Apparatus and method for fermionic quantum computing

Abstract

The present disclosure relates to a fermionic quantum computing apparatus comprising a first laser system configured for generating, inside a vacuum chamber, an array of double-well traps for trapping and manipulating a plurality of fermionic quantum particles, and a second laser system comprising a first local addressing unit for selectively and locally modifying one or more selected double-well traps of the array of double-well traps. The fermionic quantum computing apparatus also comprises a control system for controlling, based on a set of instructions of a quantum algorithm, the first laser system and the second laser system to perform local quantum tunneling operations of fermionic quantum particles in the selected double-well traps to implement quantum gate operations of the quantum algorithm.

Description

September 10, 2024Max-Planck-Gesellschaft zur Forderung M175617WO ANE / Bmn der Wissenschaften e.V.APPARATUS AND METHOD FOR FERMIONIC QUANTUM COMPUTINGFIELD OF INVENTION

[0001] The present disclosure relates to methods, apparatuses, and computer programs for implementing a fermionic quantum computer. Aspects of the present invention may benefit the fidelity and the scalability of digital and / or hybrid digitalanalog quantum computation, including systems with fermionic exchange statistics.INTRODUCTION

[0002] Quantum computing and quantum simulation are considered key pillars of quantum technologies, promising a quantum advantage over classical computation methods for applications such as factorization and optimization e.g., used in material science, drug discovery, stock option pricing, traffic simulations, etc. In particular, the ability to use quantum computers to simulate fermionic systems is considered a highly desired goal. This is because typical quantum many-body systems of interest (for example, electrons occupying the energy states of molecules) are fermionic. At the same time, the potential behind using classical numerics for computing fermionic quantum systems is limited, for example, as known in the art, due to the so-called sign problem. Further, a fermionic quantum computer based on fermionic particles may naturally respect fermionic exchange statistics, the implementation of which on a quantum computer based on bosonic particles (spins) requires a large computational overhead. As disclosed below, another key advantage of fermionic quantum computing is the conservation of particle number and spin projection that can be used for efficient error mitigation.

[0003] Various platforms have been developed for realizing quantum computers and quantum simulators, including superconducting qubits and ions and other effectively spin-based architectures. Generally, the fidelity of the quantum computation, an important indicator for the overall performance of the quantumcomputation device, is reduced by a variety of experimental imperfections. In this light, ultracold atoms have been a leading platform for the analogue simulation of fermionic quantum systems because of their ability to directly implement the coherent motion of fermions as well as the fermionic particle statistics. For example, placing neutral fermionic atoms in an optical lattice naturally realizes the Fermi-Hubbard model. Quantum simulations of the Fermi-Hubbard model have achieved a level of precision that challenges the best available numerical methods, and they can realize fermionic many-body systems with unparalleled control over system size, interactions and dynamics. In an optical lattice, all sites are created from the same pairs of laser beams, tunnelling rates are automatically homogeneous and different sites have stable degenerate energies. Such a high-quality optical lattice potential benefits the quantum simulations of quantum many-body systems with hundreds to tens of thousands of atoms.

[0004] As known in the art, realizing a universal quantum computer requires a basic set of quantum gates, including single-qubit rotations and a two-qubit entangling gate. Implementing such gates on a plurality of quantum particles requires, inter alia, local control of the quantum state of individual particles as well as control of the quantum state of a modifiable set of pairs of particles. Using these building blocks, it may be possible to implement any quantum algorithm. The application of logical qubits to quantum computations of practical interest is still considered a challenging task. In particular, for fermionic problems, a central difficulty is that digital bosonic quantum computers, e.g. based on ions or superconductors, have to decompose the Hamiltonian into a sequence of quantum gates which include the antisymmetric exchange statistics of fermions. This is expensive in terms of circuit depth and has limited digital quantum simulations to a good dozen particles. A fermionic quantum computer would work with fermionic particles and thus save the associated computational overhead.

[0005] In the context of fermionic atoms in optical lattices, implementing the equivalent of single- and two-qubit gates requires local control over individual atoms and pairs of atoms and their quantum state. The state of the art includes a variety of techniques which, in principle, could enable such local control. For example, spatial light modulators, SLMs, such as digital mirror devices, DMDs, may be used to shape laser beams in real space or Fourier space and flip the spin of individual atoms in theoptical lattice. Further, SLMs allow to manipulate the tunnelling dynamics in the optical lattice, to tune the density of atoms and controlling the spin-interaction in lattices without additional heating. In addition, by combining optical lattices with optical tweezers, atoms can be individually moved in an optical lattice. However, this motion is classical and not a unitary gate. Further, ground-state s-wave interaction between atoms can be used to realize so-called collisional gates, possibly changing the two-qubit quantum state of a pair of atoms and creating entanglement. For instance, such gates have been demonstrated in tweezers and optical lattices and recently created fully entangled clusters of up to ten bosonic atoms.In this context, D.Gonzalez-Cuadra et al. Fermionic quantum processing with programmable neutral atom arrays, PNAS, 2023, proposes to use fermionic atoms in programmable optical tweezer arrays to implement nonlocal gates based on Rydberg- mediated interaction gates, guaranteeing Fermi statistics at the hardware level. This allows to find efficient circuit decompositions for digital and variational quantum simulation algorithms, illustrated for molecular energy estimation.Further, A. Impertro et al. arXiv:23i2.i3268vi, relates to local readout and control of bosonic current and kinetic energy operators in optical lattices.Further background is given by:P. Preiss et al. Strongly correlated quantum walks in optical lattices, Science, 13 MARCH 2015, VOL 347, ISSUE 6227,C. Weitenberg et al. Single-spin addressing in an atomic Mott insulator, Nature, 17 March 2011, Vol 471 relates to directly monitoring and addressing the tunnelling quantum dynamics of single atoms in an optical lattice,- WO 2022 / 072087 At relates to a method for preparing states on a quantum computer with given particle number and total spin squared quantum numbers by means of parametrized gates. Explicit decompositions of such gates are given for an embodiment where fermions are mapped to the computational units of the quantum computer by means of a Jordan-Wigner mapping,- T. Chalopin et al. arXiv: 2405.19322vi, relates to an optical superlattice for engineering Hubbard couplings in quantum simulation,EP 4120 145 Al relates to a system for performing quantum operations comprising an optical superlattice and a plurality of optical tweezers.SUMMARY

[0006] When it comes to quantum computing of fermionic systems, such as electron problems from multi-band materials and quantum chemistry, using a platform based on fermionic particles, such as fermionic atoms in optical lattices, could reduce the computational complexity and increase the fidelity of the quantum computation process. However, so far, the controllability of fermionic atoms, possibly for the purpose of realizing single-qubit or two-qubit quantum gates, in optical lattices has been limited, while even small molecules or moderately complex band structures need hundreds or thousands of parameters to even define the quantum problem. For example, controlling the motion of fermionic atoms in the optical lattice locally, i.e. to link two sites while keeping all other tunnelling couplings zero, is an open challenge because, for example, the naive approach to lower a tunnel barrier with a focused attractive beam is technically unfeasible as it would require nanometer-scale pointing precision. Further, using Floquet engineering in interacting systems is known to cause heating and decrease the coherence time of the qubits. Therefore, there is a perpetual need for reaching high local control over the both the motion and the spin of fermionic particles in a quantum computer while overcoming at least some of the discussed limitations.

