WO2023169680A1 - Procédés et dispositifs de calcul quantique temporel continu - Google Patents

Procédés et dispositifs de calcul quantique temporel continu Download PDF

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WO2023169680A1
WO2023169680A1 PCT/EP2022/056178 EP2022056178W WO2023169680A1 WO 2023169680 A1 WO2023169680 A1 WO 2023169680A1 EP 2022056178 W EP2022056178 W EP 2022056178W WO 2023169680 A1 WO2023169680 A1 WO 2023169680A1
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hamiltonian
quantum
time
subspace
control
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Michele Reilly
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Michele Reilly
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    • G—PHYSICS
    • G06—COMPUTING; CALCULATING OR COUNTING
    • G06N—COMPUTING ARRANGEMENTS BASED ON SPECIFIC COMPUTATIONAL MODELS
    • G06N10/00—Quantum computing, i.e. information processing based on quantum-mechanical phenomena
    • G06N10/60—Quantum algorithms, e.g. based on quantum optimisation, quantum Fourier or Hadamard transforms
    • G—PHYSICS
    • G06—COMPUTING; CALCULATING OR COUNTING
    • G06N—COMPUTING ARRANGEMENTS BASED ON SPECIFIC COMPUTATIONAL MODELS
    • G06N10/00—Quantum computing, i.e. information processing based on quantum-mechanical phenomena
    • G06N10/40—Physical realisations or architectures of quantum processors or components for manipulating qubits, e.g. qubit coupling or qubit control
    • G—PHYSICS
    • G06—COMPUTING; CALCULATING OR COUNTING
    • G06N—COMPUTING ARRANGEMENTS BASED ON SPECIFIC COMPUTATIONAL MODELS
    • G06N3/00—Computing arrangements based on biological models
    • G06N3/02—Neural networks
    • G06N3/04—Architecture, e.g. interconnection topology
    • G06N3/0475—Generative networks
    • G—PHYSICS
    • G06—COMPUTING; CALCULATING OR COUNTING
    • G06N—COMPUTING ARRANGEMENTS BASED ON SPECIFIC COMPUTATIONAL MODELS
    • G06N3/00—Computing arrangements based on biological models
    • G06N3/02—Neural networks
    • G06N3/08—Learning methods
    • G06N3/09—Supervised learning

Definitions

  • the present disclosure relates to methods and devices for performing quantum computations.
  • a quantum computer is a device that processes information stored on quantum mechanical degrees of freedom, such as atoms, electrons, pho- tons and superconducting qubits.
  • Quantum information partakes in the strange and counterintuitive features of quantum mecanics: quantum bits can exist in superpositions of 0 and 1 , and two or more quantum degrees of freedom can be entangled, exhibiting what Einstein called ‘spooky action at a distance’.
  • Quantum computers employ the strange and counterintuitive features of quantum mechanics to process information in ways that classi- cal computers can’t, thereby achieving substantial potential speedups over classical computers for a variety of problems.
  • quantum computers are usually digital computers storing information in arrays of two state systems or qubits. With each qubit spanning a two dimensional Hilbert space, the total computational Hilbert space of an N-qubit register is 2 N , which is the dimen- sion of the product space of the Hilbert spaces of the individual qubits. While it is possible to also use three or more state systems, i.e. qutrits, ququads etc., in place or in addition to qubits, these offer at most practical but no fundamental benefits. It is also possible to perform quantum in- formaton processing on systems such as harmonic oscillators or modes of the electromagnetic field that possess continuous degrees of freedom.
  • a quantum algorithm that achieves some quantum computional task such as Grover’s search algorithm, Shor’s fac- torization algorithm or quantum Fourier transform, usually also includes preparation and measurement steps, but in between these projective oper- ations a pure unitary evolution takes place.
  • any classical computa- tion can be broken down into a sequence of basic logic operations or gates on the classical bits of a classical digital computer, it was established early during quantum computation research that any unitary operation can be ar- bitrarily closely realized by sequences of one- and two-qubit gates taken from a small set of gates acting on the qubits of the quantum computer.
