Quantum control and quantum computation

Abstract

A time-dependent Hamiltonian, H(t), of the quantum system comprising time-dependent terms representing the control signal is identified. A control problem is defined for evolving one or more time-dependent operators, I(t), from one or 10 more initial operators into one or more respective target operators under the dynamics of an equation of motion given by the time-dependent Hamiltonian, wherein each eigenstate of the one or more time-dependent operators is a solution to the time-dependent Schrödinger equation given by the time-dependent Hamiltonian, allowing a control signal based on the control problem to be determined. The time dependent operator may be a Hermitian operator with a vanishing total time-derivative, and the time dependent operator’s eigenstates may be non-degenerate, and represented by tensor networks or matrix product operators.

Description

Field The present disclosure relates to control schemes for quantum systems. In particular, the optimization of control signals for implementing quantum states, gates or other properties of the quantum system. Background The availability of well-controlled quantum systems offers the prospect towards a broad range of applications, including quantum simulation and quantum computation. The exponential growth of the Hilbert space in such systems suggests that even moderately small quantum systems could be employed to achieve tasks that would be impractical to perform classically. While the benefits of this scaling have inspired tremendous efforts towards the development of quantum technological devices, the scaling also has disastrous implications on the ability to simulate the dynamics of quantum systems. Devising control schemes for quantum systems requires the ability to set the system in a desired state and / or define the gates controlling the evolution of the quantum systems. Therefore, due to the difficulty in simulating quantum systems, most attempts to devise control schemes of quantum systems are restricted to exact analyses performed on small systems with potential extensions to larger systems via approximate techniques, specific systems that admit efficient descriptions, or analyses that are approximate to start with. While this exponential scaling seems an insurmountable obstacle in general, there are specific instances of quantum states that can be described with an effort that scales more favourably than exponentially. An efficient approach to describing some quantum states is given in terms of the operator (or operators) to which a state is an eigenstate. There is, for example, a large variety of quantum many-body Hamiltonians with highly complicated ground states. The Hamiltonian is typically specified very efficiently in terms of the interaction geometry and the type of interaction mechanism. Even though the explicit specification of the ground state of such a Hamiltonian is hardly ever efficient, the implicit specification via the Hamiltonian can be. Summary Aspects of the disclosure relate to determining and applying control signals to a quantum system. According to a first aspect, a computer-implemented method for determining a control signal for controlling a quantum system is provided. The method comprises: Identifying a time-dependent Hamiltonian, H(t), of the quantum system comprising time-dependent terms representing the control signal; defining a control problem for evolving one or more time-dependent operators, I(t), from one or more initial operators into one or more respective target operators under the dynamics of an equation of motion given by the time-dependent Hamiltonian, wherein each eigenstate of the one or more time-dependent operators is a solution to the time-dependent Schrodinger equation given by the time-dependent Hamiltonian; determining the control signal based on the control problem. Aspects of this disclosure reduce the dimensionality of the control problem while being applicable to a wide range of practical problems. Reducing the dimensionality of the control problem reduces the computational resources required (e.g., exponential savings in memory usage can be achieved) and increases the speed at which computations can be completed (e.g., exponential speed-up in computing time can be achieved). The advantages of reduced dimensionality of the problem become more pronounced as the qubit count increases. Furthermore, by reducing the memory requirements and the computation time larger quantum systems can be analysed and effectively controlled. Optionally, each of the one or more time-dependent operators is a hermitian operator with a vanishing total time-derivative and the time-dependent operator's eigenstates are non-degenerate. Optionally, the equation of motion given by the Hamiltonian is defined as = + i[H(t), / (t)] = 0. Optionally, the time-dependent Hamiltonian is written as H(t) = spanned by a set of operators t)k with scalar coefficients hk and wherein at least one hk is time-dependent, the time-dependent operators are written as / = spanned by the operators a7 with time-dependent scalar coefficients a / f and the operators a7 are closed under commutation with any of the operators ljk. Optionally, the relation [a7,l)fc] = -i holds for all commutators [a7,t;fc] and all A are scalars. Optionally, the equation of motion given by the Hamiltonian is written as a = K^a, where the elements of the matrix are Ky(t) = £k=1hfc(£)2y, and where hk(t) are the scalar coefficients of the time-dependent Hamiltonian and a is a vector of the time-dependent scalar coefficients aj of the timedependent operators. Optionally, the time-dependent operators are represented by tensor networks or matrix product operators. Optionally, the control signal determined based on the control problem evolves the quantum system from an eigenstate of a first Hamiltonian to an eigenstate 1^2) of a second Hamiltonian, wherein the first and the second Hamiltonian are each representable by the time-dependent Hamiltonian. Optionally, defining the control problem comprises the control problem fulfilling the conditions: the one or more initial operators is one initial operator, 1(0), with an initial eigenstate corresponding to the eigenstate |iPi> of the first Hamiltonian, the initial eigenstate of 1(0) having a nondegenerate eigenvalue y; the one or more target operators is one target operator, It, with a target eigenstate corresponding to the eigenstate |U»2> of the second Hamiltonian, the target eigenstate of It having the same eigenvalue y; and the target operator Jr being reachable from the initial operator 1(0) within the dynamics of the equation of motion. Optionally, the control signal determined based on the control problem applies a unitary gate Ut to the quantum system. Optionally, defining the control problem comprises the control problem fulfilling the conditions: the one or more target operators It; being defined based on the respective one or more initial operators 1,(0) and the unitary gate Ut as ITj = ( / 7. / 7(0)( / ^ the initial operators Ij(0) and target operators Itj defining the unitary gate Ut uniquely; and each target operator being reachable within the dynamics of the equation of motion starting from its respective initial operator. Optionally, determining a control signal based on the control problem comprises minimizing or maximizing an objective function. Optionally, determining a control signal based on the control problem comprises determining a control signal that evolves the initial operators at a time t=0 into the final operators at a time t=T, wherein the final operators at time t=T are sufficiently close to the respective target