Method, system, medium and program product for measuring critical exponents of quantum phase transitions

Abstract

The present disclosure relates to a method, a system, a medium, and a program product for measuring critical exponents of quantum phase transitions, and relates to the field of quantum computing and quantum simulation. The method includes: for each candidate value of the critical exponent, determining multiple value combinations of the change rate of the system parameters and the system size of the quantum system such that the candidate value, the change rate, and the system size satisfy a first condition; for each value combination of the change rate and the system size: linearly changing the system parameters at the change rate such that the quantum system undergoes a quantum phase transition; measuring a first correlation function of the quantum system; performing a scale transformation on the first correlation function that satisfies a second condition to obtain a second correlation function; for each candidate value, comparing the differences between the second correlation functions; and selecting the candidate value that minimizes the difference as the measurement result of the critical exponent. This method can overcome the problems of short coherence time and limited system size of the quantum system and improve the measurement accuracy.

Description

Technical Field

[0001] The present disclosure relates to the field of quantum computing and quantum simulation, and particularly to a method and system for measuring critical exponents during the phase transition of a quantum system. Background Art

[0002] Quantum phase transition ( Quantum Phase Transition, QPT ) is a phase change caused by changing certain physical parameters under conditions of absolute temperature or near absolute temperature. Different from classical thermal phase transitions, quantum phase transitions are mainly driven by quantum fluctuations. In a quantum phase transition, critical exponents are used to characterize the scaling behavior ( Scaling behavior ) of a quantum system near the critical point, reflecting the universality principle in quantum mechanics, that is, phase transitions under different microscopic mechanisms can have similar critical behaviors. Specifically, near the phase transition critical point, the scaling behavior of the order parameter in a quantum system can be described by critical exponents ( Critical exponent ). Measuring critical exponents through experimental means is of great significance for studying the nature of quantum phase transitions and the universality of the system, and can also provide guidance for practical applications such as quantum material design and quantum phase transition control. For example, some critical exponents describe the rate at which the energy gap of a substance changes as the magnetic field or interaction strength approaches the critical point. The success rate of the quantum annealing algorithm highly depends on the size of the energy gap at the phase transition point. In this case, the measured critical exponents can be used to judge the time complexity of the algorithm.

[0003] However, current measurements of the critical exponents of quantum phase transitions face difficulties such as short coherence times of quantum systems and limited system sizes, resulting in certain errors and low accuracy in the measurement results. Summary of the Invention

[0004] The object of the present invention is to propose a method and system for measuring the critical exponents of quantum phase transitions, which can overcome the problems of short coherence times of quantum systems and limited system sizes and improve the measurement accuracy.

[0005] According to one aspect of the present disclosure, the present disclosure provides a method for measuring the critical exponents of the quantum phase transition of a quantum system, including: for each candidate value μ of the critical exponent μ' , determining multiple value combinations of the change rate s of the system parameters of the quantum system and the system size L of the quantum system, such that the candidate value μ' , the change rate s and the system size L satisfy the first condition; for each value combination of the change rate s and the system size L , performing a quantum phase transition measurement, including: at the change rates linearly change the system parameters of a quantum system having corresponding system sizes in the value combinations L such that the quantum system crosses a phase transition point from a disordered phase into an ordered phase; measure a first correlation function of local order parameters at two positions in the quantum system with respect to the spatial distance between the two positions g in the ordered phase; and perform a scaling transformation on the first correlation function r satisfying a second condition to obtain a second correlation function G G wherein the first condition and the second condition are obtained according to a modified scaling law of the first correlation function G at finite system sizes; for each of the candidate values ' compare the differences between the second correlation functions G at the multiple value combinations of the change rate μ' and the system size s ; and select the candidate value L that minimizes the difference as the measurement result of the critical exponent G' According to another aspect of the present disclosure, there is provided a system for measuring a critical exponent of a quantum phase transition of a quantum system, including: a processor; and a control device coupled to the processor, wherein the processor is configured to: for each candidate value μ' of the critical exponent μ determine multiple value combinations of the change rate

[0006] of the system parameters of the quantum system and the system size μ of the quantum system such that the candidate value μ' , the change rate s and the system size L satisfy a first condition; the control device is configured to: for each value combination of the change rate μ' and the system size s perform a quantum phase transition measurement, including: linearly changing the system parameters L of a quantum system having a corresponding system size in the value combination at the change rate s such that the quantum system crosses a phase transition point from a disordered phase into an ordered phase; and measuring a first correlation function L of local order parameters at two positions in the quantum system with respect to the spatial distance between the two positions s in the ordered phase; the processor is further configured to: for each value combination of the change rate L and the system size g perform a scaling transformation on the first correlation function r of local order parameters at two positions in the quantum system with respect to the spatial distance between the two positions G ; and for each value combination of the change rate s and the system size L ​G Perform a scaling transformation that satisfies the second condition to obtain a second correlation function G' , where the first condition and the second condition are obtained according to the modified scaling law under a finite system size; for each of the candidate values μ' , compare the second correlation functions G s under multiple value combinations of the changing speed L and the system size ' ; and select a candidate value such that the difference satisfies a certain condition μ' as the measurement result of the critical exponent μ .

[0007] According to another aspect of the present disclosure, there is provided a computer-readable storage medium storing computer program instructions, which when executed by one or more processors, cause the one or more processors to execute the foregoing method.

[0008] According to another aspect of the present disclosure, there is provided a computer program product including computer-executable instructions, which when executed by one or more processors, cause the one or more processors to execute the foregoing method.

[0009] Through the following detailed description of the exemplary embodiments of the present invention with reference to the accompanying drawings, other features and advantages of the present invention will become clearer. BRIEF DESCRIPTION OF THE DRAWINGS

[0010] The drawings forming a part of the specification depict embodiments of the present disclosure and, together with the specification, are used to explain the principles of the present disclosure.

