Quantum simulation of a bosonic system

Abstract

The invention refers to a method for simulating a large-scale quantum mechanical bosonic system comprising the steps of: transforming a Hamiltonian describing the bosonic system into a qubit (14) Hamiltonian describing a corresponding system of qubits (14), comprising at least one boson creation and / or annihilation operator; simulating the evolution of the system of qubits (14) based on the at least one boson creation and / or annihilation operator and based on the qubit Hamiltonian by applying a swap network to the system of qubits (14) to read out a parameter related to the evolution of the system; and determining properties of the bosonic system based on the parameter. The invention further relates to a corresponding quantum computer and computer program.

Description

The present invention relates to quantum computers (QCs). QCs are engi- neered and well-controlled quantum devices whose computations are based on utilizing the laws of quantum mechanics. In conventional QCs the fundamental unit of the QC is the qubit (quantum bit). Quantum computers (QCs) can be used to calculate properties of quantum mechanical systems, such as atoms, molecules, or solid materials. This task is often referred to as quantum simula- tion. Presently, QCs can simulate only small systems because the effect of noise is strong and restricts the length of computations that can be run.There are multiple technical challenges in building a large-scale QC. One huge difficulty is the isolation of a QC from a noisy environment. This is a problem for qubits as well as for oscillators. The effect of noise on quantum simulation is that the results are randomized after a certain characteristic decoherence time, leading to finite computation depths. This practically means that only small com- putational tasks of material properties can be executed, since encoding of all processes of large systems is not possible without long and time-consuming gate sequences.In digital quantum simulations, the system time evolution is programmable and created through rapid control pulses, called gates, to physical qubits or qudits, which hold the information of the system state. The gate sequence is designed in such a way that it reproduces the system time evolution operator in a certain satisfactory approximation.WO 2019 / 070228 A1 discloses methods, systems, and apparatus for simulating a physical system. In one aspect, a method includes transforming a Hamiltonian describing the physical system into a qubit Hamiltonian describing a corre- sponding system of qubits, the qubit Hamiltonian comprising a transformed ki- netic energy operator; simulating the evolution of the system of qubits under the qubit Hamiltonian, comprising simulating the evolution of the system of qubits under the transformed kinetic energy operator by applying a fermionic swap network the system of qubits; and using the simulated evolution of the system of qubits under the qubit Hamiltonian to determine properties of the physical system.It is an object of the present invention to provide a possibility to perform quantum computation of large quantum mechanical systems and to compute or simulate models that cannot be computed by classical computation (normal computers) or by conventional QCs in the presence of noise. In particular, it is an object of the present invention to compute models that include bosonic modes. The re- sults can preferably be used to predict properties of various atoms, molecules, and materials that are of high industrial interest.In a first aspect of the present invention a method for simulating a large-scale quantum mechanical bosonic system is presented that comprises a step of transforming a Hamiltonian describing the bosonic system into a qubit Hamilto- nian that describes a corresponding system of qubits, wherein the Hamiltonian comprises at least one bosonic creation and / or annihilation operator. The method further comprises a step of simulating the bosonic system based on the at least one boson creation and / or annihilation operator and based on the qubit Hamiltonian by applying a swap network to the system of qubits to read out a parameter related to the evolution of the system of qubits. The method further comprises a step of determining properties of the bosonic system based on the parameter.In another aspect the present invention relates to a quantum computer for car- rying out the method as described above, wherein the quantum computer com- prises preferably a qubit being coupled to only one oscillator and up to two near- est-neighbor qubits and a swap network for swapping information of spins be- tween neighboring qubits during computation.In yet further aspects of the present invention, there is provided a corresponding computer program which comprises program code means for causing a quan- tum computer to perform the steps of the method disclosed herein when said computer program is carried out on a quantum computer.Preferred embodiments of the invention are defined