An optimal stochastic controller for an open quantum system

Abstract

The invention is in the field of a method of quantum computing, that is, information processing based on quantum-mechanical phenomena, in particular a stochastic controller for an open quantum system, a method of stochastic optimal control of an open quantum system, and a computer program for stochastic optimal control of an open quantum system.

Description

[0001] AN OPTIMAL STOCHASTIC CONTROLLER FOR AN OPEN QUANTUM SYSTEM FIELD OF THE INVENTION

[0002] The invention is in the field of a method of quantum computing, that is, information processing based on quantum-mechanical phenomena, in particular a stochastic controller for an open quantum system, a method of stochastic optimal control of an open quantum system, and a computer program for stochastic optimal control of an open quantum system.

[0003] RELATED APPLICATIONS

[0004] The present application claims the benefit of priority from European Patent Application EP 24207671.9, filed on October 20, 2024, in the name of Stichting Radboud Universiteit, The Netherlands.

[0005] The entire contents of the above-referenced applications and of all priority documents referenced in the Application Data Sheet filed herewith are hereby incorporated by reference for all purposes.

[0006] BACKGROUND OF THE INVENTION

[0007] Quantum computation may be defined as a computation using laws of quantum physics, in particular the Schrödinger equation. Concretely, quantum computation performs a sequence of operations on an experimentally prepared initial quantum state to compute a desirable final or ground state. The problem is known in the literature as quantum state preparation. The key problem in quantum computing is to find the sequence of operations that realizes a given final state from a given initial state. This problem can be formalized as an optimal control problem: Given an initial quantum state and a cost that quantifies the quality of the final state as well as the cost of the computation, find the sequence of control operations that minimizes this cost. However, these methods are often difficult to implement and do not scale well to large problem instances.

[0008] In quantum computing, a quantum algorithm is considered to relate to an algorithm that runs on a realistic model of quantum computation. An often-used model is a quantum circuit model of computation. In comparison, a classical algorithm is a sequence of instructions, or a step-by-step procedure, for solving a problem. Therein, each step or instruction can be performed on a classical computer. Similarly, a quantum algorithm may also be considered a step-by-step procedure, where each of the steps can be performed on a quantum computer. The quantum algorithm is considered to be mostly or fully inherently of a quantum nature.

[0009] In quantum computing so-called cold atoms may be used. In the last several years, Rydberg-atom quantum processors have emerged to become one of the most promising platforms for quantum computing. These quantum processors typically rely on neutral atoms as qubits that can be and are well isolated from the environment and prepared in large systems of hundreds or thousands of qubits with different geometries, e.g. using laser cooling and trapping techniques. Quantum information is encoded in the internal states of atoms with interactions mediated via their highly electronically excited Rydberg states. In combination with coherent laser excitation, Rydberg atoms can realize strong and widely tunable interactions that can extend over several pm. Moreover, they offer significant advantages over other technologies, including the possibility for long coherence times (» 1 second, for ground state encoding), fast and switchable interactions, allowing for the generation of highly entangled states, and high-fidelity entangling operations. Accordingly, ultracold Rydberg quantum systems have proven very successful for studying many -body physics and analog quantum simulation, and are now ready to advance to the forefront among other platforms for scalable quantum computing

[0010] Quantum algorithms are usually described, in the commonly used circuit model of a quantum system of quantum computation, by a quantum circuit, that acts on some input qubits, characterizing an initial quantum state of the quantum system, and terminates with a measurement on the final quantum state. A quantum circuit may consist of simple quantum gates, each of which acts on some finite number of qubits. Quantum algorithms can be categorized by the main techniques involved in the algorithm. Some commonly used techniques / ideas in quantum algorithms include phase kick-back, phase estimation, the quantum Fourier transform, quantum walks, variational quantum eigensolver (VQE), amplitude amplification and topological quantum field theory.

[0011] Most quantum computing paradigms nowadays are based on digital quantum circuits, which typically contain a fixed set of predefined quantum gates that are tailored to the physical device that may be available as such. An ansatz is made for the shape of the quantum circuit, e.g. in terms of the pattern of gates. Furthermore, a computation time required is fully determined by the gate sequence and cannot be easily decreased to minimize unwanted effects of decoherence; Methods, such as the variational quantum eigensolver (VQE), have been designed to find the ground state of quantum Hamiltonians, which has direct application for a large class of combinatoric optimization problems and problems in quantum chemistry.

[0012] A variational quantum eigensolver (VQE) is a quantum algorithm that may be used for specific purposes, such as for quantum chemistry, for quantum simulations, and optimization problems. It is a hybrid algorithm as it uses both classical computers, and quantum computers, in order to resolve the ground state of a quantum system representing e.g. a given physical system. Therein a quantum processor or quantum algorithm calculates an expectation value of the system with respect to an observable, often the target Hamiltonian, given a guess or ansatz of the parameters of the variational quantum circuit, and a classical optimizer is used to improve the parameters. The objective of the quantum algorithm, and in particular of the VQE, is to find a set of quantum operations that prepares the lowest energy state (or minima) of a target Hamiltonian, or a close approximation thereof. While the only strict requirement for the representation of an observable is that it is efficient to estimate its expectation values, it is often simplest if that operator has a compact or simple expression e.g. in terms of Pauli operators or tensor products of Pauli operators.

[0013] In contrast to these gate-based methods, e.g. (laser) pulse-based methods aim to directly optimize the pulses interacting with individual qubits, giving rise to analog quantum circuits. Optimal control is a natural way to optimize the pulse sequence. This method has been used to optimize laser pulses for quantum gate implementations, but has only recently been proposed for the full optimization of analog quantum circuits in VQEs. Pulse-based methods offer several advantages over gate-based methods: they allow for faster state preparation, simpler implementation and yield more expressive circuits. Recently such quantum-programs-at-the-pulse-level are provided by several companies and academic research groups, which relates to a programming environment that provides pulse-level access to quantum circuits. These programs provide direct access to the elementary components of the device, thus enabling programming of applications for analog quantum circuits.

[0014] To some extent the present invention also relates to a Quadratic unconstrained binary optimization (QUBO), also known as unconstrained binary quadratic programming (UBQP). This is considered to relate to a combinatorial optimization problem. It may have a wide range of applications, from finance and economics to machine learning. Moreover, due to its close connection to Ising models, QUBO constitutes a central problem class for adiabatic quantum computation, where it is solved through a physical process called quantum annealing.

[0015] To some extent the present invention also relates to Quantum annealing (QA), which is an optimization process for finding a global minimum of discrete optimization problems, e.g. the ground state, of a given objective function by gradually changing the quantum Hamiltonian from an initial Hamiltonian whose ground state can be easily prepared to a final Hamiltonian that is equal to the target Hamiltonian and whose ground state solution is the solution of the optimization problem, using the effect of quantum tunneling.

[0016] Evans et al. (https: / / doi.org / 10.48550 / arXiv.2111.09896) recite that High fidelity state preparation represents a fundamental challenge in the application of quantum technology. While the majority of optimal control approaches use feedback to improve the controller, the controller itself often does not incorporate explicit state dependence. Here, we present a general framework for training deep feedback networks for open quantum systems with quantum non-demolition measurement that allows a variety of system and control structures that are prohibitive by many other techniques and can in effect react to unmodeled effects through nonlinear filtering. We demonstrate that this method is efficient due to inherent parallelizability, robust to open system interactions, and outperforms landmark state feedback control results in simulation. Glaser et al. (DOI: 10.1140 / epjd / e2015-60464-1) recite that control that turns scientific knowledge into useful technology: in physics and engineering it provides a systematic way for driving a dynamical system from a given initial state into a desired target state with minimized expenditure of energy and resources. As one of the cornerstones for enabling quantum technologies, optimal quantum control keeps evolving and expanding into areas as diverse as quantum-enhanced sensing, manipulation of single spins, photons, or atoms, optical spectroscopy, photochemistry, magnetic resonance (spectroscopy as well as medical imaging), quantum information processing and quantum simulation. In this communication, state-of-the-art quantum control techniques are reviewed and put into perspective by a consortium of experts in optimal control theory and applications to spectroscopy, imaging, as well as quantum dynamics of closed and open systems.

