A method and an algorithm for modeling a particle wave function using qubits on a quantum computer

Abstract

A method for modeling a particle wave function on a quantum computer using qubits. The method comprises spatially dividing the wave function of the particle into a system comprising at least two discrete elements represented respectively by a qubit; and using the at least two qubits in a quantum circuit to obtain a model of the particle wave function in such a way that in all of combinations of first and second states of the at least two qubits that the system may have, there is only one of the at least two qubits that has a first state which is different than the second state of each one of the other of the at least two qubits; or there is only one group of states of the at least two qubits that form a first state which is distinguishable with the second state of each one of the other of the at least two qubits. The each one of the states of the at least two qubits or group of states represents either a presence or an absence of the particle in the spatial discrete location.

Description

[0001] A METHOD AND AN ALGORITHM FOR MODELING A PARTICLE WAVE

[0002] FUNCTION USING QUBITS ON A QUANTUM COMPUTER

[0003] Technical field

[0004] The invention is in the field of quantum computing and more specifically in the field of modeling a particle wave function using qubits.

[0005] Background Art

[0006] The following prior art documents offering a general overview of quantum computing, quantum circuits, and quantum theory / quantum mechanics and may considered as textbooks on these subjects:

[0007] On the theory and fundamentals of quantum computing and existing algorithms: Title: Dancing with Qubits. Author: Robert S. Sutor. ISBN:978-l-83882-736-6.

[0008] On implementing quantum circuits on real quantum computers: Title: Quantum computing with Python and IBM. Author: Robert Loredo. ISBN: 978-1-83898-100-6.

[0009] On quantum theory: Title: Introduction to Quantum Mechanics. Authors: David S. Griffiths and Darrell F. Schroeter. ISBN: 978-1-107-18963-8

[0010] Summary of the invention

[0011] In a first aspect, the invention provides a method for modeling a particle wave function on a quantum computer using qubits. The method comprises spatially dividing the wave function of the particle into a system comprising at least two discrete elements represented respectively by a qubit; and using the at least two qubits in a quantum circuit to obtain a model of the particle wave function in such a way that in all of combinations of first and second states of the at least two qubits that the system may have, there is only one of the at least two qubits that has a first state which is different than the second state of each one of the other of the at least two qubits; or there is only one group of states of the at least two qubits that form a first slate which is disti nguishable with the second state of each one of the other of the at least two qubits. The each one of the states or group of states of the at least two qubits represents either a presence or an absence of the particle in the spatial discrete location.

[0012] In a preferred embodiment, the first state is 11 > and the second state is 10).

[0013] In a further preferred embodiment, the first state is representative of the presence of the particle and the second state is representative of the absence of the particle.

[0014] In a further preferred embodiment, the first state is 10) and the second state is 11).

[0015] In a further preferred embodiment, the group creates a pattern of 10) and 11 ) and the second state is 10).

[0016] In a further preferred embodiment, the group creates a pattern of 10) and 11) and the second state is 11). In a further preferred embodiment, the step of using comprises ary one of the items in a list comprising entangling, superposing, causing interference between.

[0017] In a second aspect, the invention provides a quantum circuit configured to implement an algorithm, the circuit comprising at least two qubits, whereby only one of the at least two qubits has a first state different than a second state of each one of all the others of the at least two qubits.

[0018] In a preferred embodiment of the quantum circuit, a qubits probability amplitude distribution corresponds to the probability density of an arbitrary wave function that is being modeled, whereby time and space evolution of the arbitrary wave function as well as interactions of several wave functions can be evaluated by applying an appropriate Hamiltonian to the quantum circuit in order to predict behavior of physical systems of particles.