in the dependent claims. It shall be understood that the claimed quantum computer has similar and / or iden- tical preferred embodiments as the claimed method, in particular as defined in the dependent claims and as disclosed herein.By providing a Hamiltonian comprising at least one bosonic creation and / or an- nihilation operator, a design of a quantum computer can be introduced, which allows to solve a particular quantum mechanical problem in a highly efficient way. This gives the possibility to carry out simulations of larger quantum sys- tems. The physical state of oscillators can be used to store the state of the sim- ulated system, wherein preferably the physical qubit oscillator coupling is di- rectly usable for a computation of the model. Preferably, the physical layout of the quantum computer is designed in such a way that it optimally executes an efficient quantum algorithm, preferably a specifically tailored swap algorithm. The quantum computer is easy to scale because the gate sequences preferably have no significant complexity compared to conventional quantum computers.In a preferred embodiment, the oscillators of all qubit-oscillator gates are in res- onance with corresponding qubits which facilitates the calibration procedure. With the presented method and design, an arbitrary number of qubits and oscil- lators can be used simultaneously for calculations with the quantum computer.In a preferred embodiment, simulating the evolution of this system comprises a step of mapping bosonic modes to hardware oscillators of a quantum computer. Thus, with the disclosed method bosonic modes can be included in a simulation.In a further preferred embodiment, simulating the evolution of this system com- prises a step of computing an evolution of a quantum state of spins or electrons coupled to bosonic modes. Thus, with the disclosed method an evolution of bosonic modes can be included in a simulation.In a further preferred embodiment, the step of simulating the evolution of the system comprises setting up a set of two-level systems or spins coupled to bosonic modes which satisfy the Hamiltonian H = - Σi σiz / 2 + Σij vijσix(b†j + bj) + Σj wjb†j bj.Here the Pauli matrices σix / y describe the spin i, and the creation (or annihila- tion operator) b†j (bj) describes the bosons in the mode j. The coupling can also be longitudinal, i.e. proportional to σiz, or it can also be a product of multiple different spin operators. The spins can also be replaced by electronic levels. By providing a simulation that is performed by evolving a quantum state which sat- isfies the above Hamiltonian a plurality of systems can be modelled by the dis- closed approach.In a preferred embodiment, the step of simulating the evolution of the system comprises a step of computing a bosonic system which propagates in the ro- tated reference frame. In the method of the present invention the spins are not transformed in the rotating frame for the time propagation of the quantum sys- tem. With this amendment simpler gate sequences can be used when perform- ing a computation with multiple oscillators and qubits.In a preferred embodiment computing a bosonic system which propagates in the co-rotated reference system is described by the Hamiltonian H = - Σi σiz / 2 + Σij vijσix(b†j eiwjt + bj e-iwjt).In this case the modelled bosonic-mode frequencies wj appear only as a phase of the coupling terms. In this transformation the physical oscillator frequencies are usually not tunable, and thus the information of wjb†j bj -type terms in the quantum simulation should be done indirectly, for example by using such phases.Preferably, the simulation of the evolution of the system comprises a step of computing a Jaynes-Cummings-Hamilton operator in the form of Hjc = g(σ_b† + σ+b). By choosing a Jaynes-Cummings-Hamilton operator, a Jaynes-Cummings gate can be used for the computation. This physical interac- tion can be implemented on various quantum computer platforms. Thus, the method is suitable for a vast number of platforms and can be implemented in short time.Preferably, a physical frequency of the qubit is the same as the physical fre- quency of the oscillator. Thus, a calibration can be done in a straightforward manner.In a preferred embodiment, simulating the evolution of the system comprises a step of carrying out an interaction for a predetermined time tgate to effectively perform the qubit oscillator gate, that preferably satisfies Ujc(tgateg) = exp[-itgateHjc].The interaction for the predetermined time can be carried out, for example, by bringing a far-detuned qubit in resonance for the time period tgate. With this pre- ferred step the simulation can be carried out under known conditions. A calcu- lation rate can be set to a preferred period of time. Furthermore, if a simulated time step τ is fixed, the gate time is also fixed. Thus, the effective gate interac- tion time can be calibrated to a predefined value to allow small values for errors in the quantum simulation (Totter expansion).Preferably, the simulation