Control hamiltonian for a quantum gate to be applied to a quantum state

Abstract

A computer implemented method for determining a control Hamiltonian for a quantum gate G to be applied to at least one quantum state, is provided. Quantum systems driven according to the control Hamiltonian are provided.

Description

CONTROL HAMILTONIAN FOR A QUANTUM GATE TO BE APPLIED TO A QUANTUM STATE

[0001] This disclosure relates to a computer implemented method for performing analgorithm for determining a control Hamiltonian such that quantum gates or quantum logicgates can be applied to a quantum state. The disclosure also relates to quantum systems for implementing such quantum gates and their corresponding control Hamiltonians to quantum states. BACKGROUND

[0002] Currently, in quantum information processing, it is sought to apply a unitary operationsuch as a logical quantum gate ^ to a quantum state e.g., |^^^, |^^^, ..., |^^^ to evolve suchquantum state to a target quantum state e.g., respectively ^|^^^, ^|^^^, .. ., ^|^^^.

[0003] Engineering a suitable Hamiltonian that implements a given unitary gate operation isessential in quantum information processing. This problem is commonly addressed using optimal quantum control theory. In most approaches to quantum control, however, the elapsed time over which the evolution is performed does not enter as a parameter to optimizeand is typically preset before applying any optimization algorithm. This approach usuallyresults in quantum systems where a fidelity of the quantum gate is low, or with a high fidelitybut a high protocol time ^ (i.e., the protocol time is such that the evolution of the quantumstate is slow). Additionally, having a control protocol with a high protocol time ^ (i.e., slowquantum system) entails decoherence of the quantum states. Decoherence rapidly degradesthe quality of quantum states in quantum information processing.

[0004] Determining the time-dependent control Hamiltonian that generates maximumspeed of evolution of the quantum state is in general a difficult task. Restrictions typicallycome from the available forms of the control Hamiltonian, which may yield a difficult-to-solve boundary-value problem.

[0005] Even if the form of the control Hamiltonian can be relaxed, limitations may arise incases where the system is immersed in an external field that cannot be altered by thecontroller of the quantum system. For example, the external field may be fixed to avoid anexcessive energy cost, that may not be assumed, for implementing the quantum gate. Also,or simply because a complete elimination of external fields would destroy the physical conditions under which the computational basis for the information processing is defined.

[0006] Examples of the present disclosure seek to reduce at least partially one or more ofthe aforementioned problems.SUMMARY

[0007] The present disclosure provides a general solution for determining a controlHamiltonian for a quantum gate ^ to be applied to a quantum state. In an aspect of thedisclosure, there is presented a computer implemented method for determining a controlHamiltonian for a quantum gate ^ to be applied to at least one quantum state, the methodcomprising: providing a set of parameters ^,⃗ the set of parameters ^⃗ defining a quantum state|^(^⃗ )^ in an N-dimensional Hilbert space, providing at least one set ^ of parameters ^^^^⃗ , eachset ^ of parameters ^^^^⃗ specifically defining a corresponding quantum state)^ in the N-dimensional Hilbert space; providing a target quantum state= ^|^(^^^^⃗ )^; providing aHermitian drift Hamiltonian ^^(^); and providing either a finite energy resource ^^(^) of thecontrol Hamiltonian, or a protocol time ^ and a functional form of ^^(^) with respect to time t;iteratively equating an equation expressed as ∫^ ^ ^^(^)^^ = arccos ^^^(^^^^⃗ )wherein, ineach iteration, values of either a protocol time ^ or values of a finite energy resource ^^(^) ofthe control Hamiltonian, are modified until both sides of the equation are at least substantiallyequal; obtaining thereby either a protocol time ^ or a finite energy resource ^^(^) of the controlHamiltonian, constructing the control Hamiltonian in an interaction picture with respect to^^^^(^), ^^ (^, ^) according^ are defined throughand parameters represent,each: ^ modulus; ^ phase;the target quantum statein the interaction picture, with= ^ ^^(^)^^^^, where ^^(^) = ^exp ^−^∫^ ^ ^^(^)^^^ and ^ defines a time orderingoperator; and † means conjugate transpose. The computer implemented method furthercomprises determining the control Hamiltonian in the Schrödinger picture as(^, ^) =^^^^(^)^^ (^, ^)^ ^^(^); calculating a fidelity ^^ of the quantum gate ^ where ^^(^^⃗ , ^) =∫ ^^⃗ ^^^(^⃗)^^ ^^ ^(^^⃗ , ^)G^^(^⃗)^^ and ^^(^^^^⃗ , ^) = ^exp ^−^∫^  ^^(^) + ^^(^, ^)^^^; andselecting a specific set ^^ from the at least one set ^ and / or a protocol time ^^,wherein atleast for the specific set ^^, or wherein for the specific set ^^ and / or for the protocol time ^^,the fidelity ^^ of the quantum gate ^ is the highest in a plurality of fidelities as a function ofparameters ^^⃗ and the protocol time ^, the selected specific set ^^ and / or protocol time ^^defining a corresponding control Hamiltonian ^^^^ (^, ^^), wherein the corresponding controlHamiltonian ^^^^ (^, ^^) is to be used for implementing the quantum gate ^ to any state

[0008] In the present disclosure, any reference to ^^ may be understood as the protocol timeselected by the computer-implemented method with which the implementation of thequantum gate may be performed. The protocol time ^^ is an optimized protocol time whichmaximizes the gate fidelity for a given set ^^of parameters. In the present disclosure, anyreference to ^ may be understood as any or a generic protocol time. In some examples thegate fidelity for a specific set ^^ may be independent of ^, and therefore ^^ = ^ may be anyor a generic protocol time. By selecting ^^, the highest fidelity ^^ in combination with theminimum energy resource ^^(^^) may be obtained.

[0009] In other words, the computer-implemented method may either determine the controlHamiltonian ^^^^ (^, ^) with the specific set ^^ or determine the control Hamiltonian^^^^ (^, ^^) with the specific set ^^ and protocol time ^^. The computer-implemented methodmay further comprise outputting the control Hamiltonian ^^^^ (^, ^) , or the control Hamiltonian^^^^ (^, ^^), and / or outputting the specific set ^^ and / or outputting the protocol time ^^. In thepresent disclosure, any reference to the protocol time ^ may be understood as any protocoltime.

[0010] In examples, the computer-implemented method may further comprise determiningor computing the control Hamiltonian ^^^^ (^, ^^) with the specific set ^^ and protocol time ^^.The computer-implemented method may further comprise outputting the control Hamiltonian ^^^^ (^, ^^), and / or outputting the specific set ^^ and outputting the protocol time ^^.

[0011] The set of parameters ^⃗ defines a quantum state in a given N-dimensional Hilbertspace. For example, in a two-dimensional Hilbert space, and considering e.g., therepresentation of the Bloch sphere, the parameters ^⃗ are the angles ^ and ^. The set ofparameters ^⃗ includes all the values of angles ^ and ^, where each specific value of theangles ^ and ^ defines a specific quantum state.