operators. Optionally, determining a control signal that evolves the initial operators at a time t=0 into the final operators at a time t=T comprises propagating the one or more initial operators according to the equation of motion. Optionally, the control signal is a control pulse and determining the control pulse based on the control problem comprises applying a numerical pulse shaping algorithm that realizes the evolution of initial operators into final operators. Optionally, the final operators at time t=T being sufficiently close to the respective target operators comprises an infidelity being below a threshold. Optionally, the threshold is the experimentally achievable infidelity of the quantum system. Optionally, the operators spanning the one or more time-dependent operators are defined in a vector space that is smaller than the Hilbert space of the Hamiltonian. Optionally, the quantum system is a quantum computer comprising a plurality of qubits. Optionally, the control signal comprises any one of a control pulse, a magnetic field, or an electric field. Optionally, subsequent to determining the control signal, the control signal is transmitted to a controller, the controller configured to apply the control signal to the quantum system. According to a second aspect of the disclosure, a computer-readable medium is provided that is configured to store the computer-implemented method of the first aspect. According to a third aspect of the disclosure, a computing system is provided that is configured to execute the computer-implemented method of the first aspect. According to a fourth aspect of the disclosure, a system for applying a control signal to a quantum system is provided. The system comprises: a signal calculator that determines the control signal according to the computer-implemented method of the first aspect of the disclosure and a controller that applies the control signal to the quantum system. Optionally, the controller comprises a microwave generator that applies the control signal as a microwave pulse. Optionally, the controller comprises a laser generator that applies the control signal as a laser pulse. Brief description of the figures Figure 1 shows a system for determining and applying control signals to a quantum system. Figure 2 shows a method for determining a control signal. Figure 3 shows an example of numerical results of state and gate infidelities achieved with the methods disclosed herein. Figure 4 shows two different driven spin chains. Figures 5 and 6 show several spin configurations extending beyond the onedimensional arrangement. Detailed description A common way to control quantum systems is to apply control signals to the quantum system to set the system into a quantum state, apply quantum gates to the system or implement other properties of the system. The state being set may be the total quantum state of the system or the states of individual qubits or other subcomponents of the system. Similarly, a gate may be applied to the entire quantum system ora subset of the quantum system. In some implementations, quantum systems are controlled by applying a control pulse to the system. In such implementations, any method of generating control pulses may be used. In some implementations, control pulses can be applied using well-known technologies such as microwave engineering technology or laser technology. In some implementations, quantum systems are controlled by applying a field such as a magnetic field or an electric field to the quantum system. Methods of applying such fields are well-known in the art. Any control signal suitable for controlling a quantum system may be used. Figure 1 shows a system 100 for determining and applying a control signal to a quantum system 130. In addition to the quantum system 130, the system 100 comprises a signal calculator 110 for determining a control signal according to aspects of this disclosure and a controller 120 for applying a control signal determined by the signal calculator 110 to the quantum system 130. The signal calculator receives, obtains or identifies information about the quantum system 130 and the desired states, gates or other properties of the quantum system and determines a control signal that implements the desired state, gate or other properties in the quantum system. The control signal may be written, encoded or formatted in any way that is readable by the controller 120. The controller 120 receives the determined control signal and applies the control signal to the quantum system 130. The signal calculator 110 may be any computer or computing system capable of determining control signals according to the present disclosure. Such a computer may be a general-purpose computer, such as a desktop computer or a laptop, or a specialpurpose computer. The signal calculator 110 may comprise a computing cluster utilizing multiple computers or a single computer. A computer or computing system functioning as a signal calculator 110 need not be limited to determining control signals and may also implement other computer programs. Further, a network of computers functioning as a signal calculator may comprise both classical computers, quantum computers and any other type of computer. The signal calculator 110 may be remote for quantum system 130 and controller 120 and need not send the determined control signal to the controller 120 directly. In some implementations, the signal calculator 110 may send the determined control signal to a control signal library that stores a number of control signals and controller 120 may retrieve a desired control signal from the control signal library. In other implementations, the signal calculator 110 determines a control signal and sends it to the controller 120 directly. In some implementations, the controller 120 may use a control signal determined by the signal calculator 110 several times without the signal calculator re-calculating the control signal. For example, the controller 120 may repeatedly use the same control signal determined by the signal calculator to set a qubit from a ground state into a first excited state (often referred to as the |0> and 11> states for computational purposes). In some implementations, the signal calculator 110 may be integrated into the controller 120. In other implementations, the signal calculator 110 may be remote from the controller 120, e.g., in a different geographical location. As long as the signal calculator 110 has a means of communicating the control signal to the controller 120, either directly or indirectly, any distance between signal calculator 110 and controller 120 is permitted. In some implementations, signal calculator 110 may send control signals to more than one controller 120 that either control different quantum systems 130 or some or all controllers 120 may jointly control one or more quantum systems 130. Controller 120 controls quantum system 130. Controller 120 comprises a means of generating a physical control signal and applying it to the quantum system. In some implementations, controller 120 comprises a pulse generator that generates a physical pulse to be applied to the quantum system 130 based on the control pulse determined by the signal calculator. In some implementations, the controller comprises a microwave generator that applies a microwave pulse to the quantum system. In other implementations, the controller 120 comprises a laser generator to apply laser pulses to control the quantum system. Any type of signal or pulse generator that generates an electromagnetic of any other type of pulse or signal suitable for controlling the relevant quantum system 130 may be used. In some implementations, controller 