[0017] The present invention therefore relates to an improved method for quantum computing which e g. scales better with large problems to be solved, and further aspects thereof, which overcome one or more of the above disadvantages, without jeopardizing functionality and advantages.

[0018] SUMMARY OF THE INVENTION

[0019] In this patent a path integral control method is applied to control an open quantum system (quantum path integral control, QPIC) using a pulse-based optimal control framework. The control problem is to optimize the sequence of pulses such that the quantum circuit transforms a given initial state into a desired final state (as fast as possible). An open quantum system is considered to relate to a quantum -mechanical system that interacts with an external quantum system, which is known as the environment or a bath. In general, these interactions significantly change the dynamics of the system and result in quantum dissipation, such that the information contained in the system may be lost to its environment. The optimal control can be an open loop controller where the control does not depend on the state of the quantum system, and / or it may be a state dependent feedback controller. For open loop control, typically only the measurement of the cost of the final state is required. Feedback control typically requires in addition measurement of the quantum state for all times throughout the computation. QPIC control has the following advantages over the state of the art: The optimal control is computed directly from simulations of the stochastic dynamics. There is no need to solve a Bellman equation or PMP equations, as is currently done. As a result, the computational complexity is exponentially reduced from O(22n) to O(2n) with n the number of qubits. The optimal control may be computed off-line, using a simulator of the quantum circuit, or may be computed on-line using the quantum device directly. This latter possibility is considered unique for the proposed method that is not possible using any of the existing control methods. It typically has two important advantages over existing methods: 1) the control is adjusted to the particular device without the need to model noise and decoherence separately. 2) Using a quantum simulator requires explicit storage of the quantum state which is of size O(2n) and therefore restricted to relatively small n. By using the quantum device instead, the control can be computed in principle for arbitrary large number of qubits. For classical problems, such as arises in robotics, the PI control method is very efficient and provides robust solutions for high dimensional non-linear stochastic problems where other methods fail. However, the application of the path integral control method to quantum computing is non-trivial for the following reason: PI control works in a stochastic setting only, while quantum computation is deterministic following a unitary evolution. To solve this issue inventors proceed as follows. First, one considers an open quantum system that explicitly models the interaction of the quantum device and the quantum computation with the environment. The dynamics of this open quantum system is given by the so-called Lindblad equation, which is a deterministic equation that models the evolution of the quantum density operator. Subsequently, one defines a so-called stochastic unraveling that is constructed such that its average behavior coincides with the Lindblad equation. In general, the stochastic dynamics that is defined by such an unraveling is not suitable to define a PI control problem, because PI control only works for a particular class of dynamical systems. However, the unraveling of a given open quantum system is not unique and for certain classes of open quantum control problems one can use this freedom to transform the unraveling to a PI control form. This then allows to define a QPIC method for open loop control. In order to apply the PI control method to compute a state feedback controller, one must measure the state throughout the quantum evolution. However, measurement cannot be done on a quantum system without changing the quantum state. Therefore, special care must be taken. One can develop a so-called hybrid dynamics that describes the joint evolution of both the quantum state and the measurement outcome. Within this framework one can indirectly monitor the quantum state from the measurement outcomes. In simulation the applicability of this idea to control a circuit of n qubits is demonstrated to reach an arbitrary final state at minimal cost. In addition to quantum state preparation, the invention may also have application to control of qubits in quantum hardware.

[0020] The present invention relates in a first aspect to a device for quantum computing (1), in particular a control architecture, for control of and optionally comprising an open quantum system with Quantum Path Integral Control (QPIC) for providing, that is computing an accurate approximation of, a ground quantum state from an initial quantum state of a quantum system, comprising a digital controller (10), an analog quantum device (20), and a detector (30), wherein the detector (30) is configured to perform at least one measurement x(t) on the analog quantum device at a point in time t and to provide said at least one measurement to the digital controller (10), the analog quantum device (20) is configured to receive input u(t) from the digital controller (10), and wherein the analog quantum device is configured for quantum computing, the digital controller (10) comprises a filter (12), a stochastic controller (11), and an adder (15), wherein the filter is configured to receive input x(t) from the detector and is configured to estimate a distribution over current quantum states (t) at time t based on the past measurement outcomes x(0:t) and the past controls u(0:t-dt) in the form of a probability distribution P('P,t|x(0:t), u(0:t-dt)), and to provide the probability distribution PQP,t|x(0:t)) to stochastic controller 11, wherein said probability distribution is represented by a number of weighted samples QP1, wi) to (Tk, wk) with weights wi to Wk, wherein k is the number of samples, and wherein the stochastic controller at time t is configured to receive the weighted samples ('P1)to ('Pk) and to compute an optimal control LJf'H1) to 11,^), wherein the adder is configured to perform the weighted sum u(t) of the optimal controls U(t, T1) to U(t, 'H'1) with weights wi to Wk and to provide output (u(t) to the analog quantum device and to the filter (see e.g. fig. 1). Fig. 1 shows a functional diagram. It consists of a quantum device, a detector and a digital controller. The quantum device can be any analog quantum computing system e.g. consisting of (any type of) qubits. The quantum elements change dynamically in time according to the laws of quantum physics, typically under dynamic Hamiltonian H, time-dependent disturbances from the environment, and time dependent input provided by the controller u(t) as provided by the controlled stochastic quantum dynamics for hybrid systems, e.g. Eqs. 24 and 17, and subject to the path integral control cost, e.g. Eqs. 2-3. The state of the quantum system at time t is denoted as (t). The detector can be any type of detector that measures partial and noisy information of the quantum state denoted as x(t). The digital controller consists of two parts: a filter and a control law. The filter estimates the current quantum state T (t) based on the past measurement outcomes x(0:t) and the past controls u(O:t-dt). Since the quantum state cannot be known exactly, this estimate takes the form of a probability distribution P('F,t|x(0:t), u(0:t-dt)). In practice this probability distribution cannot be represented exactly and is represented by a number of weighted samples 'F1,... 'Fkwith weights wi,... Wk (e.g. using particle filtering). The control law at time t computes the optimal controls U(t? F') to U(t,lFk. The output of the digital controller u(t) = Li Wi U('F') is the weighted sum of these controls and depends in principle on the entire measurement history x(0:t) and past controls u(0:t-dt. As an approximation one may consider the special case that u(t) does not depend on x(0:t), called an open loop controller. Otherwise, it is called a feedback controller. Fig. 1 shows details of the digital controller featuring both the filter (12) and the stochastic controller (11). Both filter and stochastic controller use a physical model of the quantum device. The physical model is detailed e.g. in Fig. 1 and specifies the stochastic dynamics of the quantum device in terms of dynamic Hamiltonian H, including the time dependent external control input u(t), time-dependent disturbances from the environment C, dxi(t). The physical model also specifies the stochastic relation between the quantum state T (t) and the measurement outcome x(t) (fqand dW(t)). The physical model provides a complete description how an initial state of the quantum system 'F(O) at time t=0 evolves to a final state (T) at time t=T. The stochastic controller consists of the physical model, the initial intermediate quantum state (t) at time t and the control objective. The intermediate control objective at time t at the intermediate quantum state 'Fis a function J(' / ,t) that depends on the future quantum state trajectory T (t: T) and the control trajectory u(t: T). The control objective is to prepare a final state that is close to the ground state of a target quantum Hamiltonian from the initial intermediate quantum state 'F1, while minimizing computation costs. The physical model, the initial state, and the control objective define a stochastic optimal control problem, which is to find the feedback control function u(t: T, 'F^tH)) or open loop control function u(t: T) that minimizes the control objective J. The stochastic optimal control problem is solved using path integral control methods. The solution provides the control solution U(t,lF) for each sample T” with i=l to k. The path integral control computation involves sampling trajectories using a state and time dependent sampling control function u. The sampling procedure can be optimized by using better sampling controllers to give more accurate control law solutions (iterative importance sampling, publication available). The iterative importance sampling can be initialized with any sampling controller, for instance u=0 or (provided by the solution of the deterministic control problem. The control solution U(t,1) is computed for all samples T*1toT^ and the weighed sum is performed by the adder (15) to yield the control u(t) at time t. 1 Overview

[0021] A summary of path integral control (PI and i-PI variants), the unraveling of the Lindblad equation both in the case of no measurement and open loop control and the case of measurement (hybrid dynamics), the control of open quantum system in the absence of continuous measurement (only at the end time) including some initial numerical results and proposed research, ducethe problem of stochastic optimal feedback control an open quantum system using continuous measurement of observables that give only partial information about the quantum state, are detailed It is shown that the partially observed control problem separates in a latent (quantum) state estimation problem that can be estimated using particle filtering and a control problem that can be solved with path integral methods.