[0007] To address such and similar problems, the present disclosure provides, in a first aspect, a fermionic quantum computing apparatus, comprising a first laser system configured for generating, inside a vacuum chamber, an array of double-well traps for trapping and manipulating a plurality of fermionic quantum particles, and a second laser system comprising a first local addressing unit for selectively and locally modifying one or more selected double-well traps of the array of double-well traps. The apparatus further comprises a control system for controlling, based on a set of instructions of a quantum algorithm (e.g., received via a network from a remote user device), the first laser system and the second laser system to perform one or more local quantum tunneling operations of fermionic quantum particles in the one or more selected double-well traps to implement one or more quantum gate operations of thequantum algorithm. In some implementations, the fermionic quantum particles can be fermionic neutral atoms such as6Li.

[0008] The array of double well traps (e.g., a 2D or 3D array of double well traps with adjustable array connectivity) may correspond to an array of lattice sites in which strong and weak coupling alternate periodically such that, possibly, an array of pairs of strongly coupled sites emerges with weak or essentially zero coupling in between (see Fig. 2). Such an array of double-well traps may be realized by a periodic superlattice potential generated by a standing wave of laser radiation comprising two frequency components and f2which may differ by a factor of essentially two, i.e. f2= 2 . Thus, a superlattice potential maybe the sum of a long lattice (created by laser radiation of frequency fi) and a short lattice (created by laser radiation of frequency f2)-

[0009] In some implementations, the first laser system may generate at least two laser beams of a frequency fa and at least two beams of a different frequency fa which is different from fa by a factor of two. Next, the four beams may be combined into two pairs of beams each of which comprises a beam of frequency fa and a beam of frequency fa. The two pairs of beams may then be directed at the region of the fermionic neutral atoms, such that they interfere and an array of double wells is generated in the region of the fermionic neutral atoms (see Fig. la for details). In some implementations, the first laser system may comprise a delay line and / or a frequency modulation device, allowing to control the relative phase (pSLbetween the short and the long lattice potential (see Fig. ib for details). For certain values of the relative phase, corresponding to integer multiples of n, the double wells may be symmetric, i.e. there is essentially no energy offset between the first and the second site comprised by the double well. Tuning the relative phase away from such symmetric points may induce a tilt in all double wells of the array, corresponding to an energy difference between the first and the second site comprised by the double well. Importantly, that the ability to control the relative phase of the superlattice may allow to switch the dimerization of the array of double wells. This may mean that the symmetric double well configuration with double wells formed by eveiy 2t-th and (21 + 1) -th site may be converted into a different symmetric double well configuration in which the double wells are formed by every 2t-th and eveiy (21 — l)-th lattice site, where i denotes the site index.[ooio] Placing neutral atoms into an optical superlattice may naturally realize the Superlattice-Femi-Hubbard model, a version of the Fermi-Hubbard model known in the art, described by the Hamiltonian:4- where j denotes the site index, c. and cydenote the creation and annihilation operator, respectively, associated with site j, o e {I, T} denotes the internal degree of freedom of the particle (e.g., spin), t is the tunnelling rate between the first and the second site comprised by each double well, t0is the tunnelling rate between two adjacent double wells and U is the on-site interaction energy between particles of different spin, see Fig. 2 for details. More than one particle with the same spin occupying the same site is not allowed due to the Pauli exclusion principle.[oon] In some implementations, the particles may have additional internal degrees of freedom (such as additional spin states), and o e {I, T}, “spin up” and “spin down” may refer to a selection of them. For example, if Lie atoms are used as fermionic quantum particles, the hyperfine states (F = 3 / 2 and F = 1 / 2) in the electronic ground state can be used as spin states.

[0012] In some implementations, the tunnelling strengths t and t0may be controlled by adjusting the depth of the long and / or short lattice potentials of the superlattice. The on-site interaction strength U maybe controlled by adjusting an external magnetic field and taking advantage of a Feshbach resonance and / or by adjusting the depths of the long and / or short lattice potential of the superlattice and / or choosing the spin states of the atoms.

[0013] In some implementations, tilting a double well comprising the two adjacent sites j and j + 1 may correspond to adding a potential offset ftj = / z on site j and an offset ftj+1= ~n on site j + 1. In some implementations, all double wells may be tilted by the same amount by adjusting the relative phase between the short and the long lattice as described above. In some implementations, the local addressing unit comprised by the second laser system maybe a spatial light modulator, such as a DMD.A laser beam generated by a laser source may be shaped by the DMD and then focused onto the array of double wells using a high-resolution microscope objective. The second laser system may be used to selectively and locally modify one or more selected doublewell traps in the array of double-well traps. For instance, such local addressing may be used to induce a local potential shift i^j = / z on an arbitrary site j. This may correspond to locally tilting the double well comprising the sites, for instance,; and j + 1.

[0014] The state of a single fermionic particle with two spin states o e {I, T} occupying a double well maybe described in the basis { | T 0), 0), |0 T), |0 I)}. In particular, the fermionic atom may occupy the first or the second site comprised by the double well and may be either in the spin up or in the spin down state. Thus, one may combine pairs of spin orbitals into spin qubits (with spin qubit states |T 0), |l 0) or |0 T), |0 I)) and (effective, fermionic) orbital qubits (with qubit states |T 0), |0 T) or 11 0), |0 I)), see Fig. 3 for details and Bloch sphere representations. As known in the art, such a qubit description may allow to define single-qubit gates in the form of unitaiy rotation operations (i.e., in quantum state space). For the (conventional, bosonic) spin qubit, we may define Px(0), Ry(0), Pz(0) for rotations of phase 9 around the x, y and z axis of the spin qubit Bloch sphere, respectively. For the orbital qubit, we may define fx(0), Ty(0), fz(0) for rotations of phase 0 around the x, y and z axis of the orbital qubit Bloch sphere, respectively. Note that, for instance, TX(TT) may correspond to a tunnelling operation of the fermionic particle in a specific spins state from one to the other site comprised in the double well, and that RX(TT) may describe a spin flip operation of a particle at a specific site from, e.g., spin up to spin down. In the following, Txand Ty, may also be referred to as tunnelling operations.

[0015] In some implementations, the double well array generated by the first laser system in combination with local addressing provided by the second laser system, may be used to realize local single-qubit phase gates for the orbital qubit. This may mean that such a gate is performed for a selection of orbital qubits (corresponding to a selection of double wells in the array of double wells) while all other orbital qubits remain unaffected. As shown in Fig. 4, such a local quantum gate may, for example, be achieved by performing a first global tunnelling operation, e.g., fx(^). Performing such a tunnelling operation may comprise turning the tunnelling t between the two sites comprised in each double well on for a particular amount of time which corresponds to the phase 0 = n / 2 of the desired orbital qubit rotation. Second, the second laser systemmay be used to locally address and locally tilt the selection of double wells corresponding to the selection of orbital qubits. Doing so for a particular amount of time corresponds to inducing a relative quantum phase (p for a superposition state of orbital qubit states and may thus be used to locally implement an orbital qubit rotation rz( ).

[0016] Next, a second global tunnelling operation, e.g., Tx(y) may be performed similar to the first global tunnelling operation. As shown in Fig. 4, at the end of such a protocol, all double wells not addressed by the second laser system may return to the initial state (e.g., up to a known or inconsequential phase factor, etc.), while only the addressed double-well traps may be in a different orbital state, possibly depending on the locally imprinted phase (p. As an example, for (p = n, the local phase gate may correspond to a local tunnelling process from the first to the second site comprised in the addressed double well (see Fig. 4 for details). In some implementations of such a quantum gate, the first global tunnelling operation may be preceded by an additional local orbital qubit rotation Tz(p') and / or the second global tunnelling operation may be succeeded by a second additional orbital qubit rotation Tz((p”). Such additional local qubit rotations enhance flexibility of the local tunnelling process and allow to implement a larger class of local quantum gates with three tunable phase angles.