of the evolution of the system comprises a step of decomposing a coupling in the considered system by using a Jaynes-Cum- mings gate that is surrounded by qubit X-gates. By this approach, decomposi- tion can be implemented in a straightforward manner in order to provide a quan- tum simulation method that can be used with a variety of quantum computer platforms.In an advantageous embodiment, the simulation of the evolution of the system comprises a step of sequentially applying Hjc and σxHjcox to implement a quantum-Rabi gate. With this approach, bosonic mode frequencies can be eas- ily computed via added phases. Since the system is time propagated in the ro- tating frame of the bosonic modes, the Hamiltonian preferably comprises no specific boson frequency operators. Instead, the coupling terms have phases that increase linearly in simulated time. This can be implemented by additional Z-rotations around the Jaynes-Cummings gate. However, this construction is also true, for example in the final quantum-Rabi gate. Herein, additional phases can be applied as laser or microwave phases. This can be done in a straight- forward manner.In a preferred embodiment, the simulation of the evolution of the system com- prises a step of creating model couplings of the form σz (b† + b) with a corre- sponding gate operation Uz(tgateg) = exp[-itgategoz (b† + b)], preferably by surrounding a Jaynes-Cummings gate with Hadamard gates. With this ap- proach a broad and straightforward possibility is given to implement other forms of model interactions in the quantum simulation as well. In an alternative em- bodiment a qubit and / or a quantum-Rabi gate can be surrounded by Hadamard gates.In a further preferred embodiment, the step of simulating the evolution of the system comprises a step of computing arbitrary bosonic frequencies, preferably by surrounding a Jaynes-Cummings gate with Z-rotations. Consequently, a broad and multi-purpose model can be built that allows the simulation of arbi- trary bosonic frequencies.In a further preferred embodiment, the step of simulating the evolution of the system comprises a step of computing a coupling of bosons to a product of multiple spin operators, preferably by surrounding a Jaynes-Cummings or Rabi gate by qubit-qubit entangling gates, such as CNOT gates.In yet another preferred embodiment, the time step τ is fixed so that the gate interaction time is also fixed. Since the gate needs to satisfy the condition tgateg = τυ, where v is the spin boson coupling in the model to be computed or simulated by the quantum processor, the effective gate interaction time is cali- brated to tgate = τυ / g. During the time period of the effective gate interaction time the gate is activated and performs. Thus, the gate interaction time can be used to at least partly control the computation.The present invention is based on the idea that a digital quantum simulator can be used to solve large-scale quantum mechanical problems. A functional em- bodiment of a simulator is disclosed that operates with arbitrary numbers of qubits and oscillators. Its physical layout is designed in such a way that it opti- mally executes an efficient quantum algorithm, preferably a specifically tailored swap algorithm.The presented approach is easily scalable to large sizes. The computation via the simulator is very efficient, not only due to its layout, but also because there is no essential complexity in the gate sequences in comparison to conventional QCs because the physical interaction between qubits and oscillators is directly in the desired form. In other words, the native qubit-oscillator interaction can be directly used for computation. Furthermore, a rather simple calibration routine for the qubit-oscillator gates that are preferably always on-resonance can be applied. Thus, an overhead due to qubit calibrations is omitted while a simulta- neous calculation with arbitrary numbers of qubits and oscillators is still possi- ble.The invention uses a multi-step approach to establish a quantum simulator for a desired problem. It involves a suitable definition of the problem, a determina- tion of required computation sequences, and a corresponding calibration of qubit-oscillator gates.Furthermore, a specific physical layout of the qubits and oscillators is disclosed, which is easy to fabricate and is optimized to execute a specific quantum algo- rithm tailored for system-boson problems.These and other aspects of the invention will be apparent from and elucidated with reference to the embodiment(s) described hereinafter. In the following drawingsFigure 1 shows a basic workflow to establish a quantum simulation;Figure 2 schematically shows a suitable model;Figure 3 schematically shows a determined gate sequence;Figure 4 schematically shows the step of calibration of qubit oscillator gates;Figure 5 shows a computation and read-out scheme;Figure 6 shows an example layout of a quantum simulator; and Figures 7-9 show a full multi-step approach of a quantum simulation.In Figure 1 a basic workflow to