[0022] 2 Path integral control

[0023] There exists a class of control algorithms that can solve a large class of non-linear stochastic optimal control problems, known as the path integral (PI) control methods. An advantage is that for problems of this type, the optimal control solution can be expressed in closed form as a path integral, without the need to solve a Bellman equation or the PMP equations. The path integral can be estimated relatively efficiently through sampling. The sampling is optimized using a procedure called adaptive importance sampling which estimates gradients based on self-generated quantum trajectories and is well suited for parallel computing with virtually no overhead. PI control has been very succesfully applied to many high dimensional non-linear stochastic optimal control problem with real-time requirements that occur in robotics where all other methods fail. Inventors showed that PI control theory can produce state-of-the art control solutions for a challenging stochastic quantum control problem from NMR physics.

[0024] The idea goes back to the seminal work by Fleming and Mitter. Consider the stochastic dynamics

[0025]

[0026] = f(t, X“)dt + g(t, X“)(u(t, X“)dt + d Wt) ( 1 ) with X“ the state of the system at time t under control u and Wtis a m-dimensional Brownian motion with (dWt) = 0 and dWtd W ) - vdt and v a covariance matrix, f.g are arbitary non-linear functions making this a quite rich class of dynamical systems, but the control acts in a particular way. Define the cost

[0027] Su(xt,t) = < D(X“) + ds (v(s, X“) + ^u(s, X“)'Ru(s, X“)j + ' u(s, X^)'RdWs(2)

[0028]

[0029] The symbol ' denotes transpose and R a positive m X m symmetric matrix such that R = / lv with A > 0 a constant. V and O are the state-dependent path and end cost, respectively. The control problem is to find the function u that minimizes

[0030] J(t0, VQ) = min (Su(x0, t0))u(3)

[0031]

[0032] where ()Hdenotes the average with respect to the Brownian motion dWto tiusing control u. The function u that minimizes Eq. 3 is called the optimal control. Eqs. 1 and 2 define the class of path integral control problems.

[0033] The generic approach to solve a stochastic optimal control problem is to derive a partial differential equation for J(t, x) which is known as the Bellman equation from which one computes the optimal control. This approach is in general infeasible for high dimensional problems. Instead, for the class of control problems defined above, the solution is explicitly given in terms of a path integral:

[0034] J(t0, x0) = -Alog{e-SV(MA)v(4)

[0035] ^ ^

[0036] u(t0,x0) = v(t0, v0) + lim — (5) di,()

[0037]

[0038] dt (es

[0039] The control v in the rhs of Eqs. 4 and 5 is called the sampling control and is also a function of v, t. Therefore the J and u can be estimated by sampling using a sampling control v. Any sampling control v is an unbiased estimate of the quantity of interest, but the quality of the estimate in terms of variance strongly depends on the choice of v. It was shown that the optimal sampling controller has zero variance and is identical to the optimal control solution v = u. In addition, better control solutions v (solutions with lower control cost (Sv(to)v) are better sampling controls (samplers with lower variance).

[0040] In principle one could use v = 0 to sample the optimal control solution, but in practice this seldom works. Instead one needs a resonably good sampling control to accurately compute the optimal control, for instance by computing the open loop control solution of the corresponding deterministic optimal control problem. A generic alternative is to use adaptive importance sampling, where the sampling control v, at iteration / is used to estimate the optimal control M(- and v,+1= ui is used as sampling control in the next iteration. When both u and v are parametrized with a common parametrization u(x, t|0) this results in an interative update 9l- ff+1, providing a bootstrapping procedure starting with some inital control solution 0O- Examples of parametrizations are a grid, or a weigthed sum of basis functions or a neural network. Adaptive importance sampling has been formalized as the so-called PICE (Path Integral Cross Entropy) method where the importance sampling problem is formulated as a cross entropy minimization problem which results in a gradient descent algorithm for 9 / pdtM^e-s^a}ndw\= ff +V- - / -

[0041]

[0042] \e 5 ( 0)l)u The gradient is estimated by sampling trajectories X“ using the current controller w(x, t|0,). This method can compute the optimal control function u(x, t\G) for all x, t.

[0043] 2.1 Iterative path integral control

[0044] The path integral method was generalized as iterative path integral method. It applies to the more general stochastic optimal control problem with dynamics of the form

[0045] dX" = f(t, X“)dt + g(t, X“)u(t, X^dt + h t, Xu, t)dWt(J and cost Eq. 2. Define

[0046] H(J, u) = dtJ + (J + gu)'XJ + |Tr(WV2)J + V + (8)

[0047]

[0048] When u = -R~lg' J, the PDE I J. u) = 0 is the HJB equation for the optimal control problem define by Eqs. 7 and 2.

[0049] One proposes an iterative algorithm. In each iteration, a linear parabolic PDE

[0050] HU,Ml-i) = 0 (9) is defined where M;_I = -R ' g'^Ji

[0051]

[0052] and is the solution of the previous iteration. H(Ji, M(_I) = 0 admits a path integral solution

[0053] I rTi, \ Ji(t, x) - ( (V + —ut rRui_i))~ds + <[>(XT) \ (10)

[0054]

[0055] \Jt 2 I with xtare samples from the SDE Eq. 7 with u = u^i.

[0056] Define the errors

[0057] 6 / = H(JhUi-^y di = H(Ji, Ui) (11) Then e, is an estimate of the quality of the solution of the / th iteration and b, quantifies to what extent the solution Ji,, solves the HJB of the original control problem. One can prove the convergence of the i-PI method to the optimal control solution by showing that if e' — > 0 asymptotically, then <5' — > 0 asymptotically almost surely.

[0058] 3 Unraveling of the Lindblad equation and particle filtering

[0059] 3.1 Unravelings without measurements

[0060] The Lindblad equation is the most used approximate description of a quantum system in interaction with an environment. It is assumed to be valid when the coupling between the system and the environment is sufficiently weak the environment is sufficiently large. In this approximation, the quantum state is given by a density matrix p that evolves over time according the density evolution

[0061] p = —i[H,p] + JL(p) £C, D(P) = Du(ckpC* - ^{C / C^p)) (12)

[0062]

[0063] The first term H describes the unitary dynamics of the closed quantum system with Hamiltonian H. The term ■ c. JP) describes the influence of the environment on the evolution of p in terms of K Lindblad operators Ck, k = 1,..., K and D a real symmetric semi-definite matrix. Without loss of generality, we restrict to real symmetric matrices. Examples of Lindblad operators are measurement operators in which case Ckis Hermitian, or dissipation operators such as cr±for a single qubit.

[0064] One can show that the stochastic Schrödinger equation (SSE)

[0065] dift - -iHif / dt - -Dki(cj Ck- 2ckCt+ ckc^ if / dt + (Ck- ck)if / d^k(13)

[0066]

[0067] or equivalently the stochastic master equation (SME)

[0068] dP = -i[H, P]dt + £c, D(P)dt + [(Ck- ck)Pd^k+ h.c.] (14) with P = i / iff define an unraveling of the Lindblad equation, meaning that the expectation p = (P) satisfies the Lindblad equation Eq. 12. In Eq. 13 d^kis a real- valued Wiener process with fd^k) = 0 and dfdlf) = Dkidt, Ckare the Lindblad operators that appear in Eq. 12 and ck-

[0069]

[0070] (Ck+ Cj) if / real-valued. Note, that the SSE and SME are non-linear in if / and P, respectively. This is a necessary consequence of the fact that a non-unitary norm preserving dynamics is necessarily non-linear. The unravelings are not unique. One can also use linear unravelings (ck- 0) that preserve the norm of if / on average d (TrP) = 0. One can also define linear transformations of the Ckthat do not affect the Lindblad equation but change the unraveling.