  • the circuit model of quantum computation has, however, the downside of being, in general, far from resource optimal. Since all quantum information in a quantum computer is subject to some degree of decoherence and the quantum gates of a computer also have a certain failure probability, error correction techniques are far more important in quantum computing than in classical computing: whereas in the latter there is usually a large energy barrier seperating two distinct states of a classical bit, even a small influ- ence can change the state of a qubit, in case of the phase information even without costing any energy (if the computational basis states are also en- ergy eigenstates of the native system Hamiltonian, as is usually the case). In circuit model quantum computing intricate error correction schemes have therefore been devised to cope with this problem. They all achieve this by using logical qubits encoded in multiple physical qubits, thereby adding re- dundancy such that a single or a small number of errors in the physical qubits does not compromise the stored logical quantum information.
  • Error correction adds a significant overhead. For current devices and de- pending on the architecture, it is required for computations using on the or- der of a thousand gates or more. However, even without error correction the circuit model is suboptimal, since the stringent adherence to realizing a certain ideal operation as precisely as possible is often not well suited to the quantum systems used to implement the quantum computer. Whether these are superconducting qubits, colour centers in crystals or trapped ions, there are usually interactions present, both internal with other qubits and external with the outside world, that, though possibly small, cannot be turned off completely or even sufficiently to be negligible and which cause both gate errors and decoherence.
  • optimal control is now to find a set of time varying control pulses that will, together with the inherent time evolution of the system, yield a certain desired overall unitary time evolution of the system, thus re- alizing a desired quantum computation.
  • the circuit model is likened to go- ing from A to B by following only some limited number of prescribed roads and streets, optimal control is like taking a shortcut, ideally by flying across country, i.e. taking the shortest possible route.
  • the circuit model may still be used to figure out the desired unitary or, in the above analogy, the destination B, but it does not straightjacket the reali- zation of that unitary. In that way, substantial improvements can be achieved, in particular the time taken for a computation can be reduced but also the size of a problem instance that can be handled on a given quan- tum computing hardware set, i.e. the memory/qubit requirements.
  • QSVT Quantum Singular Value Transformation
  • Embodiments of the invention overcome these and other problems.
  • it is an object of the invention to find quantum computing devices and methods that offer a significant speed-up over known methods and devices based on quantum circuits and their implementations.
  • the latter object is achieved by a quantum computation device according to one of the claims 1 - 9 and a correspond- ing method for performing a quantum computation according to claims 10 - 18.
  • the Hamiltonian of a quantum system is the i-multiple of the anti-hermitian operator describing the time evolution of the system for an infinitesimal time interval dt.
  • the Hamiltonian is time-dependent, but it can always be decomposed into a sum of time-independent hermitian operators each multplied by a time-varying skalar field, that capture the time dependence of the Hamiltonian.
  • this disclosure proposes to perform a preliminary computation using an optimization algorithm such as a gradient descent or an evolution- ary algorithm or a combination of the two.
  • the parameter values thus ob- tained are then used by the classical control system during the actual im- plementation of the method according to this aspect of the invention on the quantum system in order to generate the desired effective Hamiltonian H eff having the input matrix A as off-diagonal block.
  • the preliminary computation may itself be performed on a quantum com- puter.
  • this requires that the cost function to be optimized can be computed without encoding the input matrix A and might therefore be re- served to special problems or problem instances.
  • a classical computer is used to perform the preliminary com- putation. This can be done efficiently, depending on the dimension D 1 D 2 of the input matrix A, if the control Hamiltonians describing the effect of the different control fields H 1 , H 2 , ..., H k and consequently monomials in the time evolution expansion, are sparse matrices and can be efficiently represented in a classical computer. This can usually be safely assumed if the quantum system is a composite system of several, well iso- lated subsystems, such as an array of qubits since in that case the control fields usually affect only of a small number of the subsystems, most often one or two.
  • the preliminary computation involves using a neural network to determine the N control parameters ⁇ k,n .
  • the desired monomial coefficients of the /-th order expansion or the elements of the matrix A itself are fed into the first or input layer of the neu- ral network.
  • the network may be trained to find suita- ble control parameters for different matrices A.
  • a Generative Neural Network is used where the nodes/neurons of the input layer are fed some standard values.
  • a different neural network is used for each problem instance, i.e. each different matrix A. In either case, the excitation of the last or output layer represent the control parameter settings. This im- plies that the output layer of the neural network contain at least N nodes.