[0012] The at least one set ^ of parameters ^^^^⃗ specifically defines a corresponding quantumstate |^(^^^^⃗ )^ in the same N-dimensional Hilbert space than the one defined by the set ofparameters ^.⃗

[0013] Providing at least one set ^ of parameters ^^^^⃗ may comprise in some examplesproviding specific angles defining a corresponding quantum state |^(^^^^⃗ )^ in the Bloch sphereas illustrated below. Providing a quantum state may comprise in some examples providing an initial quantum state of a system to be evolved to a target state. Providing a quantum statemay comprise in some examples providing an intermediate quantum state between an initialstate and a target state.

[0014] Given the quantum gate ^, given the set of parameters ^,⃗ and given the set ofparameters ^^^^⃗ , the target quantum statemay be calculated by thecomputer-implemented method.

[0015] The term gate fidelity in quantum computing usually refers to the accuracy of quantumoperations such as the accuracy of a gate implementation within a set of gates. When aquantum gate ^ operation is applied, there may be a calculated ideal result, and there maybe an actual result from real hardware. The fidelity ^^of the quantum gate ^, may be defined as or comprise a measure of how close the calculated ideal result and the actual result fromreal hardware are. Calculating the fidelity ^^ comprises ^^(^^⃗ , ^) =∫where the integral over the set of parameters ^⃗ is a regularmultidimensional integral over each parameter contained in ^⃗ according to the Haarmeasure. For example, in a given N-dimensional Hilbert space, when N=2, the parametersare angles ^, Φ and the fidelity ^^ is a two-dimensional integral sweeping over ^, Φ, or ∫ ^^⃗ =∫^ ^sin(^) ^^ ∫^^ ^ .

[0016] As defined previously, the method comprises providing either a finite energy resource^^(^) of the control Hamiltonian, or a protocol time ^ and a functional form of ^^(^) with respectto time t. The method further comprises selecting a specific set ^^and / or a protocol time ^^,from the at least one set ^, wherein for the specific set ^^ and / or for the protocol time ^^, thefidelity ^^ of the quantum gate ^ is the highest in a plurality of fidelities as a function ofparameters ^^⃗ and the protocol time ^. The method may therefore comprise selecting a specific set ^^and a protocol time ^^. The method also may therefore comprise selecting a specific set ^^.

[0017] For example, in cases where a finite energy resource ^^(^) is provided, the computer-implemented method may comprise selecting the specific set ^^and selecting the protocoltime ^^. The computer-implemented method may comprise outputting the selected ^^ andthe selected protocol time ^^ . In such examples, the implementation or operation of aquantum gate or a quantum system may be governed by the selected set ^^and by the selected protocol time ^^.

[0018] For example, in cases where a protocol time ^ and a functional form of ^^(^) withrespect to time t are provided, the method may comprise selecting the specific set ^^. In suchexamples, the implementation or operation of a quantum gate or a quantum system may begoverned by the selected the specific set ^^ and by the provided protocol time ^ or anyprotocol time ^. The computer-implemented method may comprise outputting the selected^^ and the provided protocol time ^ or any protocol time ^. In some examples, fidelity may beindependent of ^.

[0019] For example, in cases where a protocol time ^ and a functional form of ^^(^) withrespect to time t are provided, the method may comprise selecting the specific set ^^andselecting the protocol time ^^. In such examples, the implementation or operation of a quantum gate or quantum system may be governed by the selected set ^^and by theselected protocol time ^^. The computer-implemented method may comprise outputting theselected ^^ and the selected protocol time ^^. The provided protocol time ^ is said to bediscarded by the computer-implemented method. The computer-implemented method maycomprise outputting the selected ^^and the selected protocol time ^^.

[0020] The computer implemented method may be implemented by a computer.

[0021] According to this aspect, the determined control Hamiltonian ^^^ (^, ^) for generatingor computing a quantum gate ^ to be applied to a quantum state may be used in one or moreof quantum technologies, such as quantum communication, or quantum simulation, orquantum operations, or quantum sensing, or quantum metrology, or quantum computing.Furthermore, the control Hamiltonian ^^ (^, ^) = ^ (^)^^^(^, ^) ^^ ^ ^ ^^(^) is solved such that thefidelity ^^ of the quantum gate ^ is maximized for a specific set ^^ in a time-optimal manner(i.e., with highest fidelity and a low protocol time ^). Since the determined control Hamiltoniancomplies with a time-optimal version of quantum control theory, maximum fidelityunitary transformations are sought in the least possible time to reduce the impact of decoherence.

[0022] Given at least one set ^ of parametersa quantum state, and a HermitianHamiltonian ^^(^), where each set of parameters specifically defines a correspondingquantum state |^(^^^^⃗ )^, a control Hamiltonian ^^^ (^, ^) is sought such that the totalHamiltonian(^, ^), when implemented, drives the state |^(^^^^⃗ )^ totarget quantum state ^|^(^^^^⃗ )^. Advantageously, in a (^^ *100) % of the cases, the totalHamiltonian ^(^) = ^ (^) + ^^^, when implemented, drives any state |^(^⃗)^ to aquantum state ^|^(^⃗)^, in average. The evolution of the quantum state is according to thetime-dependent Schrödinger equation, ^(^ / ^^)|^(^)^ = ^(^)|^(^)^, where the totalHamiltonian ^(^) = ^ ^^(^) + ^^ (^, ^) (atomic units are used throughout).

[0023] In some examples, the Hermitian drift Hamiltonian ^^(^) may be either timedependent or time independent.

[0024] The form of the Hermitian drift Hamiltonian is not manipulable, while the controlHamiltonian is constrained to fulfill two conditions: (i) it has a finite energy resource or a givenprotocol time ^ and a functional form of ^^(^) with respect to time t, and (ii) it is restricted toa subspace of Hermitian operators. The control Hamiltonianprovided in thedisclosure allows, in a (^^ *100) % of the cases in average, when applied to any quantumstate |^(^⃗ )^ , evolving the quantum state |^(^⃗ )^ to the target quantum state ^|^(^⃗)^ wherethe quantum gate ^ has been applied with a maximized fidelity ^^ of the quantum gate. Theprovided control Hamiltonian , when applied, allows applying a common controlHamiltonian for any quantum state, i.e.: regardless of the quantum state to evolve, the samecontrol Hamiltonian is applied, while ensuring: 1) that a fidelity of the quantum gate is high,e.g., above a predetermined value or threshold such as 0.999, and 2) that the quantum state|^(^⃗)^ is satisfactorily evolved to the target quantum state ^|^(^⃗)^, in average with a fidelityabove a predetermined value or threshold.

[0025] In some examples, the fidelity of the quantum gate is high such that the fidelity of thequantum gate is above a predetermined value or threshold such as 0.9990, and more specifically between 0.9995 and 1.0000. In some examples, the fidelity of the quantum gate is greater than 0.9999. It may be understood that these fidelities of the quantum gate are theoretical. When implemented, depending on the quantum system, the fidelity of the quantum gate may be lower than the aforementioned theoretical fidelities of the quantum gate.

[0026] In some examples, the computer implemented method may be provided with either afinite energy resource ^^(^) of the control Hamiltonian or a protocol time ^; ^^(^) and ^ are^ related through ∫^ ^^(^)^^ = arccos ^^^(^^^^⃗ )

[0027] In examples when a given protocol time ^ is lower than a threshold, the threshold, forexample, being 100 microseconds, the protocol time ^ may advantageously minimize theimpact of decoherence on the quantum process in which the quantum gate ^ is applied to aquantum state |^(^⃗)^. In other words, for a given coherence time Tc, reducing the protocoltime ^ for performing a quantum process (i.e., ^ << ^^) has the clear advantage ofminimizing the effect of decoherence.