[0017] In some implementations, the fermionic quantum apparatus may comprise a third laser system comprising a second local addressing unit for selectively and locally changing an internal state (e.g. spin state) of one or more selected quantum particles in the array of double wells, e.g., via a local two-photon Raman transition. Such a second local addressing unit comprised by the second laser system may be a spatial light modulator, such as a DMD. A laser beam of the frequencies generated by one or two laser sources may be shaped by the DMD and then focused onto the array of double wells using a high-resolution objective. The third laser system may thus be used to selectively and locally modify the spin states of individual atoms and perform rotations for the corresponding spin qubits (see Fig. 3a), such as Rx(0), Ry(0), Rz(0), as described above, e.g., via locally driving Raman transitions from one spin state to another.

[0018] In a possible implementation, the fermionic quantum apparatus may further comprise an internal state transition unit for globally changing the internalstate (e.g. spin state) of the fermionic particles in the array of double-well traps. The internal state transition unit may comprise a microwave generator and a microwave antenna for generating microwave radiation and directing it at the atoms in the array of double wells. As known in the art, a pulse of MW radiation may then be used in order to coherently manipulate all particles exposed to the microwave radiation. Such a sweep may, for example, be used to flip the spin state of all fermionic atoms in the array of double wells, therefore implementing a global spin rotation Rx( ).

[0019] In some implementations, the fermionic quantum apparatus may further comprise an inter-particle interaction control unit configured to induce a controlled inter-particle interaction for pairs of fermionic quantum particles in the array of double-well traps. The inter-particle interaction strength may correspond to U in Eq. 1 and may apply globally to all particles in the array of double wells. The inter-particle interaction control unit may comprise magnetic field coils (e.g., a pair of Helmholtz coils) positioned in vicinity of the vacuum chamber and a power source for controlling electric current flowing through the coils. The inter-particle interactions may be adjusted by taking advantage of a Feshbach resonance which allows to tune the interparticle interactions on doubly occupied trap array sites via an external magnetic field (for details see: Chin et al: Feshbach Resonances in Ultracold Gases; arXiv:o8i2.i496). The strength of the magnetic field may be proportional to the current flowing through the coils. Further, in some implementations, the inter-particle interactions may also be controlled by adjusting the depth trap array, e.g., by modulating the intensity of the laser radiation generated by the first laser system.

[0020] The single-qubit rotations described herein may relate to double wells occupied by a single fermionic neutral atom. In a possible case of two fermionic neutral atoms, possibly of different spin, occupying the same double well, the first laser system and inter-particle interaction control unit may be used to realize a global SWAPaoperation. For example, if, initially, two fermionic neutral atoms occupy a first site within the double well, applying such a SWAPaoperation, as used herein, may correspond to a pair-tunnelling process, at the end of which, depending on the tunneling time, the two fermionic neutral atoms occupy the second site within the same double well. Herein, the index a is used to denote the duration of the pair-tunneling tunneling process (e.g., o=i indicates a complete tunneling evolution from one double well site to the other, and 0=1 / 2 maximally entangles the two particles). As anotherexample, if, initially, the first fermionic neutral atom occupies the first site within the double well and the second fermionic neutral atom occupies the second site within the double well, then applying a SWAPaoperation may lead to a swapping of the particles, such that the first fermionic particle occupies the second site within the double well and the second fermionic particle occupies the first site.

[0021] In general, not all double wells in the array of double wells may be occupied by two fermionic neutral atoms of different spin. Thus, it may be beneficial, in particular for complex quantum algorithms operating on a large array of quantum particles that the laser parameters used to implement the SWAPaoperations as disclosed herein are optimized such that in single-occupied double wells the singleparticle states are left essentially unaffected, e.g., by realizing an identity tunneling operation. Such an optimization may comprise employing numerical optimization methods suitable for quantum optimal control, such as gradient ascent pulse engineering, GRAPE.

[0022] In practical implementations, e.g., due to a finite interaction strength U, the SWAPaoperations, as disclosed herein, may not fully preserve a pair state. As a consequence, the value of the interaction strength U, as well as the time-dependent profile of the tunnelling strength t between the two sites of the double wells, may be optimized, such that the pair of fermionic neutral atoms in a SWAPaoperation keeps tunnelling as a pair, i.e. admixtures from states describing the two fermionic neutral atoms on different sites are suppressed, e.g., by setting the ratio t / U to some fine-tuned point where pairs do not break up into single particles at the end of a the gate time corresponding to a, or by temporally shaping the t parameter to achieve the same effect. Such an optimization may comprise employing numerical optimization methods, such as gradient ascent pulse engineering, GRAPE.

[0023] In order to realize a set of quantum gates sufficient for digital fermionic quantum computing, a two-qubit interaction gate may be required, and such a two- qubit interaction gate may correspond to local operations in pairs of fermionic neutral atoms in the array of double wells. In the following, we may assume that the double wells in the array of double wells maybe occupied either by a single fermionic neutral atom or a pair of fermionic neutral atoms. As described below, some of the components and operations described above may be combined in order to realize alocal pair tunnelling gate and / or a local spin exchange gate. Realizing such interaction gates may comprise implementing a local SWAP operation (as opposed to a global SWAP operation described above). For realizing such a local SWAP operation, one may execute a sequence of operations comprising at least one local phase gate, as described above, see Fig. 4 for details, and at least two global SWAP operations, as described above. For a detailed description of such sequences, see, for example, Fig. 5 and Fig. 6. Note that the set of quantum gates comprising the single-qubit operations (in the orbital qubit and / or the spin qubit), the local phase gate and the interaction gate (for example, the pair tunnelling gate), see Fig. 9 for an overview, in combination with the ability to change the dimerization of the array of double wells by changing the phase of the superlatticeSL, maybe sufficient for realizing a universal digital fermionic quantum computer.

[0024] In some implementations, the fermionic quantum apparatus may further comprise a fourth laser system for cooling the fermionic neutral atoms, such that they occupy a definitive motional state in the array of double-well traps. The definite motional state maybe the lowest band of the lattice potential. Cooling the atoms may comprise employing at least one cooling method known in the art, such as Raman sideband cooling, (bright or dark) molasses cooling, polarization gradient cooling, evaporation cooling or Doppler cooling. The laser system may comprise a plurality of laser beams necessary to carry out the respective cooling methods. The fermionic neutral atoms may be cooled to the definitive state in preparation for executing a sequence of quantum gates comprised by a quantum algorithm. The atoms may further be cooled during an imaging process which is part of reading out the quantum state of the plurality of fermionic neutral atoms in the array of double wells after completion of the execution of the operations which are part of the quantum algorithm, possibly in order to increase the amplitude of an imaging signal.

[0025] In a further aspect, the present invention discloses a method for fermionic quantum computing obtaining a set of instructions of a quantum algorithm, loading a plurality of fermionic quantum particles into an array of double-well traps inside a vacuum chamber; and performing, based on the set of instructions of the quantum algorithm one or more local quantum tunneling operations of fermionic quantum particles in the one or more selected double-well traps to implement one or more quantum gate operations of the quantum algorithm. In some implementations of themethod, the fermionic quantum particles maybe ultracold atoms, ions or molecules with fermionic exchange statistics, such as6Li. The goal of the quantum computation implemented by the quantum algorithm may be to approximate the equilibrium state of a fermionic many-body system such as the plurality of electrons bound to a molecule or a composition of multiple molecules, such as a protein.

