of overlap product is compared as of a single eigenstate of U with the normalized probability dependence max[Pdef (t)] as function of ©def i. The comparison is made by taking a single pair of states that maximize the overlap with a set of indices M = {nmax}, as well as two pairs of states M = {n max, Hsecond max }. The results presented in Figure 6 indicate that while M — {nmax} the selection overestimates the region for ideal searching parameters. Whereas, when M = {nmax, nmax2} the selection very precisely describes the high search probability peak.

[0083] To further check this correspondence between the evolution described by two pairs of states with maximal overlap and the max probability distribution, different marking parameters as ©def 1,2 = (5TT / 8, TT / 2) were chosen. Then the same analysis was performed, and the resulting graph is plotted in Figure 8. Thereby proving that this method can be employed in different settings as well. The correspondence for a test system size with = 40 shows that efficient search for such split-step QRW is mostly determined by the appearance of two pairs of trapped states having large overlap with both the marked element and the initial state.

[0084] To further exemplify the localization behavior of the search algorithm, the time evolution defined by two pairs of trapped states that maximize overlap criteria with the evolution of the probability of finding the marked element is shown in Figure 9.

[0085] Here it is observed that U has the same particle-hole symmetry as U(0i,2) because it is a real matrix. In addition, by calculating the overlap products denoted as wa±,j = (d, cr|j><j|i> for each spin o =f, | and pair of trapped states j at the marked element separately, a phase difference w4,j = iwT+,j and corresponding w<,j = -iwT.,j.

[0086] This symmetry is only present at the marked element site and is a property of the trapped states localizing around the marked element. Thus, the evolution of both spin components at the marked element generated by the trapped states is described as follows where the particle-hole symmetry is taken into account in by the relation wa-,j = (wa+,j)*. It is important to emphasize that after calculating the probability at a marked element, one finds that the two spin components exactly cancel oscillating parts of each other for a single value of j due to the different phase of oscillations between sin and cos in the latter two equations, resulting in with sign(o) = ±. Thus, the contribution of each single pair of trapped states is constant and due to the initial condition of the uniformly distributed state over the entire system, it could not describe the peak in probability. However, the crucial contribution comes from the superposition between two pairs of trapped states and is manifested in the last oscillating term in the total density evolution at the marked element:

[0087] The period of this oscillation is different from the fast oscillations of each trapped states and defined by the energy difference between the positive energy levels of the two pairs of trapped states (Ei - E2). The evolution of the probability on the marked element shows that such simple the structure of oscillations very well.

[0088] In Figure 9 two graphs are shown representing one specific set of random walk parameters 10 in a quantum system having a marked element and not having a marked element. Another way to check the agreement of the predicted period of the oscillations can be obtained through Fourier transform. The agreement with period defined by (E1 - E2) has been shown to have an error below 2% on the timescale of evolution T = 5000 for a quantum system with > / N = 40. Note that such definition of a period through the difference of eigenstates of operator is different from the common one used in Grover’s search algorithm. Specifically, the period definition used here is between states from different particle-hole symmetric pairs rather that between states from single pair.

[0089] Lastly in Figure 10, different 2D lattices for the quantum system are presented, which could be selected from a list not limited to, triangular, cubic, hexagonal, oblique, rhombic, or rectangular. These different lattices will show different topological properties and therefore will also obtain different optimal quantum random walk parameters. The lattice is determined by a plurality of translation vectors vx200.

[0090] The search method presented in this disclosure allows the operator to perform a quantum search on at least one marked element to be performed fast and efficiently through the use of topological split-step quantum random walks. The proposed concrete implementation of this search consists of a few parts, namely

[0091] • Checking the appearance of pair of trapped states near a marked element and overlap product maximization criteria for a small system;

[0092] • applying the same encoding of the searched site in the large system with the same walker parameters;

[0093] • initializing a uniform quantum state in the large system; • examining the probability density peak as a function of time to reveal the searched element.

[0094] This method admittedly relies on the proper tuning of the walker and the marked element parameters. At the same time, this property also allows for a protection against disorder with 0i parameter values being not within efficient search interval.

[0095] Finally, This search method demonstrates the best achievable scaling of the search time independent of system size, compared to any known in the art methods for searching unstructured databases.

[0096] Interestingly, in the search algorithm we present here the ‘structure’ is uncovered in a quantum way, i.e. it is not known a-priori where the marked elements are localized. The method of the disclosure is also suitable for reading out fast detectors, where marked elements are detector hits. Our search algorithm, as formulated in this work, can be immediately tested in photonic and synthetic lattice quantum walk experiments that realize topological quantum random walks.