[0028] In some quantum systems, for example quantum systems based on nitrogenvacancies, the given protocol time ^ may be lower than a second threshold, the secondthreshold for example being 700 nanoseconds for minimizing the effect of decoherence. Athird threshold of 30 nanoseconds may be fixed for quantum systems based onsuperconducting qubits.

[0029] In these examples, when the given protocol time is lower than the thresholds, theevolution of the quantum state is fast.

[0030] In examples where a finite energy resource ^^(^) of the control Hamiltonian isprovided, then such energy may be constant in time or dependent on time. If a protocol time^ is given, and the finite energy resource is searched and dependent on time, then afunctional form of ^^(^) with respect to time is necessary for solving the equation to obtain^^(^). For example, ^^(^) = Amplitude ∗ ^^^ (^ ⋅ t), where a is to be found by solving theequation. In the case where a protocol time ^ is given, and the finite energy resource issearched and constant in time, then no functional form depending on time is necessary for solving the equation to obtain ^^(^).

[0031] The resolution of either ^ or ^^(^) is performed iteratively until a condition is reached,for example, given a provided ^^(^), values for ^ may be modified by predefined steps or byrandom modifications for each iteration until both sides of the equation are equal within a certain tolerance, for example, both sides of the equation may differ by a tolerance or by a difference which is smaller than a tolerance. Any iterative method may solve the equation todetermine either ^ or ^^(^). Any modification, for example, increasing values of ^ or ^^(^) orrandom values of ^ or ^^(^) may solve the equation. The control Hamiltonian is constructedin an interaction picture with respect to ^^(^).

[0032] A consequence of using the control Hamiltonian(^, ^) according to the presentdisclosure is that, the fidelity ^^ of the quantum gate ^ is maximized for a specific set ^^and / or for a specific protocol time ^^, where the determined control Hamiltonian ^^^^is to beused for implementing the quantum gate ^ to any state |^(^⃗ )^.

[0033] The term “implementing” may refer to setting up the parameters relative to thedetermined control Hamiltonian for operating a quantum gate to a quantum state such thatthe state |^(^⃗ )^ is driven to the target quantum state ^|^(^⃗)^, with an average fidelity ^^ .

[0034] In some examples, the computer implemented method may comprise applying thequantum gate ^ to any quantum state |^(^⃗ )^, or to the quantum state |^(^⃗ )^, driving thequantum state |^(^⃗)^ according to the time-dependent Schrödinger equation ^(^ / ^^)|^(^)^ =-atomic units are used throughout- where ^(^) is the total Hamiltonian with thetotal Hamiltonian ^(^) being ^(^) = ^^(^) + ^^^^(t,^). In examples, applying the quantumgate ^ implies driving any / the state |^(^⃗ )^ to the target quantum state ^|^(^⃗)^, with anaverage fidelity ^^.

[0035] In some examples, the quantum gate ^ may be a J-qubit gate or an arbitrary J-qubitgate and / or comprise a Pauli unitary operation. The J-qubit gate may comprise any one ofthe following unitary operations: Hadamard, T, phase shift, arbitrary rotation, controlled NOT,SWAP, Toffoli, or any combination thereof. Depending on the J-qubit gate, the J-qubit gatemay comprise any suitable unitary operation. In the present disclosure, J defines the type ofqubit gate, for example, J=2 defines a 2-qubit gate.

[0036] In some examples, selecting the specific set ^^ and / or a protocol time ^^, maycomprise iteratively calculating the fidelity ^^ of the quantum gate ^ for each set ^ ofparameters and / or a protocol time ^. The fidelity ^^ of the quantum gate ^ depends on thetime evolution operator ^^(^^⃗ , ^), and at each iteration the control Hamiltonianis replacedby another control Hamiltonian defined by one of a remaining set of parameters of the atleast one set of parameters and / or a protocol time ^ or remaining protocol times ^. Theplurality of fidelities ^^ of the quantum gate ^ is thereby obtained.

[0037] Using one of the remaining set of parameters of the at least one set of parametersrules out sets already used in previous iterations, i.e., at each iteration the controlHamiltonianis defined by a different set of parameters that has not yet been used, forexample, given a plurality of 3 sets set ^^ , set ^^ , set ^^, if set ^^ has already been used forcalculating the fidelity, then the remaining sets are set ^^ , set ^^ When sets ^^ , ^^ have beenused, the remaining set is set ^^.

[0038] In some examples, additionally or alternatively to iteratively changing the controlHamiltonian by a different set of parameters ^, the control protocol time ^ may be iterativelychanged for defining a different control Hamiltonian which depends on the protocol time ^ .

[0039] In some examples, the set ^ of parameters ^^^^⃗ and / or the protocol time ^ mayiteratively be changed to a remaining set of parameters until the specific set ^^ and / or theprotocol time ^^ are selected (i.e., where the fidelity ^^ of the quantum gate ^ is the highest).In some examples, the set ^ of parameters ^^^^⃗ may iteratively be changed to a remaining setof parameters until all the sets have been used for calculating the fidelity, and the setscomplying with the condition “the fidelity ^^ of the quantum gate ^ is the highest” is selectedas the specific set ^^.

[0040] In some examples, the parameters ^⃗ may be (^, Φ), i.e., ^⃗ = (^, Φ ) and theparameters ^^^^⃗ may be (^ ^ , Φ^), i.e., ^^^^⃗ = ( ^ ^ , Φ^). (^, Φ ) and ( ^ ^ , Φ^) are angles defininga corresponding quantum state of a qubit (N=2) in the Bloch sphere. The parameters ^^^^⃗ maycomprise specific angles (i.e., specific values ^ ^ , Φ^) defining the corresponding quantumstateBloch sphere.

[0041] In these examples, iteratively calculating the fidelity ^^ comprises solving ^^(^^⃗ , ^) =

[0042] In some examples, the methods of the present disclosure comprise establishing aone-to-one correspondence between the interaction of one or more time dependent controlfields ^⃗ with an N level system and the control Hamiltonian. These examples allow obtainingcontrol fields ^⃗ to control the interaction of one or more fields ^ ^^^^^^⃗ with the N level system, asexemplified below.

[0043] In some examples, the N level system is a 2- level system, i.e., N is 2, and the one-to-one correspondence between the interaction of one or more time dependent control fields^⃗ with an N level system and the control Hamiltonian comprises the interaction of at leastone field ^ ^^^^^⃗ with a 2-level system and the control Hamiltonian, and the one-to-onecorrespondence is established as:where ∑^ (^^^^^^⃗ (^) ⋅ σ^⃗ ) = ∑^ (^^(^)^^ − ^^(^)^^ + ^^(^)^^) represents the sum of the possiblefields ^ ^^^^^⃗ , where ^⃗ = ∑^ (^^^^^^⃗ (^)), and σ^⃗ = (^^, ^^ , ^^), is the vector of Pauli matricescorresponding to the system.

[0044] In a further aspect, the present disclosure defines a computer implemented methodfor implementing a quantum gate ^, the method comprising: implementing the quantum gate^ under the control Hamiltonian selected by the method of the present disclosure.