establish, perform and read out a quantum sim- ulation is shown.In a first step S10 an original bosonic system of interest is regarded and studied.In a following second step S20 a physical system model, psm, of the system regarded in step S10 is created. The psm has multiple bosonic modes.In a third step S30 the physical system model, psm, is derived in a rotating frame.In a following fourth step S40 the psm is mapped to a qubit-oscillator model, qom. Thus, a Hamiltonian describing the physical system, i.e. a bosonic system, is transformed into a qubit Hamiltonian describing a corresponding system of qubits. The Hamiltonian comprises at least one boson creation and / or annihila- tion operator.After that, the qom interactions are decomposed to qubit-qubit and / or qubit-os- cillator gates in a fifth step S50.Then in a sixth step S60 a simulation of the qom is performed on a specific layout comprising qubit swaps and local qubit-oscillator gates. Thus, the evolu- tion of the system of qubits is simulated based on the qubit Hamiltonian, by applying a swap network to the system of qubits, to read out a parameter related to the evolution of the system of qubits. The evolution of the system of qubits is simulated also based on the at least one boson creation and / or annihilation op- erator.In a following seventh step S70 the results of the observables of the qom on hardware are measured.In a final eighth step S80 the results are converted to physical system proper- ties, preferably by a classical central processing unit. Thus, properties of the bosonic system are determined based on the parameter.Figure 2 schematically shows a suitable model 10 of the boson system, on which the simulation can be based.In an exemplary embodiment, a time evolution of a quantum state of spins or electrons that are coupled to bosonic modes is considered. A simple example is a set of two-level systems, for example spins, coupled to bosonic modes, whose quantum mechanical description is defined by the Hamiltonian H = - Σi σiz / 2 + Σij vijσix(b†j + bj) + Σj wjb†j bj.Here, the Pauli matrices σix / y describe the spin i and the creation (or annihila- tion) operator b†j describes the bosons in the mode j. The coupling can also be longitudinal, i.e. proportional to σiz, or a product of multiple spin operators.The spins can also be replaced by electronic levels. Obviously, a variety of sys- tems can be simulated by such models. In the proposed digital quantum simu- lation with oscillators, the quantum state is time propagated in the rotating frame of the bosonic modes, where the Hamiltonian looks like H = - Σi σiz / 2 + Σi=1,2 j=1,2 vijσix(b†j eiwjt + bj e-iwjt).The modeled bosonic-mode frequencies wj appear only as phases of the cou- pling terms. This transformation is preferred since physical oscillator frequen- cies are usually not tunable.An implementation of wjb†j bj -type terms in the digital quantum simulation can be done indirectly, for example through the use of such phases. In this applica- tion the spins are not or not necessarily transformed in the rotating frame.Figure 3 shows exemplarily determined sequences 12 of gate 20 operations needed to time-propagate the quantum state, preferably including the qubit-os- cillator gates UJc.A gate 20 between two qubits 14 or between an oscillator 18 and a qubit 14, is implemented via a direct natural interaction or coupling 16 between two quan- tum mechanical systems that establish the qubit / oscillator, or possibly via a na- tive interaction with an external party.The interaction, or several interactions, are turned-on when the gate 20 is being applied, and turned-off when the desired effect (gate) has been reached. The physical realization of "turning-on" or "turning-off" varies between different hard- ware implementations, and is in principle known in the art.The Jaynes-Cummings Hamiltonian, Hjc = g(σ_b† + σ+b), can be used to im- plement needed qubit-oscillator gates 20. Here g is a physical qubit-oscillator coupling 16.This physical interaction can be implemented on various QC platforms. It de- scribes, for example, the interaction between superconducting qubits 14 and microwave resonators. Here, the physical frequencies of the qubit 14 and oscil- lator 18 can be chosen to be the same, since this simplifies the calibration significantly.The interaction can be turned "on" for time tgate, which can be achieved for ex- ample by bringing a far-detuned qubit on-resonance for this period of time. Thus, the qubit-oscillator gate Ujc(tgateg) = exp[-itgateHjc] can be effectively executed.In the circuit description of gate sequences 12 as shown in Figure 3, this gate is marked with 20. Here "q" refers to the qubit and "o" to the oscillator. This gate 20 can be used to create model coupling between spins (or electrons) and bosonic modes.The chosen gate time tgate will be given by the used time propagation time step τ. It then needs to be calibrated to some given value, or values. It is to be un- derstood that computation related to spin terms only (Hamiltonian