[0071] 3.2 Unravelings with measurements: the hybrid quantum-classical dynamics

[0072] The above unraveling formalism can be generalized to include measurements. The starting point is to postulate a hybrid classical stochastic dynamics for the quantum state if / and the classical measurement outcomes x:

[0073] dif / = fdt + gkd£kdx = Adt + dW (15) where f,gk, A are functions of if / ,x. The noise terms d and dW are correlated: fd^kdW) = Vkdt. The time evolution of Eq. 15 can be described by a time-dependent classical probability density O / x, t) that satisfies the Fokker-Planck equation

[0074] d^P 1 1 - ’W - VT.4P) + -Tr ^gg’P) + -Tr (N2XP) + Tr (^gVP) (16)

[0075]

[0076] Define the hybrid quantum-classical density matrix p(x, t) = f difPPlif / , x, tfif / if / ^. The dynamics of p(x. t) follows directly from Eq. 16. This dynamical equation is in general not linear in p(x, t), nor is it autonomous in p (the dynamics depends on higher moments of if / ).

[0077] Sofar, this is all general. To connect to quantum physics, we now demand that the evolution equation for p(x, t) is linear. The question why quantum mechanics should be linear is non-trivial and has a long history. Linearity can be achieved by choosing f,gkin agreement with the unraveling Eq. 13 and choose Alf, x, t) = fc(x, t) + 2

[0078]

[0079] Tkckwith ck= iff'C^if and fc(x, t) a purely classical force, so that

[0080] dx = (fc+ 2Ft) f + dW (17) The dynamical equation for p(x, t) becomes autonomous and linear in p:

[0081] = —i[H,p\ + £C, D(P) - Vx (Pk(Ckp + h.c.) + fcp) + ^VAp (18)

[0082]

[0083] dt 2 with fc.ifp) the Lindblad operator (see Eq. 12). It could be called a quantum Fokker-Planck equation and succinctly captures ’everything’ about the coupled classical-quantum system. Note that the linearity condition applies to the dynamics of the expected quantity p(x, t) while the stochastic equations Eqs. 13 and 17 are non-linear.

[0084] When integrating over x, Eq. 18 reduces to the Lindblad equation Eq. 12 that describes the evolution of the unconditioned quantum density matrix pit = f dxp(x. t) = f dif Plif / , t)if / iff = (iff / ^' when we do not observe the measurements x. When we take the trace, p(x, t) reduces to the classical probability density p(x, t) - f dif Plif / , x, t) that describes the evolution of the classical system in interaction with the quantum system according to the classical Fokker-Planck equation:

[0085] ^ - -Vx[( / c+ / ?)p] + ^p(19)

[0086]

[0087] with fq(x, t) = 2r< Ti' c^pfx, t)^ the force that the quantum system exerts on the classical system. Eqs. 13 and 17 have the form of a standard hidden Markov model (fig. 4), with a prior process over quantum states ifj Eq. 13 and an observation model that affects the measurement x given if. The coupling between classical and quantum degrees of freedom is given by I, the correlation between the classical and quantum noise.

[0088] In the absence of observations, i evolves according to the prior stochastic dynamics Eq. 13 with probability distribution P(iko-t - The measurement process Eq. 17 changes this distribution to the posterior P( ot |xo:t) using Bayes’ rule:

[0089] P^.t)'P(x0-t\tl / Qt) WoJxte) = (20)

[0090]

[0091] P(x0[t) with P( o:f|^o:f) = 11 '=o P(xs\iks, xs-dt) the observation model Eq. 17. The posterior marginal distribution, known as the filtering distribution, is given by

[0092] ? Wfl*O:f) = X 'P(^Q-.t\X0-.t) (21)

[0093]

[0094] One can estimate the filtering distribution by a process called particle filtering or sequential Monte Carlo methods. One samples k trajectories, i = 1,...,k from the prior process Eq. 13 and weight each trajectory with the likelihood of the observations:

[0095]

[0096] oc nts=o pF / ks#?) and - 1, where p(dxs|i / <®) is given by the Gaussian measurement model Eq. 17. The particle filtering estimate of Eq. 21 is

[0097] k Pt(^\xo,t) ~ X ^(i) (22)

[0098]

[0099] ;=1 In the quantum case, the filtering distribution reduces to the posterior density matrix pt(xo-t) = d^Pt^xopM^ with Pt(k\xo-t the filtering distribution. Its particle filtering estimate is

[0100] Pf(X0:f) ~ (23)

[0101]

[0102] i=l As above, it is the postulate of the linearity of quantum mechanics that implies that this quantity, rather than Pt(i / / \xo:t) is sufficient to describe the posterior quantums state.

[0103] 4 Control of open quantum systems in the absence of measurement

[0104] For a control problem, H in the stochastic unraveling of the open quantum system Eq. 13 generically takes the form H = HQ + with HQ an uncontrolled part of the Hamiltonian and Hk control Hamiltonians that each operate on part of the quantum state (one or two qubits or bosons, typically). Then, Eq. 13 is of the form Eq. 7 and with Eq. 3 define the general class of control problems in the absence of measurement that is of the i-PI form and that inventors solve.

[0105] The PI control problems are a special case with g = h for which efficient numerical methods have been developed such as adaptive importance sampling. Quantum control problems in the absence of measurement are of the PI control form, provided that we can match each of the control terms UkH^dt with one of the stochastic terms Ck - Ck)i / / dWk- This is possible, provided that the non-linear terms Ck = 0, which requires Ck to be anti-Hermitian. One can define linear transformations Ck, D — > Ck, D that leave the Lindblad equation invariant. When there exists a transformation that makes all anti-Hermitian and D > 0, the resulting unraveling becomes linear (in if / ) and is of the path integral form Eq. 1:

[0106] dip - -iHipkdt - -DkiHkH^dl - iHpditkdl + dWk) (dWkdWi) - Dkidt (24)

[0107]

[0108] with Ck = iH the transformed Lindblad operators and D the transformed noise covariance matrix. Note that this is a non-linear control problem due to the coupling term iikd

[0109] Here inventors review some initial open loop control results for open quantum systems using path integral methods. Open loop control is the most common control problem that is currently being addressed in the community for closed and open quantum systems. 4.1 Control of a noisy qubit

[0110] Consider a single qubit with HQ = 0 and control operators Hi = axand H2-y. The dissipation part is given by the two non-Hermitian operators Ci - cr+and C2= cr_. Using a transformation that leaves the Lindblad equation invariant, inventors transform Ci, C2— » ZHI, Z'H2. The stochastic optimal control problem is of the path integral form

[0111] dip -Dij / dt - i ytak^iukdt + d'Wk)

[0112] k=x,y

[0113] C(ψ₀, u) = ⟨-Q / 2 Tr(ψ_T ψ_T† φφ†)⟩ + f 22 uk(t,^t)2dt

[0114]

[0115] ' ® k=x,yjwith D - (

[0116]

[0117] dW^j = (dWy^, R, Q free parameters. The initial state is ipo = X and the target state <(> = Y, with X, Y eigen states of respectively. Control of X — > Y requires < TZrotation, while the control only controls <j-xydirection. The control therefore requires a coordinated time-dependent combination of &xyto obtain the <rzrotation. Inventors compute both an open loop and a feedback control solution. Although state feedback requires measurement of the quantums state, which is not realistic and cannot be performed experimentally, we nevertheless wish to demonstrate how state information can improve the quality of the control solution. The more realistic scenario is presented in section 5. The open loop and feedback controllers is taken piecewise constant by discretizing time and (Hilbert) space and are computed using the PICE algorithm Eq. 6. The result is given in fig. 5.

[0118] 4.2 Control of n = 12 qubits

[0119] To show the scalability of the present method to large instances, inventors construct an artificial but realistic example of quantum state preparation for a chain of n - 12 qubits. For simplicity, set HQ - 0 and

[0120] n n— 1HC =

[0121]

[0122] 2;=21 a 2=x2,y + 2;=21 a, 2b=2x,y (25) where the controls M(C1act on spin i, and controls on the pair i, i + 1. The open quantum system interacts with the environment through single qubit dissipation operators <r=for each qubit with strength D\ and pair dissipation operators for each adjacent pair of qubits with strength £>2- The control cost is defined as I j1n n _ 1 t C = + | f ^22 22 22 22u^bdtl <26)

[0123]

[0124] ' ' / = l a=x,y i=l a,b=x,y1The control problem can be written as a path integral control problem with stochastic quantum dynamics n n-1 dip - -i ZE (uiadt + dWia)(r“ + 2 E (Uiabdt + dW;afe)cr“cr*+1^dZ - (n£>i + 2(n - 1)Z?2) pdt (27)

[0125]

[0126] z=l a=x,y z=l a,b=x,y Equations 26 and 27 define a path integral control problem. The initial state is 1 / 7 > = |0>” and the target state < / > - (^)”=1<f>i with f>i a random 1 -qubit state. This constitutes a non-trivial target state as it avoids the search of symmetrical solutions when the <pi are all the same. The same argument applies also to random Haar states. The results are shown in Fig. 6.