[0026] Further details of the apparatuses, and methods described above, and a related computer program are discussed in the following with reference to exemplary implementations illustrated by the drawings. The foregoing broadly outlines the features and technical advantages of examples in accordance with the present disclosure in order that the detailed description that follows may be better understood. Additional features and advantages will be described hereinafter. The conception and specific examples disclosed may be readily utilized as a basis for modifying or designing other structures for carrying out the same purposes of the present disclosure.Characteristics of the concepts disclosed herein, both their organization and method of operation, together with associated advantages will be better understood from the following description when considered in connection with the accompanying figures. Each of the figures is provided for the purposes of illustration and description, and not as a definition of the limits of the claims.BRIEF DESCRIPTION OF THE DRAWINGS

[0027] Fig. 1 illustrates components of a fermionic quantum computing apparatus for the generation of an array of double wells in a vacuum chamber using optical superlattices according to an exemplary implementation of the present disclosure.

[0028] Fig. 2 illustrates the Superlattice-Fermi-Hubbard model realized by placing fermionic neutral atoms in an optical superlattice potential according to an exemplary implementation of the present disclosure.

[0029] Fig. 3 shows a spin qubit and an orbital qubit using a Bloch sphere representation according to an exemplary implementation of the present disclosure.

[0030] Fig. 4 illustrates a protocol for realizing local tunneling gates for orbital qubits (e.g., local orbital phase gates) in an array of double wells according to an exemplary implementation of the present disclosure.

[0031] Fig. 5 illustrates a sequence for realizing local pair-tunnelling gates in an array of double wells according to an exemplary implementation of the present disclosure.

[0032] Fig. 6 illustrates a sequence for realizing local spin-exchange gates in an array of double wells according to an exemplary implementation of the present disclosure.

[0033] Fig.7 shows a block diagram of an exemplary fermionic quantum computing apparatus according to a possible implementation of the present disclosure.

[0034] Fig. 8 illustrates a method for fermionic quantum computing according to aspects of the present disclosure, e.g. by using an apparatus as described herein.

[0035] Fig. 9 illustrates an overview of local and global quantum gate operations that can be realized with the fermionic quantum computing apparatuses and methods as disclosed herein.DETAILED DESCRIPTION OF EXEMPLARY EMBODIMENTS

[0036] Various aspects of the present disclosure are described in more detail hereinafter with reference to the accompanying drawings. The present disclosure may, however, be implemented in many different forms and should not be construed as limited to any specific structure or function presented herein. Rather, these aspects are provided so that this disclosure will be thorough and complete, and will fully convey the scope of the present disclosure to those skilled in the art. Based on the teachings herein one skilled in the art should appreciate that the scope of the present disclosure is intended to cover any aspect of the present disclosure disclosed herein, whether implemented independently of or combined with any other aspect of the present disclosure. For example, an apparatus, a device or a system maybe implemented, or amethod may be practiced using any number of the aspects set forth herein. In addition, the scope of the present disclosure is intended to cover such a device, apparatus, system or method which is practiced using other structure, functionality, or structure and functionality in addition to or other than the various aspects of the present disclosure set forth herein. Any aspect of the present disclosure disclosed herein may be implemented by one or more elements of a claim. While specific feature combinations are described in the following with respect to certain aspects of the present disclosure, it is to be understood that not all features of the discussed examples must be present for realizing the technical advantages of the devices, apparatuses, systems, methods and computer programs disclosed herein. Disclosed aspects may be modified by combining certain features of one aspect with one or more features of other aspects. A skilled person will understand that features, steps, components and / or functional elements of one aspect can be combined with compatible features, steps, components and / or functional elements of any other aspect of the present disclosure.

[0037] Several aspects of trapping and manipulating (e.g., imaging, gate operations, spectroscopy, etc.) of neutral atoms will now be presented with reference to various devices, apparatuses, systems and methods that are described in the following detailed description and illustrated in the accompanying drawings by various blocks, modules, components, circuits, steps, processes, algorithms, and / or the like (collectively referred to as “elements”). These elements maybe implemented using hardware, software, or combinations thereof. Whether such elements are implemented as hardware and / or software depends upon the particular application and design constraints imposed on the overall system. Further, the apparatuses and methods disclosed herein can be part of complex quantum technology systems such as neutral atom quantum computers. The skilled person will appreciate that in the following several components of such systems, such as laser sources, imaging equipment, experiment control and timing units, etc. are not explicitly described.

[0038] FIG. 1 illustrates several components such as optical setups for the generation of an array of double wells using optical superlattices according to aspects disclosed herein. Panel a of Fig. 1 shows a typical laser beam and optics configuration with respect to a vacuum chamber 105 comprising a region 110 where fermionic quantum particles (e.g., fermionic neutral atoms) can be trapped inside an array ofdouble wells. In general, the array of double wells may be two-dimensional or three- dimensional and may be created by two perpendicular superlattices. The plurality of fermionic neutral atoms in the array of double wells may act as a quantum register of fermionic quantum computing (for example, see Fig. 7 and Fig. 9 for details). The array of double wells in the in the y-direction may be created by a first pair of bi chromatic laser beams 125a (SL-x) comprising laser radiation of two different frequencies which are possibly separated by a factor of two. The beams 125a interfere at the position 110 under a non-zero angle leading to a standing electromagnetic wave in the y-direction. Such a standing electromagnetic wave may act as a periodic superlattice potential comprising a short and a long lattice with different spatial frequencies separated by a factor of two (see Fig. 2).

[0039] Similarly, the array of double wells in the in the x-direction is created by a second pair of bichromatic beams 125b (SL-y). The wavelengths of the laser radiation comprised by the bichromatic beam may be 532 nm and 1064 nm or 767 nm and 1534 nm, or any other suitable combination of laser sources. In some implementations, the maximum depth of the superlattice potential may be insufficiently deep for performing fluorescence imaging methods for detecting the state of the fermionic atoms in the array of double wells, possibly after the execution of the quantum algorithm. As a consequence, a different optical lattice with a much larger maximum depth may be used to pin the atoms in place during imaging. For example, a pinning lattice could be created by monochromatic laser beam 115 which may penetrate the vacuum chamber and the region of the array of double wells 110 in a bow tie geometry, generating a pinning lattice in the target region of the apparatus (e.g., as described in applicants own PCT / EP2023 / 074768, incorporated herein in its entirety).

[0040] The pinning lattice may have a spacing which is considerably smaller compared to the spacing of the superlattice which creates the array of double wells. Then, after completion of the execution of the quantum algorithm, the pinning lattice may be turned on. Possibly at the same time, the atoms may be illuminated with resonant or near-resonant light in order to produce fluorescence photons. The fluorescence photons may be collected by a high-resolution microscope objective 120 and be collected via a CCD camera. The resolution of the microscope objective 120 may be sufficient for resolving the spacing of the superlattice, such that occupation of eachsite in the array of double wells can be detected with high fidelity, possibly in order to read out the quantum state of the plurality of fermionic neutral atoms in the double well array. The optical setup may contain additional optical elements not shown in the schematic, including lenses, mirrors and other optical surfaces as known to the skilled person.