120 comprises control mechanisms for applying a biasing voltage, an electric field, or a magnetic field to the quantum system 130. Controller 120 may comprise any suitable mechanism for generating a control signal for controlling quantum systems. In some implementations, controller 120 may comprise multiple generators for generating different types of signals to apply to quantum system 130. In some implementations, one controller 120 may control multiple independent quantum systems. Quantum system 130 may be any quantum system of interest. In some implementations, quantum system 130 may be a quantum computer. In some implementations, a quantum computer comprises a plurality of (physical) qubits and at least one gate. Any desired quantum system 130 may be used. Quantum system 130 may be used for a range of implementations such as quantum sensing, quantum communication, quantum simulation or other applications. Quantum system 130 may be implemented in any suitable platform. For example, quantum system 130 may be implemented using superconducting qubits, trapped ions, Rydberg atoms, semiconductor qubits, Majorana qubits, photonic qubits or any other suitable quantum system. System 100 may comprise further components that are not depicted. In some implementations, the system may comprise further controllers for implementing various aspects of quantum system control. In some implementations, outputs from the quantum system 130 are sent to classical computers or quantum computers for display or further processing. In some implementations, the system 100 may be part of a network of quantum systems and classical computers. Aspects of the present disclosure may not require the entire system 100 to function. For example, methods for determining the control signal only require the signal calculator 110. In another example, a system for determining and applying control signals may only comprise signal calculator 110 and controller 120. While in either of these cases the quantum system 130 is not required, the signal calculator may receive information about the quantum system 130 of interest, such as the quantum system Hamiltonian and the desired final Hamiltonian, states and gates. Aspects of the present disclosure deal with methods for determining a control signal for controlling a quantum system with a time-dependent Hamiltonian. One technique for determining the shape of control signals is quantum optimal control (also referred to as optimal control herein). In quantum optimal control, the control signal is determined based on an (optimal) control problem. While ideas to characterize quantum states in terms of operators to which they are eigenstates exist, optimal control primarily focuses on explicit descriptions of state vectors. The present disclosure relates to a method that realizes optimal control without explicit construction of quantum states. This method makes control problems numerically accessible whose realization based on explicit state vector descriptions is prohibitively expensive. The underlying idea of the present disclosure is inspired by the framework of quantum invariants (also referred to as invariants) [J. Lewis, H. R. and W. B. Riesenfeld, An Exact Quantum Theory of the Time-Dependent Harmonic Oscillator and of a Charged Particle in a Time-Dependent Electromagnetic Field, Journal of Mathematical Physics 10, 1458 (2003); R. S. Kaushal and S. C. Mishra, Dynamical algebraic approach and invariants for time-dependent Hamiltonian systems in two dimensions, Journal of Mathematical Physics 34, 5843 (1993); H. Korsch, Dynamical invariants and timedependent harmonic systems, Physics Letters A 74, 294 (1979)], i.e. hermitian operators I(t) with a vanishing total time-derivative, i.e. di dl -7-= —+ i[H(t), / (t)] =0. at at (1) Every instantaneous eigenvector of a non-degenerate invariant, i.e. a vector satisfying the eigenvalue relation I(t)| ^(t)) = A\W(t)), specifies a solution of the time-dependent Schrodinger equation with the time-dependent Hamiltonian H(t), where the Hamiltonian comprises terms encoding the control signal. Therefore, instead of solving the Schrodinger equation explicitly for a time-dependent state vector, one can also solve Eq. (1) fora time-dependent operator I(t). Following the eigenvalue relation, this provides all information on the desired state vector, but as long as I(t) is not diagonalised, this information remains implicit. Since the relation between operators and their eigenstates is nonlinear, working with the operators I(t) instead of a state vector implies a non-linear transformation of the problem. As such, defining the control problem in terms of operators I(t) as disclosed herein significantly reduces the dimensionality of the problem. Further, this approach is applicable to a wide range of practical situations. This approach is fundamentally different to the exploitation of symmetries (e.g. translational symmetry) that allows one to consider dynamics in a sub-space of the full Hilbert space (given by a certain quasi-momentum in the case of translational symmetry). Such a mapping from the full Hilbert space to a subspace is necessarily a linear mapping. Consider a Hamiltonian = 'Ljhfij spanned by a set of operators t)7 with scalar expansion coefficients hj, some of which can be time-independent and fixed (as in a drift Hamiltonian), but some should be time-dependent and tuneable (as in a control Hamiltonian). In addition, consider a hermitian operator I = £^=1a7a7 spanned by a set of operators a7 with time-dependent scalar expansion coefficients a7. In order for I to satisfy the equation of motion Eq. (1), it is necessary that the set of operators a7 is closed under commutation with any of the operators ljk, i.e. that any commutator [a7,tjfc] can be expanded in terms of the operators ajf such that d Md =      ■ 1=1 (2) In any finite-dimensional system, this condition can always be satisfied, if the set of operators is chosen to have sufficiently many elements, in the following, set {a,} is selected such that it is sufficiently small to obtain computation time speed up and reduce memory requirements. If the sets of operators {a7} and {l)k} coincide, the present discussion reduces to the common structure of a Lie algebra, but in order to find small sets of operators {a,} it can be beneficial to consider situations beyond Lie algebras. Given Eq. (2), the equation of motion for an invariant (Eq. (1)) reads d = K(t)a (3) in terms of the vector a of coefficients aj, and the matrix K(t) with elements d = £ . k=l (4) The matrix K is anti-hermitian and the dynamics of the vector a with elements aj is unitary. The dimension of the differential equation underlying the control problem is thus given by the size d of the set of operators a7, and not by the dimension of the underlying Hilbert space as it is the case for the Schrodinger equation. In general, d can be quadratically larger than the size of the Hilbert space, but as exemplified in the following, there are several instances of practical importance in which d is much smaller. Therefore, control problems formulated with respect to the operator I(t) in these cases benefit from this reduced dimensionality of the control problem, achieving up to exponential speed up in computation time and up to exponential saving in memory. As a consequence, the number of qubits that can be simulated on available hardware increases and the determination of control signals for larger systems of qubits becomes possible. We now define the framework for diabatic transitions between eigenstates of two Hamiltonians H1 and H2. In order to properly define the corresponding control problem in terms of an operator I(t), the