[0127] 4.3 Application to NMR physics

[0128] Nuclear Magnetic Resonance (NMR) quantum computing has gained significant popularity in recent years as a testbed technology for various experimental and theoretical research. Here inventors show that the present approach also works well for such a realistic example. NMR systems are commonly defined through a total Hamiltonian H = HQ + Hc, with

[0129] H0= ^Vi<ri+22H< = 22 “tvCrf + UiyC^i

[0130]

[0131] 1=1 i<j n-1 and with dissipation <r= operators acting on individual spins with strength D. The control cost is defined as

[0132] c - +? P E E ^adt\ (29)

[0133]

[0134] \2 2do I The control problem can be written as a path integral control problem with stochastic quantum dynamics n di / j - -iHivl / dl — i x z (jiiadt + d'Wia)oJfydt - nDtydt (30)

[0135]

[0136] i-i a=x,y Equations 29 and 30 define a path integral control problem. The initial state is |0)". The target state is a maximally entangled state \GHZn):= -3= |0' / ' + 11)”.

[0137] In Fig. 7 (Left) cost and fidelity curves and in fig. 7 (Middle) the converged open loop control solution to prepare a Diethyl fluoromalonate molecule are shown, modelled as two qubits that are weakly coupled to a thermal bath, into a fully entangled GHZ state. The open loop controller reaches a mean fidelityavg= 0.984. Despite the noise present in the open quantum system, the average fidelities that can be reached using the PI control method are similar to the fidelities that are reported for the closed quantum system.

[0138] 4.4 Further research (WP1)

[0139] • Iterative path integral control. The generic algorithm for i-PI control has been proven to converge, but its use in practice has not be explored sofar. The i-PI approach is different from the PI approach, because it does not use the log transformation which leads to the exponentials in Eqs. 4 and 5 The exponentials make the sampling problem in the PI case hard at low noise (small / I). yielding poor statistical estimates with large variance. This does not occur in the i-PI path integral formul Eq. 10. Therefore, sampling the path integrals in the iterative i-PI scheme is expected to be much easier than in the PI scheme and this may potentially turn out to be huge advantage of i-PI over the PI approach, even for cases where h = g where both methods can be applied. Inventors design a learning control strategy based on self-generated date and using the cross entropy method used previously to develop the PICE algorithm for the normal PI control problem. Inventors developed an extension of the adaptive importance sampling method for PI control (PICE) to a new iterative learning strategy using the cross entropy method.

[0140] • Parellel computing. The core of the PI control computation is to learn a model on the basis of self generated data (sample trajectories) and this procedure is iterated many times using improved controllers. The PICE computation Eq. 6 can be very efficiently parallelized by noting that the gradient consists of a sum over the data and that the computation of this sum can be distributed over multiple machines so that 9t+i= dt + S"=i d(^\ No large volumes of simulated raw data need to be transferred between machines, only the parameter updates. See fig. 3. This results in close to linear speedup.

[0141] 5 State feedback control, measurement and the control of open partially observable quantum systems

[0142] The feedback controller requires information about the quantum state through measurement. While projective measurement produces a large uncontrollable change in (collapse of) the wave function, one instead considers continuous measurement. The continuous collapse of the quantum state in continuous quantum measurement means that inventors execute real-time quantum feedback control before the quantum state collapses to a completely classical state. That is the starting point of continuous measurement-based feedback control. Measurement can be elegantly formalized by the so-called hybrid dynamics, which describes the simultaneous time evolution of the quantum state and the classical observation(s) as outlined in section 3.2.

[0143] Control of the hidden Markov process is known the control literature as a partial observable control problem. Inventors do not observe i / / , directly. Instead, inventors know about if / tonly indirectly through the measurement outcomes xo-tin terms of the filtering distribution 'Pt(if / \xo.t) Eq. 21. The optimal control solution of the partially observable control problem is given by the average of the optimal control solution at t with respect to the filtering distribution:

[0144] ; 1 \'1d^P^xo-t^ulyj, 0 ~ E 0 (31)

[0145]

[0146] where the last step is the particle filtering estimate and with t) the optimal control for the control problem on the remaining time interval [t, T\ starting at ft(lThis will be our basic approach to compute feedback controls of open quantum systems.1

[0147] 5.1 Further research

[0148] • Integrating estimation and control. Note, that for path integral control problems (for both PI and i-PI versions), the optimal control u(i / / t, f) is also estimated through weighted sampling. There exists a large literature how the sampling of the filtered distribution (particle filtering) can be optimized for instance using resampling as well as path integral control methods to optimize the particle trajectories. Since the filter should produce that are ’good’ from the point of control, the design of the optimal particle filter and the importance sampling approach for the control computation are clearly dependent. There exist close similarities between the particle filtering problem and the path integral control problem. In particular, the particle filtering problem can be viewed as a path integral control problem where the cost is given by the log likelihood of the observations. Both are solved by sampling and both sampling procedures can be optimized. For the control of quantum systems as discussed here, where both filtering and control problem need to be solved, inventors investigate methods to integrate and optimize these two sampling subproblems.

[0149] • On-line control computation. In concrete experimental setups, the feedback control should be computed in real time while the quantum process is executed. The control at time t can be computed either online or offline. In the online case, the control is computed incrementally using the control at time t - dt as follows. Suppose the computation has proceeded to time t - dt, xo-t-dt has been observed and N particles ^o r-dr have been simulated. The computation at time t consists of two steps: The particle trajectories are updated if / p1-

[0150]

[0151] + d using Eq. 13 and the trajectories are reweighted using the new measurement dxt. For each i, the optimal control w(^'\t) is estimated, which can be warm started from available optimal control solution u(if^d, t - dt), and the average Eq. 31 is computed; The control signal is sent to the quantum device in time before it reaches time t + dt and the loop is repeated. Clearly, for infinitesimal small dt this entire control computation may not be fast enough. However, one generalize this idea by replacing dt by a larger time interval At using the same theoretical concepts. While the method described in this patent application requires infinitesimal small time steps dt, it is possible to generalize to larger At. This is important when dt is too short to compute the feedback control in real time. Inventors investigate methods that generalize the PI control method to also work for larger At.

[0152] 6 Analog quantum circuits

[0153] Most quantum computing paradigms today are based on digital quantum circuits, which contain a fixed set of predefined quantum gates that are tailored to the physical device that is available. An ansatz is made for the shape of the quantum circuit in terms of the pattern of gates, thus making them subjected to strong limitations. Furthermore, the total computation time is fully determined by the gate sequence and cannot be easily decreased to minimize unwanted effects of decoherence.

[0154] In contrast, in analog quantum circuits electromagnetic signals used to control qubits and their interaction. Analog circuits offer several advantages over digital circuits: they allow for faster state preparation, simpler implementation and yield more expressive circuits. Optimal control is a natural way to optimize analog quantum circuits. The potential large impact of this approach was shown: On four transmon qubits a reduction in state preparation time of rougly three orders of magnitude compared to gate based strategies was demonstrated. Sofar these approaches consider closed quantum systems and open loop control. In collaboration with several partners, we will explore the application of the stochastic optimal control methods to compute feedback controls for open analog quantum circuits.