  • the preliminary computation involves a gra- dominant descent carried out on the weights of the neural network.
  • the fitness or cost function J( ⁇ k,m ⁇ ) to be minimized in this gradient descent can be a matrix norm applied to the difference of the input matrix A and the off-diag- onal block A( ⁇ k,m ⁇ ) connecting the first and second subspace for the given control parameter settings:
  • the preliminary computation can be performed by another computing de- vice and at any time before the actual encoding the matrix A is to be carried out by a the quantum computing method and device according to this first aspect of the invention.
  • the preliminary computation is carried out by the classical control system of the quantum computing device itself.
  • the classical control system comprises classical information processing hardware controlling the control field generators according to, ultimately, user inputs.
  • This information processing hardware may also be used to run the optimization algorithm employed in finding optimal or at least sufficiently accurate control parameter values. If the information pro- cessing hardware comprises a general purpose classical computer, this may be achieved simply by running appropriate software on that general purpose computer. Alternatively or additionally dedicated hardware de- signed or configured specifically to carry out the optimization algorithm may be used.
  • quantum computing ar- chitecture employing registers of qubits.
  • Specific quantum computing archi- tectures that are contemplated and preferred by this disclosure are super- conducting qubits, trapped ions and qubits, such as solid state qubits, placed in optical or microwave cavities (cavity QED) in order to utilize the coupling between the internal degrees of freedom of the qubit and the mode of the cavity (qumode).
  • Preferred embodiments of the device and/or method according to this first aspect comprise a subsystem described by the Hamiltonian H sc .
  • a fundamentally different kind of quantum computing devices are those, where the qubit degrees of freedom are coupled to a harmonic oscillator mode (qumode). This may in fact also be done for superconducting qubits, in which case microwave cavities matched to the energy differences usually appearing in SC-qubits are used.
  • qumode appears for trapped ions, where the internal states of the ions are coupled to their vibrational modes in the trap.
  • Solid state qubits that can be initialized and read out op- tically are placed in optical cavities. Each qubit may be assigned to its own cavity, or alternatively some cavities may contain more than one qubit.
  • the total Hamiltonian is then Preferred embodiments of the device and/or method according to this first aspect comprise a subsystem described by the Hamiltonian H QED . Unlike for superconducting qubit systems described above, this control Hamiltonian does not contain any inherent two qubit couplings. However, as to their best knowledge it was first recognized by the inventors, an effec- tive coupling of the form X i X k mediated by one of the qumodes, e.g. the j-th qumode, may be obtained by a sequence of pulses with a proper choice of the phase angles.
  • Effecting this coupling in the device and/or method according to this first as- pect involves applying a time varying control Hamlitonian acting on the qubits i and k and the qumode j in a sequence of four timesteps, each of duration ⁇ t.
  • the phase angle ⁇ ij is set to zero, i.e.
  • the coupling between qubit i and mode j in step 3 is set to be the negative of step 1 , i.e.
  • the techniques presented here in conjunction with superconducting qubit or cavity QED systems may be generalized to any quantum computing sys- tem that is described by a Hamiltonian of the same form as H SC and H QED .
  • the first aspect was concerned with the ability to “engineer” a de- sired effective Hamiltonian encoding a given input matrix as an off-diagonal block or Hamiltonian engineering for short. In the embodiments of the first aspect described in the following, this ability is utilized in order to perform further quantum computational processing with and/or of this input matrix.
  • the classical control system in order to obtain a unitary time evolution having an arbitrary function f(A) of the input matrix A as off-diagonal block connecting the first and second subspace, i.e. implementing a hamiltonian quantum singular value transformation (HQSVT), the classical control system is con- figured to alternatingly apply at least a first effective Hamiltonian and a sec- ond effective Hamiltonian.
  • the first effective Hamiltonian is one in which the input matrix A is encoded as an off-diagonal block connecting a first and a distinct second subspace of the computational Hilbert space, with other blocks having the same row or column indices being arbitrary within the contraint that the overall Hamiltonian be hermitian.
  • the dimension of the first subspace is D 1 , that of the second subspace D 2 .
  • Such an effective Hamiltonian may for instance be obtained by the techniques described hereinabove in connection with the first aspect of the invention.