[0011] Referring to the accompanying drawings, the present disclosure can be more clearly understood from the following detailed description, where:

[0012] Figure 1 shows a method for measuring the critical exponent of a quantum phase transition of a quantum system according to some embodiments of the present disclosure;

[0013] Figure 2 shows a schematic diagram of a one-dimensional blockade chain composed of Rydberg atoms according to some embodiments of the present disclosure;

[0014] Figure 3 shows a more detailed flowchart of the quantum phase transition measurement according to some embodiments of the present disclosure;

[0015] Figure 4 shows the relationship between the ground state correlation length and the control parameter of a quantum system in some examples;

[0016] Figure 5Schematic diagrams of the second correlation function for combinations of values of different change speeds and system sizes in some examples are shown;

[0017] Figure 6 Schematic diagrams showing the relationship between the differences between the second correlation functions and the candidate values of the critical exponents in some examples are shown;

[0018] Figure 7 Shows an exemplary system that can be used to implement embodiments according to the present disclosure. Detailed implementation manners

[0019] Various exemplary embodiments of the present disclosure will be described in detail below with reference to the accompanying drawings. It should be noted that: unless otherwise specifically stated, the relative arrangements of components and steps, numerical expressions and values set forth in these embodiments do not limit the scope of the present disclosure.

[0020] The following description of at least one exemplary embodiment is merely illustrative in nature and is in no way intended to limit the present disclosure, its application, or uses. That is, the structures and methods herein are shown in an exemplary manner to illustrate different embodiments of the structures and methods in the present disclosure. However, those skilled in the art will understand that they merely illustrate exemplary ways in which the present disclosure can be implemented, rather than exhaustive ways. In addition, the drawings are not necessarily drawn to scale, and some features may be enlarged to show details of specific components.

[0021] Techniques, methods, and devices known to those of ordinary skill in the relevant art may not be discussed in detail, but where appropriate, the techniques, methods, and devices should be regarded as part of the specification.

[0022] On current artificial quantum platforms, the measurement of quantum phase transition critical exponents faces the difficulty of the limited coherence time of quantum systems. Measuring the quantum phase transition critical exponents using the quantum Kibble-Zurek effect solves the difficulty of short coherence time through rapid dynamical evolution. The quantum Kibble-Zurek effect requires the system parameters of the quantum system to linearly cross the quantum phase transition point at a fixed change speed.

[0023] Near the phase transition point, physical processes satisfy various scaling laws, and the exponents can be given by the dimensions of the physical quantities. Define the dimension of the spatial distance (length scale) r to be 1, the dimension of the time scale t to be z , and the dimension of the parameter g to be -1 / υ . Since the change speed s of the system parameter is the system parameter g divided by the time t , its dimension is -1 / υ - z . Generally, its dimension is defined as- 1 / μ , obtain . The scaling law refers to the behavior of a physical quantity remaining invariant under a scaling transformation , , where b is an arbitrary scaling factor, b and the exponent of μ is the dimension of the corresponding physical quantity. To measure the critical exponent G , the two-point correlation function G of the quantum system can be measured. This two-point correlation function G is the joint correlation average function of the local order parameters measured at two positions in the quantum system. Since the two-point correlation function G is a function of the spatial distance r between two positions and the change speed s , the following can be obtained:

[0024] Equation (1).

[0025] According to this scaling law, the method for measuring the critical exponent μ includes the following steps:

[0026] Step S01: Determine each candidate value of the critical exponent μ and determine different values of the change speed s ;

[0027] Step S02: For each value of the change speed s , perform a quantum phase transition measurement, including:

[0028] Sub-step (1): Linearly change the system parameter s of the quantum system at the change speed g to make the quantum system cross the phase transition point from the disordered phase into the ordered phase;

[0029] Sub-step (2): Measure the two-point correlation function r of the local order parameters at two positions in the quantum system with respect to the spatial distance G(r) between the two positions;

[0030] Sub-step (3): Perform a scale transformation on the two-point correlation function G(r) to obtain G(s μ r) ;

[0031] Step S03: For each candidate value of the critical exponent μ , calculate s at different values of the change speed G(sμ r) the difference between;

[0032] Step S04: Select a candidate value that makes the difference satisfy certain conditions as the critical exponent μ of the measurement result.

[0033] However, the scaling law equation (1) of the quantum Kibble-Zurek effect was proposed for the case where the system size is infinite. In actual measurements, the size of the quantum system is finite. When the system size L is finite, the relationship of the above equation (1) will no longer be accurate, and errors will be introduced in the measurement μ resulting in inaccurate critical exponents of the measurement.

[0034] Some embodiments of the present disclosure are based on the following new understanding: For a finite-size system, the system size L should also be the independent variable of the two-point correlation function G and needs to satisfy to ensure that the two-point correlation function G remains unchanged, that is, at this time the two-point correlation function G satisfies the modified scaling law under a finite system size:

[0035] Equation (2).

[0036] Therefore, when changing the speed s , the system size L should be changed accordingly μ to obtain the correct scaling law. According to the above equation (2), embodiments of the present disclosure propose that for each possible candidate value of the critical exponent L and s , different s μ L are selected, but G(s μ r) is fixed as a constant, and the difference between the two-point correlation functions at the rescaled distances μ is measured, and the candidate value that makes the difference satisfy certain conditions is used as the measurement result of the critical exponent

[0037] It can be seen that the method for measuring the critical exponent of quantum phase transition in the present disclosure also adopts a dynamic measurement method based on the quantum Kibble-Zurek effect, but takes into account the influence of the finite system size on the scale invariance. Under the finite system size, it is necessary to correct the traditional scaling law to obtain the scaling law under the finite system size. Specifically, when the change rate of the system parameters changes, the system size also needs to change accordingly to obtain the correct scaling law. Designing the measurement process based on the corrected scaling law can reduce the error of the measured critical exponent.

[0038] The following will specifically introduce a method for measuring the critical exponent of quantum phase transition of a quantum system according to some embodiments of the present disclosure in conjunction with Figure 1 Specifically introduce a method for measuring the critical exponent of quantum phase transition of a quantum system according to some embodiments of the present disclosure.

[0039] The measurement method 100 starts at step 110. In step 110, for each candidate value μ of the critical exponent μ' , a plurality of value combinations of the change rate s of the system parameters of the quantum system and the system size L of the quantum system are determined, such that the candidate value μ' , the change rate s and the system size L satisfy the first condition.