[0041] Panel b of Fig. 1 shows a possible optical setup for implementing at least a part of the first laser system. The first laser system may create an array of double wells by, inter alia, generating an optical superlattice potential with controllable phase between the long and short lattice comprised in the superlattice potential. A laser source (not shown) may generate an input laser beam 130 which may be split up into two parts. The first part 135a may pass through a frequency modulation device 140, an amplifier 145, and a second harmonic generation, SHG, device 150. The frequency modulation device 140 may be an acousto-optical modulator driven by a radiofrequency source or any other suitable modulator. The amplifier 145 may increase the intensity of the laser radiation passing through. The SHG device 150 may convert the frequency of at least part of the laser beam from v to 2v. Both parts of the input laser beam 135a and 135b may be coupled into optical fibers 155 (e.g., for mode filtering) and, subsequently, overlapped using an optical combination device 160, such as a D-shaped mirror, dichroic mirror or a beam splitter. The combined beam may then be bichromatic and comprises both frequencies f and 2f . The combined beam is split up by an optical splitting device 170 into two lattice beams which are guided in a way, such that they interfere at the position of the atoms in the target region of the quantum computer and create an array of double wells 180. At least one of the two lattice beams may pass through an adjustable delay line 175 which changes the phase of the laser radiation passing through. The phase between the short and the long lattice may be controlled by changing the length AL of the delay line 175 and / or by changing the frequency of the first part of the input laser beam 135a using the frequency modulation device 140. In order to reduce the impact of environmental conditions, such as ambient air pressure and / or ambient temperature fluctuations, on the phase (pSL of the superlattice potential, the beam path of the combined beam as well as the delay line 175 may be located inside a vacuum chamber 165 isolating the optics from the environment. Further, the optical setup may contain additional intensity modulators (not shown) for controlling the intensity of the two parts of the input laser beam 135aand 135b, in order to control the relative strength of the short and the long lattice in the superlattice potential, e.g., to control the quantum tunneling rate t in the double wells.

[0042] Fig. 2 illustrates the Superlattice-Fermi-Hubbard model realized by placing fermionic neutral atoms in an optical superlattice potential 205. Panel a of Fig.1 illustrates the Fermi-Hubbard Hamiltonian of Eq. 1. It may be characterized by a tunnelling rate t between the two sites of each double well, by a tunnelling rate t0between double wells, by an onsite-interaction strength U and, possibly, by additional energy shifts i^j on particular sites with index j. Typically, the tunneling rate rate t0may be adjusted to be much smaller than t, effectively resulting an array of non-coupled double wells.

[0043] The additional energy shifts may, for example, be created by using the phase-controllability of the superlattice to induce (parallel) tilts in each double well. A tilt corresponds to a positive energy shift z on ever second site of the superlattice and a negative energy shift - / z on every other site of the superlattice. As another example, local energy shifts may be created by illuminating one or more sites in the superlattice using the first local addressing unit comprised by the second laser system. Fermionic neutral atoms trapped in the superlattice may have one of two internal states, designated as spin up (up arrow) and spin down (down arrow). The interaction strength U may refer to the energy cost associated with a pair of fermionic neutral atoms (with different spin) occupying the same site in the superlattice.

[0044] Panel b of Fig. 2 illustrates the ability to locally manipulate the superlattice potential, possibly for the purpose of implementing local quantum gates for fermionic quantum computing. As an example, the second laser system comprising the first local addressing unit maybe used to illuminate one or more selected sites 210 of the superlattice and introduce local energy shifts Hj. These may be used in order to locally tilt an arbitrary subset of double wells in the array of double wells, for example as part of implementing local phase gates (see, for example, Fig. 4). In addition, the third laser system comprising the second local addressing unit may be used to change the spin state of an arbitrary subset of the plurality of particles 215 trapped in the array of double wells.

[0045] Fig- 3 shows a spin qubit (panel a) and an orbital qubit (panel b) using a Bloch sphere representation. Panel a shows a Bloch sphere, illustrating the Hilbert space of a spin qubit realized by, for example, a fermionic neutral atom with, for example, two spin states (spin up | T) and spin down | I)) trapped in an arbitrary site of the superlattice. The particle can be in either spin state or in an arbitrary superposition of the two spin states as known in the art (see for example M. Nielsen & I. Chuang: Quantum Computation and Quantum Information: 10th Anniversary Edition). Panel b shows a corresponding Bloch sphere, illustrating the Hilbert space of an orbital qubit realized by, for example, a single fermionic neutral atom in an arbitrary spin state (for example, spin up | T) or spin down | I)), trapped in a double well of the double well array. The particle can be in either located in the left site of the double well (e. g. , | T 0) for a spin up particle) or in the right site of the double well (e. g. , 10 T) for a spin up particle) or in an arbitrary superposition of the two orbital states. As disclosed herein, and illustrated for a simple example in Fig. 4, arbitrary rotations in the Hilbert space of the orbital qubit in selected double wells can be realized by a combination of global tunneling processes and local tilting / local energy offsets. The insets depict illustrations of the wave function in the double well for orbital quantum states associated with various positions on the corresponding Bloch sphere. Here, “Re” and “Im” indicate the real and the imaginary part of the wavefunction.

[0046] Fig. 4 illustrates an exemplary protocol for realizing a local (orbital) tunneling gate in an array of double wells. As an example, each double well in the array of double wells may be occupied by a single fermionic neutral atom. As illustrated by the Bloch sphere for the orbital qubit in Fig. 3, such a fermionic neutral atom can be either in the left site of the double well or in the right site of the double well or in an arbitrary superposition of such states (e.g., the state of the atom in the double well comprising lattice sites 2 and 3). As described above, realizing a local phase gate (of phase ) may comprise, for example, three operations: a first global tunnelling operation fx(^), a local rotationhere performed on site with index 6, and a second global tunnelling operation Tx(y)-

[0047] For example, consider the double well comprising site 6 and 7: Applying the tunnelling operation Tx(“) to the double well may yield a superposition with relative phase it / 2 of the fermionic neutral atoms occupying the left and the right site of thedouble well. Then, applying, for example, the operation fz(< >6= TT) may invert the sign of the superposition to - n / 2. Next, applying the tunnelling operation Tx(y) to such a superposition may yield the orbital state which is different from the initial state. Thus, for example, a fermionic neutral atom originally localized on the left side of the double well maybe localized on the right side of the double well after applying the tunnelling operations. Note that due to the locality of the second operation fz(< >6= zr) , at the end of the protocol only the orbital state of double well comprising sites 6 and 7 is different compared to the initial state. All other (unaddressed) double wells return the initial state upon completion of the protocol.Fig. 5 illustrates a sequence for realizing local pair tunnelling gates in an array of double wells based on global quantum pair-tunnelling operations (SWAP“ operations) and inducing local quantum phases.The schematic in the top right of Fig. 5 shows a possible effect of such a pair tunnelling gate on single and two-particle quantum states in one or more selected double wells subject to local addressing for a=l: A pair of fermionic neutral atoms of opposite spin on the same site of a double well may tunnel to the other site of the double well (pair tunnelling). In contrast, states describing two fermionic neutral atoms of different spin and on different sites on the double well and quantum state describing a single fermionic neutral atom localized on any of the two sites of the double well, may remain unchanged by the completion of the quantum gate. The possible spin states of the fermionic neutral atoms (spin up and spin down) are indicated by the up arrows and down arrows.The center panel of Fig. 5 shows an exemplary time-evolution of three sequence parameters implementing such a quantum gate, including the global on-site interaction strength U, the tunnelling strength t associated with the coupling between the right and the left site of each double well, and the local energy offsetapplied to one of the two sites in one or more selected double well traps in the array of double wells. As described above, the inter-particle interaction strength may be controlled by the interparticle interaction control unit. The tunneling strength t may be controlled globally by adjusting the depth of the short lattice potential, as described above. A local offset on one or more selected double wells may be induced using local addressing. In a possibleimplementation, the pair-tunnelling gate may be realized with a sequence of three pulses: A first global pair tunnelling operation 510, corresponding to a global SWAP1 / 2operation, as disclosed herein, followed by a first local rotation Tz(< >2) 52o, corresponding to inducing a quantum phase between the orbital states of pair of fermionic neutral atoms in the one or more selected double well traps. Just as for the local phase gate described above, such a local rotation many be achieved by effectively tilting the one or more selected double wells using local addressing. The local rotation may be then followed by a second global pair tunnelling operation 530, corresponding to a global SWAP3 / 2operation, as disclosed herein.In some implementations, the sequence may comprise a second local rotation Tz( i) 540, corresponding to inducing a quantum phase between the orbital states of pair of fermionic neutral atoms in the one or more selected double well traps, prior to the first global pair-tunneling operation. Further, the sequence may comprise a second local rotation Tz(< >3) 550, corresponding to inducing a quantum phase between the orbital states of pair of fermionic neutral atoms in the one or more selected double well traps, after the second global pair-tunneling operation. Such additional local rotations may enhance the flexibility of the local pair tunnelling process.Due to the inter-particle interactions, the quantum state describing the fermionic neutral atoms in the double wells of the array of double wells may pick up undesirable interaction-induced phases. Thus, the sequence may comprise an additional interaction gate 560 which comprises adjusting the interaction strength to a selected value for a selected amount of time and, therefore, compensating for interaction phases created by other parts of the quantum gate.