[0019] Brief description of the drawings

[0020] The invention will be understood in view of the detailed description of preferred embodiments and in reference to the drawings, wherein

[0021] Figure 1 contains an example of a plot of a gaussian wave packet function of an electron,

[0022] Figure 2 contains a table of different states for 5 qubits,

[0023] Figure 3 contains an example of a simple quantum circuit according to prior art,

[0024] Figure 4 contains a graph of a probability distribution of different states that can be created after measurement, according to prior art,

[0025] Figure 5 contains an illustration of an example quantum algorithm for filtering according to the invention,

[0026] Figure 6 contains a table of states after the filtering by the quantum algorithm of Figure 5,

[0027] Figure 7 contains a table with states remaining after the filtering by the quantum algorithm of Figure 5,

[0028] Figure 8 contains a graph of probabilities for computational bases states,

[0029] Figure 9 contains an illustration of a further example quantum algorithm for filtering with the introduction of RY gates at the entry of the circuit and a modifying of the wave function amplitudes,

[0030] Figure 10 contains a graph of probabilities following from the circuit of Figure 9,

[0031] Figure 11 contains an illustration of a further example quantum algorithm making use of an X gate and a SWAP gate,

[0032] Figure 12 contains a graph of probabilities resulting from the circuit of Figure 11, and

[0033] Figure 13 contains a flowchart illustrating an example embodiment of a method for modeling a particle wave function on a quantum computer using qubits.

[0034] Same references may be used to reference same or similar features illustrated throughout the Figures.

[0035] Detailed description of preferred embodiments of the invention

[0036] The following starts by presenting some basic knowledge about particle physics. Wave function and the Schrodinger equation

[0037] In quantum mechanics, a particle is entirely defined by its wave function which is denoted . It is a function that belongs to the Hilbert space of complex numbers (the space of complex functions) and it evolves in space and time and is therefore denoted W (x, t). By definition, if we multiply W by itself, the result is a real function that gives the probability of finding a particle in a space S. Hence: where!F is the complex conjugate of and psis called the probability density of finding the particle in space S. By definition: meaning the particle is by definition somewhere. This condition adds restrictions to the wave function and it is called normalization process. We can then say that the wave function W is normalized. Figure 1 contains an example of a plot of a gaussian wave packet function W (x) of an electron as a function of a one-dimensional x- coordinate.

[0038] The key in predicting the behavior of a particle lies in calculating its wave function which obeys to the Schrodinger equation:

[0039] Knowing the evolution of the wave function of a particle enables the calculation of the different observables of the system. This has enabled the prediction of the behavior of physical systems with great accuracy. Knowing the wave function of a particle, we can calculate the expectation value of its position: or its momentum:

[0040] These are expected values meaning that when we make a measurement, the wave function "collapses" in a single point. It’s only when making several measurements that we can reconstitute the shape of the probability density.

[0041] Finding the wave function and solving the Schrodinger equation

[0042] Solving the Schrodinger equation in the case of a two particles system like the Hydrogen atom is a mathematically intense process but that ultimately leads to a very accurate solution that corresponds to observations (orbitals, energy levels, spectrum etc.).

[0043] However, solving the Schrodinger equation for a system with more than two particles is not possible with current mathematical tools because of the complexity created by the multiple interactions in the system. Applied quantum physics theory uses different methods and approximations to solve the Schrodinger equation (Timeindependent perturbation theory, the variational principle, the WKB approximation, the Bom Approximation etc.). Even though most of the time limited to ID problems, these methods have successfully led to applications in electronics, optics, chemistry etc.

[0044] Spin and entanglement

[0045] Using the definition of angular momentum of a particle, quantum theory also predicts the existence of an intrinsic characteristic of every elementary particle called spin s. Particles that make up ordinary matter (protons, neutrons and electrons) have a spin of s = When measured, the spin of these particles has two eigenstates: either + which we call spin up or — which we call spin down. Before being measured, these particles are in superposition of these two states, meaning they are simultaneously in both states with a certain probability for each state.

[0046] Elementary particles can also be entangled, with for example the spin of a particle defining the opposite spin of the other particle it is entangled with, even if the two particles are distantly far away. By definition, the spin of a particle is encoded within the wave function V of the particle and fully obeys the Schrodinger equation.