[0045] According to this aspect, the quantum gate ^ under the control Hamiltonian selected,is implemented to any quantum state.

[0046] In some examples, the method may comprise providing one or more qubits to beoperated by the quantum gate ^.

[0047] In a further aspect, the present disclosure defines a quantum system comprising:- one or more drift fields; the fields may comprise magnetic, electric or optical, or similarfields; -one or more system, such as particles, immersed in the one or more drift fields;- at least one control field generator configured to generate one or more timedependent control fields ^⃗; -wherein the generator is configured to drive the system to a target quantum stateat time ^ = ^ by means of the one or more time dependent control fields ^⃗, whereinthe one or more time dependent control fields ^⃗ are characterized in that ^⃗ =∑^ (^^^^^^⃗ (^))with ^⃗ determined by a method according to the present disclosure.

[0048] Advantageously, a system according to the aspect of the present disclosure allowsto satisfactorily drive a system (e.g., a particle or atom) from the quantum state |^(^⃗)^ to thetarget quantum state ^|^(^⃗)^, where the fidelity ^^ of the quantum gate ^ is maximized.

[0049] In examples of the quantum system according to the present disclosure:- the one or more drift fields comprise at least a unidirectional magnetic field ^^^(^);- the one or more systems comprise one particle characterized by a spin ½ , a mass^, a g-factor ^ and a charge ^; the one or more system is immersed in theunidirectional magnetic field ^^^(^); wherein the one particle is in a quantum state |^(^⃗)^; in examples the one particle is in a quantum state |^(^⃗)^defined as the ground(eigen) state of the drift Hamiltonian ^^(^) at time ^ = 0;- where the unidirectional magnetic field ^^^(^) and the one particle are defined by atime dependent Hermitian drift Hamiltonian ^^(^) = −where ^ = ^^ / 2^;- the at least one field generator comprises at least one magnetic field generatorconfigured to generate a three-dimensional time dependent magnetic field ^^⃗ (^)defined by the following components-wherein the at least one magnetic field generator is configured to drive the particle toa target quantum stateat time ^ = ^ whereby means of the oneor more time dependent control fields ^⃗, wherein the one or more time dependent control fields ^⃗ comprises a time dependent magnetic field ^^⃗ (^), wherein the timedependent magnetic field ^^⃗ (^) is characterized in that^^(^)^^ + ^^(^)^^^, determined by a method according to any one of the methods ofthe present disclosure.

[0050] In examples of the quantum system according to the present disclosure:- the one or more drift fields comprise at least an optical lattice, the optical latticecomprising one or more laser sources; -the one or more systems comprise one or more atoms trapped in the optical lattice anddefined by a time dependent Hermitian two-level drift Hamiltonian ^^(^) = Γ^^(^)^^ andthe eigenstates of ^^(^ = 0) define the computational basis;- the at least one field generator comprises at least one or more laser sources configuredto drive the one or more atoms trapped in the optical lattice to a target quantum state |^ ^ at time ^ = ^ where= ^ |^(^) ^, wherein the at least one or moresources are characterized in that ^(^) = ^ (^) + ^ ^^with aHamiltonian ^^^^ (^, ^^ ) = Γ^(^)^ ^^^^ + Γ^(^)^^ + Γ^(^)^^ = ^^(^)^^ (^, ^^) ^^^(^), where (^, ^ ) determined by a method according to any one of the methods of the presentdisclosure. FIGURES

[0051] Fig. 1 schematically shows a block diagram of a computer implemented method fordetermining a control Hamiltonian for a quantum gate ^ to be applied to at least one quantumstate, according to an example of the present disclosure.

[0052] Figure 2 represents non-limiting examples of quantum gates.

[0053] Figure 3 represents a computer 300 configured to implement the method steps of thedisclosure.

[0054] Figure 4 represents a quantum system 400 comprising the computer 300 of thepresent disclosure.

[0055] Figure 5 represents a quantum system 400 comprising a field generator 500 and thecomputer 300.

[0056] Figure 6 represents a quantum system 400 comprising a gate 600, the computer 300and the field generator 500.

[0057] Figure 7 represents a quantum system 400 comprising a gate 600, the computer 300and the field generator 500 in operation. DETAILED DESCRIPTION

[0058] A quantum gate ^, when implemented, results in a unitary transformation of aquantum state according to the time-dependent Schrödinger equation, ^(^ / ^^)|^(^)^ =^(^)|^(^)^ (atomic units are used throughout) where ^(^) is the total Hamiltonian. The totalHamiltonian ^(^) is equal to ^(^)where ^^(^) is a, possibly timedependent, Hermitian drift Hamiltonian and ^^(^) is a control Hamiltonian.

[0059] Since the form of the Hermitian drift Hamiltonian ^^(^) is not manipulable, the controlHamiltonian ^^(^) has to be determined such that to allow, when applied to any quantumstate, evolving the quantum state to a target quantum state where the quantum gate ^ hasbeen applied.

[0060] In this disclosure, the control Hamiltonian (as will be explained below) is determinedfor allowing to evolve any quantum state to a target quantum state where the quantum gate^ has been applied to the quantum state with a maximized fidelity ^^ of the quantum gate.Furthermore, the disclosed control Hamiltonian can be determined to implement anyquantum gate ^.

[0061] Figure 1 represents, in a block diagram, a computer implemented method 100according to the disclosure. Figure 1 represents, in block 101, providing: -a set of parameters ^,⃗ the set of parameters ^⃗ defining a quantum state |^(^⃗ )^ inan N-dimensional Hilbert space,- at least one set ^ of parameters ^^^^⃗ , each set ^ of parameters ^^^^⃗ specifically defininga corresponding quantum state |^(^^^^⃗ )^;- a target quantum state-a Hermitian drift Hamiltonian ^^(^); and- either:- a finite energy resource ^^(^) of the control Hamiltonian, or- a protocol time ^ and a functional form of ^^(^) with respect to time t.

[0062] Figure 1 represents, in block 102, iteratively evaluating an equation expressed as^ ∫^ ^^(^)^^ = arccos ^^^(^^^^⃗ ) ∣wherein, in each iteration, values of either a protocol time^ or values of a finite energy resource ^^(^), are modified until both sides of the equation areequal within a certain tolerance; obtaining thereby either a protocol time ^ or a finite energyresource ^^(^). For example, both sides of the equation may differ by a tolerance or by adifference which is smaller than a tolerance.

[0063] Figure 1 represents, in block 103, constructing a control Hamiltonian in an interactionpicture with respect to ^ (^),accordingw ^ and ^ are defined throughparameters in the interaction picture represent, each: ^ modulus; ^ phase;thethe interaction picture, withwhere ^ (^)and ^ defines a time orderingand † means conjugate transpose.

[0064] Figure 1 represents, in block 104, determining the control Hamiltonian in theSchrödinger picture

[0065] Figure 1 represents, in block 105, calculating a fidelity ^^ of the quantum gate ^and ^^(^^⃗ , ^) = ^exp ^−^∫^ ^ ^ (t) +

[0066] Figure 1 represents, in block 106, selecting a specific set ^^ and / or a protocol time^^, from the at least one set ^, wherein for at least the specific set ^^, or in other words,wherein for the specific set ^^ and / or for the protocol time ^^, the fidelity ^^ of the quantumgate ^ is the highest in a plurality of fidelities as a function of parameters ^^⃗ and the protocoltime ^, the selected specific set ^^and / or protocol time ^^defining a corresponding control Hamiltonian ^^ (^, ^^), wherein the corresponding control Hamiltonianis to beused for implementing the quantum gate ^ to any state |^(^⃗)^.