frequencies, fields, couplings) can be implemented in exactly the same way as in common quantum computing.The bare qubit-oscillator or Jaynes-Cummings gate 20 implements system- boson couplings of the form υ(σ_b† + σ+b). If the coupling in the considered system-boson model is not of this form, it can be decomposed into a set of qubit- oscillator gates.One possible form of the model coupling is σx(b† + b), which appears espe- cially in quantum optics. To implement this, the Jaynes-Cummings Hamiltonian can be complemented by counter-rotating terms, g(σ_b† + σ_b). Since this is equal to σxHjcox, the needed computational processes can be implemented on the quantum processor by surrounding the Jaynes-Cummings gate with qubit X-gates.For the implementation of the quantum-Rabi gate, UQR (tgateg) = exp[-itgategσx(b† + b)] a computation sequence 12 that sequentially applies Hjc and σxHjcox can be used.This computation has an error that can be estimated to be proportional to υτ, or higher powers of it. Thus, small values for τ are desirable.In the light of this idea other forms of model interactions can also be created. One common form of the model couplings is σz(b† + b), with the corresponding gate operation Uz (tgateg) = exp[-itgategoz (b† + b)].The computational sequence needed to create this can correspond to standard Hadamard gates applied to qubits.Furthermore, multi-spin interactions can also be created. For example, time evolution according to a Fröhlich-type coupling UF (tgateg) = exp[-itgateg(σzσ_b† + σzσ+b)] can be implemented by the gate sequence as shown on the right hand side of Figure 3, which now also includes entangling CNOT-operations between two qubits.Figure 4 shows a calibration of the required qubit-oscillator gates 20. Each os- cillator 18 couples to one qubit 14 only.The system time evolution can be created by implementing the Trotterized time- evolution operator 30 U(t) , wherein U(t) ≈ ρ-ίτΗ[(n-1)τ] ...ρ-ίτΗ(τ)ρ-ίτΗ(0) Here the total simulation time t is given by t = nt. At each step of the expansion, the unitary operation, i.e. time evolution over τ, can be implemented via a series of physical operations e-itH(mt) ≈ Πke-itHk(mt), where m is the number of the Trotter step.This corresponds to decomposing the model couplings to Jaynes-Cummings gates 32. There are also other possible decompositions. The size of the time step τ, which controls the error in this expansion, is important, see for example reference "Theory of trotter error with commutator scaling" Andrew M. Childs, Yuan Su, Minh C. Tran, Nathan Wiebe, and Shuchen Zhu Phys. Rev. X 11, 011020 Published 1 February 2021 https: / / doi.org / 10.1103 / PhysRevX.11.011020.The error, and thereby also the time step τ, can be fixed beforehand. This also defines how the qubit-oscillator gates 20 should be calibrated.If the simulated time step τ is fixed, then the gate time is also fixed, since the applied gate 20 needs to satisfy tgateg = τυ, where v is the spin-boson coupling in the model to be computed by the quantum processor. In other words, the effective gate interaction time can be calibrated to tgate = τυ / g. This is schemati- cally shown in graph 28.Furthermore, when one bosonic mode j couples to multiple spins i, with different coefficients vij, the gate time of oscillator j can be calibrated to all values tgate = τυij / g. In other words, for N spins, each qubit-oscillator gate can be calibrated to N different interaction times at most.Figure 5 shows computation and readout 24 of the simulation, which is per- formed by executing an algorithm that optimally fits the considered layout of the qubits 14 and oscillators 18. In the circuit, only lines of qubits 14 explicitly shown. Local qubit-oscillator gates 20 are marked as vij, referring to Ujc (tvij). The time evolution is schematically shown as arrow 26.At the end of the quantum simulation the result is read out, i.e. a measurement of the state of the qubits and (possibly) oscillators is carried out. A steady state and / or dynamical properties of the system can be of interest. These can corre- spond to simple averages of single spins and correlations between different spins. The mean value corresponds to repeated statistical averaging. The eval- uation of the two-time correlation function may require the usage of additional ancilla qubits. The reached steady-state values define, for example, the re- duced-density matrix of the system (RDM), which can be useful in itself or as input to other material simulation methods.If the oscillator state can also be measured, the population of the bosonic modes and / or the system-bath interaction can be monitored or the full state of the oscillators (Wigner function) can possibly be recovered as shown in reference "Synthesizing arbitrary quantum states in a superconducting resonator" Max Hofheinz, H. Wang, M. Ansmann, Radoslaw C. Bialczak, Erik Lucero, M. Neeley, A. D. O'Connell, D. Sank, J. Wenner, John M. Martinis & A. N. Cleland https: / / www.nature.com / articles / nature08005. The study of oscillator states may