[0040] In the present disclosure, a quantum system refers to a many-body system operating based on the principles of quantum mechanics, and its behavior is characterized by quantum phenomena such as quantum states, quantum superposition, quantum entanglement, and quantum decoherence. In a quantum system, the states of microscopic particles (such as electrons, photons, atoms, etc.) can be described by quantum states, and these states can be manipulated through quantum operations. Quantum systems can be realized according to different underlying physical mechanisms. In some embodiments, a quantum system can include any one of a superconducting quantum system, an ion trap quantum system, a photonic quantum system, a neutral atom quantum system, or a semiconductor quantum system. The measurement method of the present disclosure can be applied to quantum platforms of various technical routes. The quantum system of the present disclosure is a quantum system that can be applied to fields such as quantum computing, quantum communication, quantum sensing, and quantum simulation.

[0041] In the present disclosure, a quantum phase transition refers to a phase transition phenomenon that occurs at absolute zero (or near absolute zero). Different from classical phase transitions that are achieved by changing temperature, quantum phase transitions are achieved by changing system parameters at absolute zero temperature. Quantum phase transitions describe the abrupt change of the ground state of a many-body system due to quantum fluctuations. In some embodiments, according to the universality principle, quantum phase transitions may include various types of second-order quantum phase transitions. In some embodiments, the critical properties of quantum phase transitions can be studied using various lattice statistical models. Common lattice statistical models include, but are not limited to, the Ising model, the XY model, or the Bose-Hubbard model, etc.

[0042] In the present disclosure, the critical exponent of a quantum phase transition reflects the power-law relationship between the change rate of the order parameter and the system parameters near the phase transition point. Near the critical point of a quantum phase transition, quantum fluctuations drive the divergence of the correlation length of the quantum system. Therefore, for different length scales, the quantum system satisfies a unified power-law relationship, and its critical exponent remains unchanged. A quantum phase transition is characterized by the change of the order parameter. The order parameter of the same quantum phase transition process is the same. In some cases, the same quantum phase transition process corresponds to a unique system parameter. That is, the quantum phase transition process can only be achieved by changing this unique system parameter. In other cases, the same quantum phase transition process corresponds to different system parameters. That is, the same quantum phase transition process can be achieved by changing different system parameters. Different system parameters can result in different critical exponents. The critical exponent to be measured in the present disclosure is the critical exponent related to the change rate of the system parameters, that is, the dynamic critical exponent of the system parameters (the critical exponent described above μ ), rather than the static critical exponent of the system parameters themselves (the critical exponent described above υ ).

[0043] Different quantum systems or quantum phase transition processes may have different system parameters and order parameters. For example, for a quantum system that satisfies the two-dimensional Bose-Hubbard model, its quantum phase transition process may be to change the relative magnitudes of the tunneling and interaction of particles (bosons) in the lattice, such that the system undergoes a quantum phase transition from the Mott insulating phase to the superfluid phase. The system parameters may be the tunneling amplitude or the interaction strength of the bosons. The order parameter is the density of the superfluid condensate.

[0044] In some embodiments, the quantum system includes a Rydberg atom array. A Rydberg atom array is a quantum system based on Rydberg atoms, where the atoms are excited to an excited state (Rydberg state) with a high principal quantum number of the electron (the principal quantum number n is much greater than 1, such as 50, 60, 70, 80, 90 or other values), and are arranged in optical tweezers or an optical lattice to form an array. Rydberg atoms have strong interactions and long lifetimes. When the distance between two Rydberg atoms is close enough, the excitation of one atom will prevent the excitation of the other atom, and this phenomenon is called Rydberg blockade. In a Rydberg atom array, by precisely controlling the positions and excitation states of the atoms, a highly programmable qubit array can be realized. Depending on the spatial arrangement of the optical tweezers or the optical lattice, the Rydberg atom array can be a one-dimensional, two-dimensional or three-dimensional array.

[0045] The atoms in the Rydberg atom array can be selected from alkali metal atoms (such as rubidium atoms, cesium atoms, etc.) or alkaline earth metal atoms (such as strontium atoms and ytterbium atoms, etc.).

[0046] In some further embodiments, the quantum system is a one-dimensional blockade chain composed of Rydberg atoms. A one-dimensional blockade chain is a special Rydberg atom array. In a one-dimensional blockade chain, the distances between adjacent atoms can be equal or unequal. The distance between adjacent atoms is less than the Rydberg blockade radius, but the distance between two atoms separated by one atom is greater than the Rydberg blockade radius. Therefore, it is impossible for adjacent atoms to simultaneously exhibit Rydberg excited states, that is, the simultaneous excitation of two adjacent atoms is "blocked". The atoms in the chain show an alternating distribution pattern of "ground state", "Rydberg state", "ground state", "Rydberg state"... as Figure 2 shown.

[0047] In the present disclosure, a system parameter is any physical parameter that can be changed to make the quantum system cross the quantum phase transition critical point. In some embodiments, when the quantum system is a Rydberg atom array, the system parameter can include the frequency of the excitation light applied to the quantum system. A Rydberg atom array can be regarded as a two-level many-body system. When excitation light is applied to this system, the photons interact with the atoms, driving the energy level transitions of the atoms in the system. Set an appropriate laser intensity and adjust the excitation light frequency so that the frequency detuning between the excitation light frequency and the energy level difference (corresponding frequency) between the two energy levels of the two-level system changes. When the detuning changes from a negative value (red detuning) to a positive value (blue detuning), the many-body system crosses the phase transition point from a disordered phase (all atoms are in the ground state) to an ordered phase (for example, a state where "ground state" and "Rydberg state" are alternately distributed in a one-dimensional blockade chain).

[0048] In addition to directly selecting the frequency of the excitation light, in some embodiments, the system parameters may further include other frequency-related physical quantities. For example, it may be the detuning amount of the frequency of the excitation light.

[0049] The change rate of the system parameter s represents the change of the system parameter per unit time. For example, if the system parameter is the frequency or the frequency detuning amount, then the change rate s is the change of the frequency or the frequency detuning amount per unit time.

[0050] System size L represents the linear length of the quantum system. For example, in a one-dimensional atomic array system, the system size L can be the macroscopic length of the atomic array. However, since the number of atoms is proportional to the length, in some cases, the number of atoms N can also be used as the representation of the system size L For a two-dimensional or three-dimensional atomic array system, the system size L can be any physical quantity representing the linear length, such as the length, width, height, or diagonal length of the atomic array. Similarly, the number of atoms in the corresponding dimension of the linear length N can also be used for representation. In some cases, the system size can also be characterized by the number of qubits in the quantum system L . For example, in a superconducting quantum system, the system size can be characterized by the number of qubits. Each qubit corresponds to a Josephson junction.