[0048] Fig. 6 illustrates a sequence for realizing local spin exchange gates in an array of double wells based on global quantum pair-tunnelling operations (SWAP operations) and inducing local quantum phases.The schematic in the top right of Fig. 6 shows a possible effect of such a spin exchange tunnelling gate on single and two-particle quantum states in one or more selected double wells subject to local addressing: Quantum states describing two fermionic neutral atoms of different spin and on different sites on the double well may beswapped (spin exchange). In contrast, quantum states describing a pair of fermionic neutral atoms on the same site of the double well, as well as quantum states describing a single fermionic neutral atom localized on any of the two sites of the double well, may remain unchanged by the completion of the quantum gate. The possible spin states of the fermionic neutral atoms (spin up and spin down) are indicated by the up arrows and down arrows.The center panel of Fig. 5 shows an exemplary time-evolution of three sequence parameters implementing such a quantum gate, including the global on-site interaction strength U, the tunnelling strength t associated with the coupling between the right and the left site of each double well, and the local energy offset applied to one of the two sites in one or more selected double well traps in the array of double wells. As described above, the inter-particle interaction strength may be controlled by the inter-particle interaction control unit. The tunneling strength t may be controlled globally by adjusting the depth of the short lattice potential, as described above. A local offset on one or more selected double wells may be induced using local addressing. In a possible implementation, the pair-tunnelling gate may be realized with a sequence of three pulses: A first global spin exchange operation 610, corresponding to a global SWAP1 / 2operation, as disclosed herein, followed by a first local spin rotation / ?z(< >2) 620, corresponding to inducing a quantum phase between the spin states of pair of fermionic neutral atoms in the one or more selected double well traps. The local rotation may be then followed by a second global spin exchange operation 630, corresponding to a global SWAP3 / 2operation, as disclosed herein. In Fig.5, the local spin rotations R may also be demoted S.In some implementations, the sequence may comprise a second local spin rotation Rz(Pi) 640, corresponding to inducing a quantum phase between the spin states of pair of fermionic neutral atoms in the one or more selected double well traps, prior to the first global spin exchange operation. Further, the sequence may comprise a second local spin rotation / ?z(< >3) 650, corresponding to inducing a quantum phase between the spin states of pair of fermionic neutral atoms in the one or more selected double well traps, after the second global spin exchange operation. Such additional local rotations may enhance the flexibility of the local spin exchange process.Due to the inter-particle interactions, the quantum state describing the fermionic neutral atoms in the double wells of the array of double wells may pick up undesirable interaction-induced phases. Thus, the sequence may comprise an additional interaction gate 660 which comprises adjusting the interaction strength to a selected value for a selected amount of time and, therefore, compensating for interaction phases created by other parts of the quantum gate.

[0049] Fig. 7 shows a typical implementation of a fermionic quantum computer 700 comprising an array of double well traps 720 inside a vacuum chamber 715 for realizing a fermionic quantum register e.g. formed by a plurality of fermionic neutral atoms trapped in the array of double well traps 720. The apparatus 700 further comprises a control system 745, an inter-particle interaction control system 7 and, a state readout system 755 and a spin state transition unit 765. The illustrated implementation further comprises a first laser system 725, a second laser system 730, a third laser system 735 and a fourth laser system 740. In some implementations, the fermionic quantum computer 700 can be controlled, e.g., by a (remote) user device 705, e.g., via sending a set of instructions of a quantum algorithm, possibly via a network 710. In some implementations, the first laser system 725 maybe configured for generating, inside the vacuum chamber 715, an array of double-well traps 720 for trapping and manipulating a plurality of fermionic quantum particles which act as a quantum register for the fermionic quantum computer 700.

[0050] In some implementations, the second laser system 730 may comprise a first local addressing unit for selectively and locally modifying one or more selected double-well traps of the array of double-well traps 720. In some implementations, the third laser system 735 may comprise a second local addressing unit for selectively and locally changing an internal state of one or more selected fermionic neutral atoms in the array of double-well traps. In some implementations, the fourth laser system 740 may be configured for cooling the fermionic quantum particles to occupy a definitive motional state of a respective in the array of double-well traps 720. For example, the first laser system 725 and / or the second laser system 730 and / or the third laser system 735 and / or the fourth laser system 740 may comprise one or more lasers, which provide one or more laser beams to the fermionic neutral atoms in the quantum register.

[0051] In some implementations, the first laser system 725 and / or the second laser system 730 and / or the third laser system 735 and / or the fourth laser system 740 may further comprise a set of optical elements configured to direct laser radiation generated by a laser source onto the plurality of fermionic neutral atoms in the array of double traps 720. Ins some implementation, the control system 745 may be configured for controlling, based on a set of instructions of a quantum algorithm, the first laser system 725 and the second laser system 730 to perform one or more local quantum tunneling operations of fermionic quantum particles in the one or more selected double-well traps to implement one or more quantum gate operations of the quantum algorithm. In some implementation, the inter-particle interaction control unit 760 may be configured to induce a controlled inter-particle interaction for pairs of fermionic quantum particles in the array of double-well traps 720. In some implementation, the fermionic quantum computer 700 may further comprise an internal state transition unit for globally changing an internal state of the fermionic neutral atoms in the array of double-well traps 720.

[0052] In some implementations, the state readout system 755 may be configured to collect and / or detect photons generated by qubits (e.g., during state readout procedures). The optics system may comprise one or more optical elements (e.g., lenses, mirrors, waveguides, fiber optics cables, and / or the like) and one or more photodetectors. In various embodiments, the photodetectors may be photodiodes, photomultipliers, charge-coupled device (CCD) sensors, complementary metal oxide semiconductor (CMOS) sensors, Micro-Electro-Mechanical Systems (MEMS) sensors, and / or other photodetectors that are sensitive to light at an expected fluorescence wavelength of the qubits of the quantum computer. In various embodiments, the detectors may be in electronic communication with the control system 745.