make the use of ancilla qubits necessary.In Figure 6 an exemplary layout 40 of a quantum simulator is shown. The lay- out 40 is such that it optimally executes a swap algorithm specifically tailored for the models considered in this application. Here "q" refers to the qubit and "o" to the oscillator.Each qubit is coupled to only one oscillator and up to two nearest-neighbor qubits as shown in Figure 5. This design is easy to fabricate and allows for modelling all interactions between arbitrary spins and bosons when combined with the tailored swap algorithm.The swap algorithm tailored to the system-boson model is presented in refer- ence "A quantum algorithm for solving open system dynamics on quantum com- puters using noise" Juha Leppäkangas, Nicolas Vogt, Keith R. Fratus, Kirsten Bark, Jesse A. Vaitkus, Pascal Stadler, Jan-Michael Reiner, Sebastian Zanker, Michael Marthaler Phys. Rev. A 108, 062424 – Published 22 December 2023 https: / / doi.org / 10.1103 / PhysRevA.108.062424. The swap algorithm can also efficiently simulate fermionic systems, see reference "Quantum simulation of electronic structure with linear depth and connectivity" Ian D. Kivlichan, Jarrod McClean, Nathan Wiebe, Craig Gidney, Alán Aspuru-Guzik, Garnet Kin-Lic Chan, and Ryan Babbush Phys. Rev. Lett. 120, 110501 - Published 13 March 2018 https: / / doi.org / 10.1103 / PhysRevLett.120.110501. In this case all swap gates 22 will be replaced by so-called fswap gates which swap the state of a fermionic orbital while preserving the fermionic symmetry of the wave function.The basic idea of this swap network approach is to exchange, i.e. swap, the information of spins between neighboring qubits. By swapping the information on certain qubits following some systematic way, e.g. alternating swapping of spins on even and odd numbered qubits, it is achieved that each saved spin data physically neighbors each oscillator, preferably at least once. Thus, all sys- tem-boson interactions can efficiently be computed, with a linear depth even though simple restricted connectivity is applied.A second-order Trotter circuit for a model with two spins and two bosonic modes with transverse coupling between arbitrary spins, and bosons is shown in Fig- ure 6 where the "crossing lines" operations mark the swap gates 22 and boxes with Rz mark Z-rotations 38. The time evolution operator that accounts for arbi- trary spin-spin interactions 34 is shown as a box marked with Us. This approach can be directly generalized to an arbitrary number of qubit-oscillator pairs. A more detailed description of the final gate sequences is shown in Figures 7 to 9.Figures 8,9 represent together one circuit, which is equivalent to the circuit of Figure 7. Figures 8,9 describe more detailed the form of the introduced phase- dependent Rabi gates 36 in Figure 7. Here, gates 40 correspond to qubit X- gates realized physically with additional microwave- or laser-phases θ of the qubit drives.With respect to Figures 7 to 9, a full multi-step approach to establish a digital quantum simulation of a desired problem is described.As a concrete and simple example, the Dicke model, is chosen including two bosonic modes, which can describe, for example, creation of excitations in or- ganic solar cells under irradiation. The according Hamiltonian is H = Σi=1,2 εi / 2 ·σiz + Σi=1,2 Σj=1,2 vijσix(b†j + bj ) + Σj=1,2 wjb†j bj.With the assumption that all vij ≠ 0.In the proposed digital quantum simulation with oscillators, the quantum state is time propagated in the rotating frame of the bosonic modes, where the Hamil- tonian looks like H = - Σi σiz / 2 + Σi=1,2 j=1,2 vijσix(b†j eiwjt + bj e-iwjt).The modeled bosonic-mode frequencies wj appear (only) as phases of the cou- pling terms. These will be implemented by phases of the qubit drives.The decomposition for each oscillator follows the description given above for the quantum Rabi (QR) model. Here, each quantum-Rabi gate 36 is decom- posed from two Jaynes-Cummings gates 32.The simulation time step τ is chosen such that the Trotter error stays small, which means here that it satisfies τυij < 1. After fixing the simulation time step τ to a certain small value, the calibration of the qubit-oscillator gates can be performed.This means fixing the physical qubit-oscillator interaction time periods to tgate = τυij / g. Since a two-qubit / two-oscillator processor is considered in the example of Figure 5, the calibration looks as follows:1. The first qubit-oscillator pair is calibrated to two possible interaction times: τυ11 / g and τυ21 / g.2. The second qubit-oscillator pair is calibrated to two possible interaction times: τυ12 / g and τυ22 / g.Of course, a (standard) set of single-qubit and two-qubit gates is also calibrated. Particularly, single-qubit X-gates with arbitrary drive phases are used as well as some set of two-qubit gates that can be used to decompose the two-qubit swap- gate.To time propagate the system, a second-order Trotter formula is considered, as illustrated in reference "Theory of trotter error with commutator