  • the second effective Hamiltonian Z may likewise be obtained by employing the Hamiltonian engineering techniques described in connection with the first aspect, where as input matrix A the matrix of all Os is used and the control parameters are furthermore chosen such that the on-diagonal blocks are identity matrices.
  • the second effective Hamiltonian Z has the ef- fect of inducing a phase rotation in two dimensional subspaces spanned by each pair of left and right singular vectors.
  • the right singu- lar vectors belong to the first subspace and the left singular vectors to the second subspace.
  • the second effective Hamiltonian is applied in order to average out / refocus the time evolution of the first subspace steming from the non-zero on-diagonal blocks.
  • the steps a-c are repeated with different settings of the parameters ⁇ and t, i.e. with values ( ⁇ 2 ,t 2 ), ( ⁇ 3 ,t 3 ) and so on up to some positive integer L.
  • the classical control system is suitably configured to repeat the three steps a - c to compute the unitary
  • the unitary time evolution re- sulting from such a sequence of L repetitions can in general be expressed as with each of unitary components U j has the form acts on the two dimensional subspace spanned by where P() and Q() are polynomials, P() having a degree less or equal to L and parity L mod 2 and Q() having a degree less or equal to L-1 and parity L-1 mod 2 and from the constraint that each Uj be unitary it follows that
  • 2 1.
  • the parameters ⁇ ( ⁇ / ,t / ) ⁇ are preferably determined in an optimization step such that the transformation approximates the desired operator function f(A) as accurately as possible or at least to within a given error ⁇ .
  • a second pre- computation or pre-processing is performed in order to find optimized val- ues of the parameters ⁇ ( ⁇ j ,t j ) ⁇ .
  • f(x) x
  • f(A) A
  • the input matrix A appears as off-diagonal block of the unitary time evolu- tion.
  • the method according to these embodiments of the first aspect of the invention therefore allows to apply an arbitrary matrix A to a subspace of a quantum system. If the state of the quantum system projected onto that subspace is for instance the method of the second aspect allows com- puting the product
  • arbitrary powers of A may be ap- plied, e.g. by n repetitions the state may be computed.
  • the first aspect of the invention was concerned with techniques, methods and devices to encode a given input matrix as an off-diagonal block of an effective Hamiltonian.
  • the ability to effect such an encoding is utilized in order to perform further quantum computa- tional processing with and/or of this input matrix. In a sense, these further aspects thus provide a useful application of the aforedescribed first aspect.
  • the quantum computing device comprises a quantum system, for instance and preferably one or more qubits, providing a compu- tational Hilbert space that is controlled by a classical control system.
  • the classical control system achieves this, as in the case of the quantum computing device according to the first aspect by means of controlling one or more control field generators each generating a semi-classical control field to control the quantum system or a subsystem.
  • the classical control system is configured to carry out the following method steps:
  • step i a sampling algorithm is used to sample from times twith probabili- ties determined by the Fourier transform g(.) of the given function f(.).
  • the probability (density) distribution p(t) according to which the sampling is done is obtained by normalizing the Fourier transform g(t) according to where integration is over the domain D of the Fourier transform g() or a suitable finite subset, such as when f() is a delta-function, in which case g() is a constant.
  • step ii For each time instant t selected in the sampling step i, in step ii the matrix element is computed by the quantum computing device by means of conventional Hamiltonian evolution or simulation.
  • the steps i and ii are carried out for a number of samples R. This may be done by repeating the steps R times, or, alternatively carrying out each step for a number of samples in parallel.
  • the first approach has the advantage to be more memory efficient, yet potentially takes more time, while the latter may save time, if computation results may be reused in step ii. Irrespective of the approach taken, the results obtained are then used in step iii to calculate the integral which is equal to the desired matrix element
  • the accuracy of the method according to this third aspect of the invention scales as the square root of the number of samples R times the time it takes to evaluate the matrix element
  • the canonical ensemble partition function tr is evaluated by means of the method.
  • the Lorenzian that can be turned into a probability distribution by normalization, i.e. dividing through by, ⁇ the partition function can be evaluated preparing the quan- tum system in a fully mixed state and coupling it to an ancilla qubit that “stores” the result of the partition function in the amplitude of one of the ba- sis states, say Measuring the ancilla qubit will then yield the value of the partition function for a given inverse temperature ⁇ .