[0067] In a particular example of block 101 in Fig.1, the quantum state may be, for example,an initial quantum state or an intermediate quantum state between an initial state and a targetstate. The quantum state is defined by a set ^ of parameters ^^^^⃗ , and, correspondingly, thequantum state is)^. The target quantum state |^^^ =is the quantumafter the quantum gate ^ has been applied.

[0068] Given a Hilbert space, the number of parameters ^⃗ depends on the dimension of theHilbert space. For example, in a two-dimensional Hilbert space, and using e.g., therepresentation of the Bloch sphere, the parameters ^⃗ comprises ^ and ^ where ^ and ^ areangles defining a corresponding quantum state |^(^⃗ )^ in the Bloch sphere.

[0069] For the at least one set ^ of parameters ^^^^⃗ , the parameters ^^^^⃗ may comprise specificangles (i.e., specific values ^ ^ , Φ^) defining a corresponding quantum state^ in theBloch sphere.

[0070] For determining the control Hamiltonian, the quantum state)^ and thetarget quantum state |^^^ =are respectively defined in the interaction picture as|^(^^^^⃗ )′^ = |^(^^^^⃗ )^where ^^(^) = ^exp ^−^∫^ ^ ^^(^)^^^ and ^defines a usual time ordering operator.

[0071] For a ^-dimensional setting, or a N level system, adopting an approach in theprojective Hilbert spaceand building on the approach in Garcia et al. “HighlyAdiabatic Time-Optimal Quantum Driving at Low Energy Cost” DOI: 10.1103 / PhysRevLett.129.180402, it may be proven that the control Hamiltonian that bringsthe quantum state)^ to the target quantum state)^ may bewhere ^ is modulus, ^ is the phase, and ^ (^)represents the "velocity". Thisparticular example of block 103 of Fig. 1.

[0072] By constructing a control Hamiltonian in an interaction picture with respect to ^^(^),according to Eq. 1, the control Hamiltonian in the Schrödinger picture can bedetermined as in Eq.2:Eq.(^). This a particular example of block 104 of Fig.

[0073] After determining, the control Hamiltonian^), a fidelity ^^ of the quantum^ is calculated for each given quantum state and / or time ^ or time ^^. Each of these givenquantum states are defined by its set ^ of parameters ^^^^⃗ .

[0074] The fidelity ^^ of the quantum gate ^ is calculated as follows in Eq. 3This a particular example of block 105 of Fig. 1.

[0075] In some examples, the fidelity ^^ of the quantum gate ^ may be calculated as followsin Eq.4where ^ is the protocol time, † means conjugate transpose and tr means the trace.

[0076] In examples, for a specific set ^^ and a protocol time ^^, the parameters of the set ^^and the protocol time ^^ are such that the fidelity ^^ is maximized (e.g., the highest). Thespecific set ^^and the protocol time ^^may, in some examples, be selected or determinediteratively by calculating the fidelity ^^ of the quantum gate ^ for each protocol time ^ and foreach set ^. The specific set ^^ may, in some examples, be selected or determined by definingthe set ^^based on eigenstates of the unitary operatorThis is a particular exampleof block 106 in Fig.1.

[0077] Referring to selecting the specific set ^^ and the protocol time ^^ by iterativelycalculating the fidelity ^^, in some examples, a plurality of sets (e.g., ^^, ^^, ^^ ), with theircorresponding parameters ^^^^^^^⃗ , ^^^^^^^⃗ , ^^^^^^^⃗ and a discretized set of protocol times^^,^^, ^^…^^^^ may be provided. Each set of parameters specifically defines a correspondingquantum state |^(^^^^^^^⃗ )^, |^(^^^^^^^⃗ )^, and |^(^^^^^^^⃗ )^. The fidelity ^^ of the quantum gate ^ isiteratively calculated for each value of the protocol time ^ and for each value of the quantumstateof each sets ^^, ^^, ^^, with their correspondingparameters ^^^^^^^⃗ , ^^^^^^^⃗ , ^^^^^^^⃗ . When the fidelity ^^ is iteratively calculated, at each iteration thecontrol Hamiltonian(^, ^) is defined by a different remaining set of parameters that hasnot yet been used in previous iteration(s). For example, given the plurality of sets ^^, ^^, ^^,if set ^^has already been used for calculating the fidelity ^^in a first iteration, then theremaining sets are sets ^^, ^^. When sets ^^ , ^^ have been used for calculating the fidelity ^^in a first iteration and in a second iteration respectively, the remaining set is set ^^. For eachset ^^, ^^, ^^, the iteration(s) also run over all possible values of the protocol time ^^, ^^,^^, ^^…^^^^ .

[0078] In the example where 3 sets ^^, ^^, ^^, of parameters are provided and 100 protocoltimes ^^, ^^, ^^, ^^…^^^^ are provided, 3*100=300 fidelities, e.g., ^^^^^^^^^^⃗ , ^^ = 24 ^^^,^^ = 30 ^^^… may be obtained depending on the quantumstate and the set.

[0079] The fidelities ^^^, ^^^, and ^^^ are average fidelities of the quantum gate applied toall possible input quantum statesas defined in Eq. 5

[0080] For example, the fidelity= 24 ^^^ has been calculated via an integral whichsweeps over all the quantum states, |^(^⃗)^,and is, therefore, an average.

[0081] The average fidelity ^^^ comprises values of fidelities of each quantum state wherethe control operator ^^(^^⃗ , ^) = ^exp+ ^^^(^, ^) ^^^ is set with a specific set ofparameters. In the example of = 24 ^^^, the control Hamiltonian ^ ^^^ (^, ^1 = 24 ^^)(which depends on the specific parameters ^^^^^^^⃗ and on the protocol time = 24 ^^) togetherwith ^^(t) define the control operator ^^.

[0082] As a result, the fidelity ^^^^^^^^^^⃗ , ^^ = 24 ^^^ is the average of the fidelity of the quantumgate ^ for all possible input quantum states |^(^⃗)^, when the control operator ^^ is set to5 ^^^^^^^^^^⃗ , ^^^ = ^exp ^−^∫^^^  ^^(t) + ^ ^^^ (^, ^^)^^^.

[0083] Referring to the fidelity ^^^^^^^^^^⃗ , ^^ = 28 ^^^, the fidelity ^^^^^^^^^^⃗ , ^^ = 28 ^^^ is theaverage of the fidelity of the quantum gate applied to all possible input quantum states|^(^⃗)^when the control operator ^^ is set to ^^^^^^^^^^⃗ , ^^^ = ^exp ^−^∫ ^^^  ^^(t) + ^ ^^^ (^, ^^)^^^.

[0084] Referring to the fidelity ^^^^^^^^^^⃗ , ^^ = 30 ^^^, the fidelity ^^^^^^^^^^⃗ , ^^ = 30 ^^^ is the10 average of the fidelity of the quantum gate applied to all possible input quantum states|^(^⃗)^,when the control operator ^^ is set to ^^^^^^^^^^⃗ , ^^^ = ^exp ^−^∫^^^  ^^(t) +^^^^ (^, ^^)^^^.