[0155] 6.1 Neutral atoms

[0156] In the last several years, Rydberg-atom quantum processors have emerged to become one of the most promising platforms for quantum computing. These quantum processors rely on neutral atoms that can be well isolated from the environment and prepared in large systems of hundreds or thousands of particles with different geometries using laser cooling and trapping techniques. Quantum information is encoded in the internal states of atoms with interactions mediated via their highly electronically excited Rydberg states. In combination with coherent laser excitation, Rydberg atoms can realize strong and widely tuneable interactions that can extend over several pm. Moreover, they offer significant advantages over other technologies, including the possibility for long coherence times (» 1 second, for ground state encoding), fast and switchable interactions, allowing for the generation of highly entangled states, and produce high fidelity entangling operations. Accordingly, ultracold Rydberg quantum systems have proven very successful for studying many-body physics and analog quantum simulation, and are now ready to advance to the forefront among other platforms for scalable quantum computing.

[0157] Inventors use strongly-interacting quantum gases and neutral atom quantum computing. Therein pulse based optimal control methods (a variant of GRAPE) are applied to prepare a chain of neutral atoms in the ground state of a quantum Hamiltonian, demonstrating that the analog circuit requires much less time than an equivalent digital circuit. We aim to extend this work to open quantum systems and the feedback control using the path integral methods. We will demonstrate our findings on molecular electronic-structure problems. This problem lies at the heart of quantum chemistry and molecular physics and is a natural application of quantum computing. Due to the curse of dimensionality, exact classical computing methods can only tackle very small molecules, or obtain approximate solutions when correlations in the system are sufficiently small.

[0158] 6.2 Analog quantum computing on Amazon Braket

[0159] Amazon Braket also supports neutral atom quantum computing at the QPU Aquila from QuEra Computing, with up to 256 qubits. This is a special purpose device designed for solving optimization problems and simulating quantum phenomena in nature. The QuEra QPU operates by trapping atoms with lasers, arranging them in programmable one or two-dimensional layouts, and inducing interatomic interactions via van der Waals forces. The flexible atom arrangements and tunability of optical controls enable Aquila to realize a rich class of Hamiltonians and explore the static and dynamic properties of quantum states under these Hamiltonians by adiabatic or diabatic quantum evolution. Hamiltonians realized by the QuEra QPU have been used to study several scientific questions of interest in condensed matter and quantum many-body physics.

[0160] Amazon recently launched Braket Pulse, a programming environment that provides pulse-level access to quantum circuits, currently from two hardware providers (Rigetti Computing and Oxford Quantum Circuits (OQC)). Braket Pulse gives direct access to the elementary components of the device, thus enables the programming of applications for analog quantum circuits. In addition, Braket has developed a prototype library for programming digital quantum circuits with real-time classical control and mid circuit measurement Au-toQASM. Although there is no real-time classical feedback on the qubit yet, this is expected (by almost all hardware developers) in the near future.

[0161] Inventors use the Braket environment for new algorithms for the optimization of analog quantum circuits, using the path integral control methods. In particular, open loop optimal control solutions are computed in software and tested on the quantum hardware. Inventors develop methods to optimize the control solutions directly on the hardware. There is currently no real-time classical feedback on the qubit lifetime, but this is a focus for almost all hardware developers. Feedback controllers rely on real-time continuous weak measurement, which can be simulated using the mid-circuit measurement feature of the AutoQASM environment and ancilla qubits. The separation of the partially observable problem into an estimation problem and an observable control problem assumes that the minimum of the expected control cost is equal to the expectation of the minimal cost. This is true when the minimization is with respect to all state and time dependent functions. It is not true when the minimization is restricted to, for instance, open loop controls.

[0162] In a second aspect the present invention relates to a method of operating a device according to the invention, comprising (1) providing an initial state of the analog quantum device (20), and a detector (30), (2) providing a physical model of the initial state of the analog quantum device (20), (3) providing a digital controller (10) which comprises a fdter (12), a stochastic controller (11), and an adder (15) configured to provide output u(t) to the analog quantum device (20) and to the filter (12), wherein the stochastic controller (11) formulates a control problem, the control problem comprising a control objective in view of the initial state and desired final state of the analog quantum device, and (4) configuring the filter to receive input x(t) from the detector, (5) configuring the filter to estimate a distribution T (t) over current quantum states at time t based on the past measurement outcomes x(0:t) and the past controls u(0:t-dt) in the form of a probability distribution P(T,t|x(0:t) u(0:t-dt)), wherein said probability distribution is represented by weighted samples (1, wi)to (Tk, wk) with weights wi to Wk, wherein k is the number of samples, (6) configuring the stochastic controller to receive the weighted samples (1) to (k(7) computing an optimal control U(

[0163]

[0164] to UQ,^) using a path integral control method, (8) configuring the adder to compute the optimal control u(t) as a weighted sum of w; and U(t, T1), wherein u(t)=sum {i=l to k} Wi LJ(t,'Tl), (9) providing output u(t) from the adder to the analog quantum device and to update the filter, and (10) evolving the analog quantum device into it’s ground state. In particular, the present method can be applied to the molecular electronic-structure problem. The molecular electronic -structure problem, which lies at the heart of quantum chemistry and molecular physics, is considered to be a vital source of scientific breakthroughs, and a natural application of quantum computing. It may be considered to be a problem of accurately predicting the electronic structure of atoms and molecules. On paper, these structures are fully determined by the Schrödinger equation. Affected with the curse of dimensionality, current classical computing power is only capable of resolving the electronic-structure problem for very small molecules. This obstruction has led to the design and development of numerous computationally tractable approximate models, most notably correlated wave-function methods and Density Functional Theory. Despite the successes of these methods, they fail, for instance, to accurately describe the electronic structure of strongly correlated systems. Various approaches have been proposed to overcome such limitations in the confines of classical computing, but none have been satisfying and the search continues.

[0165] In a third aspect the present invention relates to a computer program comprising instructions, the instructions when loaded and running on a computer causing the computer to carry out: providing the device for quantum computing according to the invention, providing a controller (11), bringing the analog quantum device into an initial state, providing quantum instruction to the analog quantum device in its initial state, and evolving the quantum device into it’s ground state. The quantum instruction may relate to quantum state preparation, to non-linear systems, such as robotics, qubit control, quantum chemistry, quantum physics, e.g. NMR physics, quantum mechanics, material science, drug design, biochemistry, etc

[0166] In a fourth aspect the present invention relates to a digital controller (10) for operating a device for quantum computing, in particular for control of an open quantum system with Quantum Path Integral Control (QPIC) for providing a ground quantum state from an initial quantum state of a quantum system, comprising a fdter (12), a stochastic controller (11), and an adder (15), wherein the fdter is configured to receive input x(t) from the detector and is configured to estimate a distribution (t) over current quantum states of the analog quantum device at time t based on the past measurement outcomes x(0:t) and the past controls u(0:t-dt) in the form of a probability distribution P('P,t|x(0:t), u(0:t-dt)), wherein said probability distribution is represented by a number of weighted samples (1, wi) to (Ψk, Wk) with weights wi to Wk, wherein k is the number of samples, and wherein the stochastic controller at time t is configured to receive the samples CP1) to (k) and to compute an optimal control U(t,'P1) to 11(1,^),, wherein the adder is configured to perform the weighted sum u(t) of the optimal controls U(t,'P1) to LJ(t, H^) with weights wi to Wk and to provide output u(t) to the analog quantum device and to the filter, in particular wherein the detector (30) is configured to perform at least one measurement x(t) on the analog quantum device at a point in time t and to provide said at least one measurement to the digital controller (10), the analog quantum device (20) is configured to receive input u(t) from the digital controller (10), and wherein the analog quantum device is configured for quantum computing.

[0167] Thereby the present invention provides a solution to one or more of the above-mentioned problems.

[0168] Advantages of the present invention are detailed throughout the description.

[0169] DETAILED DESCRIPTION OF THE INVENTION

[0170] The present invention relates in a first aspect to a device according to claim 1.

[0171] In an exemplary embodiment of the present device for quantum computing the analog quantum device is configured to change dynamically in time according to the laws of quantum physics under dynamic Hamiltonian H, time-dependent disturbances from the environment, and time dependent input provided by the controller u(t). The analog quantum device is therefore configured for continuous time calculation of the quantum device.

[0172] In an exemplary embodiment of the present device for quantum computing the detector is configured to measure partial and noisy information of the quantum state of the analog quantum device denoted as x(t), in particular wherein the detector comprises a probe laser for providing an input signal, an electromagnetic arrangement, such as magnetic coils, for measuring a change in quantum state, and a detector for measuring a change in quantum state.