following conditions need to be given: (i) The initial condition 1(0) needs to be specified such that the initial state |tPi> (an eigenvector of Hi) is an eigenstate of 1(0) with a non-degenerate eigenvalue yi. (ii) The control target It needs to be chosen such that the target state |UH) (an eigenvector of Hz) is an eigenstate of It with the same eigenvalue yi. (iii) The control target needs to be reachable with the dynamics of Eq. (3). A suitable choice for either (i) or (ii) can be taken by requiring that either 1(0) or It coincide with the Hamiltonian Hi or Hz, but condition (iii) would typically prevent both choices from being possible simultaneously (e.g. if Hi and Hz have different spectra). If the choice 1(0) = Hi is taken, one can consider an adiabatic transition from Hi to Hz and simulating the adiabatic evolution in Eq. (3) yields a final operator, i.e. It, that is reachable with the available dynamics and that has |UH) as eigenstate. Once the target operator It is constructed in this way, one can use any numerical pulse shaping algorithm [for examples see: C. P. Koch et al., Quantum optimal control in quantum technologies. Strategic report on current status, visions and goals for research in Europe, EPJ Quantum Technology 9, 1 (2022)] or any other appropriate method to design a time-dependent Hamiltonian (based on Eq. (3)) that realizes a dynamics from 1(0) to I(T) at a final time T that is close to It within a diabatic time window. Defining a control problem for the realization of a control signal that implements the dynamics of a target unitary gate Ut, e.g. a quantum gate that is a function of a Hamiltonian Hz, and corresponding propagators according to the present method requires a set of initial operators Ij(O) to be propagated. In contrast to the above state transfer protocol, there is no freedom in defining the corresponding targets, but they are given by lTj = UTIj(0)ltf.; as such, they can also be constructed efficiently in terms of Eq. (3). The initial operators Ij(0) and target operators It define the target unitary L / t uniquely. This may be achieved by using a complete set of initial operators on the algebra, however, in many cases a smaller set is sufficient to define the target unitary. In this case, a numerical optimization would target a time-dependent Hamiltonian such that all operators Ij(t) resultant from the initial conditions Ij(0) are sufficiently close to their respective targets Itj at the final time T. The above-outlined approach is applicable to any set of operators {a7} and {t)k} satisfying Eq. (2), that is any set of operators for spanning the Hamiltonian and the time-dependent operator I(t) that satisfy Eq. (1). Figure 2 provides an example of a method 200 for determining a control signal according to aspects of this disclosure. In step 210, a time-dependent Hamiltonian of the quantum system is obtained, identified or received. The time-dependent Hamiltonian comprises time-dependent terms that represent the control signal. In other words, the impact of the control signal on the quantum system is encoded in the Hamiltonian. The control signal may be any signal suitable for controlling a respective quantum system. For example, the control signal may be a control pulse, a field (e.g., a magnetic field, electric field or any other field), a biasing voltage or any other signal. The control signal may comprise a combination of multiple different control signals e.g., a magnetic field and a biasing voltage. In one example, the time-dependent control Hamiltonian may be the system Hamiltonian given in the example of a spin chain in Equation (5). In this Hamiltonian the terms including fj^ and h(t) encode, among other things, the impact of the control signal. Other possible time-dependent Hamiltonians are the respective system Hamiltonians of the other examples provided below. In general, the time-dependent Hamiltonian is dependent on the quantum system and the mechanisms available to control the quantum system. In step 220, a control problem is defined for evolving one or more time-dependent operators, I(t), from one or more initial operators into one or more respective target operators under the dynamics of an equation of motion given by the time-dependent Hamiltonian H(t). Each eigenstate of the one or more time-dependent operators is a solution to the time-dependent Schrodinger equation given by the time-dependent Hamiltonian. In some implementations, the control problem consists of determining a control signal for preparing an eigenstate 11½} of a target Hamiltonian starting from an eigenstate \Wi) of an initial Hamiltonian. The initial Hamiltonian and the target Hamiltonian are representable by the time-dependent Hamiltonian. In such implementations, the control problem may be defined by an initial operator 1(0) with the eigenstate \Wi) and a non-degenerate eigenvalue y, as well as a target operator It whose eigenstate is the target state | ^2) with the same eigenvalue y. In some implementations, the eigenstate \Wi) of an initial Hamiltonian is an eigenstate that is easily prepared in the quantum system. In some implementations, the control problem consists of determining a control signal for defining a target gate to be applied to the quantum system. In such implementations, the definition of the control problem may use a set of initial operators Ij(0) and target operators 4? that define the unitary gate Ut uniquely, wherein the target operators are defined as ITj = UT / 7(0)U^. In both of these implementations, the target operators must be reachable from their respective initial operators within the dynamics of the equation of motion. In further implementations, optimal control problems may be defined for any appropriate property of the system. An example of an optimal control problem for preparing an eigenstate and an optimal control problem for applying a target gate is given for each of the target Hamiltonians Hg, He and Hd below. In some implementations, the equation of motion may be represented by Equation (1). In some implementations, the eigenstates of each of the one or more timedependent operators I(t) are non-degenerate. In some implementations, the each of the one or more time-dependent operators is a hermitian operator with a vanishing total time-derivative. In some implementations, the time-dependent Hamiltonian and time-dependent operators I(t) are spanned by the sets of operators {t)7} and {aj}, respectively. The operators {a7} are closed under commutation with the operators {t>7}. In some implementations, this relationship between the sets of operators {h7} and {a7} is given by Equation (2). In some implementations, the equation of motion may be represented by Equation (3). Examples of sets of operators {l)7} and {a7} are given for various quantum systems in Equations (13), (15) and (16) to (28). In some implementations the Hamiltonian and time-dependent operators I(t) are represented by tensor networks, i.e., matrix product operators. In step 230, the control signal is determined based on the control problem. That is, the form of the time-dependent terms representing the control signal is determined such that the equation of motion given by the time-dependent Hamiltonian evolves the initial operators to the final operators, where the final operators are numeric approximations of the target operators or the target operators themselves. For example, in the driven spin chain of Equation (5) specific forms of the functions / 7(t) and h(f) are determined such that Equation (3) evolves the initial operators to the final operators. In some implementations, determining the control signal based on the control problem comprises minimizing