[0051] The first condition is obtained according to the modified scaling law under a finite system size. The modified scaling law under a finite system size is described by Equation (2) above. According to this equation, to satisfy scale invariance, the change rate s and the system size L need to satisfy:

[0052] s μ L = C , Equation (3)

[0053] where C is a constant.

[0054] For each candidate value of the critical exponent μ , μ' , C being a constant means that the values of the change rate s and the system size L are correlated with each other. When the change rate s changes, the system size L also needs to change accordingly, thus forming a relationship between the change rate s and the system size LMultiple value combinations. For each value combination, the change speed s has a certain specific value, while the system size L has a corresponding value. And for the critical exponents of the same quantum system μ each different candidate value μ' , C is also the same constant. In some embodiments, C can be any number of the order of magnitude from 1 to 10. C reflects the distance of the quantum system from the phase transition point. C The smaller the value of s , the smaller the value of the change speed C , the closer it is to the adiabatic state, and the closer the quantum system is to the phase transition point. For different quantum systems or quantum phase transition processes, the value of the constant

[0055] can be the same or different. Here, different quantum systems refer to quantum systems based on different physical technical routes, such as the superconducting quantum system, ion trap quantum system, photon quantum system, semiconductor quantum system, neutral atom quantum system, etc. mentioned above. s and the system size L for each value combination are measured for quantum phase transition. The quantum phase transition measurement uses the dynamic measurement method based on the quantum Kibble-Zurek effect introduced above. Specifically, it includes:

[0056] In sub-step 121, the system parameters s of the quantum system with the corresponding system size L in the value combination are linearly changed at the change speed g , so that the quantum system crosses the phase transition point from the disordered phase into the ordered phase.

[0057] Changing the system parameters linearly at the change speed means that the system parameters change uniformly with time. The value range formed by the initial value and the end value of the system parameters covers the phase transition critical point of the quantum system.

[0058] When the change speed s changes, the system size L also changes. For example, it may change from L 1 to L 2 . At this time, the system size of the quantum system needs to be changed from L 1 to L 2 . In some embodiments, sub-step 121 includes according to the change speed s and the system size Lthe system size in each value combination L Prepare a quantum system. For example, it can be according to the changed new system size L 2 Prepare a quantum system. Taking a Rydberg atom array as an example of the quantum system, if L 2 < L 1 , that is, the system size decreases, then a corresponding number of atoms can be discarded from the atom array. For example, the optical tweezers or optical lattice where these atoms are located can be turned off so that the atoms are no longer trapped in the optical potential well. If L 2 > L 1 , that is, the system size increases, then a corresponding number of atoms can be added according to the original atomic spacing of the atom array, so that the system size is enlarged in proportion. For example, the corresponding optical tweezers or optical lattice can be added proportionally according to the arrangement of the original optical tweezers or optical lattice, so that the corresponding number of added atoms are captured therein.

[0059] The disordered phase of a quantum system refers to the state where the quantum system is in a state of relatively chaotic arrangement or state of microscopic particles and has no long-range regularity. In the disordered phase, the system symmetry is not spontaneously broken. In the case where the quantum system is a Rydberg atom array, the disordered phase of this many-body system can be that the atoms in the atom array are all in the ground state.

[0060] The ordered phase of a quantum system is the state where the arrangement or state of microscopic particles in the quantum system shows certain regularity on the macroscopic scale. In the ordered phase, the system symmetry has spontaneous breaking.

[0061] The order parameter is used to describe the degree of order of a system. In the ordered phase, the order parameter is non-zero, while in the disordered phase, the order parameter is zero. Therefore, it is possible to determine whether a quantum system undergoes a phase transition into the ordered phase by detecting the order parameter. In some quantum systems, such as in the system of Rydberg atom arrays, the atoms at specific positions in the system (e.g., the lattice sites formed by optical tweezers or optical lattices) can be selected to be in the ground state or in the Rydberg state as the local order parameter. For example, the fluorescence detection technique can be used to detect whether the atoms at specific positions are in the ground state or the excited state, and then the local order parameter can be determined. The atoms in the excited state will return to a lower energy level or the ground state within a certain time and release photons simultaneously, emitting fluorescence. The state of the atoms is determined by detecting the intensity, wavelength, or distribution of the photons released by the atoms, and then the local order parameter is determined. After determining the local order parameter, the local order parameters at various positions in the system are combined as the overall order parameter of the system. For example, first detect whether the atoms at each position are in the ground state or the excited state. Then, for a many-body system, if the atoms at each position are all in the ground state, the corresponding order parameter is zero, while if there are specific positions where the atoms are not in the ground state, the corresponding overall order parameter is a non-zero value. In some other quantum systems, the local order parameter measured at any position is directly used as the overall order parameter. For example, for a quantum system that conforms to the Bose-Hubbard model, the density of the superfluid condensate at any position can be directly measured as the order parameter or the local order parameter.

[0062] In sub-step 122, the first correlation function of the local order parameters at two positions in the quantum system with respect to the spatial distance between the two positions is measured in the ordered phase. r of the G .

[0063] The first correlation function G is the joint statistical average function of the local order parameter at the first position taking the first specific value and the local order parameter at the second position taking the second specific value among the two positions. The first correlation function G is related to the spatial distance r between the two positions, and thus can also be expressed as G(r) . In the Rydberg atom array, if the distance between atoms is constantly d , then the value of the spatial distance r can be 0 , d , 2d , 3d , etc.

[0064] Each detection of the local order parameter is equivalent to a destructive measurement that collapses the wave function, i.e., the superposition state of the quantum system collapses to an eigenstate corresponding to the measurement result. Denote the atom at a certain position in the ground state as, for example, "-1" and in the excited state as, for example, "+1" for each detection. By performing multiple detections, calculate the correlation average of a certain combination of the occurrences of -1 and 1 at any two positions (e.g., i position and j position), i.e., <Z i Z j > , where Z i is a random variable of -1 or +1 at the i point, and Z j is a random variable of -1 or +1 at the j point, and the angle brackets <*> represent the statistical average.