  • the quantum computing device and method ac- cording to this second aspect of the invention is combined with the ability provided by the techniques, methods and devices of the first aspect that al- low encoding an arbitrary input matrix A as an off-diagonal block of an ef- fective Hamiltonian, into a method to efficently implement a variety of opti- mization techniques.
  • control field gener- ators may comprise one or more RF pulse generators able to emit radio fre- quency pulses of desired shape, frequency (including chirp) and polariza- tion.
  • one or more lasers can be used to control quantum sys- tems the energy spectrum of which has transitions in the optical, UV or IR ranges of the electromagnetic spectrum. Control parameters of a laser in- clude polarization, amplitude/pulse shape and, for some lasers, also fre- quency.
  • control fields are electrodes for generating static or quasi- static electric fields or magnetic coils, such as wire loops or solenoids, for generating static or quasi static magnetic fields.
  • Quasi-static fields are time varying fields where the relative change is small on the time scales of the respective quantum system to be controlled.
  • the classical control system may comprise a classical general purpose computer running a suitable control software. Alternatively or in addition it may also comprise or even consist entirely of purpose built hardware into which the control routines and algorithms are hard-wired.
  • the advantage over the general purpose computer with software combination is decreased processing time and thus increased reaction and control speed.
  • FIG. 1 A schematic illustration of a generic quantum computing device according to the aspects of the invention.
  • FIG. 2 An illustration in the form of a pseudo-code procedure of a pre- ferred embodiment of the Hamiltonian Engineering method accord- ing to the first aspect of the invention.
  • FIG. 3 An illustration in the form of a pseudo-code procedure of a pre- ferred embodiment of the Fourier Sampling method of evaluating Hamiltonian matrix elements according to the second aspect of the invention.
  • FIG. 1 illustrates schematically the principal components of a quantum computing devices according to the various aspects of the invention.
  • the quantum system 2 which may for instance be a register of qubits, such as a number of trapped ions, superconducting qubits or solid state qubits.
  • the quantum system 2 ’s inherent time evolution, i.e. its time evolution in the absence of further con- trol fields is governed by the Hamiltonian H 0 .
  • control field generators 3, 31 , 32 each generating a con- trol field which may be described semi classically as a scalar time depend- ent function ⁇ k (t) coupled to a corresponding control Hamiltonian H k .
  • Some control field generators 31 may each affect only one or a few subsystems of the quantum system 2, other control field generators 32 may affect several or the entire quantum system 2 at once.
  • two disjoint subspaces 21 , 22 may be identified.
  • the goal of the first aspect of the in- vention is to encode an arbitrary input matrix A as an off-diagonal block of an effective Hamiltonian, the off-diagonal block connecting the first sub- space 21 with the second subspace 22.
  • a classical control system 4 controls the control field generators according to some user input I as well as, in general measurement results obtained by detectors 5.
  • FIG. 2 gives an overview of a preferred embodiment of the Hamiltonian en- gineering method according to the first aspect of the invention in the form of a pseudocode procedure.
  • the procedure “Hengineering” takes as inputs a D 1 xD 2 matrix A, a time independent native Hamiltonian H 0 of a quantum system with an Hilbert space of at least dimension D1+D2, K Time inde- pendent control Hamiltonians H k each coupling to a time dependent semi- classical control field ⁇ k (t), a positive real number T corresponding to a time interval over which the system is to be evolved and a positive integer M correlating to the desired precision of the result.
  • any unitary operator/matrix may be expressed as an exponential of an anti-hermitian operator/matrix.
  • this effective Hamiltonian can be written as shown in the figure with the off-diagonal block which depends on the choice of control paramters As indicated in the figure through denoting them with a "*", the diagonal blocks are, in principle, arbitrary. However, most further quantum information processing tasks performed using this ef- fective Hamiltonian are simplified and/or sped up if the diagonal blocks are as small as possible, i.e. if, ideally, they vanish.
  • This is partularly useful for, com- paratively, strong native system Hamiltonians H 0 .
  • stands for some matrix norm e.g. the 1 -norm or the 2- norm.
  • the goal of the Hamiltonian engineering procedure is to find values for pa- rameters ⁇ k, m ⁇ such that the distance J( ⁇ k, m ⁇ ) is minimal.