[0053] In some implementations, the remote user device 705 is configured to allow a user to provide input to the quantum computer 700 and receive, view, and / or the like output from the quantum computer 700. The user device may be in communication with the control system 745 of the quantum computer 700 via one or more wired or wireless networks 710 and / or via direct wired and / or wireless communications. In an example embodiment, the remote user device 705 may translate, configure, format, and / or the like information / data, quantum computingalgorithms and / or circuits, and / or the like into a computing language, executable instructions, command sets, and / or the like that the control system 745 can understand and / or implement.

[0054] In some implementations, the control system 745 may be configured to control, inter alia, the first laser system 725 and / or the second laser system 730 and / or the third laser system 735 and / or the fourth laser system 740 and / or the state readout system 755. For example, the control system 745 maybe configured to cause a controlled evolution of quantum states of one or more fermionic neutral atoms within the quantum register to execute a quantum circuit and / or algorithm. In various embodiments, the fermionic neutral atoms within the quantum register are used as qubits of the fermionic quantum computer 700. Further, the fermionic quantum computer 700 may also comprise a spin state transition unit 765 such as a MW source and MW antenna as well as a source of cold fermionic particles, e.g., a laser cooling system, e.g., comprising a Zeeman slower and / or one or more 2D and / or 3D magneto optical traps (MOTs) as known in the art.

[0055] Fig. 8 illustrates a method 800 for fermionic quantum computing, comprising: obtaining 810 a set of instructions of a quantum algorithm, loading 820 a plurality of fermionic quantum particles into an array of double-well traps inside a vacuum chamber, and performing 830, based on the set of instructions of the quantum algorithm, one or more local quantum tunneling operations of fermionic quantum particles in the one or more selected double-well traps to implement one or more quantum gate operations of the quantum algorithm. In some examples, the quantum algorithm may be configured for approximating an equilibrium state of a fermionic many-body system, e.g., for approximating an equilibrium state of an interacting multielectron system such as the electronic ground-state of a multi-atom molecule.

[0056] In some examples, obtaining the set of instructions of the quantum algorithm may comprise receiving from a remote user device via a network, data indicative of the set of instructions of the quantum algorithm. Further, the method 800 may further comprises detecting, optionally via internal state selective imaging, a state of the plurality of fermionic particles in the array of double-well traps and sending, to the remote user device via the network, data indicative of a result of the quantumalgorithm, based on the detected state of the plurality of fermionic particles in the array of double-well traps. In this manner, a fermionic quantum computing apparatus or system as disclosed herein may be used to perform, via the network, quantum computing as a service.

[0057] In some examples, the method 800 may further comprise determining, based on the detected state of the plurality of fermionic particles in the array of doublewell traps, whether a global quantum number of the plurality of fermionic particles has been conserved during execution of a part of the quantum algorithm, and determining the result of the quantum algorithm, based at least in part on whether a global quantum number of the plurality of fermionic particles has been conserved. For example, when simulating, via execution of the quantum algorithm, fermionic systems relevant for chemistry, statistical physics, or condensed matter physics, one is usually interested in obtaining states with given quantum numbers, such as a fixed number of spin up / down electrons and global spin, and when the Hamiltonian is real, its eigenstates are also manifestly real. In such situations, the ability to prepare states from the respective quantum number sector in a targeted way is desirable. This can offer additional advantages, such as the ability to use post-selection on the correct particle number for error mitigation.As the skilled person understands, the method 800 may further include additional steps and modifications as described above for the fermionic quantum computing apparatus which will not be repeated here to avoid redundancies.

[0058] Fig. 9 illustrates, an overview of local and global quantum gate operations that can be realized with the fermionic quantum computing apparatuses and methods as disclosed herein. Global dimer gate operations include global tunneling and global tilts, e.g., combined with a global change of double well connectivity / dimerization, e.g., from a first connectivity configuration to a second double well configuration. As discussed above, global and local spin state rotations can be implemented via MW radiation and / or via local Raman transitions. Local, programmable phase gates can be implemented for spin qubits and / or orbital qubits using local addressing (e.g., via using ab SLM such a DMD). For a local spin phase gate, a spin dependent light shift may be used to induce a relative quantum phase betweenspin states of a spin superposition state. Local orbital phase gates can be implemented via locally modifying / tilting the trapping potential of one or more selected double well potentials (also see Fig. 4). As explained above, global tunneling and local orbital phase gates can be used to implement arbitrary local orbital qubit rotations, providing full access to the orbital qubit Hilbert space represented by the orbital Bloch sphere.

[0059] Finally, multi-particle entangling can be realized by global inter-particle / collision gate operations, e.g., controllable via a Feshbach resonance and / or by global pair-tunneling and spin-exchange tunneling gate operations (also denoted SWAPaoperations herein).

[0060] Even though particular combinations of features are recited in the claims and / or disclosed in the specification, these combinations are not intended to limit the disclosure of various aspects. In fact, many of these features maybe combined in ways not specifically recited in the claims and / or disclosed in the specification. Although each dependent claim listed below may directly depend on only one claim, the disclosure of various aspects includes each dependent claim in combination with every other claim in the claim set. A phrase referring to “at least one of’ a list of items refers to any combination of those items, including single members. As an example, “at least one of: a, b, or c” is intended to cover a, b, c, a-b, a-c, b-c, and a-b-c, as well as any combination with multiples of the same element (e.g., a-a, a-a-a, a-a-b, a-a-c, a-b-b, a- c-c, b-b, b-b-b, b-b-c, c-c, and c-c-c or any other ordering of a, b, and c).

[0061] No element, act, or instruction used herein should be construed as critical or essential unless explicitly described as such. Also, as used herein, the articles “a” and “an” are intended to include one or more items, and may be used interchangeably with “one or more.” Furthermore, as used herein, the terms “set” and “group” are intended to include one or more items (e.g., related items, unrelated items, a combination of related and unrelated items, and / or the like), and may be used interchange-ably with “one or more.” Where only one item is intended, the phrase “only one” or similar language is used. Also, as used herein, the terms “has,” “have,” “having,” and / or the like are intended to be open-ended terms.

[0062] AS used herein, the phrase “based on” shall not be construed as a reference to a closed set of information, one or more conditions, one or more factors, or the like. In other words, the phrase “based on A” (where “A” may be information, a condition, a factor, or the like) shall be construed as “based at least on A” unless specifically recited differently.

[0063] As used herein, the term “or” is an inclusive “or” unless limiting language is used relative to the alternatives listed. For example, reference to “X being based on A or B” shall be construed as including within its scope X being based on A, X being based on B, and X being based on A and B. In this regard, reference to “X being based on A or B” refers to “at least one of A or B” or “one or more of A or B” due to “or” being inclusive. Similarly, reference to “X being based on A, B, or C” shall be construed as including within its scope X being based on A, X being based on B, X being based on C, X being based on A and B, X being based on A and C, X being based on B and C, and X being based on A, B, and C. In this regard, reference to “X being based on A, B, or C” refers to “at least one of A, B, or C” or “one or more of A, B, or C” due to “or” being inclusive. As an example of limiting language, reference to “X being based on only one of A or B” shall be construed as including within its scope X being based on A as well as X being based on B, but not X being based on A and B.

[0064] Further, process diagrams such as Fig. 8 do not necessarily indicate a particular order or sequence of steps. For example, steps may also be performed in a different order or, if hardware capabilities allow it, simultaneously, without deviating from the scope of the present disclosure.