[0047] Quantum computing background Quantum computers and qubits Classical computers are based on the binary nature of their electronic components and a bit is defined by being either 0 or 1. Recently, quantum computers have been elaborated using not only the binary but also the superposition nature of the spin. Quantum computers define themselves by the use of qubits which are particles in a superposition of states of spin up (| 1)) and spin down (| 0)). Qubits are said to be 0 and 1 simultaneously. A quantum computer consists in manipulating a certain number of qubits through different gates before making a measurement of their spin. These manipulations are mostly a combination of superposition, entanglement and interference. Typically, when put in a superposition and entangled, a set of n qubits can simultaneously be in a state of all the different combinations of Os and Is that can be made (2ntotal combinations) while a classical computer is only at one state at a time at each internal clock signal. This is why quantum computers show a competitive advantage (quantum supremacy) compared to their classical counterparts.

[0048] For example, if we consider a circuit with n = 5 qubits, we can simultaneously create the following 2n= 32 different states shown in Figure 2. Quantum circuits and gates

[0049] Quantum circuits consist in different steps of manipulation of the qubits that are called gates. In most quantum circuits, all the qubits are usually initialized to 10 ) (general rule). One gate to create a superposition state is called the Hadamard gate denoted H. It puts the qubit |0) in a superposition of |0) and | 1) with 50% probability. It is written as:

[0050] If a measurement is then made after this gate, there is a 50% chance that we find that the qubit is in a 10) state and 50% chance that we find that the qubit is in a 11) state. Other gates like the RY gate enable superposition if we want to define another probability repartition between the two states |0) and |1).

[0051] To create entanglement, the CNOT or a conditional gate can be used. This gate enables a qubit to control the state of another qubit in the circuit by flipping its state.

[0052] Example of a quantum circuit

[0053] Figure 3 contains an example of a simple circuit in which 5 qubits qo - q< are put in superposition and are entangled.

[0054] After measurement, as shown in Figure 4, we get a probability distribution of the different states that can be created with the qubits.

[0055] The probability is equally distributed among the 32 states and is therefore = 0.03125.

[0056] The model

[0057] Objectives and applications

[0058] Referring to Figure 13, the objective is to model an arbitrary wave function 1300 of a particle V spread over a distance d using n qubits with the intention of simulating systems that have at least one or multiple particles without directly solving the Schrodinger equation. This could significantly improve our capacity to generate models of physical systems based on an accurate description of their behavior at the quantum level and lead to advances in physics, optics, chemistry, electronics etc.

[0059] In a preferred embodiment the invention provides a method for modeling the particle wave function 1300 on a quantum computer using qubits. The method comprises steps of spatially dividing 1301 the wave function of the particle into a system 1302 comprising at least two discrete elements represented respectively by a qubit; and using 1303 the at least two qubits in a quantum circuit to obtain a model 1304 of the particle wave function in such a way that in all of combinations of first and second states of the at least two qubits that the system may have, there is only one of the at least two qubits that has a first state which is different than the second state of each one of the other of the at least two qubits. The each one of the states of the at least two qubits represents either a presence or an absence of the particle in the spatial discrete location. Further details for preferred embodiments around the model are provided herein below.

[0060] First step: The model

[0061] The first step is spatially dividing (discretizing) the wave function V into n spatial elements, each element being represented by a qubit and representing a portion of the distance d in a ID system (in 2D, it would be a portion of surface S and in 3D a portion of a volume F). The probability amplitude in an element of the wave function is given by the probability amplitude of the qubit. As we have seen previously, when entangled and put into superposition, these n qubits create a string of Os and Is. We then define that if the state of the qubit is |0), then it means the particle is not in this portion of position (surface, or volume), and otherwise if the state is |1) it means the particle is in this position (the opposite can also be done, just that we need to do the definition inversion in everything that follows). By definition, the particle can only be in one qubit / portion of space at a time when measured. However, if we use the previous circuit, we see that it is possible with this nomenclature to find more than one particle in different positions after the measurement.