[0065] Therefore, sub-step 122 may include: step 122a, measuring the values of the local order parameters at two positions in the quantum system in the ordered phase multiple times; and step 122b, calculating the joint correlation average for one of the combinations of the measured values of the local order parameters to obtain the first correlation function. Since each measurement of the local order parameter causes the state of the quantum system to collapse, to perform multiple measurements, it is necessary to execute step 121 after each measurement while keeping the current change rate s and the system size L unchanged. That is, after experiencing multiple cycles of "phase transition - measurement -... - phase transition - measurement", the first correlation function can be calculated.

[0066] Next, in sub-step 123, perform a scaling transformation on the first correlation function G that satisfies the second condition to obtain the second correlation function G' .

[0067] The second condition is obtained according to the modified scaling law for the finite system size. In some embodiments, the second condition includes determining G(r) about s μ' r the variation law. According to equation (2) above, for each candidate value μ of the critical exponent μ' , when the change rate s and the system size L change, perform an independent variable scaling transformation on the first correlation function G(r) obtained each time, while maintaining G(r) and rWith the mapping relationship between them remaining unchanged, the independent variable r is transformed into s μ' r to obtain G(r) Regarding s μ' r The variation law, expressed as the second correlation function G'(s μ' r) .

[0068] According to the corrected scaling law, if the candidate value μ' is the true critical exponent, then for different change rates s and system sizes L obtained under the combination of G'(s μ' r) should be equal in theory. Therefore, the critical exponent μ should be the candidate value s and system sizes L under the combination of different change rates G'(s μ' r) with the smallest difference μ' .

[0069] Figure 3 FIG. shows a more detailed flowchart of step 120, which takes into account the process of re-preparing the quantum system and realizing the phase transition due to the state collapse caused by measurement. Steps 301-step 303 are repeated a predetermined number of times. In each repetition (i.e., one round of loop), in step 301, the quantum system is first re-prepared according to the current value of the system size L , and then in step 302, the control parameter s is changed at the current value of the change rate g to cause the quantum system to undergo a quantum phase transition. Next, in step 303, the value of the local order parameter in the ordered phase is measured. In response to determining that the predetermined number of repetitions has been reached, the loop is exited and step 304 is entered. Since multiple values of the local order parameter have been obtained at this time, the first correlation function can be calculated based on the local order parameter, and then the second correlation function is obtained in step 305 based on the first correlation function. Steps 302, 303, 304, and 305 respectively correspond to steps 121, 122a, 122b, and 123 introduced above, and the relevant content will not be elaborated here.

[0070] Return to Figure 1 , in step 130, for each candidate value μ' , compare the second correlation function under multiple value combinations of the change rate s and the system size L ​G' The difference between.

[0071] For example, for each candidate value μ' , change the speed s and system size L Multiple value combinations of can be [ s 1 , L 1 ]、[ s 2 , L 2 ]……[ s m , L m ],in m is an integer greater than 1, then for each combination [ s i , L i ], 1≤ i ≤ m , the second correlation function can be measured G i ' , and compare multiple G i ' The difference between.

[0072] There are many ways to compare the differences between multiple functions. As a non-limiting example, if the function is a continuous function, each function can be plotted as a continuous curve, and the differences in the ordinates of the multiple curves under the same horizontal coordinate are compared. Statistical analysis is performed on the differences in the multiple ordinates to obtain the differences. For example, one method is to add the differences in the ordinates corresponding to all the horizontal coordinates. Other methods may include taking the average or calculating the variance. As another non-limiting example, if the function is a discrete value (data point), the discrete data points can be converted to a continuous function through curve fitting, and then the aforementioned method can be used. As another non-limiting example, if the function is a discrete value (data point), the horizontal coordinates of the data points of different functions can be made consistent by interpolating the data points of multiple functions, facilitating the comparison of the differences in the vertical coordinates. Specifically, the horizontal coordinate of a data point of one function is obtained by interpolation, and the corresponding data point in another function is obtained. Various other methods known in the art can also be used to compare differences, which will not be described in detail here.

[0073] In step 140, the candidate value that minimizes the difference is selected. μ' As a critical index μ The measurement results of the quantum phase transition of the quantum system have been achieved.

[0074] Advantages of embodiments according to the present disclosure lie in that, in the process of dynamically measuring the critical exponent of a quantum phase transition according to the quantum Kibble-Zurek effect, the influence of the finite system size on scale invariance is considered. Under finite system size, it is necessary to correct the traditional scaling law to obtain the scaling law under finite system size. Specifically, when the change rate of system parameters changes, the system size also needs to change accordingly to obtain the correct scaling law. Designing the measurement process based on the corrected scaling law can reduce the error of the measured critical exponent.

[0075] In addition, for the previous Kibble-Zurek measurement method (as described in the previous steps S01 - S04), there are great limitations on g the change rate s of system parameters, which cannot be too fast or too slow. If it is too slow, it will enter the adiabatic region (equilibrium state), and a long waiting time is required, which cannot be satisfied by the current coherence time of the quantum system, resulting in a large error in the measurement result. However, the measurement method according to the embodiments of the present disclosure overcomes the finite size effect to a certain extent and is insensitive to the adiabatic region. Therefore, good measurement accuracy can still be obtained when the change rate s is low.

[0076] It should be recognized that the above advantages do not need to be all realized in one or some specific embodiments, but can be partially distributed in different embodiments according to the present disclosure. Embodiments according to the present disclosure can have one or some of the above advantages, or alternatively or additionally have other advantages.

[0077] The following illustrates the measurement method according to some embodiments of the present disclosure through numerical simulations.

[0078] The selected quantum system is a one-dimensional blockade chain composed of Rydberg atoms. The control parameter g is the frequency detuning of the excitation light. The quantum phase transition at this time belongs to the Ising transition universality class. The critical exponent g related to the control parameter υ = 1. The relativistic z = 1, that is, the scaling of space and time is the same. According to the previous derivation of dimensions, the critical exponent s related to the change rate

[0079] .