or maximizing an objective function. In some examples, the objective function may be an objective functional. Examples of such objective functionals are the functionals J and G(U,UT) given in the example of a driven spin chain below. In some implementations, a control signal is determined that evolves initial operators at a time t=0 into final operators at a time t=T. The final operators at time t=T are sufficiently close to the respective target operators. In some implementations, to determine the control signal based on the control problem, the one or more timedependent initial operators are propagated according to the equation of motion of a quantum invariant using the time-dependent control Hamiltonian (e.g. Equation (1) or (3)) thereby evolving into respective one or more final operators. In some implementations,_the propagation of the operators is obtained by using the Suzuki-Trotter decomposition of the propagator. In some implementations, the propagation of the operators is obtained by contracting the propagator into the tensor network of the operator. In some implementations, control signal is obtained numerically. In some implementations, where the control signal is a control pulse, a numerical pulse shaping algorithm is applied to determine a control pulse implementing the evolution of the initial operators to the final operators. In other implementations, control signal is determined analytically. In some implementations, the time-dependent operators at time t=T being sufficiently close to the respective target operators comprises the time-dependent operators at time t=T corresponding to the target operators. In some implementations, the time-dependent operators at time t=T being sufficiently close to the respective target operators comprises an infidelity being below a threshold. In some implementations, the threshold is the experimentally achievable infidelity of the quantum system. These thresholds can be quantum system-dependent. In particular, the experimentally achievable infidelity depends on the quantum system in question. In the driven spin chain example below, infidelities of the operators I(t) are determined. Optionally, at step 240 the control signal is transmitted to a controller. The controller is configured control to the quantum system by applying a control signal to the quantum system. Examples of control signals are a control pulse, an electric or magnetic field, any other field, and / or a biasing voltage. Any control signal suitable for controlling the quantum systems may be used. A control signal may comprise a combination of signals, e.g., a microwave pulse and a biasing voltage. In some implementations, the signal calculator 110 of system 100 can execute steps 210 to 240. Controller 120 applies the control signal to quantum system 130 as discussed with respect to Fig. 1. In some implementations, the controller applies a control pulse to the quantum system. In some implementations, the controller applies a time-dependent field (e.g., a magnetic or electric field) to the quantum system. In some implementations, any other suitable control signal is applied. In some implementations, calculator 110 and controller 120 are one unit and no transmittal is necessary. The above methods can be used to determine control signals for a multitude of quantum systems with time-dependent Hamiltonians. In the following, a number of non-limiting examples of obtaining eigenstates and unitary gates for several target (and / or final) Hamiltonians are provided. Driven spin chain An example of a (driven) spin chain can be defined by the system Hamiltonian n n— 1 W) = £ + g XjXj+1 + / 1(0(^ + Xn} . ;=i ;=i (5) Equation (5) contains nearest-neighbour interactions in terms of the Pauli X operators, and single-spin energies in terms of the Pauli Z operators and the Pauli X operators on the end-spins of the chain. Without the single-qubit X-terms, the Hamiltonian would correspond to the regular Ising model. The parity conservation of the Ising model, however, imposes limits on the achievable control. The single-qubit X-terms break this invariance and thus enrich the accessible dynamics. This case can be treated well with the operator(s) I(t) expanded in terms of the Lie algebra generated by nested commutators of the individual terms in the Hamiltonian (Eq. (5)). It is comprised of the basis elements Zi nr=i^ XjZjkXj+k+i Yj ^jk Yj+k+1 ZjZjicYj+k+i ZqiXi+i ■^n—(n—l~)l Yn—(n—I) I with i e [l,n], j e [l,n - k - 1], k e [0,n - 2], I e [0,n - 1] and the shorthand notation Zjk = n!=i With this Lie algebra that contains d = 2n2 + 3n + 1 terms, the present framework reduces a control problem in an exponentially large Hilbert space to a problem in a vector space of quadratic growth. Therefore, solving this control problem for the timedependent operator I(t) speeds up computation time and reduces memory cost. Thanks to the quadratic scaling it is possible to design optimal control protocols for spin chains of lengths that are inaccessible for approaches based on quantum states. The following examples are motivated by typical problems of quantum simulation and state preparation or stabilisation, and they exploit the fact that the Lie algebra given in Eq. (6) contains three-body interactions such as XjZjkXj+k+1 or even n-body interactions that can not be realised by static means. The ground state of the Hamiltonian n-1 n hg = -^^- [jz,, j=l i=l (7) is a GHZ state (Greenberger-Horne-Zeilinger state), which holds significant importance in the fields of quantum sensing, quantum communication, and macroscopic quantum mechanics, attracting great experimental interest. Another interesting example is the Hamiltonian n—2 He = + £ XjZj+rXj+2 + Xn_±Zn )=1 (8) whose ground state is the one-dimensional n-qubit cluster state for measurementbased quantum computation. Since the Hamiltonians Hg and He admit an analytic construction of a target operator, the following discussion also includes the example Hd = Hc + XjXj+1 to demonstrate that no analytic solutions are necessary preconditions for the present approach. The goal in the subsequent examples will be the identification of a suitable timedependence of the functions fj(t) and h(t) for the preparation of the ground states of Hg, Hc and Hd, as well as for the realisation of stroboscopic dynamics induced by these Hamiltonians. In other words, the fj(t) and h(t) encode, among other things, the impact of the control signal on the quantum system's time-dependent Hamiltonian. Furthermore, in these examples, Hg, He and Hd are the target Hamiltonians under investigation. In all these three cases one can choose / „ = ^jZj as the initial condition, and indeed, its ground state is a product state that can be prepared efficiently. The control targets can be constructed numerically in terms of adiabatic dynamics of Eq. (3) as discussed above. In the cases of Hg and He, one can construct suitable control targets lG = HG and Ic = Hc even analytically. In the case of Hd, the control target Id can be obtained numerically via adiabatic evolution and cannot be written in a simple form. For example, a target Id with the same ground state as Hd can be found by evolving the initial operator Io under an adiabatic Hamiltonian of the form (n-1 \ j j=i ) (9) where Ti is a sufficiently long duration such that the ground state of the initial Hamiltonian, £7Z7, evolves to that of the final Hamiltonian, Hd, at t = Ti. The inclusion of the third term is necessary to eliminate real crossings in the ground state energy curve, ensuring an adiabatic transition to the ground state of the final Hamiltonian. In the d-dimensional space of the Lie algebra, Hi(t) corresponds to (n-1 \ + K^ + K^ + K- , i i=i J (10) where Kf, KDf K™ and Kf are the dxd adjoint representation matrices of Zj, Hd, XjXj+i and Xj, respectively (see Eq. (4)). Starting with the initial operator Ia = one propagates with Ki(t) a(t) = (11) where a(t) is the coefficient vector of I(t). Since the adiabatic evolution induces a transition from the ground state of £;Z7 to that of Hd, and the ground state of / 0 = XjZj, must evolve to the ground state of the control target Id, it follows that Id has the same ground state as Hd in the adiabatic limit. Once any such control target It is given, the infidelity J = 1 - tr( / (T) / r) / tr( / p) can be defined as the objective functional. Since this is readily expressed as J = 1 - a(T) • ar / ||ar|| with the vector ar of the control target It, it is indeed possible to perform all required calculations in the d-dimensional space defined by the construction of the operator I(t), and no reference to the exponentially large Hilbert space of the system is necessary. For optimizations of propagators, one can define Fj(U,UT) = tr(U UTIj(Sf)U^ / tr( / / (0)) for any initial condition Ij(0) and an objective functional G(U,Ur) = X™ iFj(U, UT) / m, where m denotes the number of initial conditions. The choice of the initial conditions depends on the Hamiltonian that generates the dynamics. The objective functional G(U,UT) with the set of initial conditions [{Zk},{XkXk+1},X1 + Xn] is maximised exactly for U = Ut. An optimisation with this would, however, imply evaluating 2n functions Fj(U, Ut) at any step in the optimisation. It can be preferable to use the objective functional G(U,l / r) with only five initial conditions {EkZ2k+i, ZkZ2k< ^^+1^+2^^^+1^1 + ^)- Even though it can in principle be maximised for propagators different than Ut, this does not seem to happen in practice. Since verification requires only a single propagation, it is numerically advantageous to use G(U,UT) for optimisation, and eventually G(U,UT) for verification. The control targets Ig, Ic and Id defined above can be used to design the timedependent functions fj(t) and h(t) in the system Hamiltonian Eq. (5) such that the system evolves towards the ground state of Hg, He and Hd, respectively. Fig. 3a depicts the infidelities for Ig (solid diamonds), Ic (solid squares), and Id (solid triangles) obtained with such optimised protocols as a function of the number n of spins in a chain with n ranging from 5 to 50. The optimisations for Ig and Ic are intentionally terminated when infidelities below 10-6 are reached. The optimisation for Id, on the other hand, requires longer running time due to the difficulty of realising the ground state of a gap-less model, and it is terminated once a maximum number of iterations is reached. In all cases, the infidelities are well below the limits imposed by system imperfections of current or foreseeable quantum devices. Fig. 3b depicts infidelities 1 - G(U(T),UC) (solid diamonds), 1 - G(U(T),Uc) (solid squares) and 1 - G(U(T),UD) (solid triangles) for the three target dynamics UG = exp (—in / BHG), Uc = exp (-in / QHc), UD = exp (-in / B HD). Similarly, to the task of state preparation, the realisation of desired propagators is achieved with infidelities far below what could be experimentally resolved with existing technology. Because of the constant interaction term in the system Hamiltonian (Eq. (5)), there is a minimal time required for all of the above control problems, also referred to as the quantum speed limit. In order to obtain infidelities as low as in Fig. 3, the duration of the controlled dynamics needs to increase with the system size, but a modest approximately linear increase (T~rm / 2g) with the qubit number, and the timescale 1 / g defined by the interaction constant g in Eq. (5) is sufficient for the ground state transfer. For Uc and Ud, the required duration T-it / 2g is independent of the system size. For Ug the required duration scales linearly in the number of qubits, T~2nn / g due to the n-body interaction term. The solid shapes in Fig. 3 depict infidelities of target operators, but the actual objects of interest are quantum states and propagators. Continuity arguments imply that a low infidelity for a target operator leads also to a low infidelity for states or propagators, and a quantitative formulation is given by the inequality ^0\U\T}HtU(T)\^0)-E0 El Eq (12) in terms of the target Hamiltonian Hr, its lowest two energies Eo and Ei, and the initial state |iPo>. Since constructing the operator U\T)HtU (T) requires simulations within the Lie algebra only, this bound can be readily evaluated. For Hg and He the lowest two energies are given by Eo(n) = -n and Ei(n) = -n + 2. The empty diamonds and squares in Fig. 3a depicting the bound in Eq. (12) for the states realised with the optimised protocols confirm the accurate state preparation. Hd, on the other hand, corresponds to a gap-less model whose energy gap Ei - Eo diminishes with increasing qubit numbers, implying that the bound in Eq. (12) gets looser. The state infidelity, in this case, is thus estimated from time-dependent simulation based on matrix product states (MPS) for qubit numbers n <30, as depicted by the empty triangles in Fig. 3a, again confirming the accuracy of the optimisation protocol. Despite the efficient matrix-product representation of the initial and final states, the states at in-between times created by the controlled dynamics for n >35 require bond dimensions exceeding 500, rendering time-dependent simulations impractical. The empty shapes in Fig. 3b show gate infidelities, 1 - |tr(l / t(7’)I / r)|2 / 22”, obtained with exact exponentiation; here, the exponential scaling of the Hilbert space prohibits simulations for chains exceeding n ~ 10. Within what is numerically achievable, however, the gate infidelities do confirm the success of the control protocols designed with the present framework. Figure 4 shows two examples of a driven spin chain. The same set of operators (13) as those of the Hamiltonian given in Eq. (5) act on the spin chain as depicted in Fig. 4a. Therefore, the set of operators defines the same Ue algebra as Eq. (6) with the elements flij = Zj, Q,2j = XjZjkXj+k+lf O,3 j = YjZjkYj+k+1, Q,4j = XjZjkYj+k+1, j = YjZjkXj+k+li fl™ = ZomXm+lr ^71 = 2omfm+i, fl™ = Xn_mZ(n_m^m, H91 = ^10 = 117=1¾ (14) with m e[0, n-1], k e[0, n-2] and j e[1, n-k-1]. As there are n(n - 1) / 2 elements of the form j and along with n elements like fl17, fl™, (1^ / He1, and the individual element n10, the size of the algebra scales as d = 2n2+3n+l, i.e. quadratic in the number of spins. This quadratic scaling is independent of any system parameters, such as interaction constants in the Hamiltonian, and is applicable to disordered systems. It also extends to Hamiltonians that include terms of Eq. (14) in addition to the terms in Eq. (13). Since for k = 0, the elements fl$7, fl^7, fl|>7 read ^,j = W+i, fl° • = X7 / +1, fl°• = / X7+1, this quadratic scaling applies to a large variety of interacting spin chains (see e.g. Fig. 4b). An even more favourable, linear scaling is obtained for the standard Ising model with periodic boundary conditions, excluding the Xi and Xn terms. Defining the optimal control problem in a smaller vector space than the Hilbert space of the Hamiltonian allows a reduction of computation time and memory cost for determining control signals. Driven spin comb While the favourable scaling discussed with respect to Figure 4 relies on the onedimensional geometry of a spin chain, similarly favourable scaling can also be identified in two-dimensional interaction geometries, such as the comb structures depicted in Fig. 5a. It is defined in terms of the set of operators III {^j,l^j,2’^2j-l,2^2j,2’^2j,2^2j + l,2} With j 6 [l,n / 2] (15) for an even number of spins n. The resultant Lie algebra contains (n2 - n) / 2 elements, i.e. it scales quadratically with the number of spins. To see this, it is helpful to notice that the Lie algebra defined by the YY interactions and the ZZ interactions (as a special case of the spin chains discussed with respect to Fig. 4) contains (n2 - 2n) / 8 elements. Including the Xi, 1X1,2 interaction expands the resulting Lie algebra by n / 2 terms and adding the interactions Xk,iXk,2 keeps augmenting the Lie algebra. If these interaction terms are added in increasing order in k, the addition of the term Xk,iXk,2 augments the Lie algebra by (n + k) / 2 terms. Including all the XX interactions thus augments the Lie algebra by (3n2 - 2n) / 8 terms resulting in a Lie Algebra with the (n2 - 2n) / 2 elements mentioned above. The additional driving (Zj^)^ on the spins of the lower chain (Fig. 5b) generates a richer algebra, of size (9n2 - 6n) / 8 and, in particular, that contains the mutually commuting 4-body operators of the form {Zz^Ym^Zz^^ which, through appropriate relabelling, correspond to the Plaquette operators characteristic of the toric code, namely {Zz^^zk-^Xz^Zz^}, Without the driving terms {Z^}^ but with the inclusion of additional interactions like and Z2),i^2)+i,i between spins in the lower chain, as depicted in Fig. 5c, the algebra expands even further. Each of these interactions doubles the Lie algebra, such that a comb structure with a small number of interactions in the lower chain still has a sub-exponential scaling. Driven spin ladder While the previous examples demonstrate that certain Lie-algebras, generated by the Hamiltonian operators {!),}, have dimensions that scale polynomially with the number of qubits n, the following example illustrates a systematic approach for identifying sets of operators {aj that form a small algebra under the action of the Hamiltonian, even if the Lie-algebra associated to the Hamiltonian scales unfavourably. Consider the spin ladder depicted in Fig. 6 with the set of Hamiltonian operators 1^2fc+l'^2k—1^2fc'^2k^2fc+2} With k € [T ,, 1] (16) n and open boundary conditions. The corresponding Lie algebra has 2z(n2 -n) / 8 elements, i.e. it is exponentially large. It is, however, possible to find a set of operators {aj with linear scaling that satisfies the required commutation relation [ct / ,l>fc] = -iSii^aj. The construction of such sets can be done recursively starting from the operator (n—2) / 4 al = ^1¾ k=0 (17) where denotes the symmetric sum over all permutations of the product of 2k pairs Z2;Z2j+i of Pauli Z operators on spins 2 to n. For instance, for n = 10, those read Zos = l (18) Zj = z2z3z4z5 + z2z3z6z7 + z2z3z8z9 + z4z5z6z7 + z4z5z8z9 + z6z7z8z9 (19) XI = Z2Z3Z4Z5Z6Z7Z8Z9 (20) In general, ax is a sum over 2(n-4) / 2 individual operators. The second operator can be readily constructed noting that a4 commutes with all the elements ¢ / of the Hamiltonian except forZiZg. In particular, one can define a2 « k (21) which now commutes with all the elements in the Hamiltonian except for Z4Z3 and / 2 / 3. This defines the third element a3 oc [a2, / 2 / 3] (22) that, since a3 anti-commutes with / 2 / 3, Z2Z4 and Z3Z5 defines the fourth element a4 oc [a3,Z2Z4] = [a3,Z3Z5] (23) In turn, a4 anti-commutes with Z2Z4, Z3Z5 and / 4 / 5 and determines ®5 K (24) This process can be repeated until the last element is derived: an K [an-l’Zn-2Zn] (25) Since this operator only anti-commutes with Zn_2Zn, it follows that the algebra {a1( ...,an} is closed under the Hamiltonian's action and, specifically, is of n size. Within the set of elements {aj, the n-body operator (after the appropriate relabelling) L(n-4) / 4j ^1(Z4 / c+2 / n_4 / c_4)sZn, fc=O (26) with being the symmetric sum over all permutations of the product of 2k pairs X2jX2j+1 and (n - 2k - 4) pairs / 2 / / 2 / +1 is composed of mutually commuting operators and of interest in the implementation of surface codes. For n = 10, those read (x2Y6y = x2x3y4 - y9 + ^3X4^6 - Yg + y2 - r5x6x7r8r9 + y2 -r7x8x9, (27) (x6Y2y = y2y3x4-x9+x2x3y4y5x6-x9 + x2-x5y6y7x8x9 + x2-x7y8y9 . (28) This shows that the vector space spanned by {aj grows linearly in n whereas the Hilbert space of the Hamiltonian grows exponentially, i.e. the vector space spanned by {aj is generally smaller than the Hilbert space. The above-described methods and examples provide the following advantages, amongst others: The dimensionality of the control problem can be reduced, which reduces the computational resources required (such as exponential savings in memory usage) and increases the speed at which computations can be completed (reaching up to exponential speed-ups in computing time). This advantage becomes more pronounced the larger the qubit count. For example, the realization of a multi-qubit entangled state for quantum sensing, where conventional methods are limited to qubit counts below 20, whereas the methods of the present disclosure can be applied to systems including hundreds of qubits. In quantum systems with control over (qubit) interaction geometries in various platforms such as superconducting qubits, trapped ions and Rydberg atoms the methods according to this disclosure can help to fully exploit the qubit count in current devices (ranging up to 256) and future device (which may have even higher qubit counts) without the need of long gate sequences that are conflicting with decoherence and the accumulation of individual gate errors. Control signals determined according to aspects of this disclosure implementing multi-qubit gates for systems with tuneable qubit interactions provide significant speed up in the implementation of multi-qubit gates in the quantum system compared to decomposing the multi-qubit gate into a quantum circuit of one-qubit and two-qubit gates. While verification of the functionality of a quantum device is a challenging problem, the description of quantum dynamics in terms of time-dependent operators according to the disclosed methods can also be used to make predictions for the targeted dynamics that can be experimentally tested. With three-body and four-body interactions naturally arising in the dynamics induced by the spin Hamiltonian given in Eq. (5), the disclosed methods and systems of this disclosure can also support the realisation of error-correcting codes and topologically protected quantum information processing. With the advent of highly controllable quantum systems that pose control problems beyond the limitations of classical simulations, there is a growing need for control techniques with good scaling behaviour. Optimisations based on time-dependent operators as described in the foregoing can help to avoid or support costly experimental optimisations. The resulting 5 ability to create quantum states that allow for sensing at the Heisenberg limit makes the present approach a valuable tool for the development of quantum technological applications. Methods for determining a control signal disclosed herein can be implemented in a 10 computing system comprising one or more computing devices. In some implementations, a computing device may be a classical computing device. In other implementations, a computing device may be a quantum computing device. In further implementations, a computing system may comprise both classical and quantum computing devices or components. Further, methods for determining a control signal 15 disclosed herein may be stored on a computer-readable medium configured such, that when read by a computing device (or computing system) the methods can be executed by the computing device (or computing system).