[0062] Second step: Entanglement intensification and filtering

[0063] The second step is to intensify the entanglement of this circuit in order to "eliminate" through a filtering process all the states where it is possible to find more than one particle. One way to do the filtering is through an example quantum algorithm shown in Figure 5.

[0064] The algorithm takes a first qubit and puts it in superposition 501. On a next gate 502, a CNOT gate is used such that depending of the state of the qubit the algorithm decides: if the state is 10) it will leave the other qubits unchanged. if the state is 11) it will cany and repeat this operation on the next qubit.

[0065] If the last qubit is reached, it means that its state must be | 1) as the state of all previous qubits was |0). The second part of the algorithm 503 is made of state inversions in order to keep the consistency in the definitions from the first step.

[0066] After the filtering stage, the strings that can be created by the qubits only have a “1” once in the string and all the other elements of the string (qubits) are “0”. This circuit also eliminates the string |00000) because we assume there is a particle. Even though before measurement, the particle’s position can be in any of the qubits, once the measurement is done, there is only one qubit that is measured in a 11) state so we only find one particle in a given position. If we take the previous example with n = 5 qubits, it means that this quantum algorithm only retains the states in bold as shown in the table of Figure 6, as the others states probability becomes equal to 0. We therefore only have the remaining states contained in the table of Figure 7.

[0067] Each state defines a spatial position of the wave function portion. We can then order the qubits depending on the position of 1 in the string in order to represent a position in space as shown in a graph of probabilities for computational basis states in Figure 8. Third step: adjusting the probability amplitudes

[0068] The previous circuit can be further enhanced to model an arbitrary wave function of a particle, as its probability density distribution is given by the construction of the algorithm. A third step illustrated in Figure 9 is to introduce RY gates (or similar method) instead of the Hadamard H gate at the entry of the circuit in order to adjust the probability densities on some qubits, e.g., 901 and 902, whereby the probability densities are as shown in Figure 10 and match an arbitrary wave function. Due to the entanglement of the qubits, modifying the probability density of one qubit automatically changes the probability density of the other qubits as they intrinsically have to adjust in order to keep the wave function normalized.

[0069] Introducing X and SWAP gates and modifying the wave function shape.

[0070] It is also possible to give negative values to the wave function by using an X (or flip) gate (see 1101) and / or a different shape by using the SWAP gate (see 1102) as illustrated in Figure 11. Figure 12 shows resulting probabilities.

[0071] Error correction and robustness

[0072] One of the issues of current quantum computers is the stability of the qubits. Indeed, due to decoherence and the difficulty to maintain a well-defined quantum state, it happens that (for no apparent reason), a qubit changes its state during the process of running the algorithm.

[0073] To solve this, quantum algorithms sometimes integrate error correction routines. One known technique to test the stability of quantum algorithms inspired by classical computing is to introduce redundance or patterns on the qubits and make measurement overtime to test the stability of the system.

[0074] Applied to the current invention, one way to reduce errors would be for example instead of coding the state of the presence of a particle 11), one could replace it by the state 111) or 1101) representing the particle over more qubits, and expect that an error has occurred in the algorithm if these states cannot be identified in the string obtained after the measurement process.

[0075] In the case of the current invention, one way to increase the robustness would be to describe the particle using several qubits that are spatially grouped representing a particle. The state of this group of qubits would form a redundance or a pattern and the algorithm made in such a way that the redundance or pattern can be identified at the measurement stage of the algorithm.

[0076] Although it would be less efficient in terms of the number of qubits used, errors during the processing of the algorithm would be identified and the solution rejected if the redundance or pattern describing the particle at the end of the algorithm is not recognized, thus making the algorithm more robust and error resistant.

[0077] Here are examples of such redundances or patterns:

[0078] • The redundance 111) could be representing the particle. This would allow the description of 4 possible positions when using 5 qubits.

[0079] • The patern 1101) could be representing the particle. This would allow the description of 3 possible positions when using 5 qubits.