[0080] Figure 4 shows the relationship between the ground state correlation length of the quantum system and the control parameter in this example. In Figure 4 - 5 , the number N of atoms in the atomic array system is used to L。If the system size of the quantum system L is infinite and the corresponding number of atoms N is also infinite, then the ground state correlation length of the many-body system ξ tends to diverge when the value of the control parameter g is close to the critical value (at this time the frequency detuning is 0). When the system size L is a finite value and the corresponding number of atoms N is also a finite value, the ground state correlation length ξ no longer tends to diverge near the critical value, but reaches a certain finite maximum value, and the ground state correlation length ξ no longer seems to satisfy the scale invariance with respect to the control parameter g However, if a scale transformation is performed, as can be seen from the small graph in the upper right corner Figure 4 , ξ / N with respect to g·N 1 / υ still satisfies the scale invariance, that is, it satisfies ξ / N = F(g·N 1 / υ ) , where F( ) is a function that satisfies the scale transformation invariance. This means that when N changes, the corresponding g and ξ also change proportionally, resulting in the separation of the curves in the figure. However, if a scaling transformation is performed, ξ / N with respect to g·N 1 / υ maintains the same power-law relationship for different system sizes, so the three curves coincide. This shows that for a quantum system with a finite system size L , the scaling law still holds, but it needs to be corrected.

[0081] Perform the dynamic numerical simulation of this quantum system according to Method 100 of the present disclosure. Figure 5 Shows the second correlation function μ obtained under different combinations of the change speed s and the system size μ' when the candidate value of the critical exponent L is taken as 0.5, where G' , s and L satisfy s μ' L = C , C is a constant. It can be seen that at this time, the differences between the three groups of G' curves are small and the degree of coincidence is high.

[0082] Figure 6 Shows a schematic diagram of the relationship between the difference between the second correlation functions G' and the candidate values μ' . It can be seen that whenμ' When the value is 0.524, the second correlation function G' has the smallest difference, that is, the second correlation function G' can best maintain scale invariance at this time. Therefore, the μ' value can be used as the measurement result of the critical exponent μ .

[0083] System parameters g The final state value in the ordered phase will also affect the scaling law. The modified scaling law of the previous equation (2) applies to the case where the final state value of the system parameter g is an infinite value. When the final state value is a finite value, the modified scaling law of equation (2) needs to be further adjusted to obtain:

[0084] Equation (4).

[0085] At this time, in addition to s and L satisfying s μ L = C , C being a constant, it is also necessary to satisfy: g υ L = D , D being a constant. Accordingly, some steps in method 100 need to be further restricted.

[0086] In some cases, the relativity of the quantum system z is known, then another candidate value of the critical exponent μ' can be calculated from υ , and the final state value of the system parameter g is determined accordingly according to υ' υ g L = D g . e .

[0087] In other cases, the relativity z is unknown. At this time, step 110 includes: for each candidate value μ' and each another candidate value of another critical exponent υ υ' , determine multiple combinations of values of the change speed s , the system size L and the final state value g e such that the candidate value μ' , the change speed s and the system size L satisfy the first condition, and another candidate value​υ' and the final state value g e and the system size L satisfy the third condition, which is obtained from a scaling law further adjusted according to the corrected scaling law under a finite system size.

[0088] critical exponent μ is associated with the change speed s while another critical exponent υ is associated with the final state value g of the system parameter g e The scaling law further adjusted according to the corrected scaling law under a finite system size is expressed as Equation (4) above. In some embodiments, the third condition includes g e υ' L = D , D where is a constant.

[0089] For each candidate value υ of the critical exponent υ' , D being a constant means that the final state value g e and the system size L are interrelated in their values. When the system size L changes, the final state value g e also needs to change accordingly. And according to the first condition, when the change speed s changes, the system size L also needs to change accordingly, thus forming multiple value combinations of the change speed s , the system size L and the final state value g e . When the candidate value μ' and the candidate value υ' are respectively specific values, for each value combination, the change speed s has a certain specific value, while the system size L and the final state value g e have corresponding values. And for each candidate value υ of the critical exponent υ' of the same quantum system D , D is also the same constant. In some embodiments D can be any number with an order of magnitude of 1 to 10. D reflects the distance of the quantum system from the phase transition point. The smaller the value ofD The values of can be the same or different. Here, different quantum systems refer to quantum systems based on different physical technical routes, such as the superconducting quantum system, ion trap quantum system, photon quantum system, semiconductor quantum system, neutral atom quantum system, etc. mentioned above.

[0090] Step 120 includes: For the changing speed s , system size L and final state value g e perform quantum phase transition measurements for each combination of values. Here, the quantum phase transition measurement still includes steps 121 to 123, but it should be noted that the end point of the quantum phase transition is restricted at this time. Specifically, in step 121, after changing the control parameter g to make the quantum system undergo a phase transition, the value of the control parameter g in the ordered phase is g e .

[0091] Step 130 includes: For each combination of each candidate value μ' and each candidate value υ' , compare the differences between the second correlation functions under multiple combinations of the changing speed s , system size L and final state value g e . Each candidate value μ' can correspond to multiple candidate values υ' , and for each combination of μ' and υ' , a series of second correlation functions can be obtained and the differences between them can be compared.

[0092] Step 140 includes: Select the candidate value μ' and the candidate value υ' in the combination that makes the difference the smallest, and use the candidate value μ' as the measurement result of the critical exponent μ .

[0093] Figure 7 shows a block diagram of the composition of the system 700 according to an embodiment of the present disclosure. The system 700 can be used to execute the method for measuring the critical exponent of quantum phase transition according to an embodiment of the present disclosure, such as method 100. As Figure 7 shown, the system 700 may include a processor 701 and a control device 702.

[0094] The control device 702 is used to perform various controls on the quantum subsystem to be measured and measure / monitor it. The control device 702 may include, but is not limited to, a CPU, a hardware microprocessor, a hardware processor, a multi-core processor, a single-core processor, a microcontroller, an application-specific integrated circuit (ASIC), a DSP, or other similar processing devices, and is capable of executing any type of instructions, algorithms, or software for performing operations and functions for controlling and measuring a quantum system according to the embodiments described in the present disclosure. The control device 702 may be various implementations of a digital circuit system, an analog circuit system, or a mixed-signal (combination of analog and digital) circuit system that performs functions in a computing system. The control device 702 may include, for example, an integrated circuit (IC), a part or circuit of a single processor core, an entire processor core, a single processor, a programmable hardware device such as a field-programmable gate array (FPGA), and / or a system including multiple processors. To cooperate in implementing the control and / or measurement functions of the quantum system, the control device 702 may be communicatively coupled to the quantum system through various actuators and / or sensors, such that the quantum system can ultimately be controlled or measured by the control device 702. For example, in the case where the quantum system is a Rydberg atom array, the control device 702 may control an optical tweezer or an optical lattice to achieve the capture and / or manipulation of atoms; and / or the control device 702 may control the radiation of an excitation light beam, including intensity, frequency, duration, etc.; and / or the control device 702 may control a fluorescence detection system to take pictures of the atom array, etc.