  • fur- ther side conditions might have to be taken into account, for instance some maximum control field strength for some or all the control fields, which could arise from practical limitations of the quantum computing device on which the method according to the first aspect is im- plemented, e.g. a maximum power output of the control field generators or a maximum allowable power input rate into the quantum system.
  • GNN Generative Neural Network
  • This GNN comprises an input layer, one ore more hidden layers and an output layer.
  • the num- ber of nodes of the input and hidden layers may be chosen relatively freely, as long as there is a sufficient number of them to provide enough freedom for the network to adapt to the control complexity demanded by the quan- tum system.
  • the number of nodes in the output layer is prefera- bly choosen to equal the number of control parameters N, such that each output represents the value of one of the control parameters ⁇ km .
  • the optimization itself is performed by repeatedly updating the weights ⁇ nm using a gradient descent algorithm GD, that is handed over the cost func- tion J and the GNN weights in the current step s and returns updated GNN weights
  • the step counter s is then incremented and, before the loop is entered again for the next pass, the stopping criterion is evalu- ated. If it has become TRUE during or due to the recent optimization step, the do-loop is quit and the result of the optimization is returned.
  • this result may be the final control parameters the corre- sponding off-diagonal block of the Hamiltonian A eff connecting the first and second subspace or the corresponding final error or any combination of these three and other quantities computed during the execution of the pro- cedure.
  • Useful stopping criteria can be the actual error dropping below a desired er- ror threshold, the step counter s reaching or exceeding a chosen maximum number of optimization steps, a computation time exceeding a preset time limit or any combination of these.
  • the procedure may alternatively be handed a different parameter cor- related to the final accuracy ⁇ . For instance, it could be handed a target/de- sired accuracy ⁇ 0 . Then the number M would be determined by/within the procedure itself such that this desired accuracy is achieved, e.g. by starting from some low default value and successfully increasing M until a solution having the target or better accuracy is found.
  • a target computation time and/or a maximum value for M and/or a maximum num- ber of times that M may be incremented may also be handed over in order to ensure execution of the procedure finishes eventually.
  • Hamiltonian engineering By providing an efficient way to encode arbitrary input matrices in the Ham- iltonian of a quantum system, Hamiltonian engineering according to the first aspect of the invention enables the Hamiltonian Quantum Singular Value Transformation (HQSVT) method mentioned earlier which in turn allows ef- ficient linear algeabra on arbitrary input matrices and thereby also input states.
  • HQSVT Hamiltonian Quantum Singular Value Transformation
  • FIG. 3 a preferred embodiment of the second aspect of the invention is illustrated again in the form of a procedure written in pseudo code, here dubbed “FourierSampling”.
  • the second aspect of the in- vention provides an efficient and, to the best knowledge of the inventors, novel way to compute matrix elements of arbitrary functions of Hamiltoni- ans.
  • the inputs to this procedure are some Hamiltonian H of a quantum system, i.e. some square hermitian matrix, two vectors from the Hilbert space of the quantum system, i.e. two D-dimensional complex valued vec- tors if the Hilbert space of the quantum system has dimension D, a function f(.) that is to be applied to H and a positive integer R corresponding to the number of samples on which the (statistical) computation of the matrix ele- ment is to be based.
  • H of a quantum system i.e. some square hermitian matrix
  • two vectors from the Hilbert space of the quantum system i.e. two D-dimensional complex valued vec- tors if the Hilbert space of the quantum system has dimension D
  • f(.) that is to be applied to H
  • R a positive integer corresponding to the number of samples on which the (statistical) computation of the matrix ele- ment is to be based.
  • the procedure makes use of g(.) the Fourier transform of f(.). which may be computed by the procedure itself, forming a dynamic subroutine or alternatively handed over to the procedure along with the other inputs.
  • N the integral over g() over its entire do- main is used. In case this is not converging such as in case of g() being a constant (i.e. f(x) being a delta distribution), a suitable finite restriction may be used.
  • the desired matrix element is then approximated by the steps stated in the “body” of the procedure.
  • First the output variable res is initialized with 0, as is the step counter k. Then follows a straightforward for-loop in each pass of which the step counter k is incremented by 1 until it reaches the desired number of samples R.