[0173] In an exemplary embodiment of the present device for quantum computing the stochastic controller (11) is configured to approximate u(t) as not to depend on x(0:t) and therefore the digital controller is configured as an open loop controller, or wherein the digital controller is configured to approximate u(t) as to depend on x(0:t) and therefore the digital controller is configured as a feedback controller.

[0174] In an exemplary embodiment of the present device for quantum computing the stochastic controller (11) and / or the filter (12) are configured for using a physical model of the analog quantum device, in particular wherein the physical model provides a description how an initial state of the analog quantum device Ψ(0) at time t=0 evolves to a final state 'P(T) at time t=T using optimal control sequence u(0: T).

[0175] In an exemplary embodiment of the present device for quantum computing the stochastic controller (11) is configured for applying a control objective on the control trajectory (0: T) of the analog quantum device including the initial and final state and the control trajectory u(0: T), and wherein the control objective is to prepare a final state of the analog quantum device, that is close to the ground state of a target quantum Hamiltonian, while minimizing computation costs.

[0176] In an exemplary embodiment of the present device for quantum computing the stochastic controller (11) is configured to define a stochastic optimal control problem, which stochastic optimal control problem is to provide a control sequence u(0: T) that minimizes the control objective J, in particular in terms of computation costs and ground state provided, wherein the stochastic optimal control problem is solved using a path integral control method,. It is noted that 0: T in u(0: T) means the continuous sequence of times 0: T that is approximated using a fine time discretisation step dt from t=0 to t=T.

[0177] In an exemplary embodiment of the present device the path integral control method is configured to reiterate such that during a subsequent iteration the control law is improved, and / or in particular wherein the path integral control method is configured for parallelised computing wherein each iteration generates trajectories with control u(0p) and based on the trajectories computes changes in parameters dΘpwith the path integral control method, wherein parallel computed dΘpare added to ΘP+1, and are used for a next iteration at p+1 as input to generate trajectories.

[0178] In an exemplary embodiment of the present device for quantum computing the digital controller is configured to be implemented on a digital (classical) processor or computer. The term “classical” is used to refer to a prior art processor or computer being readily available.

[0179] In an exemplary embodiment of the present device for quantum computing the analog quantum device comprises quantum elements.

[0180] In an exemplary embodiment of the present device for quantum computing quantum elements are selected from qubits, bosonic states, and neutral atoms.

[0181] In an exemplary embodiment of the present device for quantum computing the analog quantum device comprises m quantum elements, such as 2 to 106elements, and / or wherein the controller (11) is configured for control of a plurality of quantum elements. In an exemplary embodiment of the present method (2) providing the initial state of the analog quantum device is selected from producing the initial state of the analog quantum device, and providing a model of the initial state of the analog quantum device.

[0182] In an exemplary embodiment the present method comprises configuring the stochastic controller (11) for performing a sequence of quantum operations from an initial quantum state of the analog quantum device to a ground quantum state of the analog quantum device using a minimal cost function, wherein the cost-function qualifies the ground quantum state and includes costs of computation.

[0183] In an exemplary embodiment the present method comprises measuring the quantum state of the analog quantum system during the quantum operations by the detector (30), and providing feedback to the digital controller (10), in particular to the filter (12).

[0184] In an exemplary embodiment the present method comprises configuring the controller (11) to provide hybrid dynamics for determining j oint evolution of both the quantum state of the open analog quantum device and of a measurement outcome.

[0185] In an exemplary embodiment the present method comprises configuring the controller (11) for modelling an interaction of the analog quantum device and of the sequence of quantum operations with an environment, in particular an evolution of the quantum density matrix, such as using a Lindblad method.

[0186] In an exemplary embodiment of the present method the hybrid dynamics is such that an average behavior thereof coincides with the interaction of the analog quantum device and of the sequence of quantum interactions with an environment.

[0187] In an exemplary embodiment of the present method analog quantum device is an open quantum device, wherein the initial state of the analog quantum device is representative of a formalistic scientific problem, and wherein the ground state of the analog quantum device is representative of a solution to said formalistic problem, in particular wherein the problem is selected from a chemical problem, from a device, such as a chemical device.

[0188] The invention is further detailed by the accompanying figures and examples, which are exemplary and explanatory of nature and are not limiting the scope of the invention. To the person skilled in the art, it may be clear that many variants, being obvious or not, may be conceivable falling within the scope of protection, defined by the present claims.

[0189] SUMMARY OF THE FIGURES

[0190] Figures 1-7 show details of the invention.

[0191] DETAILED DESCRIPTION OF THE FIGURES

[0192] In the figures:

[0193] 10 Digital controller

[0194] 11 Stochastic controller

[0195] 12 filter

[0196] 15 adder

[0197] 16 feedback path 20 analog quantum device

[0198] 21 Quantum element, e.g. QUBIT

[0199] 30 detector

[0200] Figure 1 shows a functional diagram as detailed above with fig.3 iterations.

[0201] Figure 2 shows a prior art quantum optics illustration of measurement feedback control through continuous measurement as an instance of a partial observable control problem.

[0202] Figure 3: PICE uses importance sampling iterations (labelled by p). During each iteration, samples are generated using control functions uQIQlOp). These samples are used to compute new parameters 0p+i. These two steps can be distributed on many machines with minimal communication overhead, since no large volumes of simulated raw data need to be transferred between machines, only the parameter updates.

[0203] Figure 4: Hidden Markov model that represents the prior quantum process Eq. 13 (blue) and the classical process Eq. 17 (red). The coupling between these is proportional to Tk = (d k dW), the correlation between the classical and quantum noise. Given observations xo:t, the latent quantum states are updated using Bayes’ rule, which can be estimated using particle filtering.

[0204] Figure 5: Controlled quantum trajectories for the noisy qubit. X: Initial state. Y: target state. D = 0:01, R = 1, Q = 10, number of trajectories ntraj = 100, maximum number of importance sampling (IS) steps nismax = 200 and time step dt = 0:01. (Left) Four plots show the evolution of the importance sampling of the path integral control method for the open loop case, with (top left) the effective sample size versus importance sampling iteration showing the bootstrapping from an initial control u = 0 to the optimal control solution after 200 IS iterations (bottom left). (Top right) Optimal control cost versus IS iterations. (Bottom right) Mean fidelity (blue) and worse case fidelity (red) on ntraj samples versus IS iterations. Quantum trajectories on the Bloch sphere using optimal open loop control (Middle) and state feedback control (Right).

[0205] Figure 6: Control of a n = 12 qubit 1-D spin chain. The parameters are K = 50, T = 4, L = 100, R=0.1, D = 0:001, Q=1000, nISmax=1000, and ntraj=400. Average fidelity reached F∞= 0:990 + / - 0:001. We show only 2 controls, the rest are similar. Runtime 11:5hr.

[0206] Figure 7: Open loop control of a small open NMR quantum system, consisting of a Diethyl flu-oromalonate molecule, modelled as two qubits that are weakly coupled to a thermal bath. (Left) Cost (J), mean fidelity (Favg) and worse case fidelity (Fmin) vs IS steps and (Right) Time dependent (open loop) optimal control solution uix(t); uiy(t)

[0207] The figures are further detailed in the description and examples below.

[0208] EXAMPLE S / EXPERIMENTS

[0209] The above relates to examples of the present invention.