[0095] The processor 701 is used to receive data from the control device 702, perform data processing operations, and transmit to the control device 702 the parameters required for it to control the quantum system. The processor 701 may be any type of processor for performing classical computing and may include, but is not limited to, one or more general-purpose processors or dedicated processors (such as dedicated processing chips).

[0096] The processor 701 and the control device 702 may cooperate to execute instructions to implement the various methods (including method 100) or their steps and processes according to the embodiments of the present disclosure described above. Specifically, the processor 701 may be configured to execute sub-step 123 in step 110, step 120, and steps 130 - 140; the control device 702 may be configured to execute sub-steps 121 - 122 in step 120. It should be recognized that the functions or operations of the processor 701 and the control device 702 may also be partially or fully implemented in either the processor 701 or the control device 702.

[0097] System 700 may also include or be connected to a memory 703, which can be any non-transitory storage device capable of implementing data storage, and can include, but is not limited to, disk drives, optical storage devices, solid-state memories, floppy disks, flexible disks, hard disks, magnetic tapes, or any other magnetic medium, compact disks, or any other optical medium, cache memories, and / or any other storage chips or modules, and / or any other medium from which a computer can read data, instructions, and / or code. The memory 703 may store various dynamic and static instructions and / or data for the processor 701 and / or the control device 702 to read and / or execute.

[0098] The processor 701 and the control device 702 may be physically separated from each other but communicatively coupled by wired or wireless means, or may exist as different modules or components of the same entity device. In the latter case, the processor 701 and the control device 702 may each be connected to or communicate with the bus 704 via one or more interfaces. The bus 704 may include, but is not limited to, an Industry Standard Architecture (ISA) bus, a Micro Channel Architecture (MCA) bus, an Enhanced ISA (EISA) bus, a Video Electronics Standards Association (VESA) local bus, and a PCI bus or a PCI-e bus, etc.

[0099] System 700 may also include I / O devices and / or network interfaces connected to or communicating with the bus 704. Examples of I / O devices may include, but are not limited to, keyboards, touchpads, mice, joysticks, or other pointing devices, microphones, speakers, displays, or printers, etc. The network interface may be any type of device or system capable of enabling communication with external devices and / or networks, and may include, but is not limited to, modems, network cards, infrared communication devices, wireless communication devices, and / or chip sets (such as Bluetooth™ devices, WiFi devices, WiMax devices, cellular communication facilities, etc.). The I / O devices and / or network interfaces may also be communicatively coupled to the processor 701 and / or the control device 702 and / or the memory 703 via the bus.

[0100] The present disclosure may be implemented as any combination of a device, a system, an integrated circuit, and a non-transitory computer-readable medium or a computer program product. One or more processors or control devices may be implemented as an integrated circuit (IC), an application-specific integrated circuit (ASIC), or a large-scale integrated circuit (LSI), a system LSI, a super LSI, or a super LSI component that executes some or all of the functions described in the present disclosure.

[0101] This disclosure includes the use of software, applications, computer programs, or algorithms. The software, applications, computer programs, or algorithms can be stored on a non-transitory computer-readable medium or computer program product to cause a computer, such as one or more processors, to perform the steps described above and in the figures. For example, one or more memories store the software or algorithm as executable instructions, and one or more processors can execute a set of instructions associated with the software or algorithm to provide various functions according to the embodiments described in this disclosure.

[0102] Software and computer programs (which may also be referred to as programs, software applications, applications, components, or code) include machine instructions for a programmable processor and can be implemented in a high-level procedural language, an object-oriented programming language, a functional programming language, a logic programming language, or an assembly or machine language. The term "computer-readable medium" refers to any computer program product, apparatus, or device that provides machine instructions or data to a programmable data processor, such as a magnetic disk, an optical disk, a solid state storage device, a memory, and a programmable logic device (PLD), including a computer-readable medium that receives the machine instructions as a computer-readable signal.

[0103] By way of example, a computer-readable medium can include a dynamic random access memory (DRAM), a random access memory (RAM), a read-only memory (ROM), an electrically erasable read-only memory (EEPROM), a compact disc read-only memory (CD-ROM), or other optical disk storage, a magnetic disk storage, or other magnetic storage devices, or any other medium that can be used to carry or store the desired computer-readable program code in the form of instructions or data structures and that can be accessed by a general or special purpose computer or a general or special purpose processor. As used herein, a disk or disc includes a compact disc (CD), a laser disc, an optical disc, a digital versatile disc (DVD), a floppy disk, and a Blu-ray disc, where disks typically reproduce data magnetically, while discs reproduce data optically by laser. Combinations of the above are also included within the scope of computer-readable media.

[0104] The subject matter of this disclosure is provided as an example of apparatus, systems, methods, and programs for performing the features described in this disclosure. However, other features or variations are also contemplated in addition to the above features. It is contemplated that the implementation of the components and functions of this disclosure can be accomplished with any emerging technology that may replace any of the above-described implementations.

[0105] In addition, the above description provides examples and does not limit the scope, applicability, or configuration set forth in the various embodiments of the present disclosure. Changes may be made to the function and arrangement of the elements discussed without departing from the spirit and scope of the present disclosure. Various embodiments may appropriately omit, substitute, or add various processes or components. For example, features described with respect to certain embodiments may be incorporated in other embodiments.

[0106] Similarly, although operations are depicted in the drawings in a particular order, this should not be construed as requiring that the operations be performed in the particular order shown or in sequential order, or that all illustrated operations be performed to achieve the desired result. In some cases, multitasking and parallel processing may be advantageous.