Claims

September 10, 2024 Max-Planck-Gesellschaft zur Fbrderung M175617WO ANE / Bmn der Wissenschaften e.V.Claims1. Fermionic quantum computing apparatus (700), comprising: a first laser system (725) configured for generating, inside a vacuum chamber (715), an array of double-well traps (720) for trapping and manipulating a plurality of fermionic quantum particles; a second laser system (730) comprising a first local addressing unit for selectively and locally modifying one or more selected double-well traps of the array of double-well traps; and a control system (745) for controlling, based on a set of instructions of a quantum algorithm, the first laser system and the second laser system to perform one or more local quantum tunneling operations of fermionic quantum particles in the one or more selected double-well traps to implement one or more quantum gate operations of the quantum algorithm.

2. Fermionic quantum computing apparatus of claim 1, wherein, to perform the one or more local quantum tunneling operations, the control system is configured for controlling the first laser system and the second laser system: to perform a global quantum tunneling operation for the fermionic quantum particles in each occupied double-well trap of the array of double-well traps; and to locally and selectively induce a relative quantum phase between orbital quantum states of fermionic quantum particles in the one or more selected double-well traps.

3. Fermionic quantum computing apparatus of claim 2, wherein, to perform the one or more local quantum tunneling operations, the control system is configured for controlling the first laser system and the second laser system: to perform a first global quantum tunneling operation for the fermionic particles in the array of optical double-well traps;to induce a first relative quantum phase between the orbital quantum states of the fermionic quantum particles in the one or more selected double-well traps; to perform a second global quantum tunneling operation for the fermionic particles in the array of optical double well traps; and optionally: to induce a second relative quantum phase between the orbital quantum states of the fermionic quantum particles in the one or more selected double-well traps prior to the first global tunneling operation, and to induce a third relative quantum phase between the orbital quantum states of the fermionic quantum particles in the one or more selected double-well traps after the second global tunneling operation.

4. Fermionic quantum computing apparatus of claim 3, wherein the first global quantum tunneling operation and the second global tunneling operation are performed such that an orbital state of fermionic quantum particles in non-selected double-well traps is essentially left unchanged.

5. Fermionic quantum computing apparatus of any of claims 1 to 4, further comprising: a third laser system (735) comprising a second local addressing unit for selectively and locally changing an internal state of one or more selected quantum particles in the array of double-well traps; and / or a fourth laser system (740) configured for cooling the fermionic quantum particles to occupy a definitive motional state of a respective in the array of double-well traps; and / or an inter-particle interaction control unit (760) configured to induce a controlled inter-particle interaction for pairs of fermionic quantum particles in the array of double-well traps; and / or an internal state transition unit (765) for globally changing an internal state of the fermionic particles in the array of double-well traps.

6. Fermionic quantum computing apparatus of any of claims 1 to 5, wherein the control system is configured to control the first laser system and the second laser system, and optionally, the inter-particle interaction control unit toperform a local pair-tunneling gate for pairs of interacting fermionic quantum particles in the one or more selected double-well traps.

7. Fermionic quantum computing apparatus of any of claims 1 to 6, wherein the control system is configured to control the first laser system and the second laser system, and optionally, the inter-particle interaction control unit to perform a local spin-exchange gate for pairs of interacting fermionic quantum particles in the one or more selected double-well traps.

8. Fermionic quantum computing apparatus of claim 6, wherein the control system is configured to control the first laser system and / or the inter-particle interaction control unit: to perform a first global quantum pair-tunneling operation (510) for pairs of fermionic particles in the array of optical double-well traps; to induce a first relative quantum phase (520) between the orbital quantum states of pairs of fermionic quantum particles in the one or more selected double-well traps; to perform a second global quantum pair tunneling operation (530) for pairs of fermionic particles in the array of optical double well traps; and optionally, to induce a second relative quantum phase (540) between the orbital quantum states of pairs of fermionic quantum particles in the one or more selected double-well traps prior to the first global pair-tunneling operation, and optionally, to induce a third relative quantum phase (550) between the orbital quantum states of the fermionic quantum particles in the one or more selected double-well traps after the second global pair-tunneling operation, and optionally, to globally induce a fourth relative quantum phase (560) between quantum states of pairs of fermionic quantum particles on double occupied sites of the array of double well traps.

9. Fermionic quantum computing apparatus of claim 8, wherein the first global pair-tunneling operation (510) and the second global pair-tunneling operation (530) are performed such that: an orbital state of pairs of fermionic quantum particles in non-selected doublewell traps is essentially left unchanged; andan orbital state of fermionic quantum particles in single occupied double-well traps is essentially left unchanged.

10. Fermionic quantum computing apparatus of any of claims 1 to 9, wherein the first laser system comprises a first laser source operating at a first laser wavelength and a second laser source operating at a second laser wavelength to generate the array of double-well potentials by generating an optical superlattice for the fermionic quantum particles inside the vacuum chamber; and / or wherein the first laser system comprises one or more laser intensity modulators for controlling a laser intensity of laser radiation generated the first laser system to control a tunneling rate for the fermionic quantum particles in the array of generate the array of double-well potentials; and / or wherein the first laser system comprises one or more laser phase and / or frequency modulators to modify a connectivity of the array of double-well traps and / or to induce a tilt for each double-well trap.

11. Fermionic quantum computing apparatus of any of claims 1 to 10, further comprising: a particle state readout system (755), operably connected to the control system, and configured for detecting, optionally via internal state selective imaging, a state of the plurality of fermionic particles; and / or a network interface, operably connected to or integrated into the control system, and configured for receiving, from a remote user device (705) via a network (710), data indicative of the set of instructions of the quantum algorithm and / or for sending to the remote user device, via the network, data indicative of a result of the quantum algorithm.

12. Method (800) for fermionic quantum computing, comprising: obtaining (810) a set of instructions of a quantum algorithm; loading (820) a plurality of fermionic quantum particles into an array of doublewell traps inside a vacuum chamber; and performing (830), based on the set of instructions of the quantum algorithm, one or more local quantum tunneling operations of fermionic quantum particles in oneor more selected double-well traps to implement one or more quantum gate operations of the quantum algorithm.

13. Method of claim 12, wherein the quantum algorithm is configured for approximating an equilibrium state of a fermionic many-body system, preferably for approximating an equilibrium state of an interacting multi-electron system.

14. Method of claim 12 or 13, wherein obtaining the set of instructions of the quantum algorithm comprises receiving from a remote user device via a network, data indicative of the set of instructions of the quantum algorithm, and wherein the method further comprises: detecting, optionally via internal state selective imaging, a state of the plurality of fermionic particles in the array of double-well traps; sending to the remote user device via the network, data indicative of a result of the quantum algorithm, based on the detected state of the plurality of fermionic particles in the array of double-well traps.

15. Method of claim 14, further comprising: determining, based on the detected state of the plurality of fermionic particles in the array of double-well traps, whether a global quantum number of the plurality of fermionic particles has been conserved during execution of a part the quantum algorithm; and determining the result of the quantum algorithm, based at least in part on whether a global quantum number of the plurality of fermionic particles has been conserved.

16. Method of any of claims 12 to 15, further comprising one or more of: selectively and locally changing, based on the set of instructions of the quantum algorithm, an internal state of one or more selected fermionic quantum particles; controlling, based on the set of instructions of the quantum algorithm, an interparticle interaction strength.17- Method of any of claims 12 to 16, further comprising one or more manipulation steps corresponding to the manipulations that can be carried out by the apparatus of any of claims 3 to 11.

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