[0085] As a result, each fidelity ^^^^^^^^^^⃗ , ^^ = 24 ^^^, ^^^^^^^^^^⃗ , ^^ = 28 ^^^, and ^^^^^^^^^^⃗ , ^^ =30 ^^^ has a given value (i.e., average value of the fidelity of the quantum gate ^ applied to15 all the quantum states). For example, if a fidelity of the quantum gate is 1, it means that theaverage fidelity of the quantum gate ^ applied to each state is 1.

[0086] The set ^ (e.g., ^^, ^^, ^^) for each possible protocol time ^ defining the controloperator ^ for which the fidelity ^^ (e.g., ^^^ = 28 ^^^,= 30 ^^^) is the highest among all the calculated fidelities, is the selected set ^20

[0087] The selected set ^ and ^ define a corresponding control Hamiltonia ^^^ ^ n ^^ (^, ^^) tobe used for implementing the quantum gate ^ to any state

[0088] Referring to selecting or defining the set ^^ based on eigenstates of the unitaryoperator ^^^(^)^, in some examples, for a N level system with N=2, the set ^^ of parameters^^^^^^^^⃗ is defined as follows in Eq.6where ^^^, ^^are eigenstates of the unitary operator ^^(^)^ and ^ can take any real value.In this example the gate fidelity is independent from ^, or in other words, the fidelity is maximised for any ^ or independently of ^. In this case ^^ may correspond to ^ or ^^ = ^.

[0089] For N=2, when the set ^ is based on eigenstates of the unitary operator30 gate fidelity ^^ is independent of the protocol time ^ and is maximized as defined in Eq.7

[0090] In some examples, given a protocol time ^, the quantum state |^^^ ^ defines the controlHamiltonian in the interaction picture.

[0091] For valuesthat are belowconsidered to be equal to exp( ^^ / 2), i.e., e^^ = exp ^^ / 2.

[0092] In the othe cases, where the values^ are greater than 10(i.e.,

[0093] Consequently, the control Hamiltonian in the interaction picture is as follows in Eq. 8where ^ is modulus, ^ is the phase, where ^ (^) = ^^^ ^(^, ^),is theform

[0094] The corresponding expression in the Schrödinger picture can be expressed as in Eq.9

[0095] Fig.2 represents non-limiting examples of quantum gates which may be used in anyof the methods and / or systems of the present disclosure. Fig. 2 shows the name of the operator to which each gate corresponds, the symbol / circuit representation of each gate and the mathematical matrix which the gate reproduces in operation. Figure 2 represents someof the quantum gates which may be used in any of the methods and / or systems of the presentdisclosure. Any of the quantum gates disclosed in the present disclosure may be physically implemented and a corresponding schematic diagram of implementation or circuit of any gate may comprise, for example, microwave pulse generator, electro-optic modulators, or otherelements known in the art. In some examples, the quantum gate ^ may comprise one ormore Pauli gates 210 i.e. one or more of or a combination of Pauli-X gate 212, Pauli-Y gate214, Pauli-Z gate 216.

[0096] In some examples, the quantum gate ^ may comprise one or more superpositiongates 220. The superposition gates 220 may include one or more of or a combination of aHadamard gate 222, a S gate 224, a S† gate 226. The Hadamard gate 222 maps |0^ and |1^quantum states to target quantum states that are between |0^ and |1^, namely (|0^ + |1^) / √2and (|0^ − |1^) / √2. The S gate 224 and S† gate 226 may be used to add 90° phase rotationsto superpositions.

[0097] In some examples, the quantum gate ^ may comprise one or more controlled gates230. Controlled gates 230 may comprise one or more of a controlled Z (CZ) gate 232, or acontrolled NOT (CNOT) gate 234, or any combination thereof. Controlled gates 230 act on two or more qubits. For example, with the CZ gate 232, the state of one of the qubitsdetermines whether a Pauli Z operation is applied to the other qubit. With the CNOT gate 234, the state of one of the qubits determines whether a Pauli-X operation is applied to the other qubit causing the other qubit to be flipped.

[0098] In some examples, the quantum gate ^ may comprise non-Clifford gates 240. Non-Clifford gates are used to approximate arbitrary unitary transformations. For example, therepresented non-Clifford T gate 242 and T† gate 244 may add 45° phase rotations.

[0099] It may be noted that Z, S, S†, CZ, T, and T† gates are diagonal-unitary gates, whichis to say that their Dirac matrix representations are diagonal, with the off-diagonal terms beingzero, as shown in Fig.2.

[0100] In some examples, the quantum gate ^ may comprise one or more universal quantumgates. Universal quantum gates may comprise any set of gates to implement an operation ina quantum process. Universal quantum gates may comprise a finite sequence of a unitaryoperation (e.g., gates).

[0101] A unitary operation may comprise one of the following: Hadamard, phase shift,rotation, controlled, SWAP, Toffoli, or any combination thereof.

[0102] Some universal quantum gate sets may comprise any of:- the rotation operators ^^(^), ^^(^), ^^(^),the phase shift gate ^(φ) and CNOT gate;or -the Clifford set {CNOT, H, S} + T gate; or- the Toffoli gate and Hadamard gate.

[0103] In examples where the quantum gate ^ comprises more than one unit operation, i.e.,more than one of the disclosed gates, the fidelity of the quantum gate is defined as the fidelityof the whole sequence of unitary operations / gates.

[0104] Any of the quantum states disclosed in the present disclosure refer to a quantumstate of a system, e.g. a particle, or an atom, or an electron. A quantum system of the controlHamiltonian determined by the computer implemented methods of the present disclosuredepends on the system, e.g. particle or atom, or electron, under consideration. A quantumsystem comprises one or more drift fields; one or more systems (e.g. particles or atoms orelectrons) immersed in the one or more drift fields; and at least one control field generatorconfigured to generate one or more time dependent control fields ^⃗. The control fieldgenerator is configured to drive the one or more systems (e.g. particles or atoms) to acorresponding target quantum stateat time ^ = ^ by means of the one or more timedependent control fields ^⃗. The one or more time dependent control fields ^⃗ arecharacterized in that ^⃗ = ∑^ (^^^^^^⃗ (^)) with ^⃗ determined by a computer implemented methodaccording to the disclosure.

[0105] In some examples, the system is a 2- level system, i.e., N is 2, and the one-to-onecorrespondence between the interaction of one or more time dependent control fields ^⃗ withan N level system and the control Hamiltonian comprises the interaction of at least one field^ ^^^^^⃗ with a 2-level system and the control Hamiltonian, and the one-to-one correspondence isestablished as:corresponding to the system.

[0106] Given the form of the time dependent Hermitian drift Hamiltonian ^^(^), acomputational basis is defined. For example, when the Hermitian drift Hamiltonian ^^(^) is amagnetic field, the magnetic field is used as a reference. Consequently, the spin of thesystem can be either associated with a quantum state |0^ or a quantum state |1^ dependingon the spin immersed in the magnetic field used as a reference.

[0107] In some examples, the dependent control fields ^⃗ may comprise a continuous orpulsed micro-wave field.