  • the heart of the procedure is, however, the three steps of first selecting a time t, which may in general be negative, then evaluating and finally adding the result of this evaluation to the output variable res .
  • the first step i.e the selection of t
  • the likelyhood of choosing a certain value t is given by a probability distribution p(t) that is proportional to the Fourier transform g(t), i.e. it is the Fourier transform normalized to unit weight.
  • p(t) is proportional to the Fourier transform g(t)
  • the evaluation of the matrix element of the application of the Hamiltonian H for time t i.e. the unitary time evolution for the input states may be performed by conventional Hamiltonian evolution on the quantum computing device or its simulation on a classical device.
  • a precondition for an efficient implementation of this embodiment on a quantum system is therefore having an efficient way to implement the desired Hamiltonian H on the quantum system.
  • Hamiltonian enginieering according to the first aspect of the invention may be employed, for instance as described in reference to the preferred embodiment shown in FIG. 1 .
  • the output of the procedure “FourierSampling” is returned, namely the variable res, in which the evaluated matrix elements for the different selected times have been accumulated, devided by Rthe number of samples and multiplied with N, the normalization constant con- necting p(t) and g(t), i.e. the integral over the full domain of g(t): N
  • the practical applications of this procedure and the second aspect of the invention in general, are fast matrix inversion allowing the efficient solution of linear equations, linear and nonlinear differential equations on high di- mensional spaces as well as constrained quadratic optimization. Further applications comprise the evaluation of partition functions and the finding of ground states, thereby providing a new approach to combinatorial optimiza- tion.
  • the second aspect of the invention provides a quantum method for performing the powerful multiplicative weight optimization method as well as dynamic programming.
  • Industrial applications of the second aspect are all those, where the afore- described mathematical applications appear. To name just a few these are in particular modelling of industrial processes, trajectory optimization for self driving cars and inverse methods for oil and gas exploration.
  • the methodd and device of the second aspect of the invention make it possible to efficiently employ a variety of optimization techniques.
  • a particu- lar benefit is the exponential advantage Fourier Sampling enjoys over known techniques for approximate Bayesian inference

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Abstract

La présente divulgation concerne des procédés et des dispositifs de calcul quantique, un calcul correspondant à une certaine transformation unitaire souhaitée de l'espace de Hilbert d'un système quantique, tel qu'un ensemble de bits quantiques, étant mis en œuvre par application sélective d'impulsions de commande afin de réaliser un calcul quantique de temps continu plutôt que par décomposition de la transformation unitaire en des opérations prises à partir d'un ensemble fixe de portes. De cette manière, une matrice d'entrée A est codée dans l'hamiltonien global agissant sur le système quantique sous la forme d'un bloc hors diagonale connectant un premier sous-espace à un second sous-espace. Ceci est obtenu au moyen d'une "ingénierie hamiltonienne" qui utilise efficacement des champs de commande semi-classiques variant dans le temps, les valeurs des K réglages de champ de commande au niveau d'un ensemble de n instances temporelles fournissant un nombre N = n K paramètres de commande qui, à l'aide d'un algorithme d'optimisation exécuté dans une étape de pré-calcul, sont choisis de telle sorte qu'un hamiltonien efficace ayant le bloc diagonal souhaité A est obtenu. L'hamiltonien effectif ainsi obtenu est alors soit appliqué alternativement avec un hamiltonien de repos afin d'obtenir une évolution temporelle correspondant à une fonction f donnée appliquée à la matrice d'entrée A, soit son évolution temporelle est exécutée et la moyenne d'ensemble canonique échantillonnée afin de calculer des éléments matriciels arbitraires de la fonction donnée appliquée à l'hamiltonien effectif H. Dans tous les cas, les impulsions de commande requises sont optimisées au moyen d'une technique de compression classique telle qu'un réseau neuronal d'apprentissage profond afin de maximiser, pendant la phase de chargement d'informations du calcul, la quantité d'informations classiques chargées dans le système, et, pendant la phase de calcul, le nombre de transformations quantiques temporelles continues élémentaires, respectivement.
PCT/EP2022/056178 2022-03-10 2022-03-10 Procédés et dispositifs de calcul quantique temporel continu WO2023169680A1 (fr)

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