scaling" Andrew M. Childs, Yuan Su, Minh C. Tran, Nathan Wiebe, and Shuchen Zhu Phys. Rev. X 11, 011020 Published 1 February 2021 https: / / doi.org / 10.1103 / PhysRevX.11.011020. When applied to the considered Hamiltonian, the gate sequence of the Trotter step m has the form as shown in Figure 7.This can be recast to the form as shown in Figures 8 and 9, where the angle θi = Μτωi / 2 and Rφ(π) = Rz(φ)Rx(π)Rz(-φ) corresponds to the X-gate driven with (microwave or laser) phase φ.It is important to note that when using gate sequencing of this form, the compu- tation can be made in the same manner for an arbitrary number of qubits and oscillators.A person skilled in the art learns from the disclosure herein that in its given task the introduced digital quantum simulator is drastically more efficient than con- ventional QCs. This assertion can be made more concrete by making estimates for the hardware-component count and entangling-gate count when implement- ing the spin-boson operator exp [iφσz(b† + b)] using standard unary and bi- nary coding and including a number of d bosonic mode levels. The results are given in the table below.code included energy levels needed hard- ware compo- nents needed entan- gling gatesqubit-oscillator arbitrary 1 qubit, 1 oscilla- tor 2unary d 1 + d qubits O(d)binary d 1 + log2(d) qubits O[d² log(d)]For large values of d either the qubit count (unary coding) or the entangling- gate count (binary coding) becomes large. On the other hand, in the qubit-os- cillator quantum simulation, the same operation needs only two entangling gates, for arbitrary values of d.Since the fabrication of oscillators is not more complicated than that of qubits, it is obvious that the qubit-oscillator quantum simulator can be drastically more efficient than regular QCs. The entangling-gate count is based on the analysis made in reference "Resource-efficient digital quantum simulation of d-level sys- tems for photonic, vibrational, and spin-s Hamiltonians" Sawaya, N.P.D., Menke, T., Kyaw, T.H. et al. npj Quantum Inf 6, 49 (2020). https: / / doi.org / 10.1038 / s41534-020-0278-0.If a system coupling to bosons is considered, with a low number of excitations in the bosonic modes, it is possible to replace the resonator with qubits. In this case the overall layout and the algorithm stay basically the same. Primarily the gate Ujc (tgateg) = exp[-itgateHjc] has to change by e.g. changing Hjc to Hjc = g(σ_oresonator + σ+oresonator), where σ_oresonator are Pauli matrices acting on the qubit that is replacing the resonator.The main differences / improvements of the presented approach, when com- pared to reference "Experimentally simulating the dynamics of quantum light and matter at deep-strong coupling" Langford, N.K., Sagastizabal, R., Kounala- kis, M. et al. Nat Commun 8, 1715 (2017). https: / / doi.org / 10.1038 / s41467-017- 01061-x, are the following:Computation with multiple oscillators instead of a single oscillator.Specific physical layout, which simplifies engineering but still provides efficient quantum simulations.Time propagation of the state is done in a different frame.The calibration of the device is simplified (oscillators are always on resonance).Decomposing of arbitrary models is sketched.Thus, the foregoing discussion discloses and describes merely exemplary em- bodiments of the present disclosure. As will be understood by those skilled in the art, the present disclosure may be embodied in other specific forms without departing from the spirit or essential characteristics thereof. Accordingly, the description is intended to be illustrative, but not limiting the scope of the disclo- sure, as well as other claims. The disclosure, including any readily discernible variants of the teachings herein, defines, in part, the scope of the foregoing claim terminology such that no inventive subject matter is dedicated to the pub- lic.In the claims, the word "comprising" does not exclude other elements or steps, and the indefinite article "a" or "an" does not exclude a plurality. A single element or other unit may fulfill the functions of several items recited in the claims. The mere fact that certain measures are recited in mutually different dependent claims does not indicate that a combination of these measures cannot be used to advantage.The elements of the disclosed devices, circuitry and system may be imple- mented by corresponding hardware and / or software elements, for instance ap- propriated circuits. A circuit is a structural assemblage of electronic components including conventional circuit elements, integrated circuits including application specific integrated circuits, standard integrated circuits, application specific standard products, and field programmable gate arrays. Furthermore, a circuit includes central processing units, graphics processing units, and microproces- sors which are programmed or configured according to software code. A circuit does not include pure software, although a circuit includes the above-described hardware executing software.