[0210] 10 Digital controller

[0211] 11 Stochastic controller

[0212] 12 filter

[0213] 15 adder feedback path

[0214] analog quantum device Quantum element, e.g. QUBIT detector

Claims

CLAIMS1. Device for quantum computing (1), in particular a control architecture, for control of an open quantum system with Quantum Path Integral Control (QPIC) for providing a ground quantum state from an initial quantum state of a quantum system, comprisinga digital controller (10), an analog quantum device (20), and a detector (30),whereinthe detector (30) is configured to perform at least one measurement x(t) on the analog quantum device at a point in time t and to provide said at least one measurement to the digital controller (10),the analog quantum device (20) is configured to receive input u(t) from the digital controller (10), and wherein the analog quantum device is configured for quantum computing,the digital controller (10) comprises a filter (12), a stochastic controller (11), and an adder (15), wherein the filter is configured to receive input x(t) from the detector and is configured to estimate a distribution over current quantum states (t) at time t based on the past measurement outcomes x(0:t) and the past controls u(0:t-dt) in the form of a probability distribution P(Ψ,t|x(0:t)) and to provide the probability distribution Pt|x(0:t)) to stochastic controller (11), wherein said probability distribution is represented by weighted samples (Ψ1, wi)to (Ψk, wk) with weights wi to Wk, wherein k is the number of samples, and wherein the stochastic controller is configured to receive the weighted samples (Ψ1, wi)to (Ψk, wk) and to compute an optimal control U(Ψi) for each weighted sample and to provide optimal control U(Ψi) for each weighted sample to adder (15), wherein the path integral control method is configured for parallelised computing wherein each iteration generates trajectories with control U(Θp) and based on the trajectories computes changes in parameters dΘpwith the path integral control method, wherein parallel computed dΘpare added to ΘP+1, and are used for a next iteration at p+1 as input to generate trajectories, wherein the adder is configured to sum the optimal controls and to provide output u(t) to the analog quantum device (20) and to the filter (12).

2. The Device for quantum computing according to claim 1, wherein the analog quantum device is configured to change dynamically in time according to the laws of quantum physics under dynamic Hamiltonian H, time-dependent disturbances from the environment, and time dependent input provided by the controller u(t).

3. The Device for quantum computing according to any of claims 1-2, wherein the detector is configured to measure partial and noisy information of the quantum state of the analog quantum device denoted as x(t), in particular wherein the detector comprises a probe laser for providing an input signal, an electromagnetic arrangement, such as magnetic coils, for measuring a change in quantum state, and a detector for measuring a change in quantum state.

4. The Device for quantum computing according to any of claims 1-3, wherein the stochastic controller (11) is configured to approximate u(t) as not to depend on x(0:t) and therefore the digital controller is configured as an open loop controller, or wherein the digital controller is configured to approximate u(t) as to depend on x(0:t) and therefore the digital controller is configuredas a feedback controller.

5. The Device for quantum computing according to any of claims 1-4, wherein the stochastic controller (11) and / or the filter (12) are configured for using a physical model of the analog quantum device, in particular wherein the physical model provides a description how an initial state of the analog quantum device Ψ(0) at time t=0 evolves to a final state (T) at time t=T using optimal control sequence u(0:t).

6. The Device for quantum computing according to any of claims 1-5, wherein the stochastic controller (11) is configured for applying a control objective on the control trajectory T (0: T) of the analog quantum device including the initial and final state and the control trajectory u(0: T-dt), and wherein the control objective is to prepare a final state of the analog quantum device.

7. The Device for quantum computing according to any of claims 1-6, wherein the stochastic controller (11) is configured to define a stochastic optimal control problem, which stochastic optimal control problem is to provide a control sequence u(0: T-dt) that minimizes the control objective J, wherein the stochastic optimal control problem is solved using a path integral control method, and wherein the solution of the stochastic optimal control problem provides the control law U(t), in particular wherein the path integral control method is configured to reiterate such that during a subsequent iteration the parametrized control law is U(t, Ψ) =U(t, Ψ,Θp) improved,.

8. The Device for quantum computing according to any of claims 1-7, wherein the digital controller is configured to be implemented on a digital processor or computer, such as a classical processor or computer, and / orWherein the analog quantum device comprises quantum elements, and / or wherein quantum elements are selected from qubits, bosonic states, and neutral atoms, and / orWherein the analog quantum device comprises m quantum elements, such as 2 to 106elements, and / or wherein the controller (11) is configured for control of a plurality of quantum elements.

9. A method of operating a device according to any of claims 1-8, comprising(1) providing an initial state of the analog quantum device (20), and a detector (30), (2) providing a physical model of the initial state of the analog quantum device (20), (3) providing a digital controller (10) which comprises a filter (12), a stochastic controller (11), and an adder (15) configured to provide output u(t) to the analog quantum device (20) and to the filter (12),, wherein the stochastic controller (11) formulates a control problem, the control problem comprising a control objective in view of the initial state of the analog quantum device, and(4) configuring the filter to receive input x(t) from the detector,(5) configuring the filter to estimate a distribution (t) over current quantum states at time t based on the past measurement outcomes x(0:t) and the past controls u(0:t-dt) in the form of a probability distribution P('P,t|x(0:t)(u(0:t-dt)), wherein said probability distribution is represented by weighted samples (Ψ1, wi)to (Ψk, wk) with weights wi to wk, wherein k is the number of samples,(6) configuring the stochastic controller to receive the weighted samples (Ψ1, wi)to (Ψk,Wk),(7) computing an optimal control U(t, Y‘) for each sample using a path integral control method,(8) configuring the adder to perform the weighted sum of the optimal controls,(9) providing output from the adder to the analog quantum device and to the filter, and (10) evolving the analog quantum device into its ground state.

10. The method according to claim 9, wherein (2) providing the initial state of the analog quantum device is selected from producing the initial state of the analog quantum device, and providing a model of the initial state of the analog quantum device.

11. The method according to claim 9 or 10, comprisingconfiguring the stochastic controller (11) for performing a sequence of quantum operations from an initial quantum state of the analog quantum device to a ground quantum state of the analog quantum device minimizing a cost function, wherein the cost-function qualifies the ground quantum state and includes costs of computation, and / orcomprising measuring the quantum state of the analog quantum system during the quantum operations by the detector (30), andproviding feedback to the digital controller (10), in particular to the filter (12).

12. The method according to any of claims 9-11, comprisingconfiguring the stochastic controller (11) to provide hybrid dynamics for determining joint evolution of both the quantum state of the open analog quantum device and of a measurement outcome, and / orconfiguring the stochastic controller (11) for modelling an interaction of the analog quantum device and of the sequence of quantum operations with an environment, in particular an evolution of the quantum density operator, in particularwherein the hybrid dynamics is such that an average behavior thereof coincides with the interaction of the analog quantum device and of the sequence of quantum interactions with an environment.

13. The method according to any of claims 9-12, wherein the analog quantum device is an open quantum device, wherein the initial state of the analog quantum device is representative of a formalistic scientific problem, and wherein the ground state of the analog quantum device is representative of a solution to said formalistic problem, in particular wherein the problem is selected from a chemical problem, from a device, such as a chemical device.

14. A computer program comprising instructions, the instructions when loaded and running on a computer causing the computer to carry out:bringing the analog quantum device of the device for quantum computing according to any of claims 1-8 into an initial state,providing quantum instruction to the analog quantum device in its initial state, such as for a molecular electronic-structure problem, andevolving the quantum device into its ground state.

15. A digital controller (10) for operating a device for quantum computing, in particular for control of an open quantum system with Quantum Path Integral Control (QPIC) for providing a ground quantum state from an initial quantum state of a quantum system, comprising a filter (12), a stochastic controller (11), and an adder (15), wherein the filter is configured to receive input x(t) from a detector and is configured to estimate a distribution (t) over current quantum states of an analog quantum device at time t based on the past measurement outcomes x(0:t) and the past controls u(O:t-dt) in the form of a probability distribution P(Ψ,t|x(0:t)) and to provide the probability distribution P('P,t|x(0:t),u(0:t-dt)) to stochastic controller (11), wherein said probability distribution is represented by weighted samples (1, wi)to (Ψk, wk) with weights wi to wk, wherein k is the number of samples, and wherein the stochastic controller at time t is configured to receive the samples1tok)and to compute an optimal control U(t,Ψ1) ) to U(t,k), and to provide optimal control U(Ψi) for each weighted sample to adder (15), wherein the path integral control method is configured for parallelised computing wherein each iteration generates trajectories with control u(Θp) and based on the trajectories computes changes in parameters dΘpwith the path integral control method, wherein parallel computed dΘpare added to ΘP, and are used for a next iteration at p+1 as input to generate trajectories, wherein the adder is configured to perform the weighted sum u(t) of the optimal controls U(t,vP1) to U(t,k) with weights wi to Wk and to provide output u(t) to the analog quantum device and to the filter, in particular wherein the detector (30) is configured to perform at least one measurement x(t) on the analog quantum device at a point in time t and to provide said at least one measurement to the digital controller (10),the analog quantum device (20) is configured to receive input u(t) from the digital controller (10), and wherein the analog quantum device is configured for quantum computing.

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