[0108] In some examples, it is considered a spin 1 / 2 immersed in a time-varying magneticfield ^^⃗ (^). The spin-field interaction Hamiltonian can be written as the Eq.10Eq. 10:where ^ = ^^ / 2^, with ^ the g-factor and ^ and ^ the mass and the charge of the particle,respectively. A one-to-one correspondence between the components of the magnetic fieldand the control Hamiltonian are established, resulting in

[0109] In some examples, considering a Bose-Einstein condensate (BEC) in an opticallattice, under appropriate conditions the wavefunction of the BEC in the periodic potential of the optical lattice can be approximated by considering only the two lowest energy bands, and realize a Landau-Zener Hamiltonian of the form in Eq.11 Eq.11: ^^^(^) = Γ(^)^^ + ^^^,where terms accompanying ^^ and ^^ can be controlled through the quasimomentum ^ andthe depth ^^of the optical lattice, respectively. To make a given evolution of a two-levelsystem introducing the control Hamiltoniananinteraction term corresponding to a ^^Pauli matrix is added. The interaction term corresponding to a ^^may be implemented by introducing an additional interaction into the system, for example, through an extra laser or microwave field.

[0110] In the case of atoms in an optical lattice, ^^ components can be realized by adding asecond optical lattice shifted with respect to the first by ^^ / 4, where ^^is the lattice spacing. The interaction term corresponding to a ^^may be implemented through an appropriatetransformation Γ → Γ^ andsuch that no extra field is necessary. This transformation,which may be independent of the physical system under consideration (and could be considered as well for the spin in a magnetic field case), means that the resulting protocol is intrinsically more stable, as there will be no problems associated, for example, with phase fluctuations between the fields.

[0111] Figure 3 represents a computer 300 configured to determine a control Hamiltonianto be used for implementing a quantum gate, by implementing any one of themethods or computer-implemented methods of the present disclosure.

[0112] Figure 4 represents a quantum system 400 comprising the computer 200 of thepresent disclosure.

[0113] Figure 5 represents a quantum system 400 comprising a field generator 500. Thefield generator 500 may comprise, in non-limiting examples, a magnetic field generator, for example comprising a solenoid, or the field generator 500 may comprise, in non-limiting examples, an optical lattice. Figure 5 further represents the computer 200. The field generator 500 may be configured based on ^^^^ (^, ^^) provided or determined by thecomputer or by the computer-implemented methods of the present disclosure. The communication represented by double arrow in figure 5 between the computer 300 and the field generator may be a direct or indirect communication, e.g. a communication through an intermediate entity.

[0114] Figure 6 represents a quantum system 400 comprising a gate 600, the computer 200and the field generator 500.

[0115] Figure 7 represents a quantum system 400 comprising the gate 600 the computer200, and the field generator 500 generating a control field 501. The computer provides acontrol Hamiltonian control Hamiltonian ^^^^ (^, ^^) by implementing any one of the methodsor computer-implemented methods of the present disclosure. In the example of figure 7, thecomputer outputs (not shown) a selected operating protocol time ^^ indicating the operationof one or more of the elements of the quantum system (e.g., the operation time of, forexample, the field generator). The selected operating protocol time ^^ is an optimizedprotocol time which maximizes the gate fidelity for a given set ^^of parameters. In someexamples, the computer outputs no protocol time ^^ and the operation of the quantum systemmay be governed by the energy resource of the control Hamiltonian, which has a one-to-one correspondence with the protocol time given by the equation that relates them or may begoverned by other parameters. In the present disclosure, any reference to the protocol time^ may be understood as any or a generic protocol time. The gate 600 operates according tothe control Hamiltonian ^^^^ (^, ^^), transforming a qubit in an arbitrary state to a targetquantum state. The gate 600 may comprise any type of gate or any one or more of the gates described for figure 2. The gate 600 may operate under the influence or under the action ofthe field generator 500. The field generator may generate the control field 501 that is usedfor immersing the system (e.g., atom, or particle, or electron) according to examples of the present disclosure.

[0116] In a 2-level system a quantum bit is a system whose state lives in a two dimensionalHilbert space and is therefore described as a linear combination ^|0^ + β|1^ , where |0^ and|1^ are two basis states, and ^ and ^ are complex numbers, which satisfy |^|^+ |^|^= 1.Figure 7 represents the evolution of the initial state ^^|0^ + β^|1^ to a target state ^^|0^ +β^|1^ when implementing the gate 600.

[0117] In examples, the present disclosure presents a computer implemented method fordetermining a control Hamiltonian for a quantum gate ^ to be applied to at least one quantumstate, the method comprising:- providing:- at least one set of parameters ^,⃗ the set of parameters ^⃗ defining a quantum state|^(^⃗ )^ in an N-dimensional Hilbert space,- at least one set ^ of parameters ^^^^⃗ , each set ^ of parameters ^^^^⃗ specifically defininga corresponding quantum state |^(^^^^⃗ )^ in the N-dimensional Hilbert space;- a target quantum state-a Hermitian drift Hamiltonian ^^(^); and- either:- a finite energy resource ^^(^) of the control Hamiltonian, or- a protocol time ^ and a functional form of ^^(^) with respect to time t;- iteratively equating an equation expressed as^ ∫^ ^^(^)^^ = arccos ^^^(^^^^⃗ ) ∣wherein, in each iteration, values of either a protocol time^ or values of the finite energy resource ^^(^), are modified until both sides of the equationare at least substantially equal; obtaining thereby either the protocol time ^ or the finite energyresource ^^(^);- constructing the control Hamiltonian in an interaction picture with respect to ^^(^),^^^^ (^, ^) according towhere ^ and ^ are defined throughparameters in the interaction picture represent, each:^ modulus;^ phase;^^ ^^ ^ the target quantum statein the interaction picture, withwhereand ^ defines a time ordering operator;and † means conjugate transpose;- determining the control Hamiltonian in the Schrödinger picture as-calculating a fidelity ^^ of the quantum gate ^ where ^^(^^⃗ , ^) =^exp ^−^∫^ ^ ^^(t) + ^^^(^, ^)^^^; and- selecting a specific set ^^ from the at least one set ^, and a protocol time ^^, wherein forthe specific set ^^ and for the protocol time ^^, the fidelity ^^ of the quantum gate ^ isthe highest in a plurality of fidelities as a function of parameters ^^⃗ and the protocol time ^, the selected specific set ^^and protocol time ^^defining a corresponding control Hamiltonian ^^^^ (^, ^^), wherein the corresponding control Hamiltonianis tobe used for implementing the quantum gate ^ to any state

[0118] All of the features disclosed in this specification, all of the examples, including anyaccompanying claims, abstract and drawings, and / or all of the steps of any method or process so disclosed, may be combined in any combination, except combinations where at least some of such features and / or steps are mutually exclusive.