Claims

1. Method for simulating a large-scale quantum mechanical bosonic system comprising the steps of: transforming (S40) a Hamiltonian describing the bosonic system into a qubit (14) Hamiltonian describing a corresponding system of qubits (14), comprising at least one boson creation and / or annihilation operator; simulating (S60) the evolution of the system of qubits (14) based on the at least one boson creation and / or annihilation operator and based on the qubit Hamiltonian by applying a swap network to the system of qubits (14) to read out a parameter related to the evolution of the system of qubits; and determining (S80) properties of the bosonic system based on the parameter.

2. Method according to claim 1, wherein the step of simulating the evolution of the system comprises a step of mapping bosonic modes to hardware oscillators (18) of a quantum computer.

3. Method according to claim 1 or 2, wherein the step of simulating the evo- lution of the system comprises a step of computing an evolution of a quan- tum state of spins or electrons coupled to bosonic modes.

4. Method according to claim 3, wherein the step of simulating the evolution of the system comprises a step of setting up a set of two-level systems coupled to bosonic modes which preferably satisfy the Hamiltonian H = - Σi σiz / 2 + Σij vijσix(b†j + bj) + Σj wjb†j bj .

5. Method according to one of the preceding claims, wherein the step of sim- ulating the evolution of the system comprises a step of computing a bosonic system which propagates in a co-rotated reference system which is preferably described by the Hamiltonian H = - Σi σiz / 2 + Σij vijσix(b†j eiwjt + bj e-iwjt).

6. Method according to one of the preceding claims, wherein the step of sim- ulating the evolution of the system comprises a step of computing a Jaynes-Cummings-Hamilton operator in the form of Hjc = g(σ_b† + σ+b), wherein a physical frequency of the qubit (14) is preferably the same as a physical frequency of the oscillator (18).

7. Method according to one of the preceding claims, wherein the step of sim- ulating the evolution of the system comprises a step of carrying out an interaction for a predetermined time tgate, to effectively perform the qubit- oscillator gate (20), that preferably satisfies Ujc(tgateg) = exp[-itgateHjc].

8. Method according to one of the preceding claims, wherein the step of sim- ulating the evolution of the system comprises a step of decomposing a coupling (16) in the system by using a Jaynes-Cummings gate (32) that is surrounded by qubit (14) X-gates.

9. Method according to one of the preceding claims, wherein simulating the evolution of the system comprises a step of sequentially applying Hjc and σxHjcox to implement a quantum-Rabi gate (36).

10. Method according to one of the preceding claims, wherein the step of sim- ulating the evolution of the system comprises a step of creating model couplings (16) of the form σz (b† + b), with a corresponding gate operation Uz(tgateg) = exp[-itgategoz (b† + b)], preferably by sur- rounding a Jaynes-Cummings gate (32) with Hadamard-gates (20).

11. Method according to one of the preceding claims, wherein the step of sim- ulating the evolution of the system comprises a step of computing arbitrary bosonic mode frequencies, preferably by surrounding a Jaynes-Cum- mings gate (32) with Z-rotations (38).

12. Method according to one of the preceding claims, wherein an effective gate interaction time is calibrated to tgate = τυ / g.

13. Method according to one of the preceding claims, wherein the step of sim- ulating the evolution of the system comprises a step of computing a cou- pling of bosons to multi-spin processes, preferably by surrounding a Jaynes-Cumming gate and / or a Rabi gate with qubit-qubit entangling gates, particular preferably with CNOT gates.

14. Quantum computer for simulating a large-scale quantum mechanical bosonic system comprising: a transformation unit for transforming a Hamiltonian describing the bosonic system into a qubit (14) Hamiltonian describing a corresponding system of qubits (14), comprising at least one boson creation and / or an- nihilation operator; a simulation unit for simulating the evolution of the system of qubits (14) based on the at least one boson creation and / or annihilation operator and based on the qubit Hamiltonian by applying a swap network to the system of qubits (14) to read out a parameter related to the evolution of the sys- tem of qubits; and a determination unit for determining properties of the bosonic system based on the parameter, wherein the quantum computer preferably includes a qubit (14) being coupled to only one oscillator (18) and up to two nearest-neighbor qubits (14) and a swap network for swapping information of spins between neighboring qubits (14) during computation.

15. Computer program which comprises program code means for causing a quantum computer to perform the steps of the method of claims 1 to 13, when said computer program is carried out on a quantum computer.

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