Claims

1. A computer-implemented method for determining a control signal for controlling a quantum system comprising:identifying a time-dependent Hamiltonian, H(t), of the quantum system comprising time-dependent terms representing the control signal;defining a control problem for evolving one or more time-dependent operators, I(t), from one or more initial operators into one or more respective target operators under the dynamics of an equation of motion given by the time-dependent Hamiltonian, wherein each eigenstate of the one or more time-dependent operators is a solution to the time-dependent Schrodinger equation given by the time-dependent Hamiltonian;determining the control signal based on the control problem.

2. The method of claim 1, wherein each of the one or more time-dependent operators is a hermitian operator with a vanishing total time-derivative and the timedependent operator's eigenstates are non-degenerate.

3. The method of claim 2, wherein the equation of motion given by the Hamiltonian is defined as= ^ + i[WV(t)] = 0 .

4. The method of claim 3, whereinthe time-dependent Hamiltonian is written as H(t) = Xk^k spanned by a set of operators ^k with scalar coefficients hk and wherein at least one hk is timedependent;the time-dependent operators are written as / = y,‘j=1ajaj spanned by the operators a7 with time-dependent scalar coefficients af, andthe operators a, are closed under commutation with any of the operators Dk-5. The method of claim 4, wherein the relation = -i holds for allcommutators [a,,l)fc] and all A are scalars.

6. The method of claim 5, wherein the equation of motion given by the Hamiltonian is written asa = ,where the elements of the matrix K(t) are Ktj(t) =Ek=iht(t)4, andwhere hk(t) are the scalar coefficients of the time-dependent Hamiltonian and a is a vector of the time-dependent scalar coefficients at of the time-dependent operators.

7. The method of any one of claims 2 to 6, wherein the time-dependent operators are represented by tensor networks or matrix product operators.

8. The method of any preceding claim, wherein the control signal determined based on the control problem evolves the quantum system from an eigenstate |iPi> of a first Hamiltonian to an eigenstate |IP2) of a second Hamiltonian and wherein the first and the second Hamiltonian are each representable by the time-dependent Hamiltonian.

9. The method of claim 8, wherein defining the control problem comprises the control problem fulfilling the conditions:the one or more initial operators is one initial operator, 1(0), with an initial eigenstate corresponding to the eigenstate | tPi> of the first Hamiltonian, the initial eigenstate of 1(0) having a non-degenerate eigenvalue y;the one or more target operators is one target operator, It, with a target eigenstate corresponding to the eigenstate | Uh) of the second Hamiltonian, the target eigenstate of It having the same eigenvalue y; andthe target operator It being reachable from the initial operator 1(0) within the dynamics of the equation of motion.

10. The method of any one of claims 1 to 7, wherein the control signal determined based on the control problem applies a unitary gate Ur to the quantum system.

11. The method of claim 10, wherein defining the control problem comprises the control problem fulfilling the conditions:the one or more target operators hj being defined based on the respective one or more initial operators Ij(0) and the unitary gate Ut as lTj = UtIj^U^.;the initial operators Ij(0) and target operators hj defining the unitary gate Ut uniquely; andeach target operator being reachable within the dynamics of the equation of motion starting from its respective initial operator.

12. The method of any preceding claim, wherein determining a control signal based on the control problem comprises minimizing or maximizing an objective function.

13. The method of any preceding claim, wherein determining a control signal based on the control problem comprises:determining a control signal that evolves the initial operators at a time t=0 into the final operators at a time t=T; andwherein the final operators at time t=T are sufficiently close to the respective target operators.

14. The method of claim 13, wherein determining a control signal that evolves the initial operators at a time t=0 into the final operators at a time t=T comprises: propagating the one or more initial operators according to the equation of motion.

15. The method of any one of claims 13 to 14, wherein the control signal is a control pulse and determining the control pulse based on the control problem comprises applying a numerical pulse shaping algorithm that realizes the evolution of initial operators into final operators.

16. The method of any one of claims 13 to 15, wherein the final operators at time t=T are sufficiently close to the respective target operators comprises an infidelity being below a threshold.

17. The method of claim 16, wherein the threshold is the experimentally achievable infidelity of the quantum system.

18. The method of any preceding claim, wherein the operators spanning the one or more time-dependent operators are defined in a vector space that is smaller than the Hilbert space of the Hamiltonian.

19. The method of any preceding claim, wherein the quantum system is a quantum computer comprising a plurality of qubits.

20. The method of any preceding claim, wherein the control signal comprises any one of a control pulse, a magnetic field or an electric field.

21. The method of any preceding claim, wherein subsequent to determining the control signal, the control signal is transmitted to a controller, the controller configured to apply the control signal to the quantum system.

22. A computer-readable medium configured to store any one of the methods of claims 1 to 21.

23. A computing system configured to execute any one of the methods of claims 1 to 21.

24. A system for applying a control signal to a quantum system comprising: a signal calculator that determines the control signal according to any one of claims 1 to 21,a controller that applies the control signal to the quantum system.

25. A system according to claim 24, wherein the controller comprises a microwave generator that applies the control signal as a microwave pulse.

26. A system according to any one of claims 24 to 25, wherein the controller comprises a laser generator that applies the control signal as a laser pulse.

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