[0097] Reference Numbers

[0098] 10 First set of random walker parameters / Walker 1

[0099] 20 Second set of random walker parameters / Walker 2

[0100] 100 node 200 translation vector

[0101] 200i, 2002, 2OO3 first, second, third, etc. translation vector

[0102] 500 topological phase

[0103] 501 topological phase separation line

[0104] 1000 at least one marked element

Claims

CLAIMS1. A method for searching at least one locally marked element defined by marking parameters within a 2D lattice of a quantum system, wherein the quantum system is operated in discrete time and wherein the 2D lattice is arranged for supporting quantum random walks which are defined by quantum random walk parameters, the method having a pre-set time comprises the steps of:1) calibrating the quantum system by: i) initializing a quantum state in the quantum system on at least one node; ii) performing one step of the split-step quantum random walk on the quantum state; iii) obtaining probability density information of the quantum state through overlap with eigenstates of a unitary operator; iv) repeating steps ii) and iii) to determine quantum random walk parameters based on the highest probability in the probability density information of the quantum state;2) searching the quantum system for the at least one marked element by: v) tuning the marking parameters of the at least one locally marked element; vi) initializing a uniformly distributed quantum state on the entire quantum system; vii) performing a plurality of split-step quantum random walks for the uniformly distributed quantum state; viii) after the pre-set time has lapsed, converting the uniformly distributed quantum state of step vii) into an observable.

2. The method according to claim 1 , wherein step 1) of calibrating the quantum system is performed on a processor simulating a physical quantum system and wherein step 2) of searching the quantum system is performed on the actual physical quantum system.

3. The method according to any of the previous, wherein the marking parameters of step v) are selected based on the highest search probability.

4. The method according to any of the claims 1 or 2, wherein the marking parameters of step v) are selected based on a-priori knowledge of the quantum system.

5. The method according to claim 4, wherein the marked element is configured with localized marking parameters as a localized lattice marked element node.

6. The method according to any of the previous claims, wherein the localized marking parameters are fixed during steps iii) and iv), preferable the localized marking parameters are also fixed during steps vi) through viii).

7. The method according to any of the previous, wherein the localized marking parameters and split-step quantum random walk parameters form distinct topological phases.

8. The method according to any of the previous claims, wherein the split-step quantum random walk includes a set of shift and spin flip operations applied sequentially to each state of the system at each time step.

9. The method according to any of the previous claims, wherein the at least one locally marked element comprises a plurality of locally marked elements, wherein the marking parameters further comprise a hyperparameter which indicates the relationship between each of the locally marked elements.

10. The method according to any of the previous claims, wherein the observable of step viii) of converting the uniformly distributed quantum state into an observable, is selected from a list not limited to spin, magnetic, or photonic.

11. The method according to any of the previous claims, wherein the 2D lattice of a quantum system is selected from a list not limited to, triangular, cubic, hexagonal, oblique, rhombic, or rectangular.

12. The method according to any of the previous claims, wherein step viii) of converting the uniformly distributed quantum state into an observable is performed by classical traditional quantum state tomography.

13. The method according to any of the claims 1-11 , wherein step viii) of converting the uniformly distributed quantum state into an observable is performed by shadow tomography.

14. The method according to any of the previous claims, wherein step iv) of repeating steps ii) and iii) is performed until a further pre-set time has lapsed.

15. The method according to any of the claims 1-13, wherein step iv) of repeating steps ii) and iii) is performed until the probability density information of the quantum state of successive rounds of repeating steps ii) and iii) has changed less than 1 %.

16. A method for calibrating a search of at least one locally marked element defined by marking parameters a quantum system, wherein the quantum system is operated in discrete time and wherein the quantum system is arranged for supporting quantum random walks which are defined by quantum random walk parameters, the method having a pre-set time comprising the steps of:- initializing a quantum state in the quantum system on at least one node;- performing one step of the split-step quantum random walk on the quantum state;- obtaining probability density information of the quantum state through overlap with eigenstates of a unitary operator;- repeating the steps of performing one step of the split-step quantum random walk and the step of obtaining probability density information, such that quantum random walk parameters can be determined based on the highest probability in the probability density information of the quantum state.

17. A method for searching at least one locally marked element defined by marking parameters a quantum system, wherein the quantum system is operated in discrete time and wherein the quantum system is arranged for supporting quantum random walks which are defined by quantum random walk parameters, the method having a pre-set time, the method comprising the steps of:- tuning marking parameters of the at least one locally marked element;- initializing a uniformly distributed quantum state on the entire quantum system;- performing a plurality of split-step quantum random walks for the uniformly distributed quantum state;- after the pre-set time has lapsed, converting the by quantum random walk affected uniformly distributed quantum state into an observable.

18. The method of calibrating according to claim 16 or the method of searching according to claim 17, wherein quantum system comprises a 2D-lattice.

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