Claims

1. A method for measuring the critical exponent of a quantum phase transition of a quantum system, characterized in that The method includes: For each candidate value μ of the critical exponent μ' , determine multiple combinations of values of the rate of change s of the system parameters of the quantum system and the system size L of the quantum system, such that the candidate value μ' , the rate of change s and the system size L satisfy a first condition; For changing the speed s and system size L perform quantum phase transition measurements for each value combination, including: at the said change rate s linearly change the system parameters of a quantum system having a corresponding system size in the said combination of values L such that the quantum system crosses a phase transition point from a disordered phase into an ordered phase g ; Measure a first correlation function of local order parameters at two positions in the quantum system with respect to the spatial distance between the two positions in the ordered phase r ; and G ; and Perform a scaling transformation on the first correlation function G that satisfies the second condition to obtain a second correlation function G' , where the first condition and the second condition are obtained according to the modified scaling law of the first correlation function G under a finite system size; For each of the candidate values μ' , compare the differences between the second correlation function s at the multiple value combinations of the change speed L and the system size G' ; and Select the candidate value that minimizes the said difference μ' as the measurement result of the said critical exponent μ ​ Among them, the first condition is s μ' L = C , C is a constant. The scale transformation of the first correlation function G to satisfy the second condition includes: determining the variation law of the first correlation function G relative to s μ' r .

2. The method according to claim 1, wherein System parameters g The final value in the ordered phase is an infinite value.

3. The method according to claim 1, characterized in that, The final value of the system parameter g in the ordered phase is a finite value g e , Among them, for each candidate value μ of the critical exponent μ' , determine the change rate s of the system parameters of the quantum system L and multiple value combinations of the system size μ' of the quantum system, such that the candidate value s , the change rate L and the system size satisfy the first condition, including: For each candidate value μ' and each other candidate value υ of another critical exponent υ' , determine multiple combinations of values of the change rate s , the system size L and the final state value g e such that the candidate value μ' , the change rate s and the system size L satisfy a first condition, and the other candidate value υ' , the final state value g e and the system size L satisfy a third condition, where the third condition is obtained according to a scaling law adjusted based on a modified scaling law at a finite system size Among them, for changing the speed s and the system size L Performing quantum phase transition measurements for each value combination of For each combination of values of the change speed s, the system size L and the final state value g e perform the quantum phase transition measurement Among them, for each of the candidate values μ' , compare the differences between the second correlation functions s at the multiple value combinations of the change speed L and the system size G' , including: For each of the candidate values μ' and each of the candidate values υ' , compare the differences between the second correlation functions s at the change speed L , system size g e for the multiple value combinations of G' . Among them, a candidate value that minimizes the difference is selected μ' as the measurement result of the critical exponent μ including: Select the candidate value that minimizes the difference μ' and the candidate value υ' in the combination of μ' as the measurement result of the critical exponent μ .

4. The method according to claim 3, wherein The third condition is g e υ' L = D , D is a constant.

5. The method according to claim 1, characterized in that The quantum system is any one of a superconducting quantum system, an ion trap quantum system, a photonic quantum system, a neutral atom quantum system, or a semiconductor quantum system.

6. The method according to claim 1, characterized in that The quantum system includes a Rydberg atom array, the system parameter includes the frequency of the excitation light applied to the quantum system, the local order parameter includes the characterization of whether the atoms at each lattice point in the Rydberg atom array are in the ground state or the Rydberg state, and all the atoms in the quantum system are in the ground state in the disordered phase.

7. The method according to claim 1, wherein Measuring a first correlation function r of local order parameters at two positions in the quantum system with respect to the spatial distance between the two positions in the ordered phase r comprising: G including: Measuring the values of the local order parameters at the two positions in the quantum system in the ordered phase multiple times; and Calculate a joint correlation average for one of the combinations of the measured values of the local order parameter to obtain the first correlation function G .

8. The method according to claim 1, wherein Compare the second correlation function under the multiple value combinations of the changing speed s and the system size L The differences between G' include: Plot the second correlation function G' as a continuous curve; and Comparing the differences in the ordinates of the continuous curves corresponding to the multiple value combinations at the same abscissa.

9. The method according to claim 1, wherein For changing the speed s and system size L performing quantum phase transition measurements for each value combination, further comprising: According to the system dimensions in each of the value combinations L Prepare the quantum system.

10. A system for measuring the critical exponent of a quantum phase transition of a quantum system, characterized in that The system includes: A processor; and A control device coupled to the processor, wherein the processor is configured to: For each candidate value μ of the critical exponent μ' , determine the change rate s of the system parameters of the quantum system L and multiple value combinations of the system size μ' of the quantum system, such that the candidate value s , the change rate L and the system size satisfy the first condition; The control device is configured to: For changing the speed s and system size L perform quantum phase transition measurements for each value combination, including: at the rate of change s linearly change the system parameters of a quantum system having a corresponding system size in the value combination L such that the quantum system crosses a phase transition point from a disordered phase into an ordered phase; and g ​ Measure the first correlation function r of the local order parameters at two positions in the quantum system with respect to the spatial distance between the two positions in the ordered phase r G ,​ The processor is further configured to: For each combination of values of the change in speed s and the system size L perform a scale transformation on the first correlation function G to satisfy the second condition, obtaining a second correlation function G' , where the first condition and the second condition are obtained according to the modified scaling law under a finite system size; For each of the candidate values μ' , compare the differences between the second correlation functions s at the multiple value combinations of the change speed L and the system size G' ; and Select the candidate value that minimizes the said difference μ' as the measurement result of μ the said critical exponent Among them, the first condition is s μ' L = C , C is a constant. Performing a scaling transformation on the first correlation function G to satisfy the second condition includes: determining the variation law of the first correlation function G with respect to s μ' r .

11. The system according to claim 10, wherein The quantum system is any one of a superconducting quantum system, an ion trap quantum system, a photonic quantum system, a neutral atom quantum system, or a semiconductor quantum system.

12. The system according to claim 10, characterized in that, The quantum system includes a Rydberg atom array, the system parameter includes the frequency of the excitation light applied to the quantum system, the local order parameter includes the characterization of whether the atoms at each lattice point in the Rydberg atom array are in the ground state or the Rydberg state, and all the atoms in the quantum system are in the ground state in the disordered phase.

13. A computer-readable storage medium storing computer program instructions, characterized in that, When executed by one or more processors, the computer program instructions cause the one or more processors to execute the method according to any one of claims 1-9.

14. A computer program product, comprising computer-executable instructions, characterized in that, When executed by one or more processors, the computer-executable instructions cause the one or more processors to execute the method according to any one of claims 1-9.

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