[0080] Advantages of the method

[0081] This way of using qubits to model a wave function has several advantages compared to the current approximation methods which aim at directly solving the Schrodinger equation:

[0082] 1. It is not necessary to solve the Schrodinger equation, the objective is to let the qubits which are intrinsically quantum objects go through the interactions. A Hamiltonian can be applied to the particles / qubits in order to model their time and space evolution.

[0083] 2. It is not necessary to normalize the modeled wave function as it is automatically done by the quantum circuit.

[0084] 3. We can model a complex wave function as the qubits can be manipulated in the Bloch sphere to have complex or negative amplitudes.

[0085] 4. Using proper interaction functions in the quantum algorithm, it is possible to model the interaction of a system that has more than two particles. We can for example include electrostatic forces or electromagnetic fields.

[0086] 5. It is also possible to model obstacles or give specific boundary conditions by setting certain values of the probability amplitude at defined locations.

[0087] 6. We can model systems that evolve in 2D or 3D.

[0088] 7. With the development of quantum computers, the accuracy of the modeled wave function can be increased with the number of qubits.

[0089] 8. We do not need to worry about the "collapse" of the wave function as this is also intrinsically performed by the qubits after the measurement.

[0090] 9. The Heisenberg uncertainty is insured by the intrinsic properties given by the qubits.

[0091] 10. Interactions of several particles can be modelled by adding more qubits representing additional particles using a similar method. In a first step, constraints of interaction of the different particles (electrostatic forces, state differentiation, Pauli’s exclusion etc.) will be added to the circuit and in a second phase, the current algorithm is also to be applied to the additional particles.

[0092] Industrial applications One important goal in the field of physics and chemistry is to precisely solve the Schrodinger equation that defines physical properties of particles and their interactions. With the rapid development of quantum computers, this method of modeling a wave function could be used to accurately model the physical and chemical properties of atoms and molecules, and predict molecular structure, physical properties, chemical reactions etc. This would potentially reduce development times of new drugs in the medical industry, for example. Modeling electron tunneling could also lead to advances in the design, development and reliability of electronic systems.

Claims

Claims1. A method for modeling a particle wave function (1300) on a quantum computer using qubits, the method comprising: spatially dividing (1301) the wave function of the particle into a system (1302) comprising at least two discrete elements represented respectively by a qubit; using (1303) the at least two qubits in a quantum circuit to obtain a model (1304) of the particle wave function in such a way that in all of combinations of first and second states of the at least two qubits that the system may have, there is only one of the at least two qubits that has a first state which is different than the second state of each one of the other of the at least two qubits; or there is only one group of states of the at least two qubits that form a first state which is distinguishable with the second state of each one of the other of the at least two qubits; whereby the each one of the states or group of states of the at least two qubits represents either a presence or an absence of the particle in the spatial discrete location.

2. The method of claim 1, in which the first state is 11) and the second state is 10).

3. The method of claim 2, in which the first state is representative of the presence of the particle and the second state is representative of the absence of the particle.

4. The method of claim 1 in which the first state is 10) and the second state is |1).

5. The method of claim 1 in which the group creates a pattern of 10) and 11) and the second state is 10).

6. The method of claim 1 in which the group creates a pattern of 10) and and 11) and the second state is 11 ).

7. The method of any one of claims 1 to 4, wherein the step of using comprises any one of the items in a list comprising entangling, superposing, causing interference between.

8. A quantum circuit configured to implement an algorithm, the circuit comprising at least two qubits, whereby only one of the at least two qubits has a first state different than a second state of each one of all the others of the at least two qubits.

9. The quantum circuit of claim 8, wherein a qubits probability amplitude distribution corresponds to the probability density of an arbitrary wave function that is being modeled, whereby time and space evolution of the arbitrary wave function as well as interactions of several wave functions can be evaluated by applying an appropriate Hamiltonian to the quantum circuit in order to predict behavior of physical systems of particles.

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