Claims

CLAIMS1. A computer implemented method for determining a control Hamiltonian for aquantum gate ^ to be applied to at least one quantum state, the method comprising:- providing:- at least one set of parameters ^,⃗ the set of parameters ^⃗ defining a quantumstate |^(^⃗ )^ in an N-dimensional Hilbert space,- at least one set ^ of parameters ^^^^⃗ , each set ^ of parameters ^^^^⃗ specificallydefining a corresponding quantum state |^(^^^^⃗ )^ in the N-dimensional Hilbertspace; -a Hermitian drift Hamiltonian ^^(^); and- either:- a finite energy resource ^^(^) of the control Hamiltonian, or- a protocol time ^ and a functional form of ^^(^) with respect to time t;- iteratively equating an equation expressed as^ ∫^ ^^(^)^^ = arccos ^^^(^^^^⃗ ) ∣wherein, in each iteration, values of either a protocoltime ^ or values of the finite energy resource ^^(^), are modified until both sides of theequation are at least substantially equal; obtaining thereby either the protocol time ^ orthe finite energy resource ^^(^); -constructing the control Hamiltonian in an interaction picture with respect to ^^(^),^^^^ (^, ^) according towhere ^ and ^ are defined through ^^(^^^ ^^⃗ ) ∣ ^^ ^ = ^^(^^^^⃗ )^^ ^^(^)^^^^ = ^ ^^^, and parameters in the interaction picture represent, each:^ modulus;^ phase;the target quantum statein the interaction picture, withwhere ^^(^) = ^exp ^−^∫^ ^ ^^(^)^^^ and ^ defines a time ordering operator;and † means conjugate transpose;- determining the control Hamiltonian in the Schrödinger picture as- calculating a fidelity ^^ of the quantum gate ^ where ^^(^^⃗ , ^) =and ^^(^^⃗ , ^) = ^exp ^−^∫^ ^ ^^(t) + ^^^(^, ^)^^^;and- selecting a specific set ^^ from the at least one set ^, and / or a protocol time ^^,wherein for the specific set ^^ and / or for the protocol time ^^, the fidelity ^^ of thequantum gate ^ is the highest in a plurality of fidelities as a function of parameters^^⃗ and the protocol time ^, the selected specific set ^^ and / or protocol time ^^defining a corresponding control Hamiltonian ^ ^^^ (t, ^^), wherein thecorresponding control Hamiltonian ^^^^ (^, ^^) is to be used for implementing thequantum gate ^ to any state2. The computer implemented method of claim 1, the method comprising applying thequantum gate ^ to any quantum state |^(^⃗)^, driving the quantum state |^(^⃗)^according to the time-dependent Schrödinger equation ^(^ / ^^)|^(^)^ = ^(^)|^(^)^(atomic units are used throughout) where ^(^) is the total Hamiltonian with the totalHamiltonian ^(^) being ^(^) = ^ ^^^(^) + ^^ (^, ^^).

3. The computer implemented method of claim 1 or claim 2, wherein selecting the specific set ^^and the protocol time ^^comprises iteratively calculating the fidelity ^^of the quantum gate ^ for each set ^ of parameters and protocol time ^, wherein ateach iteration the control Hamiltonian ^ ^^^ (^, ^^) is replaced by another controlHamiltonian defined by one of a remaining set of parameters of the at least one set ofparameters and / or protocol times ^; obtaining thereby the plurality of fidelities ^^ of thequantum gate ^.

4. The computer implemented method of claim 1 or claim 2, wherein the quantum gate^ comprises a Pauli unitary operation.

5. The computer implemented method of any one of claims 1 to 3, wherein the quantumgate ^ is a J-qubit gate.

6. The computer implemented method of any of claims 3 or claim 5,wherein the parameters ^⃗ = (^, Φ ) and the parameters ^^^^⃗ = ( ^ ^ , Φ^), where (^, Φ )and ( ^ ^ , Φ^) are angles defining a corresponding quantum state in the Bloch sphere;wherein iteratively calculating the fidelity ^^ comprises solving ^^(^^⃗ , ^) =wherein at each iteration a set ^ of parameters ^^ , Φ^ , and a control time ^ changes toa remaining set of parameters of the two or more sets of parameters.

7. A computer implemented method for implementing a quantum gate ^, the method comprising: implementing the quantum gate ^ under the control Hamiltonian selected by themethod of any one of claims 1 to 6.

8. The computer implemented method of claim 7, the method comprising providing one or more qubits to be operated by the quantum gate ^.

9. A computer implemented method according to any one of claims 1 to 8 further comprising establishing a one-to-one correspondence between the interaction of one or more time dependent control fields ^⃗ with an N level system and the controlHamiltonian.

10. A computer implemented method according to claim 9 wherein N is 2 and the one- to-one correspondence between the interaction of one or more time dependent control fields ^⃗ with an N level system and the control Hamiltonian comprises the interactionof at least one field ^ ^^^^^⃗ with a 2-level system and the control Hamiltonian, and the one-to-one correspondence is established as:of thepossible fields^^ , ^^), is the vector of Paulimatrices corresponding to the system.

11. The computer implemented method of any one of claims 1 to 10, wherein in a given selecting set ^^ comprises determining ^^ by establishingwhere ^^, ^^are eigenstates of the unitary operatorand ^ is a real value.

12. A quantum system comprising: -one or more drift fields;- one or more system immersed in the one or more drift fields;- at least one control field generator configured to generate one or more timedependent control fields ^⃗; -wherein the generator is configured to drive the system to a target quantumstate |^^^ at time ^ = ^ by means of the one or more time dependent controlfields ^⃗, wherein the one or more time dependent control fields ^⃗ arecharacterized in that ^⃗ = ∑^ (^^^^^^⃗ (^));with ^⃗ determined by a method according to claim 9 or 10.

13. A quantum system according to claim 12 further comprising a quantum gate ^configured to be implemented by a control Hamiltonian ^ ^^^ (^, ^^ ) determined by acomputer implemented method according to any one of claims 1 to 8. system according to claim 12 or 13, wherein: or more drift fields comprise at least a unidirectional magnetic field-the one or more systems comprise one particle characterized by a spin ½ , amass ^, a g-factor ^ and a charge ^; the one or more system is immersed inthe unidirectional magnetic field ^^^(^); wherein the one particle is in a quantum state |^^^; -where the unidirectional magnetic field ^^^(^) and the one particle are definedby a time dependent Hermitian drift Hamiltonian ^^(^) = −where ^= ^^ / 2^;- the at least one field generator comprises at least one magnetic field generatorconfigured to generate a three-dimensional time dependent magnetic field ^^⃗ (^)defined by the following components-wherein the at least one magnetic field generator is configured to drive theparticle to a target quantum state at time ^ = ^ where= ^|^(^⃗)^means of the one or more time dependent control fields ^⃗, wherein the one or more time dependent control fields ^⃗ comprises a time dependent magneticfield ^^⃗ (^), wherein the time dependent magnetic field ^^⃗ (^) is characterized inthatwith a control Hamiltonian ^^^^ (^, ^^)^^(t) ^^ + ^^(t) ^^) determined by a method according to any one of claims 1 to 11.

15. A quantum system according to claim 12 or 13 comprising: -the one or more drift fields comprise at least an optical lattice, the optical latticecomprising one or more laser sources; -the one or more systems comprise one or more atoms trapped in the opticallattice and defined by a time dependent Hermitian two-level drift Hamiltonian ^^(^) = Γ^^(^)^^ and the eigenstates of ^^(^ = 0) define the computationalbasis; -the at least one field generator comprises at least one or more laser sourcesconfigured to drive the one or more atoms trapped in the optical lattice to a target quantum stateat time ^ = ^ where= ^ |^(^) ^, wherein the atone or more laser sources are characterized in thatwith a control Hamiltonian ^^(^) = Γ^(^)^^ + Γ^(^)^^ + Γ^(^)^^ =is determined by a method according any one of claims 1 to 11.

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